DIFFRACTION OF SINGULARITIES FOR THE WAVE EQUATION ON MANIFOLDS WITH CORNERS

RICHARD MELROSE, ANDRÁS VASY, AND JARED WUNSCH

Abstract. We consider the fundamental solution to the wave equation on a manifold with corners. If the initial pole of the solution is appropriately situated, we show that the singularities which are diffracted by the corners (i.e., loosely speaking, are not propagated along limits of transversely reflected rays) are smoother than the main singularities of the solution. More generally, we show that subject to a hypothesis of nonfocusing, diffracted wavefronts of any solution to the wave equation are smoother than the incident singularities.

Contents

1. Introduction 2 1.1. The problem and its history 2 1.2. The hypothesis 5 1.3. Structure of the proof 6 2. Geometry: metric and Laplacian 7 3. Bundles and bicharacteristics 10 3.1. b-Cotangent bundle 10 3.2. Edge-smooth cotangent bundle 14 3.3. Edge-b cotangent bundle 17 3.4. Bicharacteristics 20 3.5. A summary 31 4. Edge-b calculus 31 5. Differential-pseudodifferential operators 33 5.1. The calculus 33 5.2. Dual spaces and adjoints 39 5.3. Domains 43 5.4. The wave operator as an element of x^{−2} Diff_{es,♯}^{2}(M) 45 6. Coisotropic regularity and non-focusing 52 7. Edge propagation 58 7.1. Radial points in the interior of the edge 59 7.2. Propagation into radial points over the boundary of the edge 59 7.3. Propagation at hyperbolic points within the edge 73 7.4. Propagation at glancing points within the edge 80 8. Propagation of fiber-global coisotropic regularity 85 9. Geometric theorem 87 Appendix A. Some functional analysis 95 Appendix B. The edge-b calculus 101 References 107

Date: March 18, 2009.

1. Introduction

1.1. The problem and its history. Let X_{0} be a manifold with corners, of dimension n, i.e., a manifold locally modeled on (ℝ^{+})^{f+1}×ℝ^{n−f−1}, endowed with an incomplete metric, smooth and non-degenerate up to the boundary. We consider the wave equation

(1.1)  □u ≡ D_{t}^{2}u − Δu = 0 on M_{0} = ℝ × X_{0},

where D_{t} = ı^{−1}(∂/∂t) and Δ is the nonnegative Laplace-Beltrami operator; we will impose either Dirichlet or Neumann conditions at ∂X_{0}. As is well known, by the classic result of Duistermaat-Hörmander^{1} (see [4]), that the wavefront set of a solution u propagates along null-bicharacteristics in the interior. However, the behavior of singularities striking the boundary and corners of M_{0} is considerably subtler.

Indeed the propagation of singularities for the wave equation on manifolds with boundary is already a rather subtle problem owing the the difficulties posed by “glancing” bicharacteristics, those which are tangent to the boundary. Chazarain [1] showed that singularities striking the boundary transversely simply reflect according to the usual law of geometric optics (conservation of energy, tangential momentum, hence “angle of incidence equals angle of reflection”) for the reflection of bicharacteristics. This result was extended in [23] and [24] by showing that, at glancing points, singularities may only propagate along certain generalized bicharacteristics. The continuation of these curves may fail to be unique at (non-analytic) points of infiniteorder tangency as shown by Taylor [31]. Whether all of these branches of bicharacteristics can carry singularities is still not known.

As was shown initially in several special examples (namely those amenable to separation of variables) the interaction of wavefront set with a corner gives rise to new, diffractive phenomena, in which a single bicharacteristic carrying a singularity into a corner engenders singularities releasing from the corner along a whole family of bicharacteristics. For instance, a ray carrying a singularity transversely into a codimension-two corner will in general produce singularities on the entire cone of rays reflected in such a way as to conserve both energy and momentum tangent to the corner (see Figure 1) The first diffraction problem to be rigorously treated was that of the exterior of a wedge,^{2} which was analyzed by Sommerfeld [30]; Figure 1. A ray carrying a singularity may strike a corner of codimension two and give rise to a whole family of diffracted singularities, conserving both energy and momentum along the corner.

1This result, viewed in the context of hyperbolic equations, built on a considerable body of work prior to the introduction of the wavefront set; see especially [12], [15].

2This is not in fact a manifold with corners, but is quite closely related.

subsequently, many related examples were analyzed by Friedlander [5], and more generally the case of exact cones was worked out explicitly by Cheeger-Taylor [2], [3] in terms of the functional calculus for the Laplace operator on the cross section of the cone. All of these explicit examples reveal that generically a diffracted wave arises from the interaction of wavefront set of the solution with singular strata of the boundary of the manifold; this has long been understood at a heuristic level, with the geometric theory of diffraction of by Keller [11] describing the classes of trajectories that ought to contribute to the asymptotics of the solution in various regimes.

Subsequent work has been focused primarily on characterizing the bicharacteristics on which singularities can propagate, and on describing the strength and form of the singularities that arise. The propagation of singularities on a wide class of manifolds including manifolds with boundary was first understood in the analytic case by Lebeau [13], [14]. In the C^{∞} setting employed here, the special case of manifolds with conic singularities was studied by Melrose-Wunsch [26] and on edge manifolds (i.e., cone bundles) by Melrose-Vasy-Wunsch [25]. Vasy [34] obtained results analogous to Lebeau’s in the case of manifolds with corners, and it is the results of this work that directly bear on the situation studied here.

While the foregoing results characterize which bicharacteristics may carry singularities for solutions to the wave equation, they ignore the question of the regularity of the diffracted front. In general, we expect a singularity measured microlocally with respect to Sobolev spaces an so in WF^{s} to propagate strongly, i.e. certain “geometric” bicharacteristics through that point are also in WF^{s}; in simple cases, these are known to be those which are locally approximable by bicharacteristics that miss the corners and reflect Figure 2. Geometric optic rays hitting a corner F, emanating from a point o. The rays labelled G are geometric at F, while those labelled NG are non-geometric at F. The leftmost geometric ray is a limit of rays like the unlabelled one shown on the figure that just miss F. The blown up version of the picture, i.e. where (r, θ) are introduced as coordinates at the origin, is shown on the right. The front face (i.e., the lift of the corner) is denoted ff . The reflecting line indicates the broken geodesic of length π induced on ff given by r = 0, θ ∈ [0, θ_{0}]. The total length of the three segments shown on ff is π; this can be thought of as the sum of three angles on the picture on the left: namely the angles between the incident ray and the right boundary (corresponding to the first segment), the right and left boundaries, finally the left boundary and the emanating reflected ray.

transversely off boundary hypersurfaces. More generally, we can define geometric bicharacteristics as follows: To begin, we blow up the corner, i.e. introduce polar coordinates around it; this serves to replace the corner with its inward-pointing normal bundle, which fibers over the corner with fiber given by one orthant of a sphere, S^{f} ∩ (ℝ^{+})^{f+1}. We will define geometric broken bicharacteristics passing through the corner as those that lift to the blown-up space to enter and leave the lift of the corner at points connected by generalized broken geodesics of length π with respect to the naturally defined metric on S^{f} ∩(ℝ^{+})^{f+1}, undergoing specular reflection at its boundaries and corners.^{3} (The fiber metric depends on the ambient metric and will not in general be the standard metric on the sphere). Bicharacteristics that enter and leave the corner at points in a fiber that are not at distance-π in this sense are referred to as “diffractive.” See Figure 2.

It turns out that subject to certain hypotheses of nonfocusing, the singularities propagating along diffractive bicharacteristics emanating from the corner will be weaker than those on geometric bicharacteristics. In particular, the fundamental solution satisfies the nonfocusing condition, hence one consequence of our main theorem is as follows:

3The actual definition is considerably complicated by the existence of glancing rays, and is discussed in detail in §3.4.

Theorem 1.1. Consider the fundamental solution u_{o} to the wave equation, with pole o sufficiently close to a corner, W, of codimension k with a correspondingly short geodesic from o to Y meeting each boundary face containing Y transversally. While u_{o} lies microlocally in ^{4} H^{−n/2+1−0}, it is less singular by (k − 1)/2 derivatives along diffractive bicharacteristics emanating from W, that is it lies microlocally in H^{(−n+k+1−0)/2}. A more precise version of this result (with “sufficiently close… ” eludicated) appears in Corollary 9.6.

A more general theorem on regularity of the diffracted wave subject to the nonfocusing condition is the central result of this paper. See §1.2 for a rough statement of the nonfocusing condition and §6 for technical details; the main theorem is stated and proved in §9.

There are a few related results known in special case. Gérard-Lebeau

[7] explicitly analyzed the problem of an analytic conormal wave incident on an analytic corner in ℝ^{2}, obtaining a 1/2-derivative improvement of the diffracted wavefront over the incident one. The first and third authors [26] obtained the results for manifolds with conic singularities, which the authors subsequently generalized to the case of edge manifolds [25].

1.2. The hypothesis. We now describe the nonfocusing hypothesis in more detail, in the context of the simplest geometric situation to which our results apply.

It is easily seen from the explicit form of the fundamental solution that it is not in general true that diffracted rays are more regular than incident singularities. For example, take Δ to be the Dirichlet Laplacian in a sectorial domain {r ∈ [0, ∞), θ ∈ [0, θ_{0}]} in ℝ^{2}, and consider the solution

(1.2)  \frac{sin t\sqrt{Δ}}{\sqrt{Δ}}ϕ(θ)δ(r − r_{0}),

where ϕ ∈ C_{c}^{∞}((0, θ_{0})) is supported close to some value θ′. This solution is manifestly locally in H^{1/2−0} by energy conservation. On the other hand one may see from the explicit form of the propagator in [2], [3] after convolution with ϕ(θ) that a spherical wave of singularities emanates from the corner at time t = r_{0}, with regularity H^{1/2−0}, hence the same as the overall Sobolev regularity of the solution. The bicharacteristics along which singularities propagate are, for short time, just the lifts of the straight lines r = r_{0} ± t, θ ∈ supp ϕ, hence travelling straight into or out of the vertex. Perturbing these slightly to make them miss the vertex, we see that in fact there are two “geometric” continuations^{5} for each bicharacteristic, depending on whether we approximate it by geodesics passing to the left or to the right of the vertex (see Figure 2). Thus, the geometric continuations of the rays on which singularities strike the vertex are close to the two possible continuations of the single ray θ = θ′, hence do not include all points on the spherical wave. So we have an example in which there are “non-geometric” singularities of full strength.

4Here and henceforth we emply the notation s − 0 to mean s − ϵ for all ϵ > 0.

5What a geometric continuation of a bicharacteristic is in general will be elucidated in §3.4.

The nonfocusing condition serves exactly to rule out such situations. The above example has the property that applying negative powers of (Id +D_{θ}^{2}) does not regularize the short-time solution (or the initial data) as it is already C^{∞} in the θ direction. In this simple example, the nonfocusing condition says precisely that the solution is regularized by negative powers of (Id +D_{θ}^{2}), or, equivalently, that it can be written

(Id +D_{θ}^{2})^{N}v, v ∈ H^{s}

for some s exceeding the overall Sobolev regularity. For instance, the fundamental solution

u = \frac{sin t\sqrt{Δ}}{\sqrt{Δ}}δ(r − r′)δ(θ − θ′)

looks, after application of a sufficiently negative power of (Id +D_{θ}^{2}), like a distribution of the form

\frac{sin t\sqrt{Δ}}{\sqrt{Δ}}δ(r − r′)f(θ)

with f ∈ C^{M}, M ≫ 0, hence we can write

u ∈ (Id +D_{θ}^{2})^{N}H^{1/2−0},

for some N ≫ 0, at least locally, away from the boundary. We also observe that the example (1.2) enjoys a property which is essential dual to the nonfocusing condition, to whit, fixed regularity under repeated application of D_{θ}. We refer to this property as “coisotropic regularity” (the terminology will be explained in §6) and it plays an essential role in our proof.

The nonfocusing condition and coisotropic regularity in a more general setting are subtler owing to their irreducibly microlocal nature: the operator D_{θ} has to be replaced by a family of operators characteristic along the flowout of the corner under consideration.

1.3. Structure of the proof. We now describe the logical structure of the proof, as it is somewhat involved. The heart of the argument is a series of propagation of singularities results, obtained by positive commutator methods. In order to be able to distinguish between “geometric” and “diffractive” bicharacteristics at a corner of M_{0}, we begin by performing a blowup of the corner, i.e. introducing polar coordinates centered at it, to obtain a new manifold with corners M. The commutants that we employ in our commutator argument almost lie in a calculus of pseudodifferential operators, the edge-b calculus, that behaves like Mazzeo’s edge calculus [16] near the new boundary face introduced by the blowup (henceforth, the edge) and like Melrose’s b calculus [19, 21, 22] at the remaining boundary faces. The complication, as in the previous work of Vasy [34] on propagation of singularities, is that in order to control certain error terms we in fact must employ a hybrid differential-pseudodifferential calculus, in which we keep track of certain terms involving differential operators normal to the boundary faces other than W̃ .

Even this propagation result alone is insufficient to obtain our result, as it does not allow regularity of greater than a certain degree to propagate out of the edge, with the limitation in fact not exceeding the a priori regularity of the solution. What it does allow for, however, is the propagation of coisotropic regularity of arbitrarily high order, suitably microlocalized in the edge-b sense. This allows us to conclude that given a ray γ leaving the edge, if the solution enjoys coisotropic regularity along all rays incident upon the edge that are geometrically related to γ, then we may conclude coisotropic regularity along γ as well. (If some of these incident rays are glancing, i.e. tangent to the boundary, we require as our hypothesis actual differentiability globally at all incoming glancing rays rather than coisotropic regularity, which no longer makes sense; the version of the commutator argument that deals with these rays is the most technically difficult aspect of the argument.) In particular, then, global coisotropic regularity together with C^{∞} regularity at glancing rays implies global coisotropic regularity leaving the edge away from glancing. We are then able to dualize this result to show that the nonfocusing condition propagates as well.

Consequently, we show that subject to the nonfocusing condition, in the model case of the sector considered above, if γ is an outgoing ray such that the solution is C^{∞} along all incoming rays geometrically related to it,

u ∈ (Id +D_{θ})^{N}H^{s} along γ,

where s = (−n + k + 1)/2 − 0 for the fundamental solution near a general codimension-k corner on an n-manifold, i.e. is 1/2−0 for our sector. On the other hand the microlocal propagation of coisotropic regularity shows that

D_{θ}^{k}u ∈ H^{s_{0}} for all k along γ

where s_{0} is the overall regularity of the solution (−n/2 + 1 − 0 for the fundamental solution). An interpolation argument then yields

u ∈ H^{s−0} along γ,

proving the theorem.

The authors acknowledge the support of the National Science Foundation under grants DMS-0408993 (RM), DMS-0733485 and DMS-0801226 (AV) and DMS-0700318 (JW). The second author was also supported by a Chambers Fellowship from Stanford University. All three authors are grateful to MSRI for both support and hospitality in the fall of 2008, during which the bulk of this manuscript was written.

2. Geometry: metric and Laplacian

Let X_{0} be a connected n-dimensional manifold with corners. We work locally, near a given point in the interior of a corner Y of codimension f + 1.

Thus, we have local coordinates x_{1}, …, x_{f+1}, y_{1}, …, y_{n−f−1} in which Y is given by x_{1} = … = x_{f+1} = 0. Suppose that g_{0} is a smooth Riemannian metric on X_{0}, non-degenerate up to all boundary faces. We may always choose local coordinates in which it takes the form

(2.1)  g_{0} = ∑ a_{ij} dx_{i} dx_{j} + ∑ b_{ij} dy_{i} dy_{j} + 2 ∑ c_{ij} dx_{i} dy_{j}

with c_{ij}|_{Y} = 0. This can be arranged by changing the y variables to

y_{j} = y′_{j} + ∑ x_{k}Y_{jk}(y′)

while keeping the x_{j} unchanged. The the cross-terms then become

2 ∑ c_{ij} dx_{i} dy′_{j} + 2 ∑ b_{ij} Y_{jk} dy′_{i} dx_{k},

which can be made to vanish by making the appropriate choice of the Y_{jk}, using the invertibility of {b_{ij}}.

Let X = [X_{0}; Y], be the real blowup of Y in X_{0} (see [21, 20]) and let Ỹ denote the front face of the blowup, which we also refer to as the edge face. Recall that the blow-up arises by identifying a neigborhood of Y in X_{0} with the inward-pointing normal bundle N^{+}Y of Y in X_{0} and blowing up the origin in the fibers of the normal bundle (i.e. introducing polar coordinates in the fibers). Since the normal bundle is trivialized by the defining functions of the boundary faces, a neighborhood of Ỹ in X is globally diffeomorphic to

[0, ∞) × Y × Z, where Z = S^{f} ∩ [0, ∞)^{f+1}.

We use coordinates z_{1}, …, z_{f} in Z; near a corner of Z of codimension k, these are divided into z′_{1}, …, z′_{k} ∈ [0, ∞) and z′′_{k+1}, …, z′′_{f} ∈ ℝ. There is significant freedom in choosing the identification of a neighborhood of Y and the coordinates on the fibers of the normal bundle but the naturality of the smooth structure on the blown up manifold, [X_{0}; Y], corresponds to the fact that these are smoothly related.

The metric g_{0} identifies NY as a subbundle of T_{Y} X_{0}. This corresponds to coordinates (x_{i}, y_{j}) as above with the dy_{j} orthogonal to dx_{i} at Y. In the blow-up polar coordinates are introduced in the x_{i} but the y_{j} are left unchanged. It is convenient to think of these as polar coordinates induced by ∑_{ij} a_{ij} dx_{i} dx_{j}. In particular, we choose

x = ( ∑_{ij} a_{ij}(0, y)x_{i}x_{j})^{1/2}

as the ‘polar variable’ which is the defining function of the front face. With this choice, the metric takes the form

(2.2) g = dx^{2} + h(y, dy) + x^{2}k(x, y, z, dz) + xk′(x, y, z, dx, dy, x dz).

More generally, one can simply consider the wider class of manifolds with corners with metrics of the form (2.2), we refer to these as ‘edge metrics’ for brevity. Note, however, that there are no results currently available in this wider setting that limit propagation of singularities to generalized broken Figure 3. A manifold X_{0} with corners of codimension two (below) and its blowup X (top). A geodesic hitting the codimension-two corner is shown, together with its lift to the blown-up space X, which then strikes the front “edge” face of the blowup.

bicharacteristics. Despite this, the results in §7 remain valid in this more general context.

Now, set

M = ℝ_{t} × X, M_{0} = ℝ × X_{0}, W̃ = ℝ × Ỹ ,

where W̃ now represents the space-time “edge.”

Let Diff_{es}^{∗}(X) denote the filtered algebra of operators on C^{∞}(X) generated by those vector fields that are tangent to the fibers of the front face Ỹ produce by the blow up; xD_{x}, xD_{y_{j}}, D_{z_{j}} form a local coordinte basis of these vector fields. See §3 and §5 for further explanation of this algebra and of our terminology. The same definition leads to the algebra of operators Diff_{es}^{∗}(M) on C^{∞}(M) with local generating basis xD_{t}, xD_{x}, xD_{y_{j}}, D_{z_{j}}.

Lemma 2.1. The Laplace operator Δ ∈ x^{−2} Diff_{es}^{2}(X) on X is of the form

Δ ∈ D_{x}^{2} + \frac{f}{ıx}D_{x} + \frac{1}{x^{2}} Δ_{Z} + Δ_{Y} + x^{−1} Diff_{es}^{2}(X)

where Δ_{Z} is the Laplace operator in Z with respect to the metric h (and hence depends parametrically on x and y) and Δ_{Y} is the Laplacian on Y with respect to the metric k.

In particular, □ = D_{t}^{2} − Δ_{X} ∈ x^{−2} Diff_{es}^{2}(M).

3. Bundles and bicharacteristics

In this section, we discuss several different geometric settings in which the propagation problem for □ on M_{0} may be viewed. Somewhat loosely, each of these corresponds to a choice of a Lie algebra of vector fields with different boundary behavior; these then lead to distinct bundles of covectors, with the corresponding descriptors used as section headings here. The first, the “b”-bundle, can be considered either on M_{0} or M. Indeed, the bundle of b-covectors on M_{0} is the setting for the propagation results of [34]: these results are, as will be seen below, necessarily global in the corner, and do not distinguish between diffractive rays and geometric subset of them as defined below. In order to discuss the improvement in regularity which can occur for propagation along the geometric rays, two more bundles of covectors, lying over the blown-up space M, are introduced. These, the “edge-b” and “edgesmooth” bundles, keep track of local information in the fibers of the blowup of W in M_{0}, and allow us to distinguish the diffractive rays from geometric ones. The distinction between the edge-b and edge-smooth bundles comes only at the boundary of W̃ , and the tension between the two presentations of the problem gives rise to reflection of singularities off boundary faces, uniformly up to the edge W.

In order to alleviate some of the notational burden on the reader, a table §3.5 is include in which the various bundles, their coordinates, their sections, and some of the maps among them are reviewed. The standard objects for a manifold with corners, Q, correspond to uniform smoothness up to all boundary faces, so V(Q) denotes the Lie algebra of smooth vector fields, T Q, the tangent bundle, of which V(Q) forms all the smooth sections, T^{∗}Q is its dual, etc.

3.1. b-Cotangent bundle. Let V_{b}(Q) ⊂ V(Q) denote the Lie subalgebra of those smooth vector fields on the general manifold with corners Q, which are tangent to each boundary face. The local vector fields

x_{1}∂_{x_{1}}, …, x_{f+1}∂_{x_{f+1}}, ∂_{y_{1}}, …, ∂_{y_{n−f−1}};

form a basis over smooth functions, so V_{b}(Q) is the space of C^{∞}-sections of a vector bundle, denoted

bT Q.

The dual bundle ^{b}T^{∗}Q therefore has sections spanned by

\frac{dx_{1}}{x_{1}} , …, \frac{dx_{f+1}}{x_{f+1}} , dy_{1}, …, dy_{n−f−1}.

The natural map

(3.1)  π_{s→b} : T^{∗}Q ⟶ ^{b}T^{∗}Q

is the adjoint of the bundle map ι : ^{b}T Q → T Q corresponding to the inclusion of V_{b}(Q) in V(Q).

Canonical local coordinates on T^{∗}M_{0} correspond to decomposing a covector in terms of the basis as

τ^{s} dt + ∑_{j} η_{j}^{s} dy_{j} + ∑_{j} ξ_{j}^{s} dx_{j},

and elements of ^{b}T^{∗}M_{0} may be written

τ^{b} dt + ∑_{j} η_{j}^{b} dy_{j} + ∑_{j} ξ_{j}^{b} \frac{dx_{j}}{x_{j}} ,

so defining canonical coordinates. The map (3.1) then takes the form

π_{s→b}(x, y, t, ξ^{s}, η^{s}, τ^{s}) = (x, y, t, xξ^{b}, η^{b}, τ^{b}) = (x, y, t, xξ^{s}, η^{s}, τ^{s}),

with xξ^{s} = (x_{1}ξ_{1}^{s}, …, x_{f+1}ξ_{f+1}^{s}).

The setting for the basic theorem on the propagation of singularities in [34] is ^{b}T^{∗}M_{0}. In particular, generalized broken bicharacteristics, or GBB ’s, are curves in ^{b}T^{∗}M_{0}. In order to analyze the geometric improvement, spaces that will keep track of finer singularities are needed. Before introducing these, we first recall the setup for GBB’s . Note that at W, π_{s→b}|_{Y} maps N^{∗}W onto the zero section over W, and is injective on its complement, so we may make the identification

π_{s→b}|_{Y} (N^{∗}W) ≅ T^{∗}W.

We also recall that it is convenient to work on cosphere bundles. Since it is linear, π_{s→b} intertwines the ℝ^{+}-actions, but it does not induce a map on the corresponding cosphere bundles since it maps part of T^{∗}M_{0} \ o into the zero section of ^{b}T^{∗}M_{0}. However, on the characteristic set of □ this map is better behaved. Let

p_{0} = σ_{2}(□) ∈ C^{∞}(T^{∗}M_{0} \ o)

be the standard principal symbol of □ ∈ Diff^{2}(M_{0}); it is of the form

p_{0} = (τ^{s})^{2} − (∑ A_{jk}ξ_{j}^{s}ξ_{k}^{s} + ∑ B_{jk}η_{j}^{s}η_{k}^{s} + 2 ∑ C_{jk}ξ_{j}^{s}η_{k}^{s})

with A_{jk}, B_{jk}, C_{jk} ∈ C^{∞}(M_{0}), A_{jk} = A_{kj} and B_{jk} = B_{kj}, C_{jk}|_{x=0} = 0. Let

(3.2)  ^{s}Σ_{0} = p_{0}^{−1}({0})/ℝ^{+} ⊂ S^{∗}M_{0}

be the spherical image of the characteristic set of □. This has two connected components, ^{s}Σ_{0,±}, corresponding to τ^{s} ≷ 0 since {τ^{s} = 0} ∩^{s}Σ_{0} = ∅. Now, N^{∗}W ⊂ {τ^{s} = 0}, so W ∩ p^{−1}(0) = ∅, meaning W is non-characteristic for □. Since N^{∗}W is the null space of π_{s→b}, there is an induced map on the sphere bundles { \atop π_{s→b}} : ^{s}Σ_{0} ⟶ ^{b}S^{∗}M_{0}; the range is denoted

(3.3)  ^{b}Σ_{0} = { \atop π_{s→b}}(p^{−1}(0))/ℝ^{+} ⊂ ^{b}S^{∗}M_{0}.

Again, ^{b}Σ_{0} has two connected components corresponding to the sign of τ^{s} in ^{s}Σ_{0} and hence the sign of τ^{b}. These will be denoted ^{b}Σ_{0,±}.

We use τ^{s}, resp. τ^{b}, to obtain functions homogeneous of degree zero on T^{∗}M_{0} \ o inducing coordinates on S^{∗}M_{0} near ^{s}Σ_{0} :

x, y, t, ξ̂^{s} = ξ^{s}/|τ^{s}|, η̂^{s} = η^{s}/|τ^{s}|.

Note also that these coordinates are global in the fibers of S^{∗}M_{0}∩^{s}Σ_{0,±} → M_{0} for each choice of sign ±.

τ̂^{s} = sgn τ^{s}

lifts to a constant function ±1 on ^{s}Σ_{0,±}. There are similar coordinates on bS^{∗}M_{0} near ^{b}Σ_{0}.

In these coordinates, (3.4) bΣ_{0} ∩^{b}S_{W}^{∗} M_{0} = {(x, y, t, ξ̂^{b}, η̂^{b}) : x = 0, ξ̂^{b} = 0, ∑ B_{jk}η̂_{j}^{b}η̂_{k}^{b} ≤ 1} ⊂ S^{∗}W.

We also remark that with H_{p_{0}} denoting the Hamilton vector field of p_{0},

H_{s} = |τ^{s}|^{−1}H_{p_{0}}

is a homogeneous degree zero vector field near p_{0}^{−1}({0}), thus can be regarded as a vector field on S^{∗}M_{0}.

Now we define the b-hyperbolic and b-glancing sets by

(3.5)  G_{W,b} = {q ∈ ^{b}S_{W}^{∗} M_{0} : |(π_{s→b})^{−1}(q) ∩^{s}Σ_{0}| = 1}

and

(3.6)  H_{W,b} = {q ∈ ^{b}S_{W}^{∗} M_{0} : |(π_{s→b})^{−1}(q) ∩^{s}Σ_{0}| ≥ 2},

These are thus also subsets of S^{∗}W. In local coordinates^{6} they are given by

G_{W,b} = {(x, y, t, τ̂^{b}, ξ̂^{b}, η̂^{b}) : x = 0, τ̂^{b} ∈ {±1}, ξ̂^{b} = 0, ∑ B_{jk}η̂_{j}^{b}η̂_{k}^{b} = 1}
H_{W,b} = {(x, y, t, τ̂^{b}, ξ̂^{b}, η̂^{b}) : x = 0, τ̂^{b} ∈ {±1}, ξ̂^{b} = 0, ∑ B_{jk}η̂_{j}^{b}η̂_{k}^{b} < 1}.

(3.7)

Note that for q ∈ ^{b}S_{W}^{∗} M_{0}, at the unique point q_{0} in (π_{s→b})^{−1}(q) ∩^{s}Σ_{0}, we have ξ̂^{s} = 0, and correspondingly H_{s}(q_{0}) is tangent to W, explaining the “glancing” terminology.

Now we discuss bicharacteristics.

Definition 3.1. A generalized broken bicharacteristic, or GBB , is a continuous map γ : I → ^{b}Σ_{0} such that for all f ∈ C^{∞}(^{b}S^{∗}M_{0}), real-valued,

(3.8)  lim inf_{s→s_{0}} \frac{(f ∘ γ)(s) − (f ∘ γ)(s_{0})}{s − s_{0}}

(3.9) ≥ inf{H_{s}(π_{s→b}^{∗}f)(q) : q ∈ π_{s→b}^{−1}(γ(s_{0})) ∩^{s}Σ_{0}}.

6The discrete variable τ̂^{b} is not, of course, part of the coordinate system, but serves to identify which of two components of the characteristic set we are in.

Remark 3.2. Replacing f by −f, we deduce that the inequality

(3.10)  lim sup_{s→s_{0}} \frac{(f ∘ γ)(s) − (f ∘ γ)(s_{0})}{s − s_{0}}

(3.11) ≤ sup{H_{s}(π_{s→b}^{∗}f)(q) : q ∈ π_{s→b}^{−1}(γ(s_{0})) ∩^{s}Σ_{0}}.

also holds.

We recall an alternative description of GBB’s , which was in fact Lebeau’s definition [14]. (One could use this lemma as the defining property of GBB ; the equivalence of these two possible definitions is proved in [32, Lemma 7].)

Lemma 3.3. (See [32, Lemma 7].) Suppose γ is a GBB . Then

(1) If γ(s_{0}) ∈ G_{W,b}, let q_{0} be the unique point in the preimage of γ(s_{0}) under { \atop π_{s→b}} = π_{s→b}|_{^{s}Σ_{0}}. Then for all f ∈ C^{∞}(^{b}S^{∗}M_{0}) real valued, f ∘ γ is differentiable at s_{0}, and

\frac{d(f ∘ γ)}{ds}|_{s=s_{0}} = H_{s}π_{s→b}^{∗}f(q_{0}).

(2) If γ(s_{0}) ∈ H_{W,b}, lying over a corner given in local coordinates by x_{j} = 0, j = 1, …, f + 1, there exists ϵ > 0 such that x_{j}(γ(s)) = 0 for s ∈ (s_{0} − ϵ, s_{0} + ϵ) if and only if s = s_{0}. That is, γ does not meet the corner {x_{1} = … = x_{f+1} = 0} in a punctured neighborhood of s_{0}.

Remark 3.4. It also follows directly from the definition of GBB (by combining (3.8) and (3.10)) that, more generally, if the set

(3.12)  {H_{s}(π_{s→b}^{∗}f)(q) : q ∈ π_{s→b}^{−1}(γ(s_{0})) ∩^{s}Σ_{0}}

consists of a single value (for instance, if π_{s→b}^{−1}(γ(s_{0}))∩^{s}Σ_{0} is a single point), then f ∘ γ must be differentiable at s_{0}, with derivative given by this value. This is indeed how Lemma 3.3 is proved. The first part of the lemma follows because π_{s→b}^{−1}(γ(s_{0})) ∩^{s}Σ_{0} is a single point, giving differentiability. On the other hand, the second half follows using f = ∑ ξ̂_{j}^{b}, for which the single value in (3.12) is −(1 − ∑ B_{ij}η̂_{i}^{b}η̂_{j}^{b}) < 0, for γ(s_{0}) ∈ H_{W,b}. Thus, f is locally strictly decreasing. Since f(q′) = 0 if q′ ∈ ^{b}S_{W}^{∗} M_{0} ∩ ^{b}Σ_{0}, in particular at γ(s_{0}), it is non-zero at γ(s) for nearby but distinct values of s—so in particular for such s, γ(s) ∉ ^{b}S_{W}^{∗} M_{0} ∩^{b}Σ_{0}, showing that γ leaves W instantaneously. In fact, this argument also demonstrates the following useful lemma.

Lemma 3.5. Let U be a coordinate neighborhood around some p ∈ W, K a compact subset of U. Let ϵ_{0} > 0. Then there exists an δ > 0 with the following property. Suppose that γ is a GBB and γ(s_{0}) ∈ ^{b}S_{K}^{∗}M_{0}. If ∑_{j=1}^{f+1} ξ̂_{j}^{b}(γ(s_{0})) > 0 and 1 − h(y(γ(s_{0}), η̂^{b}(γ(s_{0})) > ϵ_{0} then γ|_{[s_{0},s_{0}+δ]} ∩ bS_{W}^{∗} M_{0} = ∅, while if ∑_{j=1}^{f+1} ξ̂_{j}^{b}(γ(s_{0})) < 0 and 1 − h(y(γ(s_{0}), η̂^{b}(γ(s_{0})) > ϵ_{0} then γ|_{[s_{0}−δ,s_{0}]} ∩^{b}S_{W}^{∗} M_{0} = ∅.

Proof. Let U_{1} ⊂ U be open such that K ⊂ U_{1}, \overline{U_{1}} ⊂ U. GBB’s are uniformly Lipschitz, i.e. with Lipschitz constant independent of the GBB , in compact sets (thus are equicontinuous in compact sets), so it follows that there is an δ_{1} > 0 such that γ(s_{0}) ∈ ^{b}S_{K}^{∗}M_{0} implies that γ(s) ∈ ^{b}S_{U_{1}}^{∗}M_{0} for s ∈ [s_{0} − δ_{1}, s_{0} + δ_{1}]. Now the uniform Lipschitz nature of the function 1 − h(y(γ(s)), η̂^{b}(γ(s))) shows that there exists δ_{2} ∈ (0, δ_{1}] such that for |s−s_{0}| ≤ δ_{2}, 1 − h(y(γ(s)), η̂^{b}(γ(s))) > ϵ_{0}/2. Now let f = ∑ ξ̂_{j}^{b}. Then

H_{s}(π_{s→b}^{∗}f)|_{^{s}Σ_{0}} = − ∑ A_{ij}ξ̂_{i}^{b}ξ̂_{j}^{b}+∑ x_{j}F_{1j} = −(1−∑ B_{ij}ζ̂_{i}^{b}ζ̂_{j}^{b})+∑ x_{j}F_{2j},

with F_{1j}, F_{2j} ∈ C^{∞}(S^{∗}M_{0}), so there exist δ_{3} > 0 and c > 0 such that if x_{j} < δ_{3} for j = 1, …, f + 1, then H_{s}(π_{s→b}^{∗}f)|_{^{s}Σ_{0}} ≤ −c. Now if x_{j}(γ(s_{0})) > δ_{3}/2 for some j, the uniform Lipschitz character of x_{j} ∘ γ shows the existence of δ′ > 0 (independent of γ) such that x_{j}(γ(s_{0})) ≠ 0 for |s − s_{0}| < δ′. On the other hand, if x_{j}(γ(s_{0}) ≤ δ_{3}/2 for all j, then the uniform Lipschitz character of x_{j} ∘γ shows the existence of δ′′ ∈ (0, δ_{2}] such that x_{j}(γ(s_{0})) < δ_{3} for |s − s_{0}| < δ′′, so f(γ(s)) is strictly decreasing on [s_{0} − δ′′, s_{0} + δ′′]. In particular, if f(γ(s_{0})) > 0, then f(γ(s)) > 0 for s ∈ [s_{0} − δ′′, s_{0}], so γ(s_{0}) ∉ ^{b}S_{W}^{∗} M_{0}, and if f(γ(s_{0})) < 0, then f(γ(s)) < 0 for s ∈ [s_{0}, s_{0} + δ′′], so γ(s_{0}) ∉ ^{b}S_{W}^{∗} M_{0} again. This completes the proof of the lemma. □

We now recall the following statement, due to Lebeau.

Lemma 3.6. (Lebeau, [14, Proposition 1]) If γ is a generalized broken bicharacteristic, s_{0} ∈ I, q_{0} = γ(s_{0}), then there exist unique q̃_{+}, q̃_{−} ∈ ^{s}Σ_{0} satisfying π_{s→b}(q̃_{±}) = q_{0} and having the property that if f ∈ C^{∞}(^{b}S^{∗}M_{0}) then f ∘ γ is differentiable both from the left and from the right at s_{0} and

(\frac{d}{ds}) (f ∘ γ)|_{s_{0}±} = H_{s}π_{s→b}^{∗}f(q̃_{±}).

Definition 3.7. A generalized broken bicharacteristic segment γ, defined on [0, s_{0}) or (−s_{0}, 0], γ(0) = q ∈ H_{Y,b} is said to approach W normally as s → 0 if for all j,

lim_{s→0±} \frac{x_{j}(γ(s))}{s} ≠ 0;

this limit always exists by [14, Proposition 1].

Remark 3.8. If γ approaches W normally then there is s_{1} > 0 such that γ(s) ∈ S^{∗}M_{0}^{∘} for s ∈ (0, s_{1}) or s ∈ (−s_{1}, 0) since x_{j}(γ(0)) = 0, and the one-sided derivative of x_{j} ∘ γ is non-zero.

While the actual derivatives depend on the choice of the defining functions x_{j} for the boundary hypersurfaces, the condition of normal incidence is independent of these choices.

3.2. Edge-smooth cotangent bundle. We now discuss another bundle, ultimately in order to discuss the refinement of GBB’s that allows us to obtain a diffractive improvement. Let β : M → M_{0} be the blow-down map.

Let V_{es}(M) denote the set of vector fields that are tangent to the fibers of β|W̃ : W̃ → W (hence to W̃). This is a C^{∞}(M)-module, with sections locally spanned by

x∂_{x}, x∂_{y_{j}}, ∂_{z′_{j}}, ∂_{z′′_{j}}.

(In fact, one can always use local coordinate charts without the z′′ variables in this setting.) Under the blow-down map β : M → M_{0}, elements of V(M_{0}) lift to vector fields of the form x^{−1}V, V ∈ V_{es}(M), where x is a defining function of the front face, W̃. Conversely, x^{−1}V_{es}(M) is spanned by the lift of elements of V(M_{0}) over C^{∞}(M), i.e.

(3.13)  x^{−1}V_{es}(M) = C^{∞}(M) ⊗_{C^{∞}(M_{0})} β^{∗}V(M).

Let ^{es}T M denote the “edge-smooth” tangent bundle of M, defined as the bundle whose smooth sections are elements of V_{es}(M); such a bundle exists by the above description of a local spanning set of sections. Let ^{es}T^{∗}M denote the dual bundle. Thus in the coordinates of §2, sections of ^{es}T^{∗}M are spanned by

(3.14)  \underline{τ} \frac{dt}{x} + \underline{ξ} \frac{dx}{x} + \underline{η} · \frac{dy}{x} + \underline{ζ}′ · dz′ + \underline{ζ}′′ · dz′′.

By (3.13), taking into account that dt^{2} − g_{0} is a Lorentz metric on M_{0}, we deduce that its pull-back g to M is a Lorentzian metric on x^{−1}^{es}T^{∗}M, i.e. that x^{−2}g is a symmetric non-degenerate bilinear form on ^{es}T M with signature (+, −, …, −). Correspondingly, the dual metric G has the property that x^{2}G is a Lorentzian metric on ^{es}T^{∗}M. Note that G is the pull-back of G_{0} = σ_{2}(□) ∈ C^{∞}(T^{∗}M_{0} \ o). We thus conclude that σ_{2}(□) ∈ C^{∞}(T^{∗}M_{0} \ o) lifts to an element of x^{−2}C^{∞}(^{es}T^{∗}M \ o); let

p = σ_{es,2}(x^{2}□) ∈ C^{∞}(^{es}T^{∗}M \ o)

be such that x^{−2}p is this lift, so

p|_{x=0} = \underline{τ}^{2} − (\underline{ξ}^{2} + h(y, \underline{η}) + k(y, z, \underline{ζ})).

Let ^{es}Σ ⊂ ^{es}S^{∗}M denote the characteristic set of x^{2}□, i.e., the set

esΣ = p^{−1}({0})/ℝ^{+} = {σ_{es,2}(x^{2}□) = 0}/ℝ^{+}.

Thus, using the coordinates

(3.15)  x, y, t, z, ˆ \underline{ξ} = \underline{ξ}/|\underline{τ}|, ˆ \underline{η} = \underline{η}/|\underline{τ}|, ˆ \underline{ζ} = \underline{ζ}/|\underline{τ}|, \underline{σ} = |\underline{τ}|^{−1},

on ^{es}T^{∗}M, valid where \underline{τ} ≠ 0, hence (outside the zero section) near where p = 0, and dropping \underline{σ} to obtain coordinates on ^{es}S^{∗}M,

(3.16) ^{es}Σ∩^{es}S∗ ; W̃ M = \{(x = 0, t, y, z, ˆ \underline{ξ}, ˆ \underline{η}, ˆ \underline{ζ}) : ˆ \underline{ξ}^{2} +h(y, ˆ \underline{η})+k(y, z, ˆ \underline{ζ}) = 1\}.

The rescaled Hamilton vector field

H_{es} = |\underline{τ}|^{−1}H_{p}

is homogeneous of degree 0, and thus can be regarded as a vector field on esS^{∗}M which is tangent to ^{es}Σ. (Note that while H_{es} depends on the choice of x, and the particular homogeneous degree −1 function, |\underline{τ}|^{−1}, used to re-homogenize H_{p}, these choices only change H_{es} by a positive factor, so its direction is independent of the choices—though our choices are in any case canonical.)

With the notation of [25, Section 7] (where it is explained slightly differently, as the underlying manifold is not a blow-up of another space), corresponding to the edge fibration

β : W̃ → W = Y × ℝ_{t},

there is a natural map

ϖ_{es} : ^{es}T ∗ ; W̃ M → T^{∗}W.

In fact, in view of (3.13), the bundle x^{−1}^{es}T M (whose sections are x^{−1} times smooth sections of ^{es}T M) can be identified with β^{∗}T M_{0}, so one has a natural map x^{−1}^{es}T M → T M_{0}. Dually, x^{es}T^{∗}M can be identified with β^{∗}T^{∗}M_{0}, so one has a natural map x^{es}T^{∗}M → T^{∗}M_{0}. Multiplication by x maps ^{es}T^{∗}M to x^{es}T^{∗}M, and π_{s→b} : T^{∗}M_{0} → ^{b}T^{∗}M_{0} restricts to the quotient map T_{W}^{∗} M_{0} → T^{∗}W = T^{∗}M_{0}/N^{∗}W over W, so ϖ_{es} is given by the composite map

\begin{aligned}^{es}T ∗ ; W̃ M ∋ α ↦ xα ∈ x^{es}T ∗ ; W̃ M & ↦ β_{∗}(xα) ∈ T_{W}^{∗} M_{0} \\ & ↦ [xα] ∈ T^{∗}W ⊂ ^{b}T^{∗}M_{0}, \end{aligned}

which in local coordinates (3.14) is given by

ϖ_{es}(x = 0, y, t, z, \underline{ξ}, \underline{η}, \underline{τ}, \underline{ζ}) = (y, t, \underline{η}, \underline{τ}).

The fibers can be identified with ℝ_{ξ} ×T^{∗}Z. In view of the ℝ^{+}-action on ^{es}Σ, this gives rise to a map ϖ_{es} : ^{es}Σ → S^{∗}W, which is a fibration over H_{W,b} (where 1 − h(y, η̂^{b}) > 0) with fiber

\begin{aligned}ϖ_{es}^{−1}(y, t, η̂^{b}) = \{(x = 0, y, t, z, ˆ \underline{ξ}, ˆ \underline{η}, ˆ \underline{ζ}) : & ˆ\underline{η} = η̂^{b}, \\ & |ˆ\underline{ξ}|^{2} + k(y, z, ˆ \underline{ζ}) = 1 − h(y, η̂^{b})\}; \end{aligned}

the fibers degenerate at G_{W,b}. Then H_{es} is tangent to the fibers of ϖ_{es}. In fact, as computed in [25, Proof of Lemma 2.3] (which is directly valid in our setting), using coordinates (3.15) on ^{es}T^{∗}M,

\begin{matrix}− \frac{1}{2}H_{es} = −\frac{1}{2}\underline{σ}H_{p} \\ = ˆ \underline{ξ}(x∂_{x} − \underline{σ}∂\underline{σ} − ˆ \underline{ζ} · ∂ˆ\underline{ζ}) + K^{ij}ˆ\underline{ζ}_{i}∂_{z_{j}} + K^{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j}∂ˆ\underline{ξ} − \frac{1}{2} \frac{∂K^{ij}}{∂z_{k}} ˆ \underline{ζ}_{i}ˆ\underline{ζ}_{j}∂_{ˆ\underline{ζ}_{k}} + xH′, \end{matrix} (3.17)

with H′ tangent to the boundary, hence as a vector field on^{es}S^{∗}M, restricted to ^{es}S∗ ; W̃ M, H_{es} is given by

(3.18)  −\frac{1}{2}H_{es} = −ˆ\underline{ξ}ˆ\underline{ζ}∂ˆ\underline{ζ} + K^{ij}ˆ\underline{ζ}_{i}∂_{z_{j}} + K^{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j}∂ˆ\underline{ξ} − \frac{1}{2} \frac{∂K^{ij}}{∂z_{k}} ˆ \underline{ζ}_{i}ˆ\underline{ζ}_{j}∂_{ˆ\underline{ζ}_{k}}.

It is thus tangent to the fibers given by the constancy of y, t, ˆ \underline{η}. Notice also that H_{es} is indeed tangent to the characteristic set, given by (3.16), and in esS∗ ; W̃ M, it vanishes exactly at ˆ \underline{ζ} = 0. We let

R_{es} = \{q ∈ ^{es}Σ ∩^{es}S∗ ; W̃ M : H_{es}(q) = 0\} = \{(t, y, z, ˆ \underline{ξ}, ˆ \underline{η}, ˆ \underline{ζ}) ∈ ^{es}Σ : ˆ \underline{ζ} = 0\}

be the es-radial set.

3.3. Edge-b cotangent bundle. Finally, we construct a bundle ^{eb}T^{∗}M over M that behaves like ^{b}T^{∗}M away from W̃, and behaves like ^{e}T^{∗}M near the interior of W̃. Before doing so, we remark that the pullback of ^{b}T^{∗}M_{0} to M is ^{b}T^{∗}M, so β : M → M_{0} induces a map

β_{♯} : ^{b}T^{∗}M → ^{b}T^{∗}M_{0},

such that

β_{♯}|_{^{b}T_{w}^{∗}M} → ^{b}T_{β(w)}^{∗}M_{0}, w ∈ M,

is an isomorphism. It commutes with the ℝ^{+}-action, hence induces a map

β̂_{♯} : ^{b}S^{∗}M → ^{b}S^{∗}M_{0},

such that

β̂_{♯}|_{^{b}S_{w}^{∗}M} → ^{b}S_{β(w)}^{∗}M_{0}, w ∈ M,

is an isomorphism.

More precisely, ^{eb}T^{∗}M arises from the lift of vector fields on M_{0} which are tangent to all faces of M_{0} and vanish at W. (The set Ṽ of such vector fields is a C^{∞}(M_{0})-module, but is not all sections of a vector bundle over M_{0}— unlike its analogue, V(M_{0}), in the construction of V_{es}(M); locally x_{j}∂_{x_{j}}, j = 1, …, f + 1, and x_{i}∂_{y_{j}}, i = 1, …, f + 1, j = 1, …, n − f − 1, give a spanning list.)

Definition 3.9. Let V_{eb}(M) consist of vector fields whose sections are tangent to all of ∂M and to the fibers of W̃ .

This is again a C^{∞}(M)-module, and locally x∂_{x}, x∂_{y_{j}}, z′_{j}∂_{z′_{j}} and ∂_{z′′_{j}} give a spanning set; in fact

V_{eb}(M) = C^{∞}(M) ⊗_{C^{∞}(M_{0})} β^{∗}Ṽ.

Thus, there is a vector bundle, called the “edge-b” tangent bundle of M, denoted ^{eb}T M, whose sections are exactly elements of V_{eb}(M). Let ^{eb}T^{∗}M denote the dual bundle. Thus in the coordinates of §2, sections of ^{eb}T^{∗}M are spanned by

τ \frac{dt}{x} + ξ \frac{dx}{x} + η · \frac{dy}{x} + ∑ ζ′_{i} \frac{dz′_{i}}{z′_{i}} + ζ′′ · dz′′

In particular, we point out that the lift of ∑ x_{j}D_{x_{j}} from M_{0} to M by β is xD_{x}, up to xV_{eb}(M), hence considering their principal symbols gives

∑_{j} β^{∗}ξ_{j}^{b} = ξ at x = 0.

Dividing by β^{∗}|τ^{b}| = x^{−1}|τ| yields

(3.19)  ∑_{j} β^{∗}ξ̂_{j}^{b} = xξ̂+ O(x^{2}), ξ̂ = ξ/|τ|.

There exists a natural bundle map

π_{es→eb} : ^{es}T^{∗}M → ^{eb}T^{∗}M,

analogous to the bundle map π_{s→b} : T^{∗}M_{0} → ^{b}T^{∗}M_{0} of (3.1). In canonical coordinates, this maps

(\underline{τ}, \underline{ξ}, \underline{η}, \underline{ζ}′, \underline{ζ}′′) ↦ (τ = \underline{τ}, ξ = \underline{ξ}, η = \underline{η}, ζ′_{i} = \underline{ζ}′_{i}z′_{i}, ζ′′ = \underline{ζ}′′).

This map commutes with the ℝ^{+}-action of dilations in the fibers, and maps p^{−1}({0}) ⊂ ^{es}T^{∗}M \ o into the complement of the zero section of ^{eb}T^{∗}M, so it gives rise to a map

{ \atop π_{es→eb}} : ^{es}Σ → ^{eb}S^{∗}M.

Let

ebΣ = { \atop π_{es→eb}}(^{es}Σ) ⊂ ^{eb}S^{∗}M.

In coordinates

x, y, t, z, ξ̂ = ξ/|τ|, η̂ = η/|τ|, ζ̂ = ζ/|τ|,

on ^{eb}S^{∗}M, and analogously defined coordinates on ^{es}S^{∗}M,

{ \atop π_{es→eb}}(x, y, t, z, ˆ \underline{ξ}, ˆ \underline{η}, ˆ \underline{ζ}′, ˆ \underline{ζ}′′) = (x, y, t, z, ξ̂ = ˆ \underline{ξ}, η̂ = ˆ \underline{η}, ζ̂′_{i} = ˆ \underline{ζ}′_{i}z′_{i}, ζ̂′′ = ˆ \underline{ζ}′′),

so for w ∈ W̃, z′_{1}(w) = … = z′_{p}(w) = 0, z′_{p+1}(w) ≠ 0, …, z′_{k}(w) ≠ 0, with p ≥ 1,

ebΣ ∩^{eb}S_{w}^{∗}M = {(ξ̂,η̂, ζ̂) ∈ ^{eb}S_{w}^{∗}M : ζ̂′_{1} = … = ζ̂′_{p} = 0,
1 ≥ ξ̂^{2} + h(y, η̂) + k(y, z, (0, …, 0, ζ̂′_{p+1}, …, ζ̂′_{k}), ζ̂′′)}.

We again also obtain a map ϖ_{eb} : ^{eb}Σ ∩ ^{eb}S∗ ; W̃ M → S^{∗}W analogously to ϖ_{es} which is a fibration over H_{W,b}; in local coordinates (on S^{∗}W near the projection of ^{eb}Σ, (y, t, η̂) are local coordinates, η̂ = η/|τ|)

(3.20)  ϖ_{eb}(0, y, t, z, ξ̂, η̂, ζ̂) = (y, t, η̂ = ˆ \underline{η}).

More invariantly we can see this as follows. As discussed in [25, Section 7] in the setting where the fibers on W̃ have no boundaries, one considers the map

x· : ^{eb}T^{∗}M → ^{b}T^{∗}M

given by multiplication of the covectors by x away from W̃, which extends to a C^{∞} map as indicated, namely

\begin{matrix}x· : τ \frac{dt}{x} + ξ \frac{dx}{x} + η · \frac{dy}{x} + ∑ ζ′_{i} \frac{dz′_{i}}{z′_{i}} + ζ′′ · dz′′ \\ ↦ τ dt + xξ \frac{dx}{x} + η dy + ∑ xζ′_{i} \frac{dz′_{i}}{z′_{i}} + xζ′′ · dz′′. \end{matrix}

Note that at x = 0, this gives

\begin{matrix}x · (α) = τ dt + η dy, \\ α = τ \frac{dt}{x} + ξ \frac{dx}{x} + η · \frac{dy}{x} + ∑ ζ′_{i} \frac{dz′_{i}}{z′_{i}} + ζ′′ · dz′′ ∈ ^{eb}T_{w}^{∗}M, w ∈ W̃ . \end{matrix} ; (3.21)

In particular, as the image under (x·) ∘ π_{es→eb} of p^{−1}({0}) ⊂ ^{es}T^{∗}M \ o is disjoint from the zero section, and since multiplication by x commutes with the ℝ^{+}-action in the fibers, β̂_{♯} ∘ (x·) descends to a map

ϖ_{eb} : ^{eb}Σ → ^{b}Σ_{0},

and away from W̃ it is given by the restriction of the natural identificantion of ^{eb}S_{M\W̃}^{∗} M with ^{b}S_{M_{0}\W}^{∗} M_{0}, while at W̃, as (3.21) shows, is given by

(3.20), where we consider S^{∗}W ⊂ ^{b}S^{∗}M_{0}, cf. (3.4).

We now introduce sets of covectors that are respectively elliptic, glancing, and hyperbolic with respect to the boundary faces of M_{0} meeting at the corner W; these sets are thus of covectors over the boundary of W̃ :

E = ^{eb}S_{∂W̃}^{∗} M \^{eb}Σ = \{q ∈ ^{eb}S_{∂W̃}^{∗} M\o : ({ \atop π_{es→eb}})^{−1}(q) = ∅\},
G = \{q ∈ ^{eb}S_{∂W̃}^{∗} M\o : |{ \atop π_{es→eb}}^{−1}(q)| = 1\},
H = \{q ∈ ^{eb}S_{∂W̃}^{∗} M\o : |{ \atop π_{es→eb}}^{−1}(q)| ≥ 2\},

so ^{eb}Σ ∩^{eb}S_{∂W̃}^{∗} M = G ∪ H.

In coordinates, note that, for instance, for

w ∈ W̃ , z′_{1}(w) = … = z′_{p}(w) = 0, z′_{p+1}(w) ≠ 0, …, z′_{k}(w) ≠ 0,

with p ≥ 1, (3.22)

E ∩^{eb}S_{w}^{∗}M = {∃j, 1 ≤ j ≤ p, ζ̂′_{j} ≠ 0}
∪ {1 < ξ̂^{2} + h(y, η̂) + k(y, z, (0, …, 0, ζ̂′_{p+1}, …, ζ̂′_{k}), ζ̂′′)},
G ∩^{eb}S_{w}^{∗}M = {ζ̂′_{1} = … = ζ̂′_{p} = 0,
1 = ξ̂^{2} + h(y, η̂) + k(y, z, (0, …, 0, ζ̂′_{p+1}, …, ζ̂′_{k}), ζ̂′′)},
H ∩^{eb}S_{w}^{∗}M = {ζ̂′_{1} = … = ζ̂′_{p} = 0,
1 > ξ̂^{2} + h(y, η̂) + k(y, z, (0, …, 0, ζ̂′_{p+1}, …, ζ̂′_{k}), ζ̂′′)}.

Remark 3.10. The set G_{W,b} defined in (3.5) represents rays that are glancing with respect to the corner W, i.e., are tangent to all boundary faces meeting at W, while G defined above describes the rays that are glancing with respect to one or more of the boundary faces meeting at W (see Figure 4). The sets G_{W,b} and H_{W,b} live in S^{∗}W ⊂ ^{b}S_{W}^{∗} M_{0}. This can be lifted to ^{b}S^{∗}M by β (since ^{b}T^{∗}M = β^{∗}^{b}T^{∗}M_{0}), but in this picture G_{W,b} and H_{W,b} are global in the fibers of β, i.e., live over all of W̃, not merely over its boundary.

Figure 4. Glancing rays. The ray depicted at top, in M (projected down to X), terminates at a point in G. The ray depicted at bottom, in M_{0} (projected down to X_{0}), terminates at a point in G_{W,b}.

3.4. Bicharacteristics. We now turn to bicharacteristics in^{eb}Σ, which will be the dynamical locus of the geometric improvement for the propagation result. Taking into account that H_{es} is tangent to the fibers of ϖ_{es}, one expects that over W̃, these bicharacteristics will lie in a single fiber of the related map ϖ_{eb}, i.e. y, t, η̂ will be constant along these. The fibers of ϖ_{es} and ϖ_{eb} have a rather different character depending on whether they are over a point in G_{W,b} or in H_{W,b}. Namely, over G_{W,b} the fibers of ϖ_{es} resp. ϖ_{eb} are ξ̂^{b} = 0, ζ̂^{b} = 0 resp. ξ = 0, ζ = 0 i.e. they are the zero section. By contrast over α = (t, y, η̂) ∈ H_{W,b}, the fiber of ϖ_{es} is

H_{es→α,b} = \{(t, y, z, ˆ \underline{ξ}, ˆ \underline{η} = η̂, ˆ \underline{ζ}) ∈ ^{es}S^{∗}M : ˆ \underline{ξ}^{2} + k(y, z, ˆ \underline{ζ}) = 1 − h(y, ˆ \underline{η})\}

while that of ϖ_{eb} is

H_{eb→α,b} = {(t, y, z, ξ̂,η̂, ζ̂) ∈ ^{eb}S^{∗}M : ζ̂′_{1} = … = ζ̂′_{p} = 0,

ξ̂^{2} + k(y, z, (0, …, 0, ζ̂′_{p+1}, …, ζ̂′_{k}), ζ̂′′) ≤ 1 − h(y, η̂)}.

The geometric improvement will take place over H_{W,b}, so from now on we concentrate on this set. Now, for α = (t, y, η̂) ∈ H_{W,b}

R_{es} ∩ H_{es→α,b} = \{(t, y, z, ˆ \underline{ξ}, ˆ \underline{η} = η̂, ˆ \underline{ζ} = 0) ∈ ^{es}S^{∗}M : ˆ \underline{ξ}^{2} = 1 − h(y, ˆ \underline{η})\},

hence has two connected components which we denote by

R_{es,α,I/O} = \{(t, y, z, ˆ \underline{ξ}, ˆ \underline{η} = η̂, ˆ \underline{ζ} = 0) ∈ ^{es}S^{∗}M : ˆ \underline{ξ} = ± sgn(\underline{τ})\sqrt{1 − h(y, ˆ \underline{η})}\},

with sgn(\underline{τ}) being the constant function ±1 on the two connected components of ^{es}Σ.

Here the labels “I/O”, stand for “incoming/outgoing.” This is explained by

−\frac{1}{2}H_{es}t = −ˆ\underline{τ}, −\frac{1}{2}H_{es}x = ˆ \underline{ξ}x,

so in a neighborhood of R_{es,α,I}, H_{es}t and H_{es}x have the opposite signs, i.e. if t is increasing, x is decreasing along H_{es}, just as one would expect an ‘incoming ray’ to do; at outgoing points the reverse is the case.

We also let

R_{eb,α,I/O} = { \atop π_{es→eb}}(R_{es,α,I/O})
= \{(t, y, z, ξ̂, η̂, ζ̂ = 0) ∈ ^{eb}S^{∗}M : ξ̂ = ± sgn(τ)\sqrt{1 − h(y, η̂)}\},

and

R_{eb,S,I/O} = ∪_{α∈S}R_{eb,α,I/O}

for S ⊂ H_{W,b}.

Definition 3.11. An edge generalized broken bicharacteristic, or EGBB , is a continuous map γ : I → ^{eb}Σ such that for all f ∈ C^{∞}(^{eb}S^{∗}M), real-valued,

(3.23)

lim inf_{s→s_{0}} \frac{(f ∘ γ)(s) − (f ∘ γ)(s_{0})}{s − s_{0}}
≥ inf\{H_{es}({ \atop π_{es→eb}}^{∗}f)(q) : q ∈ { \atop π_{es→eb}}^{−1}(γ(s_{0})) ∩^{es}Σ\}.

Lemma 3.12. (1) An EGBB outside ^{eb}S∗ ; W̃ M is a reparameterized GBB (under the natural identification of ^{b}S_{M_{0}\W}^{∗} M_{0} with ^{eb}S_{M\W̃}^{∗} M), and conversely.

(2) If a point q on an EGBB lies in ^{eb}S∗ ; W̃ M, then the whole EGBB lies in ^{eb}S∗ ; W̃ M, in ϖ_{eb}^{−1}(ϖ_{eb}(q)), i.e. in the fiber of ϖ_{eb} through q.

(3) The only EGBB through a point in R_{eb,α,I/O} is the constant curve.

(4) For α ∈ H_{W,b}, an EGBB in H_{eb→α,b}\R_{eb,α,I/O} projects to a reparameterized GBB in ^{b}T^{∗}Z, hence to geodesic of length π in W̃ .

Proof.

(1) As H_{s} and H_{es} differ by an overall factor under the natural identification ι : S_{M_{0}\W}^{∗} M_{0} → ^{es}S_{M\W̃}^{∗} M, namely

ι_{∗}H_{s} = |τ^{s}|^{−1} |\underline{τ}| x^{−2}H_{es} = x^{−1}H_{es},

we obtain this immediately.

(2) The tangency of H_{es} to the fibers of ϖ_{es} means that if we set f equal to any of ±y_{j}, ±t, ±η̂_{j}^{b}, H_{es}f = 0. By (3.23), then (f ∘ γ)′(s_{0}) = 0 for all s_{0}, and for each of these choices. This ensures that γ remains in the fiber.

(3) H_{es} vanishes at the unique q ∈ { \atop π_{es→eb}}^{−1}(γ(s_{0})) ∩ ^{es}Σ if γ(s_{0}) ∈ R_{eb,α,I/O}. Moreover the function ξ̂∘ γ is in C^{1}, as

H_{es}ˆ\underline{ξ} = 2K(y, t, z, ˆ \underline{ζ}) = 2(1 − h(y, ˆ \underline{η}) − ˆ \underline{ξ}^{2})

on ^{es}Σ. Thus, (3.23) entails that if ˆ \underline{ξ} = ±\sqrt{1 − h(y, ˆ \underline{η})} at some point on an EGBB , then it is constant.

(4) This follows from a reparameterization argument, as in [25], taking into account that H_{es} is tangent to the fibers of ^{es}T ∗ ; W̃ M, hence can be considered as a vector field on ℝˆ\underline{ξ} × T^{∗}Z. (In fact, a completely analogous argument takes place in [33, Section 6] in the setting of N-body scattering.) □

Suppose now that γ : [0, δ_{0}] → ^{b}Σ_{0} is a GBB with γ(0) = α ∈ H_{W,b}. Thus, assuming δ_{0} > 0 is sufficiently small, by Lemma 3.5, γ|_{(0,δ_{0}]} ∩^{b}S_{W}^{∗} M_{0} = ∅. Since ^{b}S_{M_{0}\W}^{∗} M_{0} is naturally diffeomorphic to ^{eb}S_{M\W̃}^{∗} M, we can lift γ|_{(0,δ_{0}]}

to a curve γ̃ : (0, δ_{0}] → ^{eb}S^{∗}M in a unique fashion. It is natural to ask whether this lifted curve extends continuously to 0, which is a question we now address.

The following is easily deduced from Lebeau, [14, Proposition 1] (stated here in Lemma 3.6) and its proof:

Lemma 3.13. Suppose that α ∈ H_{W,b}. There exists δ_{0} > 0 with the following property.

Suppose γ : [0, δ_{0}] → ^{b}S^{∗}M_{0} is a GBB with γ(0) = α. Let γ̃ : (0, δ_{0}] → ebS^{∗}M be the unique lift of γ|_{(0,δ_{0}]} to ^{eb}S^{∗}M. Then γ̃ (uniquely) extends to a continuous map γ̃ : [0, δ_{0}] → ^{eb}S^{∗}M, with γ̃(0) ∈ R_{eb,α,O}.

In addition, γ approaches W normally if and only if

γ̃(0) ∉ ^{eb}S_{∂W̃}^{∗} M ∩ R_{eb,α,O} = G ∩ R_{eb,α,O}.

The analogous results hold if [0, δ_{0}] is replaced by [−δ_{0}, 0] and R_{eb,α,O} is replaced by R_{eb,α,I}.

Remark 3.14. The proof in fact shows that δ_{0} can be chosen independent of α as long as we fix some K ⊂ H_{W,b} ⊂ ^{b}S_{W}^{∗} M_{0} compact and require α ∈ K.

Remark 3.15. The special case of a normal GBB segment γ, which lifts to a curve γ̃ : [0, δ_{0}] → ^{eb}S^{∗}M starting at W̃^{∘}, follows directly by the description of geodesic in edge metrics from [25], since normality implies that for sufficiently small δ_{0} > 0, γ|_{(0,δ_{0}]} has image disjoint from x_{j} = 0 for all j, i.e. the boundaries can be ignored, and one is simply in the setting of [25]. This argument also shows that given α ∈ H_{W,b} and p ∈ R_{eb,α,O} \ G, for sufficiently small δ_{0} > 0, there is a unique GBB γ : [0, δ_{0}] → ^{b}S^{∗}M_{0} with γ(0) = α such that the lift γ̃ of γ satisfies γ(0) = p.

Proof. Let α = (y_{0}, t_{0}, η̂_{0}^{b}). First, by Lemma 3.5, γ|_{(0,δ_{0}]} ∩^{b}S_{W}^{∗} M_{0} = ∅ for δ_{0} > 0 sufficiently small, hence the lift γ̃|_{(0,δ_{0}]} exists and is unique. Lebeau proves in [14, Proof of Proposition 1] (with our notation) that

lim_{s→0} ξ̂(γ̃(s)) = \sqrt{1 − h(y_{0}, η̂_{0}^{b})} and \frac{dx(γ(s))}{ds}|_{s=0} = 2\sqrt{1 − h(y_{0}, η̂_{0}^{b})} > 0.

This implies that

sup\{|ˆ\underline{ζ}(q)| : q ∈ ({ \atop π_{s→b}})^{−1}(γ(s))\} → 0 as s → 0+,

since

∑ K_{ij}(y, z)ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j} = 1 − h(y, ˆ \underline{η}) − ˆ \underline{ξ}^{2} + xG ≤ 1 − h(y, ˆ \underline{η}) − ˆ \underline{ξ}^{2} + Cx

on ^{es}Σ, and 1 − h(y(γ̃(s)), ˆ \underline{η}(γ̃(s))) − ˆ \underline{ξ}(γ̃(s))^{2} + Cx(γ̃(s)) → 0. It remains to show that the coordinates z_{j} have a limit as s → 0. But by Lemma 3.6, (dx_{j} ∘ γ/ds)|_{s=0} = 2ξ̂_{j}^{s}(0) exists, and ∑ A_{ij}ξ̂_{i}^{s}(0)ξ̂_{j}^{s}(0) = 1 − h(y_{0}, η̂_{0}^{b}) > 0. Thus, considering z_{j}(γ(s)) = x_{j}(γ(s))/x(γ(s)), L’Hôspital’s rule shows that

lim_{s→0+} z_{j}(γ(s)) = ξ̂_{j}^{s}(0)/\sqrt{1 − h(y_{0}, η̂_{0}^{b})} exists, finishing the proof of the first claim. The second claim follows at once from the last observation regarding lim_{s→0+} z_{j}(γ(s)). □

We also need the following result, which is a refinement of Lemma 3.13, insofar as Lebeau’s result only deals with a single GBB emanating from the corner W of M_{0} : the following lemma extends Lemma 3.13 uniformly to GBB’s starting close to but not at the corner. For simplicity of notation, we only state the results for the outgoing direction.

Lemma 3.16. Suppose that α ∈ H_{W,b}, p ∈ R_{eb,O,α}, p_{n} ∈ ^{eb}S_{M\W̃}^{∗} M,

and p_{n} → p in ^{eb}S^{∗}M. Suppose δ_{0} > 0 is sufficiently small (see following remark). Let γ_{n} : [0, δ_{0}] → ^{b}S^{∗}M_{0} be GBB such that γ_{n}(0) = p_{n}. For n sufficiently large, let γ̃_{n} : [0, δ_{0}] → ^{eb}S^{∗}M be the unique lift of γ_{n} to a map [0, δ_{0}] → ^{eb}S^{∗}M. Then for N sufficiently large, {γ̃_{n}}_{n≥N} is equicontinuous.

Remark 3.17. As p_{n} → p, ξ̂(p_{n}) → ξ̂(p) > 0, so there exists N > 0 such that

∑_{j=1}^{f+1} ξ̂_{j}^{b}(p_{n}) = x(p_{n})ξ̂(p_{n}) + O(x(p_{n})^{2}) > 0

for n ≥ N; cf. (3.19). Thus, by Lemma 3.5, there exists δ_{0} > 0 such that γ_{n}|_{[0,δ_{0}]} ∩ ^{b}S_{W}^{∗} M_{0} = ∅ for n ≥ N—this is the δ_{0} in the statement of the lemma. Hence, for n sufficiently large, γ_{n} has a unique lift γ̃_{n} to ^{eb}S^{∗}M, since ^{eb}S^{∗}M and ^{b}S^{∗}M_{0} are naturally diffeomorphic away from W̃, resp. W as previously noted.

Proof. Note first that {γ_{n}}_{n∈ℕ} is equicontinuous by Lebeau’s result [14, Corollaire 2] (see also the proof of [14, Proposition 6])—indeed, this follows directly from our definition of GBB . This implies that {γ̃_{n}}_{n∈ℕ} is equicontinuous at all s_{0} ∈ (0, δ_{0}], for given such a s_{0}, there exists K_{0} ⊂ M_{0} compact disjoint from W such that γ_{n}|_{[s_{0},δ_{0}]} has image in ^{b}S_{K_{0}}^{∗}M_{0}, which is canonically diffeomorphic to ^{eb}S_{β^{−1}(K_{0})}^{∗}M. Thus, it remains to consider equicontinuity at 0.

For sufficiently large n, all γ_{n} have image in ^{b}S_{K}^{∗}M_{0} where K is compact and K ⊂ O for a coordinate chart O on M_{0}. Thus, by the equicontinuity of γ_{n}, the coordinate functions

x_{j} ∘ γ_{n}, t_{j} ∘ γ_{n}, y_{j} ∘ γ_{n}, ξ̂_{j}^{b} ∘ γ_{n}, η̂_{j}^{b} ∘ γ_{n}

are equicontinuous. We need to show that for the lifted curves, γ̃_{n}, the coordinate functions

x ∘ γ̃_{n}, t_{j} ∘ γ̃_{n}, y_{j} ∘ γ̃_{n}, z_{j} ∘ γ̃_{n}, ξ̂∘ γ̃_{n}, η̂ ∘ γ̃_{n}, ζ̂_{j} ∘ γ̃_{n}

are equicontinuous at 0. By the above description, and y_{j} ∘ γ̃_{n} = y_{j} ∘ γ_{n}, t_{j} ∘ γ̃_{n} = t_{j} ∘ γ_{n} and η̂_{j} ∘ γ̃_{n} = η̂_{j}^{b} ∘ γ_{n} are equicontinuous, as is x ∘ γ̃_{n} in view of x = (∑ a_{ij}x_{i}x_{j})^{1/2}. Thus, it remains to consider ξ̂∘γ̃_{n}, z_{j} ∘γ̃_{n} and ζ̂_{j} ∘γ̃_{n}.

Let p = (0, y_{0}, z_{0}, ξ̂_{0}, η̂_{0}, 0), and write μ = ξ̂_{0} > 0. Thus,

μ = \sqrt{1 − h(y_{0}, η̂_{0})}.

Let ϵ_{1} > 0. One can show easily, as in the proof of Lebeau’s [14, Proposition 1], that for all n sufficiently large (so that p_{n} sufficiently close to p) and s_{0} > 0 sufficiently small,

(3.24)  s ∈ [0, s_{0}] ⟹ ξ̂∘ γ̃_{n}(s) ∈ [μ − ϵ_{1}, μ + ϵ_{1}].

Indeed, H_{s}ξ̂ = 2x^{−1} ∑ K_{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j} + F with F smooth, so H_{s}ξ̂ ≥ −C_{0} over the compact set K, hence

(3.25)  ξ̂∘ γ̃_{n}(s) ≥ ξ̂(p_{n}) − C_{0}s.

On the other hand, on ^{es}Σ,

ˆ\underline{ξ}^{2} = 1 − h(y, ˆ \underline{η}) − ∑ K_{ij}(y, z)ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j} + xG ≤ 1 − h(y, ˆ \underline{η}) + C_{1}x,

hence on ^{eb}Σ,

ξ̂^{2} ≤ 1 − h(y, η̂) + C_{1}x.

Let

Φ(x, y, η̂^{b}) = \sqrt{1 − h(y, η̂^{b}) + C_{1}x};

this is thus a Lipschitz function on a neighborhood of α in ^{b}S^{∗}M_{0}, hence there is s′_{0} > 0 such that Φ ∘ γ_{n}|_{[0,s′_{0}]} is uniformly Lipschitz for n sufficiently large. Thus,

|ξ̂(γ̃_{n}(s))| ≤ Φ(γ_{n}(s)) ≤ Φ(α) + |Φ(p_{n}) − Φ(α)| + |Φ(γ_{n}(s)) − Φ(p_{n})|
≤ \sqrt{1 − h(y_{0}, η̂_{0})} + |Φ(p_{n}) − Φ(α)| + C′s.

DIFFRACTION ON MANIFOLDS WITH CORNERS 25

Thus, for sufficiently large n (so that p_{n} is close to p),

(3.26)  |ξ̂(γ̃_{n}(s))| ≤ \sqrt{1 − h(y_{0}, η̂_{0})} + ϵ_{1}/2 + C′s.

Combining (3.25) and (3.26) gives (3.24).

Now consider the function

Θ = 1 − h(y, η̂) − ξ̂^{2},

so π_{s→b}^{∗}Θ|_{^{s}Σ_{0}∩S^{∗}M_{0}} = ∑ K_{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j}. This satisfies

H_{s}Θ = −2ξ̂H_{s}ξ̂+ F_{1} = −4x^{−1}ξ̂∑ K_{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j} + F_{2} = −4x^{−1}ξ̂Θ + F_{3} with F_{j} smooth. Now,

x(p_{n}) + (μ − ϵ_{1})s ≤ x ∘ γ_{n}(s) ≤ x(p_{n}) + (μ + ϵ_{1})s,

so

\frac{d}{ds}Θ ∘ γ_{n} + 4x^{−1}ξ̂Θ ≤ C

implies that

\frac{d}{ds}Θ_{n} + \frac{4(μ − ϵ_{1})}{x(p_{n}) + (μ + ϵ_{1})s}Θ_{n} ≤ C,

where we write Θ_{n} = Θ ∘ γ_{n}. Multiplying through by

(x(p_{n}) + (μ + ϵ_{1})s)^{4(μ−ϵ_{1})/(μ+ϵ_{1})}

gives

\begin{matrix}\frac{d}{ds}((x(p_{n}) + (μ + ϵ_{1})s)^{4(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}) \\ ≤ C(x(p_{n}) + (μ + ϵ_{1})s)^{4(μ−ϵ_{1})/(μ+ϵ_{1})}. \end{matrix} ; (3.27)

Integration gives

\begin{matrix}(x(p_{n}) + (μ + ϵ_{1})s)^{4(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}(s) − x(p_{n})^{4(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}(0) \\ ≤ C′((x(p_{n}) + (μ + ϵ_{1})s)^{1+4(μ−ϵ_{1})/(μ+ϵ_{1})} − x(p_{n})^{1+4(μ−ϵ_{1})/(μ+ϵ_{1})}). \end{matrix} (3.28)

Thus, (3.29) Θ_{n}(s) ≤ (1 + (μ + ϵ_{1})s/x(p_{n}))^{−4(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}(0)

+ C′((x(p_{n}) + (μ + ϵ_{1})s) − x(p_{n})(1 + (μ + ϵ_{1})s/x(p_{n}))^{−4(μ−ϵ_{1})/(μ+ϵ_{1})}). Since

(1 + (μ + ϵ_{1})s/x(p_{n}))^{−4(μ−ϵ_{1})/(μ+ϵ_{1})} < 1,

this yields

(3.30)  Θ_{n}(s) ≤ Θ_{n}(0) + C′(x(p_{n}) + (μ + ϵ_{1})s)

On the other hand, as on ^{s}Σ_{0}, Θ = ∑ K_{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j} + xF, with F smooth, so Θ ≥ −Cx, we deduce that

Θ_{n}(s) ≥ −C(x(p_{n}) + (μ + ϵ_{1})s).

Thus,

−C(x(p_{n}) + (μ + ϵ_{1})s) ≤ Θ_{n}(s) ≤ Θ_{n}(0) + C′(x(p_{n}) + (μ + ϵ_{1})s).

Suppose now that ϵ > 0 is given. As p_{n} → p, there is an N such that for n ≥ N, Cx(p_{n}) + Θ_{n}(0), C′x(p_{n}) ≤ ϵ/2. Moreover, let s_{0} > 0 such that C(μ + ϵ_{1})s_{0}, C′(μ + ϵ_{1})s_{0} < ϵ/2. Then for n ≥ N, s ∈ [0, s_{0}], −ϵ ≤ Θ_{n}(s) − Θ_{n}(0) ≤ ϵ, giving the equicontinuity of Θ_{n} at 0 for n ≥ N. In view of the definition of Θ_{n} and the already known equicontinuity of y ∘ γ̃_{n} and η̂ ∘ γ̃_{n}, it follows that (ξ ∘ γ̃_{n})^{2}, hence ξ ∘ γ̃_{n} are equicontinuous. As on ^{es}Σ, |ˆ\underline{ζ}|^{2} ≤ C|Θ| + C′x, we also have |ζ̂|^{2} ≤ C|Θ| + C′x there, so

|ζ̂ ∘ γ̃_{n}(s) − ζ̂(p_{n})| ≤ |ζ̂(p_{n})| + |ζ̂ ∘ γ̃_{n}(s)| ≤ |ζ̂(p_{n})| + C|Θ_{n}(s)| + C′x(γ̃_{n}(s)).

Given ϵ > 0, by the equicontinuity of Θ_{n} and x ∘ γ̃_{n}, there is s_{0} such that for s ∈ [0, s_{0}], C|Θ_{n}(s)| + C′x(γ̃_{n}(s)) < ϵ/2. As ζ̂(p_{n}) → 0 due to p_{n} → p, for n sufficiently large, |ζ̂(p_{n})| < ϵ/2, so for n sufficiently large and s ∈ [0, s_{0}], |ζ̂ ∘ γ̃_{n}(s) − ζ̂(p_{n})| ≤ ϵ, giving the equicontinuity of ζ̂ ∘ γ̃_{n} at 0.

It remains to check the equicontinuity of Z_{n} = z ∘ γ̃_{n}. But

|\frac{dZ_{n}}{ds} | ≤ C sup\{x(q)^{−1}|ˆ\underline{ζ}(q)| : q ∈ ^{s}Σ_{0}, π_{s→b}(q) = γ_{n}(s)\},

and for such q, by (3.29),

x^{−2}|ˆ\underline{ζ}|^{2} ≤ Cx^{−2}(|Θ| + x)
≤ C(x(p_{n}) + (μ − ϵ_{1})s)^{−2}(1 + (μ + ϵ_{1})s/x(p_{n}))^{−4(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}(0)
+ C(x(p_{n}) + (μ + ϵ_{1})s),

so

x^{−1}|ˆ\underline{ζ}|

≤ C(x(p_{n}) + (μ − ϵ_{1})s)^{−1}(1 + (μ + ϵ_{1})s/x(p_{n}))^{−2(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}(0)^{1/2}

+ C\sqrt{x(p_{n}) + (μ + ϵ_{1})s}

≤ Cx(p_{n})^{−1}(1 + (μ − ϵ_{1})s/x(p_{n}))^{−1−2(μ−ϵ_{1})/(μ+ϵ_{1})}Θ_{n}(0)^{1/2}

+ C\sqrt{x(p_{n}) + (μ + ϵ_{1})s}.

Thus, integrating the right hand side shows that

|Z_{n}(s) − Z_{n}(0)| ≤ C′Θ_{n}(0)^{1/2}((1 + (μ − ϵ_{1})s/x(p_{n}))^{−2(μ−ϵ_{1})/(μ+ϵ_{1})} − 1)
+ C′s\sqrt{x(p_{n}) + (μ + ϵ_{1})s}
≤ C′Θ_{n}(0)^{1/2} + C′s\sqrt{x(p_{n}) + (μ + ϵ_{1})s}.

An argument as above gives the desired equicontinuity for n sufficiently large, completing the proof of the lemma. □

Corollary 3.18. Suppose that α ∈ H_{W,b}, p ∈ R_{eb,O,α}, p_{n} ∈ ^{eb}S_{M\W̃}^{∗} M, and

p_{n} → p in ^{eb}S^{∗}M. Let γ_{n} : [0, δ_{0}] → ^{b}S^{∗}M_{0} be GBB such that γ_{n}(0) = p_{n}. Then there is a GBB γ : [0, δ_{0}] → ^{b}S^{∗}M_{0} and γ_{n} has a subsequence, {γ_{n_{k}}}, such that γ_{n_{k}} → γ uniformly, the lift γ̃ : [0, δ_{0}] → ^{eb}S^{∗}M of γ satisfies γ̃(0) = p, and the lift γ̃_{n_{k}} of γ_{n_{k}} converges to γ̃ uniformly.

Proof. As p_{n} → p, it follows that there is a compact set K_{0} ⊂ M_{0} such that γ_{n}(s) ∈ ^{b}S_{K_{0}}^{∗}M_{0} for all n and all s ∈ [0, δ_{0}]. Then by the compactness of the set of GBB’s with image in ^{b}S_{K}^{∗}M_{0} in the topology of uniform convergence, [14, Proposition 6], γ_{n} has a subsequence, γ_{n_{k}}, uniformly converging to a GBB γ : [0, δ_{0}] → ^{b}S^{∗}M_{0}. In particular, γ(0) = lim_{k} γ_{n_{k}}(0) = lim_{k} ϖ_{eb}(p_{n_{k}}) = ϖ_{eb}(p) = α. By Lemma 3.13, γ lifts to a curve γ̃ : [0, δ_{0}] → ^{eb}S^{∗}M. We claim that γ̃(0) = p—once we show this, the corollary is proved.

Let γ̃_{n} : [0, δ_{0}] → ^{eb}S^{∗}M be the lift of γ_{n}. By Lemma 3.16, {γ̃_{n_{k}}}_{k∈ℕ} is equicontinuous. Since for δ > 0 γ_{n_{k}}|_{[δ,δ_{0}]} → γ uniformly, and these curves all have images in ^{b}S_{K_{1}}^{∗}M_{0} for some K_{1} compact, disjoint from W, where bS_{K_{1}}^{∗}M_{0} and ^{eb}S_{β^{−1}(K_{1})}^{∗}M are canonically diffeomorphic, we deduce that γ̃_{n_{k}}|_{[δ,δ_{0}]} → γ̃|_{[δ,δ_{0}]} uniformly; in particular {γ̃_{n_{k}}|_{[δ,δ_{0}]}} is a Cauchy sequence in the uniform topology.

Let d be a metric on ^{eb}S^{∗}M giving rise to its topology. Given ϵ > 0 let δ > 0 be such that for 0 ≤ s ≤ δ and for all n, one has d(γ̃_{n}(s), γ̃_{n}(0)) = d(γ̃_{n}(s), p_{n}) < ϵ/3—this δ exists by equicontinuity. Next, let N be such that for k, m ≥ N, d(p_{n_{k}}, p_{n_{m}}) < ϵ/3 and for k, m ≥ N, δ ≤ s ≤ δ_{0}, d(γ̃_{n_{k}}(s), γ̃_{n_{m}}(s)) ≤ ϵ/3; such a choice of N exists by the uniform Cauchy statement above, and the convergence of {p_{n}}. Thus, for k, m ≥ N and 0 ≤ s ≤ δ,

d(γ̃_{n_{k}}(s), γ̃_{n_{m}}(s)) ≤ d(γ̃_{n_{k}}(s), p_{n_{k}}) + d(p_{n_{k}}, p_{n_{m}}) + d(p_{n_{m}}, γ̃_{n_{m}}(s)) ≤ ϵ.

Since we already know the analogous claim for δ ≤ s ≤ δ_{0}, it follows that {γ̃_{n_{k}}} is uniformly Cauchy, hence converges uniformly to a continuous map γ̂ : [0, δ_{0}] → ^{eb}S^{∗}M. In particular, γ̂(0) = lim_{k} γ̃_{n_{k}}(0) = lim_{k} p_{n_{k}} = p. But γ̃_{n_{k}}|_{[δ,δ_{0}]} → γ̃|_{[δ,δ_{0}]} uniformly for δ > 0, so γ̃|_{[δ,δ_{0}]} = γ̂|_{[δ,δ_{0}]}. The continuity of both γ̃ and γ̂ now shows that γ̃ = γ̂, and in particular γ̃(0) = p as claimed. □

Now we are ready to introduce the bicharacteristics that turn out in general to carry full-strength, rather than weaker, diffracted, singularities.

Definition 3.19. A geometric GBB is a GBB γ : (−s_{0}, s_{0}) → ^{b}Σ_{0} with q = γ(0) ∈ H_{W,b} such that there is an EGBB ρ : ℝ → ^{eb}T ∗ ; W̃ M with

lim_{s→−∞} ρ(s) = lim_{t→0^{−}} γ̃_{−}(t),
lim_{s→+∞} ρ(s) = lim_{t→0^{+}} γ̃_{+}(t),

with γ̃_{+}, resp. γ̃_{−}, denoting the lifts γ|_{[0,δ_{0}]}, resp. γ|_{[−δ_{0},0]}, δ_{0} > 0 sufficiently small, to ^{eb}S^{∗}M.

We say that two points w, w′ ∈ ^{b}Σ_{0} are geometrically related if they lie along a single geometric GBB .

Let T be a large parameter, fixed for the duration of the this paper.

Definition 3.20. For p ∈ H_{W,b} the flowout of p, denoted F_{O,p}^{W}, is the union of images γ((0, T]) of GBB’s γ : [0, T] → ^{b}Σ_{0} with γ(0) = p.

For p ∈ H_{W,b}, the regular part of the flowout of p, denoted F_{O,p,reg}^{W}, is the union of images γ((0, s_{0})) of normally approaching (or regular) GBB’s γ : [0, s_{0}) → ^{b}Σ_{0} with γ(0) = p and γ(s) ∈ T^{∗}M^{∘} for s ∈ (0, s_{0}).

The regular part of the flowout of a subset of H_{W,b} is the union of the regular parts of the flowouts from the points in the set.

We let

F_{O,p,sing}^{W}

denote the union of images γ(0, T] of non-normally-approaching GBB’s γ, i.e. those GBB’s γ with γ(0) ∈ G ∩ R_{eb}.

Incoming flowout and its regular part are defined correspondingly and denoted

F_{I,p}^{W}, F_{I,p,reg}^{W}.

We let F_{I/O}^{W} denote the union of the flowouts of all p ∈ H_{W,b}.

We also need to define the flowout into or out of a single hyperbolic point q ∈ R_{eb,α,I/O} (i.e. for p ∈ H_{W,b} as above, we will consider the flow in/out to a single point in a fiber q ∈ R_{eb,p,I/O}). By Remark 3.15, given such a q, there is a unique EGBB γ(s) with lim_{s→−∞} γ(s) = q.

Definition 3.21. For q ∈ R_{eb,I/O}\^{eb}S_{∂W̃}^{∗} M, let F_{I/O,q} denote the image γ((0, T]) where γ is the unique EGBB with lim_{s→±∞} γ(s) = q. Let F_{I/O,q,reg} be defined as the union of γ((0, s_{0})) with γ(s) ∈ T^{∗}M^{∘} for all s ∈ (0, s_{0}). Additionally, let F_{I/O} denote the union of all flowouts of q ∈ R_{eb,I/O}\^{eb}S_{∂W̃}^{∗} M, and let F = F_{I} ∪ F_{O}.

One needs some control over the intervals on which normally approaching GBB do not hit the boundary of M:

Lemma 3.22. Suppose K ⊂ W̃^{∘} is compact, K ⊂ H_{W,b} is compact, ϵ_{0} > 0. Then there is δ_{0} > 0 such that if γ : [0, ϵ_{0}] → ^{b}S^{∗}M_{0} a GBB with lift γ̃, γ̃(0) ∈ R_{eb,α,O} ∩^{eb}S_{K}^{∗}M for some α ∈ K, then γ̃((0, δ_{0})) ∩^{eb}S_{∂M}^{∗}M = ∅.

Remark 3.23. Another proof could be given that uses the description of the edge bicharacteristics in [25], since the GBB covered are normally incident.

Proof. First, by Lemma 3.5 there is a δ′_{0} > 0 such that any GBB γ with γ(0) ∈ K satisfies γ|_{(0,δ′_{0}]} disjoint from ^{b}S_{W}^{∗} M_{0}.

Suppose now that there is no δ_{0} > 0 as claimed. Then there exist GBB’s γ_{j} : [0, ϵ_{0}] → ^{b}S^{∗}M_{0}, p′_{j} ∈ ^{eb}S_{K}^{∗}M ∩ R_{eb,α_{j},O}, α_{j} ∈ K, and δ_{j} > 0, δ_{j} → 0, such that γ_{j}(δ_{j}) ∈ ^{b}S_{∂M_{0}}^{∗}M_{0}, and the lift γ̃_{j} of γ_{j} satisfies γ̃_{j}(0) = p′_{j}. We may assume that δ_{j} < ϵ_{0}/2 and δ_{j} < δ′_{0} for all j, hence γ_{j}(δ_{j}) ∉ ^{b}S_{W}^{∗} M_{0}. By passing to a subsequence, using the compactness of K and of K, hence of ebS_{K}^{∗}M ∩ R_{eb,K,O}, we may assume that {α_{j}} converges to some α ∈ K, and {p′_{j}} converges to some p ∈ ^{eb}S_{K}^{∗}M ∩ R_{eb,α,O}. Using the continuity of γ̃_{j} for each j, we may then choose some 0 < ϵ_{j} < δ_{j} such that p_{j} = γ̃_{j}(ϵ_{j}) → p as well; note that p_{j} ∉ ^{eb}S∗ ; W̃ M. (We introduce ϵ_{j} to shift the argument of γ_{j} by ϵ_{j}, namely to ensure that γ_{j}(. + ϵ_{j}) at s = 0 is outside ^{b}S_{W}^{∗} M, so Corollary 3.18 is applicable.) Thus, we can apply Corollary 3.18 to conclude that γ_{j}(.+ϵ_{j}) : [0, ϵ_{0}/2] → ^{b}S^{∗}M_{0} has a subsequence γ_{n_{j}} such that γ_{n_{j}}(.+ϵ_{n_{j}}) converges uniformly to a GBB γ, the lifts γ̃_{n_{j}}(.+ϵ_{n_{j}}) also converge uniformly to the lift γ̃, and γ̃(0) = p. Thus, γ̃_{n_{j}}((δ_{n_{j}} − ϵ_{n_{j}}) + ϵ_{n_{j}}) → γ̃(0) = p since δ_{n_{j}} − ϵ_{n_{j}} → 0. As γ̃_{n_{j}}((δ_{n_{j}} − ϵ_{n_{j}}) + ϵ_{n_{j}}) ∈ ^{eb}S_{∂M\W̃}^{∗} M and ^{eb}S_{∂M\W̃^{∘}}^{∗}M

is closed, it follows that p ∈ ^{eb}S_{∂M\W̃^{∘}}^{∗}M, contradicting p ∈ ^{eb}S_{K}^{∗}M. This proves the lemma. □

Corollary 3.24. Suppose U ⊂ W̃^{∘} is open with Ū ⊂ W̃^{∘} compact, U ⊂ H_{W,b} is open with Ū ⊂ H_{W,b} compact. Then there is δ_{0} > 0 such that the set O of points p ∈ ^{eb}S^{∗}M for which there is a GBB γ with lift γ̃ such that γ̃(0) ∈ ^{eb}S_{U}^{∗}M ∩R_{eb,U,O} and γ(s) = p for some s ∈ [0, δ_{0}) is a C^{∞} coisotropic submanifold of ^{eb}S^{∗}M transversal to ^{eb}S∗ ; W̃ M.

Proof. By Lemma 3.22, with K = Ū, K = Ū, there is a δ_{0} > 0 as in the lemma, hence the set O consists of points p for which the GBB γ only meet ∂M at s = 0, so (taking into account part (2) of Lemma 3.12 as well) O is a subset of the edge flow-out studied in [25] (e.g. by extending the edge metric g smoothly across the boundary hypersurfaces other than W̃). In particular, the properties of the flowout of such an open subset being C^{∞}, coisotropic^{7} and transversal to ^{eb}S∗ ; W̃ M follow from Theorem 4.1 of [25]. □

We now turn to properties of the singular flowout.

Lemma 3.25. The singular flowout, F_{sing}, is closed in ^{eb}S_{M\W̃}^{∗} M.

Proof. Suppose p_{n} ∈ F_{sing}, and let γ_{n} be such that the lift γ̃_{n} of γ_{n} satisfies γ̃_{n}(0) ∈ G ∩ R_{eb}, and γ_{n}(s_{n}) = p_{n}, s_{n} ∈ (0, T]. Suppose that p_{n} → p ∈ ebS_{M\W̃}^{∗} M. Then there exists a compact subset K of M such that γ_{n}(s) ∈

bS_{K}^{∗}M for all n and all s ∈ [0, T]. By passing to a subsequence we may assume that s_{n} → s; as p ∉ ^{eb}S∗ ; W̃ M, s ≠ 0. By passing to yet another subsequence we may also assume that γ_{n}(0) → q ∈ G ∩ R_{eb}. Let ϵ_{n} > 0, ϵ_{n} → 0, so γ_{n}(ϵ_{n}) ∉ ^{eb}S∗ ; W̃ M and γ_{n}(ϵ_{n}) → q. By Corollary 3.18 we conclude that γ_{n}(.+ϵ_{n}) : [0, T] → ^{b}S^{∗}M_{0} has a subsequence γ_{n_{j}} such that γ_{n_{j}}(.+ϵ_{n_{j}}) converges uniformly to a GBB γ, the lifts γ̃_{n_{j}}(.+ϵ_{n_{j}}) also converge uniformly to the lift γ̃, and γ̃(0) = q. In particular, as γ_{n_{j}}((s_{n_{j}}−ϵ_{n_{j}})+ϵ_{n_{j}}) = γ_{n_{j}}(s_{n_{j}}) = p_{n_{j}} → p, and s_{n_{j}} − ϵ_{n_{j}} → s, γ(s) = p, so p ∈ F_{sing} as claimed. □

7In [25], being coisotropic is considered as a property of submanifolds of a symplectic manifold, ^{e}T^{∗}M \ o, M being an edge manifold. Conic submanifolds of ^{e}T^{∗}M \ o can be identified with submanifolds of ^{e}S^{∗}M, and conversely, thus one can talk about submanifolds of ^{e}S^{∗}M being coisotropic. Alternatively, this notion could be defined using the contact structure of ^{e}S^{∗}M, but for the sake of simplicity, and due to the role of symplectic structures in classical microlocal analysis, we did not follow this route in [25], necessitating making the connection via homogeneity here.

Lemma 3.26. Suppose K ⊂ W̃^{∘} is compact, K ⊂ H_{W,b} is compact. Then K has a neighborhood U in M and there is ϵ_{0} > 0 such that if γ : [0, ϵ_{0}] → bS^{∗}M_{0} is a GBB with lift γ̃, γ̃(0) ∈ R_{eb,O} ∩ G, γ(0) ∈ K then γ̃(s) ∉ ^{eb}S_{U}^{∗}M for s ∈ (0, ϵ_{0}].

Proof. Let ϵ_{0} > 0 be such that any GBB γ with γ(0) ∈ K satisfies γ|_{(0,ϵ_{0}]} disjoint from ^{b}S_{W}^{∗} M_{0}; such ϵ_{0} exists by Lemma 3.5.

Now suppose that no U exists as stated. Then there exist GBB γ_{n} and s_{n} ∈ (0, ϵ_{0}] such that the lifts γ̃_{n} of γ_{n} satisfies γ̃_{n}(0) ∈ R_{eb,O} ∩ G, γ_{n}(0) ∈ K and π(γ̃_{n}(s_{n})) → q, q ∈ K, where π : ^{eb}S^{∗}M → M is the bundle projection.

By the compactness of K and the compactness of ∪_{α∈K}R_{eb,O,α} ∩ G we may pass to a subsequence (which we do not indicate in notation) such that γ_{n}(0) converges to some α ∈ K and γ̃_{n}(0) converges to some p ∈ R_{eb,O} ∩ G. We may further pass to a subsequence such that s_{n} → s_{0} ∈ [0, ϵ_{0}], and still further (taking into account the compactness of the fibers of ^{eb}S^{∗}M → M) that γ̃_{n}(s_{n}) → p̃ ∈ ^{eb}S_{K}^{∗}M. Choose^{8} ϵ_{n} ∈ (0, s_{n}) sufficiently small such that ϵ_{n} → 0 and γ_{n}(ϵ_{n}) → p. By Corollary 3.18 γ_{n}(. + ϵ_{n}) has a convergent subsequence γ_{n_{k}} such that γ_{n_{k}}(. + ϵ_{n_{k}}) converge uniformly to a GBB γ and the lifts γ̃_{n_{k}}(. + ϵ_{n_{k}}) converge uniformly to the lift γ̃ and γ̃(0) = p. Thus, γ̃_{n_{k}}(s_{n_{k}} + ϵ_{n_{k}}) → γ̃(s_{0}), so γ̃(s_{0}) = p̃ ∈ ^{eb}S_{K}^{∗}M. But by the definition of ϵ_{0}, γ(s_{0}) ∉ ^{b}S_{W}^{∗} M_{0} if s_{0} > 0, while s_{0} = 0 is impossible as γ̃(0) = p ∈ ^{eb}S_{∂W̃}^{∗} M, while K ⊂ W̃^{∘}. This contradiction shows that the claimed U exists, proving the lemma. □

Corollary 3.27. Suppose K ⊂ W̃^{∘} is compact, K ⊂ H_{W,b} is compact. Then K has a neighborhood U in M and there is ϵ_{0} > 0 such that if o ∈ U \ W̃ and γ is a GBB with γ(0) ∈ ^{b}S_{o}^{∗}M_{0} then for s ∈ [−ϵ_{0}, 0], γ(s) ∈ K implies γ is normally incident.

In particular, if q ∈ W̃^{∘}, α ∈ H_{W,b} and γ_{0} is a GBB with γ_{0}(0) = α and lift γ̃_{0}(0) ∈ ^{eb}S_{q}^{∗}M then there is δ_{0} > 0 such that s ∈ (0, δ_{0}], γ_{0}(s) ∈ ^{b}S_{o}^{∗}M_{0} implies that every GBB γ with γ(0) ∈ ^{b}S_{o}^{∗}M_{0}, γ(s) = α, s ∈ [−ϵ_{0}, 0], is normally incident.

Proof. Let U and ϵ_{0} be as in Lemma 3.26. If o ∈ U, γ is a GBB with γ(0) ∈ ^{b}S_{o}^{∗}M_{0}, s_{0} ∈ [−ϵ_{0}, 0], γ(s_{0}) ∈ K and γ is not normally incident, then the lift γ̃ of γ satisfies γ̃(s_{0}) ∈ R_{eb,O} ∩ G by Lemma 3.13. Thus, with γ_{0}(s) = γ(s − s_{0}), so γ̃_{0}(0) ∈ R_{eb,O} ∩ G, γ_{0}(0) ∈ K, Lemma 3.26 shows that γ̃_{0}(s) ∉ ^{eb}S_{U}^{∗}M for s ∈ (0, ϵ_{0}], contradicting γ̃_{0}(−s_{0}) ∈ ^{b}S_{o}^{∗}M_{0}.

8Again, we do this so that Corollary 3.18 is applicable; cf. the proof of Lemma 3.22.

The second half follows by taking K = {α}, K = {q}. □

3.5. A summary. The following table summarizes a number of the most useful facts about the bundles that we have introduced above.

Manifold  M_{0}  M_{0}  M  M
Bundle  b  s  eb  es
Vector fields x_{i}∂_{x_{i}}, ∂_{y_{k}} ∂_{x_{j}}, ∂_{y_{k}} x∂_{x}, x∂_{y}, z′_{i}∂_{z′_{i}}, ∂_{z′′_{j}} x∂_{x}, x∂_{y}, ∂_{z_{i}}
Dual coords  ξ_{j}^{s}, η_{j}^{s}  ξ_{j}^{b}, η_{j}^{b}  ξ, η, ζ′, ζ′′  \underline{ξ}, \underline{η}, \underline{ζ}
Char. set  ^{b}Σ_{0}  ^{s}Σ_{0}  ^{eb}Σ  ^{es}Σ

(We have omitted time coordinates and their duals, as they behave just like y variables, and the notation follows suit.)

We also employ a number of maps among these structures, the most common being:

π_{s→b} : T^{∗}M_{0} → ^{b}T^{∗}M_{0},
π_{es→eb} : ^{es}T^{∗}M → ^{eb}T^{∗}M,
ϖ_{es} : ^{es}T ∗ ; W̃ M → T^{∗}W,
ϖ_{eb} : ^{eb}T ∗ ; W̃ M → T^{∗}W.

Recall that hats over maps indicate their restrictions to the relevant characteristic set.

4. Edge-b calculus

Recall from Definition 3.9 that V_{eb}(M) is the space of smooth vector fields that are tangent to all of ∂M and tangent to the fibration of W̃ ⊂ ∂M given by blowdown. Thus, in local coordinates, V_{eb}(M) is spanned over C^{∞}(M) by the vector fields

x∂_{x}, x∂_{t}, x∂_{y}, z′_{i}∂_{z′_{i}}, ∂_{z′′}

Definition 4.1. The space Diff_{eb}^{∗}(M) is the filtered algebra of operators over C^{∞}(M) generated by V_{eb}(M).

Recall also that V_{eb}(M) = C^{∞}(M;^{eb}T M), and ^{eb}T^{∗}M is the dual bundle of ^{eb}T M. In Appendix B the corresponding pseudodifferential operators are constructed.

Theorem 4.2. There exists a pseudodifferential calculus Ψ_{eb}^{∗}(M) microlocalizing Diff_{eb}^{∗}(M).

The double space M_{eb}^{2} on which the kernels are defined is such that the quotient x/x′ of the same boundary defining function on the left or right factor, lifts to be smooth except near the ‘old’ boundaries at which the kernels are required to vanish to infinite order. It follows that x/x′ is a multiplier (and divider) on the space of kernels. This corresponds the action by conjugation of these defining fucntions, so it is possible to define a weighted version of the calculus. Set

Ψ_{eb}^{m,l}(M) = x^{−l}Ψ_{eb}^{m}(M).

Proposition 4.3. Ψ_{eb}^{∗,∗}(M) is a bi-filtered calculus.

Now, Ψ_{eb}^{∗}(M) has all the properties (I–VII) of [25, Section 3], where V in [25, Section 3] is replaced by eb . Since the multiplier x/x′ is identically equal to one on the lifted diagonal, the symbol is unaffected by this conjugation and hence the principal symbol map extends to

σ_{eb,m,l} : Ψ_{eb}^{m,l}(M) → x^{−l}S_{hom}^{m}(^{eb}T^{∗}M),

with the standard short exact sequence – see properties (III–IV). There are edge-b-Sobolev spaces, H_{eb}^{s}(M), defined via the elliptic elements of Ψ_{eb}^{s}(M), and on which the elements define bounded maps

A ∈ Ψ_{eb}^{m}(M) ⟹ A : H_{eb}^{s}(M) → H_{eb}^{s−m}(M).

see property (VII).

The symbol of the commutator of A ∈ Ψ_{eb}^{m,l}(M) and B ∈ Ψ_{eb}^{m′,l′}(M) is given by

σ_{eb,m+m′−1,l+l′}(ı[A, B]) = H_{eb,σ_{eb,m,l}(A)}(σ_{eb,m′,l′}(B)).

In local coordinates the edge-b Hamilton vector field becomes

(4.1) H_{eb,f} = \frac{∂f}{∂ξ} x∂_{x} − (x\frac{∂f}{∂x} + η · \frac{∂f}{∂η} )∂_{ξ} + x\frac{∂f}{∂η} ∂_{y} + (\frac{∂f}{∂ξ} η − x\frac{∂f}{∂y} ) · ∂_{η}

+ ∑ (\frac{∂f}{∂ζ′_{j}} z′_{j}∂_{z′_{j}} − z′_{j} \frac{∂f}{∂z′_{j}} ∂_{ζ′_{j}}) + ∑ (\frac{∂f}{∂ζ′′_{j}} ∂_{z′′_{j}} − \frac{∂f}{∂z′′_{j}} ∂_{ζ′′_{j}}).

In particular,

(4.2)  x^{−k}H_{eb,x^{k}a} = ka∂_{ξ} + H_{eb,a}.

In the space-time setting, where one of the y variables, t, is distinguished (and we still write y for the rest of the base variables), it is useful to rewrite this using the re-homogenized dual variables η̂ = η/|τ|, ξ̂ = ξ/|τ|, ζ̂ = ζ/|τ|, σ = |τ|^{−1}, valid near ^{eb}Σ, this becomes

(4.3) σ^{−1}H_{eb,f} = \frac{∂f}{∂ξ̂}x∂_{x} − (x\frac{∂f}{∂x} − σ \frac{∂f}{∂σ} − ζ̂ · \frac{∂f}{∂ζ̂})∂ξ̂ + x\frac{∂f}{∂η̂}∂_{y}

− x(σ \frac{∂f}{∂σ} + η̂ · \frac{∂f}{∂η̂} + ξ̂\frac{∂f}{∂ξ̂} + ζ̂ · \frac{∂f}{∂ζ̂})∂_{t} − \frac{∂f}{∂ξ̂}(σ∂_{σ} + ζ̂ · ∂ζ̂)

+ x\frac{∂f}{∂t} · (σ∂_{σ} + η̂ · ∂η̂ + ξ̂∂ξ̂ + ζ̂ · ∂ζ̂) − x\frac{∂f}{∂y} · ∂η̂

+ ∑ (\frac{∂f}{∂ζ̂′_{j}} z′_{j}∂_{z′_{j}} − z′_{j} \frac{∂f}{∂z′_{j}} ∂ζ̂′_{j}) + ∑ (\frac{∂f}{∂ζ̂′′_{j}} ∂_{z′′_{j}} − \frac{∂f}{∂z′′_{j}} ∂ζ̂′′_{j}).

This is tangent to the fibers of ϖ_{eb} : ^{eb}Σ ∩ ^{eb}S∗ ; W̃ M → S^{∗}W, in fact to its natural extension to a neighborhood of ^{eb}Σ ∩^{eb}S∗ ; W̃ M in ^{eb}S∗ ; W̃ M, so if b ∈ C^{∞}(^{eb}S∗ ; W̃ M) with b|_{^{eb}S∗ ; W̃ M} constant along the fibers of this extension,

then σ^{μ−1}H_{eb,f}b ∈ xC^{∞}(^{eb}S^{∗}M) for f homogeneous degree μ.

The fact that the operators are defined by kernels which are conormal means that there is an operator wave front set WF′_{eb} for the eb-calculus, i.e. for A ∈ Ψ_{eb}^{∗}(M), WF′_{eb}(A) ⊂ ^{eb}S^{∗}M, with the properties (A)-(F) of [25, Section 3], so in particular algebraic operations are microlocal, see properties (A)-(B), and there are microlocal parametrices at points at which the principal symbol is elliptic, see property (E). These parametrices have error terms with which are smooth on the double space, but they are not compact.

As is the case for the b-calculus, Ψ_{eb}^{∗}(M) has the additional property that the radial vector fields V_{j} for all boundary hypersurfaces {ρ_{j} = 0} other than W̃, [A, V_{j}] ∈ ρ_{j}Ψ_{eb}^{m}(M) if A ∈ Ψ_{eb}^{m}(M), i.e. there is a gain of ρ_{j} over the a priori order. This follows from the analysis of the normal operators, see [25, Section 3]. In local coordinates a radial vector field for z′_{j} = 0 is given by z′_{j} ∂_{z′_{j}}; V_{j} being a radial vector field for z′_{j} = 0 means that V_{j} − z′_{j} ∂_{z′_{j}} ∈ z_{j}V_{eb}(M). This latter requirement can easily be seen to be defined independently of choices of coordinate systems.

5. Differential-pseudodifferential operators

5.1. The calculus. We start by defining an algebra of operators which includes □. First, recall that V_{es}(M) is the Lie algebra of vector fields that are tangent to the front face and to the fibers of the blow down map restricted to the front face, β|W̃ : W̃ → W (but are not required to be tangent to other boundary faces). Thus, elements V of V_{es}(M) define operators V : Ċ^{∞}(M) → Ċ^{∞}(M) and also V : C^{∞}(M) → C^{∞}(M).

Definition 5.1. Let Diff_{es}(M) be the filtered algebra of operators (acting either on Ċ^{∞}(M) or C^{∞}(M)) over C^{∞}(M) generated by V_{es}(M).

We also let Diff_{es}^{k,l}(M) = x^{−l} Diff_{es}^{k}(M); this is an algebra of operators acting on Ċ^{∞}(M), and also on the space of functions classical conormal to W̃, ∪_{s∈ℝ}x^{−s}C^{∞}(M).

Remark 5.2. Note that the possibility of the appearance of boundary terms requires care to be exercised with adjoints, as opposed to formal adjoints. See for instance Lemma 5.18.

We also remark that Diff_{es}^{k}(M), hence x^{−l} Diff_{es}^{k}(M), is closed under conjugation by x^{−r} where x is a defining function for W. This follows from the fact that Diff_{es}^{1}(M) is so closed; the key property is that

x^{r}(x∂_{x})x^{−r} = (x∂_{x}) − r ∈ Diff_{es}^{1}(M).

We will require, for commutator arguments that involve interaction of singularities with ∂M\W̃ , a calculus of mixed differential-pseudodifferential operators, mixing edge-b-pseudodifferential operators with these (more singular) edge-smooth differential operators.

Definition 5.3. Let

Diff_{es}^{k} Ψ_{eb}^{m}(M) = {∑ A_{α}B_{β} : A_{α} ∈ Diff_{es}^{k}(M), B_{β} ∈ Ψ_{eb}^{m}(M)}

Proposition 5.4. ∪_{k,m} Diff_{es}^{k} Ψ_{eb}^{m}(M) is a filtered C^{∞}(M)-module, and an algebra under composition; it is commutative to top eb-order, i.e. for P ∈ Diff_{es}^{k} Ψ_{eb}^{m}(M), Q ∈ Diff_{es}^{k′} Ψ_{eb}^{m′}(M),

[P, Q] ∈ Diff_{es}^{k+k′} Ψ_{eb}^{m+m′−1}(M).

The key is the following lemma.

Lemma 5.5. If A ∈ Ψ_{eb}^{m}(M) and Q ∈ V_{es}(M), then

(5.1)  [A, Q] = ∑ Q_{j}A_{j} + B, [A, Q] = ∑ A′_{j}Q′_{j} + B′_{j}

where B, B′ ∈ Ψ_{eb}^{m}(M), A_{j}, A′_{j} ∈ Ψ_{eb}^{m−1}(M) and Q_{j}, Q′_{j} ∈ V_{es}(M).

Proof. As both V_{es}(M) and Ψ_{eb}^{m}(M) are C^{∞}(M)-modules, we can use a partition of unity, and it suffices to work locally and with a spanning set of vector fields. Since xD_{x}, xD_{y_{j}}, D_{z′′_{j}} ∈ V_{eb}(M), the conclusion is automatic for Q chosen from among these vector fields since then B = [A, Q] ∈ Ψ_{eb}^{m}(M). Thus it only remains to consider the Q = D_{z′_{j}} where z′_{j} is a defining function

for one of the other boundary faces. Then for Q̃ = z′_{j}Q = z′_{j}D_{z′_{j}} ∈ Diff_{eb}^{1}(M),

[A, Q̃] ∈ Ψ_{eb}^{m}(M)). The normal operator at z′_{j} = 0 satisfies N_{j}([A, Q̃]) = [N_{j}(A), N_{j}(Q̃)], and N_{j}(Q̃) is scalar, and hence commutes with N_{j}(A). Thus N_{j}([A, Q̃]) = 0, so [A, Q̃] ∈ z′_{j}Ψ_{eb}^{m}(M). Consequently,

(5.2)  −[A, Q] = [Q, A] = (z′_{j})^{−1}[Q̃, A] + ([(z′_{j})^{−1}, A]z′_{j})(z′_{j})^{−1}Q̃,

with the first term on the right hand side in Ψ_{eb}^{m}(M), the second of the form ÃQ, Ã ∈ Ψ_{eb}^{m−1}(M). This proves the first half of the lemma. The other part is similar. □

Proof of Proposition 5.4. The algebra properties follow immediately from the lemma. It only remains to verify the leading order commutativity.

As the bracket is a derivation in each argument, it suffices to consider P, Q lying in either V_{es}(M) or Ψ_{eb}^{∗}(M). If both operators are in Ψ_{eb}^{∗}(M), the result follows from the symbol calculus. If P, Q ∈ V_{es}(M), we have [P, Q] = R ∈ V_{es}(M). We need to write R as a sum of elements of Diff_{es}^{2}(M) times elements of Ψ_{eb}^{−1}(M). To this end, let Λ be an elliptic element of Ψ_{eb}^{2}(M) given by a sum of square of vector fields in V_{eb}(M), e.g. in local coordinates

Λ = (x∂_{x})^{2} + (x∂_{t})^{2} + ∑(x∂_{y})^{2} + ∑(z′_{i}∂_{z′_{i}})^{2} + ∑ ∂_{z′′_{j}}^{2}.

We write Λ = ∑ V_{j}^{2} for brevity. Let Υ ∈ Ψ_{eb}^{−2}(M) be an elliptic parametrix for Λ. Then we may write

Id = ∑ V_{j}(V_{j}Υ) + E,

with E ∈ Ψ_{eb}^{−∞}(M). Now since V_{eb}(M) ⊂ V_{es}(M), we certainly have V_{j} ∈ V_{es}(M) for each j, hence RV_{j} ∈ Diff_{es}^{2}(M). Moreover V_{j}Υ ∈ Ψ_{eb}^{−1}(M). Thus,

R = ∑(RV_{j})(V_{j}Υ) + RE,

and we have shown that R ∈ Diff_{es}^{2} Ψ_{eb}^{−1}(M).

Finally, if P ∈ V_{es}(M) and Q ∈ Ψ_{eb}^{m}(M) (or vice-versa) then using the lemma (and its notation) we may write

[P, Q] = ∑ Q_{j}A_{j} + B.

Using the same method as above to write B = ∑ RV_{j}(V_{j}Υ) + BE we find that [P, Q] ∈ Diff_{es}^{1} Ψ_{eb}^{m−1}(M). □

The above proof also yields the following useful consequence

Lemma 5.6. For all m, l ∈ ℝ, and k ∈ ℕ,

Diff_{es}^{m} Ψ_{eb}^{l}(M) ⊂ Diff_{es}^{m+k} Ψ_{eb}^{l−k}(M).

We note the following consequence of (5.2):

Lemma 5.7. Let A ∈ Ψ_{eb}^{m}(M), a = σ_{eb,m}(A). Then

ı[x^{−1}D_{z′_{j}}, A] = A_{1}x^{−1}D_{z′_{j}} + x^{−1}A_{0}

where A_{0} ∈ Ψ_{eb}^{m}(M), A_{1} ∈ Ψ_{eb}^{m−1}(M),

σ_{eb,m}(A_{0}) = \frac{∂a}{∂z′_{j}} , σ_{eb,m−1}(A_{1}) = \frac{∂a}{∂ζ′_{j}} + \frac{∂a}{∂ξ}.

Note that this is exactly what one would expect from computation at the level of edge-b symbols: the Hamilton vector field of ζ′_{i}/(xz′_{i}) is (ζ′_{i}/(xz′_{i}))(∂_{ζ′_{i}}+ ∂_{ξ}) + x^{−1}∂_{z′_{i}}.

Proof. This follows immediately from writing

[x^{−1}D_{z′_{j}}, A] = [x^{−1}, A]D_{z′_{j}} + x^{−1}[D_{z′_{j}}, A].

We then use (5.2) together with the following principal symbol calculations in Ψ_{eb}^{∗}(M), see (4.1):

ıσ_{eb,m}([Q′, A]) = z′_{j}∂_{z′_{j}}a,
ıσ_{eb,m−1}([(z′_{j})^{−1}, A]z′_{j}) = ∂_{ζ′_{j}}a,
ıσ_{eb,m−1}([x^{−1}, A]) = x^{−1}∂_{ξ}a,

as well as [x^{l}Ψ_{eb}^{k}(M), x^{l′}Ψ_{eb}^{k′}(M)] ⊂ x^{l+l′}Ψ_{eb}^{k+k′−1}(M), which allows one to exchange factors after the previous steps without affecting the computed principal symbols. □

We now define the edge-smooth Sobolev spaces. It is with respect to these base spaces that we will measure regularity in proving propagation of edge-b wavefront set.

Definition 5.8. For s ≥ 0 integer,

H_{es}^{s,l−(f+1)/2}(M) = {u ∈ x^{l}L_{g}^{2}(M) : A ∈ Diff_{es}^{s}(M) ⇒ Au ∈ x^{l}L_{g}^{2}(M)}.

The norm in H_{es}^{s,l−(f+1)/2}(M), up to equivalence, is defined using any finite number of generators A_{j} for the finitely generated C^{∞}(M)-module Diff_{es}^{s}(M) by

∥u∥_{H_{es}^{s,l−(f+1)/2}(M)} = (∑_{j} ∥x^{−l}A_{j}u∥_{L_{g}^{2}(M)}^{2})^{1/2} .

The space H_{es,0}^{s,l−(f+1)/2}(M) is the closure of Ċ^{∞}(M) in H_{es}^{s,l−(f+1)/2}(M).

Remark 5.9. The orders above are chosen so that setting s = 0, l = 0, we obtain L_{g}^{2}(M) = H_{es}^{0,−(f+1)/2}(M). Thus x^{(f+1)/2}L_{g}^{2}(M) = L^{2}(M, x^{−(f+1)} dg) is the L^{2}-space corresponding to densities that are smooth up to all boundary hypersurfaces of M except W̃, and that are b-densities at the interior of W̃, meaning that x(x^{−(f+1)} dg) is actually a smooth non-degenerate density on M. This convention keeps the weights consistent with [25].

Note also that the subspace C of C^{∞}(M) given by

C = x^{∞}C^{∞}(M)

is dense in H_{es}^{s,l−(f+1)/2}(M) for all s and l; one could even require supports disjoint from W̃. Thus, the difference between H_{es,0}^{s,l−(f+1)/2}(M) and H_{es}^{s,l−(f+1)/2}(M) corresponds to the behavior at the boundary hypersurfaces of M other than W̃, i.e. those arising from the boundary hypersurfaces of M_{0}, where the boundary conditions are imposed. Thus, this difference is similar to the difference between H^{s}(Ω) and H_{0}^{s}(Ω) for domains Ω with smooth boundary in a manifold.

The boundedness of Ψ_{eb}^{0}(M) on H_{es}^{1,1−(f+1)/2}(M) is an immediate consequence of the commutation property in Lemma 5.5.

Theorem 5.10. Ψ_{eb}^{0}(M) is bounded on both H_{es}^{1,1−(f+1)/2}(M) and on the closed subspace H_{es,0}^{1,1−(f+1)/2}(M).

Remark 5.11. The more general case of H_{es}^{1,l−(f+1)/2}(M) with arbitrary l follows from the case of l = 1 using x^{−l}Ax^{l} ∈ Ψ_{eb}^{0}(M) for A ∈ Ψ_{eb}^{0}(M).

In fact, reduction to l = 0 would make the proof below even more transparent.

The case of H_{es}^{s,l−(f+1)/2}(M) can be proved similarly, but we do not need this here.

Proof. As Ψ_{eb}^{0}(M) : Ċ^{∞}(M) → Ċ^{∞}(M), the second statement follows from the first and the definition of H_{es,0}^{1,1−(f+1)/2}(M).

As above, let C be the subspace of C^{∞}(M) consisting of functions vanishing to infinite order at W̃ , which is thus dense in H_{es}^{1,1−(f+1)/2}(M). Let A ∈ Ψ_{eb}^{0}(M). As Ψ_{eb}^{0}(M) : C → C, and A is bounded on L_{g}^{2}(M), one merely needs to check that for Q ∈ Diff_{es}^{1}(M) there exists C > 0 such that for u ∈ C,

∥x^{−1}QAu∥_{L_{g}^{2}} ≤ C∥u∥_{H_{es}^{1,1−(f+1)/2}(M)}.

But

x^{−1}QAu = ([x^{−1}, A]x)(x^{−1}Qu) + x^{−1}[Q, A]u + A(x^{−1}Qu).

By Lemma 5.5, [Q, A] = ∑ A_{j}Q_{j} + B, B ∈ Ψ_{eb}^{0}(M), A_{j} ∈ Ψ_{eb}^{−1}(M), Q_{j} ∈ Diff_{es}^{1}(M), hence x^{−1}[Q, A] = ∑(x^{−1}A_{j}x)(x^{−1}Q_{j}) + (x^{−1}Bx)x^{−1},

x^{−1}QAu = ([x^{−1}, A]x)(x^{−1}Qu) + ∑(x^{−1}A_{j}x)(x^{−1}Q_{j}u)
+ (x^{−1}Bx)(x^{−1}u) + A(x^{−1}Qu),

so the desired conclusion follows from

∥x^{−1}Qu∥_{L_{g}^{2}(M)}, ∥x^{−1}Q_{j}u∥_{L_{g}^{2}(M)}, ∥x^{−1}u∥_{L_{g}^{2}(M)} ≤ C∥u∥_{H_{es}^{1,1−(f+1)/2}(M)}, u ∈ C,

and additionally [x^{−1}, A]x, x^{−1}A_{j}x ∈ Ψ_{eb}^{−1}(M) ⊂ Ψ_{eb}^{0}(M) (which are thus bounded on L_{g}^{2}(M), just as A, x^{−1}Bx ∈ Ψ_{eb}^{0}(M) are). □

We can now define the eb-wave front set relative to a given Hilbert (or even Banach) space, which in practice will be either the Dirichlet form domain or a weighted edge-smooth Sobolev space serving as a stand-in for the Neumann form domain.

Definition 5.12. Let 𝔛 denote a Hilbert space on which operators in Ψ_{eb}^{0}(M) are bounded, with the operator norm of Op(a) depending on a fixed seminorm of a.

Let q ∈ ^{eb}S^{∗}M, u ∈ 𝔛. For m ≥ 0, r ≤ 0, we say that q ∉ WF_{eb,𝔛}^{m,r}(u) if there exists A ∈ Ψ_{eb}^{m,r}(M) elliptic at q such that Au ∈ 𝔛. We also define q ∉ WF_{eb,𝔛}^{∞,r}(u) if there exists A ∈ Ψ_{eb}^{0,r}(M) elliptic at q such that P Au ∈ 𝔛 for all P ∈ Diff_{eb}^{∗}(M).

There is an inclusion

WF_{eb,𝔛}^{m,r} u ⊂ WF_{eb,𝔛}^{m′,r′} u

if

m ≤ m′, r ≤ r′.

Remark 5.13. We could alter this definition to allow u a priori to lie in the larger space

∑ A_{j}(𝔛)

with A_{j} ∈ Ψ_{eb}^{∞,0}(M); this would allow us to give a non-trivial definition of WF_{eb,𝔛}^{m,r} u even for m < 0.

The restriction to r ≤ 0 is more serious: operators in Ψ_{eb}^{∗,0}(M) would in general fail to be microlocal with respect to a putative WF_{eb,𝔛}^{m,r}(M) with r > 0.

Note also that if 𝔛′ is a closed subspace of 𝔛, with the induced norm, and if elements of Ψ_{eb}^{0}(M) restrict to (necessarily bounded) maps 𝔛′ → 𝔛′, then for u ∈ 𝔛′,

(5.3)  WF_{eb,𝔛}^{m,r}(u) = WF_{eb,𝔛′}^{m,r}(u).

In particular, this holds with 𝔛 = H_{es}^{k,l}(M) and 𝔛′ = H_{es,0}^{k,l}(M).

The eb-wave front set captures eb-regularity:

Lemma 5.14. If u ∈ 𝔛, r ≤ 0, m ≥ 0 and WF_{eb,𝔛}^{m,r}(u) = ∅, then u ∈ H_{eb,𝔛,loc}^{m,r}(M), i.e. for all A ∈ Ψ_{eb}^{m,r}(M) with compactly supported kernel, Au ∈ 𝔛.

Proof. This is a standard argument (see e.g. [34, Lemma 3.10]): For each q ∈ ^{eb}S^{∗}M there is B_{q} ∈ Ψ_{eb}^{m,r}(M) elliptic at q such that B_{q}u ∈ 𝔛. By compactness, ^{eb}S^{∗}M can be covered by ⋃_{j} ell(B_{q_{j}}) for finitely many points q_{j}. Now choose Q ∈ Ψ_{eb}^{−m,−r}(M) elliptic, and set B = ∑ QB_{q_{j}}^{∗}B_{q_{j}}. Then B is elliptic and Bu ∈ 𝔛. As B has a parametrix G ∈ Ψ_{eb}^{−m,−r}(M) with GB − Id ∈ Ψ_{eb}^{−∞,0}(M),

Au = AG(Bu) + (A(Id −GB))u, and A(Id −GB) ∈ Ψ_{eb}^{−∞,r}(M) ⊂ Ψ_{eb}^{0,0}(M),

shows the claim. □

Pseudodifferential operators are microlocal, as follows by a standard argument:

Lemma 5.15. (Microlocality) If B ∈ Ψ_{eb}^{s,l}(M) then for r, r − l ≤ 0, u ∈ 𝔛,

WF_{eb,𝔛}^{m−s,r−l}(Bu) ⊂ WF′(B) ∩ WF_{eb,𝔛}^{m,r}(u).

In particular, if WF′(B) ∩ WF_{eb,𝔛}^{m,r}(u) = ∅ then Bu ∈ H_{eb,𝔛,loc}^{m−s,r−l}(M).

Proof. We assume m ≥ s and m ≥ 0 in accordance with the definition above; but the general case is treated easily by the preceeding remarks.

If q ∈ ^{eb}S^{∗}M, q ∉ WF′(B), let A ∈ Ψ_{eb}^{m−s,r−l}(M) be elliptic at q such that WF′(A)∩WF′(B) = ∅. Thus AB ∈ Ψ_{eb}^{−∞,r}(M) ⊂ Ψ_{eb}^{0,0}(M), hence ABu ∈ 𝔛, so q ∉ WF_{eb,𝔛}^{m−s,r−l}(Bu).

On the other hand, if q ∈ ^{eb}S^{∗}M, q ∉ WF_{eb,𝔛}^{m,r}(u), then there is C ∈ Ψ_{eb}^{m,r}(M) elliptic at q such that Cu ∈ 𝔛. Let G be a microlocal parametrix for C, so G ∈ Ψ_{eb}^{−m,−r}(M), and q ∉ WF′(GC − Id). Let A ∈ Ψ_{eb}^{m−s,r−l} be elliptic at q and such that WF′(A) ∩ WF′(GC − Id) = ∅. Then

ABu = ABGCu + AB(Id −GC)u,

and AB(Id −GC) ∈ Ψ_{eb}^{−∞,r}(M) ⊂ Ψ_{eb}^{0,0}(M) since WF′(A) ∩ WF′(Id −GC) = ∅, so the second term on the right hand side is in 𝔛. On the other hand, Cu ∈ 𝔛 and ABG ∈ Ψ_{eb}^{0,0}(M), so ABG(Cu) ∈ 𝔛 as well, proving the wave front set containment.

The final claim follows immediately from this and Lemma 5.14. □

There is a quantitative version of the lemma as well. Since the proof is similar, cf. [34, Lemma 3.13], we omit it.

Lemma 5.16. Suppose that K ⊂ ^{eb}S^{∗}M is compact, U is a neighborhood of K, K̃ ⊂ M compact.

Let Q ∈ Ψ_{eb}^{s,l}(M) elliptic on K with WF′(Q) ⊂ U and the Schwartz kernel of Q supported in K̃ × K̃.

If B is a bounded family in Ψ_{eb}^{s,l}(M) with Schwartz kernel supported in K̃ × K̃ and with WF′(B) ⊂ K then for r, r − l ≤ 0, there is C > 0 such that for u ∈ 𝔛 with WF_{eb,𝔛}^{m,r}(u) ∩ U = ∅,

∥Bu∥_{𝔛} ≤ C(∥u∥_{𝔛} + ∥Qu∥_{𝔛}).

5.2. Dual spaces and adjoints. We now discuss the dual spaces. For simplicity of notation we suppress the loc and c subscripts for the local spaces and compact supports. In principle this should only be done if M is compact, but, as this aspect of the material is standard, we feel that this would only distract from the new aspects. See for instance [34, Section 3] for a treatment where all the compact supports and local spaces are spelled out in full detail.

Recall now from Appendix A that if 𝔛 is a dense subspace of L_{g}^{2}, equipped with an inner product ⟨., .⟩_{𝔛} in which it is a Hilbert space and the inclusion map ι into L_{g}^{2} is continuous, then there is a linear injective inclusion map L_{g}^{2} → 𝔛^{∗} with dense range, namely

ι^{∗} = ι^{†} ∘ j ∘ c : ℌ → 𝔛^{∗}

where ι^{†} : (L_{g}^{2}(M))^{∗} → 𝔛^{∗} is the standard adjoint map, j : L_{g}^{2}(M) → L_{g}^{2}(M)^{∗} the standard conjugate-linear identification of a Hilbert space with its dual, and c is pointwise complex conjugation of functions. In particular, one has the chain of inclusions 𝔛 ⊂ L_{g}^{2}(M) ⊂ 𝔛^{∗}, and one considers 𝔛^{∗}, together with these inclusions, the dual space of 𝔛 with respect to L_{g}^{2}(M).

Definition 5.17. For s ≥ 0, the dual space of H_{es}^{s,l}(M) with respect to the L_{g}^{2}(M) inner product is denoted Ḣ_{es}^{−s,−l−(f+1)}(M).

For s ≥ 0, the dual space of the closed subspace

H_{es,0}^{s,l}(M) ≡ Ḣ_{es}^{s,l}(M)

is denoted H_{es}^{−s,−l−(f+1)}(M); this is a quotient space of Ḣ_{es}^{−s,−l−(f+1)}(M). We denote the quotient map by

ρ : Ḣ_{es}^{−s,−l−(f+1)}(M) → H_{es}^{−s,−l−(f+1)}(M).

The standard characterization of these distribution spaces, by doubling across all boundary faces of M except W̃, is still valid—see [10, Appendix B.2] and [34, §3]. Note that for all s, l, elements of Ḣ_{es}^{−s,−l−(f+1)}(M) are in particular continuous linear functionals on C, which in turn is a dense subspace of H_{es}^{s,l}(M). In particular, they can be identified as elements of the dual C′ of C. Thus, were it not for the infinite order vanishing imposed at W̃ for elements of C, these would be “supported distributions”—hence the notation with the dot. On the other hand, elements of H_{es}^{−s,−l−(f+1)}(M) are only continuous linear functionals on Ċ^{∞}(M) (rather than on C), though by the Hahn-Banach theorem can be extended to continuous linear functionals on C in a non-unique fashion.

If P ∈ x^{−r} Diff_{es}^{k}(M), then it defines a continuous linear map

P : H_{es}^{k,l}(M) → H_{es}^{0,l−r}(M).

Thus, its Banach space adjoint (with respect to the sesquilinear dual pairing) is a map

P^{∗} : (H_{es}^{0,l−r}(M))^{∗} = Ḣ_{es}^{0,r−l−(f+1)}(M) → (H_{es}^{k,l}(M))^{∗} = Ḣ_{es}^{−k,−l−(f+1)}(M),

⟨P^{∗}u, v⟩ = ⟨u, P v⟩, u ∈ Ḣ_{es}^{0,r−l−(f+1)}(M), v ∈ H_{es}^{k,l}(M).

(5.4)

In principle, P^{∗} depends on l and r. However, the density of C in these spaces shows that in fact it does not.

There is an important distinction here between considering P^{∗} as stated, or as composed with the quotient map, ρ ∘ P^{∗}.

Lemma 5.18. Suppose that P ∈ x^{−r} Diff_{es}^{k}(M). Then there exists a unique Q ∈ x^{−r} Diff_{es}^{k}(M) such that ρ ∘ P^{∗} = Q. However, in general, acting on C, P^{∗} ≠ Q.

If, on the other hand, P ∈ x^{−r} Diff_{eb}^{k}(M), then there exists a unique Q ∈ x^{−r} Diff_{eb}^{k}(M) such that P^{∗} = Q.

Proof. For the first part we integrate by parts in ⟨u, P v⟩ using u, v ∈ Ċ^{∞}(M) (noting that Ċ^{∞}(M) is dense in H_{es,0}^{k,l}(M)). Thus, one can localize. In local coordinates the density is dg = Jx^{f} dx dy dz, with J ∈ C^{∞}(M), so for a vector field V ∈ V_{es}(M), noting the lack of boundary terms due to the infinite order vanishing of u and v, one has (with the first equality being the definition of V^{∗})

\begin{aligned}⟨V^{∗}u, v⟩ & = ⟨u, V v⟩ = ∫ u \overline{V v} Jx^{f} dx dy dz \\ & = ∫ (J^{−1}x^{−f}V^{†}(Jx^{f}u)) \overline{v} Jx^{f} dx dy dz, \end{aligned}

where for V = a(xD_{x}) + ∑ b_{j}(xD_{y_{j}}) + ∑ c_{j}D_{z_{j}}, with a, b_{j}, c_{j} ∈ C^{∞}(M),

V^{†} = D_{x}x\overline{a} + ∑ xD_{y_{j}} \overline{b_{j}} + ∑ D_{z_{j}} \overline{c_{j}} ∈ Diff_{es}^{1}(M).

Conjugation of V^{†} by Jx^{f} still yields an operator in Diff_{es}^{1}(M). This shows the existence (and uniqueness!) of the desired Q, namely

Q = J^{−1}x^{−f}V^{†}Jx^{f}.

The density of Ċ^{∞}(M) in H_{es}^{0,r−l−(f+1)}(M) now finishes the proof of the first claim when P = V ∈ V_{es}(M), since this means that ⟨P^{∗}u, v⟩ = ⟨Qu, v⟩ for all u ∈ H_{es}^{0,r−l−(f+1)}(M), v ∈ H_{es,0}^{k,l}(M). The general case follows by induction and adding weight factors (recalling Remark 5.2).

The same calculation works even if u, v ∈ C provided that V ∈ V_{eb}(M): in this case D_{z_{j}} is replaced by vector fields tangent to all boundary faces, i.e. D_{z′′_{j}} and z′_{j}D_{z′_{j}}, for which there are no boundary terms—in the second case

due to the vanishing factor z′_{j}. This proves the claim if P ∈ x^{−r} Diff_{eb}^{k}(M).

Note, however, that this calculation breaks down if u, v ∈ C and V ∈ V_{es}(M): the D_{z′_{j}} terms gives rise to non-vanishing boundary terms in general, namely

∑_{j} ∫_{H_{j}}(−ı)\overline{c_{j}} u \overline{v} Jx^{f} dx dy dẑ_{j} = ∑_{j}⟨−ı\overline{σ_{es,1}(V)(dx_{j})}u, v⟩_{H_{j}},

where H_{j} is the boundary hypersurface z′_{j} = 0, dẑ_{j} shows that dz′_{j} is dropped from the density, and on H_{j} one uses the density induced by the Riemannian density and dz′_{j}. This completes the proof of the lemma. □

We now define an extension of Diff_{es}(M) as follows.

Definition 5.19. Let x^{−r} Diff_{es,†}^{k}(M) denote the set of Banach space adjoints of elements of x^{−r} Diff_{es}^{k}(M) in the sense of (5.4).

Also let x^{−2r} Diff_{es,♯}^{2k}(M) denote operators of the form

∑_{j=1}^{N} Q_{j}P_{j}, P_{j} ∈ x^{−r} Diff_{es}^{k}(M), Q_{j} ∈ Diff_{es,†}^{k}(M).

For M non-compact, the sum is taken to be locally finite.

Thus, if P ∈ x^{−2r} Diff_{es,♯}^{2k}(M), P_{j}, Q_{j} as above, and Q_{j} = R_{j}^{∗}, R_{j} ∈ x^{−r} Diff_{es}^{k}(M), then

⟨P u, v⟩ = ∑_{j=1}^{N}⟨P_{j}u, R_{j}v⟩.

We are now ready to discuss Dirichlet and Neumann boundary conditions for P ∈ x^{−2r} Diff_{es,♯}^{2k}(M).

Definition 5.20. Suppose P ∈ x^{−2r} Diff_{es,♯}^{2k}(M). By the Dirichlet operator associated to P we mean the map

ρ ∘ P : H_{es,0}^{k,l}(M) → H_{es}^{−k,l−2r}(M),

where ρ : Ḣ_{es}^{−k,l−2r}(M) → H_{es}^{−k,l−2r}(M) is the quotient map. For f ∈ H_{es}^{−k,l−2r}(M) we say that u ∈ H_{es,0}^{k,l}(M) solves the Dirichlet problem for P u = f if ρ ∘ P u = f. We also say in this case that P u = f with Dirichlet boundary conditions.

Similarly, for f ∈ Ḣ_{es}^{−k,l−2r}(M) we say that u ∈ H_{es}^{k,l}(M) solves the Neumann problem for P u = f if P u = f. We also say in this case that P u = f with Neumann boundary conditions. Correspondingly, for the sake of completeness, by the Neumann operator associated to P we mean P itself.

Remark 5.21. As noted in Lemma 5.18, when considering the action of Diff_{es}(M) on Ċ^{∞}(M), Diff_{es}(M) is closed under adjoints (which thus map to C^{−∞}(M), i.e. extendible distributions), so one can suppress the subscript ♯ on Diff_{es,♯}(M). Thus, the subscript’s main role is to keep the treatment of the Neumann problem clear—without such care, one would need to use quadratic forms throughout, as was done in [34].

We now turn to the action of Ψ_{eb}^{m,l}(M) on the dual spaces. Note that any A ∈ Ψ_{eb}^{m,l}(M) maps C to itself, and that Ψ_{eb}^{m,l}(M) is closed under formal adjoints, i.e. if A ∈ Ψ_{eb}^{m,l}(M) then there is a unique A^{∗} ∈ Ψ_{eb}^{m,l}(M) such that ⟨Au, v⟩ = ⟨u, A^{∗}v⟩ for all u, v ∈ C —cf. Diff_{eb}(M) in Lemma 5.18. We thus define A : C′ → C′ by ⟨Au, v⟩ = ⟨u, A^{∗}v⟩, u ∈ C′, v ∈ C. Since C is (even sequentially) dense in C′ endowed with the weak-* topology, this definition is in fact the only reasonable one, and if u ∈ C, the element of C′ given by this is the linear functional induced by Au on C.

Next, for subspaces of C′ we have improved statements. In particular, most relevant here, dually to Theorem 5.10, any A ∈ Ψ_{eb}^{0}(M) is bounded on Ḣ_{es}^{−1,l}(M) and on H_{es}^{−1,l}(M).

We now turn to an extension of Diff_{es} Ψ_{eb}(M). First, taking adjoints in Lemma 5.5, we deduce:

Lemma 5.22. If A ∈ Ψ_{eb}^{m}(M) and Q ∈ Diff_{es,†}^{1}(M), then [A, Q] = ∑ Q_{j}A_{j}+ B, B ∈ Ψ_{eb}^{m}(M), A_{j} ∈ Ψ_{eb}^{m−1}(M), Q_{j} ∈ Diff_{es,†}^{1}(M).

Similarly, [A, Q] = ∑ A′_{j}Q′_{j} + B′_{j}, B′ ∈ Ψ_{eb}^{m}(M), A′_{j} ∈ Ψ_{eb}^{m−1}(M), Q′_{j} ∈ Diff_{es,†}^{1}(M).

Proof. The proof is an exercise in duality; we only spell out to emphasize our definitions. We have for u ∈ C′, v ∈ C,

⟨[A, Q]u, v⟩ = ⟨(AQ − QA)u, v⟩ = ⟨u, (Q^{∗}A^{∗} − A^{∗}Q^{∗})v⟩ = ⟨u, [Q^{∗}, A^{∗}]v⟩

where Q^{∗} ∈ Diff_{es}^{1}(M), A^{∗} ∈ Ψ_{eb}^{M}. Thus, by Lemma 5.5 (applied with V_{es}(M) replaced by Diff_{es}^{1}(M)), there exist Ã_{j} ∈ Ψ_{eb}^{m−1}(M), B̃ ∈ Ψ_{eb}^{m}(M), Q̃_{j} ∈ Diff_{es}^{1}(M) such that [Q^{∗}, A^{∗}] = −[A^{∗}, Q^{∗}] = ∑ Q̃_{j}Ã_{j} + B̃. Thus,

⟨[A, Q]u, v⟩ = ⟨u, (∑ Q̃_{j}Ã_{j} + B̃)v⟩ = ⟨(Ã_{j}^{∗}Q̃_{j}^{∗} + B̃^{∗})u, v⟩,

with Ã_{j}^{∗} ∈ Ψ_{eb}^{m−1}(M), B̃^{∗} ∈ Ψ_{eb}^{m}(M), Q̃_{j}^{∗} ∈ Diff_{es,†}^{1}(M). This proves the second half of the lemma. The first half is proved similarly, using the second half of the statement of Lemma 5.5 rather than its first half. □

In fact, the analogue of Lemma 5.7 also holds with D_{z′_{j}} replaced by D_{z′_{j}}^{∗} ∈

Diff_{es,†}^{1}(M):

Lemma 5.23. Let A ∈ Ψ_{eb}^{m,l}(M), a = σ_{eb,m}(A). Then

ı[x^{−1}D_{z′_{j}}^{∗}, A] = A_{1}x^{−1}D_{z′_{j}}^{∗} + x^{−1}A_{0}

where A_{0} ∈ Ψ_{eb}^{m,l}(M), A_{1} ∈ Ψ_{eb}^{m−1,l}(M),

σ_{eb,m}(A_{0}) = \frac{∂a}{∂z′_{j}} , σ_{eb,m−1}(A_{1}) = \frac{∂a}{∂ζ′_{j}} + \frac{∂a}{∂ξ}.

We thus make the following definition:

Definition 5.24. Let

Diff_{es,♯}^{k} Ψ_{eb}^{m}(M) = {∑ A_{α}B_{β} : A_{α} ∈ Diff_{es,♯}^{k}(M), B_{β} ∈ Ψ_{eb}^{m}(M)}.

Using Proposition 5.4 and duality, as in the previous lemma, we deduce the following:

Proposition 5.25. Diff_{es,♯}^{k} Ψ_{eb}^{m,l}(M) is a Ψ_{eb}^{0}(M)-bimodule, and

P ∈ Diff_{es,♯}^{k} Ψ_{eb}^{m,l}(M), A ∈ Ψ_{eb}^{s,r}(M) ⇒

P A, AP ∈ Diff_{es,♯}^{k} Ψ_{eb}^{m+s,l+r}(M), [P, A] ∈ Diff_{es,♯}^{k} Ψ_{eb}^{m+s−1,l+r}(M).

(5.5)

5.3. Domains. In this section, we discuss the relationship between Dirichlet and Neumann form domains of Δ and the scales of weighted Sobolev that we have introduced. First, we identify the Dirichlet quadratic form domain in terms of the edge-smooth Sobolev spaces.

The Friedrichs form domain of Δ with Dirichlet boundary conditions on X_{0} is

H_{0}^{1}(X_{0}),

also denoted by Ḣ^{1}(X_{0}) (see [10, Appendix B.2]); we may also view this space as the completion of C_{c}^{∞}(X_{0}) in the H^{1}(X_{0})-norm,

∥u∥_{H^{1}(X_{0})} = ∥u∥_{L_{g_{0}}^{2}(X_{0})} + ∥du∥_{L_{g_{0}}^{2}(X_{0};T^{∗}X_{0})}.

Equivalently in terms of “doubling” X_{0} across all boundary hypersurfaces, H_{0}^{1}(X_{0}) consists of H^{1}-functions on the “double” supported in X_{0}.

Lemma 5.26. On Ċ^{∞}(X) = β^{∗}Ċ^{∞}(X_{0}), the norms

∥u∥_{H^{1}(X_{0})} = (∥u∥_{L_{g_{0}}^{2}}^{2} + ∥du∥_{L_{g_{0}}^{2}}^{2} )^{1/2}

and

∥u∥_{H_{es}^{1,1−(f+1)/2}(X)}

are equivalent.

Proof. Multiplication by elements of C^{∞}(X_{0}) is bounded with respect to both norms (with respect to H_{es}^{1,1−(f+1)/2}(X) even C^{∞}(X) is bounded), so one can localize in X_{0}, or equivalently in X near a fiber β^{−1}(p), p ∈ W, of W̃, and assume that u is supported in such a region.

Elements of V(X_{0}) lift under β to span x^{−1}V_{es}(X) as a C^{∞}(X)-module by (3.13). In particular, merely since β^{∗}V(X_{0}) ⊂ x^{−1}V_{es}(X), we obtain^{9}

(5.6)  ∥u∥_{H_{0}^{1}(X_{0})} ≲ ∥u∥_{H_{es}^{1,1−(f+1)/2}(X)}, u ∈ Ċ^{∞}(X).

We now prove the reverse inequality. By the spanning property, we have

(5.7)  ∥x^{−1}Au∥_{L_{g}^{2}} ≲ ∥u∥_{H^{1}(X_{0})}

for any A ∈ V_{es}(X) as C^{∞}(X) is bounded on L_{g}^{2}(X) = L_{g_{0}}^{2}(X). As V_{es}(X) together with the identity operator generates Diff_{es}^{1}(X), we only need to prove

∥x^{−1}u∥_{L_{g}^{2}(X)} ≲ ∥u∥_{H^{1}(X_{0})},

for u ∈ Ċ^{∞}(X) supported near a fiber β^{−1}(p), p ∈ Y, of Ỹ. However, this follows easily from identifying a neighborhood of β^{−1}(p) with [0, ϵ)_{x}×O_{y}×Z_{z}, where O ⊂ ℝ^{n−f−1}, and using the Poincaré inequality in Z, namely that

∥u(x, y, .)∥_{L^{2}(Z)} ≤ C∥(d_{Z}u)(x, y, .)∥_{L^{2}(Z)}, u ∈ Ċ^{∞}(X_{0}).

Multiplying the square of both sides by x^{−2+f} and integrating in x, y, yields

∥x^{−1}u(x, y, .)∥_{L_{g}^{2}(X)} ≤ C∥(x^{−1}d_{Z}u)(x, y, .)∥_{L_{g}^{2}(X)} ≤ C′∥u∥_{H_{0}^{1}(X)}

by (5.7). □

In view of the definition of H_{es,0}^{1,1−(f+1)/2}(X) as the closure of Ċ^{∞}(X) in

H_{es}^{1,1−(f+1)/2}(X), we immediately deduce:

Proposition 5.27. The Dirichlet form domain of Δ is given by

(5.8)  H_{0}^{1}(X_{0}) = H_{es,0}^{1,1−(f+1)/2}(X)

in the strong sense that the natural (up to equivalence) Hilbert space norms on the two sides are equivalent. In particular, for u ∈ H_{0}^{1}(X_{0}), we have

∥x^{−1}Qu∥_{L_{g}^{2}} ≤ C∥u∥_{H^{1}(X_{0})}

for all Q ∈ Diff_{es}^{1}(X).

For Neumann boundary conditions the quadratic form domain is H^{1}(X_{0}), whose lift is not quite so simple in terms of the edge-smooth spaces. However, we have the following lemma, which suffices for the edge-b propagation results below (with a slight loss).

9We use the notation that a ≲ b if there exists C > 0 such that a ≤ Cb. Usually a and b depend on various quantities, e.g. on u here, and C is understood to be independent of these quantities.

Lemma 5.28. We have H_{es}^{1,1−(f+1)/2}(X) ⊂ H^{1}(X_{0}) ⊂ H_{es}^{1,−(f+1)/2}(X), with all inclusions being continuous.

Proof. The first inclusion is an immediate consequence of (5.6) holding for u ∈ C, C as in Remark 5.9 (thus dense in H_{es}^{1,l−(f+1)/2}(X) for all l), using again that elements of V(X_{0}) lift under β to span (and in particular lie in) x^{−1}V_{es}(X) as a C^{∞}(X)-module by (3.13).

For the second inclusion, we need to prove that ∥Au∥_{L_{g}^{2}(X)} ≤ C∥u∥_{H^{1}(X_{0})} for A ∈ Diff_{es}^{1}(X). As this is automatic for A ∈ C^{∞}(X), we are reduced to considering A ∈ V_{es}(X). But (5.7) still holds for u ∈ C, so ∥Au∥_{L_{g}^{2}(X)} ≤ C′∥x^{−1}Au∥_{L_{g}^{2}} ≤ C∥u∥_{H^{1}(X_{0})} for A ∈ V_{es}(X). This finishes the proof of the lemma. □

5.4. The wave operator as an element of x^{−2} Diff_{es,♯}^{2}(M). For f ∈ C, in local coordinates,

df = (x∂_{x}f) \frac{dx}{x} + ∑_{j}(x∂_{y_{j}}f) \frac{dy_{j}}{x} + ∑_{j}(∂_{z_{j}}f) dz_{j}.

Thus, the exterior derivative satisfies

d ∈ Diff_{es}^{1}(M; \underline{ℂ},^{es}T^{∗}M),

with \underline{ℂ} denoting the trivial bundle. As the dual Riemannian metric is of the form x^{−2}G, where G is a smooth fiber metric on ^{es}T^{∗}X, and Δ = d^{∗}d, we deduce that □ ∈ x^{−2} Diff_{es,♯}^{2}(M). However, we need a more precise description of □ for our commutator calculations.

So suppose now that U is a coordinate chart near a point q at ∂W̃ with coordinates (x, y, z′, z′′) centered at q, and recall from (2.2) that the Riemannian metric has the form

(5.9) g = dx^{2} + h(y, dy) + x^{2}k(x, y, z, dz) + xk′(x, y, z, dx, dy, x dz).

By changing z′′ if necessary (while keeping x, y, z′ fixed), cf. the argument of § 2 leading to (5.10), we can arrange that the dual metric K of k have the form

\begin{aligned}K(0, y, z) = ∑_{i,j=1}^{k} k_{1,ij}(0, y, z)\underline{ζ}′_{i}\underline{ζ}′_{j} & + ∑_{i=1}^{k} ∑_{j=k+1}^{f} k_{3,ij}(0, y, z)\underline{ζ}′_{i}\underline{ζ}′′_{j} \\ & + ∑_{i,j=k+1}^{f} k_{2,ij}(0, y, z)\underline{ζ}′′_{i}\underline{ζ}′′_{j},  k_{3}|_{C} = 0. \end{aligned} (5.10)

We deduce the following lemma:

Lemma 5.29. Let U be a coordinate chart near a point with x = 0 and z′ = 0, and suppose that we have arranged that at

C = {x = 0, z′ = 0},

the vector spaces

sp{dz′_{i}, i = 1, …, k} and sp{dz′′_{j}, j = k + 1, …, f}

are orthogonal with respect to K. With Q_{i} = x^{−1}D_{z′_{i}}, the wave operator satisfies

(5.11) □ = ∑_{i,j} Q_{i}^{∗}κ_{ij}Q_{j} + ∑_{i}(x^{−1}M_{i}Q_{i} + Q_{i}^{∗}x^{−1}M′_{i}) + x^{−2}H̃ on U

with

κ_{ij} ∈ C^{∞}(M), M_{i}, M′_{i} ∈ Diff_{eb}^{1}(M), H̃ ∈ Diff_{eb}^{2}(M)

σ_{eb,1}(M_{i}) = m_{i} = σ_{eb,1}(M′_{i}), h̃ = σ_{eb,2}(H̃),

κ_{ij}|W̃ = −k_{1,ij}(y, z), m_{i}|_{C} = 0, m_{i}|W̃ = −\frac{1}{2} ∑_{j=k+1}^{f} k_{3,ij}ζ′′_{j},

h̃|W̃ = τ^{2} − ξ^{2} − h(y, η) − ∑_{i,j=k+1}^{f} k_{2,ij}(y, z)ζ′′_{i}ζ′′_{j}.

(5.12)

We next note microlocal elliptic regularity.

Proposition 5.30. Let u ∈ 𝔛 ≡ H_{es}^{1,l}(M), and suppose that

□u ∈ Ḣ_{es}^{−1,l−2}(M) = 𝔜

with Dirichlet or Neumann boundary conditions. Then

WF_{eb,𝔛}^{m,0}(u) ⊂ ^{eb}Σ ∪ WF_{eb,𝔜}^{m,0}(□u).

In particular, if □u = 0, then WF_{eb,𝔛}^{∞,0}(u) ⊂ ^{eb}Σ.

Proof. The proof goes along the same lines as Proposition 4.6 of [34] and Theorem 8.11 of [25]; we thus provide a sketch. An essential ingredient the top-order commutativity of x^{l} Diff_{es,♯}^{2} Ψ_{eb}^{m}(M), which allows us to treat all commutators as error terms. The key estimate is stated in Lemma 5.31 below.

Given the lemma, one proceeds by an inductive argument, showing that if WF_{eb,𝔛}^{s−1/2,0}(u) ⊂ ^{eb}Σ ∪ WF_{eb,𝔜}^{s−1/2,0}(□u) (which is a priori known for s = 1/2, starting our inductive argument) then WF_{eb,𝔛}^{s,0}(u) ⊂ ^{eb}Σ ∪ WF_{eb,𝔜}^{s,0}(□u). In

order to show this, one takes A ∈ Ψ_{eb}^{s,l+(f−1)/2}(M), with WF′(A) ∩^{eb}Σ = ∅, WF′(A) ∩ WF_{eb,𝔜}^{s,0}(□u) = ∅. Λ_{γ} uniformly bounded in Ψ_{eb}^{0,0}(M), γ ∈ (0, 1], with Λ_{γ} ∈ Ψ_{eb}^{−1,0}(M) for all γ,

σ_{eb,0,0}(Λ_{γ}) = (1 + γ(|ξ|^{2} + |τ|^{2} + |η|^{2} + |ζ|^{2}))^{−1},

so A_{γ} = Λ_{γ}A is uniformly bounded in Ψ_{eb}^{m,l+(f−1)/2}(M) and A_{γ} → A in Ψ_{eb}^{m+δ,l+(f−1)/2}(M) (δ > 0 fixed) as γ → 0. One then concludes by Lemma 5.31 that for all ϵ ∈ (0, 1],

|⟨∂_{t}A_{γ}u, ∂_{t}A_{γ}u⟩ − ⟨d_{X}A_{γ}u, d_{X}A_{γ}u⟩| ≤ ϵ∥A_{γ}u∥_{H_{es}^{1,1−(f+1)/2}(M)}^{2} + Cϵ^{−1},

with C uniformly bounded, independent of γ. The microlocal ellipticity of the Dirichlet form, i.e.

∥A_{γ}u∥_{H_{es}^{1,1−(f+1)/2}(X)}^{2} ≤ C′|⟨∂_{t}A_{γ}u, ∂_{t}A_{γ}u⟩ − ⟨d_{X}A_{γ}u, d_{X}A_{γ}u⟩|,

now gives that for sufficently small ϵ ∈ (0, 1], ϵ∥A_{γ}u∥_{H_{es}^{1,1−(f+1)/2}(M)}^{2} can

be absorbed in (C′)^{−1}∥A_{γ}u∥_{H_{es}^{1,1−(f+1)/2}(M)}^{2}, and then one concludes that

∥A_{γ}u∥_{H_{es}^{1,1−(f+1)/2}(M)} is uniformly bounded independent of γ. As A_{γ} → A

strongly, one concludes by a standard argument Au ∈ H_{es}^{1,1−(f+1)/2}(M). Thus, x^{l+(f−1)/2}Au ∈ H_{es}^{1,l}(M), hence (as 𝔛 = H_{es}^{1,l}(M))

ell(A) ∩ WF_{eb,𝔛}^{s,0}(u) = ∅,

completing the iterative step. □

As already mentioned, the key ingredient is the following lemma.

Lemma 5.31. For Neumann boundary conditions, let 𝔛 = H_{es}^{1,l+1}(M), 𝔜 = Ḣ_{es}^{−1,l−1}(M); for Dirichlet boundary conditions let 𝔛 = H_{es,0}^{1,l+1}(M), 𝔜 = H_{es}^{−1,l−1}(M). Let K ⊂ ^{eb}S^{∗}M be compact, U ⊂ ^{eb}S^{∗}M open, K ⊂ U. Suppose that A is a bounded family of ps.d.o’s in Ψ_{eb}^{m,l+(f+1)/2}(M) with WF′(A) ⊂ K, such that for A ∈ A, A ∈ Ψ_{eb}^{m−1,l+(f+1)/2}(M) (but the bounds for A in Ψ_{eb}^{m−1,l+(f+1)/2}(M) are not necessarily uniform in A!). Then there exist G ∈ Ψ_{eb}^{m−1/2,0}(M), G̃ ∈ Ψ_{eb}^{m,0}(M) with WF′ G, WF′ G̃ ⊂ U and C_{0} > 0 such that for ϵ > 0, A ∈ A, one has in both the Neumann and the Dirichlet case, with the respective spaces 𝔛 and 𝔜,

u ∈ 𝔛, WF_{eb,𝔛}^{m−1/2,0}(u) ∩ U = ∅, WF_{eb,𝔜}^{m,0}(□u) ∩ U = ∅ ⇒
|⟨∂_{t}Au,∂_{t}Au⟩ − ⟨d_{X}Au, d_{X}Au⟩|
≤ ϵ∥Au∥_{H_{es}^{1,1−(f+1)/2}(M)}^{2} + C_{0}(∥u∥_{𝔛}^{2} + ∥Gu∥_{𝔛}^{2}
+ ϵ^{−1}∥□u∥_{𝔜}^{2} + ϵ^{−1}∥G̃□u∥_{𝔜}^{2}).

(5.13)

Remark 5.32. Recall that v ∈ H_{es}^{1,1−(f+1)/2}(M) is equivalent to d_{X}u ∈ L_{g}^{2}(M), ∂_{t}u ∈ L_{g}^{2}(M) and x^{−1}v ∈ L_{g}^{2}(M), so ϵ∥Au∥_{H_{es}^{1,1−(f+1)/2}(M)}^{2} on the

right hand side of (5.13) is comparable to the terms ⟨∂_{t}Au, ∂_{t}Au⟩ and ⟨d_{X}Au, d_{X}Au⟩. However, if A is supported away from ^{eb}Σ, the Dirichlet form is microlocally elliptic, so this term can be absorbed into the left hand side, as was done in Proposition 5.30.

The hypotheses in (5.13) assure that the other terms on the right hand side are finite, independent of A ∈ A.

Proof. Again, this follows the argument as Lemma 4.2 and 4.4 of [34] and Lemma 8.8 and 8.9 of [25], so we only sketch the proof. We sketch the Neumann argument; the Dirichlet case needs only simple changes. We have

⟨∂_{t}u, ∂_{t}A^{∗}Au⟩ − ⟨d_{X}u, d_{X}A^{∗}Au⟩ = ⟨□u, A^{∗}Au⟩

for all u ∈ 𝔛 and A ∈ Ψ_{eb}^{m−1,l+(f−1)/2}(M) since A^{∗}Au ∈ H_{es}^{1,−l−(f−1)}(M), which is mapped by □ into Ḣ_{es}^{−1,−l−(f+1)}(M) = (H_{es}^{1,l}(M))^{∗}. Modulo commutator terms, one can rewrite the left hand side as

⟨∂_{t}Au, ∂_{t}Au⟩ − ⟨d_{X}Au, d_{X}Au⟩,

which is the left hand side of (5.13). The commutator terms can be estimated by the second and third terms (which do not depend on ϵ) on the right hand side of (5.13). The other terms on the right hand side arise by estimating (using that the dual of H_{es}^{1,l}(M) is Ḣ_{es}^{−1,−l−(f+1)}(M))

|⟨□u, A^{∗}Au⟩| ≤ ∥A□u∥_{Ḣ_{es}^{−1,−1−(f+1)/2}(M)} ∥Au∥_{H_{es}^{1,1−(f+1)/2}(M)}
≤ ϵ^{−1}∥A□u∥_{Ḣ_{es}^{−1,−1−(f+1)/2}(M)}^{2} + ϵ∥Au∥_{Ḣ_{es}^{−1,−1−(f+1)/2}(M)}^{2}
= ϵ^{−1}∥x^{−l−(f+1)/2}A□u∥_{Ḣ_{es}^{−1,l−1}(M)}^{2} + ϵ∥Au∥_{Ḣ_{es}^{−1,−1−(f+1)/2}(M)}^{2},

and as x^{−l−(f+1)/2}A is uniformly bounded in Ψ_{eb}^{m,0}(M), with wave front set in K, ∥x^{−l−(f+1)/2}A□u∥_{𝔜}^{2} can be estimated by a multiple of ∥□u∥_{𝔜}^{2}+∥G̃□u∥_{𝔜}^{2} in view of Lemma 5.16. This completes the proof. □

The following is analogous to Lemma 7.1 of [34] and Lemma 9.8 of [25] and states that near G the fiber derivatives x^{−1}D_{z′_{i}} of microlocalized solutions Au to the wave equation can be controlled by a small multiple of the time derivative, modulo error terms (note that G is lower order than A by 1/2). The theorem mentions a δ-neighborhood of a compact set K ⊂ G (for δ < 1); by this we mean the set of points of distance < δ from K with respect to the distance induced by some Riemannian metric on ^{eb}S^{∗}M. Note that if d and d′ are the distance functions of two such Riemannian metrics, then there is a constant c > 0 such that every δ-neighborhood of K with respect to d is contained in a cδ-neighborhood with respect to d′, and vice versa, with c independent of δ (as δ → 0), so up to possibly rescaling, the choice of the Riemannian metric is not important. In particular, G is defined by x = 0, z′ = 0, 1 − h(y, η̂) − ξ̂^{2} − k(y, z, ζ̂′ = 0, ζ̂′′) = 0, so the set given by

x < C′δ, |z′| < C′δ, |1 − h(y, η̂) − ξ̂^{2} − k(y, z, ζ̂′ = 0, ζ̂′′)| < C′δ,

is contained in a C′′δ-neighborhood of G for some C′′ > 0, with C′′ independent of δ (as long as C′ is bounded).

Lemma 5.33. For Dirichlet or Neumann boundary conditions let 𝔛 and 𝔜 be as in Lemma 5.31.

Let K ⋐ G. There exists δ_{0} ∈ (0, 1) and C_{0} > 0 with the following property.

Let 0 < δ < δ_{0}, and δ > 0, and let U be a δ-neighborhood of K in ebS^{∗}M. Suppose A is a bounded family of ps.d.o’s in Ψ_{eb}^{m,l+(f+1)/2}(M) with WF′(A) ⊂ U, such that for A ∈ A, A ∈ Ψ_{eb}^{m−1,l+(f+1)/2}(M). Then there exist G ∈ Ψ_{eb}^{m−1/2,0}(M), G̃ ∈ Ψ_{eb}^{m,0}(M) with WF′ G, WF′ G̃ ⊂ U and C̃ = C̃(δ) > 0 such that for A ∈ A, one has in both the Neumann and the Dirichlet case, with the respective spaces 𝔛 and 𝔜, that

u ∈ 𝔛, WF_{eb,𝔛}^{m−1/2,0}(u) ∩ U = ∅, WF_{eb,𝔜}^{m,0}(□u) ∩ U = ∅

implies

\begin{aligned}∑ & ∥x^{−1}D_{z′_{i}}Au∥ & 2 \\ & ≤ C_{0}δ∥D_{t}Au∥^{2} + C̃(∥u∥_{𝔛}^{2} + ∥Gu∥_{𝔛}^{2} + ∥□u∥_{𝔜}^{2} + ∥G̃□u∥_{𝔜}^{2}). \end{aligned}

Proof. This is an analogue of Lemma 7.1 of [34] and Lemma 9.8 of [25], so we only indicate the main idea. By Lemma 5.31 one has control of the Dirichlet form in terms of the second through fifth terms on the right hand

side, so it suffices to check that ∑ ∥x^{−1}D_{z′_{i}}Au∥^{2} can be controlled by the

Dirichlet form and δ∥D_{t}Au∥^{2}. This uses that K ⋐ G, D_{t} is elliptic on ^{eb}Σ, and ⟨H̃Au, Au⟩ is small as WF′(A) ⊂ U; see the aforementioned Lemma 7.1 of [34] and Lemma 9.8 of [25] for details. □

Corollary 5.34. For Dirichlet or Neumann boundary conditions let 𝔛 and 𝔜 be as in Lemma 5.31. Let K ⋐ G, δ > 0.

Then there exists a neighborhood U of K in ^{eb}S^{∗}M with the following property. Suppose that A is a bounded family of ps.d.o’s in Ψ_{eb}^{m,l+(f+1)/2}(M) with WF′(A) ⊂ U, such that for A ∈ A, A ∈ Ψ_{eb}^{m−1,l+(f+1)/2}(M). Then there exist G ∈ Ψ_{eb}^{m−1/2,0}(M), G̃ ∈ Ψ_{eb}^{m,0}(M) with WF′ G, WF′ G̃ ⊂ U and C̃ = C̃(δ) > 0 such that for A ∈ A, one has in both the Neumann and the Dirichlet case, with the respective spaces 𝔛 and 𝔜, that

u ∈ 𝔛, WF_{eb,𝔛}^{m−1/2,0}(u) ∩ U = ∅, WF_{eb,𝔜}^{m,0}(□u) ∩ U = ∅

implies

\begin{aligned}∑ & ∥x^{−1}D_{z′_{i}}Au∥ & 2 \\ & ≤ δ∥D_{t}Au∥^{2} + C(∥u∥_{𝔛}^{2} + ∥Gu∥_{𝔛}^{2} + ∥□u∥_{𝔜}^{2} + ∥G̃□u∥_{𝔜}^{2}). \end{aligned}

Proof. Fix a Riemannian metric on ^{eb}S^{∗}M. Let δ_{0}, C_{0} be as in Lemma 5.33, and let δ′ = min(δ_{0}/2, δ/C_{0}). Applying Lemma 5.33 with δ′ in place of δ gives the desired conclusion, if we let U be a δ′-neighborhood of K. □

Recall now that C = {x = 0, z′ = 0} denotes one boundary face of W̃ in local coordinates, and that as a vector field on ^{es}T^{∗}M tangent to W̃ (but

50 RICHARD MELROSE, ANDRÁS VASY, AND JARED WUNSCH

not necessarily the other boundary faces), restricted to ^{es}T ∗ ; W̃ M, H_{es} is given by

−\frac{1}{2}H_{es} = ˆ \underline{ξ}x∂_{x} − ˆ \underline{ξ}\underline{σ}∂\underline{σ} − ˆ \underline{ξ}ˆ\underline{ζ}∂ˆ\underline{ζ} + K^{ij}ˆ\underline{ζ}_{i}∂_{z_{j}} + K^{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j}∂ˆ\underline{ξ} − \frac{1}{2} \frac{∂K^{ij}}{∂z_{k}} ˆ \underline{ζ}_{i}ˆ\underline{ζ}_{j}∂_{ˆ\underline{ζ}_{k}};

see (3.17)-(3.18). We can expand the K^{ij} terms by breaking them up into z′ and z′′ components at C, using (5.10). This becomes particularly interesting at a point q ∈ ^{es}Σ which is the unique point in the preimage of p ∈ ^{eb}S_{C}^{∗}M ∩G under { \atop π_{es→eb}}. At such points ˆ \underline{ζ}′ = 0, so many terms vanish. One thus obtains

−\frac{1}{2}H_{es}(q) = ˆ \underline{ξ}x∂_{x}−ˆ\underline{ξ}\underline{σ}∂\underline{σ}−ˆ\underline{ξ}ˆ\underline{ζ}′′∂ˆ\underline{ζ}′′ +k_{2,ij}ˆ\underline{ζ}′′_{i}∂_{z′′_{j}} +k_{2,ij}ˆ\underline{ζ}′′_{i}ˆ\underline{ζ}′′_{j}∂ˆ\underline{ξ}− \frac{1}{2} \frac{∂k_{2,ij}}{∂z′′_{k}} ˆ \underline{ζ}′′_{i}ˆ\underline{ζ}′′_{j}∂ˆ\underline{ζ}′′_{k}.

Pushing forward under { \atop π_{es→eb}}, we obtain

\begin{aligned}({ \atop π_{es→eb}}_{∗}H_{es})(p) = & − 2ξ̂x∂_{x} + 2ξ̂σ∂_{σ} + 2ξ̂ζ̂′′∂ζ̂′′ \\ & − 2k_{2,ij}ζ̂′′_{i}∂_{z′′_{j}} − 2k_{2,ij}ζ̂′′_{i}ˆ\underline{ζ}′′_{j}∂ξ̂ + \frac{∂k_{2,ij}}{∂z′′_{k}} ζ̂′′_{i}ζ̂′′_{j}∂ζ̂′′_{k}. \end{aligned}

Below, this appears as the vector field |τ|V_{0}, and will give the direction of propagation at glancing points in Theorem 7.7. Lemma 5.35. Let Q_{i} = x^{−1}D_{z′_{i}}, κ_{ij}, m_{i}, h be as in Lemma 5.29. For A ∈ Ψ_{eb}^{m,l}(M),

(5.14) ı[□, A^{∗}A] = ∑ Q_{i}^{∗}L_{ij}Q_{j} + ∑(x^{−1}L_{i}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i}) + x^{−2}L_{0},

with L_{ij} ∈ Ψ_{eb}^{2m−1,2l}(M), L_{i}, L′_{i} ∈ Ψ_{eb}^{2m,2l}(M), L_{0} ∈ Ψ_{eb}^{2m+1,2l}(M), σ_{eb,2m−1}(L_{ij}) = 2aV_{ij}a, where V_{ij} = κ_{ij}(∂_{ζ′_{i}} + ∂_{ζ′_{j}} + 2∂_{ξ}) + H_{eb,κ_{ij}}, σ_{eb,2m}(L_{i}) = σ_{eb,2m}(L′_{i}) = 2aV_{i}a, where

V_{i} = ∑_{j} κ_{ij}∂_{z′_{j}} + \frac{1}{2}(m_{i}∂_{ξ} + H_{eb,m_{i}}) + \frac{1}{2}m_{i}(∂_{ξ} + ∂_{ζ′_{i}}),

σ_{eb,2m+1}(L_{0}) = 2aV_{0}a, V_{0} = 2h̃∂_{ξ} + H_{eb,h̃} + ∑_{i} m_{i}∂_{z′_{i}},

WF′_{eb}(L_{ij}), WF′_{eb}(L_{i}), WF′_{eb}(L′_{i}), WF′_{eb}(L_{0}) ⊂ WF′_{eb}(A).

(5.15)

In particular, for f ∈ C^{∞}(^{eb}S^{∗}M) with f|W̃ = ϖ_{eb}^{∗}ϕ for some ϕ ∈ S^{∗}W,

(5.16)  V_{ij}f|W̃ = 0, V_{i}f|W̃ = 0, V_{0}f|W̃ = 0.

Moreover, as smooth vector fields tangent to ^{eb}T ∗ ; W̃ M (but not necessarily tangent to the other boundaries),

\begin{matrix}V_{0}|_{C} = −2ξ x∂_{x} − 2(ξ^{2} + ∑_{ij} k_{2,ij}ζ′′_{i}ζ′′_{j})∂_{ξ} − 2ξ(τ ∂_{τ} + η ∂_{η}) \\ − 2 ∑_{i,j} k_{2,ij}ζ′′_{i}∂_{z′′_{j}} + ∑_{ℓ,i,j}(∂_{z′′_{ℓ}}k_{2,ij})ζ′′_{i}ζ′′_{j}∂_{ζ′′_{ℓ}}, \\ V_{ij}|_{C} = −k_{1,ij}(∂_{ζ′_{i}} + ∂_{ζ′_{j}} + 2∂_{ξ}) + ∑_{ℓ}(∂_{z′′_{ℓ}}k_{1,ij})∂_{ζ′′_{ℓ}}, V_{i}|_{C} = − ∑_{j} k_{1,ij}∂_{z′_{j}}, \end{matrix} (5.17)

and

\begin{matrix}(|τ|V_{0}ξ̂)|W̃ = −2 ∑_{ij} k_{2,ij}(0, y, z)ζ′′_{i}ζ′′_{j} \\ (|τ|V_{i}ξ̂)|W̃ = − ∑_{j} k_{3,ij}(0, y, z)ζ′′_{j}, (|τ|V_{ij}ξ̂)|W̃ = −2k_{1,ij}(0, y, z), \\ (|τ|^{−s−1}x^{−r}V_{0}(|τ|^{s}x^{r}))|W̃ = −2(r + s)ξ̂, (|τ|^{−s}x^{−r}V_{i}(|τ|^{s}x^{r}))|W̃ = 0, \\ (|τ|^{−s+1}x^{−r}V_{ij}(|τ|^{s}x^{r}))|W̃ = 0, \end{matrix} (5.18)

Remark 5.36. This is the main commutator computation that we use in the next section. We stated explicitly the results we need. First, equation (5.16) shows that functions of the “slow variables” do not affect the commutator to leading order at W̃, hence they are negligible for all of our subsequent calculations.

Next, (5.17) gives the form of the commutator explicitly at C; this is what we need for hyperbolic or glancing propagation within W̃, i.e. at points of H, resp. G away from radial points. These are sufficiently local that we only need the explicit calculation at C, rather than at all of W̃.

Finally, (5.18) contains the results we need at radial points in G: there the construction is rather global in W̃, so it would be insufficient to state these results at C only. On the other hand, localization in ζ̂ is accomplished by localizing in ξ̂, the “slow variables” and the characteristic set, so fewer features of V_{ij}, etc., are relevant.

Proof. By Lemma 5.29,

[□, A^{∗}A] = ∑_{i,j}([Q_{i}^{∗}, A^{∗}A]κ_{ij}Q_{j} + Q_{i}^{∗}κ_{ij}[Q_{j}, A^{∗}A] + Q_{i}^{∗}[κ_{ij}, A^{∗}A]Q_{j})

+ ∑_{i}(x^{−1}M_{i}[Q_{i}, A^{∗}A] + [x^{−1}M_{i}, A^{∗}A]Q_{i}

+ [Q_{i}^{∗}, A^{∗}A]x^{−1}M′_{i} + Q_{i}^{∗}[x^{−1}M′_{i}, A^{∗}A])

+ [x^{−2}H̃, A^{∗}A] on U.

The three terms on the first line of the right hand side are the only ones contributing to L_{ij}; in the case of the third term, via

ıσ_{eb,2m−1}([κ_{ij}, A^{∗}A]) = H_{eb,κ_{ij}}a^{2} = 2aH_{eb,κ_{ij}}a,

while in the case of the first two terms by evaluating the commutators using Lemma 5.7 and taking only the A_{1}-terms, with the notation of the lemma. The A_{0}-terms of the first two commutators on the first line of the right hand side (with the notation of Lemma 5.7) contribute to L_{i} or L′_{i}, as do the second and fourth terms on the second line and the A_{1}-term of the first and third terms on the second line. Finally, the expression on the third line, as well as the A_{0}-term of the first and third terms on the second line contribute to L_{0}. We also use (4.2) to remove the weight from H_{eb,x^{−1}m_{i}} and H_{eb,x^{−2}h̃}, e.g. x^{2}H_{eb,x^{−2}h̃} = −2h̃∂_{ξ} + H_{eb,h̃}.

The computation of the Hamilton vector fields at C then follows from Lemma 5.29 and (4.1) (recalling that t is one of the y-variables). □

6. Coisotropic regularity and non-focusing

In this section we recall from [25] the notion of coisotropic regularity and, dually, that of nonfocusing. We will be working microlocally near F_{reg} and in particular, away from the difficulties of the glancing rays in F_{sing}. Consequently all the results in this section have proofs identical to those in [25, Section 4], where the fiber Z is without boundary.

Let K be a compact set in R_{eb}^{∘}. By Lemma 3.25, there exists an open set U ⊂ ^{eb}S^{∗}M such that K ⊂ U and \overline{U} ∩ F ⊂ F_{reg}. Recall from Corollary 3.24 that in this case F ∩ U is a coisotropic submanifold of ^{eb}S^{∗}M—recall from Footnote 7 that a submanifold of ^{eb}S^{∗}M is coisotropic if the corresponding conic submanifold of ^{eb}T^{∗}M \ o is such.

In what follows, we let U be an arbitrary open subset of ^{eb}S^{∗}M satisfying \overline{U} ∩ F ⊂ F_{reg}, thus U ∩ F is a C^{∞} embedded coisotropic submanifold of ebS^{∗}M; the foregoing remarks establish that such subsets are plentiful.

Definition 6.1. Given U as above, let M denote the module (over Ψ_{eb}^{0,0}(M)) of operators A ∈ Ψ_{eb}^{1,0}(M) such that

• WF′ A ⊂ U,

• σ_{eb,1}(A)|_{F_{reg}} = 0.

Let A be the algebra generated by M with A^{k} = A ∩ Ψ_{eb}^{k,0}(M).

Let 𝔛 be a Hilbert space on which Ψ_{eb}^{0,0}(M) acts, and let K ⊂ U. We say that^{10} u has coisotropic regularity of order k relative to 𝔛 in K if there exists Q ∈ Ψ_{eb}^{0,0}(M), elliptic on K, such that

A^{k}Qu ∈ 𝔛.

10Note that our choice of U containing K does not matter in the definition.

We say that u satisfies the nonfocusing condition of degree k relative to 𝔛 on K if there exists Q ∈ Ψ_{eb}^{0,0}(M), elliptic on K, such that

Qu ∈ A^{k}(𝔛).

We say that u is nonfocusing resp. coisotropic of degree k relative to 𝔛 on an arbitrary open subset S of F if for every open O ⊂ S with closure disjoint from F_{sing}, it is nonfocusing resp. coisotropic on O of degree k with respect to 𝔛.

Remark 6.2.

(1) u is coisotropic on K if and only if u is coisotropic at every p ∈ K, i.e. on {p} for every p ∈ K. This can be seen by a partition of unity and a microlocal elliptic parametrix construction, as usual.

(2) The conditions of coisotropic regularity and nonfocusing should be, loosely speaking, considered to be dual to one another; a precise statement to this effect appears in the proof of Theorem 9.4 below.

(3) Coisotropy and nonfocusing are only of interest on F_{reg} itself: off of this set, to be coisotropic of order k with respect to 𝔛 means merely to be microlocally in H_{eb,𝔛}^{k} while to be nonfocusing means to be microlocally in H_{eb,𝔛}^{−k}.

(4) Certainly, away from W̃ , σ(□) vanishes on F_{reg}, as the latter lies in the characteristic set Σ by definition. Splitting Σ into component according to the sign of τ, and letting Π_{±} be pseudodifferential operators over M^{∘} microlocalizing near each of these components, we thus have

□Π_{±} = Q_{±}A_{±} + R

with A_{±} in M, Q_{±} elliptic of order 1, and R smoothing.

As a consequence of coisotropy, we have:

Lemma 6.3. The module M is closed under commutators, and is finitely generated, i.e., there exist finitely many A_{i} ∈ Ψ_{eb}^{0} with σ_{eb,1}(A_{i})|_{F_{reg}} = 0 such that

M = {A ∈ Ψ_{eb}^{1}(U) : ∃Q_{i} ∈ Ψ_{e}^{0}(U), A = ∑_{i=0}^{N} Q_{i}A_{i}}.

Moreover we may take A_{N} to have symbol |τ|^{−1}σ_{eb,2,0}(x^{2}□) and A_{0} = Id. We thus also obtain

(6.1)  A^{k} = { ∑_{|α|≤k} Q_{α} ∏_{i=1}^{N} A_{i}^{α_{i}}, Q_{α} ∈ Ψ_{eb}^{0}(U)}

where α runs over multiindices α : {1, …, N} → ℕ_{0} and |α| = α_{1}+⋯+α_{N}.

Corollary 6.4. If u is coisotropic of order k on K relative to 𝔛 then there exists U open, K ⊂ U such that for Q̃ ∈ Ψ_{eb}^{0,0}(M), WF′(Q̃) ⊂ U implies Q̃A^{α}u ∈ 𝔛 for |α| ≤ k.

Conversely, suppose U is open and for Q̃ ∈ Ψ_{eb}^{0,0}(M), WF′(Q̃) ⊂ U implies Q̃A^{α}u ∈ 𝔛 for |α| ≤ k. Then for K ⊂ U, u is coisotropic of order k on K relative to 𝔛.

Proof. Suppose first that u is coisotropic of order k on K relative to 𝔛. By definition, there exists Q elliptic on K such that A^{k}Qu ⊂ 𝔛. Let U be such that Q is elliptic on \overline{U}, K ⊂ U, and let S ∈ Ψ_{eb}^{0,0}(M) be a microlocal parametrix for Q, so WF′(R) ∩ \overline{U} = ∅ where R = SQ − Id.

We prove the corollary by induction, with the case k = 0 being immediate as one can write Q̃u = Q̃SQu + Q̃Ru, Q̃S ∈ Ψ_{eb}^{0,0}(M) is bounded on 𝔛, Qu ∈ 𝔛, Q̃R ∈ Ψ_{eb}^{−∞,0}(M) (for they have disjoint WF′), so Q̃Ru ∈ 𝔛.

Suppose now that k ≥ 1, and the claim has been proved for k − 1. By Lemma A.1, applied with Q_{n} = Q (i.e. there is no need for the subscript n, or for uniformity),

A^{α}Q = QA^{α} + ∑_{|β|≤|α|−1} C_{β}A^{β}.

Thus, for |α| = k,

Q̃A^{α}u = Q̃SQA^{α} + Q̃RA^{α}u

and

Q̃SQA^{α} = Q̃SA^{α}Q − ∑_{|β|≤|α|−1} Q̃SC_{β}A^{β}

together with the induction hypothesis (due to which and to Q̃SC_{β} ∈ Ψ_{eb}^{0,0}(M) with WF′(Q̃SC_{β}) ⊂ U, Q̃SC_{β}A^{β}u ∈ 𝔛) and Q̃R ∈ Ψ_{eb}^{−∞,0}(M) imply Q̃A^{α}u ∈ 𝔛, providing the inductive step.

The proof of the converse statement is similar. □

We now specialize to the case

ℌ = L_{g}^{2}(I × X_{0})

where I is a compact interval. We additionally introduce another Hilbert space 𝔛 ⊂ ℌ, given by H_{0}^{1}(I × X_{0}) or H^{1}(I × X_{0}) with I an interval and the 0 denoting vanishing at I × ∂X_{0}. Note that Id +Δ : 𝔛 → 𝔛^{∗} is an isometry.

Suppose K is compact. For N ≥ k + r we let 𝔜_{K} denote the subspace of ℌ

𝔜_{K} = {u ∈ 𝔛^{∗} : WF_{b,ℌ^{∗}}^{N}(u) ⊂ K, u is coisotropic of order k w.r.t.
H_{b,𝔛^{∗}}^{r} on K}.

Let

ℨ_{K} = {ϕ ∈ 𝔛 : u is coisotropic of order k w.r.t. H_{b,𝔛}^{r} on K}.

Lemma 6.5. Suppose that K ⊂ O, K compact, O open with compact closure, and Q ∈ Ψ^{0}(M) such that WF′(Id −Q) ∩ K = ∅, WF′(Q) ⊂ O. Let

(6.2)  𝔜 = {u ∈ 𝔛^{∗} : T_{N}(Id −Q)u ∈ 𝔛^{∗}, |α| ≤ k ⇒ T_{r}A^{α}Qu ∈ 𝔛^{∗}}

and

(6.3)  ℨ = {u ∈ 𝔛 : |α| ≤ k ⇒ T_{r}A^{α}Qu ∈ 𝔛}

Then

𝔜_{K} ⊂ 𝔜 ⊂ 𝔜\overline{O}.

and

ℨ_{K} ⊂ ℨ ⊂ ℨ\overline{O}.

Proof. If u ∈ 𝔜_{K}, then WF_{b,ℌ^{∗}}^{N}(u) ⊂ K implies that T_{N}(Id −Q)u ∈ 𝔛^{∗}. Moreover, since u is coisotropic on K, it is coisotropic on a neighborhood O′ of K; we construct Q′ ∈ Ψ^{0}(M) with WF Q′ ⊂ O′, WF(Id −Q′) ∩ K = ∅. Then

T_{r}A^{α}Qu = T_{r}A^{α}Q′Qu + T_{r}A^{α}(Id −Q′)Qu,

and the first term is in 𝔛^{∗} by coisotropy of u on O′ while the latter is in 𝔛^{∗} by the wavefront condition on u.

On the other hand, if u ∈ 𝔜, we have T_{N}(Id −Q)u ∈ 𝔛^{∗} hence WF_{b,ℌ^{∗}}^{N}(u)∩ ell(Id −Q) = ∅, so in particular, WF_{b,ℌ^{∗}}^{N}(u) ⊂ O^{c}, since Id −Q must be elliptic on O^{c}. It remains, given p ∈ \overline{O}, to check coisotropic regularity at p. If p ∈ ell(Id −Q), it again follows from the wavefront set condition, hence it suffices to consider p ∈ ell Q ⊃ [ell(Id −Q)]^{c}; at such points coisotropic regularity follows from T_{r}A^{α}Qu ∈ 𝔛^{∗}.

The proof for ℨ works analogously. □

Corollary 6.6. Suppose K = ∩_{j}O_{j}, O_{j} open with compact closure, \overline{O_{j+1}} ⊂ O_{j}. Let 𝔜_{j}, ℨ_{j} be given by (6.2), (6.3) where Q_{j} satisfies WF′(Id −Q_{j})∩K = ∅, WF′(Q_{j}) ⊂ O. Then 𝔜_{K} = ∩_{j}𝔜_{j}, ℨ_{K} = ∩_{j}ℨ_{K}.

In particular, 𝔜_{K} and ℨ_{K} become Fréchet spaces when equipped with the 𝔜_{j}, ℨ_{j} norms.

Remark 6.7. It is easy to see that the Fréchet topology is independent of the choice of the particular O_{j}.

Proof. The fact that 𝔜_{K} ⊂ ∩_{j}𝔜_{j} follows from Lemma 6.5. For the reverse inequality, note that u ∈ ∩𝔜_{j} ⊂ ∩𝔜\overline{O_{j}} has WF_{b,ℌ^{∗}}(u) ⊂ ∩\overline{O_{j}} = K. On the other hand, as u ∈ 𝔜_{1}, |α| ≤ k ⇒ A^{α}Q_{1}u ∈ 𝔛 and Q_{1} is elliptic on K. Thus, u ∈ 𝔜_{K}.

The same holds for ℨ_{K}. □

We now note the following functional-analytic facts:

Lemma 6.8. Let Q be as above, and again let

(6.4)  𝔜 = {u ∈ 𝔛^{∗} : T_{N}(Id −Q)u ∈ 𝔛^{∗}, |α| ≤ k ⇒ T_{r}A^{α}Qu ∈ 𝔛^{∗}},

and

ℨ = {u ∈ 𝔛 : |α| ≤ k ⇒ T_{r}A^{α}Qu ∈ 𝔛}

Then the dual of 𝔜 with respect to the space ℌ (see Appendix A) is

𝔜^{∗} = {u : u = v_{0} + T_{N}(Id −Q)v_{1} + ∑_{|α|≤k} T_{r}A^{α}Qv_{α}, v_{0}, v_{1}, v_{α} ∈ 𝔛},

and the dual of ℨ with respect to ℌ is

ℨ^{∗} = {u : u = v_{0} + ∑_{|α|≤k} T_{r}A^{α}Qv_{α}, v_{0}, v_{α} ∈ 𝔛^{∗}},

Proof. First consider the dual of ℨ with respect to ℌ. We apply the discussion of Appendix A leading to (A.7). More precisely, with the notation of the Appendix, we take ℌ = L_{g}^{2}(I×X_{0}), and 𝔛 = H^{1}(I×X_{0}), resp. 𝔛 = H_{0}^{1}(I×X_{0}), as set out earlier. We also let D = C^{∞}(I × X_{0}), resp. D = Ċ^{∞}(I × X_{0}) (with the dot indicating infinite order vanishing at I × ∂X_{0}). We define the operators B_{k} in (A.6) as follows: we take B_{k}, k = 1, …, N, to be a collection of C^{∞} vector fields on X_{0} which span V(X) over C^{∞}(X), B_{0} = Id, and define the 𝔛-norm on D by

(6.5)  ∥u∥_{𝔛}^{2} = ∥B_{0}u∥_{ℌ}^{2} + ∑_{k=1}^{N} ∥B_{k}u∥_{ℌ}^{2},

cf. (A.4); then 𝔛 is the completion of D. Then we take the collection of A_{j} in defining the space 𝔜 in Appendix A, with the norm (A.3), to be T_{r}A_{α}Q, |α| ≤ k. Then our claim about ℨ^{∗} follows from (A.7) and (A.1), taking into account that the principal symbol of the conjugate a pseudo-differential operator by complex conjugation is the complex conjugate of the principal symbol of the original operator, so its vanishing on F_{reg} is unaffected.

We now consider the dual of 𝔜, with 𝔜 given by (6.4). As Id +Δ : 𝔛 → 𝔛^{∗} is an isomorphism, the norm on 𝔛^{∗} is given by

∥u∥_{𝔛^{∗}} = ∥(Id +Δ)^{−1}u∥_{𝔛} = ∑_{k=0}^{N} ∥B_{k}(Id +Δ)^{−1}u∥_{ℌ}^{2},

with B_{k} as in (6.5), we are again in the setting leading to (A.7) with 𝔛 in the Appendix given by our 𝔛^{∗}, the B_{k} in the Appendix given by B_{k}(Id +Δ)^{−1}, the space 𝔜 in the Appendix being our space 𝔜 in (6.4), and the A_{j} in the appendix given by T_{N}(Id −Q) and T_{r}A^{α}Q, |α| ≤ k. Then our claim about 𝔜^{∗} follows from (A.7) and (A.1). □

Now let

𝔜̃_{K} = {u ∈ T_{N}(𝔛) : u is nonfocusing of order k w.r.t. H_{b,𝔛}^{−r} on K},

ℨ̃_{K} = {u ∈ 𝔛^{∗} : u is nonfocusing of order k w.r.t. H_{b,𝔛^{∗}}^{−r} on K}.

Lemma 6.9. Define 𝔜, ℨ as above. Then

𝔜̃_{K} ⊂ 𝔜^{∗} ⊂ 𝔜̃\overline{O}.

and

ℨ̃_{K} ⊂ ℨ^{∗} ⊂ ℨ̃\overline{O}.

The proof follows that of Lemma 6.5 closely, using the characterization of 𝔜^{∗} and ℨ^{∗} from Lemma 6.8.

We remark that away from W̃ , we may always (locally) conjugate by an FIO to a convenient normal form: being coisotropic, locally F can be put in a model form ζ = 0 by a symplectomorphism Φ in some canonical coordinates (y, z, η, ζ), see [10, Theorem 21.2.4] (for coisotropic submanifolds one has k = n−l, dim S = 2n, in the theorem). We may moreover arrange the (z, ζ) coordinates (i.e. apply a further symplectomorphism) so that σ(□)∘Φ = qζ_{1} for some symbol q elliptic in a small open set. We now quantize Φ to a FIO T, elliptic on some small neighborhood of a w ∈ F_{reg}, which can be arranged to have the following properties:

• T □ = QD_{z_{1}}T + R where Q ∈ Ψ^{1}(M^{∘}) is elliptic near Φ(w) and R is a smoothing operator.

• u has coisotropic regularity of order k (near w) with respect to H^{s} if and only if D_{z}^{α}T u ∈ H^{s} whenever |α| < k.

• u is nonfocusing of order k (near w) with respect to H^{s} if and only if T u ∈ ∑_{|α|≤k} D_{z}^{α}H^{s}.

Let G ∈ Ψ^{−1}(M^{∘}) be a parametrix for Q. As a consequence of the above observation, □u = f implies that D_{z_{1}}T u − GT f ∈ C^{∞} microlocally near Φ(w), and if f is coisotropic of order k relative to H^{s−1}, then D_{z}^{α}GT f ∈ H^{s} for |α| ≤ k (with an analogous statement for non-focusing) hence we have now sketched the proof of the following:

Proposition 6.10. Suppose u is a distribution on M^{∘}, □u = f. If f is coisotropic of order k, resp. nonfocusing of order k, with respect to H^{s−1} then the coisotropic regularity of order k, resp. non-focusing regularity of order k, with respect to H^{s}, is invariant under the Hamilton flow over M^{∘}.

In particular, for a solution to the wave equation, coisotropic regularity of order k with respect to H^{s} and nonfocusing of order k with respect to H^{s} are invariant under the Hamilton flow over M^{∘}.

(We remark that one could certainly give an alternative proof of this proposition by positive commutator arguments similar to, but much easier than, those used for propagation of edge regularity in the following section.)

Corollary 6.11. Suppose that f is coisotropic, resp. non-focusing, of order k relative to H^{m−1}, supported in t > T. Let u be the unique solution of □u = f with Dirichlet or Neumann boundary conditions, supported in t > T. Then u is coisotropic, resp. non-focusing, of order k relative to H^{m} at p ∈ S^{∗}M^{∘} provided every^{11} GBB γ with γ(0) = p has the property that there exists s_{0} such that t(γ(s_{0})) < T, and for s ∈ [0, s_{0}] (or s ∈ [s_{0}, 0], if s_{0} < 0), γ(s) ∈ S^{∗}M^{∘}.

The analogous statements hold if f is supported in t < T, and u is the unique solution of □u = f supported in t < T, provided we replace t(γ(s_{0})) < T by t(γ(s_{0})) > T.

Proof. This is an immediate consequence of Proposition 6.10, taking into account that u is coisotropic, resp. non-focusing, in t < T, by virtue of vanishing there. □

If K ⊂ M^{∘} is compact, then there is δ > 0 such that if p ∈ S_{K}^{∗}M^{∘} and γ is a GBB with γ(0) = p, then for s ∈ (−δ, δ), γ(s) ∈ M^{∘}. As s is equivalent to t as a parameter along GBB , we deduce the following result.

Corollary 6.12. Suppose K ⊂ M^{∘} is compact. Suppose that f is coisotropic, resp. non-focusing, of order k relative to H^{m−1}, supported in t > T. Let u be the unique solution of □u = f with Dirichlet or Neumann boundary conditions, supported in t > T. Then there exists δ_{0} > 0 such that u is coisotropic, resp. non-focusing, of order k relative to H^{m} at p ∈ S_{K}^{∗}M^{∘} if t(p) < T + δ_{0}.

Of course, what happens to coisotropic regularity and nonfocusing when bicharacteristics reach W̃ is of considerable interest, and will be discussed below.

7. Edge propagation

This section contains a series of theorems that will enable us to track propagation of regularity into and back out of the edge. They are as follows:

• Theorem 7.1, which governs propagation of regularity into and out of the interior or W̃ as well as the microlocal propagation of coisotropic regularity there (i.e. iterated regularity under application of operators in A).

• Theorem 7.3, which governs propagation of regularity into W̃ along glancing rays, tangent to one or more of the boundary faces meeting at ℝ × Y (in the blown-down picture).

• Theorem 7.6, which governs the propagation of edge regularity at non-radial hyperbolic points at the boundary of the edge W̃ .

• Theorem 7.7, which governs the propagation of edge regularity at

glancing points at the boundary of the edge W̃ . These theorems will then be assembled (together with the propagation over the interior of the edge, which we may simply quote from [25]) to yield the propagation of coisotropic regularity into and out of the edge in Theorem 8.1, and this result is the key ingredient in proving the “geometric” improvement in regularity on the diffracted wave.

11The restriction of this GBB to [0, s_{0}], if s_{0} ≥ 0, or [s_{0}, 0] if s_{0} < 0, is unique under these assumptions.

7.1. Radial points in the interior of the edge. The following theorem enables us to track edge wavefront set entering and leaving the edge at radial points over its interior.

Theorem 7.1 (Propagation at radial points in the interior of the edge). (See [25, Theorem 11.1]) Let u ∈ H_{es}^{1,l}(M) solve □u = 0 with Dirichlet or Neumann boundary conditions.

(1) Let m > l + f/2. Given α ∈ H_{W,b}, and p ∈ R_{eb,α,I}^{∘}, if (F_{I,p}\∂M) ∩ WF^{m} Au = ∅, for all A ∈ A^{k} then p ∉ WF_{e}^{m,l′} Bu for all l′ < l and all B ∈ A^{k}.

(2) Let m < l + f/2. Given α ∈ H_{W,b}, p ∈ R_{eb,α,O}^{∘}, if a neighborhood U of p in ^{e}S^{∗}|_{∂M}M is such that WF_{e}^{m,l}(Au)∩U ⊂ ∂F_{O} for all A ∈ A^{k} then p ∉ WF_{e}^{m,l}(Bu) for all B ∈ A^{k}.

This theorem is literally the same theorem as [25, Theorem 11.1] as we are restricting our attention to the interior of the fibers, hence the presence of a fiber boundary in our setting is irrelevant. We thus refer the reader to [25] for the proof.

Remark 7.2. In fact, we could take u ∈ H_{es}^{−∞,l}(M) here, but the restriction on regularity will be necessary in later theorems to maintain the boundary condition at the side faces z′_{i} = 0, and we prefer to keep a uniform hypothesis. Moreover, the boundary conditions are irrelevant; again, they are stated for the sake of uniformization.

7.2. Propagation into radial points over the boundary of the edge. We now turn to the question of propagation into the edge at glancing points, i.e. at points over the boundary of the fibers of M. Note that the hypotheses of this theorem are global in the boundary of the fiber: we do not attempt to distinguish different points in the fiber boundary.

Theorem 7.3 (Propagation into radial points over the boundary of the edge). Let u ∈ 𝔛 ≡ H_{es}^{1,l}(M) solve □u = 0 with Dirichlet or Neumann boundary conditions (see Definition 5.20).

Let m > l + f/2 − 1 with m ≥ 0, l′ ≤ 0. Suppose that q ∈ H_{W,b} and there exists a neighborhood U of

R_{eb,q,I} ∩^{eb}S_{∂W̃}^{∗} M = R_{eb,q,I} ∩ G

in ^{eb}S^{∗}M such that {x > 0}∩U ∩WF_{eb,𝔛}^{m,∗} u = ∅. Then ^{eb}S_{∂W̃}^{∗} M ∩WF_{eb,𝔛}^{m,l′} u = ∅ for all l′ < 0.

Proof. Choose local coordinates on W, and let q = (y_{0}, t_{0}, τ̂_{0} ∈ {±1}, η̂_{0}) ∈ H_{W,b}. Choose ξ_{0} such that ξ̂_{0}^{2} = 1 − h(y, η̂_{0}) with sgn ξ̂_{0} = sgn τ_{0} (this is the incoming point).

One of the central issues in proving the theorem is to construct a symbol that is localized in the hypothesis region that is sufficiently close to being flow-invariant. To begin, we will need a localizer in the fiber variables. Fix any K ⋐ W̃^{∘} and fix a small number ϵ_{K} > 0. Let

Υ : ^{e}S_{K}^{∗}(M) ∩ \{|\underline{ζ}|/|\underline{ξ}| < ϵ_{K}\} → Z

be locally defined by

Υ(q′) = z(exp_{z,ˆ\underline{ζ}} s_{∞}H_{es}), s_{∞} = \begin{matrix}sgn & ˆ\underline{ξ} \\ |ˆ\underline{ζ}| & K_{q} \end{matrix} arctan \frac{|ˆ\underline{ζ}|}{ˆ\underline{ξ}}.

where q′ ∈ ^{es}S_{K}^{∗}(M) has “edge-smooth” coordinates (t, y, z, τ, ξ, η, ζ) (we are using the canonical identification of ^{e}S^{∗}M with ^{es}S^{∗}M away from ∂M\W̃). This map is well-defined provided ϵ_{K} is chosen sufficiently small (so that the flow stays away from ∂W̃). The map simply takes a point over the boundary to its limit point in the fiber variables along the forward bicharacteristic flow, hence on W̃, we certainly have Υ_{∗}(H_{es}) = 0.

We now employ Υ to create a localizer away from ∂W̃ . Fix

K′′ ⊂ U′ ⊂ K′ ⊂ U ⊂ K ⊂ W̃^{∘}

with K′′, K′, K compact and U′, U open such that

(1) α ∈ ^{es}S_{K\U}^{∗}M and |ˆ\underline{ζ}(α)/ˆ\underline{ξ}(α)| < ϵ_{K} imply Υ_{K}(α) ∈ W̃ \ K′,

(2) α ∈ ^{es}S_{K′′}^{∗}M and |ˆ\underline{ζ}(α)/ˆ\underline{ξ}(α)| < ϵ_{K} imply Υ_{K}(α) ∈ U′.

Now let χ ∈ C^{∞}(W̃) be equal to 0 on W̃ \K′ and 1 on U′. For ϵ_{K} sufficiently small, χ∘Υ_{K} vanishes on^{es}S_{K\U}^{∗}M, hence can be extended as 0 to^{es}S_{W̃ \U}^{∗}M to define a C^{∞} function. Thus, this extension of χ ∘ Υ_{K} is well defined and smooth on

\{|ˆ\underline{ζ}/ˆ\underline{ξ}| < ϵ_{K}, x < ϵ_{K}\} ∪^{es}S_{W̃ \U}^{∗}M ⊂ ^{es}S^{∗}M;

it equals 0 on the fibers over W̃ \U and 1 on those over K′′. But

\{|ζ̂/ξ̂| < ϵ′_{K}, x < ϵ′_{K}\} ∪^{es}S_{W̃ \U}^{∗}M ⊂ \{|ˆ\underline{ζ}/ˆ\underline{ξ}| < ϵ_{K}, x < ϵ_{K}\} ∪^{es}S_{W̃ \U}^{∗}M

for ϵ′_{K} > 0 sufficiently small as on U, |ˆ\underline{ζ}/ˆ\underline{ξ}| ≲ |ζ̂/ξ̂|, since |z′_{i}| are all bounded

away from 0 there. Due to the vanishing near ∂W̃, we can equivalently regard this extension of χ ∘ Υ_{K} as a C^{∞} function ρ on the following subset of the edge-b cosphere bundle:

{|ζ̂/ξ̂| < ϵ′_{K}, x < ϵ′_{K}} ∪^{eb}S_{W̃ \U}^{∗}M.

The resulting function satisfies

(7.1)  H_{es}(ρ) = O(x) on {|ζ̂/ξ̂| < ϵ′_{K}, x < ϵ′_{K}} ∪^{eb}S_{W̃ \U}^{∗}M.

and

(7.2)  ρ = 0 on ^{eb}S_{W̃ \U}^{∗}M.

It is convenient to extend ρ to all of ^{eb}S^{∗}M by defining it to be an arbitrary fixed positive constant, say 1, where it is not previously defined. Note that by (7.2), when we need to calculate derivatives of ρ in a commutator calculation, we may always assume that we are away from ∂W̃, hence use the edge-calculus Hamilton vector field result.

Now consider the function

ω = |η̂ − η̂_{0}|^{2} + |y − y_{0}|^{2} + ρ^{2} + |t − t_{0}|^{2}.

(Note that keeping ω, |ξ − ξ_{0}| and x sufficiently small on Σ_{eb} automatically means that ζ̂ is small as well.)

We now identify some appropriate neighborhoods in which to localize. First, choose ϵ_{0}, ϵ_{1} < 1 such that

|ζ̂|^{2} < ϵ_{0}, x < ϵ_{1}, ω < ϵ_{1}, |ξ − ξ_{0}|^{2} < ϵ_{1} ⟹ |ζ̂/ξ̂| < ϵ′_{K}/2.

Second, choose ϵ_{2} < ϵ_{1} such that

Σ_{eb} ∩ \{x < ϵ_{2}\} ∩ \{ω < ϵ_{2}\} ∩ \{|ξ̂− ξ̂_{0}|^{2} < ϵ_{2}\} ⊂ \{|ζ̂|^{2} < \frac{ϵ_{0}}{2} \}.

Let

K̃ = \{x ≤ ϵ_{2}\} ∩ \{ω ≤ ϵ_{2}\} ∩ \{|ξ̂− ξ̂_{0}|^{2} ≤ ϵ_{2}\} ∩ \{|ζ̂|^{2} ≤ \frac{ϵ_{0}}{2} \}.

Next, given δ > 0, which will depend on K̃, let U = U_{δ} be as in Corollary 5.34. Finally, given any β > 0 (to be specified below) we will choose ϵ = ϵ(β, δ) so that

(7.3)  ϵ, ϵ(1 + β) < ϵ_{2},

and so that

K_{ϵ} = {x ≤ ϵ, ω ≤ ϵ(1 + β), |ξ̂− ξ̂_{0}|^{2} ≤ ϵ, |ζ̂|^{2} ≤ ϵ_{0}} ⊂ U = U_{δ}.

(Note that K_{ϵ} ⊂ K̃ by (7.3).)

Let ϕ ∈ C_{c}^{∞}([0, ϵ)), ψ_{0} ∈ C_{c}^{∞}([0, ϵ_{0})), identically 1 on [0, ϵ_{0}/2], ψ_{1} ∈ C_{c}^{∞}([0, ϵ)), identically 1 on [0, ϵ/2], ψ ∈ C_{c}^{∞}((−∞, ϵ)), all non-increasing,

a = a_{ϵ} = |τ|^{s}x^{−r}ψ(ω − βx)ψ_{1}(|ξ̂− ξ̂_{0}|^{2})ϕ(x)ψ_{0}(|ζ̂|^{2})

Thus,

x ≤ ϵ, ω ≤ ϵ(1 + β), |ξ̂− ξ̂_{0}|^{2} ≤ ϵ, |ζ̂|^{2} ≤ ϵ_{0} on supp a_{ϵ}.

We usually suppress the ϵ-dependence of a below in our notation. Equation (7.3) ensures that ϵ(1 + β) < 1 on supp a, so ρ < 1, and thus (7.1) holds. We have also arranged that

|ζ̂/ξ̂| < ϵ′_{K}/2 on supp a

and that ψ_{0}(|ζ̂|^{2}) = 1 on supp(ψ(ω −βx)ϕ(x)ψ_{1}(|ξ̂−ξ̂_{0}|^{2}))∩Σ_{eb}. This latter observation means that we need never consider derivatives falling on the ψ_{0} term when computing the action of the Hamilton vector field on a. (The cutoff ψ_{0}(ζ̂) is therefore not necessary for correct localization of a, as that is achieved by the cutoffs in ω, ξ̂and x if we restrict our attention to Σ_{eb}; rather this is necessary to make a a symbol, which it would not be if independent of ζ.)

We quantize a to A ∈ Ψ_{eb}^{s,r}(M), i.e. take any A with σ_{eb,s}(A) = a. By Lemma 5.35,

(7.4) ı[□, A^{∗}A] = ∑ Q_{i}^{∗}L_{ij}Q_{j} + ∑(x^{−1}L_{i}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i}) + x^{−2}L_{0},

with

L_{ij} ∈ Ψ_{eb}^{2s−1,2r}(M), L_{i} ∈ Ψ_{eb}^{2s,2r}(M), L_{0} ∈ Ψ_{eb}^{2s+1,2r}(M),

σ_{eb,2s−1}(L_{ij}) = 2aV_{ij}a, σ_{eb,2s}(L_{i}) = 2aV_{i}a, σ_{eb,2s+1}(L_{0}) = 2aV_{0}a,

WF′_{eb}(L_{ij}), WF′_{eb}(L_{i}), WF′_{eb}(L′_{i}), WF′_{eb}(L_{0}) ⊂ WF′_{eb}(A),

(7.5)

with V_{ij}, V_{i} and V_{0} smooth vector fields tangent on ^{eb}T^{∗}M tangent to W̃ and such that for f ∈ C^{∞}(^{eb}S^{∗}M) with f|W̃ = ϖ_{eb}^{∗}ϕ for some ϕ ∈ S^{∗}W,

(7.6)  V_{ij}f|W̃ = 0, V_{i}f|W̃ = 0, V_{0}f|W̃ = 0.

In view of Corollary 5.34, we are led to regard the L_{ij} and L_{i} terms as negligible, provided that their principal symbol is bounded by a constant multiple of σ_{eb,s+1}(L_{0}) times the appropriate power of |τ| (to arrange homogeneity of the same degree). Also by Lemma 5.35,

\begin{matrix}(|τ|V_{0}ξ̂)|W̃ = −2 ∑_{ij} k_{2,ij}(0, y, z)ζ′′_{i}ζ′′_{j} \\ (|τ|V_{i}ξ̂)|W̃ = − ∑_{j} k_{3,ij}(0, y, z)ζ′′_{j}, (|τ|V_{ij}ξ̂)|W̃ = −2k_{1,ij}(0, y, z), \\ (|τ|^{−s′−1}x^{−r′}V_{0}(|τ|^{s′}x^{r′}))|W̃ = −2(r′ + s′)ξ̂, (|τ|^{−s′}x^{−r′}V_{i}(|τ|^{s′}x^{r′}))|W̃ = 0, \\ (|τ|^{−s′+1}x^{−r′}V_{ij}(|τ|^{s′}x^{r′}))|W̃ = 0, \end{matrix} (7.7)

In particular, with s′ = 0, r′ = 1, |τ|^{−1}V_{0}x = −2ξ̂x, while V_{i}x, V_{ij}x are O(x^{2}).

In computing V a for various homogeneous degree μ − 1 vector fields V on ^{eb}T^{∗}M, we will employ the following arrangement of terms:

V a =ψ(ω − βx)ϕ(x)ψ_{0}(|ζ̂|^{2})ψ_{1}(|ξ̂− ξ̂_{0}|^{2})V (|τ|^{s}x^{−r})
+ |τ|^{s}x^{−r}ψ_{0}(|ζ̂|^{2})ϕ(x)ψ_{1}(|ξ̂− ξ̂_{0}|^{2})ψ′(ω − βx)(V ω − βV x)
+ |τ|^{s}x^{−r}ψ(ω − βx)ψ_{0}(|ζ̂|^{2})ψ_{1}(|ξ̂− ξ̂_{0}|^{2})V (ϕ(x))
+ |τ|^{s}x^{−r}ψ(ω − βx)ψ_{0}(|ζ̂|^{2})ϕ(x)V (ψ_{1}(|ξ̂− ξ̂_{0}|^{2}))
+ |τ|^{s}x^{−r}ψ(ω − βx)ϕ(x)ψ_{1}(|ξ̂− ξ̂_{0}|^{2})V (ψ_{0}(|ζ̂|^{2})).

As |τ|^{−1}V_{0}x = −2ξ̂x while |τ|^{−1}V_{0}ω = xf for some f ∈ C^{∞}(^{eb}S^{∗}X), and |ξ̂| is bounded below on K̃ (which is a compact subset of ^{eb}S^{∗}M), it follows that there exists β > 0 such that −(sgn τ_{0})|τ|^{−1}(V_{0}ω −βV_{0}x) ≥ x on K̃, and thus

(7.8)  −(sgn τ_{0})|τ|^{−1}(V_{0}ω − βV_{0}x) = xc_{2}^{2}

for some smooth positive function c_{2} defined on K̃, hence on a neighborhood of supp a in ^{eb}S^{∗}M. Moreover,

V_{0}ψ_{1}(|ξ̂− ξ̂_{0}|^{2}) = −4(ξ̂− ξ̂_{0})ψ′_{1}(|ξ̂− ξ̂_{0}|^{2}) ∑_{ij} k_{2,ij}(0, y, z)ζ′′_{i}ζ′′_{j}.

Similar computations hold for the V_{i} and V_{ij} terms, with result shown below in (7.9).

We start by discussing the terms in (7.4)-(7.5) in which the vector fields V_{0}, V_{i}, V_{ij} differentiate ψ_{1}(|ξ̂− ξ̂_{0}|^{2}). These terms altogether have the form

(7.9) ∑ Q_{i}^{∗}L_{ij,1}Q_{j} + ∑(x^{−1}L_{i,1}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i,1}) + x^{−2}L_{0,1}, where

σ_{eb,2s−1}(L_{ij,1})|W̃ = −(sgn τ_{0})a_{1}k_{1,ij}(0, y, z),

σ_{eb,2s}(L_{i,1})|W̃ = σ_{eb,2s}(L′_{i,1})|W̃ = −\frac{1}{2}(sgn τ_{0})a_{1} ∑_{j} k_{3,ij}(0, y, z)ζ′′_{j},

σ_{eb,2s+1}(L_{0,1})|W̃ = −(sgn τ_{0})a_{1} ∑_{ij} k_{2,ij}(0, y, z)ζ′′_{i}ζ′′_{j},

with a_{1} = 8a(sgn τ_{0})(ξ̂− ξ̂_{0})|τ|^{s}x^{−r}ψ(ω − βx)ψ_{0}(|ζ̂|^{2})ϕ(x)ψ′_{1}(|ξ̂− ξ̂_{0}|^{2}).

On K̃ ∩ supp ψ′_{1}(|ξ̂ − ξ̂_{0}|^{2}) ∩ ^{eb}Σ, ξ̂ − ξ̂_{0} has sign − sgn ξ̂_{0} = − sgn τ_{0}, so (sgn τ_{0})(ξ̂− ξ̂_{0}) < 0 there. Thus, noting that the right hand side on the last line is a square for x sufficiently small in view of ψ′_{1} ≤ 0 when (sgn τ_{0})(ξ̂_{0} − ξ̂) > 0, it has the form

(7.10) (− sgn τ_{0})d_{Z}^{∗}x^{−1}C_{0}^{∗}C_{0}x^{−1}d_{Z} + E_{0} + E′_{0} + F_{0},

C_{0} ∈ Ψ_{eb}^{s−1/2,r}(M), E_{0}, E′_{0} ∈ Diff_{es,♯}^{2} Ψ_{eb}^{2s−1,2r+2}(M)
σ_{eb,s−1/2}(C_{0}) = (H((sgn τ_{0})(ξ_{0} − ξ))ϕ̃(x)a_{1})^{1/2} ,
WF′_{eb}(E′_{0}) ∩^{eb}Σ = ∅, WF′_{eb}(E_{0}) ⊂ {x > 0} ∩ supp a,

F_{0} ∈ Diff_{es,♯}^{2} Ψ_{eb}^{2s−2,2r+2}(M), WF′_{eb}(E′_{0}), WF′_{eb}(F_{0}) ⊂ supp a,

where H is the Heaviside step function (recall that ψ′_{1} = 0 near the origin, and ψ′ ≤ 0) and ϕ̃ ∈ C_{c}^{∞}([0, ϵ_{2})) is identically 1 near 0 and has sufficiently small support.

Next, the terms in (7.4)-(7.5) in which the vector fields V_{0}, V_{i}, V_{ij} differentiate ψ(ω − βx) have the form

∑ Q_{i}^{∗}L_{ij,2}Q_{j} + ∑(x^{−1}L_{i,2}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i,2}) + x^{−2}L_{0,2}, where
σ_{eb,s−1}(L_{ij,2}) = a_{2}^{2}f_{ij,2}, σ_{eb,s}(L_{i,2}) = σ_{eb,s}(L′_{i,2}) = a_{2}^{2}f_{i,2},
σ_{eb,s+1}(L_{0,2}) = −(sgn τ_{0})a_{2}^{2}c_{2}^{2}, c_{2} as in (7.8)
a_{2}^{2} = −2xa|τ|^{s+1}x^{−r}ψ′(ω − βx)ψ_{0}(|ζ̂|^{2})ϕ(x)ψ_{1}(|ξ̂− ξ̂_{0}|^{2}),

with f_{ij,2}, f_{i,2} smooth. Moreover, terms in (7.4)–(7.5) in which the vector fields V_{0}, V_{i}, V_{ij} differentiate ψ_{0}(|ζ̂|^{2}) have wave front set disjoint from^{eb}Σ as already discussed, while the terms in which these vector fields differentiate ϕ(x) are supported in supp a ∩ {x > 0}, where we will assume the absence of WF_{eb,𝔛}^{s−1,∗} u (the weight is indicated by an asterisk as we are away from x = 0, so it is irrelevant).

Finally, the terms in (7.4)–(7.5) in which the vector fields V_{0}, V_{i}, V_{ij} differentiate |τ|^{s}x^{−r} have the form

∑ Q_{i}^{∗}L_{ij,3}Q_{j} + ∑(x^{−1}L_{i,3}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i,3}) + x^{−2}L_{0,2}, where
σ_{eb,s−1}(L_{ij,3}) = a^{2}xf_{ij,3}, σ_{eb,s}(L_{i,3}) = σ_{eb,s}(L′_{i,3}) = a^{2}xf_{i,3},
σ_{eb,s+1}(L_{0,3}) = −a^{2}(sgn τ_{0})|τ|2(s − r)c_{3}^{2},

where c_{3}^{2}|W̃ = 4(sgn τ_{0})ξ̂ > 0.

Finally, recall that terms with ψ_{0} derivatives are supported in the elliptic set of □.

We are now ready to piece together the above information to compute the commutator [□, A^{∗}A]. First we choose a family of operators convenient for adjusting orders: pick

T_{ν} ∈ Ψ_{eb}^{ν,0}(M), σ_{eb,ν}(T_{ν}) = |τ|^{ν} near K̃.

Thus, T_{ν} are simply weights, for |τ|^{ν} is elliptic of order ν on a neighborhood of K̃.

Adding all the terms computed above, and rearranging them as needed, noting the top order commutativity in eb-order of Diff_{es,♯} Ψ_{eb}(M), we finally deduce that

\begin{matrix}−ı & (sgn τ_{0})[□, A^{∗}A] \\ = & A_{2}^{∗}(C_{2}^{∗}x^{−2}C_{2} + ∑_{i}(x^{−1}F_{i,2}Q_{i} + Q_{i}^{∗}x^{−1}F′_{i,2}) + ∑_{ij} Q_{i}^{∗}F_{ij,2}Q_{j})A_{2} \\ + A^{∗}T_{1/2}^{∗}(C_{3}^{∗}2(s − r)x^{−2}C_{3} + ∑_{i}(x^{−1}F_{i,3}Q_{i} + Q_{i}^{∗}x^{−1}F′_{i,3}) \\ + ∑_{ij} Q_{i}^{∗}F_{ij,3}Q_{j})T_{1/2}A \\ + d_{Z}^{∗}x^{−1}C_{0}^{∗}C_{0}x^{−1}d_{Z} + E + E′ + R′′ \end{matrix} ; (7.11)

with

(1) A_{2} ∈ Ψ_{eb}^{s+1/2,r−1/2}(M), σ_{eb,s+1/2}(A_{2}) = a_{2}, WF′_{eb}(A_{2}) ⊂ supp a

(2) C_{2}, C_{3} ∈ Ψ_{eb}^{0,0}(M); F_{i,2}, F′_{i,2}, F_{i,3}, F′_{i,3} ∈ Ψ_{eb}^{−1,0}(M) and F_{ij,2}, F_{ij,3} ∈ Ψ_{eb}^{−2,0}(M);

(3) On K̃, σ_{eb,0}(C_{2}) ≠ 0 and σ_{eb,0}(C_{3}) = (sgn τ_{0})ξ̂ ≠ 0,

(4) C_{0} ∈ Ψ_{eb}^{s−1/2,r}(M), WF′_{eb}(C_{0}) ⊂ supp a,

(5) E, E′ ∈ Diff_{es,♯}^{2} Ψ_{eb}^{2s−1,2r+2}(M),

(6) R′′ ∈ Diff_{es,♯}^{2} Ψ_{eb}^{2s−2,2r+2}(M) (i.e. is lower order), WF′_{eb}(R′′) ⊂ supp a,

(7) WF′_{eb} E ⊂ {x > 0} ∩ supp a (our hypothesis region),

(8) WF′_{eb} E′ ∩^{eb}Σ = ∅, WF′_{eb}(E′) ⊂ supp a.

When we pair both sides of this equation (suitably regularized) with a solution to the wave equation the terms E, E′ and R′′ will be controlled respectively by the hypothesis on u in x > 0, microlocal elliptic regularity, and an inductive hypothesis in the iterative argument in which we improve the order by 1/2 (or less) in each step. The remaining terms on the right hand side are either positive, or involve Q_{i}, and the latter terms are controlled by the former, by Corollary 5.34. Thus, save for the need to mollify to make sure that we can actually apply this commutator to u and pair it with u, and also be able to rewrite the commutator as the difference of products, this would give our positive commutator result, controlling ∥x^{−1}T_{1/2}C_{3}Au∥.

We do, however, need to mollify. Let σ > 0 (typically we take σ = 1/2, always σ ∈ (0, 1/2]) Λ_{γ} ∈ Ψ_{eb}^{−σ}(M) for γ > 0, such that {Λ_{γ} : γ ∈ (0, 1]} is a bounded family in Ψ_{eb}^{0}(M), and Λ_{γ} → Id as γ ↓ 0 in Ψϵ̃ ; eb(M), for all ϵ̃ > 0. Let the principal symbol of Λ_{γ}, considered a bounded family in Ψ_{eb}^{0}(M), be (1+γ|τ|^{2})^{−σ/2} on a neighborhood of K̃. Let A_{γ} = Λ_{γ}A. We now have A_{γ} ∈ Ψ_{eb}^{s−σ,r}(M) for γ > 0, and A_{γ} is uniformly bounded in Ψ_{eb}^{s,r}(M), A_{γ} → A in Ψ_{eb}^{s+ϵ̃,r}(M). Moreover,

(7.12)  ı[□, A_{γ}^{∗}A_{γ}] = Λ_{γ}^{∗}ı[□, A^{∗}A]Λ_{γ} + A^{∗}ı[□, Λ_{γ}^{∗}Λ_{γ}]A + R̃,

with R̃ uniformly bounded in Diff_{es,♯}^{2} Ψ_{eb}^{2s−2,2r+2}(M) (hence lower order). Now, for a vector field V on ^{eb}T^{∗}M,

V (1 + γ|τ|^{2})^{−σ/2} = −(σ/2)γ(1 + γ|τ|^{2})^{−σ/2−1}V |τ|^{2}.

Applying this, the general formula (7.4)–(7.5) with Λ_{γ} in place of A and (7.7) with r′ = 0, s′ = 2, we deduce that

−(sgn τ_{0})A^{∗}ı[□, Λ_{γ}^{∗}Λ_{γ}]A

= A^{∗}Λ_{γ}^{∗}T_{1/2}^{∗}Λ̃_{γ}^{∗}( − 2σC_{3}^{∗}x^{−2}C_{3} + ∑_{i}(x^{−1}F_{i,4}Q_{i} + Q_{i}^{∗}x^{−1}F′_{i,4})

+ ∑_{ij} Q_{i}^{∗}F_{ij,4}Q_{j})T_{1/2}Λ̃_{γ}Λ_{γ}A + R′_{γ},

(7.13) with F_{ij,4} ∈ Ψ_{eb}^{−2,0}(M), F_{i,4}, F′_{i,4} ∈ Ψ_{eb}^{−1,0}(M), Λ̃_{γ} uniformly bounded in Ψ_{eb}^{0,0}(M) with principal symbol

σ_{eb}(Λ̃_{γ}) = (γ|τ|^{2}(1 + γ|τ|^{2})^{−1})^{1/2} ≤ 1,

C_{3} ∈ Ψ_{eb}^{0,0}(M) and T_{1/2} ∈ Ψ_{eb}^{1/2,0}(M) as in (7.11) and R′_{γ} uniformly bounded in Diff_{es,♯}^{2} Ψ_{eb}^{2s−2,2r+2}(M), hence lower order. Note that this commutator has the opposite sign from (7.11), which limits our ability to regularize. However, as long as σ′ − σ > 0, we can write

2σ′ Id −2σΛ̃_{γ}^{∗}Λ̃_{γ} = B_{γ}^{∗}B_{γ}

with B_{γ} uniformly bounded in Ψ_{eb}^{0,0}(M). Thus, if s − r > σ, taking σ′ such that σ < σ′ < s − r, we deduce that

\begin{matrix}−ı & (sgn τ_{0})[□, A_{γ}^{∗}A_{γ}] \\ = & A_{2,γ}^{∗}(C_{2}^{∗}x^{−2}C_{2} + ∑_{i}(x^{−1}F_{i,2}T_{1}Q_{i}T_{−1} + T_{−1}Q_{i}^{∗}T_{1}x^{−1}F′_{i,2}) \\ + ∑_{ij} T_{−1}Q_{i}^{∗}Q_{j}F_{ij,2}T_{1})A_{2,γ} \\ + A_{γ}^{∗}T_{1/2}^{∗}(C_{3}^{∗}2(s − r − σ′)x^{−2}C_{3} \\ + ∑_{i}(x^{−1}F_{i,5}T_{1}Q_{i}T_{−1} + T^{−1}Q_{i}^{∗}T_{1}x^{−1}F′_{i,5}) \\ + ∑_{ij} T_{−1}Q_{i}^{∗}Q_{j}F_{ij,5}T_{1})T_{1/2}A_{γ} \\ + A_{γ}^{∗}T_{1/2}^{∗}C_{3}^{∗}B_{γ}^{∗}B_{γ}C_{3}T_{1/2}A_{γ} \\ + d_{Z}^{∗}x^{−1}Λ_{γ}^{∗}C_{0}^{∗}C_{0}Λ_{γ}x^{−1}d_{Z} + E_{γ} + E′_{γ} + R′′_{γ}, \end{matrix} ; (7.14)

with the terms as in (7.11), in particular F_{ij,5}, F′_{ij,5} as F_{ij,3}, etc., there, and A_{2,γ} = A_{2}Λ_{γ}, etc. Here we rewrote the terms in (7.11) somewhat, inserting T_{1} and T_{−1} in places (recall that T_{1}T_{−1} differs from Id by an element of Ψ_{eb}^{−1,0}(M) on K̃, and this difference can be absorbed in R′′_{γ}) in order to be able to use Corollary 5.34 directly below. Applying both sides of (7.14) to u and pairing with u, we claim we may integrate by parts for any γ > 0 on the right hand side of the resulting expression to obtain

\begin{matrix}−ı & (sgn τ_{0})⟨[□, A_{γ}^{∗}A_{γ}]u, u⟩ \\ = & ∥x^{−1}C_{2}A_{2,γ}u∥^{2} + 2(s − r − σ′)∥x^{−1}C_{3}T_{1/2}A_{γ}u∥^{2} \\ + ∑_{ij}⟨Q_{j}F_{ij,2}T_{1}A_{2,γ}u, Q_{i}T_{−1}^{∗}A_{2,γ}u⟩ \\ + ∑_{i} (⟨Q_{i}T_{−1}A_{2,γ}u, x^{−1}T_{1}^{∗}F_{i,2}^{∗}A_{2,γ}u⟩ + ⟨x^{−1}T_{1}F′_{i,2}A_{2,γ}u, Q_{i}T_{−1}A_{2,γ}u⟩) \\ + ∑_{i} (⟨Q_{i}T_{−1}T_{1/2}A_{γ}u, x^{−1}T_{1}^{∗}F_{i,5}^{∗}T_{1/2}A_{γ}u⟩ \\ + ⟨x^{−1}T_{1}F′_{i,5}T_{1/2}A_{γ}u, Q_{i}T_{−1}T_{1/2}A_{γ}u⟩) \\ + ∑_{ij}⟨Q_{j}T_{1}F_{ij,5}T_{1/2}A_{γ}u, Q_{i}T_{−1}^{∗}T_{1/2}A_{γ}u⟩ \\ + ∥B_{γ}C_{3}T_{1/2}Au∥^{2} + ∥C_{0}Λ_{γ}x^{−1}d_{Z}u∥^{2} + ⟨(E_{γ} + E′_{γ} + R′′_{γ})u, u⟩, \end{matrix} (7.15)

and that we may similarly expand the left side by using

(7.16)  ⟨[□, A_{γ}^{∗}A_{γ}]u, u⟩ = ⟨A_{γ}^{∗}A_{γ}u, □u⟩ − ⟨□u, A_{γ}^{∗}A_{γ}u⟩,

so that pairing with a solution to the wave equation yields identically zero.

We begin by justifying these two integrations by parts, after which we will read off the consequences. We start with the Dirichlet case. Note that the L_{g}^{2}-dual of 𝔜 = H_{es,0}^{1,l}(M) is H_{es}^{−1,−l−(f+1)}(M) (where as usual the x^{f+1} factor derives from the difference between the metric density used in the pairing and the “edge-density” used to define the norm on H_{es,0}^{·,·}(M)). We have

x^{−2} Diff_{es,♯}^{2}(M) ∋ □ : 𝔜 → H_{es}^{−1,l−2}(M) = x^{2l+(f+1)−2}(𝔜)^{∗}.

Here we suppressed the quotient map ρ : Ḣ_{es}^{−1,l−2}(M) → H_{es}^{−1,l−2}(M), i.e. the stated mapping property is, strictly speaking, for ρ ∘ □. Furthermore, the dual of H_{eb,𝔜}^{s′,r′}(M) is

(H_{eb,𝔜}^{s′,r′}(M))^{∗} = H_{eb,(𝔜)^{∗}}^{−s′,−r′}(M).

Equation (7.16) makes sense directly and naively for γ > 0 if the products of □ with A_{γ}^{∗}A_{γ} ∈ Ψ_{eb}^{2s−2σ,2r}(M) map H_{eb,𝔜}^{s′,r′}(M) to its dual, H_{eb,(𝔜)^{∗}}^{−s′,−r′}(M). We thus require

A_{γ}^{∗}A_{γ} : H_{eb,𝔜}^{s′,r′}(M) → H_{eb,𝔜}^{−s′,−r′−2l−(f+1)+2}(M)

which holds if

(7.17)  \begin{aligned}s − σ & ≤ s′, \\ r & ≤ r′ + l + (f + 1)/2 − 1. \end{aligned}

Following the same line of reasoning shows that if we are willing to settle for just (7.15), by contrast, we only require the milder hypotheses

(7.18)  \begin{aligned}s − σ & ≤ s′ + 1/2, \\ r & ≤ r′ + l + (f + 1)/2 − 1. \end{aligned}

In fact, we claim that (7.18) suffices for both (7.15) and (7.16), with the latter being obtained via the following subtler regularization.

This is best done by replacing u in the second slot of the pairing by a separate factor Λ̃_{γ′}u, where Λ̃_{γ} is constructed just as Λ_{γ}, but with the greater degree of regularization σ = 1. Thus we have a replaced the lost half of an edge derivative (on each factor) which obtains from assuming (7.18) instead of (7.17) and may again integrate by parts to obtain, for γ, γ′ > 0,

\begin{matrix}⟨[□, A_{γ}^{∗}A_{γ}]u, Λ_{γ′}u⟩ = ⟨A_{γ}^{∗}A_{γ}u, □Λ_{γ′}u⟩ − ⟨□u, A_{γ}^{∗}A_{γ}Λ_{γ′}u⟩ \\ = ⟨A_{γ}^{∗}A_{γ}u, Λ_{γ′}□u⟩ + ⟨A_{γ}^{∗}A_{γ}u, [□, Λ_{γ′}]u⟩ − ⟨□u, A_{γ}^{∗}A_{γ}Λ_{γ′}u⟩ \end{matrix} ; (7.19)

Now, Λ_{γ′} → Id strongly (but not in norm) on H_{eb,𝔜}^{s′,r′}(M) and on H_{eb,𝔜^{∗}}^{s′,r′} for all s′, r′; this takes care of the first and third terms. Furthermore, [□, Λ_{γ′}] → 0 strongly (but not in norm) as a map from H_{eb,𝔜}^{s′,r′}(M) to H_{eb,H_{es}^{−1,l}(M)}^{s′+1,r′}(M).

Thus, letting γ′ → 0 shows (7.16) just under the assumption s−σ ≤ s′+1/2, r ≤ r′ − 1 + l + (f + 1)/2.

The Neumann case is completely analogous, except that then L_{g}^{2}-dual of 𝔛 = H_{es}^{1,l}(M) is Ḣ_{es}^{−1,−l−(f+1)}(M). We have

x^{−2} Diff_{es,♯}^{2}(M) ∋ □ : 𝔛 → Ḣ_{es}^{−1,l−2}(M) = x^{2l+(f+1)−2}𝔛^{∗}.

Furthermore, the dual of H_{eb,𝔛}^{s′,r′}(M) is

(H_{eb,𝔛}^{s′,r′}(M))^{∗} = H_{eb,𝔛^{∗}}^{−s′,−r′}(M).

The rest of the argument proceeds unchanged.

Having justified our integrations by parts, we now show that we can absorb the Q_{i}-terms in (7.15) in the positive terms (uniformly as γ ↓ 0) by using Corollary 5.34. Thus, given δ > 0, let U be as in Corollary 5.34; for sufficiently small ϵ > 0, supp a ⊂ U. For instance, by Cauchy-Schwarz,

|⟨Q_{j}F_{ij,2}T_{1}A_{2,γ}u, Q_{i}T_{−1}^{∗}A_{2,γ}u⟩| ≤ ∥Q_{i}T_{−1}^{∗}A_{2,γ}u∥^{2} + ∥Q_{j}F_{ij,2}T_{1}A_{2,γ}u∥^{2}

≤ δ(∥D_{t}T_{−1}^{∗}A_{2,γ}u∥^{2} + ∥D_{t}F_{ij,2}T_{1}A_{2,γ}u∥^{2})
+ 𝟋(∥u∥_{H_{es}^{1,r+1/2−(f+1)/2}(M)}^{2} + ∥Gu∥_{H_{es}^{1,r+1/2−(f+1)/2}(M)}^{2}),

where G ∈ Ψ_{eb}^{s−1,0}(M). The the ∥Gu∥_{H_{es}^{1,r+1/2−(f+1)/2}(M)}^{2} term can be estimated

as ⟨R′′_{γ}u, u⟩ since

(7.20)  ∥Gu∥_{H_{es}^{1,r+1/2−(f+1)/2}(M)}^{2} = ∥x^{−r+1/2}d_{M}Gu∥_{L_{g}^{2}(M;ΛM)}^{2},

and (x^{−r+1/2}d_{M}G)^{∗}(x^{−r+1/2}d_{M}G) ∈ Diff_{es}^{2} Ψ_{eb}^{2s−2,2r+1}(M), hence in fact a little better than R′′_{γ}, which has weight 2r + 2. Now, for 𝟋_{0} > 0 sufficiently large, depending on K̃ but not on ϵ > 0 (as long as ϵ satisfies (7.3) and supp a ⊂ U, i.e. ϵ > 0 is sufficiently small), we have

σ_{eb,0,1}(D_{t}F_{ij,2}T_{1}) ≤ 𝟋_{0}σ_{eb,0,1}(x^{−1}C_{2})

on a neighborhood of K̃. Thus,

∥D_{t}F_{ij,2}T_{1}A_{2,γ}u∥^{2}
≤ 2𝟋_{0}∥x^{−1}C_{2}A_{2,γ}u∥^{2}
+ 𝟋′(∥x^{−r−1}u∥_{H_{es}^{1,r+1/2−(f+1)/2}(M)}^{2} + ∥G′u∥_{H_{es}^{1,r+1/2−(f+1)/2}(M)}^{2}),

with G′ ∈ Ψ_{eb}^{s−1,0}(M) (so the last term behaves like (7.20)). Thus, if we choose δ > 0 such that 8𝟋_{0}n^{2}δ < 1, the first term (for all i, j) can be absorbed in ∥x^{−1}C_{2}A_{2,γ}u∥^{2}, while the last two terms are estimated as ⟨R′′_{γ}u, u⟩. Essentially identical arguments deal with all the other terms with Q_{i} and Q_{j}. In the case where Q_{i} is present on one side of the pairing only, we write, for instance,

\begin{matrix}|⟨Q_{i}T_{−1}A_{2,γ}u, x^{−1}T_{1}^{∗}F_{i,2}^{∗}A_{2,γ}u⟩| \\ ≤ δ^{−1/2}∥Q_{i}T_{−1}A_{2,γ}u∥^{2} + δ^{1/2}∥x^{−1}T_{1}^{∗}F_{i,2}^{∗}A_{2,γ}u∥^{2}. \end{matrix}

Using Corollary 5.34 on the first term, we have an estimate as above after possibly reducing δ > 0.

Recall that uniform finiteness of ∥x^{−1}C_{3}T_{1/2}A_{γ}u∥ as γ ↓ 0 will give absence of WF_{eb,𝔛}^{s−1/2,r−l−(f+1)/2} u ∩ ell A (as always the contribution to the weight of (f + 1)/2 comes from the metric weight while l comes from the weight in the definition of the base space, 𝔛). Similarly evaluating the other terms in the pairing, we take the extreme values of s′, r′ allowed by (7.18) to obtain

(7.21)

WF_{eb,𝔛}^{s−1/2−σ,r+1−l−(f+1)/2} u ∩ WF′ A = ∅,
and WF_{eb,𝔛}^{s−1/2,r+1−l−(f+1)/2} u ∩ WF′ A ∩ {x > 0} = ∅,
and s > r + σ, σ ∈ (0, 1/2]
⟹ WF_{eb,𝔛}^{s−1/2,r+1−l−(f+1)/2} u ∩ ell A = ∅,

or, relabeling,

(7.22)

WF_{eb,𝔛}^{s,r} u ∩ WF′ A = ∅, WF_{eb,𝔛}^{s+σ,r} u ∩ WF′ A ∩ {x > 0} = ∅,
s > r + l + f/2 − 1,
⟹ WF_{eb,𝔛}^{s+σ,r} u ∩ ell A = ∅.

Recall here that a = a_{ϵ}, and

(7.23)  0 < ϵ < ϵ′ ⇒ WF′(A_{ϵ}) ∩^{eb}Σ ⊂ ell A_{ϵ′} ∩^{eb}Σ.

Finally, we show how to use (7.22) iteratively, together with an interpolation argument, to finish the proof of the theorem. A priori we have u ∈ 𝔛, i.e.

WF_{eb,𝔛}^{0,0} u = ∅,

If 0 > l + f/2 − 1, we may iteratively apply (7.22) (shrinking ϵ > 0 by an arbitratily small amount, using (7.23) to estimate the lower order error terms R′′_{γ}) starting with s = 0 and always keeping r = 0, to obtain the conclusion of the theorem. (We choose σ = 1/2 at every stage in this process, until we are applying (7.22) with s such that s + 1/2 > m, at which point we finish the iteration by choosing σ = m − s so as to retain our estimates on the wavefront set in the hypothesis region.)

However if 0 ≤ l + f/2 − 1, we may not apply (7.22) directly owing to the lack of positivity of the commutator, and we must employ an interpolation argument as follows. Applying (7.22) iteratively, this time with r = r_{0} < 0 chosen sufficiently negative that we recover 0 > r_{0} + l + f/2 − 1, shows that we obtain

(7.24)  WF_{eb,𝔛}^{m,r_{0}} u ∩ S = ∅,

with S = WF′ A_{ϵ} for some ϵ > 0, A_{ϵ} constructed as above. Let

L = sup{r′ : WF_{eb,𝔛}^{m,r′} u ∩ S = ∅, r′ ≤ 0}.

Note that the set on the right hand side is non-empty by (7.24). We aim to show that L = 0. To this end, note that if L < 0, then for any r′ < L

WF_{eb,𝔛}^{m,r′} u ∩ S = WF_{eb,𝔛}^{0,0} ∩S = ∅.

An interpolation then yields, for δ ∈ (0, 1),

WF_{eb,𝔛}^{mδ,r′δ} u ∩ S = ∅.

Note that for any δ ∈ (0, 1) fixed, the compactness of S implies that for some ϵ′ > ϵ,

WF_{eb,𝔛}^{mδ,r′δ} u ∩ WF′ A_{ϵ′} = ∅

still holds. If δ ∈ (0, 1) in addition satisfies

(7.25)  mδ > r′δ + l + f/2 − 1

then by iterating (7.22), shrinking ϵ′ in each step (but keeping it larger than ϵ), we conclude that

WF_{eb,𝔛}^{m,r′δ} u ∩ S = ∅,

providing a contradiction with the definition of L if

(7.26)  r′δ > L.

It remains to check whether δ ∈ (0, 1) satisfying both (7.25) and (7.26) exists. This is evident from Figure 5, but a proof is as follows: we have l + f/2 − 1 ≥ 0 by assumption (otherwise we are in the preceding case);

s r  (m, r′) (m, 0) (0, 0)  s = r + l + f/2 − 1 (mδ, r′δ)  (m, r′δ)

Figure 5. The interpolation argument. The figure shows the (s, r) plane, where we plot the values for which with have absence of WF_{eb,𝔛}^{m,r}(u) (i.e. microlocal regularity of this order holds). We have a priori regularity of order (0, 0) and wish to conclude regularity of order (m, l′) with l′ < 0. By (7.22) we may take a step to the right of length σ for any σ ∈ (0, 1/2] provided that our starting regularity is below the line s = r+l+f/2−1 and that our endpoint is on s ≤ m. If we know (m, r′) regularity, we know regularity by interpolation on the whole line connecting this point to the origin; then starting on the interpolation line just below its intersection with s = r +l +f/2−1 allows us to achive (m, r′δ) regularity by moving to the right, thus improving the optimal weight for which we have our estimate.

moreover m > 0 (so that the theorem is not vacuous), and l + f/2 − 1 < m by hypothesis. Thus, for any r′ < 0,

0 ≤ \frac{l + f/2 − 1}{m − r′} < \frac{m}{m − r′} < 1.

Setting

δ_{0}(r′) = \frac{l + f/2 − 1}{m − r′} ∈ [0, 1)

we see that (7.25) is an equality with δ = δ_{0}(r) and that taking δ ∈ (δ_{0}(r′), 1) yields (7.25). In particular, (7.25) is satisfied by δ = δ(r′, λ) = δ_{0}(r′)(1 + λ) for any λ > 0 sufficiently small. If L < 0, we have δ_{0}(L) < 1, hence the function of r′ and λ given by

r′δ(r′, λ) = r′δ_{0}(r′)(1 + λ)

is strictly greater than L for r′ = L, λ = 0. Hence increasing λ slightly and decreasing r′ slightly preserves this relationship by continuity, and these choice of r′ and δ yield r′δ > L as desired.

□

In order to verify the hypotheses of Theorem 7.3, which are stated at points over the edge, we will employ the following geometric result.

First note that if q ∈ H_{W,b} then R_{eb,q,I} has a neighborhood O_{1} in ^{eb}S^{∗}M and there is a δ_{0} > 0 such that any GBB γ with γ(0) ∈ O_{1} ∩ {x > 0} satisfies γ|_{[−δ_{0},0]} ∩^{b}S_{W}^{∗} M_{0} = ∅. Indeed, we simply need to take a coordinate neighborhood

\begin{aligned}O_{1} = \{p ∈ ^{eb}S^{∗}M : & ξ̂(p) < −\sqrt{1 − h(q)}/2, x(p) < ϵ_{1}, |y(p) − y(q)| < ϵ_{1}, \\ & |t(p) − t(q)| < ϵ_{1}, |η̂(p) − η̂^{b}(q)| < ϵ_{1}\}, \end{aligned}

ϵ_{1} > 0 sufficiently small, since on its intersection with {x > 0} (where^{eb}S^{∗}M is naturally identified with ^{b}S^{∗}M_{0}), ∑ ξ̂_{j}^{b} < 0, hence Lemma 3.5 gives the desired δ_{0} (cf. the argument of Remark 3.17). Thus, such GBB γ can be uniquely lifted to curves γ̃ in ^{eb}S^{∗}M.

Lemma 7.4. Suppose that q ∈ H_{W,b}. There exists ϵ_{0} > 0 with the following property.

Suppose that 0 < ϵ_{1} ≤ ϵ_{0}, and U is a neighborhood of F_{I,q,sing}^{W} ∩ {t = t(q) − ϵ_{1}}. Then there is a neighborhood O of R_{eb,q,I} ∩ G in ^{eb}S^{∗}M such that for every (maximally extended) backward GBB γ with γ(0) ∈ O ∩ {x > 0} there is an s_{0} < 0 such that γ(s_{0}) ∈ U, γ(s) ∉ ^{b}S_{W}^{∗} M_{0} for s ∈ [s_{0}, 0].

Proof. It follows from the discussion preceeding the statement of the lemma that there is a neighborhood O_{1} of R_{eb,q,I} and δ_{0} > 0 such that every GBB γ defined on [−δ_{0}, 0], with γ(0) ∈ O_{1} ∩ {x > 0} satisfies γ(s) ∉ ^{b}S_{W}^{∗} M_{0} for s ∈ [−δ_{0}, 0]. As t(γ(s)) − t(γ(0)) = 2τ̂^{b}(γ(0))s, this implies that there is an ϵ_{0} > 0 such that for t(γ(s)) ∈ [t(q) − ϵ_{0}, t(q)], γ(s) ∉ ^{b}S_{W}^{∗} M_{0}.

Suppose now for the sake of contradiction that there is no neighborhood O of R_{eb,q,I} ∩ G in ^{eb}S^{∗}M such that for every (maximally extended) backward GBB γ with γ(0) ∈ O ∩ {x > 0}, there exists s_{0} < 0 with γ(s_{0}) ∈ U. As R_{eb,q,I} ∩ G is compact, we conclude that there is a sequence of points p_{n} ∈ O_{1} ⊂ ^{eb}S^{∗}M with x(p_{n}) > 0 (so p_{n} can be regarded as a point in bS^{∗}M_{0}) and GBB γ_{n} such that

• γ_{n}(0) = p_{n},

• the image of γ_{n} disjoint from U,

• p_{n} → p ∈ R_{eb,q,I} ∩ G. By Corollary 3.18, {γ_{n}} has a subsequence {γ_{n_{k}}} converging uniformly to a GBB γ such that the lift γ̃ of γ to ^{eb}S^{∗}M satisfies γ̃(0) = p. Thus, by Lemma 3.13, γ is not normally incident, so the image of γ is in F_{I,q,sing}^{W}, and thus intersects F_{I,q,sing}^{W} ∩{t = ϵ_{1}}. As γ_{n_{k}} → γ uniformly, for large enough k, γ_{n_{k}} intersects U, providing a contradiction. Thus, there exists O such that for every (maximally extended) backward GBB γ with γ(0) ∈ O ∩ {x > 0}, there exists s_{0} < 0 with γ(s_{0}) ∈ U. We may assume that O ⊂ O_{1} by replacing O by O ∩ O_{1} if needed.

To finish the proof, we note that, provided ϵ_{0} > 0 is sufficiently small, if γ(0) ∈ O ⊂ O_{1}, t(γ(s)) ∈ [t(q) − ϵ_{0}, t(q)] implies γ(s) ∉ ^{b}S_{W}^{∗} M_{0}. □

Theorem 7.3 and this lemma immediately give the following Corollary.

Corollary 7.5. Let u ∈ 𝔛 ≡ H_{es}^{1,l}(M) solve □u = 0 with Dirichlet or Neumann boundary conditions.

Let m > l + f/2 − 1 with m ≥ 0, l′ ≤ 0. Suppose that q ∈ H_{W,b} and F_{I,q,sing}^{W} ∩ WF_{eb,𝔛}^{m,∗} u = ∅. Then ^{eb}S_{∂W̃}^{∗} M ∩ WF_{eb,𝔛}^{m,l′} u = ∅ for all l′ < 0.

Proof. Let ϵ_{0} > 0 be as in Lemma 7.4. As WF_{eb,𝔛}^{m,∗} u is closed, F_{I,q,sing}^{W} ∩{t = t(q) − ϵ_{0}/2} has a neighborhood U disjoint from WF_{eb,𝔛}^{m,∗} u. By Lemma 7.4, R_{eb,q,I} ∩ G has a neighborhood O such that every backward GBB γ with γ(0) ∈ O ∩ {x > 0} intersects U and is disjoint from ^{b}S_{W}^{∗} M_{0}. By the propagation of singularities, [34], WF_{eb,𝔛}^{m,∗}(u) ∩ O ∩ {x > 0} = WF_{b,𝔛}^{m,∗}(u) ∩ O ∩ {x > 0} = ∅. Note that this uses the fact that every backward GBB γ with γ(0) ∈ O ∩ {x > 0} intersects U and is disjoint from ^{b}S_{W}^{∗} M_{0}, for we do not assume that u lies in a b-derivative of 𝔛 as we allow arbitrary weights at W̃. Thus, by Theorem 7.3, ^{eb}S_{∂W̃}^{∗} M ∩ WF_{eb,𝔛}^{m,l′} u = ∅ for all l′ < 0. □

7.3. Propagation at hyperbolic points within the edge. Now we consider propagation within ^{eb}S∗ ; W̃ M, away from the radial points. The propagation away from ∂W̃ is given by the results in [25]: on the edge cosphere bundle over W̃^{∘}, we find that WF_{eb} u = WF_{eb,𝔛} u (with, say, 𝔛 = H_{es}^{1,l}(M), for Dirichlet boundary conditions, 𝔛 = H_{es,0}^{1,l}(M) for Neumann boundary conditions— though this is irrelevant since we are working away from ∂W̃) given by is a union of integral curves of H_{es}|_{^{es}S∗ ; W̃ M}, given by (3.18), i.e.

\frac{1}{2}H_{es} = −ˆ\underline{ξ}ˆ\underline{ζ}∂ˆ\underline{ζ} + K^{ij}ˆ\underline{ζ}_{i}∂_{z_{j}} + K^{ij}ˆ\underline{ζ}_{i}ˆ\underline{ζ}_{j}∂ˆ\underline{ξ} − \frac{1}{2} \frac{∂K^{ij}}{∂z_{k}} ˆ \underline{ζ}_{i}ˆ\underline{ζ}_{j}∂_{ˆ\underline{ζ}_{k}},

where, as beforehand, hats denote variables divided by |\underline{τ}|, hence coordinates in the edge-smooth cosphere bundle (which over W̃^{∘} is canonically identified with the edge cotangent bundle). This leaves open only the question of how bicharacteristics reaching z′ = 0 interact with those leaving z′ = 0, i.e. the problem of reflection/diffraction from the boundary faces and corners of Z. Since the propagation over the interior of W̃ can be considered as a special case of propagation at G \ R_{eb}, see Theorem 7.7 (with no z′ variables, i.e. with k = 0 in the notation of this theorem), we do not state the interior propagation result of [25] here explicitly.

Let us thus begin by considering a hyperbolic point q ∈ H given by

(7.27) x = 0, t = t_{0}, y = y_{0}, z′ = 0, z′′ = z′′_{0}, ξ̂ = ξ̂_{0}, η̂ = η̂_{0}, ζ̂′ = 0, ζ̂′′_{0},

in edge-b canonical coordinates. Thus, in addition to ζ′ = 0, we have

1 > ξ̂^{2} + h(y_{0}, η̂_{0}) + k(y_{0}, z′ = 0, z′′_{0}, ζ̂′ = 0, ζ̂′′_{0}).

In the special case that z′ is a variable in ℝ^{1}, i.e. if q lies on a codimensionone boundary face of eb, then two points in ^{eb}S∗ ; W̃ (M) lie above q and two edge bicharacteristics in ^{eb}S∗ ; W̃^{∘}(M) contain q in their closures; we denote them γ_{±} with the ± given by sgn(ζ^{s}′ · z′); we will take γ_{±} to be only the segments of these bicharacteristics in |z′| < ϵ ≪ 1 in order not to enter into global considerations. Our sign convention is such that γ_{±} tends toward q under the forward resp. backward bicharacteristic flow. What we will show in this case is that if u ∈ 𝔛 ≡ H_{es}^{1,l}(M) and □u = 0 with Dirichlet or Neumann boundary conditions then

γ_{−} ∩ WF_{eb,𝔛}^{m,0} u = ∅ ⟹ γ_{+} ∩ WF_{eb,𝔛}^{m,0} u = ∅ for any m.

More generally, we have the following result, which via standard geometric arguments (see [24]) implies the propagation along EGBB’s through p :

Theorem 7.6. For Neumann boundary conditions, let 𝔛 = H_{es}^{1,l}(M), 𝔜 = Ḣ_{es}^{−1,l−2}(M); for Dirichlet boundary conditions let 𝔛 = H_{es,0}^{1,l}(M), 𝔜 = H_{es}^{−1,l−2}(M).

Let u ∈ 𝔛 solve □u = f, f ∈ 𝔜. Let p ∈ H be given by (7.27). Let U be an open neighborhood of p in ^{eb}S∗ ; W̃ (M) let m ∈ ℝ, l′ ≤ 0, and suppose that

WF_{eb,𝔜}^{m+1,l′}(f) ∩ U = ∅. Then

U ∩ {∑ ζ̂′_{i} > 0} ∩ WF_{eb,𝔛}^{m,l′}(u) = ∅ ⟹ p ∉ WF_{eb,𝔛}^{m,l′}(u).

Thus, the hypothesis region of the theorem, in which we make a wavefront assumption lies within the points with at least one z′_{i} non-zero, i.e. away from ebS_{C}^{∗}M, where C = {x = 0, z′ = 0}, and with momenta directed toward the boundary z′ = 0.

Proof. As usual, one needs to prove that if in addition to the hypotheses above p ∉ WF_{eb,𝔛}^{m−1/2,l′}(u) then p ∉ WF_{eb,𝔛}^{m,l′}(u), with a slightly more controlled

(but standard) version if m = ∞. So we assume p ∉ WF_{eb,𝔛}^{m−1/2,l′}(u) from now on.

For a constant β to be determined later, let

(7.28)  ϕ = ∑ ζ̂′_{i} + βω

where

ω = |z′|^{2}+|z′′ − z′′_{0}|^{2}+|ζ̂′′ − ζ̂′′_{0}|^{2}+|y − y_{0}|^{2}+|t − t_{0}|^{2}+x^{2}+|η̂ − η̂_{0}|+|ξ̂− ξ̂_{0}|^{2} Then for β sufficiently small, we have

|τ|^{−1}H_{p}ϕ > 0.

Now let χ_{0} ∈ C^{∞}(ℝ) with support in [0, ∞) and χ_{0}(s) = exp(−1/s) for s > 0. Thus, χ′_{0}(s) = s^{−2}χ_{0}(s). Take χ_{1} ∈ C^{∞}(ℝ) to have support in [0, ∞), to be equal to 1 on [1, ∞) and to have χ′_{1} ≥ 0 with χ′_{1} ∈ C_{c}^{∞}((0, 1)). Finally, let χ_{2} ∈ C_{c}^{∞}(ℝ) be supported in [−2c_{1}, 2c_{1}] and be identically equal to 1 on [−c_{1}, c_{1}]. Pick δ < 1. Set

a = |τ|^{s}x^{−r}χ_{0}(M(1 − ϕ/δ))χ_{1}(∑ ζ̂′_{j}/δ + 1)χ_{2}(|ζ̂′|^{2}).

Note that on the support of a, we have

(7.29)  ∑ ζ′_{j} > −δ,

hence we also obtain

(7.30)  0 ≤ ω < 2\frac{δ}{β} .

Thus, by keeping δ and δ/β both small, we can keep the support of a within any desired neighborhood of ζ′ = 0, ω = 0.

We now quantize a to A ∈ Ψ_{eb}^{s,r}(M). We claim that

(7.31) ı[□, A^{∗}A]

= B^{∗}(∑ D_{z′_{i}}^{∗}C_{ij}D_{z′_{j}} + R_{0} + ∑(R_{i}D_{z′_{i}} + D_{z′_{i}}^{∗}R′_{i}) + ∑ D_{z′_{i}}R_{ij}D_{z′_{j}})B

+ A^{∗}W A + R′′ + E + E′

where (1) B ∈ Ψ_{eb}^{s+1/2,r+1}(M) has symbol

M^{1/2}τ^{s+1/2}x^{−r−1}δ^{−1/2}χ_{1}χ_{2}\sqrt{χ_{0}χ′_{0}}

(2) C_{ij} ∈ Ψ_{eb}^{−2,0}(M), and the symbol-valued quadratic form σ(C_{ij}) is strictly positive definite on a neighborhood of WF′_{eb} B,

(3) R, R_{i}, R′_{i}, R_{ij} are in Ψ_{eb}^{0,0}(M), Ψ_{eb}^{−1,0}(M), Ψ_{eb}^{−1,0}(M), and Ψ_{eb}^{−2,0}(M) respectively and have (unweighted) symbols bounded by multiples of \sqrt{δ}(\sqrt{β} + 1/\sqrt{β}).

(4) R′′ ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−2,0}(M), E, E′ ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−1,0}(M),

(5) E is microsupported where we have assumed regularity,

(6) W ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−1,2}(M),

(7) E′ is supported off the characteristic set. These terms arise as follows. Applying Lemma 5.35, we have (with Q_{i} = x^{−1}D_{z′_{i}})

(7.32) ı[□, A^{∗}A] = ∑ Q_{i}^{∗}L_{ij}Q_{j} + ∑(x^{−1}L_{i}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i}) + x^{−2}L_{0},

with L_{ij} ∈ Ψ_{eb}^{2s−1,2r}(M), L_{i}, L′_{i} ∈ Ψ_{eb}^{2s,2r}(M), L_{0} ∈ Ψ_{eb}^{2s+1,2r}(M), σ_{eb,2m−1}(L_{ij}) = 2aV_{ij}a, V_{ij} = κ_{ij}(∂_{ζ′_{i}} + ∂_{ζ′_{j}} + 2∂_{ξ}) + H_{eb,κ_{ij}},

σ_{eb,2m}(L_{i}) = σ_{eb,2m}(L′_{i}) = 2aV_{i}a,

V_{i} = ∑_{j} κ_{ij}∂_{z′_{j}} + \frac{1}{2}(m_{i}∂_{ξ} + H_{eb,m_{i}}) + \frac{1}{2}m_{i}(∂_{ξ} + ∂_{ζ′_{i}}),

σ_{eb,2m+1}(L_{0}) = 2aV_{0}a, V_{0} = 2h̃∂_{ξ} + H_{eb,h̃} + ∑_{i} m_{i}∂_{z′_{i}},

WF′_{eb}(L_{ij}), WF′_{eb}(L_{i}), WF′_{eb}(L′_{i}), WF′_{eb}(L_{0}) ⊂ WF′_{eb}(A).

(7.33)

with

\begin{matrix}V_{0}|_{C} = −2ξ x∂_{x} − 2(ξ^{2} + ∑_{ij} k_{2,ij}ζ′′_{i}ζ′′_{j})∂_{ξ} − 2ξ(τ ∂_{τ} + η ∂_{η}) \\ − 2 ∑_{i,j} k_{2,ij}ζ′′_{i}∂_{z′′_{j}} + ∑_{ℓ,i,j}(∂_{z′′_{ℓ}}k_{2,ij})ζ′′_{i}ζ′′_{j}∂_{ζ′′_{ℓ}}, \\ V_{ij}|_{C} = −k_{1,ij}(∂_{ζ′_{i}} + ∂_{ζ′_{j}} + 2∂_{ξ}) + ∑_{ℓ}(∂_{z′′_{ℓ}}k_{1,ij})∂_{ζ′′_{ℓ}}, V_{i}|_{C} = − ∑_{j} k_{1,ij}∂_{z′_{j}}, \end{matrix} (7.34)

First we evaluate the terms in L_{ij} coming from terms in which V_{ij} hits χ_{0}(M(1 − ϕ/δ)). The main contribution will be from the derivatives falling on ζ̂′, with the rest controlled by shrinking β; in particular,

V_{ij}(ϕ/δ) = −2k_{1,ij} + r_{ij}

with

|r_{ij}| ≤ const(β\sqrt{ω} + \sqrt{ω});

on the support of a, this is in turn controlled by a multiple of

β\sqrt{δ/β} + \sqrt{δ/β}.

Thus, from these two terms, we obtain corresponding terms in ı[□, A^{∗}A] of the forms

B^{∗}( ∑ D_{z′_{i}}^{∗}C_{ij}D_{z′_{j}})B

and

B^{∗}( ∑ D_{z′_{i}}^{∗}R_{ij}D_{z′_{j}})B

respectively.

Similarly, terms with V_{i} and V_{0} hitting χ_{0} go into the R_{i} and R_{0} terms in (7.31) respectively.

The terms arising from

V_{•}(χ_{1}(∑ ζ̂′_{j}/δ + 1))

are supported on the hypothesis region, {∑ ζ̂′_{j} < 0}, hence give commutator terms of the form x^{−2}E above.

The terms arising from

V_{•}(χ_{2}(|ζ̂′|^{2}))

lie off of the characteristic set, hence give commutator terms of the form x^{−2}E′ above.

The term arising from differentiating |τ|^{s}x^{−r} gives the commutator term A^{∗}W A.

As we are interested in edge-b wavefront set, the term D_{z′_{i}}^{∗}C_{ij}D_{z′_{j}} is slightly inconvenient, but we note that owing to strict positivity of C_{ij} we may replace it by a multiple of Δ_{z′}/x^{2} plus another positive term. Rewriting Δ_{z′}/x^{2} = (Δ_{z′}/x^{2} + □) − □, and noting that the first of these terms is in x^{−2} Diff_{eb}^{2}(M) and elliptic on the hyperbolic set, we see that we in fact have

(7.35) ı[A^{∗}A, □] = R′′′□ + B^{∗}(C^{∗}C + ∑ D_{z′_{i}}^{∗}C̃_{ij}D_{z′_{j}} + R_{0}

+ ∑ D_{z′_{i}}^{∗}R′_{i} + R_{i}D_{z′_{i}} + A^{∗}W A + ∑ D_{z′_{i}}^{∗}R_{ij}D_{z′_{j}})B + R′′ + E + E′

where C̃_{ij} ∈ Ψ_{eb}^{0,2}(M) is a positive matrix of operators (this is a priori true only at the symbolic level, but we may absorb lower-order terms in R_{ij}).

Following [34], we find that for any 𝟋 > 0,

(7.36) |⟨R_{0}w, w⟩| ≤ C(\sqrt{δ})(\sqrt{β} + 1/\sqrt{β})∥w∥^{2} + 𝟋^{−1}∥R′_{0}w∥^{2} + 𝟋∥w∥^{2},

where R′_{0} ∈ Ψ_{eb}^{−1} has the same microsupport as R_{0}, and is one order lower. Here we have employed L^{2} boundedness of Ψ_{eb}^{0,0}(M), or more specifically, the square-root argument used to prove it (cf. [34] for details, specifically the treatment following (6.18)). By the same token, we can estimate

(7.37) ⟨R_{i}D_{z′_{i}}w, w⟩ ≤ C(\sqrt{δ})(\sqrt{β} + 1/\sqrt{β})(∥T_{−1}D_{z′_{i}}w∥^{2} + ∥w∥^{2})

+ 2𝟋∥w∥^{2} + 𝟋^{−1}∥R̃_{i}D_{z′_{i}}w∥^{2},

where R̃_{i} ∈ Ψ_{eb}^{−2,0}(M) has the same microsupport as R_{i}, and is one order lower. We also compute

(7.38) |⟨R_{ij}D_{z′_{i}}w, D_{z′_{j}}w⟩| ≤ C(\sqrt{δ})(\sqrt{β} + 1/\sqrt{β})(∥T_{−1}D_{z′_{i}}w∥^{2} + ∥T_{−1}D_{z′_{j}}w∥^{2})

+ 2𝟋(∥T_{−1}D_{z′_{j}}w∥^{2} + ∥T_{−1}D_{z′_{j}}w∥^{2}) + 2𝟋∥w∥^{2}

+ 𝟋^{−1}∥R̃_{ij}D_{z′_{i}}w∥^{2} + 𝟋^{−1}∥R̃′_{ij}D_{z′_{j}}w∥^{2},

where R̃_{ij}, R̃′_{ij} ∈ Ψ_{eb}^{−3}(M) have the same microsupport as R_{ij}, and are one order lower. Although the argument is identical to that in [34], [25], we reproduce the derivation of (7.38) for the convenience of the reader; (7.36) and (7.37) follow by similar (easier) arguments. To begin, we note that T_{1}^{∗}R_{ij} ∈ Ψ_{eb}^{−1}(M) has symbol bounded by C(\sqrt{δ})(\sqrt{β} +1/\sqrt{β}), hence by the Hörmander square-root argument

(7.39)  ∥T_{1}^{∗}R_{ij}u∥^{2} ≤ C(\sqrt{δ})(\sqrt{β} + 1/\sqrt{β})∥u∥^{2} + ∥R̃_{ij}u∥^{2}

with R̃_{ij} as described above. Now write D_{z′_{j}}w = T_{1}T_{−1}D_{z′_{j}}w − F D_{z′_{j}}w; this permits us to expand

⟨R_{ij}D_{z′_{i}}w, D_{z′_{j}}w⟩ = ⟨T_{1}^{∗}R_{ij}D_{z′_{i}}w, T_{−1}D_{z′_{j}}w⟩ − ⟨R_{ij}D_{z′_{i}}w, F D_{z′_{j}}w⟩.

The first term on the right may be controlled, using (7.39) and Cauchy-Schwarz, by the RHS of (7.38); the second term may also be so estimated

by again applying Cauchy-Schwarz and absorbing ∥F D_{z′_{j}}w∥^{2} into a term

∥R̃′_{ij}D_{z′_{j}}w∥^{2} with appropriately enlarged R̃′_{ij}.

Now we turn to making our commutator argument. Let u be a solution to

□u = f

with Dirichlet or Neumann boundary conditions. Choose Λ_{γ} ∈ Ψ_{eb}^{−1}(M) converging to the identity as γ ↓ 0 as in §7.2. Note that by making Λ_{γ} ∈ Ψ_{eb}^{−1}(M), we are combining the roles of the regularizer Λ_{γ} ∈ Ψ_{eb}^{−σ}(M) in §7.2, required for obtaining an improvement over the a priori assumptions, and the regularizer Λ̃_{γ′} used to justify the pairing argument, see (7.19). Let A_{γ} = Λ_{γ}A with A constructed as above. As before, we have

(7.40)  ı[□, A_{γ}^{∗}A_{γ}] = Λ_{γ}^{∗}ı[□, A^{∗}A]Λ_{γ} + A^{∗}ı[□, Λ_{γ}^{∗}Λ_{γ}]A + R̃,

with R̃ uniformly bounded in Diff_{es,♯}^{2} Ψ_{eb}^{2s−2,2r+2}(M) (hence lower order), and where

[□, Λ_{γ}^{∗}Λ_{γ}] ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−1,2}(M)

is uniformly bounded, and in fact

[□, Λ_{γ}^{∗}Λ_{γ}] = Λ_{γ}^{∗}W̃_{γ}Λ_{γ},

with W̃_{γ} uniformly bounded in Diff_{es,♯}^{2} Ψ_{eb}^{−1,2}(M), cf. (7.13).

Now we pair A_{γ}^{∗}A_{γ} with u. Letting B_{γ} = BΛ_{γ}, provided integrations by parts can be justified, we have

(7.41) ⟨A_{γ}□u, A_{γ}u⟩ − ⟨A_{γ}, A_{γ}□u⟩

= ⟨ı[A_{γ}^{∗}A_{γ}, □]u, u⟩ = ∥CB_{γ}u∥^{2} + ∑ ⟨C̃_{ij}D_{z′_{j}}B_{γ}u, D_{z′_{i}}B_{γ}u⟩

+ ⟨R_{0}B_{γ}u, B_{γ}u⟩ + ∑ ⟨D_{z′_{i}}B_{γ}u, R_{i}B_{γ}u⟩ + ⟨R′_{i}B_{γ}u, D_{z′_{i}}B_{γ}u⟩

+ ∑ ⟨D_{z′_{j}}R_{ij}B_{γ}u, D_{z′_{i}}B_{γ}u⟩ + ⟨W_{γ}A_{γ}u, A_{γ}u⟩ + ⟨(R′′_{γ} + E_{γ} + E′_{γ})u, u⟩ where W_{γ} is uniformly bounded in Diff_{es,♯}^{2} Ψ_{eb}^{−1,2}(M) and comprises both the W term from above and the term containing [□, Λ_{γ}^{∗}Λ_{γ}]. The integrations by parts may by justified, for any γ > 0, if

(7.42)  WF_{eb,𝔛}^{s′,l′} u ∩ WF′ A = ∅

whenever

s − 1 ≤ s′, r ≤ l′ + l + \frac{(f + 1)}{2} − 1

since then the products of □ with A_{γ}^{∗}A_{γ} map H_{eb,𝔛}^{s′,l′}(M) to its dual (as

required in the Neumann setting), as well as mapping H_{eb,𝔜}^{s′,l′}(M), 𝔜 = H_{es,0}^{1,l}(M), to its dual (as required in the Dirichlet setting). We take s′ = m−

1/2, hence s = m+1/2, and r = l′+l+ \frac{(f+1)}{2} −1 here, and note that it suffices to have the microlocal assumptions (7.42) rather than global assumptions in view the microlocality of Ψ_{eb}^{(}M), see Lemma 5.15 and Lemma 5.16.

We now examine the terms on the RHS. The first two are positive. To the third, we apply (7.36), with w = B_{γ}u: if δ, δ/β, and 𝟋 are sufficiently small, we may absorb the first and third terms on the RHS of (7.36) in ∥CB_{γ}u∥, while the lower-order second is uniformly bounded by our wavefront assumptions. Likewise, applying (7.37) and (7.38), we may choose 𝟋, δ, β so as to absorb terms involving 𝟋 and (\sqrt{δ})(\sqrt{β} + 1/\sqrt{β}) in ∥x^{−1}CB_{γ}u∥^{2}; the 𝟋^{−1} terms, as they are lower order, remain bounded. Moreover, as χ′_{0}(s) = s^{−2}e^{−1/s} for s > 0 and vanishes for s ≤ 0,

M^{2}(1 − ϕ/δ)^{2}χ′_{0}(M(1 − ϕ/δ)) = χ_{0}(M(1 − ϕ/δ)).

Thus,

|τ|^{1/2}x^{−1}a
= |τ|^{s+1/2}x^{−r−1}M(1 − ϕ/δ)\sqrt{χ_{0}(M(1 − ϕ/δ))χ′_{0}(M(1 − ϕ/δ))}
χ_{1}(∑ ζ̂′_{j}/δ + 1)χ_{2}(|ζ̂′|^{2})
= M^{1/2}δ^{1/2}(1 − ϕ/δ)b ≤ 2M^{1/2}δ^{1/2}b

as ϕ ≥ ∑ ζ̂′_{i} ≥ −δ on supp a. We deduce that ∥T_{1/2}x^{−1}A_{γ}u∥ can be estimated by 4M^{1/2}δ^{1/2}∥CB_{γ}u∥ plus lower order terms, and hence, for M chosen sufficently small, we may absorb the W_{γ} term in ∥CB_{γ}u∥^{2}.

Finally we consider the LHS of (7.41). We have

(7.43) |⟨A_{γ}□u, A_{γ}u⟩| ≤ ∥(T_{−1/2})^{∗}xA_{γ}□u∥ ∥T_{1/2}x^{−1}A_{γ}u∥ + |⟨xA_{γ}□u, x^{−1}F A_{γ}u∥

≤ 𝟋_{1}^{−1}∥(T_{−1/2})^{∗}xA_{γ}□u∥^{2} + 𝟋_{1}∥T_{1/2}x^{−1}A_{γ}u∥^{2} + |⟨xA_{γ}□u, x^{−1}F A_{γ}u∥

with F ∈ Ψ_{eb}^{−1,0}(X), hence x^{−1}F A_{γ} uniformly bounded in Ψ_{eb}^{s−1,r+1}(M), xA_{γ} uniformly bounded in Ψ_{eb}^{s,r−1}(M), so as s = m+1/2, and r = l′+l+ \frac{(f+1)}{2} −1, the last term is uniformly bounded by the a priori assumptions. Similarly, ∥(T_{−1/2})^{∗}xA_{γ}□u∥ is uniformly bounded, as (T^{−1/2})^{∗}xA_{γ} is uniformly bounded in Ψ_{eb}^{s−1/2,r−1}(M), while ∥T_{1/2}x^{−1}A_{γ}u∥^{2} can be absorbed in ∥CB_{γ}u∥^{2} (for 𝟋_{1} sufficiently small) as discussed above.

The net result is that

∥x^{−1}CBu_{γ}∥^{2}

remains uniformly bounded as γ ↓ 0. Noting that CB_{γ} → CB strongly (cf. the proof of Lemma A.3), CB ∈ Ψ_{eb}^{s+1/2,r+1}(M) is elliptic at q, s = m + 1/2, and r = l′ + l + \frac{(f+1)}{2} − 1, 𝔛 = H_{es}^{1,l}(M), we can complete the proof in the standard manner. □

7.4. Propagation at glancing points within the edge. Let q ∈ G\R_{eb} be given by

x = 0, t = t_{0}, y = y_{0}, z′ = 0, z′′ = z′′_{0}, ξ̂ = ξ̂_{0}, η̂ = η̂_{0}, ζ̂′ = 0, ζ̂′′ = ζ̂′′_{0}.

As q ∈ G,

ξ̂_{0}^{2} + h(y_{0}, η̂_{0}) + k(y, z, ζ̂′ = 0, ζ̂′′_{0}) = 1.

As q ∉ R_{eb},

ξ̂_{0}^{2} + h(y_{0}, η̂_{0}) < 1,

so ζ̂′′_{0} ≠ 0, and h(y_{0}, η̂_{0}) < 1, so ϖ_{eb}(q) ∈ H_{W,b}. We will let Π locally denote the coordinate projection onto the variables

(ξ̂, z′′, ζ̂′′);

Let W be a homogeneous vector field equal to V_{0} (from (5.17)) at q, and extended in local coordinates to a constant vector field in (ξ̂, z′′, ζ̂′′).

Theorem 7.7. For Neumann boundary conditions, let 𝔛 = H_{es}^{1,l}(M), 𝔜 = Ḣ_{es}^{−1,l−2}(M); for Dirichlet boundary conditions let 𝔛 = H_{es,0}^{1,l}(M), 𝔜 = H_{es}^{−1,l−2}(M).

Let u ∈ 𝔛 solve □u = f, f ∈ 𝔜. Let q ∈ G\R_{eb} be as above, and suppose that m ∈ ℝ, l′ ≤ 0, and q ∉ WF_{eb,𝔜}^{m+1,l′}(f).

There exists δ_{0} > 0 and C_{0} > 0 such that for all δ ∈ (0, δ_{0}) and β ∈ (C_{0}δ, 1),

(7.44) (q′ ∈ Σ_{eb}, |Π(q′)−Π(q)−δW| < δβ and |z′(q′)| < δβ ⟹ q′ ∉ WF_{eb,𝔛}^{m,l′} u)

⟹ q ∉ WF_{eb,𝔛}^{m,l′} u

Proof. Below we will choose δ_{0} > 0 sufficiently small so that WF_{eb,𝔜}^{m+1,l′}(f) is disjoint from a δ_{0}-neighborhood of q (see the discussion before Lemma 5.33).

Let k be the codimension of the face over which q lies. Let ρ_{2n−2k} be the degree-zero homogeneous function with

ρ_{2n−2k}|W̃ = τ^{−2}h̃|W̃ ,

with h̃ as in Lemma 5.29; note that ρ_{2n−2k}(q) = 0 by (3.22) and dρ_{2n−2k}(q) ≠ 0 since at least one of the dζ̂′′_{k+1}(q), …, dζ̂′′_{f}(q) components of dρ_{2n−2k}(q) is non-zero, in view of the quadratic nature of τ^{2}(1 − ρ_{2n−2k}) in the fibers of the cotangent bundle and (3.22) and ζ̂′′_{0} ≠ 0 as observed above. We remark that, with V_{0} as in (5.17),

V_{0}ρ_{2n−2k}|W̃ = 0.

Note that ^{eb}S^{∗}M has dimension 2(n + 1) − 1 = 2n + 1, thus, with C = {x = 0, z′ = 0}, G ∩^{eb}S_{C}^{∗}M has dimension 2n + 1 − 2k − 2 = 2n − 2k − 1 in view of (3.22). We proceed by remarking that by (5.16),

(Wx)|W̃ = 0, (Wy_{j})|W̃ = 0, (Wt)|W̃ = 0, (Wη̂_{j})|W̃ = 0,

so t, y_{j}, η̂_{j} give 1 + 2(n − f − 1) = 2(n − f) − 1 homogeneous degree zero functions on ^{eb}T^{∗}M (or equivalently C^{∞} functions on ^{eb}S^{∗}M) whose restrictions to G ∩^{eb}S_{C}^{∗}M have linearly independent differentials at q. We let ρ_{2}, …, ρ_{2n−2f} be given by these functions, and let ρ_{1} = x. We next remark that, in the notation of (5.10),

Wξ̂(q) = −2k_{2}(x = 0, y_{0}, z′ = 0, z′′_{0}), ζ̂′′_{0}) < 0

as ζ̂_{0} ≠ 0, hence W(q) ≠ 0. Further, we let ρ_{j}, j = 2n − 2f + 1, …, 2n − 2k − 1, be degree-zero homogeneous functions on ^{eb}T^{∗}M (or equivalently C^{∞} functions on ^{eb}S^{∗}M) such that d(ρ_{2}|_{G}), …, d(ρ_{2n−2k−1}|_{G}) have linearly independent differentials at q, and such that

Wρ_{j}(q) = 0.

Such functions ρ_{j} exist as G ∩ ^{eb}S_{C}^{∗}M has dimension 2n + 1 − 2k − 2 = 2n−2k −1, so the 2n−2f −1 functions ρ_{2}, …, ρ_{2n−2f} can be complemented by some functions ρ_{2n−2f+1}, …, ρ_{2n−2k−1} to obtain 2n − 2k − 1 functions whose pullbacks to G ∩^{eb}S_{C}^{∗}M have linearly independent differentials and which are annihilated by W at q, for the space of such one-forms is 2n − 2k − 2 dimensional. Thus, by dimensional considerations (using W(q) ≠ 0), {dρ_{j}|_{G}(q) : j = 2, …, 2n − 2k − 1} spans the space of one-forms on G annihilated by W(q), and dρ_{2}, …, dρ_{2n−2k−1} together with d(ξ̂|_{G} − ξ̂_{0}) span T^{∗}G. Let

ω_{0} = ∑_{j=1}^{2n−2k−1} ρ_{j}^{2};

then keeping in mind that |x| ≤ ω_{0}^{1/2}, and with V_{0}, V_{i}, V_{ij} as in (5.17),

\begin{matrix}|τ|^{−1}|V_{0}ω_{0}| ≲ \sqrt{ω_{0}}(\sqrt{ω_{0}} + |ξ̂− ξ̂_{0}| + |z′|), \\ |V_{i}ω_{0}| ≲ \sqrt{ω_{0}} x ≤ ω_{0}, |τ| |V_{ij}ω_{0}| ≲ \sqrt{ω_{0}}, \end{matrix}

by (5.17) and (5.15). Note also that

\begin{matrix}|τ|^{−1}|V_{0}|z′|^{2}| ≲ |z′|(|z′| + x) \\ |V_{i}|z′|^{2}| ≲ |z′|, |τ||V_{ij}|z′|^{2}| ≲ |z′|^{2}. \end{matrix}

Let ω = ω_{0} + |z′|^{2}. Then

|τ|^{−1}|V_{0}ω| ≲ \sqrt{ω}(\sqrt{ω} + |ξ̂− ξ̂_{0}|),  |V_{i}ω| ≲ \sqrt{ω},  |τ| |V_{ij}ω| ≲ \sqrt{ω}. (7.45)

Let

ϕ = ξ̂− ξ̂_{0} + \frac{1}{β^{2}δ} ω

By (5.18),

\begin{matrix}||τ|^{−1} V_{0}ξ̂+ 2 ∑_{ij} k_{2,ij}ζ̂′′_{i}ζ̂′′_{j}| ≲ x ≤ ω^{1/2}, \\ |V_{i}ξ̂| ≲ (|z′| + x) ≲ ω^{1/2}, |τ| |V_{ij}ξ̂| ≲ 1. \end{matrix} ; (7.46)

In particular, as ζ̂′′_{0} ≠ 0,

|τ|^{−1} V_{0}ξ̂ ≤ −c_{0} + C′_{1}ω^{1/2},

for some c_{0} > 0, C′_{1} > 0.

Set

a = |τ|^{s}x^{−r}χ_{0}(M(2 − ϕ/δ))χ_{1}((ξ̂− ξ̂_{0} + δ)/βδ + 1)χ_{2}(|ζ̂′|^{2}).

We always assume for this argument that β < 1, so on supp a we have

ϕ ≤ 2δ and ξ̂− ξ̂_{0} ≥ −βδ − δ ≥ −2δ.

Since ω ≥ 0, the first of these inequalities implies that ξ̂ − ξ̂_{0} ≤ 2δ, so on supp a

(7.47)  |ξ̂− ξ̂_{0}| ≤ 2δ.

Hence,

(7.48)  ω ≤ β^{2}δ(2δ − (ξ̂− ξ̂_{0})) ≤ 4δ^{2}β^{2}.

Moreover, on supp dχ_{1},

(7.49)  ξ̂− ξ̂_{0} ∈ [−δ − βδ, −δ], ω^{1/2} ≤ 2βδ,

so this region lies in the hypothesis region of (7.44) after β and δ are both replaced by appropriate constant multiples.

We now quantize a to A ∈ Ψ_{eb}^{s,r}(M). By Lemma 5.35,

(7.50) ı[□, A^{∗}A] = ∑ Q_{i}^{∗}L_{ij}Q_{j} + ∑(x^{−1}L_{i}Q_{i} + Q_{i}^{∗}x^{−1}L′_{i}) + x^{−2}L_{0},

with

L_{ij} ∈ Ψ_{eb}^{2s−1,2r}(M), L_{i}, L′_{i} ∈ Ψ_{eb}^{2s,2r}(M), L_{0} ∈ Ψ_{eb}^{2s+1,2r}(M),
σ_{eb,2s−1}(L_{ij}) = 2aV_{ij}a,
σ_{eb,2s}(L_{i}) = σ_{eb,2s}(L′_{i}) = 2aV_{i}a, σ_{eb,2s+1}(L_{0}) = 2aV_{0}a,
WF′_{eb}(L_{ij}), WF′_{eb}(L_{i}), WF′_{eb}(L′_{i}), WF′_{eb}(L_{0}) ⊂ WF′_{eb}(A),

(7.51) with V_{ij}, V_{i} and V_{0} as above, given by (5.17). Thus, we obtain

(7.52) ı[□, A^{∗}A]

= B^{∗}(C^{∗}C + ∑ D_{z′_{i}}^{∗}C_{ij}D_{z′_{j}} + ∑(R_{i}D_{z′_{i}} + D_{z′_{i}}^{∗}R′_{i}) + ∑ D_{z′_{i}}^{∗}R_{ij}D_{z′_{j}})B

+ A^{∗}W A + R′′ + E + E′

where (1) B ∈ Ψ_{eb}^{s+1/2,r+1}(M) has symbol

M^{1/2}|τ|^{s+1/2}x^{−r−1}δ^{−1/2}χ_{1}χ_{2}\sqrt{χ_{0}χ′_{0}}

(2) C ∈ Ψ_{eb}^{0,0}(M), has strictly positive symbol on a neighborhood of WF′_{eb} B, given by (−V_{0}ϕ)^{1/2} near WF′_{eb} B,

(3) C_{ij} ∈ Ψ_{eb}^{−2,0}(M), (C_{ij}) positive semidefinite,

(4) R_{i}, R′_{i}, R_{ij} are in Ψ_{eb}^{−1,0}(M), Ψ_{eb}^{−1,0}(M), and Ψ_{eb}^{−2,0}(M) respectively and have (unweighted) symbols r_{i}, r′_{i}, r_{ij} with

(7.53)  |τ| |r_{i}|, |τ| |r′_{i}|, |τ|^{2} |r_{ij}| ≲ 1/β,

(5) W ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−1,−2}(M),

(6) R′′ ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−2,0}(M), E, E′ ∈ Diff_{es,♯}^{2} Ψ_{eb}^{−1,0}(M),

(7) E is microsupported where we have assumed regularity,

(8) E′ is supported off the characteristic set.

These terms arise as follows. By (7.45), (7.46), (7.47), and (7.49),

V_{0}ϕ = V_{0}(ξ̂− ξ̂_{0}) + \frac{1}{β^{2}δ} V_{0}ω

≤ −c_{0} + C′_{1}ω^{1/2} + \frac{1}{β^{2}δ} C′′_{1}ω^{1/2}(ω^{1/2} + |ξ̂− ξ̂_{0}|)

≤ −c_{0} + 2(C′_{1} + C′′_{1})(δ + \frac{δ}{β}) ≤ −c_{0}/4 < 0.

provided that δ < \frac{2}{16(C′_{1}+C′′_{1})}, \frac{β}{δ} > \frac{16(C′_{1}+C′′_{1})}{2}, i.e. that δ is small, but β/δ is not too small—roughly, β can go to 0 at most as a multiple of δ (with an appropriate constant) as δ → 0. Recall also that β < 1, so there is an upper bound as well for β, but this is of no significance as we let δ → 0. Thus, we define C to have principal symbol equal to the product of (−V_{0}ϕ)^{1/2} times a cutoff function identically 1 in a neighborhood of supp a, but with sufficiently small support so that −V_{0}ϕ > 0 on it. Thus, the L_{0}-term of (7.50) gives rise to the C^{∗}C term of (7.52), as well as contributing to the E and E′ terms (where χ_{1} and χ_{2} are differentiated), W (where the weight |τ|^{s}x^{−r} is differentiated) and the lower order term R′′.

Similarly, the L_{i}, L′_{i} and L_{ij} terms in which V_{i} or V_{ij} differentiates χ_{1} or χ_{2} contribute to the E and E′ terms, while those in which they differentiate the weight contributes to the W term, so it remains to consider when V_{i} and V_{ij} differentiate χ_{0}. As we keep β < 1,

|V_{i}ϕ| ≤ |V_{i}ξ̂| + |V_{i}ω| ≲ 1 + (β^{2}δ)^{−1}ω^{1/2} ≲ 1 + β^{−1} ≲ β^{−1}, |V_{ij}ϕ| ≲ β^{−1},

which thus proves the estimates on the terms arise this way, namely R_{i}, R′_{i}, R_{ij}, above.

We now employ Lemma 5.33 to estimate the D_{z′_{i}} terms as in the proof of Theorem 7.3. Note that we are using the finer result, Lemma 5.33, rather than its corollary here (unlike in Theorem 7.3), to obtain better control over the constant in front of the D_{z′_{i}} terms as we shrink δ and β. The important fact is that G ∩^{eb}S_{C}^{∗}M is defined by ρ_{2n−2k} = 0, x = 0, z′ = 0, and

ρ_{2n−2k}, x, |z′| ≲ ω^{1/2} ≤ 2δβ

on the wave front set of C_{ij}, R_{i}, R′_{i}, R_{ij}. Thus, we can apply Lemma 5.33 for a C_{1}βδ-neighborhood of a compact subset of G. Noting that xD_{t}T_{−1} ∈ Ψ_{eb}^{0,0}(M), we conclude that, with B_{γ} = BΛ_{γ}, and for Neumann boundary conditions,

∑ ∥D_{z′_{i}}T_{−1}B_{γ}u∥^{2} ≤ C_{0}C_{1}βδ∥B_{γ}u∥^{2} + C̃Q(u, Gu, G̃u)
Q(u, Gu, G̃u) = ∥u∥_{H_{es}^{1,r−(f−1)/2}(M)}^{2} + ∥Gu∥_{H_{es}^{1,r−(f−1)/2}(M)}^{2}

+ ∥□u∥2 ; Ḣ_{es}^{−1,r−(f+3)/2} + ∥G̃□u∥2 ; Ḣ_{es}^{−1,r−(f+3)/2},

(7.54)

where G ∈ Ψ_{eb}^{s,0}(M), G̃ ∈ Ψ_{eb}^{s+1/2,0}(M) (independent of γ) with wave front set in a neighborhood of supp a. For Dirichlet conditions we simply replace

H_{es}^{1,r−(f−1)/2}(M) by H_{es,0}^{1,r−(f−1)/2}(M)

and

Ḣ_{es}^{−1,r−(f+3)/2}(M) by H_{es}^{−1,r−(f+3)/2}(M).

Note that by (7.53) we have for all w ∈ L^{2},

∥T_{1}^{∗}R_{i}^{∗}w∥ ≤ C_{2}β^{−1}∥w∥ + ∥R̃_{i}w∥, R̃_{i} ∈ Ψ_{eb}^{−1,0}(M),

with R̃_{i} having the same microsupport as R_{i}. But

|⟨R_{i}D_{z′_{i}}B_{γ}u, B_{γ}u⟩|

≤ |⟨R_{i}T_{1}D_{z′_{i}}T_{−1}B_{γ}u, B_{γ}u⟩| + |⟨R_{i}[T_{1}, D_{z′_{i}}]T_{−1}B_{γ}u, B_{γ}u⟩|

≤ ∥D_{z′_{i}}T_{−1}B_{γ}u∥ ∥T_{1}^{∗}R_{i}^{∗}B_{γ}u∥ + |⟨R̂_{i}B_{γ}u, B_{γ}u⟩|, R̂_{i} ∈ Diff_{es}^{1} Ψ_{eb}^{−2,0}(M),

and |⟨R̂_{i}B_{γ}u, B_{γ}u⟩| can be estimated by the inductive hypothesis, while

∥D_{z′_{i}}T_{−1}B_{γ}u∥ ∥T_{1}^{∗}R_{i}^{∗}B_{γ}u∥
≤ (C_{0}C_{1}βδ)^{1/2}C_{2}β^{−1}∥B_{γ}u∥^{2} + C̃^{1/2}Q(u, Gu, G̃u)^{1/2}∥B_{γ}u∥
≤ C_{2}(C_{0}C_{1}δ/β)^{1/2}∥B_{γ}u∥^{2} + 𝟋^{−1}C̃Q(u, Gu, G̃u) + 𝟋∥B_{γ}u∥^{2}.

As 𝟋 > 0 is freely chosen, the main point is that if δ/β is sufficiently small, the first term can be absorbed into ∥CB_{γ}u∥^{2}, for the principal symbol of C is bounded below by (c_{0}/4)^{1/2} on supp a. Since the R′_{i} term is analogous, and the R_{ij} term satisfies better estimates (for one uses (7.54) directly, rather than its square root, as for R_{i}), the proof can be finished as in Theorem 7.6.

□

Finally, applying arguments that go back to [24] we may put together Theorems 7.6 and 7.7 to obtain propagation of edge-b wavefront set along EGBB’s over the edge face:

Theorem 7.8. For Neumann boundary conditions, let 𝔛 = H_{es}^{1,l}(M), 𝔜 = Ḣ_{es}^{−1,l−2}(M); for Dirichlet boundary conditions let 𝔛 = H_{es,0}^{1,l}(M), 𝔜 = H_{es}^{−1,l−2}(M).

Let u ∈ 𝔛 solve □u = f, f ∈ 𝔜. Then for all s ∈ ℝ ∪ {∞}, l′ ≤ 0,

((WF_{eb,𝔛}^{s,l′}(u) \ WF_{eb,𝔜}^{s+1,l′}(f)) ∩^{eb}S∗ ; W̃ M) ⊂ ^{eb}Σ

is a union of maximally extended EGBB’s in ^{eb}Σ \ WF_{eb,𝔜}^{s+1,l′}(f).

8. Propagation of fiber-global coisotropic regularity

We now state a microlocal result on the propagation of coisotropy. The result says that coisotropic regularity propagates along EGBB’s provided that we also have infinite order regularity along all rays arriving at radial points in G.

Theorem 8.1 (Microlocal propagation of coisotropy). Suppose that u ∈ H_{es}^{1,l}(M), □u = 0, with Dirichlet or Neumann boundary conditions (see Definition 5.20), p ∈ H_{W,b}. Suppose also that

(i) q ∈ (H_{eb→p,b} ∩ R_{eb,O}) \^{eb}S_{∂W̃}^{∗} M, (ii) u has coisotropic regularity of order k ∈ ℕ relative to H^{m} on the

coisotropic F_{I,reg}^{W} in an open set containing all points in F_{I,p,reg}^{W}∩{0 < x < δ} that are geometrically related to F_{O,q}.

(iii) WF_{b}(u) ∩ F_{p,I,sing}^{W} = ∅.

Then u has coisotropic regularity of order k relative to H^{m′} for all

m′ < min(m, l + f/2)

on F_{O,reg}^{W}, in a neighborhood of F_{O,q,reg}.

Proof. The second numbered assumption and propagation of WF_{e} through incoming radial points, Theorem 7.1 part (1), implies that along EGBB’s in the backward flow of q which pass through R_{eb,I}\G there is no WF_{eb}^{m,l̃}, with

l̃= min(l, m − f/2 − 0).

In view of Theorem 7.3, the third assumption gives the same along EGBB’s in the backward flow of q which pass through R_{eb,I} ∩G. Thus, near q, but on the EGBB’s in the backward flow of q, there is no WF_{eb}^{m,l̃} at all. Propagation of singularities through q (Theorem 7.1 part (2)) then gives no WF_{e}^{m̃,l̃}, m̃ = min(m, l̃+ f/2 − 0), on the flow-out. Substituting in l̃, we see that m̃ = min(m, l + f/2 − 0), giving no WF_{eb}^{m′,l̃}. Thus in x > 0, near the flow out, there is no WF^{m′}, which gives the case k = 0.

We now turn to the general case, k ≠ 0. To begin, note that assumption ii and Theorem 7.1 imply that in fact we have coisotropic regularity of order k relative to H_{eb}^{m,l̃} at all q′ ∈ R_{eb,p,I} that are connected to q by an EGBB .

This in turn yields absence of WF_{eb}^{m+k,l̃} in a neighborhood of each such q in ebS∗ ; W̃ M, as the operators in A are all characteristic only at the radial points over W̃ . By Theorem 7.8 followed by the second part of Theorem 7.1, we then achieve coisotropic regularity of order k relative to H_{e}^{m′,l̃} at q, hence in a neighborhood as well. □

Corollary 8.2. Suppose that u ∈ H_{es}^{1,l}(M), □u = 0, with Dirichlet or Neumann boundary conditions, p ∈ H_{W,b}, k ∈ ℕ. Suppose also that

(i) u has coisotropic regularity of order k relative to H^{m} on the coisotropic F_{I,reg}^{W} in a neighborhood of F_{I,p,reg},

(ii) WF_{b}(u) ∩ F_{p,I,sing}^{W} = ∅.

Then u has coisotropic regularity of order k relative to H^{m′} for all

m′ < min(m, l + f/2)

on F_{O,reg}^{W} in a neighborhood of F_{O,p,reg}^{W}.

Finally, we prove that the regularity with respect to which coisotropic regularity is gained in the above results is not, in fact, dependent on the weight l : Corollary 8.3. Suppose that u ∈ H^{1}(M_{0}), □u = 0, with Dirichlet or Neumann boundary conditions, p ∈ H_{W,b}, k ∈ ℕ, ϵ > 0. There exists k′ (depending on k and ϵ) such that

(i) u has coisotropic regularity of order k′ relative to H^{s} on the coisotropic F_{I,reg}^{W} in a neighborhood of F_{I,p,reg}^{W},

(ii) WF_{b}(u) ∩ F_{p,I,sing}^{W} = ∅.

implies that u has coisotropic regularity of order k relative to H^{s−ϵ} on F_{O,reg}^{W} in a neighborhood of F_{O,p,reg}^{W}.

Proof. Consider Dirichlet boundary conditions first. Then

u ∈ H_{es,0}^{1,1−(f+1)/2}(M).

Thus, by Corollary 8.2, u has coisotropic regularity of order k′ relative to^{12} H^{m−ϵ}, m < min(s, 1/2) on F_{O,reg} near F_{O,p,reg}, strictly away from ∂M.

12An improved version of the argument, using the time derivative to shift the domain, gives coisotropy of order k′ relative to H^{s−1/2−ϵ}.

On the other hand, by the propagation of singularities, [34, Corollary 8.4], u is in H^{s} along F_{O,p}^{W}. Hence the theorem follows by the interpolation result of the following lemma, Lemma 8.4.

Consider Neumann boundary conditions next. Then u ∈ H_{es}^{1,−(f+1)/2}(M), so by Corollary 8.2, u has coisotropic regularity of order k′ relative to H^{m−ϵ}, m < min(s, −1/2) on F_{O,reg}^{W} near F_{O,p,reg}^{W}, strictly away from ∂M.

Proceeding now as in the Dirichlet case, using [34, Corollary 8.4], we complete the proof. □

Lemma 8.4. Suppose that u is in H^{s} microlocally near some point q away from ∂M, and it is coisotropic of order N relative to H^{m} near q with s > m. Then for ϵ > 0 and k < \frac{ϵN}{s−m}, u is coisotropic of order k relative to H^{s−ϵ} near q.

In particular, if u is in H^{s} microlocally near some point q away from ∂M and u is coisotropic (of order ∞) relative to H^{m} near q with s > m, then u is coisotropic relative to H^{s−ϵ} for all ϵ > 0.

Proof. If Q ∈ Ψ^{0}(M) and WF′(Q) lies sufficiently close to q, then the hypotheses are globally satisfied by u′ = Qu. Moreover, being coisotropic, locally F can be put in a model form ζ = 0 by a symplectomorphism Φ in some canonical coordinates (y, z, η, ζ), by [10, Theorem 21.2.4] (for coisotropic submanifolds one has k = n − l, dim S = 2n, in the theorem). Further reducing WF′(Q) if needed, and using an elliptic 0th order Fourier integral operator F with canonical relation given by Φ to consider the induced problem for v = F u′ = F Qu, we may thus assume that v ∈ H^{s}, and D_{z}^{α}v ∈ H^{m} for all α, i.e. ⟨D_{z}⟩^{N}v ∈ H^{m}. Considering the Fourier transform v̂ of v, we then have ⟨η, ζ⟩^{s}v̂ ∈ L^{2}, ⟨η, ζ⟩^{m}⟨ζ⟩^{N} v̂ ∈ L^{2}. But this implies ⟨η, ζ⟩^{mθ+s(1−θ)}⟨ζ⟩^{Nθ}v̂ ∈ L^{2} for all θ ∈ [0, 1] by interpolation (indeed, in this case by Hölder’s inequality). In particular, taking θ = \frac{ϵ}{s−m}, ⟨η, ζ⟩^{s−ϵ}⟨ζ⟩^{k}v̂ ∈ L^{2} if k < \frac{Nϵ}{s−m}, and the lemma follows. □

9. Geometric theorem

The final essential ingredient in the proof of the geometric theorem is the dualization of the coisotropic propagation result, Corollary 8.3. We now prove such a result.

Theorem 9.1. Let u ∈ H_{b,H_{loc}^{1}}^{−∞}(M_{0}) satisfy the wave equation with Dirichlet

or Neumann boundary conditions. Let p ∈ H_{W,b}, and w ∈ F_{O,p,reg}^{W}.

Suppose k ∈ ℕ and ϵ > 0. Then there is k′ ∈ ℕ (depending on k and ϵ) such that if WF_{b}(u) ∩ F_{I,p,sing}^{W} = ∅ and u is non-focusing of order k relative to H^{s} on a neighborhood of F_{I,p,reg}^{W} in F_{I,reg}^{W} then u is non-focusing of order k′ relative to H^{s−ϵ} at w.

Proof. We assume s ≤ 0 to simplify notation; we return to the general case at the end of the argument.

Let T = t(p), and choose T_{0} < T < T_{1} sufficiently close to T. Let χ be smooth step function such that χ ≡ 1 on a neighborhood of [T, ∞] and χ ≡ 0 on a neighborhood of (−∞, T_{0}]. We find that

v ≡ χu

satisfies □v = f with f = [□, χ]u, and v vanishes on a neighborhood of (−∞, T_{0}] × X. Thus, we write

□_{+}^{−1}f = v.

By propagation of singularities, [34], only singularities of f on F_{I,p}^{W} affect regularity at w, i.e. if F_{I,p}^{W} ∩ WF_{b,Ḣ^{−1}(M_{0})}^{s−1}(f) = ∅ then w ∉ WF_{b,H^{1}(M_{0})}^{s}(u),

hence in particular w is non-focusing of order 0 relative to H^{s+1}. Thus,

Q_{0} ∈ Ψ_{b}^{0}(M_{0}), WF′(Id −Q_{0}) ∩ F_{I,p}^{W} ∩^{b}S_{supp dχ}^{∗}M_{0} = ∅
⇒ w ∉ WF_{b,H^{1}(M_{0})}(□_{+}^{−1}((Id −Q_{0})f)),

so it suffices to analyze □_{+}^{−1}(Q_{0}f). We choose WF′(Q_{0}) sufficiently small such that

q ∈ WF′(Q_{0}) ⇒ either q ∉ WF_{b}(f) or

f is non-focusing of order k at q relative to H^{s−1};

this is possible by our hypotheses. We may thus replace f by f_{0} = Q_{0}f, assume that f_{0} is the sum of a distribution that is non-focusing of order k relative to H^{s−1} and is supported in M_{0}^{∘} plus an element of H_{b,H^{1}(M_{0})}^{∞}, and show that □_{+}^{−1}f_{0} is non-focusing at w of order k′ (for some k′ to be determined) relative to H^{s−ϵ}.

Let

T′_{0} < T_{0} < T_{1} < T′_{1}.

We regard [T_{0}, T_{1}] as the time interval for analysis, but we enlarge it to [T′_{0}, T′_{1}] in order to be able to apply some b-ps.d.o’s with symbol elliptic for t ∈ [T_{0}, T_{1}] to elements of our function spaces. (The ends of the interval would be slightly troublesome.) We define a Hilbert space 𝔛 to be 𝔛 = H^{1}([T′_{0}, T′_{1}] × X_{0}) in the case of Neumann conditions, or 𝔛 = H_{0}^{1}([T′_{0}, T′_{1}] × X_{0}) in the case of Dirichlet conditions, where 0 indicates vanishing enforced at [T′_{0}, T′_{1}] × ∂X_{0} (but not at the endpoints of the time interval). Let 𝔛^{∗} be the L_{g}^{2}-dual of 𝔛.

We further let

(i)

T_{0} < T̃_{0} < t′_{0} < t_{0} < T < t_{1} < t′_{1} < T_{1}

such that supp dχ ⊂ (t_{0}, T). (ii) χ_{0} ∈ C^{∞}(ℝ) such that supp(1−χ_{0}) ⊂ (T̃_{0}, +∞), supp χ_{0} ⊂ (−∞, t′_{0}). (iii) U_{0} be an open set with \overline{U_{0}} ⊂ \{t ∈ (t′_{0}, T)\}, \overline{U_{0}} ∩ F_{I,sing}^{W} = ∅ and

WF_{b,𝔛^{∗}}(f_{0}) ⊂ U_{0}.

(iv) U_{1} be a neighborhood of w with \overline{U_{1}} ⊂ \{t ∈ (T, t′_{1})\} and \overline{U_{1}}∩F_{O,sing}^{W} =

∅.

(v) B_{0}, B_{1} ∈ Ψ_{b}^{0}(M) with

WF′(B_{j}) ⊂ U_{j}, w ∉ WF′(Id −B_{1}), WF′(Id −B_{0}) ∩ WF_{b,𝔛^{∗}}(f_{0}) = ∅,

and with Schwartz kernel supported in (t′_{0}, t′_{1})^{2} × X^{2}. (vi) A_{i}, i = 1, …, N, denote first-order pseudodifferential operators,

generating M as defined in §6, but now locally over a neighborhood of \overline{U_{0}} ∪ \overline{U_{1}} in M^{∘}, and with kernels compactly supported in M^{∘}.

(vii) T_{ν} ∈ Ψ_{b}^{ν}(M) with elliptic principal symbol on [T_{0}, T_{1}] × X_{0} with Schwartz kernel supported in (T′_{0}, T′_{1})^{2} × (X_{0})^{2}. Thus, T_{ν} can be applied to elements of 𝔛 and 𝔛^{∗}.

Now suppose that we are given r and ϵ > 0. Then, with k as in the statement of the Theorem, Corollary 8.3 gives a k′ = k′(r, ϵ, k) using s = r in the notation of that corollary. We let 𝔜 be a space of microlocally coisotropic functions on ell(B_{1}) relative to 𝔛^{∗} which are in addition extremely wellbehaved elsewhere (they will be finite-order conormal to the boundary) in {t ≥ t′_{0}}, but are merely in H_{b,𝔛^{∗}}^{1} for t near T_{0}: Let N > r > 1 + ϵ and set

\begin{aligned}𝔜 = \{ψ & ∈ 𝔛^{∗} : \\ & ∥T_{N}(Id −B_{1} − χ_{0})ψ∥_{𝔛^{∗}}^{2} + ∥T_{1}χ_{0}ψ∥_{𝔛^{∗}}^{2} + ∑_{|α|≤k′} ∥T_{r}A^{α}B_{1}ψ∥_{𝔛^{∗}}^{2} < ∞\}. \end{aligned}

Thus,

ψ ∈ 𝔜 ⇒ WF_{b,𝔛^{∗}}^{N}(ψ) ⊂ WF′(B_{1}) ∪^{b}S_{supp χ_{0}}^{∗}M_{0},

and

(9.1)  ∥ψ∥_{H_{b,𝔛^{∗}}^{1}} ≲ ∥ψ∥_{𝔜}

for ψ supported in [T_{0}, T_{1}] × X_{0} (where the T_{s} are elliptic).

Also, let ℨ be the space of microlocally coisotropic functions on ell(B_{0}) relative to 𝔛 (and just in 𝔛 elsewhere):

ℨ = {ϕ ∈ 𝔛 : ∑_{|α|≤k} ∥T_{r−1−ϵ}A^{α}B_{0}ϕ∥_{𝔛}^{2} < ∞}.

Note that as discussed in Section 6 (in particular, Lemma 6.8)

ℨ^{∗} = 𝔛^{∗} + ∑_{|α|≤k} T_{r−1−ϵ}A^{α}B_{0}𝔛^{∗},

so by our assumption on f_{0}, f_{0} ∈ ℨ^{∗}, provided −(r − 1 − ϵ) − 1 ≤ s − 1, i.e. provided^{13} r ≥ −s + 1 + ϵ. Moreover, if v_{0} ∈ 𝔜^{∗}, then

(9.2)  v_{0} ∈ 𝔛 + T_{N}(Id −B_{1} − χ_{0})𝔛 + T_{1}χ_{0}𝔛 + ∑_{|α|≤k′} T_{r}A^{α}B_{1}𝔛.

13Note that for such r, r ≥ 1 + ϵ as required above, since s ≤ 0.

In particular, as w ∉ WF′(Id −B_{1} − χ_{0}) ∪ ^{b}S_{supp χ_{0}}^{∗}M_{0}, v_{0} is non-focusing at w of order k′ relative to H^{−r+1}, hence relative to H^{s−ϵ}, if we actually choose r = −s + 1 + ϵ.

For I ⊂ [T_{0}, T_{1}], let Ḋ_{I} denote the subspace of H_{b,𝔛}^{∞} consisting of functions supported in I × X, Ė_{I} denote the subspace H_{b,𝔛^{∗}}^{∞} consisting of functions supported in I × X, so □ : Ḋ_{I} → Ė_{I} is continuous, Ḋ_{I} ⊂ ℨ, Ė_{I} ⊂ 𝔜 are dense with continuous inclusions. Also let Ḋ = Ḋ_{I}, Ė = Ė_{I} for I = (T_{0}, T_{1}). Finally we also let ℨ_{I} be space of restrictions of elements of ℨ to I, and analogously for 𝔜_{I}. Then, as we will prove in Lemma 9.2^{14}, Corollary 8.3 implies that

(9.3)  ϕ ∈ Ḋ ⇒ ∥ϕ∥_{ℨ} ≤ C∥□ϕ∥_{𝔜},

where vanishing for t near T_{1} is used. In fact, we prove a somewhat more precise statement:^{15}

Lemma 9.2. For τ ∈ [T_{0}, T_{1}), τ′ > τ,

(9.4)  ϕ ∈ Ḋ ⇒ ∥ϕ∥_{ℨ_{[τ′,T_{1}]}} ≤ C∥□ϕ∥_{𝔜_{[τ,T_{1}]}}.

Proof of Lemma: Recall first that by standard energy estimates (taking into account the vanishing of ϕ near T_{1})

(9.5) ϕ ∈ Ḋ ⇒ ∥ϕ∥_{𝔛_{[τ,T_{1}]}} ≲ ∥□ϕ∥_{𝔛_{[τ,T_{1}]}^{∗}} + ∥D_{t}□ϕ∥_{𝔛_{[τ,T_{1}]}^{∗}} ≲ ∥□ϕ∥_{𝔜_{[τ,T_{1}]}}.

Thus, we only need to prove that for |α| ≤ k′,

∥T_{r−1−ϵ}A^{α}B_{0}ϕ∥_{𝔛_{[τ′,T_{1}]}} ≲ ∥□ϕ∥_{𝔜_{[τ,T_{1}]}}.

If □ϕ is supported away from ∂M, then this follows from Corollary 6.11 and Corollary 8.3. In general, let Q ∈ Ψ^{0}(M) be such that WF′(B_{1}) ∩ WF′(Id −Q) = ∅, and Q has compactly supported Schwartz kernel in (M^{∘})^{2}. Then Q□ϕ has support away from ∂M, so

(9.6) ∥T_{r−1−ϵ}A^{α}B_{0}□_{−}^{−1}(Q□ϕ)∥_{𝔛_{[τ′,T_{1}]}} ≲ ∥Q□ϕ∥_{𝔜_{[τ,T_{1}]}} ≲ ∥□ϕ∥_{𝔜_{[τ,T_{1}]}},

where □_{−}^{−1} denotes the backward solution of the inhomogeneous wave equation. On the other hand,

∥(Id −Q)□ϕ∥_{H_{b,𝔛^{∗}}^{N}(M_{0})} ≲ ∥□ϕ∥_{𝔜_{[τ,T_{1}]}},

so by propagation of b-regularity [34],

∥□_{−}^{−1}((Id −Q)□ϕ)∥_{H_{b,𝔛}^{N−1}(M_{0})} ≲ ∥□ϕ∥_{𝔜_{[τ,T_{1}]}},

14The only reason for Corollary 8.3 combined with Corollary 6.11 not yielding the result immediately is that Corollary 8.3 is stated for the homogeneous wave equation.

This suffices for our purposes as we only require inhomogeneities that are very regular near the boundary, hence the propagation result of [34] is adequate.

15Notice that ϕ is merely supported in (T_{0}, T_{1}) here; not in (τ, T_{1}), which would be (9.3) on [τ, T_{1}], except for the loss of going from τ′ to τ.

hence the much weaker statement

(9.7)  ∥T_{r−1−ϵ}A^{α}B_{0}□_{−}^{−1}((Id −Q)□ϕ)∥_{𝔛_{[τ′,T_{1}]}} ≲ ∥□ϕ∥_{𝔜_{[τ,T_{1}]}},

also holds. Combining (9.6) and (9.7) proves (9.4). This concludes the proof of Lemma 9.2, and hence of (9.3) as well.

In particular, recalling that Ḋ = Ḋ_{(T_{0},T_{1})}, (9.3) shows that for ψ ∈ RanḊ □ there is a unique ϕ ∈ Ḋ such that ψ = □ϕ; we denote this by ϕ = □^{−1}ψ. Thus,

∥□^{−1}ψ∥_{ℨ} ≤ C∥ψ∥_{𝔜}, ψ ∈ RanḊ □.

Now consider the linear functional on RanḊ □ given by

ψ ↦ ⟨f_{0}, □^{−1}ψ⟩, ψ ∈ RanḊ □,

which satisfies

|⟨f_{0}, □^{−1}ψ⟩| ≤ ∥f_{0}∥_{ℨ^{∗}}∥□^{−1}ψ∥_{ℨ} ≤ C∥f_{0}∥_{ℨ^{∗}}∥ψ∥_{𝔜}, ψ ∈ RanḊ □.

This has a unique extension to a continuous linear functional ℓ on \overline{RanḊ □}, the closure of RanḊ □ in 𝔜.

If we used the Hahn-Banach theorem at this point to extend the linear functional ℓ further to a linear functional v_{0} on all of 𝔜, we would obtain a solution of the wave equation □v_{0} = f_{0} on (T_{0}, T_{1}), as ⟨□v_{0}, ϕ⟩ = ⟨v_{0}, □ϕ⟩ = ⟨f_{0}, ϕ⟩ for ϕ ∈ Ḋ, which is indeed non-focusing at w, but we need not just any solution, but the forward solution, □_{+}^{−1}f_{0}. So we proceed by extending the linear functional ℓ to a continuous linear functional L on

(9.8)  \overline{RanḊ_{(T_{0},t′_{1})} □} + \overline{Ė_{(T_{0},t′_{0})}},

first, in such a manner that the extension is ℓ on the first summand and vanishes on the second summand. If we actually have such an extension, then we can further extend it to all of 𝔜, then vanishing on the first summand shows that it solves the wave equation on (T_{0}, t′_{1}), while vanishing on the second summand shows that it vanishes on (T_{0}, t′_{0}), so its restriction as a distribution on (T_{0}, t′_{0}) is indeed □_{+}^{−1}f_{0}. In order to obtain such an extension we show:

Lemma 9.3.

(i) ℓ vanishes on the intersection of the two summands, so L is welldefined as a (not necessarily continuous) linear map,

(ii) The subspace (9.8) of 𝔜 is closed, and given an element ψ + ρ in the sum, there is a representation^{16} ψ̃ + ρ̃ of ψ + ρ as a sum of elements of the two summands such that one can estimate the 𝔜-norm of ψ̃ and ρ̃ in terms of ψ + ρ.

16Since the intersection of the summands is non-trivial, this can only be true for some representation, not all representations!

Proof of Lemma: We start with the statement regarding intersection of the summands in (9.8). Thus, we claim that if

supp f_{0} ⊂ [t_{0}, t_{1}] × X,

then

(9.9)  ψ ∈ \overline{RanḊ_{[T_{0},t′_{1}]} □} and supp ψ ⊂ (T_{0}, t′_{0}) ⇒ ℓ(ψ) = 0.

To see this let ψ_{j} → ψ in 𝔜, ψ_{j} = □ϕ_{j}, ϕ_{j} ∈ Ḋ_{[T_{0},t′_{1}]}. Then {ψ_{j}} is Cauchy in 𝔜, hence {ϕ_{j}} is Cauchy in ℨ by (9.3), hence converges to some ϕ ∈ ℨ. By the support condition on ϕ_{j}, supp ϕ ⊂ [T_{0}, t′_{1}]. As □ϕ_{j} → □ϕ in 𝔛^{∗} (for □ : 𝔛 → 𝔛^{∗} is continuous), and □ϕ_{j} = ψ_{j} → ψ in 𝔜, hence in 𝔛^{∗}, we deduce that ψ = □ϕ, i.e. □ϕ is supported in (T_{0}, t′_{0}). Thus, ψ_{j}|_{(t′_{0},T_{1})} → 0 in the 𝔜 topology hence ϕ_{j}|_{[t_{0},T_{1}]} → 0 in the ℨ topology using (9.4) with τ = t′_{0}, τ′ = t_{0}, so, by the support condition on f,

|⟨f_{0}, ϕ_{j}⟩| ≤ ∥f_{0}∥_{ℨ_{[t_{0},t_{1}]}^{∗}}∥ϕ_{j}∥_{ℨ_{[t_{0},t_{1}]}} → 0,

so we deduce that ⟨f_{0}, ϕ⟩ = 0 as claimed.

Next we turn to the closedness of the sum in (9.8). First, we claim that if ψ ∈ RanḊ_{(T_{0},t′_{1})} □, ρ ∈ Ė_{(T_{0},t′_{0})} then there exist ψ̃ ∈ RanḊ_{(T_{0},t′_{1})} □, ρ̃ ∈ Ė_{(T_{0},T̃_{0})}

such that

ψ + ρ = ψ̃ + ρ̃ and ∥ψ̃∥_{𝔜} ≲ ∥ψ + ρ∥_{𝔜}.

Indeed, let χ_{+} ∈ C^{∞}(ℝ) such that

supp χ_{+} ⊂ (T_{0}, +∞) and supp(1 − χ_{+}) ⊂ (−∞, T̃_{0}).

Let □_{−}^{−1}(ψ + ρ) denote the backward solution of the inhomogeneous wave equation; i.e. the unique solution of □ũ = ψ + ρ which vanishes on (t_{1}, T_{1}). Then let

ψ̃ = □(χ_{+}□_{−}^{−1}(ψ + ρ)) ∈ RanḊ_{(T_{0},t′_{1})} □,

so

ρ̃ ≡ ψ + ρ − ψ̃ = (1 − χ_{+})(ψ + ρ) − [□, χ_{+}]□_{−}^{−1}(ψ + ρ) ∈ Ė_{(T_{0},T̃_{0})}.

Moreover,

(9.10)  ψ̃ = χ_{+}(ψ + ρ) + [□, χ_{+}]□_{−}^{−1}(ψ + ρ)

satisfies

∥ψ̃∥_{𝔜} ≲ ∥ψ + ρ∥_{𝔜},

as follows by inspecting the two terms on the right hand side of (9.10): for the first this is clear, for the second this follows from ∥ψ+ρ∥_{H_{b,𝔛^{∗}}^{1}} ≲ ∥ψ+ρ∥_{𝔜},

see (9.1), hence one has a bound in 𝔛 for □_{−}^{−1}(ψ + ρ) by (9.5), and then supp[□, χ_{+}] ⊂ supp dχ_{+} ⊂ (T_{0}, T̃_{0}) gives the desired bound in 𝔜. This concludes the proof of Lemma 9.3.

Thus, if ψ_{j} ∈ RanḊ_{(T_{0},t′_{1})} □, ρ_{j} ∈ Ė_{(T_{0},T̃_{0})} and ψ_{j} + ρ_{j} converges to some

ν ∈ 𝔜 then defining ψ̃_{j} and ρ̃_{j} as above, we deduce that due to the Cauchy property of {ψ̃_{j} + ρ̃_{j}}, {ψ̃_{j}} is Cauchy in 𝔜, hence so is {ρ̃_{j}}, thus by the completeness of 𝔜 they converge to elements in ψ ∈ \overline{RanḊ_{(T_{0},t′_{1})} □}, resp.

ρ ∈ \overline{Ė_{(T_{0},T̃_{0})}} with ψ + ρ = ν. This shows that \overline{RanḊ_{(T_{0},t′_{1})} □} + \overline{Ė_{(T_{0},T̃_{0})}} is

closed, and indeed gives an estimate^{17} that if ν ∈ \overline{RanḊ_{(T_{0},t′_{1})} □} + \overline{Ė_{(T_{0},T̃_{0})}}

then there exists ψ ∈ \overline{RanḊ_{(T_{0},t′_{1})} □} and ρ ∈ \overline{Ė_{(T_{0},T̃_{0})}} such that ψ + ρ = ν and

(9.11)  ∥ψ∥_{𝔜} + ∥ρ∥_{𝔜} ≲ ∥ν∥_{𝔜}.

As mentioned earlier, this construction allows us to define a unique continuous linear functional L on

\overline{RanḊ_{(T_{0},t′_{1})} □} + \overline{Ė_{(T_{0},t′_{0})}},

in such a way that it is ℓ on the first summand and it vanishes on the second summand: uniqueness is automatic, existence (without continuity) follows from (9.9), as the two functionals agree on the intersection of the two spaces, while continuity follows from (9.11). Then we extend L by the Hahn-Banach theorem to a linear functional v_{0} on 𝔜.

Then v_{0} ∈ 𝔜^{∗} solves □v_{0} = f_{0} on (T_{0}, t′_{1}), since for ϕ ∈ Ḋ_{(T_{0},t′_{1})}

⟨□v_{0}, ϕ⟩ = ⟨v_{0}, □ϕ⟩ = ⟨f_{0}, ϕ⟩,

and v vanishes on (T_{0}, t_{0}), for it vanishes on Ḋ_{(T_{0},t_{0})}, i.e. on test functions supported there, so it is the restriction of the forward solution of the wave equation to (T_{0}, T_{1}). We have thus shown that if f_{0} ∈ ℨ^{∗} is supported in [t_{0}, t_{1}], which holds if f_{0} satisfies the support condition, is microlocally non-focusing on U_{0}, and is conormal to the boundary elsewhere, then the forward solution of □v_{0} = f_{0} is in 𝔜^{∗} (cf. Lemma 6.9), hence by (9.2) it is in particular microlocally non-focusing of order k′ relative to H^{s−ϵ} at w. This completes the proof of the theorem if s ≤ 0.

If s > 0, one could use a similar argument relative to slightly different spaces: the only reason for the restriction is that elements of 𝔜 lie in 𝔛^{∗} and a larger space (which would thus have a smaller dual relative to L^{2}) would be required to adapt the argument. However, it is easy to reduce the general case to s ≤ 0: replacing u by ũ = (1 + D_{t}^{2})^{N}u, N > s/2, ũ is non-focusing of order k relative to H^{s−2N} on a neighborhood of F_{I,p,reg}^{W} and solves the wave equation, hence it is non-focusing of order k′ relative to H^{s−2N−ϵ} at w by the already established s ≤ 0 case of this theorem, and then the microlocal ellipticity of (1 + D_{t}^{2})^{N} near the characteristic set (recall that w is over the interior of M_{0}) shows that u itself is non-focusing of order k′ relative to H^{s−ϵ} at w, as claimed. □

17This estimate follows from the open mapping theorem, given that the sum is closed, but the direct argument yields it anyway.

As a consequence of the proposition of nonfocusing, we are now able to prove our main theorem:

Theorem 9.4. Let u ∈ H_{loc}^{1}(M_{0}) satisfy the wave equation with Dirichlet or Neumann boundary conditions. Let p ∈ H_{W,b}, and w ∈ F_{O,p,reg}^{W}.

Assume

(i) u satisfies the nonfocusing condition relative to H^{s} on an open neighborhood of F_{I,p,reg}^{W} in F_{I,reg}^{W},

(ii) WF^{s} u ∩ {w′ ∈ F_{I,p,reg}^{W} : w′, w are geometrically related} = ∅, (iii) WF_{b}(u) ∩ F_{p,I,sing}^{W} = ∅.

Then

w ∉ WF^{s−0}(u).

Proof. By using a microlocal partition of unity (cf. the argument at the beginning of Theorem 9.1), we may arrange that (ii) is strengthened to

(9.12) WF^{∞} u ∩ {w′ ∈ F_{I,p,reg}^{W} : w′, w are geometrically related} = ∅,

for if a microlocal piece ũ is in H^{s} near F_{I,p}^{W}, then it is also such at w.

Let r < s. On the one hand, by conditions (i) and (iii), u satisfies the non-focusing condition (of some, possibly large, order k′) relative to H^{r} at w due to Theorem 9.1. On the other hand, by Theorem 8.1, (9.12) and condition (iii), u is microlocally coisotropic at w, i.e. there exists S ∈ ℝ such that^{18} microlocally near w

(9.13)  A^{α}u ∈ H^{S} ∀α.

We can interpolate non-focusing and (9.13) as discussed in [25] to conclude that microlocally near w, u ∈ H^{r−0}. Since r < s is arbitrary, this proves the result. □

Corollary 9.5. Let u be a solution to □u = 0 with Dirichlet or Neumann boundary conditions, and let p ∈ H_{W,b}. Suppose that for some ϵ_{0} > 0, in a neighborhood of F_{I,p}^{W}(ϵ_{0}) in ^{b}S_{M_{0}\W}^{∗} M_{0}, u is a Lagrangian distribution of order s with respect to L ⊂ T^{∗}M_{0}^{∘}, a conic Lagrangian such that L ∩ F_{I,p,sing}^{W} = ∅ and the intersection of L and F_{I,reg}^{W} is transverse at F_{I,p,reg}^{W}.

Then if w ∈ F_{O,reg}^{W} is not geometrically related to any point in L,

w ∉ WF^{−s−n/4+(k−1)/2−0} u = ∅

where k is the codimension of W.

The a priori regularity of such a solution is H^{−s−n/4−0} so this represents a gain in regularity along the diffracted wave of (k − 1)/2 − 0 derivatives.

Corollary 9.5 follows immediately from Section 14 of [25] and Theorem 9.4.

18The particular choice of S is dependent on the background regularity of the solution, which in turn can be low, depending on the order of nonfocusing relative to H^{s}.

Corollary 9.6. Let γ : (−ϵ_{0}, 0] → ^{b}Σ_{0} be a GBB normally incident at W, γ(0) = α ∈ H_{W,b}, and let \overline{γ} be its projection to M_{0}^{∘}. Given o ∈ \overline{γ}((−ϵ_{0}, 0)), let u_{o} be the forward fundamental solution of □, i.e. u_{o} = □_{+}^{−1}δ_{o}.

There exists ϵ > 0 such that if o ∈ \overline{γ}((−ϵ, 0)) then for all w ∈ F_{O,α,reg}^{W}, such that w is not geometrically related to point in ^{b}S_{o}^{∗}M_{0} ∩^{b}Σ_{0},

w ∉ WF^{(−n+k+1)/2−0} u_{o}

where k is the codimension of W.

Note that this represents a gain of (k − 1)/2 − 0 derivatives relative to the overall regularity of the fundamental solution, which lies in H^{−n/2+1−0}.

Proof. The hypotheses on the location of o ensure that, with L denoting the flowout of ^{b}S_{o}^{∗}M_{0} ∩^{b}Σ_{0}, L is disjoint from F_{I,α,sing}^{W} in view of Corollary 3.27. Thus, the microlocal setting is the same as that of [25], hence the hypotheses of Corollary 9.5 are satisfied. □

Appendix A. Some functional analysis

We often encounter the following setup. Suppose that ℌ, 𝔜 are Banach, resp. locally convex, spaces, and

ι : 𝔜 → ℌ

is a continuous injection with dense range (so one can think of 𝔜 as a subspace of ℌ with a stronger topology). Let ℌ′, 𝔜′ denote the spaces of linear functionals on ℌ, 𝔜 with the norm, resp. the ℌ-weak and 𝔜-weak topologies (which is the weak-* topology in the Banach space setting). Then the adjoint of ι is the map

ι^{†} : ℌ′ → 𝔜′, ι^{†}ℓ(v) = ℓ(ιv), ℓ ∈ ℌ′,

and ι^{†} is continuous in the respective topologies. The injectivity of ι implies that ι^{†} has dense range, while the fact that ι has dense range implies that ι^{†} is injective. Thus, one can think of ℌ′ as a subspace of 𝔜′, with a stronger topology.

If ℌ is a Hilbert space with inner product ⟨., .⟩_{ℌ} ℂ-linear in the first argument, there is a canonical (conjugate-linear) isomorphism j_{ℌ} : ℌ → ℌ′ given by j_{ℌ}(u)(v) = ⟨v, u⟩_{ℌ}. Suppose also that there is a canonical conjugate linear isomorphism

c_{ℌ} : ℌ → ℌ, c_{ℌ}^{2} = Id, ⟨u, c_{ℌ}v⟩ = ⟨v, c_{ℌ}u⟩;

if ℌ is a function space, this is usually given by pointwise complex conjugation. Thus,

T_{ℌ} = j_{ℌ} ∘ c_{ℌ} : ℌ → ℌ′

is a linear isomorphism. Thus, if A : 𝔜 → ℌ is continuous linear, then A^{†} : ℌ′ → 𝔜′ continuous linear, and

A^{♭} = A^{†} ∘ j_{ℌ} ∘ c_{ℌ} : ℌ → 𝔜′

is continuous and linear. In particular, letting A be our continuous injection,

ι^{♭} = ι^{†} ∘ j_{ℌ} ∘ c_{ℌ} : ℌ → 𝔜′

is linear, injective with dense range, so ℌ can be considered a subspace of 𝔜′ (with a stronger topology). In particular,

ι^{♭} ∘ ι : 𝔜 → 𝔜′

is also injective with dense range. One considers the triple (𝔜′, ℌ, ι) the ℌ-dual of 𝔜; we will denote this either simply by 𝔜′,or by 𝔜^{∗} if we want to emphasize the inclusion of 𝔜 into 𝔜′ via ℌ, in what follows. Note that if 𝔜 is also a Hilbert space with a canonical conjugate linear isomorphism^{19} c_{𝔜},

ι ∘ c_{𝔜} = c_{ℌ} ∘ ι, c_{𝔜}^{2} = Id,

then we have the canonical linear isomorphism T_{𝔜} = j_{𝔜} ∘ c_{𝔜} : 𝔜 → 𝔜′, and it is important to keep in mind that T_{𝔜} is (usually) different from ι^{♭}ι = ι^{†} ∘ T_{ℌ} ∘ ι:

\begin{matrix}T_{𝔜}(u)(v) = ⟨v, c_{𝔜}u⟩_{𝔜}, \\ ι^{♭}ι(u)(v) = ⟨ιv, (c_{ℌ} ∘ ι)u⟩_{ℌ} = ⟨ιv, (ι ∘ c_{𝔜})u⟩_{ℌ}, \end{matrix}

for u, v ∈ 𝔜. A simple example, when X a compact manifold with a smooth non-vanishing density ν is obtained by 𝔜 = Ċ^{∞}(X) (a Fréchet space) and ℌ = L_{ν}^{2}(X) with respect to the density ν, with ι : 𝔜 → 𝔛 the inclusion. Then ι^{♭}ι : Ċ^{∞}(X) → C^{−∞}(X) is the standard inclusion of Schwartz functions in tempered distributions: ι^{♭}ιf(ϕ) = ∫ fϕ ν.

In fact, we shall always consider a setting with D a dense subspace of ℌ, with a locally convex topology, with respect to which the inclusion map is continuous (i.e. which is stronger than the subspace topology), so using the linear isomorphism j_{ℌ} ∘ c_{ℌ} : ℌ → ℌ′, we have continuous inclusions, with dense ranges,

D ⊂ ℌ ≡ ℌ′ → D′.

Suppose now that A : D → D, hence A^{†} : D′ → D′, and suppose that A^{†} maps D, i.e. more precisely the range of ι^{♭}ι (with ι : D → ℌ the inclusion), to itself, and let

Ã = (ι^{♭}ι)^{−1}A^{†}(ι^{♭}ι) : D → D.

Then for f, ϕ ∈ D

\begin{aligned}⟨ιϕ, ιÃf⟩_{ℌ} & = (ι^{♭}ιc_{D}Ãf)(ϕ) = (c_{D′}ι^{♭}ιÃf)(ϕ) = (c_{D′}A^{†}ι^{♭}ιf)(ϕ) \\ & = \overline{(A^{†}ι^{♭}ιf)(c_{D}ϕ)} = \overline{(ι^{♭}ιf)(Ac_{D}ϕ)} = \overline{⟨ιc_{D}f, ιAc_{D}ϕ⟩} \\ & = ⟨ιc_{D}Ac_{D}ϕ, ιf⟩, \end{aligned} ; (A.1)

so Ã is the formal adjoint of c_{D}Ac_{D} with respect to the ℌ inner product.

Given a Hilbert space ℌ as above, hence an inclusion of D into D′, we shall also have to consider subspaces 𝔛 of D′ with a locally convex topology, which contain the image of D in D′ (under the ℌ-induced inclusion map), and such that the inclusion maps

19Again, pointwise complex conjugation on function spaces is a good example.

D ↪ 𝔛 ↪ D′

are continuous, with dense range, hence one has the corresponding sequence of adjoint maps, which are continuous, with dense range, when all the duals are equipped with the weak topologies. As (D′)′ = D, one obtains

D ↪ 𝔛′ ↪ D′.

If further

D ↪ 𝔛 ↪ ℌ ↪ D′

continuous, with dense ranges, then

D ↪ ℌ′ ≡ ℌ ↪ 𝔛′ ↪ D′,

and similarly if one had the reverse inclusion between 𝔛 and ℌ.

One way that subspaces such as 𝔜 arise is by considering a a finite number of continuous linear maps A_{j} : D → D, such that there exist continuous extensions A_{j} : D′ → D′ (which are then unique by the density of D in D′), hence A_{j} : 𝔛 → D′, j = 1, …, k. Then, in what essentially amounts to constructing a “joint maximal domain” for the A_{j}, and writing ι_{𝔛D′} : 𝔛 → D′ for the inclusion, let

(A.2)  𝔜 = {u ∈ 𝔛 : ∀j, A_{j}u ∈ Ran ι_{𝔛D′}}

with

(A.3)  ∥u∥_{𝔜}^{2} = ∥u∥_{𝔛}^{2} + ∑ ∥ι_{𝔛D′}^{−1}A_{j}u∥_{𝔛}^{2},

where the injectivity of ι_{𝔛D′} was used. If {u_{n}} is Cauchy in 𝔜, then it is such in 𝔛, so converges to some u ∈ 𝔛, and thus A_{j}u_{n} → A_{j}u ∈ D′. Moreover, if {u_{n}} is Cauchy in 𝔜 then ι_{𝔛D′}^{−1}A_{j}u_{n} is Cauchy in 𝔛 so converges to some v_{j} ∈ 𝔛, hence A_{j}u_{n} → ι_{𝔛D′}v_{j} in 𝔛. Thus, A_{j}u = ι_{𝔛D′}v_{j}, so A_{j}u ∈ Ran ι_{𝔛D′}, and A_{j}u_{n} → A_{j}u in 𝔛, proving that 𝔜 is a Hilbert space. We will simply write A_{j} for A_{j}|_{𝔜} : 𝔜 → 𝔛. Note that D ⊂ 𝔜, so 𝔜 is dense in 𝔛. However, D is not necessarily dense in 𝔜.

An example is given by 𝔛 = L_{g}^{2}(X) on a compact manifold with or without boundary, g a Riemannian metric, A_{j} be a finite set of C^{∞} vector fields which span all vector fields over C^{∞}(X), D either C^{∞}(X) or Ċ^{∞}(X); then 𝔜 = H_{g}^{1}(X). If D = C^{∞}(X), then D is dense in 𝔜, but if D = Ċ^{∞}(X), then this is not in general the case: it fails if the boundary of X is nonempty. Other examples are given coisotropic distributions, where the A_{j} are products of first order ps.d.o’s characteristic on a coisotropic manifold; see a general discussion below for spaces given by such ps.d.o’s.

Another way a subspace like 𝔜 might arise from continuous linear maps A_{j} : D → D is the following. In a “joint minimal domain” construction, one can define

(A.4)  ∥u∥2 ; 𝔜̃ = ∥u∥_{𝔛}^{2} + ∑ ∥A_{j}u∥_{𝔛}^{2},

as above, and let 𝔜̃ be the completion of D with respect to this norm, so 𝔜̃ is a Hilbert space. Moreover, the inclusion map ι_{D𝔛} : D → 𝔛 as well as A_{j} extend to continuous linear maps

ι̃_{D𝔛} = ι_{𝔜̃𝔛}, Ã_{j} : 𝔜̃ → 𝔛,

and ι_{𝔜̃𝔛} has dense range (for D canonically injects into the completion). In addition, with 𝔜 as above, the inclusion map ι_{D𝔜} : D → 𝔜 extends continuously to a map

ι̃_{𝔜̃𝔜} : 𝔜̃ → 𝔜

which is an isometry, and is in particular injective. This in particular shows that the inclusion map from 𝔜̃ to 𝔛 is also injective, with a dense range. For X a manifold with boundary and D = Ċ^{∞}(X), A_{j} vector fields as above, one has 𝔜 = H_{0}^{1}(X); with D = C^{∞}(X), one has 𝔜 = H^{1}(X).

Note that the closure of D in 𝔜 is 𝔜̃, so D is dense in 𝔜 if and only if 𝔜 = 𝔜̃ (i.e. ι̃_{𝔜̃𝔜} is surjective). From this point on we assume that 𝔜 = 𝔜̃. This

is true, for instance, if one is given B_{1}, …, B_{r} ∈ Ψ_{eb}^{1,0}(M), and A_{1}, …, A_{k} are up to s-fold products of these, as shown below in Lemma A.3. Thus, 𝔜′ ⊂ D′ (i.e. the inclusion map is injective).

Using the inclusion map ι_{𝔜𝔛} we can now identify the dual of 𝔜 with respect to ℌ. We start with the case ℌ = 𝔛. By the Riesz lemma, 𝔜′ = T_{𝔜}𝔜 and T_{𝔜} is unitary, where

T_{𝔜}v(u) = ⟨u, c_{𝔜}v⟩_{𝔜}.

But

\begin{aligned}T_{𝔜}(v)(u) & = ⟨u, c_{𝔜}v⟩_{𝔜} = ⟨ι_{𝔜ℌ}u, ι_{𝔜ℌ}c_{𝔜}v⟩_{ℌ} + ∑_{j}⟨A_{j}u, A_{j}c_{𝔜}v⟩_{ℌ} \\ & = j_{ℌ}ι_{𝔜ℌ}c_{𝔜}v(ι_{𝔜ℌ}u) + ∑_{j} j_{ℌ}A_{j}c_{𝔜}v(A_{j}u). \end{aligned}

Thus,

T_{𝔜}(v)(u) = ((ι_{𝔜ℌ}^{♭}ι_{𝔜ℌ} + ∑ A_{j}^{♭}c_{ℌ}A_{j}c_{𝔜})v)(u).

We conclude that

T_{𝔜}v = (ι_{𝔜ℌ}^{♭}ι_{𝔜ℌ} + ∑ A_{j}^{♭}c_{ℌ}A_{j}c_{𝔜})v,

and

(A.5)  𝔜^{∗} = (ι_{𝔜ℌ}^{♭}ι_{𝔜ℌ} + ∑ A_{j}^{♭}c_{ℌ}A_{j}c_{𝔜})𝔜.

This also shows that

𝔜^{∗} = Ran ι_{𝔜ℌ}^{†} + ∑ Ran A_{j}^{†},

for ⊃ follows from the definition of ι^{†}, etc., while ⊂ follows from (A.5). We recall here that by (A.1), A_{j}^{†} is the formal adjoint of c_{D}A_{j}c_{D}.

More generally, we do not need to assume 𝔛 = ℌ; rather assume that

(A.6)  ⟨u, v⟩_{𝔛} = ∑_{k}⟨B_{k}u, B_{k}v⟩_{ℌ},

where B_{k} : 𝔛 → ℌ are continuous linear maps (and there is no assumption on the relationship in the sense of inclusions between 𝔛 and ℌ). Then

T_{𝔜}(v)(u) = ⟨u, c_{𝔜}v⟩_{𝔜} = ⟨ι_{𝔜𝔛}u, ι_{𝔜𝔛}c_{𝔜}v⟩_{𝔛} + ∑_{j}⟨A_{j}u, A_{j}c_{𝔜}v⟩_{𝔛}

= ∑_{k} (⟨B_{k}ι_{𝔜𝔛}u, B_{k}ι_{𝔜𝔛}c_{𝔜}v⟩_{ℌ} + ∑_{j}⟨B_{k}A_{j}u, B_{k}A_{j}c_{𝔜}v⟩_{ℌ})

= ∑_{k} (j_{ℌ}B_{k}ι_{𝔜𝔛}c_{𝔜}v(B_{k}ι_{𝔜𝔛}u) + ∑_{j} j_{ℌ}B_{k}A_{j}c_{𝔜}v(B_{k}A_{j}u))

= (∑_{k} ((B_{k}ι_{𝔜𝔛})^{♭}c_{ℌ}B_{k}ι_{𝔜𝔛} + ∑_{j}(B_{k}A_{j})^{♭}c_{ℌ}B_{k}A_{j})c_{𝔜}v) (u),

so we conclude as above, using (B_{k}A_{j})^{♭} = A^{†}B_{k}^{♭}, etc., that

(A.7)  𝔜^{∗} = Ran ι_{𝔜𝔛}^{†} + ∑_{j} A_{j}^{†};

note that the same computation as in (A.1) with factors of ι omitted shows that A_{j}^{†} is the formal adjoint of c_{D}A_{j}c_{D}.

We now prove the density lemma mentioned above. We start by commuting bounded families of operators through products of first order ps.d.o’s.

Lemma A.1. Let r ≥ 1. For s ∈ ℕ, let J_{s} be the set of maps {j : {1, …, s} → {1, …, r}.

Suppose that A_{1}, …, A_{r} ∈ Ψ_{eb}^{1,0}(M), and for s ∈ ℕ and j ∈ J_{s}, let

A_{j} = A_{j_{1}} … A_{j_{s}}.

Then for k ∈ ℕ, j ∈ J_{k} and {Q_{n}} a uniformly bounded family in Ψ_{eb}^{0,0}(M),

A_{j}Q_{n} = Q_{n}A_{j} + ∑_{s≤k−1} ∑_{i∈J_{s}} C_{in}A_{i},

with {C_{in} : n ∈ ℕ} uniformly bounded in Ψ_{eb}^{0,0}(M), and the uniform bounds are microlocal (so in particular WF′({C_{in} : n ∈ ℕ}) ⊂ WF′({Q_{n} : n ∈ ℕ})).

Moreover, for ϵ > 0, if Q_{n} → Id in Ψ_{eb}^{ϵ,0}(M) then C_{in} → 0 in Ψ_{eb}^{ϵ,0}(M) as n → ∞.

Remark A.2. We do not need the microlocality of the uniform bounds below, but it is useful elsewhere.

Proof. We proceed by induction, with k = 0 being clear.

Suppose k ≥ 1, and the statement has been proved with k replaced by k − 1. Then for j ∈ J_{k},

A_{j}Q_{n} = Q_{n}A_{j} + [A_{j_{1}}, Q_{n}]A_{j_{2}} … A_{j_{k}} + … + A_{j_{1}} … A_{j_{k−1}}[A_{j_{k}}, Q_{n}].

Note that [A_{j_{m}}, Q_{n}] ∈ Ψ_{eb}^{0,0}(M) uniformly, and in a microlocal sense (and [A_{j_{m}}, Q_{n}] → 0 in Ψ_{eb}^{ϵ,0}(M) if Q_{n} → Id in Ψ_{eb}^{0,0}(M)). Thus, the first two terms are of the stated form. For the others, there are l ≤ k − 1 factors in front of the commutator, which is bounded in Ψ_{eb}^{0,0}(M) (and converges to 0 in Ψ_{eb}^{ϵ,0}(M) if Q_{n} → Id in Ψ_{eb}^{ϵ,0}(M)), so by the inductive hypothesis

A_{j_{1}} … A_{j_{l}}[A_{j_{l+1}}, Q_{n}]

can be rewritten as ∑_{s≤l} ∑_{i∈J_{s}} C_{s,in}A_{i}, hence

A_{j_{1}} … A_{j_{l}}[A_{j_{l+1}}, Q_{n}]A_{j_{l+2}} … A_{j_{k}}

is rewritten as C_{s,in}A_{i_{1}} … A_{i_{s}}A_{j_{l+2}} … A_{j_{k}} with s + (k − (l + 1)) ≤ l + k − (l + 1) = k − 1 factors of the A’s, hence is of the stated form. □

Lemma A.3. Suppose that B_{1}, …, B_{r} ∈ Ψ_{eb}^{1,0}(M), and let 𝔛 be a Hilbert space on which Ψ_{eb}^{0,0}(M) acts, with operator norm on 𝔛 bounded by a fixed Ψ_{eb}^{0,0}(M)-seminorm, and suppose that D ⊂ 𝔛 a dense subspace with a locally convex topology, and with all Q ∈ Ψ_{eb}^{−∞,0}(M), Q : 𝔛 → D continuous, while for all Q ∈ Ψ_{eb}^{m,0}(M), Q : D → 𝔛 is continuous, with bound given by a fixed Ψ_{eb}^{m,0}(M) seminorm and a fixed seminorm on D.

For k ∈ ℕ, let

𝔜 = {u ∈ 𝔛 : ∀s ≤ k, ∀j ∈ J_{s}, A_{j}u ∈ 𝔛}

with

∥u∥_{𝔜}^{2} = ∥u∥_{𝔛}^{2} + ∑_{s≤k} ∑_{j∈J_{s}} ∥A_{j}u∥_{𝔛}^{2},

Then D is dense in 𝔜.

Proof. We start by observing that if Q_{n} is a uniformly bounded family in Ψ_{eb}^{0,0}(M), Q ∈ Ψ_{eb}^{0,0}(M) and Q_{n} → Q in Ψ_{eb}^{ϵ,0}(M) for ϵ > 0, then Q_{n} → Q strongly on 𝔛. Indeed, Q_{n} is uniformly bounded on 𝔛 by the assumptions of the lemma, so it suffices to prove that for a dense subset of 𝔛, which we take to be D, u ∈ D implies Q_{n}u → Qu in 𝔛. But this is immediate, for Q_{n} → Q in Ψ_{eb}^{ϵ,0}(M), hence as a map D → 𝔛, by the assumptions of the lemma.

Now let Λ_{n} ∈ Ψ_{eb}^{−∞,0}(M) uniformly bounded in Ψ_{eb}^{0,0}(M) and Λ_{n} → Id in Ψ_{eb}^{ϵ,0}(M) for ϵ > 0, so Λ_{n} → Id strongly on 𝔛. We claim that for s ≤ k,

(A.8)  u ∈ 𝔜, j ∈ J_{s} ⇒ A_{j}Λ_{n}u → A_{j}u as n → ∞ in 𝔛.

Since Λ_{n}u ∈ D, this will prove the lemma. Note that Λ_{n}u → u in 𝔛.

By Lemma A.1, for j ∈ J_{s},

A_{j}Λ_{n} = Λ_{n}A_{j} + ∑_{l≤s−1} ∑_{i∈J_{l}} C_{in}A_{i},

with {C_{in} : n ∈ ℕ} uniformly bounded in Ψ_{eb}^{0,0}(M), C_{in} → 0 in Ψ_{eb}^{ϵ,0}(M) for ϵ > 0. Correspondingly, {C_{in} : n ∈ ℕ} is uniformly bounded as operators on 𝔛, and C_{in} → 0 strongly on 𝔛. Since A_{j}u ∈ 𝔛 and A_{i}u ∈ 𝔛 for i ∈ J_{l}, l ≤ s, we deduce that A_{j}Λ_{n}u → A_{j}u in 𝔛, completing the proof. □

Appendix B. The edge-b calculus

Let M be, for this section, a general compact manifold with corners and let W be one of its boundary hypersurfaces. In the body of the paper above, M is obtained by the blow up of a boundary face Y of a manifold with corners M_{0} and W is the front face of the blow up, i.e. the preimage of Y under the blow-down map. In fact the discussion is mostly local in the interior of Y and hence we could assume that Y has locally maximal codimension, so that it, and hence W has no boundary. We shall not make this assumption here and include the setting obtained by blow-up without actually restricting the discussion to it. Instead, we shall suppose that W is equipped with a fibration

(B.1)  [figure: Z; W; ϕ; Y.]

Since the manifolds here may have corners, this is to be a fibration in that sense, so the typical fiber, Z, is required to be a compact manifold with corners, the base, Y, is a manifold with corners and ϕ is supposed to be locally trivial in the sense that each p ∈ Y has a neighborhood U over which there is a diffeomorphism giving a commutative diagram

(B.2)  \xymatrix{ϕ^{−1}(U) \ar[rr]^{≃} \ar[dr]_{ϕ} &  & U × Z \ar[dl]^{π_{U}} \\  & U.}

Mainly for notational reasons we will also assume that Y is connected.

Let V_{eb}(M) ⊂ V_{b}(M) be the Lie subalgebra of all those smooth vector fields on M which are tangent to all boundary (hypersur-)faces and in addition are tangent to the fibers of ϕ on W. The calculus of edge-b pseudodifferential operators will be constructed in this setting, it is determined by M and ϕ and microlocalizes V_{eb}(M).

In case the ϕ has a single fiber, i.e. Z = W, corresponding to the case that W is not blown up at all, the Lie algebra V_{eb}(M) reduces to V_{b}(M) and the desired microlocalization is just the algebra of b-pseudodifferential operators on M as a manifold with corners. The construction of this algebra is discussed in [19], [22] and of course W is in no way singled out amongst the boundary hypersurfaces. The pseudodifferential operators are described in terms of their Schwartz kernels, which are the conormal distributions with respect to the resolved diagonal in a blown-up version of M^{2}, with the additional constraint of vanishing rapidly at boundary faces which do not meet the lifted diagonal. The resolved double space in this case is

(B.3)  M_{b}^{2} = [M^{2}; B], B = {B × B; B ∈ M_{1}(M)}.

Here, in generality, M_{p}(M) is the collection of all connected boundary faces of codimension p of the manifold with corners M. It is of crucial importance that the lift to M_{b}^{2} of the diagonal, is a p-submanifold – the lift in this case is the closure (in M_{b}^{2}) of the inverse image of the interior of the diagonal:

(B.4)  Diag_{b} = cl(Diag(M) ∩ int(M^{2})).

Then the operators on functions correspond to the kernels

(B.5)  Ψ^{m}(M) = {A ∈ I^{m}(M_{b}^{2}; β^{∗}Ω_{R}); A ≡ 0 at ∂M_{b}^{2} \ ff(β)}

where ff(β) is the collection of boundary faces produced by the blow-ups defining the combined blow-down map β : M_{b}^{2} ⟶ M^{2}.

The composition properties of these operators, including the fact that the ‘small calculus’ is an algebra, can be obtained geometrically from the corresponding triple space

(B.6) M_{b}^{3} = [M^{3}; B^{3}, B^{2}], B^{3} = {B^{3}; B ∈ M_{1}(M)},

B^{2} = {M × B × B, B × M × B, B × B × M; B ∈ M_{1}(M)}.

There is considerable freedom in the order of blow-ups here and this is sufficient to show that the three projections, π_{O}, from M^{3} to M^{2} lift to ‘stretched projections’ π_{O,b} : M_{b}^{3} ⟶ M_{b}^{2}, O = S, C, F, corresponding to the left, outer and right two factors respectively; these maps are b-fibrations and factor through a product of M_{b}^{2} and M in each case.

As already noted, it is crucial for the definition (B.5) that the lifted diagonal Diag_{b} be a p-submanifold, meaning that it meets the boundary locally as a product. This also turns out to be essential in the construction of M_{eb}^{2} below.

There is another extreme case in which the microlocalization of the Lie algebra V_{eb} is well-established, namely when W is the only boundary hypersurface, so M is a manifold with boundary; this is the case of an ‘edge’ alone, with no other boundaries. The construction of a geometric resolution in this case can be found in [16] and [17]. It is quite parallel to, and of course includes as a special case, the b-algebra on a manifold with boundary. In the general edge case when the fibration ϕ is non-trivial (but W itself has no boundary) the centre blown up in (B.3), which would be W^{2}, is replaced by the fiber diagonal

(B.7)

Diag_{ϕ} = {(p, p′) ∈ W^{2}; ϕ(p) = ϕ(p′)} = (ϕ × ϕ)^{−1}(Diag(F))
M_{ϕ}^{2} = [M^{2}; Diag_{ϕ}].

Similarly, the triple space is obtained by blow-up of the triple fiber product (B.8) Diag_{ϕ}^{3} = {(p, p′, p′′) ∈ W^{3}; ϕ(p) = ϕ(p′) = ϕ(p′′)} = (ϕ × ϕ × ϕ)^{−1}(Diag^{3}(F))

and then the three partial fiber diagonals, the inverse images, Diag_{ϕ}^{O}, O = S, C, F of Diag_{ϕ} under the three projections π_{O} : M^{3} ⟶ M^{2} :

(B.9)  M_{ϕ}^{3} = [M^{3}; Diag_{ϕ}^{3}; Diag_{ϕ}^{S}, Diag_{ϕ}^{C}, Diag_{ϕ}^{F}].

Again the three projections lift to b-fibrations

(B.10)  \xymatrix{M_{ϕ}^{3} \ar[r]^{π_{F,ϕ}} \ar[r]^{π_{S,ϕ}} \ar[r]|{π_{C,ϕ}} & M_{ϕ}^{2}.}

The microlocalization of V_{eb} is accomplished here by the combination of these two constructions. The diagonal, even for a manifold with boundary, is not a p-submanifold – does not meet the boundary faces in a product manner – and as already noted this is remedied by the b-resolution. Since the fiber diagonal in W^{2} is the inverse image of the diagonal in Y it too is not a p-submanifold in case Y has boundary, but then the partial b-resolution of M resolves it to a p-submanifold which can then be blown up.

More explicitly, the boundary hypersurfaces of M, other than W itself, fall into three classes according to their behavior relative to ϕ. Namely there may be some disjoint from W; these are relatively unimportant in the discussion below. Otherwise the intersection of W and such a boundary hypersurface, B, is a boundary hypersurface B ∩ W of W. The remaining two cases correspond to this being the preimage under ϕ of a boundary hypersurface of Y or, if this is not the case, then B ∩ W is the union of boundary hypersurfaces of the fibers of ϕ, corresponding to a fixed boundary hypersurface of Z. In brief, the boundary hypersurfaces B ∈ M_{1}(M) \ {W } which meet W correspond either to the boundary hypersurfaces of Y or of Z. Let B′, B(Z) and B(Y) ⊂ M_{1}(M) denote the three disjoints subsets into which M_{1}(M) \ {W } is so divided.

To define the double space on which the kernels are conormal distributions with respect to the lifted diagonal, just as in both special cases discussed above, we make one blow up for each of the boundary hypersurfaces. For those other than W, this is the same as for the b-double space for M, which is to say the corners, B × B, are to be blown up for all B ∈ M_{1}(M) \ {W }. Since these submanifolds are mutually transversal boundary faces within M^{2} they may be blown up in any order with the same final result. For W we wish to blow up the fiber diagonal, Diag_{ϕ}, in (B.7). This is certainly a manifold with corners, since it is the fiber product of W with itself as a bundle over Y, given by ϕ. However, as noted, it is not embedded as a p-submanifold if Y has non-trivial boundary. If x_{i} and y_{j} are respectively boundary defining functions and interior coordinates near some boundary point of Y, and x′, y′, x′′, y′′ are their local lifts to W^{2} under the two copies of ϕ, then Diag_{ϕ} ⊂ W^{2} is the ‘diagonal’ x′ = x′′, y′ = y′′. Near a boundary point of Y this is not a p-submanifold.

Lemma B.1. The fiber diagonal Diag_{ϕ} lifts to a p-submanifold of [M^{2}; B(Y)].

We will still denote the lifted submanifold as Diag_{ϕ} .

Proof. Since Diag_{ϕ} ⊂ W^{2} and this is the smallest boundary face of M^{2} with this property, under the blow up of other boundary faces of M^{2}, Diag_{ϕ} lifts to the subset (always a submanifold in fact) of the lift of W^{2} under the blow up of the intersection of W^{2} with the boundary face which is the center of the blow up. That is, to track the behavior of Diag_{ϕ} we need simply blow up the intersections of the elements of B(Y) with W^{2}, inside W^{2}. This corresponds to exactly the ‘boundary resolution’ of Y^{2} to Y_{b}^{2} as discussed briefly above. So the diagonal in Y lifts to be a p-submanifold. Since ϕ is a fibration over Y, it follows easily from the local description that Diag_{ϕ} lifts to a p-submanifold of the blow up, [W^{2}; B(Y) ∩ W^{2}] and hence to a p-submanifold of [M^{2}; B(Y)] as claimed. □

Thus blowing up the elements of B(Y) in M^{2} resolves Diag_{ϕ} to a psubmanifold, by resolving the diagonal in Y^{2}. The defining functions of the elements of B(Z) restrict to defining functions of boundary faces of the lift of Diag_{ϕ} so all the remaining boundary faces, in B′ ∪B(Z) are transversal to this lift. Such transversality is preserved under blow up of boundary faces, so we may define the eb-double space in several equivalent ways as regards the order of the blow-ups and in particular:-

(B.11)

M_{eb}^{2} =[M^{2}; B^{2}, Diag_{ϕ}]
≡[M^{2}; B(Y)^{2}, Diag_{ϕ}, B(Z)^{2}, (B′)^{2}],

where the ‘squares’ mean the set of self-products of the elements and the ordering within the boundary faces is immaterial.

The fibration ϕ restricts to a fibration, ϕ_{B}, of B ∩ W for each B ∈ B(Z), over the same base Y. For each B ∈ B(Y) instead ϕ restricts to B ∩ W to a fibration, again denoted ϕ_{B}, over Y (B), the corresponding boundary hypersurface of Y. Thus considering B ∈ B(Z) or B ∈ B(Y) as manifolds with corners on their own, each inherits a fibration structure as initially given on W ⊂ M on the intersection B ∩ W ∈ M_{1}(B). For the elements of B′ there is a corresponding trivial structure with no W.

Lemma B.2. The diagonal in M^{2} lifts to a p-submanifold of M_{eb}^{2} and the ‘front faces’ of M_{eb}^{2}, those boundary hypersurfaces produced by blow up, are of the form B_{eb}^{2} × [0, 1], corresponding to each B ∈ B̃ = M_{1}(M) \ {W } with its induced fibration structure, and that corresponding to Diag_{ϕ} is the pullback of the bundle [W^{2}; (B(Z) ∩ (W))^{2}], defined by blowing up the diagonal corners of the fibers, to a (closed) quarter ball bundle over Y.

Proof. These statements are all local and follow by elementary computations in local coordinates. □

Thus, the definition of the ‘small’ calculus of edge-b pseudodifferential operators is directly analogous to (and of course extends in generality) (B.5):

(B.12)  Ψ_{eb}^{m}(M) = {A ∈ I^{m}(M_{eb}^{2}; β^{∗}Ω_{R}); A ≡ 0 at ∂M_{b}^{2} \ ff(β)}

where the particular fibration ϕ is not made explicit in the notation. The fact that these kernels define operators on Ċ^{∞}(M) and C^{∞}(M) reduces to the fact that push-forward off the right factor of M, which is to say under the left projection, gives a continuous map

(B.13)  (π_{L,ϕ})_{∗} : Ψ_{eb}^{m}(M) ⟶ C^{∞}(M).

The principal symbol map is well-defined at the level of conormal distributions, taking values in the smooth homogeneous fiber-densities of the non-zero part of the conormal bundle to the submanifold in question. In this case N^{∗} Diag_{eb} = ^{eb}T^{∗}M is a natural identification and the density factors cancel as in the standard case so

(B.14)  σ_{m} : Ψ_{eb}^{m}(M) ⟶ C^{∞}(^{eb}S^{∗}M; N_{m})

where N_{m} is the bundle of functions which are homogeneous of degree −m.

The structure of the front faces leads directly to the ‘symbolic’ structure of the (small) algebra of pseudodifferential operators. Namely, there are homomorphisms to model operator algebras corresponding to each boundary face of M. For those other than W the model is a parameterized (‘suspended’) family of edge-b (or for those boundary faces not meeting W plain b-) operators corresponding to the fibrations of boundary hypersurfaces of W and for W the model is

The corresponding triple space can be defined by essentially the same modifications to the construction of M_{b}^{3} as correspond to obtaining M_{eb}^{2} in place of M_{b}^{2}.

Lemma B.3. Under the blow-down map for the partial triple b-product

(B.15) β3 ; B̃ : M 3 ; B̃ = [M^{3}; B(Y)^{3}; B(Y)^{2}; B(Z)^{3}; (B′)^{3}; B(Z)^{2}, (B′)^{2}] ⟶ M^{3}

the triple fiber diagonal and the three partial fiber diagonals

(B.16)

Diag_{ϕ}^{3} = {(p, p′, p′′) ∈ W^{3}; ϕ(p) = ϕ(p′) = ϕ(p′′)},
Diag_{ϕ}^{O} = (π_{O})^{−1}(Diag_{ϕ}), O = S, C, F,

all lift to p-submanifolds.

Proof. This reduces to the same argument as above, namely that the triple and three partial diagonals in Y^{3} are resolved to p-submanifolds in Y_{b}^{3} and the effect of the first two sets of blow-ups in (B.15) on Y^{3} is to replace it by Y_{b}^{3} and hence to resolve the submanifolds in (B.16). Under the subsequent blow-ups of boundary faces any p-submanifold lifts to a p-submanifold. □

Thus we may define the edge-b triple space to be

(B.17)  M_{eb}^{3} = [M 3 ; B̃; Diag_{ϕ}^{3}; Diag_{ϕ}^{S}; Diag_{ϕ}^{C}; Diag_{ϕ}^{F}].

Proposition B.4. The three partial diagonals lift to b-submanifolds intersecting in the lifted triple diagonal and the three projections lift to b-fibrations

(B.18)  \xymatrix{M_{eb}^{3} \ar[r]^{π_{F,eb}} \ar[r]^{π_{S,eb}} \ar[r]|{π_{C,eb}} & M_{ϕ}^{2}.}

where π_{O,eb} is transversal to the other two lifted diagonals.

Proof. The existence of the stretched projections as smooth maps follows from the possibility of commutation of blow-ups. For the sake of definiteness, concentrate on π_{F} , the projection onto the right two factors.

After the blow up of the triple fiber diagonal in (B.17), the three partial fiber diagonals are disjoint so the other two can be blown up last. When it is to be blown up, the triple fiber diagonal is a submanifold of Diag_{ϕ}^{F} so the order can be exchanged, showing that there is a composite blown-down map

(B.19)  M_{eb}^{3} ⟶ [M 3 ; B̃; Diag_{ϕ}^{F}].

The manifold with corners M 3 ; B̃ is the b-resolved triple product where the boundary hypersurface W is ignored. The commutation arguments showing the existence of a composite blow-down map M_{b}^{3} ⟶ M × M_{b}^{2} carry over directly to give an alternative construction

(B.20)  M 3 ; B̃ = [M × M 2 ; B̃; F]

where F consists of those boundary faces in (B.15) which involve a defining function on the first factor of M^{3} – so all the triple products and the double products with boundary hypersurface in the first factor. These are all transversal to Diag_{ϕ}^{F} , realized as a p-submanifold of M × M 2 ; B̃ so can be commuted past it in the blow up, giving the map π_{F,ϕ} in (B.18). That it is a b-submersion follows from its definition as a composite of blow-downs of boundary faces, together with the corresponding fact for the edge case. That it is a b-fibration follows from the fact that the image of a boundary hypsersurface is either a boundary hypersurface or the whole manifold since this is true locally in the interior of boundary hypersurfaces. □

These facts together show that the small calculus of edge-b pseudodifferential operators, as defined in (B.12) is a filtered algebra. It also follows directly that the symbol (B.14) is multiplicative as in the standard case. The extension to operators on sections of bundles is essentially notational.

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