CHAPTER 4 Wavefront set

We have now studied special distributions, the Schwartz kernels of pseudodifferential operators, to be able to apply them to the study of general distributions. Proceeding with this general idea, I shall next introduce wavefront set, a refinement of singular support, of general distributions. This notion is fundamental to the general idea of ‘microlocalization.’

4.1. Characteristic variety

If A ∈ Ψ_{∞}^{m}(ℝ^{n}), the left-reduced symbol is elliptic at (\overline{x}, \overline{ξ}) ∈ ℝ^{n} × (ℝ^{n}\\{0\}) if there exists ϵ > 0 such that

\begin{matrix}| σ_{L}(A)(x, ξ) |≥ ϵ|ξ|^{m} in \\ \{(x, ξ) ∈ ℝ^{n} × (ℝ^{n}\\{0\}) ; |x − \overline{x}| ≤ ϵ, | \frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|ξ|} |≤ ϵ, |ξ| ≥ \frac{1}{ϵ}\}. \end{matrix} (4.1)

Directly from the definition, ellipticity at (\overline{x}, \overline{ξ}) is actually a property of the principal symbol, σ_{m}(A) and if A is elliptic at (\overline{x}, \overline{ξ}) then it is elliptic at (\overline{x}, t\overline{ξ}) for any t > 0. Clearly

\{(\overline{x}, \overline{ξ}) ∈ ℝ^{n} × (ℝ^{n}\\{0\}); A is elliptic (of order m) at (\overline{x}, \overline{ξ})\}

is an open cone in ℝ^{n} × (ℝ^{n}\{0}). The complement

(4.2) Σ_{m}(A) = \{(\overline{x}, \overline{ξ}) ∈ ℝ^{n} × (ℝ^{n}\\{0\}) ; A is not elliptic of order m at (\overline{x}, \overline{ξ})\}

is therefore a closed conic subset of ℝ^{n} × (ℝ^{n}\{0}) ; it is the characteristic set (or variety) of A. From the composition properties of pseudodifferential operators

Σ_{m+m′}(A ∘ B) = Σ_{m}(A) ∪ Σ_{m′}(B). (4.3)

4.2. Wavefront set

We adopt the following bald definition:

If u ∈ C_{c}^{−∞}(ℝ^{n}) = {u ∈ S′(ℝ^{n}); supp(u) ⋐ ℝ^{n}} then

WF(u) = ⋂{Σ_{0}(A); A ∈ Ψ_{∞}^{0}(ℝ^{n}) and Au ∈ C^{∞}(ℝ^{n})}. (4.4)

Thus WF(u) ⊂ ℝ^{n} × (ℝ^{n}\{0}) is always a closed conic set, being the intersection of such sets. The first thing we wish to show is that WF(u) is a refinement of sing supp(u). Let

π : ℝ^{n} × (ℝ^{n}\{0}) ∋ (x, ξ) ⟼ x ∈ ℝ^{n} (4.5)

be projection onto the first factor.

Proposition 4.1. If u ∈ C_{c}^{−∞}(ℝ^{n}) then

(4.6) π(WF(u)) = sing supp(u).

Proof. The inclusion π(WF(u)) ⊂ sing supp(w) is straightforward. Indeed, if \overline{x} ∉ sing supp(u) then there exists ϕ ∈ C_{c}^{∞}(ℝ^{n}) with ϕ(\overline{x}) ≠ 0 such that ϕu ∈ C^{∞}(ℝ^{n}). Of course as a multiplication operator, ϕ ∈ Ψ_{∞}^{0}(ℝ^{n}) and Σ_{0}(ϕ) ∌ (\overline{x}, \overline{ξ}) for any \overline{ξ} ≠ 0. Thus the definition (4.4) shows that (\overline{x}, \overline{ξ}) ∉ WF(u) for all \overline{ξ} ∈ ℝ^{n}∖0 proving the inclusion.

Using the calculus of pseudodifferential operators, the opposite inclusion,

π(WF(u)) ⊃ sing supp(u)

is only a little more complicated. Thus we have to show that if (\overline{x}, \overline{ξ}) ∉ WF(u) for all \overline{ξ} ∈ ℝ^{n}∖0 then \overline{x} ∉ sing supp(u). The hypothesis is that for each (\overline{x}, \overline{ξ}), \overline{ξ} ∈ ℝ^{n}∖0, there exists A ∈ Ψ_{∞}^{0}(ℝ^{n}) such that A is elliptic at (\overline{x}, \overline{ξ}) and Au ∈ C^{∞}(ℝ^{n}). The set of elliptic points is open so there exists ϵ = ϵ(\overline{ξ}) > 0 such that A is elliptic on

\{(x, ξ) ∈ ℝ^{n} × (ℝ^{n}∖0); |x − \overline{x}| < ϵ, | \frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|\overline{ξ}|} |< ϵ\}. (4.7)

Let B_{j}, j = 1, …, N be a finite set of such operators associated to \overline{ξ}_{j} and such that the corresponding sets in (4.7) cover \{\overline{x}\} × (ℝ^{n}∖0); the finiteness follows from the compactness of the sphere. Then consider

B = ∑_{j=1}^{N} B_{j}^{∗}B_{j} ⟹ Bu ∈ C^{∞}(ℝ^{n}).

This operator B is elliptic at (\overline{x}, ξ), for all ξ ≠ 0. Thus if ϕ ∈ C_{c}^{∞}(ℝ^{n}), 0 ≤ ϕ(x) ≤ 1, has support sufficiently close to \overline{x}, ϕ(x) = 1 in |x − \overline{x}| < ϵ/2 then, since B has nonnegative principal symbol

B + (1 − ϕ) ∈ Ψ_{∞}^{0}(ℝ^{n})

is globally elliptic. Thus, by Lemma 3.4, there exists G ∈ Ψ_{∞}^{0}(ℝ^{n}) which is a parametrix for B + (1 − ϕ) :

Id = G ∘ B + G(1 − ϕ). (4.8)

Let ψ ∈ C_{c}^{∞}(ℝ^{n}) be such that supp(ψ) ⊂ \{ϕ = 1\} and ψ(\overline{x}) ≠ 0. Then, from the reduction formula

ψ ∘ G ∘ (1 − ϕ) ∈ Ψ_{∞}^{−∞}(ℝ^{n}).

Thus from (4.8) we find

ψu = ψG ∘ Bu + ψG(1 − ϕ)u ∈ C^{∞}(ℝ^{n}).

Thus \overline{x} ∉ sing supp(u) and the proposition is proved.

We extend the definition to general tempered distributions by setting

WF(u) = ⋃_{ϕ∈C_{c}^{∞}(ℝ^{n})} WF(ϕu), u ∈ S′(ℝ^{n}). (4.9)

Then (4.6) holds for every u ∈ S′(ℝ^{n}).

4.3. Essential support

Next we shall consider the notion of the essential support of a pseudodifferential operator. If a ∈ S_{∞}^{m}(ℝ^{N}; ℝ^{n}) we define the cone support of a by

cone supp(a) =\{(\overline{x}, \overline{ξ}) ∈ ℝ^{N} × (ℝ^{n}∖0); ∃ ϵ > 0 and ∀ M ∈ ℝ, ∃ C_{M} s.t.

|a(x, ξ)| ≤ C_{M}⟨ξ⟩^{−M} if |x − \overline{x}| ≤ ϵ, \{| \frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|\overline{ξ}|} ≤ ϵ\}^{∁}.

(4.10)

This is clearly a closed conic set in ℝ^{N} × (ℝ^{n}∖0). By definition the symbol decays rapidly outside this cone, in fact even more is true.

Lemma 4.1. If a ∈ S_{∞}^{∞}(ℝ^{N}; ℝ^{n}) then
(\overline{x}, \overline{η}) ∉ cone supp(a) ⟹
∃ ϵ > 0 s.t. ∀ M, α, β ∃ C_{M} with

| D_{x}^{α}D_{ξ}^{β}a(x, η) |≤ C_{M}⟨η⟩^{−M} if |x − \overline{x}| < ϵ, | \frac{η}{|η|} − \frac{\overline{η}}{|\overline{η}|} |< ϵ.

(4.11)

Proof. To prove (4.11) it suffices to show it to be valid for D_{x_{j}}a, D_{ξ_{k}}a and then use an inductive argument, i.e. to show that

(4.12) cone supp(D_{x_{j}}a), cone supp(D_{ξ_{k}}a) ⊂ cone supp(a).

Arguing by contradiction suppose that D_{x_{ℓ}}a does not decay to order M in any cone around (\overline{x}, \overline{η}) ∉ cone supp . Then there exists a sequence (x_{j}, η_{j}) with

\begin{cases}x_{j} ⟶ \overline{x}, | \frac{η_{j}}{|η_{j}|} − \frac{\overline{η}}{|\overline{η}|} |⟶ 0, |η_{j}| ⟶ ∞ \\ and | D_{x_{ℓ}}a(x_{j}, η_{j}) |> j⟨η_{j}⟩^{M}. \end{cases} (4.13)

We can assume that M < m, since a ∈ S_{∞}^{m} (ℝ^{n}; ℝ^{N}) . Applying Taylor’s formula with remainder, and using the symbol bounds on D_{x_{j}}^{2}a, gives

a(x_{j} + te_{ℓ}, ξ_{j}) = a(x_{j}, ξ_{j}) + it(D_{x_{j}}a)(x_{j}, ξ_{j}) + O (t^{2}⟨ξ_{j}⟩^{m}) , (e_{ℓ})_{j} = δ_{ℓj} (4.14)

providing |t| < 1. Taking t = ⟨ξ_{j}⟩^{M−m} ⟶ 0 as j ⟶ ∞, the first and third terms on the right in (4.14) are small compared to the second, so

| a (x_{j} + ⟨ξ_{j}⟩^{\frac{M−m}{2}} , ξ_{j}) |> ⟨ξ_{j}⟩^{2M−m},

contradicting the assumption that (\overline{x}, \overline{ξ}) ∉ cone supp(a). A similar argument applies to D_{ξ_{ℓ}}a so (4.11), and hence the lemma, is proved.

For a pseudodifferential operator we define the essential support by

(4.15) WF′(A) = cone supp (σ_{L}(A)) ⊂ ℝ^{n} × (ℝ^{n}∖0) .

Lemma 4.2. For every A ∈ Ψ_{∞}^{m}(ℝ^{n})

(4.16) WF′(A) = cone supp(σ_{R}(A)).

Proof. Using (4.11) and the formula relating σ_{R}(A) to σ_{L}(A) we conclude that

(4.17) cone supp(σ_{L}(A)) = cone supp(σ_{R}(A)),

from which (4.16) follows.

58 4. WAVEFRONT SET

A similar argument shows that WF′(A ∘ B) ⊂ WF′(A) ∩ WF′(B). (4.18) Indeed the asymptotic formula for σ_{L}(A ∘ B) in terms of σ_{L}(A) and σ_{L}(B) shows that (4.19) cone supp(σ_{L}(A ∘ B)) ⊂ cone supp (σ_{L}(A)) ∩ cone supp (σ_{L}(B)) which is the same thing.

4.4. Microlocal parametrices The concept of essential support allows us to refine the notion of a parametrix

for an elliptic operator to that of a microlocal parametrix.

Lemma 4.3. If A ∈ Ψ_{∞}^{m}(ℝ^{n}) and z ∉ Σ_{m}(A) then there exists a microlocal parametrix at z, B ∈ Ψ_{∞}^{−m}(ℝ^{n}) such that (4.20) z ∉ WF′(Id −AB) and z ∉ WF′(Id −BA).

Proof. If z = (\overline{x}, \overline{ξ}), \overline{ξ} ≠ 0, consider the symbol

γ_{ϵ}(x, ξ) = ϕ (\frac{x − \overline{x}}{ϵ}) (1 − ϕ)(ϵξ)ϕ ((\frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|\overline{ξ}|})/ϵ)

where as usual ϕ ∈ C_{c}^{∞}(ℝ^{n}), ϕ(ζ) = 1 in |ζ| ≤ \frac{1}{2}, ϕ(ζ) = 0 in |ζ| ≥ 1. Thus γ_{ϵ} ∈ S_{∞}^{0} (ℝ^{n}; ℝ^{n}) has support in

|x − \overline{x}| ≤ ϵ, |ξ| ≥ \frac{1}{2ϵ}, |\frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|\overline{ξ}|}| ≤ ϵ (4.21)

and is identically equal to one, and hence elliptic, on a similar smaller set

|x − \overline{x}| < \frac{ϵ}{2}, |ξ| ≥ \frac{1}{ϵ}, |\frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|\overline{ξ}|}| ≤ \frac{ϵ}{2}. (4.22)

Define L_{ϵ} ∈ Ψ_{∞}^{0}(ℝ^{n}) by σ_{L}(L_{ϵ}) = γ_{ϵ}. Thus, for any ϵ > 0,

(4.23) z ∉ WF′(Id −L_{ϵ}), WF′(L_{ϵ}) ⊂ \{(x, ξ); |x − \overline{x}| ≤ ϵ and |\frac{ξ}{|ξ|} − \frac{\overline{ξ}}{|\overline{ξ}|}| ≤ ϵ\} .

Let G_{2m} ∈ Ψ_{∞}^{2m}(ℝ^{n}) be a globally elliptic operator with positive principal symbol. For example take σ_{L}(G_{2m}) = (1 + |ξ|^{2})^{m}, so G_{s} ∘ G_{t} = G_{s+t} for any s, t ∈ ℝ. Now consider the operator J = (Id −L_{ϵ}) ∘ G_{2m} + A^{∗}A ∈ Ψ_{∞}^{2m}(ℝ^{n}). (4.24) The principal symbol of J is (1 − γ_{ϵ})(1 + |ξ|^{2})^{m} + |σ_{m}(A)|^{2} which is globally elliptic if ϵ > 0 is small enough (so that σ_{m}(A) is elliptic on the set (4.21). According to Lemma 3.1, J has a global parametrix H ∈ Ψ_{∞}^{−2m}(ℝ^{n}). Then B = H ∘ A^{∗} ∈ Ψ_{∞}^{−m}(ℝ^{n}) (4.25) is a microlocal right parametrix for A in the sense that B ∘ A − Id = R_{R} with z ∉ WF′(R_{R}) since (4.26) R_{R} = B ∘ A − Id = H ∘ A^{∗} ∘ A − Id

= (H ∘ J − Id) + H ∘ (Id −L_{ϵ})G_{2m} ∘ A and the first term on the right is in Ψ_{∞}^{−∞}(ℝ^{n}) whilst z is not in the operator wavefront set of (Id −L_{ϵ}) and hence not in the operator wavefront set of the second term.

By a completely analogous construction we can find a left microlocal parametrix. Namely (Id −L_{ϵ}) ∘ G_{2m} + A ∘ A^{∗} is also globally elliptic with parametrix H′ and then B′ = A^{∗} ∘ H′ satisfies

B′ ∘ A − Id = R_{L}, z ∉ WF′(R_{L}).

Then, as usual,

B = (B′ ∘ A − R_{L}) B = B′ (A ∘ B) − R_{L}B = B′ + B′R_{R} − R_{L}B (4.27)

so z ∉ WF′(B − B′), which implies that B is both a left and right microlocal parametrix.

In fact this argument shows that such a left parametrix is essentially unique. See Problem 4.20.

4.5. Microlocality

Now we can consider the relationship between these two notions of wavefront set.

Proposition 4.2. Pseudodifferential operators are microlocal in the sense that

WF(Au) ⊂ WF′(A) ∩ WF(u) ∀ A ∈ Ψ_{∞}^{∞}(ℝ^{n}), u ∈ C_{c}^{−∞}(ℝ^{n}). (4.28)

Proof. We need to show that

(4.29) WF(Au) ⊂ WF′(A) and WF(Au) ⊂ WF(u).

the second being the usual definition of microlocality. The first inclusion is easy. Suppose (\overline{x}, \overline{ξ}) ∉ cone supp σ_{L}(A). If we choose B ∈ Ψ_{∞}^{0}(ℝ^{n}) with σ_{L}(B) supported in a small cone around (\overline{x}, \overline{ξ}) then we can arrange

(\overline{x}, \overline{ξ}) ∉ Σ_{0}(B), WF′(B) ∩ WF′(A) = ∅.

Then from (4.18), WF′(BA) = ∅ so BA ∈ Ψ_{∞}^{−∞}(ℝ^{n}) and BAu ∈ C^{∞}(ℝ^{n}). Thus (\overline{x}, \overline{ξ}) ∉ WF(Au).

Similarly suppose (\overline{x}, \overline{ξ}) ∉ WF(u). Then there exists G ∈ Ψ_{∞}^{0}(ℝ^{n}) which is elliptic at (\overline{x}, \overline{ξ}) with Gu ∈ C^{∞}(ℝ^{n}). Let B be a microlocal parametrix for G at (\overline{x}, \overline{ξ}) as in Lemma 4.3. Thus

u = BGu + Su, (\overline{x}, \overline{ξ}) ∉ WF′(S). (4.30)

Now apply A to this identity. Since, by assumption, Gu ∈ C_{c}^{∞}(ℝ^{n}) the first term on the right in

Au = ABGu + ASu (4.31)

is smooth. Since, by (4.18), (\overline{x}, \overline{ξ}) ∉ WF′(AS) it follows from the first part of the argument above that (\overline{x}, \overline{ξ}) ∉ WF(ASu) and hence (\overline{x}, \overline{ξ}) ∉ WF(Au).

We can deduce from the existence of microlocal parametrices at elliptic points a partial converse of (8.24).

Proposition 4.3. For any u ∈ C^{−∞}(ℝ^{n}) and any A ∈ Ψ_{∞}^{m}(ℝ^{n})

WF(u) ⊂ WF(Au) ∪ Σ_{m}(A). (4.32)

Proof. If (\overline{x}, \overline{ξ}) ∉ Σ_{m}(A) then, by definition, A is elliptic at (\overline{x}, \overline{ξ}). Thus, by Lemma 4.3, A has a microlocal parametrix B, so

u = BAu + Su, (\overline{x}, \overline{ξ}) ∉ WF′(S). (4.33)

It follows that (\overline{x}, \overline{ξ}) ∉ WF(Au) implies that (\overline{x}, \overline{ξ}) ∉ WF(u) proving the Proposition.

4.6. Explicit formulations

From this discussion of WF′(A) we can easily find a ‘local coordinate’ formulations of WF(u) in general.

Lemma 4.4. If (\overline{x}, \overline{ξ}) ∈ ℝ^{n} × (ℝ^{n} ∖0) and u ∈ S′(ℝ^{n}) then (\overline{x}, \overline{ξ}) ∉ WF(u) if and only if there exists ϕ ∈ C_{c}^{∞}(ℝ^{n}) with ϕ(\overline{x}) ≠ 0 such that for some ϵ > 0, and for all M there exists C_{M} with

| ϕ̂u(ξ) |≤ C_{M}⟨ξ⟩^{M} in | \frac{ξ}{|ξ|} − \frac{ξ̄}{|ξ|} |< ϵ. (4.34)

Proof. If ζ ∈ C^{∞}(ℝ), ζ(ξ) ≡ 1 in |ξ| < \frac{ϵ}{2} and supp(ζ) ⊂ [\frac{−3ϵ}{4}, \frac{3ϵ}{4}] then

γ(ξ) = (1 − ζ)(ξ) · ζ(\frac{ξ}{|ξ|} − \frac{\overline{x}}{|\overline{x}|}) ∈ S_{∞}^{0}(ℝ^{n})

is elliptic at ξ̄ and from (4.34)

γ(ξ) · ϕ̂u(ξ) ∈ S(ℝ^{n}).

Thus if σ_{R}(A) = ϕ_{1}(x)γ(ξ) then A(ϕ_{2}u) ∈ C^{∞} where ϕ_{1}ϕ_{2} = ϕ, ϕ_{1}(\overline{x}), ϕ_{2}(\overline{x}) ≠ 0, ϕ_{1}, ϕ_{2} ∈ C_{c}^{∞}(ℝ^{n}). Thus (\overline{x}, \overline{ξ}) ∉ WF(u). Conversely if (\overline{x}, \overline{ξ}) ∉ WF(u) and A is chosen as above then A(ϕ_{1}u) ∈ S(ℝ^{n}) and Lemma 4.4 holds.

4.7. Wavefront set of K_{A}

At this stage, a natural thing to look at is the wavefront set of the kernel of a pseudodifferential operator, since these kernels are certainly an interesting class of distributions.

Proposition 4.4. If A ∈ Ψ_{∞}^{m}(ℝ^{n}) then

\begin{matrix}WF(K_{A}) = \{ & (x, y, ξ, η) ∈ ℝ^{2n} × (ℝ^{2n}∖0); \\ x = y, ξ + η = 0 and (x, ξ) ∈ WF′(A)\}. \end{matrix} (4.35)

In particular this shows that WF′(A) determines WF(K_{A}) and conversely.

Proof. Using Proposition 4.1 we know that π (WF(K_{A})) ⊂ {(x, x)} so

WF(K_{A}) ⊂ {(x, x; ξ, η)}.

To find the wave front set more precisely consider the kernel

K_{A}(x, y) = (2π)^{−n} ∫ e^{i(x−y)·ξ}b(x, ξ)dξ

where we can assume |x − y| < 1 on supp(K_{A}). Thus is ϕ ∈ C_{c}^{∞}(X) then

g(x, y) = K_{A}(x, y) ∈ C_{c}^{−∞}(ℝ^{n})

and

(4.36)  ĝ(ζ, η) = (2π)^{−n} ∫ e^{−iζx−iηy}e^{i(x−y)·ζ}(ϕb)(x, ξ)dζdxdy

= ∫ e^{−i(ζ+η)·x}(ϕb)(x, −η)dx (4.37)

= ϕ̂b(ζ + η, −η). (4.38)

The fact that ϕb is a symbol of compact support in x means that for every M

| ϕ̂b(ζ + η, −η) |≤ C_{M} (⟨ζ + η⟩)^{−M} ⟨η⟩^{m}.

This is rapidly decreasing if ζ ≠ −η, so

WF(K_{A}) ⊂ {(x, x, η, −η)} as claimed.

Moreover if (\overline{x}, \overline{η}) ∉ WF′(A) then choosing ϕ to have small support near \overline{x} makes ϕ̂b rapidly decreasing near −\overline{η} for all ζ. This proves Proposition 4.4.

4.8. Elementary calculus of wavefront sets

We want to achieve a reasonable understanding, in terms of wavefront sets, of three fundamental operations. These are

Pull-back: F^{∗}u

Push-forward: F_{∗}u and Multiplication: u_{1} · u_{2}.

In order to begin to analyze these three operations we shall first introduce and discuss some other more “elementary” operations:

Pairing: (u, v) ⟶ ⟨u, v⟩ = ∫ u(x)\overline{v(x)}dx (4.39)

(4.40) Projection: u(x, y) ⟼ ∫ u(x, y)dy

(4.41) Restriction: u(x, y) ⟼ u(x, 0)

(4.42) Exterior product: (u, v) ⟼ (u ⊠ v)(x, y) = u(x)v(y)

(4.43) Invariance: F^{∗}u, for F a diffeomorphism.

Here (4.41) and (4.43) are special cases of (4.8), (4.40) of (4.8) and (4.42) is a combination of (4.8) and (4.8). Conversely the three fundamental operations can be expressed in terms of these elementary ones. We can give direct definitions of the latter which we then use to analyze the former. We shall start with the pairing in (4.39).

4.9. Pairing

We know how to ‘pair’ a distribution and a C^{∞} function. If both are C^{∞} and have compact supports then

⟨u_{1}, u_{2}⟩ = ∫ u_{1}(x)\overline{u_{2}(x)}dx (4.44)

and in general this pairing extends by continuity to either C_{c}^{−∞}(ℝ^{n}) × C^{∞}(ℝ^{n}) or C^{∞}(ℝ^{n}) × C_{c}^{−∞}(ℝ^{n}) Suppose both u_{1} and u_{2} are distributions, when can we pair them?

62 4. WAVEFRONT SET

Proposition 4.5. Suppose u_{1}, u_{2} ∈ C_{c}^{−∞}(ℝ^{n}) satisfy

WF(u_{1}) ∩ WF(u_{2}) = ∅ (4.45) then if A ∈ Ψ_{∞}^{0}(ℝ^{n}) has (4.46) WF(u_{1}) ∩ WF′(A) = ∅, WF(u_{2}) ∩ WF′(Id −A^{∗}) = ∅ the bilinear form ⟨u_{1}, u_{2}⟩ = ⟨Au_{1}, u_{2}⟩ + ⟨u_{1}, (Id −A^{∗})u_{2}⟩ (4.47) is independent of the choice of A.

Notice that A satisfying (4.46) does indeed exist, just choose a ∈ S_{∞}^{0} (ℝ^{n}; ℝ^{n}) to be identically 1 on WF(u_{2}), but to have cone supp(a) ∩ WF(u_{1}) = ∅, possible because of (4.45), and set A = q_{L}(a).

Proof. Of course (4.47) makes sense because Au_{1}, (Id −A^{∗})u_{2} ∈ C^{∞}(ℝ^{n}). To prove that this definition is independent of the choice of A, suppose A′ also satisfies (4.46). Set

⟨u_{1}, u_{2}⟩′ = ⟨A′u_{1}, u_{2}⟩ + ⟨u_{1}, (Id −A′)^{∗}u_{2}⟩.

Then

WF′(A − A′) ∩ WF(u_{1}) = WF′((A − A′)^{∗}) ∩ WF(u_{2}) = ∅.

The difference can be written ⟨u_{1}, u_{2}⟩′ − ⟨u_{1}, u_{2}⟩ = ⟨(A − A′)u_{1}, u_{2}⟩ − ⟨u_{1}, (A − A′)^{∗}u_{2}⟩. (4.48) Naturally we expect this to be zero, but this is not quite obvious since u_{1} and u_{2} are both distributions. We need an approximation argument to finish the proof.

Choose B ∈ Ψ_{∞}^{0}(ℝ^{n}) with

WF′(B) ∩ WF(u_{1}) = WF′(B) ∩ WF(u_{2}) = ∅

WF′(Id −B) ∩ WF(A − A′) = ∅ (4.49)

If v_{n} ⟶ u_{2}, in C_{c}^{−∞}(ℝ^{n}), v_{n} ∈ C_{c}^{∞}(ℝ^{n}) then

w_{n} = ϕ[(Id −B) v_{n} + Bu_{2}] ⟶ u_{2}

if ϕ ≡ 1 in a neighbourhood of supp(u_{2}), ϕ ∈ C_{c}^{∞}(ℝ^{n}). Here Bu_{2} ∈ C^{∞}(ℝ^{n}), so (A − A′)w_{n} = (A − A′)ϕ(Id −B) · v_{n} + (A − A′)ϕBu_{2} ⟶ (A − A′)u_{2} in C^{∞}(ℝ^{n}), since (A − A′)ϕ(Id −B) ∈ Ψ_{∞}^{−∞}(ℝ^{n}). Thus

⟨(A − A′) u_{1}, u_{2}⟩ ⟶ ⟨(A − A′) u_{1}, u_{2}⟩

⟨u_{1}, (A − A′)^{∗} w_{n}⟩ ⟶ ⟨u_{1}, (A − A′)^{∗} u_{2}⟩,

since w_{n} ⟶ u_{2} in C_{c}^{−∞}(ℝ^{n}) and (A − A′)^{∗}w_{n} ⟶ (A − A′)^{∗} u_{2} in C^{∞}(ℝ^{n}). Thus

⟨u_{1}, u_{2}⟩′ − ⟨u_{1}, u_{2}⟩ = lim_{n→∞}[⟨(A − A′) u_{1}, w_{n}⟩ − ⟨u_{1}, (A − A′)^{∗} w_{n}] = 0.

Here we are using the complex pairing. If we define the real pairing by (u_{1}, u_{2}) = ⟨u_{1}, \overline{u}_{2}⟩ (4.50) then we find

4.11. PROJECTION 63

Proposition 4.6. If u_{1}, u_{2} ∈ C_{c}^{−∞}(ℝ^{n}) satisfy

(x, ξ) ∈ WF(u_{1}) ⟹ (x, −ξ) ∉ WF(u_{2}) (4.51) then the real pairing, defined by (u_{1}, u_{2}) = (Au_{1}, u_{2}) + (u_{1}, A^{t}u_{2}), (4.52) where A satisfies (4.46), is independent of A.

Proof. Notice that WF(\overline{u}) = \{(x, −ξ) ∈ ℝ^{n} × (ℝ^{n}∖0); (x, ξ) ∈ WF(u)\}. (4.53) We can write (4.51), using (4.50), as

(u_{1}, u_{2}) = ⟨Au_{1}, \overline{u}_{2}⟩ + ⟨u_{1}, \overline{A^{t}u_{2}}⟩.

Since, by definition, \overline{A^{t}u_{2}} = A^{∗}\overline{u}_{2},

(u_{1}, u_{2}) = ⟨Au_{1}, \overline{u}_{2}⟩ + ⟨u_{1}, A^{∗}\overline{u}_{2}⟩ = ⟨u_{1}, \overline{u}_{2}⟩

is defined by (4.47), since (4.51) translates to (4.45).

4.10. Multiplication of distributions The pairing result (4.52) can be used to define the product of two distributions

under the same hypotheses, (4.51).

Proposition 4.7. If u_{1}, u_{2} ∈ C_{c}^{−∞}(ℝ^{n}) satisfy (x, ξ) ∈ WF(u_{1}) ⟹ (x, −ξ) ∉ WF(u_{2}) (4.54) then the product of u_{1} and u_{2} ∈ C_{c}^{−∞}(ℝ^{n}) is well-defined by u_{1}u_{2}(ϕ) = (u_{1}, ϕu_{2}) = (ϕu_{1}, u_{2}) ∀ ϕ ∈ C_{c}^{∞}(ℝ^{n}) (4.55) using (4.52).

Proof. We only need to observe that if u ∈ C_{c}^{−∞}(ℝ^{n}) and A ∈ Ψ_{∞}^{m}(ℝ^{n}) has WF′(A) ∩ WF(u) = ∅ then for any fixed ψ ∈ C_{c}^{∞}(ℝ^{n}) ∥ψAϕu∥_{C^{k}} ≤ C∥ϕ∥_{C^{p}} p = k + N (4.56) for some N, depending on m. This implies the continuity of ϕ ⟼ u_{1}u_{2}(ϕ) defined by (4.55).

4.11. Projection Here we write ℝ_{z}^{n} = ℝ_{x}^{p} × ℝ_{y}^{k} and define a continuous linear map, which we

write rather formally as an integral C_{c}^{−∞}(ℝ^{n}) ∋ u ⟼ ∫ u(x, y)dy ∈ C_{c}^{−∞}(ℝ^{p}) (4.57) by pairing. If ϕ ∈ C^{∞}(ℝ^{p}) then π_{1}^{∗}ϕ ∈ C^{∞}(ℝ^{n}), π_{1} : ℝ^{n} ∋ (x, y) ⟼ x ∈ ℝ^{p} (4.58) and for u ∈ C_{c}^{−∞}(ℝ^{n}) we define the formal ‘integral’ in (4.57) by (∫ u(x, y)dy, ϕ) = ((π_{1})_{∗}u, ϕ) := u(π_{1}^{∗}ϕ). (4.59)

In this sense we see that the projection is dual to pull-back (on functions) under π_{1}, so is “push-forward under π_{1},” a special case of (4.8). The support of the projection satisfies

(4.60) supp ((π_{1})_{∗}u) ⊂ π_{1} (supp(u)) ∀ u ∈ C_{c}^{−∞}(ℝ^{n}),

as follows by duality from

(4.61) supp(π_{1}^{∗}ϕ) ⊂ π_{1}^{−1} (supp ϕ) .

Proposition 4.8. Let π_{1} : ℝ^{p+k} ⟶ ℝ^{p} be projection, then for every u ∈ C_{c}^{−∞}(ℝ^{p+k})

WF ((π_{1})_{∗}u) ⊂ {(x, ξ) ∈ ℝ^{p} × (ℝ^{p}\0) ;
∃ y ∈ ℝ^{k} with (x, y, ξ, 0) ∈ WF(u)}.

Proof. First notice that

(π_{1})_{∗} : C_{c}^{∞}(ℝ^{n}) ⟶ C_{c}^{∞}(ℝ^{p}). (4.62)

Combining this with (4.60) we see that

(4.63) sing supp ((π_{1})_{∗}u) ⊂ π_{1} (sing supp u)

which is at least consistent with Proposition 4.8. To prove the proposition in full let me restate the local characterization of the wavefront set, in terms of the Fourier transform:

Lemma 4.5. Suppose K ⊂⊂ ℝ^{n} and Γ ⊂ ℝ^{n}∖0 is a closed cone, then

\begin{matrix}u ∈ C_{c}^{−∞}(ℝ^{n}), WF(u) ∩ (K × Γ) = ∅, A ∈ Ψ_{∞}^{m}(ℝ^{n}), WF′(A) ⊂ K × Γ \\ ⟹ Au ∈ S(ℝ^{n}). \end{matrix} (4.64)

In particular

u ∈ C_{c}^{−∞}(ℝ^{n}), WF(u) ∩ (K × Γ) = ∅, ϕ ∈ C_{c}^{∞}(ℝ^{n}), supp(ϕ) ⊂ K

⟹ ϕ̂u(ξ) is rapidly decreasing in Γ.

(4.65)

Conversely suppose Γ ⊂ ℝ^{n} ∖0 is a closed cone and u ∈ S′(ℝ^{n}) is such that for some ϕ ∈ C_{c}^{∞}(ℝ^{n})

(4.66)  ϕ̂u(ξ) is rapidly decreasing in Γ

then

WF(u) ∩ {x ∈ ℝ^{n}; ϕ(x) ≠ 0} × int(Γ) = ∅. (4.67)

With these local tools at our disposal, let us attack (4.8). We need to show that

\begin{matrix}(\overline{x}, \overline{ξ}) ∈ ℝ^{p} × (ℝ^{p}\0) s.t. (\overline{x}, y, \overline{ξ}, 0) ∉ WF(u) ∀ y ∈ ℝ^{n} \\ ⟹ (\overline{x}, \overline{ξ}) ∉ WF ((π_{1})_{∗}u) . \end{matrix} (4.68)

Notice that, WF(u) being conic and π(WF(u)) being compact, WF(u)∩(ℝ^{n}×S^{n−1}) is compact. The hypothesis (4.68) is the statement that

\{\overline{x}\} × ℝ^{k} × 𝕊^{n−1} × \{0\} ∩ WF(u) = ∅.

Thus \overline{x} has an open neighbourhood, W, in ℝ^{p}, and (\overline{ξ}, 0) a conic neighbourhood γ_{1} in (ℝ^{n}\0) such that

(W × ℝ^{k} × γ_{1}) ∩ WF(u) = ∅.

Now if ϕ ∈ C_{c}^{∞}(ℝ^{p}) is chosen to have support in W

(4.69) { \atop (π_{1}^{∗}ϕ)u}(ξ, η) is rapidly decreasing in γ_{1}.

Set v = ϕ(π_{1})_{∗}u. From the definition of projection and the identity

v = ϕ(π_{1})_{∗}u = (π_{1})_{∗}[(π_{1}^{∗}ϕ)u],

we have

v̂(ξ) = v(e^{−ix·ξ}) = { \atop ((π_{1}^{∗}ϕ)u)}(ξ, 0). (4.70)

Now (4.69) shows that v̂(ξ) is rapidly decreasing in γ_{1} ∩ (ℝ^{p} × {0}), which is a cone around \overline{ξ} in ℝ^{p}. Since v = ϕ(π_{1})_{∗}u this shows that (\overline{x}, \overline{ξ}) ∉ WF ((π_{1})_{∗}u) , as claimed.

Before going on to talk about the other operations, let me note a corollary of this which is useful and, even more, helps to explain what is going on:

Corollary 4.1. If u ∈ C_{c}^{−∞}(ℝ^{n}) and
WF(u) ∩ {(x, y, ξ, 0); x ∈ ℝ^{p}, y ∈ ℝ^{k}, ξ ∈ ℝ^{p}\0} = ∅

then (π_{1})_{∗}(u) ∈ C_{c}^{∞}(ℝ^{n}).

Proof. Indeed, (4.8) says WF ((π_{1})_{∗}u) = ∅.

Here, the vectors (x, y, ξ, 0) are the ones “normal” (as we shall see, really conormal) to the surfaces over which we are integrating. Thus Lemma 4.5 and Corollary 4.1 both state that the only singularities that survive integration are the ones which are conormal to the surface along which we integrating; the ones even partially in the direction of integration are wiped out. This in particular fits with the fact that if we integrate in all variables then there are no singularities left.

4.12. Restriction

Next we wish to consider the restriction of a distribution to a subspace

C_{c}^{−∞}(ℝ^{n}) ∋ u ⟼ u ↾ {y = 0} ∈ C_{c}^{−∞}(ℝ^{p}). (4.71)

This is not always defined, i.e. no reasonable map (4.71) exists for all distributions. However under an appropriate condition on the wavefront set we can interpret (4.71) in terms of pairing, using our definition of products. Thus let

ι : ℝ^{p} ∋ x ⟼ (x, 0) ∈ ℝ^{n}

be the inclusion map. We want to think of u ↾ {y = 0} as ι^{∗}u. If u ∈ C_{c}^{∞}(ℝ^{n}) then for any ϕ′ ∈ C_{c}^{∞}(ℝ^{n}) the identity

ι^{∗}u(ι^{∗}ϕ′) = u (ϕ′δ(y)) (4.72)

holds.

The restriction map ι^{∗} : C_{c}^{∞}(ℝ^{n}) ⟶ C_{c}^{∞}(ℝ^{p}) is surjective. If u ∈ C_{c}^{−∞}(ℝ^{n}) satisfies the condition

WF(u) ∩ {(x, 0, 0, η); x ∈ ℝ^{p}, η ∈ ℝ^{n−p}} = ∅ (4.73)

then we can interpret the pairing

ι^{∗}u(ϕ) = u (ϕ′δ(y)) ∀ ϕ ∈ C_{c}^{∞}(ℝ^{p})
where ϕ′ ∈ C_{c}^{∞}(ℝ^{n}) and ι^{∗}ϕ′ = ϕ

(4.74) to define ι^{∗}u. Indeed, the right side makes sense by Proposition 4.7.

Thus we have directly proved the first part of

Proposition 4.9. Set R = {u ∈ C_{c}^{−∞}(ℝ^{n}); (4.73) holds} then (4.74) defines a linear restriction map ι^{∗} : R ⟶ C_{c}^{−∞}(ℝ^{p}) and

(4.75) WF(ι^{∗}u) ⊂ {(x, ξ) ∈ ℝ^{p} × (ℝ^{p}∖0); ∃ η ∈ ℝ^{n} with (x, 0, ξ, η) ∈ WF(u)}.

Proof. First note that (4.73) means precisely that

(4.76)  û(ξ, η) is rapidly decreasing in a cone around {0} × ℝ^{k}\0.

When u ∈ C_{c}^{∞}(ℝ^{n}) taking Fourier transforms in (4.72) gives

ι̂^{∗}u(ξ) = \frac{1}{(2π)^{k}} ∫ û(ξ, η)dη. (4.77)

In general (4.76) ensures that the integral in (4.77) converges, it will then hold by continuity.

We actually apply (4.77) to a localized version of u; if ψ ∈ C_{c}^{∞}(ℝ^{p}) then

{ \atop ψι^{∗}(u)}(ξ) = (2π)^{−k} ∫ ψ̂(ξ)û(ξ, η)dη. (4.78)

Thus suppose (\overline{x}, \overline{ξ}) ∈ ℝ^{p} × (ℝ^{p}\0) is such that (\overline{x}, 0, \overline{ξ}, η) ∉ WF(u) for any η. If ψ has support close to \overline{x} and ζ ∈ C_{c}^{∞}(ℝ^{n−p}) has support close to 0 this means

(4.79)  ψ̂ζu(ξ, η) is rapidly decreasing in a cone around each (\overline{ξ}, η).

We also have rapid decrease around (0, η) from (4.76) (make sure you understand this point) as

ψ̂ζu(ξ, η) is rapidly decreasing in γ × ℝ^{p}

for a cone, γ, around \overline{ξ}. From (4.77)

{ \atop ψι^{∗}(ζu)}(ξ) is rapidly decreasing in γ.

Thus (\overline{x}, \overline{ξ}) ∉ WF(ι^{∗}(ζu)). Of course if we choose ζ(y) = 1 near 0, ι^{∗}(ζu) = ι^{∗}(u) so (\overline{x}, \overline{ξ}) ∉ WF(u), provided (\overline{x}, 0, \overline{ξ}, η) ∉ WF(u), for all η. This is what (4.75) says.

Try to picture what is going on here. We can restate the main conclusion of Proposition 4.9 as follows.

Take WF(u) ∩ {(x, 0, ξ, η) ∈ ℝ^{p} × {0} × (ℝ^{n}\0)} and let Z denote projection off the η variable:

ℝ^{p} × {0} × ℝ^{p} × ℝ^{k} ⟶^{Z} ℝ^{p} × ℝ^{p} (4.80)

then

WF(ι^{∗}u) ⊂ Z(WF(u) ∩ {y = 0}).

We will want to think more about these operations later.

4.13. Exterior product

This is maybe the easiest of the elementary operators. It is always defined

(u_{1} ⊠ u_{2})(ϕ) = u_{1} (u_{2}(ϕ(x, ·)) = u_{2}(u_{1}(ϕ(·, y)). (4.81)

Moreover we can easily compute the Fourier transform:

{ \atop u_{1} ⊠ u_{2}}(ξ, η) = û_{1}(ξ)û_{2}(η). (4.82)

Proposition 4.10. The (exterior) product

C_{c}^{−∞}(ℝ^{p}) × C_{c}^{−∞}(ℝ^{k}) ⟵ C_{c}^{−∞}(ℝ^{p+k}) (4.83)

is a bilinear map such that

WF(u_{1} ⊠ u_{2}) ⊂ [(supp(u_{1}) × {0}) × WF(u_{2})]

∪ [WF(u_{1}) × (supp(u_{2}) × {0})] ∪ [WF(u_{1}) × WF(u_{2})].

Proof. We can localize near any point (\overline{x}, \overline{y}) with ϕ_{1}(x)ϕ_{2}(y), where ϕ_{1} is supported near \overline{x} and ϕ_{2} is supported near \overline{y}. Thus we only need examine the decay of

{ \atop ϕ_{1}u_{1} ⊠ ϕ_{2}u_{2}} = { \atop ϕ_{1}u_{1}}(ξ) · { \atop ϕ_{2}u_{2}}(η). (4.84)

Notice that if { \atop ϕ_{1}u_{1}}(ξ) is rapidly decreasing around \overline{ξ} ≠ 0 then the product is rapidly decreasing around any (\overline{ξ}, η). This gives (4.10).

4.14. Diffeomorphisms

We next turn to the question of the extension of F^{∗}, where F : Ω_{1} ⟶ Ω_{2} is a C^{∞} map, from C^{∞}(Ω_{2}) to some elements of C^{−∞}(Ω_{2}). The simplest example of pull-back is that of transformation by a diffeomorphism.

We have already noted how pseudodifferential operators behave under a diffeomorphism: F : Ω_{1} ⟶ Ω_{2} between open sets of ℝ^{n}. Suppose A ∈ Ψ_{∞}^{m}(ℝ^{n}) has Schwartz kernel of compact support in Ω_{1} × Ω_{1} then we define

A_{F} : C_{c}^{∞}(Ω_{2}) ⟶ C_{c}^{∞}(Ω_{2}) (4.85)

by A_{F} = G^{∗} · A · F^{∗}, G = F^{−1}. In § 3.13 we showed that A_{F} ∈ Ψ_{∞}^{m}(ℝ^{n}). In fact we showed much more, namely we computed a (very complicated) formula for the full symbols. Recall the definition of the cotangent bundle of ℝ^{n}

T^{∗}ℝ^{n} ≃ ℝ^{n} × ℝ^{n} (4.86)

identified as pairs of points (\overline{x}, \overline{ξ}), where \overline{x} ∈ ℝ^{n} and

\overline{ξ} = df(\overline{x}) for some f ∈ C^{∞}(ℝ^{n}). (4.87)

The differential df(\overline{x}) of f at \overline{x} ∈ ℝ^{n} is just the equivalence class of f(x)−f(\overline{x}) ∈ I\overline{x} modulo I2 ; \overline{x}. Here

\begin{cases}I\overline{x} = \{g ∈ C^{∞}(ℝ^{n}); g(\overline{x}) = 0\} \\ I2 ; \overline{x} = \{ ∑_{finite} g_{i}h_{i}, g_{i}, h_{i} ∈ I\overline{x}\}. \end{cases} (4.88)

The identification of \overline{ξ}, given by (4.86) and (4.87), with a point in ℝ^{n} is obtained using Taylor’s formula. Thus if f ∈ C^{∞}(ℝ^{n})

f(x) = f(\overline{x}) + ∑_{i=1}^{n} \frac{∂f}{∂x_{j}} (\overline{x})(x − \overline{x})_{j} + ∑_{i,j=1} g_{ij}(x)x_{i}x_{j}. (4.89)

The double sum here is in I2 ; \overline{x}, so the residue class of f(x) − f(\overline{x}) in I\overline{x}/I2 ; \overline{x} is the same as that of

∑_{i=1}^{n} \frac{∂f}{∂x_{j}} (\overline{x})(x − \overline{x})_{j}.

That is, d(x − \overline{x})_{j} = dx_{j}, j = 1, …, n form a basis for T ∗ ; \overline{x}ℝ^{n} and in terms of this basis

df(\overline{x}) = ∑_{i=1}^{n} \frac{∂f}{∂x_{j}} (\overline{x})dx_{j}.

Thus the entries of \overline{ξ} are just (\frac{∂f}{∂x_{1}} , … \frac{∂f}{∂x_{n}} ) for some f. Another way of saying this is that the linear functions ξ · x = ξ_{1}x_{1} + ξ_{2}x_{2} ⋯ ξ_{n}x_{n} have differentials spanning T_{x}^{∗}ℝ^{n}.

So suppose F : Ω_{1} ⟶ Ω_{2} is a C^{∞} map. Then

F^{∗} : T ∗ ; \overline{y}Ω_{2} ⟶ T ∗ ; \overline{x}Ω_{1}, \overline{y} = F (\overline{x})

is defined by F^{∗}df(\overline{y}) = d(F^{∗}f)(\overline{x}) since F^{∗} : I\overline{y} ⟶ I\overline{x}, F^{∗} : I2 ; \overline{y} ⟶ I2 ; \overline{x}. In

coordinates F (x) = y ⟹

\frac{∂}{∂x_{j}} (F^{∗}f(x)) = \frac{∂}{∂y} f(F (x)) = ∑_{k=1}^{n} \frac{∂f}{∂x_{k}} (y)\frac{∂F_{k}}{∂x_{j}}

i.e. F^{∗}(η · dy) = ξ · dx if

ξ_{j} = ∑_{k=1}^{n} \frac{∂F_{k}}{∂x_{j}} (x) · η_{k}. (4.90)

Of course if F is a diffeomorphism then the Jacobian matrix \frac{∂F}{∂x} is invertible and (4.90) is a linear isomorphism. In this case

\begin{matrix}F^{∗} : T_{Ω_{2}}^{∗}ℝ^{n} ⟷ T_{Ω_{1}}^{∗}ℝ^{n} \\ (x, ξ) ⟷ (F (x), η) \end{matrix} (4.91)

with ξ and η connected by (4.90). Thus (F^{∗})^{∗} : C^{∞}(T^{∗}Ω_{1}) ⟶ C^{∞}(T^{∗}Ω_{2}).

Proposition 4.11. If F : Ω_{1} ⟶ Ω_{2} is a diffeomorphism of open sets of ℝ^{n} and A ∈ Ψ_{∞}^{m}(ℝ^{n}) has Schwartz kernel with compact support in Ω_{1} × Ω_{2} then

σ_{m}(A_{F}) = (F^{∗})^{∗}σ_{m}(A) (4.92)

and

F^{∗} (WF′(A_{F})) = WF′(A). (4.93)

It follows that symbol σ_{m}(A) of A is well-defined as an element of S_{∞}^{m−[1]}(T^{∗}ℝ^{n}) independent of coordinates and WF′(A) ⊂ T^{∗}ℝ^{n}\0 is a well-defined closed conic set, independent of coordinates. The elliptic set and the characteristic set Σ_{m} are therefore also well-defined complementary conic subsets of T^{∗}Ω\0.

Proof. Look at the formulae.

The main use we make of this invariance result is the freedom it gives us to choose local coordinates adapted to a particular problem. It also suggests that there should be neater ways to write various formulae, e.g. the wavefront sets of push-forward and pull-backs.

Proposition 4.12. If u ∈ C_{c}^{−∞}(ℝ^{n}) has supp(u) ⊂ Ω_{2} and F : Ω_{1} ⟶ Ω_{2} is a diffeomorphism then

WF(F^{∗}u) ⊂ \{(x, ξ) ∈ ℝ^{n} × (ℝ^{n}\0); (F (x), η) ∈ WF(u), η_{j} = ∑_{i} \frac{∂F_{i}}{∂x_{j}} (x)ξ_{i}\}. (4.94)

Proof. Just use the standard definition

WF(F^{∗}u) = ⋂{Σ(A); A(F^{∗}u) ∈ C^{∞}}.

To test the wavefront set of F^{∗}u it suffices to consider A’s with kernels supported in Ω_{1} × Ω_{1} since supp(F^{∗}u ⋐ Ω_{1} and for a general pseudodifferential operator A′ there exists A with kernel supported in Ω_{1} such that A′u − Au ∈ C^{∞}(ℝ^{n}). Then AF^{∗}u ∈ C_{c}^{∞}(Ω_{1}) ⟺ A_{F} u ∈ C_{c}^{∞}(Ω_{2}). Thus

(4.95)  WF(F^{∗}u) = ⋂{Σ(A); A_{F} u ∈ C^{∞}}

= ⋂{F^{∗}(Σ(A_{F})); A_{F} u ∈ C^{∞}} (4.96)

= F^{∗} WF(u) (4.97)

since, for u, it is enough to consider operators with kernels supported in Ω_{2}×Ω_{2}.

4.15. Products

Although we have discussed the definition of the product of two distributions we have not yet analyzed the wavefront set of the result.

Proposition 4.13. If u_{1}, u_{2} ∈ C_{c}^{−∞}(ℝ^{n}) are such that
(x, ξ) ∈ WF(u_{1}) ⟹ (x, −ξ) ∉ WF(u_{2})

then the product u_{1}u_{2} ∈ C_{c}^{−∞}(ℝ^{n}), defined by Proposition 4.7 satisfies

WF(u_{1}u_{2}) ⊂ {(x, ξ); x ∈ supp(u_{1}) and (x, ξ) ∈ WF(u_{2})}
∪{(x, ξ); x ∈ supp(u_{2}) and (x, ξ) ∈ WF(u_{1})}
∪{(x, ξ); ξ = η_{1} + η_{2}, (x, η_{i}) ∈ WF(u_{i}), i = 1, 2}.

Proof. We can represent the product in terms of three ‘elementary’ operations.

u_{1}u_{2}(x) = ι^{∗}[F^{∗}(u_{1} ⊠ u_{2})] (4.98)

where F : ℝ^{2n} ⟶ ℝ^{2n} is the linear transformation

F (x, y) = (x + y, x − y)

and ι : ℝ^{n} ↪ ℝ^{n}×{0} ⊂ ℝ^{2n} is inclusion as the first factor. Thus (4.98) corresponds to the ‘informal’ notation

u_{1}u_{2}(x) = u_{1}(x + y)u_{2}(x − y) ↾ {y = 0} (4.99)

and will follow by continuity once we analyse the wavefront set properties.

We know from Proposition 4.10 that

WF (u_{1} ⊠ u_{2}) ⊂ {(X, Y, Ξ, H) ; X ∈ supp(u_{1}), Ξ = 0, (Y, H) ∈ WF(u_{2})}

∪{(X, Y, Ξ, H) ; (X, Ξ) ∈ WF(u_{1}), Y ∈ supp(u_{2}), H = 0}
∪{(X, Y, Ξ, H) ; (X, Ξ) ∈ WF(u_{1}), (Y, H) ∈ WF(u_{2})}.

Since F is a diffeomorphism, by Proposition 4.12,

WF(F^{∗}(u_{1} ⊠ u_{2})) = {(x, y, ξ, η); (F^{t}(x, y), Ξ, H) ∈ WF(u_{1} ⊠ u_{2}),
(ξ, η) = A^{t}(Ξ, H)}.

where F^{t} is the transpose of F as a linear map. In fact F^{t} = F, so

WF(F^{∗}(u_{1} ⊠ u_{2})) ⊂

\{(x, y, ξ, η); x + y ∈ supp(u_{1}), ξ + η = 0, (x − y, \frac{1}{2}(ξ − η)) ∈ WF(u_{2})\}

∪ \{(x, y, ξ, η); (x + y, \frac{1}{2}(ξ + η)) ∈ WF(u_{1}), (x − y, \frac{1}{2}(ξ − η)) ∈ WF(u_{2})\}

and so using Proposition 4.9

WF(F^{∗}(u_{1} ⊠ u_{2})) ↾ {y = 0}

⊂ {(x, 0, ξ, −ξ); x ∈ supp(u_{1}), (x, ξ) ∈ WF(u_{2})}

∪ {(x, 0, ξ, η); (x ∈ supp(u_{2}), (x, ξ) ∈ WF(u_{2})}

∪ \{(x, 0, ξ, η); (x, \frac{1}{2}(ξ + η)) ∈ WF(u_{2}), (x, \frac{1}{2}(ξ − η)) ∈ WF(u_{1})\}

Notice that

(x, 0, 0, η) ∈ WF (F^{∗}(u_{1} ⊠ u_{2})) ⟹ (x, \frac{1}{2}η) ∈ WF(u_{1}) and (x, \frac{1}{2}η) WF(u_{2})

which introduces the assumption under which u_{1}u_{2} is defined. Finally then we see that

WF(u_{1}u_{2}) ⊂ {(x,ξ); x ∈ supp(u_{1}), (x, ξ) ∈ WF(u_{2})}
∪{(x, ξ); x ∈ supp(u_{2}), (x, ξ) ∈ WF(u_{1})}
∪{(x, ξ); (x, η_{1}) ∈ WF(u_{1}), (x, η_{2}) ∈ WF(u_{2}) and ξ = η_{1} + η_{2}}.

(4.100)

which is another way of writing the conclusion of Proposition 4.13.

4.16. Pull-back

Now let us consider a general C^{∞} map

F : Ω_{1} ⟶ Ω_{2}, Ω_{1} ⊂ ℝ^{n}, Ω_{2} ⊂ ℝ^{m}. (4.101)

Thus even the dimension of domain and range spaces can be different. When can we define F^{∗}u, for u ∈ C_{c}^{−∞}(Ω_{2}) and what can we say about WF(F^{∗}u)? For a general map F it is not possible to give a sensible, i.e. consistent, definition of F^{∗}u for all distributions u ∈ C^{−∞}(Ω_{2}).

For smooth functions we have defined

F^{∗} : C_{c}^{∞}(Ω_{2}) ⟶ C^{∞}(Ω_{1})

but in general F^{∗}ϕ does not have compact support, even if ϕ does. We therefore impose the condition that F be proper

F^{−1}(K) ⋐ Ω_{2} ∀ K ⋐ Ω_{2}, (4.102)

(mostly just for convenience). In fact if we want to understand F^{∗}u near \overline{x}_{1} ∈ Ω_{1} we only need to consider u near F (\overline{x}_{1}) ∈ Ω_{2}.

The problem is that the map (4.101) may be rather complicated. However any smooth map can be decomposed into a product of simpler maps, which we can analyze locally. Set

graph(F) = {(x, y) ∈ Ω_{1} × Ω_{2}; y = F (x)} ⟶^{ι_{F}} Ω_{1} × Ω_{2}.

This is always an embedded submanifold of Ω_{1} × Ω_{2} the functions y_{i} − F_{i}(x), i = 1, …, N are independent defining functions for graph(F) and x_{1}, …, x_{n} are coordinates on it. Now we can write

F = π_{2} ∘ ι_{F} ∘ g (4.103)

where g : Ω_{1} ⟷ graph(F) is the diffeomorphism onto its range x ⟼ (x, F (x)). This decomposes F as a projection, an inclusion and a diffeomorphism. Now consider

F^{∗}ϕ = g^{∗} · ι_{F}^{∗} · π_{2}^{∗}ϕ (4.104)

i.e. F^{∗}ϕ is obtained by pulling ϕ back from Ω_{2} to Ω_{1} × Ω_{2}, restricting to graph(F) and then introducing the x_{i} as coordinates. We have directly discussed (π_{2}^{∗}ϕ) but we can actually write it as

π_{2}^{∗}ϕ = 1 ⊠ ϕ(y), (4.105)

so the result we have proved can be applied to it. So let us see what writing (4.104) as

F^{∗}ϕ = g^{∗} ∘ ι_{F}^{∗} (1 ⊠ ϕ) (4.106)

tells us. If u ∈ C_{c}^{−∞}(Ω_{2}) then

WF(1 ⊠ u) ⊂ {(x, y, 0, η); (y, η) ∈ WF(u)} (4.107)

by Proposition 4.10. So we have to discuss ι_{F}^{∗} (1 ⊠ u), i.e. restriction to y = F (x). We can do this by making a diffeomorphism:

T_{F} (x, y) = (x, y + F (x)) (4.108)

so that T_{F}^{−1}(graph(F)) = {(x, 0)}. Notice that g ∘ T_{F} = π_{1}, so

F^{∗}ϕ = ι_{{y=0}}^{∗} (T_{F}^{∗}(1 ⊠ u)) . (4.109)

Now from Proposition 4.12 we know that

(4.110) WF(T_{F}^{∗}(1 ⊠ u)) = T_{F}^{∗}(WF(1 ⊠ u))

= {(X, Y, Ξ, H); (X, Y + F (X), ξ, η) ∈ WF(1 ⊠ u),

η = H, ξ_{i} = Ξ_{i} + Σ\frac{∂F_{j}}{∂x_{i}} H_{j}\}

i.e.

WF(T_{F}^{∗}(1 ⊠ u)) = \{(x, y, ξ, η); ξ_{i} = ∑_{j} \frac{∂F_{j}}{∂x_{j}} (x)η_{j}, (F (x), η) ∈ WF(u)\}. (4.111)

So consider our existence condition for restriction to y = 0, that ξ ≠ 0 on WF(T_{F}^{∗}(1⊠ u)) i.e.

(F (x), η) ∈ WF(u) ⟹ ∑_{j} \frac{∂F_{j}}{∂x_{i}} (x)η_{j} ≠ 0. (4.112)

If (4.112) holds then, from (4.110) and Proposition 4.9

WF(F^{∗}u) ⊂ \{(x, ξ); ∃ (F (x), η) ∈ WF(u) and ξ_{j} = ∑_{j} \frac{∂F_{j}}{∂x_{i}} (x)η_{j}\}. (4.113)

We can reinterpret (4.112) and (4.113) more geometrically. The differential of F gives a map

F^{∗} : T_{F (x)}^{∗}Ω_{2} ⟶ T_{x}^{∗}Ω_{1} ∀ x ∈ Ω_{1}

(F (x), η) ⟼ (x, ξ) where ξ_{i} = Σ\frac{∂F_{j}}{∂x_{i}} η_{j}.

Thus (4.112) can be restated as:

∀ x ∈ Ω_{1}, the null space of F_{x}^{∗} : T_{F (x)}^{∗}Ω_{2} ⟶ T_{x}^{∗}Ω_{1}

does not meet WF(u)

(4.114)

and then (4.113) becomes

(4.115) WF(F^{∗}u) ⊂ ⋃_{x∈Ω_{1}} F_{x}^{∗}[WF(u) ∩ T_{F (x)}^{∗}Ω_{2}] = F^{∗}(WF(u))

(proved we are a little careful in that F^{∗} is not a map; it is a relation between T^{∗}Ω_{2} and T^{∗}Ω_{1}) and in this sense (4.114) holds. Notice that (4.113) is a sensible “consequence” of (4.114), since otherwise WF(F^{∗}u) would contain some zero directions.

Proposition 4.14. If F : Ω_{1} ⟶ Ω_{2} is a proper C^{∞} map then F^{∗} extends (by continuity) from C_{c}^{∞}(Ω_{2}) to

{u ∈ C_{c}^{−∞}(Ω_{2}); F^{∗}(WF(u)) ∩ (Ω_{1} × 0) = ∅ in T^{∗}Ω_{1}} (4.116)

and (4.115) holds.

4.17. The operation F_{∗}

Next we will look at the dual operation, that of push-forward. Notice the basic properties of pull-back:

(4.117) Maps C_{c}^{∞} to C_{c}^{∞}(if F is proper)

(4.118) Not always defined on distributions.

Dually we find

Proposition 4.15. If F : Ω_{1} ⟶ Ω_{2} is a C^{∞} map of an open subset of ℝ^{n} into an open subset of ℝ^{n} then for any u ∈ C_{c}^{−∞}(Ω_{1})

F_{∗}(u)(ϕ) = u(F^{∗}ϕ) (4.119)

is a distribution of compact support and

F_{∗} : C_{c}^{−∞}(Ω_{1}) ⟶ C_{c}^{−∞}(Ω_{2}) (4.120)

4.17. THE OPERATION F_{∗} 73

has the property:

WF(F_{∗}u) ⊂ {(y, η);y ∈ F (supp(u)), y = F (x), F_{x}^{∗}η = 0} ∪

{(y, η); y = F (x), (x, F_{x}^{∗}η) ∈ WF(u)}. (4.121)

Proof. Notice that the ‘opposite ’ of (4.117) and (4.118) hold, i.e. F_{∗} is always defined but even if u ∈ C_{c}^{∞}(Ω_{1}) in general F_{∗}u ∉ C_{c}^{∞}(Ω_{2}). All we really have to prove is (4.121). As usual we look for a formula in terms of elementary operations. So suppose u ∈ C_{c}^{∞}(Ω_{1})

\begin{aligned}F_{∗}u(ϕ) & = u(F^{∗}ϕ) ϕ ∈ C_{c}^{∞}(Ω_{2}) \\ & = ∫ u(x) ϕ(F (x)) dx \\ & = ∫ u(x)δ(y − F (x)) ϕ(y) dydx. \end{aligned} (4.122)

Thus, we see that

F_{∗}u = π_{∗}H^{∗}(u ⊠ δ) (4.123)

where δ = δ(y) ∈ C_{c}^{−∞}(ℝ^{m}), H is the diffeomorphism

H(x, y) = (x, y − F (x))

and π : ℝ^{n+m} ⟶ ℝ^{m} is projection off the first factor.

Thus (4.123) is the desired decomposition into elementary operations, since u⊠δ ∈ C_{c}^{−∞}(ℝ^{n+m}), π_{∗}H^{∗}(u⊠δ) is always defined and indeed the map is continuous, which actually proves (4.123).

So all we need to do is estimate the wavefront set using our earlier results. From Proposition 4.10 it follows that

(4.124) WF(u ⊠ δ) ⊂ {(x, 0, ξ, η); x ∈ supp(u), ξ = 0} ∪ {(x, 0, ξ, 0); (x, ξ) ∈ WF(u)}

∪ {(x, 0, ξ, η); (x, ξ) ∈ WF(u)}

= {(x, 0, ξ, η); x ∈ supp(u), ξ = 0} ∪ {(x, 0, ξ, η); (x, ξ) ∈ WF(u)}.

Then consider what happens under H^{∗}. This is a diffeomorphism so the wavefront set transforms under the pull-back:

WF(H^{∗}(u ⊠ δ)) = WF(u(x)δ(y − F (x))

= \{(x, F (x), Ξ, η); Ξ_{i} = ξ_{i} − ∑_{j} \frac{∂F_{j}}{∂x_{i}} (x)η_{j}, (x, 0, ξ, η) ∈ WF(u ⊠ δ)\}

= \{(x, F (x), Ξ, η); x ∈ supp(u), Ξ_{i} = − ∑_{j} \frac{∂F_{j}}{∂x_{i}} (x)η_{j})\}

∪ \{(x, F (x), Ξ, η); η ∈ ℝ^{m}, (x, ξ) ∈ WF(u)), Ξ_{i} = ξ_{i} − ∑_{j} \frac{∂F_{i}}{∂x_{j}} η_{j}\}.

74 4. WAVEFRONT SET

Finally recall the behaviour of wavefront sets under projection, to see that

WF(F_{∗}u) ⊂ {(y, η); ∃ (x, y, 0, η) ∈ WF(H^{∗}(u ⊠ δ))}

= {(y, η); y = F (x) for some x ∈ supp(u) and

∑_{j} \frac{∂F_{j}}{∂x_{i}} η_{j} = 0, i = 1, …, n\}

∪{(y, η); y = F (x) for some (x, ξ) ∈ WF(u) and

ξ_{i} = ∑_{j} \frac{∂F_{i}}{∂x_{i}} η_{j}, i = 1, …, n)\}.

This says

(4.125)  WF(F_{∗}u) ⊂ {(y, η); y ∈ F (supp(u)) and F_{x}^{∗}(η) = 0}

∪ {(y, η); y = F (x) with (x, F_{x}^{∗}η) ∈ WF(u)} (4.126) which is just (4.121). As usual one should note that the two terms here are “really the same”.

Now let us look at F_{∗} as a linear map, F_{∗} : C_{c}^{∞}(Ω_{1}) ⟶ C_{c}^{−∞}(Ω_{2}). (4.127) As such it has a Schwartz kernel, indeed (4.123) is just the usual formula for an operator in terms of its kernel: F_{∗}u(y) = ∫ K(y, x)u(x)dx, K(y, x) = δ(y − F (x)). (4.128) So consider the wavefront set of the kernel: WF(δ(y − F (x)) = WF(H^{∗}δ(y)) = {(y, x; η, ξ); y = F (x), ξ = F_{x}^{∗}η}. (4.129) Now changing the order of the factors we can regard this as a subset

WF′(K) = {((y, η), (x, ξ)); y = F (x), ξ = F^{∗}η} ⊂ (Ω_{2} × ℝ^{m}) × (Ω_{1} × ℝ^{n}). (4.130)

As a subset of the product we can regard WF′(K) as a relation: if Γ ⊂ Ω_{2} × (ℝ^{n}\0) set

WF′(K) ∘ Γ =
{(y, η) ∈ Ω_{2} × (ℝ^{m}\0); ∃ ((y, η)), (x, ξ)) ∈ WF′(K) and (x, ξ) ∈ Γ}

Indeed with this definition (4.131) WF(F_{∗}u) ⊂ WF′(K) ∘ WF(u), K = kernel of F_{∗}.

4.18. Wavefront relation

One serious application of our results to date is:

4.18. WAVEFRONT RELATION 75

Theorem 4.1. Suppose Ω_{1} ⊂ ℝ^{n}, Ω_{2} ⊂ ℝ^{m} are open and A ∈ C^{−∞}(Ω_{1} × Ω_{2}) has proper support, in the sense that the two projections

\xymatrix{ & supp(A) \ar[dl]^{π_{1}} \ar[dr]_{π_{2}} \\ Ω_{1} & & Ω_{2}} (4.132)

are proper, then A defines a linear map A : C_{c}^{∞}(Ω_{2}) ⟶ C_{c}^{−∞}(Ω_{1}) (4.133) and extends by continuity to a linear map

(4.134)  A : {u ∈ C_{c}^{−∞}(X); WF(u) ∩ {(y, η) ∈ Ω_{2} × (ℝ^{n}\0);

∃ (x, 0, y, −η) ∈ WF(K)} = ∅} ⟶ C_{c}^{−∞}(Ω_{1}) (4.135) for which WF(Au) ⊂ WF′(A) ∘ WF(u), (4.136) where

WF′(A) = {((x, ξ), (y, η)) ∈(Ω_{1} × ℝ^{n}) × (Ω_{2} × ℝ^{m}); (ξ, η) ≠ 0

and (x, y, ξ, −η) ∈ WF(K)}. (4.137)

Proof. The action of the map A can be written in terms of its Schwartz kernel as Au(x) = ∫ K(x, y)u(y)dy = (π_{1})_{∗}(K · (1 ⊠ u)). (4.138) Here 1 ⊠ u is always defined and

WF(1 ⊠ u) ⊂ {(x, y, 0, η); (y, η) ∈ WF(u)}.

So the main question is, when is the product defined? Our sufficient condition for this is:

(x, y, ξ, η) ∈ WF(K) ⟹ (x, y, −ξ, −η) ∉ WF(1 ⊠ u)

which is

(x, y, 0, η) ∈ WF(K) ⟹ (x, y, 0, −η) ∉ WF(1 ⊠ u)
i.e. (y, −η) ∉ WF(u)

This of course is (4.135):

Au is defined (by continuity) if {(y, η) ∈ WF(u); ∃ (x, 0, y, −η) ∈ WF(A)} = ∅.

Then from our bound on the wavefront set of a product

WF (K · (1 ⊠ u)) ⊂

{(x, y, ξ, η); (ξ, η) =(ξ′, η′) + (0, η′′) with
(x, y, ξ′, η′) ∈ WF(K) and (x, η′′) ∈ WF(u)}
∪{(x, y, ξ, η); (x,y, ξ, η) ∈ WF(K), y ∈ supp(u)}
∪{(x, y, 0, η);(x, y) ∈ supp(A)(y, η) ∈ WF(u)}.

76 4. WAVEFRONT SET

This gives the bound

(4.139) WF (π_{∗}(K · (1 ⊠ u))) ⊂ {(x, ξ); (x, y, ξ, 0) ∈ WF(K · (1 ⊠ u)) for some y}

⊂ WF′(A) ∘ WF(u). (4.140)

4.19. Applications Having proved this rather general theorem, let us note some examples and

applications.

First, for pseudodifferential operators we know that

WF′(A) ⊂ {(x, x, ξ, ξ)}

i.e. corresponds to the identity relation (which is a map). Then (4.136) is the microlocality of pseudodifferential operators. The next result also applies to all pseudodifferential operators.

Corollary 4.2. If K ∈ C^{−∞}(Ω_{1} × Ω_{2}) has proper support and WF′(K) ∩ {(x, y, ξ, 0)} = ∅ (4.141) then the operator with Scwartz kernel K defines a continuous linear map A : C_{c}^{∞}(Ω_{2}) ⟶ C_{c}^{∞}(Ω_{1}). (4.142) If WF′(K) ∩ {(x, y, 0, η)} = ∅ (4.143) then A extends by continuity to A : C_{c}^{−∞}(Ω_{2}) ⟶ C_{c}^{−∞}(Ω_{1}). (4.144)

Proof. Immediate from (4.133)-(4.144).

4.20. Problems

Problem 4.1. Show that the general definition (4.9) reduces to

WF(u) = ⋂{Σ_{0}(A); A ∈ Ψ_{∞}^{0}(ℝ^{n}) and Au ∈ C^{∞}(ℝ^{n})}, u ∈ S′(ℝ^{n}) and prove the basic result of ‘microlocal elliptic regularity:’

If u ∈ S′(ℝ^{n}) and A ∈ Ψ_{∞}^{m}(ℝ^{n}) then

WF(u) ⊂ Σ(A) ∪ WF(Au). (4.145)

Problem 4.2. Compute the wavefront set of the following distributions:

\begin{matrix}δ(x) ∈ S′(ℝ^{n}), |x| ∈ S′(ℝ^{n}) and \\ χ_{𝔹^{n}}(x) = \begin{cases}1 & |x| ≤ 1 \\ 0 & |x| > 1. \end{cases} \end{matrix}

Problem 4.3. Let Γ ⊂ ℝ^{n} × (ℝ^{n}∖0) be an open cone and define

(4.146)  C_{c,Γ}^{−∞}(ℝ^{n}) ={u ∈ C_{c}^{−∞}(ℝ^{n}); Au ∈ C^{∞}(ℝ^{n})

(4.147) ∀ A ∈ Ψ_{∞}^{0}(ℝ^{n}) with WF′(A) ∩ Γ = ∅}. Describe a complete topology on this space with respect to which C_{c}^{∞}(ℝ^{n}) is a dense subspace.

Problem 4.4. Show that, for any pseudodifferential operator A ∈ Ψ_{∞}^{m}(ℝ^{n}), WF′(A) = WF′(A^{∗}).

Problem 4.5. Give an alternative proof to Lemma 4.3 along the following lines (rather than using Lemma 3.1). If σ_{L}(A) is the left reduced symbol then for ϵ > 0 small enough

b_{0} = γ_{ϵ}/σ_{L}(A) ∈ S_{∞}^{−m} (ℝ^{n}; ℝ^{n}) .

If we choose B_{0} ∈ Ψ_{∞}^{−m}(ℝ^{n}) with σ_{L}(B_{0}) = b_{0} then

Id −A ∘ B_{0} = G ∈ Ψ_{∞}^{0}(ℝ^{n}) (4.148)

has principal symbol

σ_{0}(G) = 1 − σ_{L}(A) · b_{0}.

From (4.22)

γ_{ϵ/4}σ_{0}(G) = γ_{ϵ/4}.

Thus we conclude that if σ_{L}(C) = γ_{ϵ/4} then

G = (Id −C)G + CG with CG ∈ Ψ_{∞}^{−1}(ℝ^{n}).

Thus (4.148) becomes

Id −AB_{0} = CG + R_{1} WF′(R_{1}) ∌ z. (4.149)

Let B_{1} ∼ ∑_{j≥1}(CG)^{j}, B_{1} ∈ Ψ^{−1} and set

B = B_{0} (Id +B_{1}) ∈ Ψ_{∞}^{−m}(ℝ^{n}).

From (4.149)

(4.150)  AB = AB_{0}(I + B_{1})

= (Id −CG) (I + B_{1}) − R_{1} (Id +B_{1}) (4.151)

= Id +R_{2}, WF′(R_{2}) ∌ z. (4.152)

Thus B is a right microlocal parametrix as desired. Write out the construction of a left parametrix using the same method, or by finding a right parametrix for the adjoint of A and then taking adjoints using Problem 4.4.

Problem 4.6. Essential uniqueness of left and right parametrices.

Problem 4.7. If (x̄, ξ̄) ∈ ℝ^{n} ×(ℝ^{n}∖0) is a given point, construct a distribution u ∈ C_{c}^{−∞}(ℝ^{n}) which has

WF(u) = {(x̄, tξ̄); t > 0} ⊂ ℝ^{n} × (ℝ^{n}∖0).

Problem 4.8. Suppose that A ∈ Ψ_{∞}^{m}(ℝ^{n}) has Schwartz kernel of compact support. If u ∈ C_{c}^{−∞}(ℝ^{n}) use the four ‘elementary operations’ (and an earlier result on the wavefront set of kernels) to investigate under what conditions

κ(x, y) = K_{A}(x, y)u(y) and then γ(x) = (π_{1})_{∗}κ

make sense. What can you say about WF(γ)?

Problem 4.9. Consider the projection operation under π_{1} : ℝ^{p} × ℝ^{k} ⟶ ℝ^{p}. Show that (π_{1})_{∗} can be extended to some distributions which do not have compact support, for example

{u ∈ S′(ℝ^{n}); supp(u) ∩ K × ℝ^{k} is compact for each K ⊂⊂ ℝ^{n}} .

78 4. WAVEFRONT SET