Problem set 6

Due November 16, 2004. Problems from notes:- 61, 62, plus the following two problems also at the end of the problems in the notes as numbers 75, 76.

Restriction from Sobolev spaces. The Sobolev embedding theorem shows that a function in $H^m(\mathbb{R}^n),$ for $m>n/2$ is continuous - and hence can be restricted to a subspace of $\mathbb{R}^n.$ In fact this works more generally. Show that there is a well defined restriction map

$$H^m(\mathbb{R}^n)\longrightarrow H^{m-\frac12}(\mathbb{R}^n) m>\frac12$$ (8)

with the following properties:

1. On $\mathcal{S}(\mathbb{R}^n)$ it is given by $u\longmapsto u(0,x'),$ $x'\in\mathbb{R}^{n-1}.$
2. It is continuous and linear.

Hint: Use the usual method of finding a weak version of the map on smooth Schwartz functions; namely show that in terms of the Fourier transforms on $\mathbb{R}^n$ and $\mathbb{R}^{n-1}$

$$\widehat{u(0,\cdot)}(\xi')=(2\pi)^{-1}\int_{\mathbb{R}}\hat u(\xi_1,\xi')d\xi_1,\ \forall\ \xi '\in\mathbb{R}^{n-1}.$$ (9)

Use Cauchy's inequality to show that this is continuous as a map on Sobolev spaces as indicated and then the density of $\mathcal{S}(\mathbb{R}^n)$ in $H^m(\mathbb{R}^n)$ to conclude that the map is well-defined and unique.

Restriction by WF: From class we know that the product of two distributions, one with compact support, is defined provided they have no `opposite' directions in their wavefront set:

$$(x,\omega )\in\operatorname{WF}(u)\Longrightarrow (x,-\omega )\notin\operatorname{WF}(v) uv\in\mathcal{C}^{-\infty}_{\text{c}}(\mathbb{R}^n).$$ (10)

Show that this product has the property that $f(uv)=(fu)v=u(fv)$ if $f\in\mathcal{C}^{\infty}(\mathbb{R}^n).$ Use this to define a restriction map to $x_1=0$ for distributions of compact support satisfying $((0,x'),(\omega _1,0))\notin\operatorname{WF}(u)$ as the product

$$u_0=u\delta (x_1).$$ (11)

[Show that $u_0(f),$ $f\in\mathcal{C}^{\infty}(\mathbb{R}^n)$ only depends on $f(0,\cdot)\in\mathcal{C}^{\infty}(\mathbb{R}^{n-1}).$