Exercise 27, § 3

Let $n\in \mathbf{N}$. Let N denote the euclidean norm on $\mathbf{R}^n$ and $\mu$ the Lebesgue measure on every open subset of $\mathbf{R}^n$. Let Δ be the differential operator

viewed as an endomorphism of $\mathscr{D}'(\mathbf{R}^n)$.

For $k\in \mathbf{R}$, let $S^k(\mathbf{R}^n)$ denote the space of tempered distributions $f$ in $\mathscr{S}'(\mathbf{R}^n)$ such that $(1 + N^k)\mathscr{F}(f)$ belong to $L^2(\mathbf{R}^n)$.

Let $U\subset \mathbf{R}^n$ be an open subset of $\mathbf{R}^n$. Let $S^k_U(\mathbf{R}^n)$ denote the space of distributions $f\in \mathscr{D}'(\mathbf{R}^n)$ such that for every test function $\varphi \in \mathscr{D}(\mathbf{R}^n)$ with support contained in U, one has $\varphi f\in S^k(\mathbf{R}^n)$.

a) The space $S^k(\mathbf{R}^n)$ is a Hilbert space with scalar product

When $k\in \mathbf{N}$, one has $W^k(\mathbf{R}^n) = S^k(\mathbf{R}^n)$ and the norms of these two Banach spaces are equivalent; moreover, one then has $W^k(\mathbf{R}^n)\subset S^k_U(\mathbf{R}^n)$.

b) The dual of $S^k(\mathbf{R}^n)$ is identified with $S^{-k}(\mathbf{R}^n)$ by the bilinear form

for $(f, g)\in S^{-k}(\mathbf{R}^n)\times S^k(\mathbf{R}^n)$.

c) Suppose that U is relatively compact. One has

(Let $f\in \mathscr{D}'(\mathbf{R}^n)$; apply the results of exercise 19 to $\varphi f$, where $\varphi \in \mathscr{D}(\mathbf{R}^n)$ is equal to 1 on U.)

d) Let $k\in \mathbf{R}$ and $f\in S^k(\mathbf{R}^n)$. For every integer $i$ such that $1\leqslant i\leqslant n$, one has $\partial_if\in S^{k-1}(\mathbf{R}^n)$. Moreover, if $\Delta (f)\in S^k(\mathbf{R}^n)$, then $f\in S^{k+2}(\mathbf{R}^n)$.

e) Let $k\in \mathbf{R}$ and $f\in S^k_U(\mathbf{R}^n)$. If $\Delta (f)\in S^k_U(\mathbf{R}^n)$, then $f\in S^{k+2}_U(\mathbf{R}^n)$.

f) Let $k > n/2$ be a real number and $m\in \mathbf{N}$ an integer such that $m < k-n/2$. The space $S^k(\mathbf{R}^n)$ is contained in $\mathscr{C}^m(\mathbf{R}^n)$. (Using the CauchySchwarz inequality, verify that $x\mapsto x^{\alpha}\mathscr{F}f(x)$ belongs to $L^1(\mathbf{R}^n)$ for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant m$. Now prove that $\mathscr{F}(f)$ is differentiable on $\mathbf{R}$ by using the fourier inversion formula.)

g) Let $k > n/2$ be a real number and $m\in \mathbf{N}$ an integer such that $m < k-n/2$. The restriction to U of an element of $S^k_U(\mathbf{R}^n)$ belongs to $\mathscr{C}^m(U)$. (For each $x\in U$, verify that one obtains a well-defined function $g\in \mathscr{C}^m(U)$ by setting $g(x) = (\varphi_xf)(x)$, where $\varphi_x\in \mathscr{D}(\mathbf{R}^n)$ is a function with support in U equal to 1 in a neighbourhood of $x$; verify finally that the restriction of $f$ to U is equal to $g.$)

h) Let $g\in \mathscr{D}'(\mathbf{R}^n)$. If $f\in \mathscr{D}'(\mathbf{R}^n)$ satisfies $\Delta (f) =g$ and if the restriction of $g$ to U is in $\mathscr{C}^{\infty}$(U), then the restriction of $f$ to U is in $\mathscr{C}^{\infty}(U)$ ("elliptic regularity theorem").