Lemma 16.15.16BB3FD42D191C

Proof. (1) Fix $s$ with $0<|s|<s_0$. We claim that $\delta _ sZ(x)=\int _0^1\partial _ eZ(x+tse)\, \mathrm dt$ for almost every $x\in D'$. For smooth $Z$ this is the fundamental theorem of calculus. In general, let $K\subset D'$ be compact; the set $K'=\{ x+tse:x\in K,\ 0\leq t\leq 1\} $ is a compact subset of $D$. The mollifications $Z_\varepsilon =Z*\rho _\varepsilon $ are smooth near $K'$ for small $\varepsilon $, they converge to $Z$ in $L^1(K')$, and $\partial _ eZ_\varepsilon =(\partial _ eZ)*\rho _\varepsilon \to \partial _ eZ$ in $L^1(K')$, by proposition 2.5.3(1) and lemma 2.5.1(2). Hence the identity for $Z_\varepsilon $ passes to the limit in $L^1(K)$; for the right side we use $\int _ K\int _0^1|F(x+tse)|\, \mathrm dt\, \mathrm dx\leq \int _{K'}|F|$, by Tonelli's theorem. This proves the claim. By Hölder's inequality on $[0,1]$ and Tonelli's theorem,

Similarly, since $\delta _ sZ(x)-\partial _ eZ(x)=\int _0^1\bigl (\partial _ eZ(x+tse)-\partial _ eZ(x)\bigr )\, \mathrm dt$ for almost every $x\in D'$, Hölder's inequality and Tonelli's theorem give, with $F$ the extension of $\partial _ eZ$ by zero to $\mathbb R^ d$,

which tends to zero as $s\to 0$ by the continuity of translation in $L^ p(\mathbb R^ d)$ (lemma 2.5.1(2)).

(2) Let $\chi \in C_ c^\infty (D')$ with $0\leq \chi \leq 1$, extend $\chi Z$ by zero, and let $0<|s|<\min \{ s_1,\operatorname {dist}(\operatorname{Supp}\chi ,\mathbb R^ d\setminus D')\} $. Then $\delta _ s(\chi Z)$ vanishes outside $D'$, and on $D'$ it equals $(\tau _ s\chi )\delta _ sZ+(\delta _ s\chi )Z$. Since $D'$ is bounded and $p\geq 2$, Hölder's inequality gives

The function $\chi Z$ lies in $L^1\cap L^2$, so its $L^2$ Fourier transform is given by the integral (theorem 2.9.4), and $\widehat{\delta _ s(\chi Z)}=m_ s\widehat{\chi Z}$ with $m_ s(\xi )=(\mathrm e^{2\pi \mathrm is\xi _ e}-1)/s$, where $\xi _ e$ is the component of $\xi $ in the direction $e$. By Plancherel's theorem, $\int |m_ s|^2|\widehat{\chi Z}|^2\leq M'^2$, and as $m_ s(\xi )\to 2\pi \mathrm i\xi _ e$, Fatou's lemma gives $\int 4\pi ^2\xi _ e^2|\widehat{\chi Z}|^2\leq M'^2$. Let $F\in L^2$ be the inverse $L^2$ transform of $2\pi \mathrm i\xi _ e\widehat{\chi Z}$. For a test function $\phi $ we have $\widehat{\partial _ e\phi }=2\pi \mathrm i\xi _ e\widehat\phi $, by integration by parts, so Parseval's identity, which extends to $L^2$ because the transform is unitary, gives

Thus $\partial _ e(\chi Z)=F\in L^2$. Since $\chi $ may equal $1$ near any given compact subset of $D'$, it follows that $\partial _ eZ\in L^2_{\mathrm{loc}}(D')$.

Now let $D''\Subset D'''\Subset D'$ be open. By (1), applied with $D'''$ in place of $D$ and $p=2$, we have $\delta _ sZ\to \partial _ eZ$ in $L^2(D'')$, so along a sequence $s_ j\to 0$ the functions $\delta _{s_ j}Z$ converge to $\partial _ eZ$ almost everywhere on $D''$. Fatou's lemma gives $\| \partial _ eZ\| _{L^ p(D'')}\leq \varliminf _ j\| \delta _{s_ j}Z\| _{L^ p(D'')}\leq M$. Exhausting $D'$ by such sets $D''$ and using monotone convergence, we obtain $\| \partial _ eZ\| _{L^ p(D')}\leq M$. $\square$