Integration by parts for dual-exponent Sobolev functions

The four factors in the asserted products lie in conjugate $L^p$ spaces: $u,D_{i}v\in L^p\times L^{p^{\mathrm{\prime }}}$ and $v,D_{i}u\in L^{p^{\mathrm{\prime }}}\times L^p$. By [F3] each product is integrable, with the endpoint pairs covered by the same inequality. The Sobolev test pairing in [F1] is bilinear, so no conjugation enters the desired identity.

If $u=0$ as an a.e. class, then $D_{i}u=0$ by uniqueness of the weak derivative; both sides are zero. The same argument applies if $v=0$.

Let $g\in W^{1,r}(\mathrm{\Omega })$, $1\le r\le \mathrm{\infty }$, vanish almost everywhere outside a compact $K\subseteq \mathrm{\Omega }$. On the open set $V=\mathrm{\Omega }\setminus K$, $g=0$ as a class. By restriction in [F4], $D_{i}g{\mathrm{\mid }}_V$ is a weak derivative of zero there; zero is also such a derivative, so uniqueness in [F5] gives $D_{i}g=0$ almost everywhere on $V$. If $K=\mathrm{\varnothing }$, this proves that $g=D_{i}g=0$ as classes, and any asserted pairing involving this compact factor vanishes.

First take $1\le r<\mathrm{\infty }$. Choose $\chi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ equal to one on a neighborhood of $K$ by [F6]. Extend representatives of $g$ and $D_{i}g$ by zero to $G$ and $H$ on ${\mathbb{R}}^n$. For every $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, the functions $g{\mathrm{\partial }}_i\varphi$ and $g{\mathrm{\partial }}_i(\chi \varphi )$ agree almost everywhere: they agree on a neighborhood of $K$, and $g=0$ off $K$. Likewise $D_{i}g\varphi$ and $D_{i}g\chi \varphi$ agree almost everywhere by step 1.3. The weak identity [F2] for $g$ applied to $\chi \varphi$ therefore gives ${\int }_{{\mathbb{R}}^n}G{\mathrm{\partial }}_i\varphi ={\int }_{\mathrm{\Omega }}g{\mathrm{\partial }}_i(\chi \varphi )=-{\int }_{\mathrm{\Omega }}{D}_{i}g\chi \varphi =-{\int }_{{\mathbb{R}}^n}H\varphi \mathrm{.}$ Thus $H$ is the weak derivative of $G$ on ${\mathbb{R}}^n$, and $G,H\in L^r$.

The function $\eta$ from [F7] is integrable: it is bounded and supported on a compact set of finite measure by [F8]. Its integral $c$ is positive: the box $Q=[-1\mathrm{/}(2\sqrt{n}),1\mathrm{/}(2\sqrt{n}){]}^n$ lies in the unit ball, since for $x\in Q$ one has $\mathrm{\mid }x{\mathrm{\mid }}^2\le n(1\mathrm{/}(2\sqrt{n}){)}^2=1\mathrm{/}4<1$. By [F9], ${\lambda }_n(Q)=(1\mathrm{/}\sqrt{n}{)}^n>0$; as $\eta =1$ on $Q$ and $\eta \ge 0$, its integral is at least ${\lambda }_n(Q)$. Put $\rho =\eta \mathrm{/}c$ and use the family ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x\mathrm{/}\epsilon )$ from [F12]. By step 1.3, if $K=\mathrm{\varnothing }$, then $G$ and $H$ are zero classes and $G_{\epsilon }=0$ for every \varepsilon, so support and convergence are immediate. Assume $K$ is nonempty below. For $G_{\epsilon }={\rho }_{\epsilon }\ast G$, [F11] gives smoothness and compact support. The integral defining $G_{\epsilon }(x)$ vanishes whenever $\mathrm{dist}(x,K)>2\epsilon$, because $G=0$ off $K$ and $\mathrm{supp}{\rho }_{\epsilon }\subseteq {\bar{B}}_{2\epsilon }(0)$. For each $x\in K$, openness gives a ball $B(x,r_x)\subseteq \mathrm{\Omega }$. Compactness supplies finitely many half-radius balls covering $K$. Let $\delta$ be one quarter of the least radius in this finite list. For every $0<\epsilon <\delta$, the closed $2\epsilon$-neighborhood of $K$ lies in the union of the corresponding $B(x,r_x)$: every point $y$ in the neighborhood has some $z\in K$ with $\mathrm{\mid }y-z\mathrm{\mid }\le 2\epsilon$; a covering half-radius ball $B(x,r_x\mathrm{/}2)$ contains $z$, so $\mathrm{\mid }y-x\mathrm{\mid }<r_x\mathrm{/}2+2\epsilon <r_x$. The neighborhood is compact, being $K+{\bar{B}}_{2\epsilon }(0)$, and hence lies in $\mathrm{\Omega }$. Therefore $G_{\epsilon }$ vanishes off a compact subset of $\mathrm{\Omega }$, so $G_{\epsilon }\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Since $G,H$ are supported in $K$, they are in $L^1({\mathbb{R}}^n)$: for $r=1$ this is immediate, and for $1<r<\mathrm{\infty }$ it follows from finite measure of $K$ [F8] and Hölder [F3]. Thus the convolutions in this step are defined.

At every $x$, the weak identity [F2] for $G$ may be tested with the compactly supported smooth function $y\mapsto {\rho }_{\epsilon }(x-y)$. Since ${\mathrm{\partial }}_{y_i}{\rho }_{\epsilon }(x-y)=-{\mathrm{\partial }}_{x_i}{\rho }_{\epsilon }(x-y)$, it gives ${\mathrm{\partial }}_i{G}_{\epsilon }(x)=\int G(y){\mathrm{\partial }}_{x_i}{\rho }_{\epsilon }(x-y)dy=\int H(y){\rho }_{\epsilon }(x-y)dy=({\rho }_{\epsilon }\ast H)(x)\mathrm{.}$ By the finite-$L^r$ convergence in [F10], applied to $G$ and $H$, $\mathrm{\parallel }{G}_{\epsilon }-G{\mathrm{\parallel }}_{L^r({\mathbb{R}}^n)}+\mathrm{\parallel }{\mathrm{\partial }}_i{G}_{\epsilon }-H{\mathrm{\parallel }}_{L^r({\mathbb{R}}^n)}⟶0.$ Consequently, taking ${\epsilon }_j=\delta \mathrm{/}(j+1)$ gives restrictions in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ whose classical $i$th derivatives converge to $D_{i}g$ along with convergence to $g$ in $L^r$.

Suppose first that $u$ is compactly supported and $p<\mathrm{\infty }$. Apply steps 1.3–4.1 to $g=u$, $r=p$, obtaining $u_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $u_j\to u$ and ${\mathrm{\partial }}_i{u}_j\to D_{i}u$ in $L^p$. The weak identity [F2] for $v$ with weak derivative $D_{i}v$, tested against $u_j$, says ${\int }_{\mathrm{\Omega }}{u}_j{D}_{i}v=-{\int }_{\mathrm{\Omega }}v{\mathrm{\partial }}_i{u}_j\mathrm{.}$ By [F3], the differences between the left sides and $\int u{D}_{i}v$ are at most $\mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_p\mathrm{\parallel }{D}_{i}v{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$, while the differences between the right sides and $-\int v{D}_{i}u$ are at most $\mathrm{\parallel }v{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\mathrm{\parallel }{\mathrm{\partial }}_i{u}_j-D_{i}u{\mathrm{\parallel }}_p$. Both tend to zero, proving the claim in this case. If instead $v$ is compactly supported and $p^{\mathrm{\prime }}<\mathrm{\infty }$, apply the same approximation to $v$ in $L^{p^{\mathrm{\prime }}}$ and test the weak identity for $u$ with derivative $D_{i}u$; Hölder gives the two corresponding limits. This covers every case with a compactly supported factor of finite exponent, including $p=1$ when $u$ is compact and $p=\mathrm{\infty }$ when $v$ is compact.

It remains only the case in which the compact factor has exponent $\mathrm{\infty }$ and its partner has exponent $1$. Write the compact factor as $g\in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$, the other as $h\in W^{1,1}(\mathrm{\Omega })$, and let $K$ contain the essential support of $g$. The case $K=\mathrm{\varnothing }$ is already settled by step 1.3. Choose $\chi$ as in [F6] with $\chi =1$ on a neighborhood of $K$, and put $w=\chi h$. For any $\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, [F13] and the weak identity [F2] for $h$, tested with $\chi \varphi$, give ${\int }_{\mathrm{\Omega }}w{\mathrm{\partial }}_i\varphi ={\int }_{\mathrm{\Omega }}h{\mathrm{\partial }}_i(\chi \varphi )-{\int }_{\mathrm{\Omega }}h({\mathrm{\partial }}_i\chi )\varphi =-{\int }_{\mathrm{\Omega }}(\chi D_{i}h+h{\mathrm{\partial }}_i\chi )\varphi \mathrm{.}$ For each coordinate $j$, the same test calculation gives the weak derivative $D_{j}w=\chi D_{j}h+h{\mathrm{\partial }}_j\chi$. Each displayed candidate and $w$ are in $L^1$ and compactly supported, so the definition [F1] gives $w\in W^{1,1}(\mathrm{\Omega })$. By steps 1.3–4.1, choose $w_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $w_j\to w$ and ${\mathrm{\partial }}_i{w}_j\to D_{i}w$ in $L^1$. Test the weak identity [F2] for $g$, whose weak derivative is $D_{i}g$, with $w_j$: ${\int }_{\mathrm{\Omega }}g{\mathrm{\partial }}_i{w}_j=-{\int }_{\mathrm{\Omega }}(D_{i}g)w_j\mathrm{.}$ The limits pass by $\mathrm{\mid }\int g({\mathrm{\partial }}_i{w}_j-D_{i}w)\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{\mathrm{\partial }}_i{w}_j-D_{i}w{\mathrm{\parallel }}_1$ and the analogous estimate using $D_{i}g\in L^{\mathrm{\infty }}$. On a neighborhood of $K$, $w=h$ as a.e. classes, so restriction [F4] and uniqueness [F5] give $D_{i}w=D_{i}h$ there. Off $K$, both $g$ and $D_{i}g$ vanish almost everywhere by step 1.3. Thus the limit is ${\int }_{\mathrm{\Omega }}g{D}_{i}h=-{\int }_{\mathrm{\Omega }}h{D}_{i}g\mathrm{.}$ It gives the required orientation when $g=u,h=v$; when $g=v,h=u$, rearrange the same equality. This handles the remaining endpoint cases without any strong approximation in $W^{1,\mathrm{\infty }}$.

If $p=1$, a compactly supported $u$ is handled by step 5.1; if only $v\in W^{1,\mathrm{\infty }}$ is compactly supported, step 5.2 applies. If $p=\mathrm{\infty }$, a compactly supported $v\in W^{1,1}$ is handled by step 5.1; if only $u\in W^{1,\mathrm{\infty }}$ is compactly supported, step 5.2 applies. For $1<p<\mathrm{\infty }$, both exponents are finite, so step 5.1 applies whichever factor is compact. The zero-factor and empty-domain cases were addressed in the Statement and steps 1.2–1.3. If one starts instead from the stronger full Axiom of Choice [F15], then for each countable family of nonempty sets $(X_m{)}_{m\in \mathbb{N}}$ AC supplies a choice function on its range; composing with $m\mapsto X_m$ gives a choice function for the indexed family. Thus AC implies Countable Choice [F14], which is the only role of full AC and licenses exactly the Wkp, weak-derivative uniqueness, finite-measure, and mollification interfaces used in [F1], [F5], [F8], [F9], [F10], and [F11]. Under the Statement's Countable Choice hypothesis, no stronger choice principle is used.