Uniformly bounded difference quotients represent a weak derivative

Fix $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})$ and put $h_1:=\mathrm{dist}(\mathrm{supp}\phi ,\mathrm{\partial }\mathrm{\Omega })>0$. For every $0<\mathrm{\mid }h\mathrm{\mid }<\min (h_0,h_1)$ the integration-by-parts identity of [F2] applies (its support condition holds because $\mathrm{supp}\phi ⋐{\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathrm{\Omega }$) and gives ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\phi dx={\int }_{{\mathrm{\Omega }}_{i,h}}{\delta }_h^{i}u\phi dx=-{\int }_{{\mathrm{\Omega }}_{i,-h}}u{\delta }_{-h}^i\phi dx=-{\int }_{\mathrm{\Omega }}u{\delta }_{-h}^i\phi dx\mathrm{.}$ Here $\phi$ is extended by zero to ${\mathbb{R}}^n$; its difference quotient is globally defined and supported in ${\mathrm{\Omega }}_{i,-h}$ for these $h$, whereas ${\delta }_h^{i}u$ is integrated only where it is defined.

The test functions are dense in $L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$: given $f\in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$ and $\eta >0$, [F6] provides $v\in C^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})\cap L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$ with $\mathrm{\parallel }v-f{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}<\eta \mathrm{/}2$. Choose a compact exhaustion $K_1\subseteq K_2\subseteq \cdots$ of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ with $K_j\subseteq \mathrm{int}{K}_{j+1}$ and cutoffs ${\chi }_j\in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }};[0,1])$ equal to $1$ on $K_j$; then ${\chi }_{j}v\to v$ pointwise everywhere and $\mathrm{\mid }{\chi }_{j}v-v{\mathrm{\mid }}^{p^{\mathrm{\prime }}}\le 2^{p^{\mathrm{\prime }}}\mathrm{\mid }v{\mathrm{\mid }}^{p^{\mathrm{\prime }}}\in L^1({\mathrm{\Omega }}^{\mathrm{\prime }})$, so [F5] gives $\mathrm{\parallel }{\chi }_{j}v-v{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}\to 0$, and for $j$ large ${\chi }_{j}v\in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})$ lies within $\eta$ of $f$.

By [F3] the classes ${\delta }_{-h}^i\phi$ converge pointwise as $h\to 0$ to ${\mathrm{\partial }}_i\phi$ and are bounded in absolute value by $\mathrm{\parallel }D\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$; choose a fixed compact neighbourhood $K⋐\mathrm{\Omega }$ containing $\mathrm{supp}\phi$ and all its translates by $t{e}_i$ for sufficiently small $\mathrm{\mid }t\mathrm{\mid }$. The support of every such ${\delta }_{-h}^i\phi$ lies in $K$, and $u\in L^1(K)$ by Hölder, so ${\mathbf{1}}_K\mathrm{\mid }u\mathrm{\mid }\mathrm{\parallel }D\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$ is an integrable majorant on $\mathrm{\Omega }$, and [F5] gives $-{\int }_{\mathrm{\Omega }}u{\delta }_{-h}^i\phi dx⟶-{\int }_{\mathrm{\Omega }}u{\mathrm{\partial }}_i\phi dx=:\mathrm{\ell }(\phi )\mathrm{.}$ Thus the limit exists for every test function, and step 1.1 identifies it with ${\lim }_{h\to 0}{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\phi dx$.

For every $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})$ the hypothesis and Hölder give, for all $0<\mathrm{\mid }h\mathrm{\mid }<\min (h_0,h_1)$, $\mid {\int }_{{\mathrm{\Omega }}_{i,h}}{\delta }_h^{i}u\phi dx\mid =\mid {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\phi dx\mid \le \mathrm{\parallel }{\delta }_h^{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})}\le C\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})};$ passing to the limit along $h\to 0$ in step 2.1 yields $\mathrm{\mid }\mathrm{\ell }(\phi )\mathrm{\mid }\le C\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})}$. Hence $\mathrm{\ell }$ is a $\mathbb{K}$-linear functional on the subspace $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})$ of $L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$, bounded there with constant $C$.

By step 1.2 and the boundedness of step 3.1, $\mathrm{\ell }$ has a unique extension to a bounded linear functional $\mathrm{\Lambda }$ on $L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$ with $\mathrm{\parallel }\mathrm{\Lambda }\mathrm{\parallel }\le C$: for $f\in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$ choose test functions ${\phi }_j\to f$; the values $\mathrm{\ell }({\phi }_j)$ form a Cauchy sequence because $\mathrm{\mid }\mathrm{\ell }({\phi }_j)-\mathrm{\ell }({\phi }_k)\mathrm{\mid }\le C\mathrm{\parallel }{\phi }_j-{\phi }_k\mathrm{\parallel }$, and $\mathrm{\Lambda }(f):={\lim }_j\mathrm{\ell }({\phi }_j)$ is independent of the approximating sequence.

Apply [F7] with $q=p^{\mathrm{\prime }}$ (so that $q^{\mathrm{\prime }}=p$) to the functional $\mathrm{\Lambda }$ on $L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$: for $\mathbb{K}=\mathbb{R}$ the real duality theorem, and for $\mathbb{K}=\mathbb{C}$ the complex-linear duality lemma, provide a class $w\in L^p({\mathrm{\Omega }}^{\mathrm{\prime }};\mathbb{K})$ with $\mathrm{\Lambda }(g)={\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}wgdx(g\in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})),\mathrm{\parallel }w{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}=\mathrm{\parallel }\mathrm{\Lambda }\mathrm{\parallel }\le C\mathrm{.}$

For every $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})$ (so that $\mathrm{supp}\phi ⋐\mathrm{\Omega }$ and ${\mathrm{\partial }}_i\phi$ has the same support) steps 2.1 and 5.1 give ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}u{\mathrm{\partial }}_i\phi dx=-\mathrm{\ell }(\phi )=-\mathrm{\Lambda }(\phi )=-{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}w\phi dx\mathrm{.}$ By [F8] this is exactly the weak-derivative identity, so $w$ is the weak derivative $D_{i}u$ of $u$ on ${\mathrm{\Omega }}^{\mathrm{\prime }}$ and $\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}\le C$.

It remains to remove the test-function restriction in the convergence. Let $\phi \in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$ and $\eta >0$; by step 1.2 choose $\psi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})$ with $\mathrm{\parallel }\phi -\psi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}<\eta \mathrm{/}(2C+2\mathrm{\parallel }w{\mathrm{\parallel }}_{L^p}+1)$, and then $0<\mathrm{\mid }h\mathrm{\mid }$ small enough that $\mid {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\psi dx-{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}w\psi dx\mid <\eta \mathrm{/}2$, which is possible by steps 2.1 and 5.1. For such $h$, $\mid {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\phi -{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}w\phi \mid \le \mathrm{\parallel }{\delta }_h^{i}u{\mathrm{\parallel }}_{L^p}\mathrm{\parallel }\phi -\psi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}+\mid {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}({\delta }_h^{i}u-w)\psi \mid +\mathrm{\parallel }w{\mathrm{\parallel }}_{L^p}\mathrm{\parallel }\psi -\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}<\eta ,$ using [F4] twice and the uniform bound of the hypothesis. Hence ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\phi \to {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}w\phi$ for every $\phi \in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$, which with step 6.1 proves the weak convergence ${\delta }_h^{i}u\rightharpoonup D_{i}u$ in $L^p({\mathrm{\Omega }}^{\mathrm{\prime }})$.