The difference-quotient characterisation of $W^{1,p}$ for $1<p<\mathrm{\infty }$

Let $v\in C^{\mathrm{\infty }}(\mathrm{\Omega })\cap W^{1,p}(\mathrm{\Omega })$ and $0<\mathrm{\mid }h\mathrm{\mid }<\mathrm{dist}({\mathrm{\Omega }}^{\mathrm{\prime }},\mathrm{\partial }\mathrm{\Omega })$. Take first $h>0$ and fix $x\in {\mathrm{\Omega }}^{\mathrm{\prime }}$; by [F4] and the convexity of $t\mapsto t^p$, Jensen's inequality [F5] for the probability measure ${\mu }_h$ gives $\mathrm{\mid }{\delta }_h^{i}v(x){\mathrm{\mid }}^p=\mid \frac{1}{h}{\int }_0^h{\mathrm{\partial }}_{i}v(x+t{e}_i)dt{\mid }^p\le \frac{1}{h}{\int }_0^h\mathrm{\mid }{\mathrm{\partial }}_{i}v(x+t{e}_i){\mathrm{\mid }}^{p}dt\mathrm{.}$ For $h<0$ the same computation applies with the segment from $x+h{e}_i$ to $x$ and the average taken over $[h,0]$.

For part (2), choose a finite $h_0>0$ with $h_0<\mathrm{dist}({\mathrm{\Omega }}^{\mathrm{\prime }},\mathrm{\partial }\mathrm{\Omega })\mathrm{/}2$; when $\mathrm{\Omega }={\mathbb{R}}^n$ and the distance is infinite, take $h_0=1$; the hypothesis is exactly the uniform bound required by the weak-limit lemma [F8], which gives $u\in W^{1,p}({\mathrm{\Omega }}^{\mathrm{\prime }})$ with $\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}\le C$ and the convergence ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{\delta }_h^{i}u\phi \to {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}{D}_{i}u\phi$ for every $\phi \in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{\mathrm{\prime }})$; by [F7] the latter is precisely weak convergence ${\delta }_h^{i}u\rightharpoonup D_{i}u$ in $L^p({\mathrm{\Omega }}^{\mathrm{\prime }})$.

Integrating the estimate of step 1.1 over ${\mathrm{\Omega }}^{\mathrm{\prime }}$ and applying Fubini and the translation invariance of [F6] (every point $x+t{e}_i$ with $x\in {\mathrm{\Omega }}^{\mathrm{\prime }}$, $t$ between $0$ and $h$, satisfies $\mathrm{dist}(x+t{e}_i,\mathrm{\partial }\mathrm{\Omega })>0$ because $\mathrm{\mid }h\mathrm{\mid }<\mathrm{dist}({\mathrm{\Omega }}^{\mathrm{\prime }},\mathrm{\partial }\mathrm{\Omega })$) gives, for either sign of $h$, ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}\mathrm{\mid }{\delta }_h^{i}v{\mathrm{\mid }}^{p}dx\le \frac{1}{h}{\int }_0^h{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}\mathrm{\mid }{\mathrm{\partial }}_{i}v(x+t{e}_i){\mathrm{\mid }}^{p}dxdt\le {\int }_{\mathrm{\Omega }}\mathrm{\mid }{\mathrm{\partial }}_{i}v{\mathrm{\mid }}^{p}dx,$ so $\mathrm{\parallel }{\delta }_h^{i}v{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}\le \mathrm{\parallel }{D}_{i}v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$.

Let now $1\le p<\mathrm{\infty }$ and $u\in W^{1,p}(\mathrm{\Omega })$ and choose $v_j\in C^{\mathrm{\infty }}(\mathrm{\Omega })\cap W^{1,p}(\mathrm{\Omega })$ with $\mathrm{\parallel }{v}_j-u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}\to 0$, as [F3] permits. For fixed $h$ with $0<\mathrm{\mid }h\mathrm{\mid }<\mathrm{dist}({\mathrm{\Omega }}^{\mathrm{\prime }},\mathrm{\partial }\mathrm{\Omega })$, [F1] applied to $w=v_j-u$ gives $\mathrm{\parallel }{\delta }_h^i{v}_j-{\delta }_h^{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}\le 2\mathrm{\parallel }{v}_j-u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\mathrm{/}\mathrm{\mid }h\mathrm{\mid }\to 0$, and $\mathrm{\parallel }{D}_i{v}_j-D_{i}u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\to 0$ by [F2]; step 2.1 applied to each $v_j$ and passage to the limit (norms are continuous) yield $\mathrm{\parallel }{\delta }_h^{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}^{\mathrm{\prime }})}\le \mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$.

The coordinate argument in steps 1.1--3.1 works on any open $U$ with the segment condition: each translated set $U+th{e}_i$ lies in $\mathrm{\Omega }$, so Tonelli and change of variables bound its integral by the gradient norm on $\mathrm{\Omega }$. For $p=\mathrm{\infty }$, fix a compact $K\subset U$. The union of its segments is compact in $\mathrm{\Omega }$, so choose a bounded open tube $T⋐\mathrm{\Omega }$ containing that union. For each finite $q\ge 1$, the same coordinate argument applied to $u{\mathrm{\mid }}_T$ gives $\mathrm{\parallel }{\delta }_h^{i}u{\mathrm{\parallel }}_{L^q(K)}\le \mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\mathrm{\mid }T{\mathrm{\mid }}^{1\mathrm{/}q}$. If the quotient exceeded this essential bound by $\epsilon$ on a positive-measure subset of $K$, its $L^q$ norm would exceed $(\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_{\mathrm{\infty }}+\epsilon )\mathrm{\mid }E{\mathrm{\mid }}^{1\mathrm{/}q}$, a contradiction as $q\to \mathrm{\infty }$. Exhausting $U$ by compact sets proves the $L^{\mathrm{\infty }}$ coordinate bound.

Tangential strong convergence. For a tangential direction on $H$, smooth approximation [F3] and the fixed-$h$ translation bounds pass the segment formula to $u$: ${\delta }_h^{i}u(x)={\int }_0^1{D}_{i}u(x+th{e}_i)dt$ as $L^p(H)$ classes. Indeed Jensen and tangential change of variables bound the $L^p$ error between the averages of two gradient approximations by their $L^p(H)$ distance. Extend $D_{i}u$ by zero as an $L^p$ class on ${\mathbb{R}}^n$; tangential shifts preserve $H$, so [F9] gives $\mathrm{\parallel }{D}_{i}u(\cdot +t{e}_i)-D_{i}u{\mathrm{\parallel }}_{L^p(H)}\to 0$ uniformly for $\mathrm{\mid }t\mathrm{\mid }\le \mathrm{\mid }h\mathrm{\mid }$ as $h\to 0$. Jensen and Tonelli applied to the segment formula therefore give $\mathrm{\parallel }{\delta }_h^{i}u-D_{i}u{\mathrm{\parallel }}_{L^p(H)}^p\le {\int }_0^1\mathrm{\parallel }{D}_{i}u(\cdot +th{e}_i)-D_{i}u{\mathrm{\parallel }}_{L^p(H)}^{p}dt\to 0$.

Vector estimate. Put $z_i={\delta }_h^{i}u$ and $d_i=D_{i}u$. Step 3.1 gives $\mathrm{\parallel }{z}_i{\mathrm{\parallel }}_p\le \mathrm{\parallel }{d}_i{\mathrm{\parallel }}_p$ for each coordinate. If $1\le p\le 2$, the finite-dimensional inequalities yield $\mathrm{\parallel }z{\mathrm{\parallel }}_{L^p({\mathrm{\ell }}^2)}\le ({\sum }_i\mathrm{\parallel }{z}_i{\mathrm{\parallel }}_p^p{)}^{1\mathrm{/}p}\le ({\sum }_i\mathrm{\parallel }{d}_i{\mathrm{\parallel }}_p^p{)}^{1\mathrm{/}p}\le n^{1\mathrm{/}p-1\mathrm{/}2}\mathrm{\parallel }d{\mathrm{\parallel }}_{L^p({\mathrm{\ell }}^2)}$. If $2\le p<\mathrm{\infty }$, the triangle inequality in $L^{p\mathrm{/}2}$ gives $\mathrm{\parallel }z{\mathrm{\parallel }}_{L^p({\mathrm{\ell }}^2)}\le ({\sum }_i\mathrm{\parallel }{z}_i{\mathrm{\parallel }}_p^2{)}^{1\mathrm{/}2}\le ({\sum }_i\mathrm{\parallel }{d}_i{\mathrm{\parallel }}_p^2{)}^{1\mathrm{/}2}\le n^{1\mathrm{/}2-1\mathrm{/}p}({\sum }_i\mathrm{\parallel }{d}_i{\mathrm{\parallel }}_p^p{)}^{1\mathrm{/}p}\le n^{1\mathrm{/}2-1\mathrm{/}p}\mathrm{\parallel }d{\mathrm{\parallel }}_{L^p({\mathrm{\ell }}^2)}$. For $p=\mathrm{\infty }$, step 4.1 gives $\mathrm{\mid }{z}_i\mathrm{\mid }\le \mathrm{\parallel }{d}_i{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }d{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathrm{\ell }}^2)}$ a.e., hence $\mathrm{\parallel }z{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathrm{\ell }}^2)}\le \sqrt{n}\mathrm{\parallel }d{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathrm{\ell }}^2)}$. These finite-dimensional comparisons follow from Hölder applied to the finite sum. Thus the stated vector constant is valid, and equals one at $p=2$. Each component uses its own coordinate segment; no common translated gradient vector is asserted.

Steps 3.1--4.1 prove (1) for every $u\in W^{1,p}(\mathrm{\Omega })$, and step 1.2 proves (2) for $1<p<\mathrm{\infty }$; no step used any regularity of $\mathrm{\partial }\mathrm{\Omega }$, only the containedness ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }$ and the shrunken-domain definition of the quotients, so both assertions are identities of classes on arbitrary open sets.