Mean-zero Poincare estimate on bounded connected extension domains below the dimension

The contradiction setup. Because $\mathrm{\Omega }$ is nonempty open and bounded, it contains a ball and $0<\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }<\mathrm{\infty }$ by [F20], so the mean $u_{\mathrm{\Omega }}$ is defined for every $u\in L^p(\mathrm{\Omega };\mathbb{K})$. Suppose the asserted constant does not exist. Then for every $j\ge 1$ there is $u_j\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$ with $\mathrm{\parallel }{u}_j-(u_j{)}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}>j\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$; the left side is positive, and Countable Choice [F1] selects such a sequence. Put $v_j:=(u_j-(u_j{)}_{\mathrm{\Omega }})\mathrm{/}\mathrm{\parallel }{u}_j-(u_j{)}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$. Then $v_j\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$, $(v_j{)}_{\mathrm{\Omega }}=0$, $\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}=1$ and $\mathrm{\parallel }D{v}_j{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}<1\mathrm{/}j$.

Translation differences. Every $W\in W^{1,p}({\mathbb{R}}^n;\mathbb{K})$ and $h\in {\mathbb{R}}^n$ satisfy $\mathrm{\parallel }W(\cdot -h)-W{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }DW{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$. For smooth compactly supported $\phi$ the fundamental theorem [F9] applied to $t\mapsto \phi (x-th)$ gives $\phi (x-h)-\phi (x)=-{\int }_0^{1}D\phi (x-th)hdt$, so Minkowski [F13] and translation invariance [F8] give $\mathrm{\parallel }\phi (\cdot -h)-\phi {\mathrm{\parallel }}_p\le \mathrm{\mid }h\mathrm{\mid }{\int }_0^1\mathrm{\parallel }D\phi (\cdot -th){\mathrm{\parallel }}_{p}dt=\mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }D\phi {\mathrm{\parallel }}_p$. For general $W$, [F7] provides ${\phi }_k\in C_c^{\mathrm{\infty }}$ with ${\phi }_k\to W$ in $W^{1,p}({\mathbb{R}}^n)$; applying the smooth bound to ${\phi }_k$ and letting $k\to \mathrm{\infty }$, using translation invariance [F8] on the left and strong convergence on the right, gives the claim.

Extension and cutoff. Fix $E$ as in [F4] and put $M_0:=\mathrm{\parallel }E\mathrm{\parallel }$. The closure $\bar{\mathrm{\Omega }}$ is compact. Choose a bounded open ball $U$ containing it; [F6] gives a smooth $\chi$ equal to $1$ on $\bar{\mathrm{\Omega }}$ with closed support contained in $U$. That support is bounded and hence compact, so $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$; put $K:=\mathrm{supp}\chi$, a compact set. Define $F_j:=\chi E{v}_j$. By [F5] each $F_j$ lies in $W^{1,p}({\mathbb{R}}^n;\mathbb{K})$, has support in $K$, restricts to $v_j$ on $\mathrm{\Omega }$ (because $\chi =1$ there) and satisfies $\mathrm{\parallel }{F}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}+\mathrm{\parallel }D{F}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le (n+1)(1+\mathrm{\parallel }\chi {\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }D\chi {\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\parallel }E{v}_j{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}\le M$ with $M:=(n+1{)}^2(1+\mathrm{\parallel }\chi {\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }D\chi {\mathrm{\parallel }}_{\mathrm{\infty }})M_0$, using $\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{W^{1,p}}\le n+1$ and the elementary bound of the Euclidean gradient norm by the sum of its coordinate norms from step 1.1 and the operator bound of [F4].

Mollification and equicontinuity at one scale. Fix $0<\epsilon \le 1$. By [F10] and [F11], $F_j\ast {\rho }_{\epsilon }$ is smooth on ${\mathbb{R}}^n$ with $D(F_j\ast {\rho }_{\epsilon })=F_j\ast D{\rho }_{\epsilon }$ and support in the compact set $K^{\mathrm{\prime }}:=K+{\bar{B}}_1(0)$. Holder [F12] gives, uniformly in $j$ and $x$, $\mathrm{\mid }(F_j\ast {\rho }_{\epsilon })(x)\mathrm{\mid }\le \mathrm{\parallel }{F}_j{\mathrm{\parallel }}_{L^p}\mathrm{\parallel }{\rho }_{\epsilon }{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}\le M\mathrm{\parallel }{\rho }_{\epsilon }{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}$ and $\mathrm{\mid }D(F_j\ast {\rho }_{\epsilon })(x)\mathrm{\mid }\le M\mathrm{\parallel }D{\rho }_{\epsilon }{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}$, using step 2.1. The second bound makes the family $\{F_j\ast {\rho }_{\epsilon }{\}}_j$ equicontinuous on the compact metric space $K^{\mathrm{\prime }}$ and the first makes it pointwise bounded.

Uniform mollification error. By [F10], $F_j\ast {\rho }_{\epsilon }-F_j=\int {\rho }_{\epsilon }(y)(F_j(\cdot -y)-F_j)dy$ on ${\mathbb{R}}^n$. Choose finite-valued Borel representatives of each $F_j$ using [F21], changing them to zero on a Borel null set and outside $K$. Then $(x,y)\mapsto F_j(x-y)$ and $F_j(x)$ are Borel measurable, since subtraction and projection are continuous. The integrand is therefore product measurable, with measurable absolute section integrals by [F19]. Minkowski [F13], the translation bound of step 1.2 and $\mathrm{supp}{\rho }_{\epsilon }\subseteq {\bar{B}}_{\epsilon }(0)$ give $\mathrm{\parallel }{F}_j\ast {\rho }_{\epsilon }-F_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le \int {\rho }_{\epsilon }(y)\mathrm{\parallel }{F}_j(\cdot -y)-F_j{\mathrm{\parallel }}_{L^p}dy\le \epsilon \mathrm{\parallel }D{F}_j{\mathrm{\parallel }}_{L^p}\le \epsilon M$, the last inequality by step 2.1.

One scale at a time. Fix $0<\epsilon \le 1$. The family $\{F_j\ast {\rho }_{\epsilon }{\}}_j$ is uniformly bounded and equicontinuous on the compact set $K^{\mathrm{\prime }}$ by step 3.1, so by Arzela-Ascoli [F14] its closure in $C(K^{\mathrm{\prime }};\mathbb{C})\cong C(K^{\mathrm{\prime }};\mathbb{R}{)}^2$ is compact; applying [F14] to the real and imaginary parts componentwise and then the sequential compactness of [F15], there is a subsequence $(j_k{)}_k$ and $G_{\epsilon }\in C(K^{\mathrm{\prime }};\mathbb{K})$ with $F_{j_k}\ast {\rho }_{\epsilon }\to G_{\epsilon }$ uniformly on $K^{\mathrm{\prime }}$.

Diagonalisation over the scales. Apply step 4.1 successively to the scales ${\epsilon }_m=2^{-m}$, each time to the previously selected subsequence, and select the $m$-th extracted subsequence at stage $m$ in such a way that the diagonal sequence $(j_k)$, where $j_k$ is the $k$-th index of the $k$-th subsequence, is strictly increasing; Dependent Choice [F2] formalises the recursion. Then for every fixed $m$ the tail $(F_{j_k}\ast {\rho }_{{\epsilon }_m}{)}_{k\ge m}$ is a subsequence of the $m$-th extracted subsequence, hence converges uniformly on $K^{\mathrm{\prime }}$ and in particular is Cauchy in $L^p(K^{\mathrm{\prime }})$.

Cauchy and the $L^p$ limit. For $k,l\ge m$, step 3.2 applied at scale ${\epsilon }_m$ and the uniform convergence on $K^{\mathrm{\prime }}$ of step 5.1 give $\mathrm{\parallel }{F}_{j_k}-F_{j_l}{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le 2M{\epsilon }_m+\mathrm{\parallel }{F}_{j_k}\ast {\rho }_{{\epsilon }_m}-F_{j_l}\ast {\rho }_{{\epsilon }_m}{\mathrm{\parallel }}_{L^p(K^{\mathrm{\prime }})}$. Given $\delta >0$ choose $m$ with $2M{2}^{-m}<\delta \mathrm{/}2$, then $k,l$ large enough that the second term is below $\delta \mathrm{/}2$; hence $(F_{j_k}{)}_k$ is Cauchy in $L^p({\mathbb{R}}^n;\mathbb{K})$ and converges by [F16] to some $F\in L^p({\mathbb{R}}^n;\mathbb{K})$.

Properties of the limit. Restrict $v:=F{\mathrm{\mid }}_{\mathrm{\Omega }}$. Since $F_{j_k}{\mathrm{\mid }}_{\mathrm{\Omega }}=v_{j_k}$ by step 2.1 and $\mathrm{\parallel }v-v_{j_k}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }F-F_{j_k}{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\to 0$ by step 6.1, the limit of the norms gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}=1$, and Holder [F12] with $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }<\mathrm{\infty }$ gives $\mid {\int }_{\mathrm{\Omega }}v\mid ={\lim }_k\mid {\int }_{\mathrm{\Omega }}{v}_{j_k}\mid =0$ because every $v_{j_k}$ has mean zero (step 1.1). Thus $v\in L^p(\mathrm{\Omega };\mathbb{K})$ has unit norm and mean zero.

The weak gradient of the limit vanishes. Let $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and $1\le i\le n$. Since $v_{j_k}$ is the weak $i$-th derivative pair, ${\int }_{\mathrm{\Omega }}{v}_{j_k}{\mathrm{\partial }}_i\phi =-{\int }_{\mathrm{\Omega }}{D}_i{v}_{j_k}\phi$ for every $k$ by [F17]. Holder [F12] gives $\mid {\int }_{\mathrm{\Omega }}{D}_i{v}_{j_k}\phi \mid \le \mathrm{\parallel }{D}_i{v}_{j_k}{\mathrm{\parallel }}_{L^p}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}\le j_k^{-1}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}}\to 0$, using step 1.1, and ${\int }_{\mathrm{\Omega }}{v}_{j_k}{\mathrm{\partial }}_i\phi \to {\int }_{\mathrm{\Omega }}v{\mathrm{\partial }}_i\phi$ by step 7.1 and Holder. Hence ${\int }_{\mathrm{\Omega }}v{\mathrm{\partial }}_i\phi =0$ for every test function and every $i$, so the zero function is a weak $i$-th derivative of $v$; by uniqueness of weak derivatives [F17], $D_{i}v=0$ almost everywhere on $\mathrm{\Omega }$ and $v\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$.

The contradiction. By step 8.1 the class $v\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$ has all weak derivatives zero almost everywhere, so [F18] and the connectedness of $\mathrm{\Omega }$ give a constant $c\in \mathbb{K}$ with $v=c$ almost everywhere on $\mathrm{\Omega }$. Step 7.1 gives $0={\int }_{\mathrm{\Omega }}v=c\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$, hence $c=0$ and $v=0$ almost everywhere, contradicting $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}=1$ from step 7.1. Therefore a constant $C_P=C_P(n,p,\mathrm{\Omega },\mathbb{K})$ with the asserted property exists.