Morrey--Rellich compactness for $p>n$

Given: the Axiom of Choice, a bounded extension domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n<p<\mathrm{\infty }$, $\alpha =1-n\mathrm{/}p$, and a sequence $(u_j)$ with $M:={\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}<\mathrm{\infty }$.

Extension and cutoff. There are a bounded extension operator $E$ and a fixed $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $\eta =1$ on $\mathrm{\Omega }$; the products $v_j:=\eta E{u}_j$ lie in $W^{1,p}({\mathbb{R}}^n)$, are supported in the fixed compact set $\mathrm{supp}\eta$, satisfy $\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}\le C_{1}M$, and equal $u_j$ almost everywhere on $\mathrm{\Omega }$. (Sobolev extension domains and extension operators, A Euclidean bump for a compact set inside an open set, Weak Leibniz rule with a smooth factor, Integer-order Sobolev spaces and their norms)

Morrey's inequality on the compact set used here. Each $v_j$ in [F1] is supported in a fixed compact set $K$. Choose one ball $B(x,R)$ containing $K\cup \bar{\mathrm{\Omega }}$. The supplier's local estimate on $B(x,R)$, with $B(x,2R)\subset {\mathbb{R}}^n$, gives $[v_j^{\ast }{]}_{C^{0,\alpha }(B(x,R))}\le C_2\mathrm{\parallel }D{v}_j{\mathrm{\parallel }}_{L^p(B(x,2R))}$ for the continuous representative. Its average on $B(x,R)$ has modulus at most $\mathrm{\mid }B(x,R){\mathrm{\mid }}^{-1\mathrm{/}p}\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{L^p(B(x,R))}$, so the same oscillation estimate also bounds $\mathrm{\parallel }{v}_j^{\ast }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(B(x,R))}$ by $C_3\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}$. Continuous representatives are unique because continuous functions equal almost everywhere on an open ball are equal everywhere there. Thus both norms on $\bar{\mathrm{\Omega }}$ are bounded by $C_4\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}$, with constants depending only on $n,p,R$. (Morrey's inequality for $p>n$, Local Hölder and scaled C-two-alpha norms on balls, The space $L^p(\mu )$ as the quotient by null functions)

Arzel`a--Ascoli. A uniformly bounded, equicontinuous family of real functions on a compact metric space has a uniformly convergent subsequence; for complex-valued families apply this to real and imaginary parts. (Arzelà--Ascoli for real $C(K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded)

H"older interpolation. For a bounded function $g:K\to \mathbb{K}$ on any set $K\subseteq {\mathbb{R}}^n$, write $[g{]}_{C^{0,\gamma }(K)}:={\sup }_{x\mathrm{\ne }y\in K}\mathrm{\mid }g(x)-g(y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x-y{\mathrm{\mid }}^{\gamma }$ and $\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{0,\gamma }(K)}:=\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}+[g{]}_{C^{0,\gamma }(K)}$. For $0<\beta <\alpha$ and $\theta =\beta \mathrm{/}\alpha$, the bound $\mathrm{\mid }g(x)-g(y)\mathrm{\mid }\le \min \{2\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)},[g{]}_{C^{0,\alpha }(K)}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }\}$ gives $[g{]}_{C^{0,\beta }(K)}\le 2^{1-\theta }\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}^{1-\theta }[g{]}_{C^{0,\alpha }(K)}^{\theta },$ and hence $\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{0,\beta }(K)}\le \mathrm{\parallel }g{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}+2^{1-\theta }\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}^{1-\theta }[g{]}_{C^{0,\alpha }(K)}^{\theta }\mathrm{.}$ These follow from $\min \{A,B\}\le A^{1-\theta }{B}^{\theta }$ for $A,B\ge 0$; the seminorm and supremum conventions agree with Local Hölder and scaled C-two-alpha norms on balls.