Rellich Kondrachov and Sobolev Compactness

Given: the Axiom of Choice, $n\ge 1$, $1\le p<\mathrm{\infty }$, a scalar field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, a class $u\in W^{1,p}({\mathbb{R}}^n;\mathbb{K})$ and a vector $h\in {\mathbb{R}}^n$.

Smooth density in $W^{1,p}({\mathbb{R}}^n)$. Under Countable Choice, for every $u\in W^{1,p}({\mathbb{R}}^n;\mathbb{K})$ there are ${\phi }_m\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$ with $\mathrm{\parallel }{\phi }_m-u{\mathrm{\parallel }}_{W^{1,p}}\to 0$; Countable Choice is supplied by the assumed Axiom of Choice. (Compactly supported smooth functions are dense in W^{k,p}(R^n), The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

Fundamental theorem of calculus. For a smooth function $\phi$, $\phi (x-h)-\phi (x)=-{\int }_0^{1}D\phi (x-th)\cdot hdt$, with the complex-valued identity read componentwise. (Fundamental theorem of calculus for absolutely continuous functions)

Minkowski and translation isometry. Minkowski's integral inequality applies to the $t$-integral on $[0,1]$, and Lebesgue translation invariance gives $\mathrm{\parallel }w(\cdot -th){\mathrm{\parallel }}_p=\mathrm{\parallel }w{\mathrm{\parallel }}_p$ for every $w\in L^p$. (Minkowski's integral inequality, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, The space $L^p(\mu )$ as the quotient by null functions)

Vector gradient norm. The $L^p$ norm of the Euclidean magnitude $\mathrm{\mid }Du\mathrm{\mid }=({\sum }_i\mathrm{\mid }{D}_{i}u{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ is equivalent, with constants depending only on $n,p$, to the finite sum of the component $L^p$ norms used in Integer-order Sobolev spaces and their norms. Indeed $\mathrm{\mid }{D}_{i}u\mathrm{\mid }\le \mathrm{\mid }Du\mathrm{\mid }\le {\sum }_i\mathrm{\mid }{D}_{i}u\mathrm{\mid }$, so Minkowski bounds $\mathrm{\parallel }\mathrm{\mid }Du\mathrm{\mid }{\mathrm{\parallel }}_p$ above by ${\sum }_i\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_p$. Also $\mid \mathrm{\mid }D{u}_m\mathrm{\mid }-\mathrm{\mid }Du\mathrm{\mid }\mid \le \mathrm{\mid }D(u_m-u)\mathrm{\mid }\le {\sum }_i\mathrm{\mid }{D}_i(u_m-u)\mathrm{\mid }$, which proves convergence in $L^p$ when $u_m\to u$ in $W^{1,p}$. (Integer-order Sobolev spaces and their norms)

Weak derivatives. A locally integrable function $v$ has weak ${\mathrm{\partial }}_j$-derivative $w$ when $\int v{\mathrm{\partial }}_j\psi =-\int w\psi$ for every $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$; if $v,w\in L^p$, this places $v$ in $W^{1,p}$ in that coordinate. (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms)

Given: the Axiom of Countable Choice, $1\le p<\mathrm{\infty }$, and a relatively compact family $\mathcal{F}\subseteq L^p({\mathbb{R}}^n)$ with closure $\bar{\mathcal{F}}$.

Compact metric spaces are totally bounded. A compact metric space is totally bounded and complete. (A compact metric space is complete and totally bounded, and neither implication uses any choice principle)

Total boundedness is finite-net covering. A metric space $(X,d)$ is totally bounded when for every $\delta >0$ there are finitely many points $x_1,\dots ,x_N\in X$ with $X={\bigcup }_{i=1}^{N}B(x_i,\delta )$; a subset of a totally bounded space which is itself totally bounded as a subspace has the same property for every $\delta >0$. (Finite $\epsilon$-net and totally bounded metric space, Open ball, closed ball and sphere in a metric space, A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded)

Continuity of translation in $L^p$. For every $g\in L^p({\mathbb{R}}^n)$, $\mathrm{\parallel }{\tau }_{h}g-g{\mathrm{\parallel }}_p\to 0$ as $\mathrm{\mid }h\mathrm{\mid }\to 0$, where ${\tau }_{h}g=g(\cdot -h)$ acts on almost-everywhere classes. ($\mathrm{\parallel }{\tau }_{h}f-f{\mathrm{\parallel }}_p\to 0$ in $L^p({\mathbb{R}}^n)$ as $h\to 0$, for $1\le p<\mathrm{\infty }$, Translation of a function on ${\mathbb{R}}^n$)

Translation is an $L^p$-isometry. $\mathrm{\parallel }{\tau }_{h}g{\mathrm{\parallel }}_p=\mathrm{\parallel }g{\mathrm{\parallel }}_p$ for every $g\in L^p({\mathbb{R}}^n)$ and every $h$, because Lebesgue measure is translation invariant. (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, The space $L^p(\mu )$ as the quotient by null functions)

Given: $1\le p<\mathrm{\infty }$, a bounded set $E\subseteq {\mathbb{R}}^n$, and a family $\mathcal{F}\subseteq L^p({\mathbb{R}}^n)$ whose every member vanishes almost everywhere outside $E$.

Bounded sets lie in balls. $E\subseteq {\mathbb{R}}^n$ is bounded in the metric sense of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space if and only if there is a centre $x_0$ and a radius $r>0$ with $E\subseteq B(x_0,r)$; a ball is contained in the ball about the origin of radius $\mathrm{\mid }{x}_0\mathrm{\mid }+r$. (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, Open ball, closed ball and sphere in a metric space)

Restriction to a measurable tail. If $A\subseteq E^c$ is measurable and $f$ vanishes almost everywhere on $E^c$, then for any measurable representative $g$ the function $\mathrm{\mid }g{\mathrm{\mid }}^p{\mathbf{1}}_A$ is measurable and zero almost everywhere. (The space $L^p(\mu )$ as the quotient by null functions, Measure-null sets and almost-everywhere statements relative to a measure)

Zero integral and almost-everywhere vanishing. A nonnegative measurable $h$ satisfies $\int hdx=0$ if and only if $h=0$ almost everywhere. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere)

Given: the Axioms of Countable and Dependent Choice, $1\le p<\mathrm{\infty }$, and a family $\mathcal{F}\subseteq L^p({\mathbb{R}}^n)$ satisfying conditions (i)--(iii) of the statement; write $M:={\sup }_{f\in \mathcal{F}}\mathrm{\parallel }f{\mathrm{\parallel }}_p<\mathrm{\infty }$. For $R>0$ and $f\in \mathcal{F}$ put $f_R:=f{\mathbf{1}}_{\{\mathrm{\mid }x\mathrm{\mid }\le R\}}$, and for each $\epsilon >0$ let ${\eta }_{\epsilon }$ be the radial mollifier at scale $\epsilon$ of A radial mollifier family in Rn.

Mollification. For $f_R\in L^p\subseteq L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$, the function $f_R\ast {\eta }_{\delta }$ is smooth and ${\mathrm{\partial }}^{\alpha }(f_R\ast {\eta }_{\delta })=f_R\ast ({\mathrm{\partial }}^{\alpha }{\eta }_{\delta })$. (Convolution with a mollifier is smooth, and derivatives pass under the integral sign)

H"older's inequality. For conjugate exponents $p,p^{\mathrm{\prime }}$, $\int \mathrm{\mid }uv\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_p\mathrm{\parallel }v{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$; in particular $\mathrm{\mid }(f_R\ast {\eta }_{\delta })(x)\mathrm{\mid }\le \mathrm{\parallel }{f}_R{\mathrm{\parallel }}_p\mathrm{\parallel }{\eta }_{\delta }{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$ and $\mathrm{\mid }\mathrm{\nabla }(f_R\ast {\eta }_{\delta })(x)\mathrm{\mid }\le \mathrm{\parallel }{f}_R{\mathrm{\parallel }}_p\mathrm{\parallel }\mathrm{\nabla }{\eta }_{\delta }{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$ pointwise. (Holder's inequality for integrals, including the endpoint cases)

Minkowski's integral inequality. For measurable $F$ on a product of sigma-finite measure spaces with ${\int }_Y\mathrm{\parallel }F(\cdot ,y){\mathrm{\parallel }}_{p}d\nu (y)<\mathrm{\infty }$, $\parallel {\int }_{Y}F(\cdot ,y)d\nu (y){\parallel }_p\le {\int }_Y\mathrm{\parallel }F(\cdot ,y){\mathrm{\parallel }}_{p}d\nu (y)$. (Minkowski's integral inequality)

Translations are $L^p$-isometries. $\mathrm{\parallel }{\tau }_{h}g{\mathrm{\parallel }}_p=\mathrm{\parallel }g{\mathrm{\parallel }}_p$ for every $g\in L^p$ and every $h$, since Lebesgue measure is translation invariant. (Translation of a function on ${\mathbb{R}}^n$, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, The space $L^p(\mu )$ as the quotient by null functions)

Finite nets for equicontinuous families. An equicontinuous pointwise bounded family in $C(K,\mathbb{R})$, $K$ a compact metric space, is totally bounded for the supremum metric; the same holds after applying the statement to real and imaginary parts of a complex-valued family, and Arzelà--Ascoli for real $C(K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded records the equivalent compact-closure form. (An equicontinuous pointwise-bounded family in $C(K,\mathbb{R})$ has a finite net in the supremum metric, Arzelà--Ascoli for real $C(K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded)

Total boundedness and its closure. A metric space is totally bounded when it has a finite $\delta$-net for every $\delta >0$; total boundedness passes to subsets and to closures, and for $g,h$ supported in a compact ball $K$ one has $\mathrm{\parallel }g-h{\mathrm{\parallel }}_p\le \mathrm{\parallel }g-h{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}p}$. (Finite $\epsilon$-net and totally bounded metric space, A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded, Open ball, closed ball and sphere in a metric space)

Completeness and compactness of $L^p$. $L^p({\mathbb{R}}^n)$ is complete; a closed subspace of a complete metric space is complete, and a complete and totally bounded metric space is compact under Countable Choice; compactness and sequential compactness of a metric space are equivalent under Countable and Dependent Choice. (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, Closed subspaces of complete metric spaces are complete; the converse under countable choice, A complete, totally bounded metric space is compact, proved from countable choice used exactly once, For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Given: the Axiom of Choice, $n\ge 1$, a bounded open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $1\le p<\mathrm{\infty }$, and a sequence $(u_j)$ with $M:={\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W_0^{1,p}(\mathrm{\Omega })}<\mathrm{\infty }$.

Zero extension. For each $j$ the zero extension ${\tilde{u}}_j$ of $u_j$ lies in $W^{1,p}({\mathbb{R}}^n)$ with $D_i{\tilde{u}}_j$ the zero extension of $D_i{u}_j$ and $\mathrm{\parallel }{\tilde{u}}_j{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}=\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$; the extension vanishes outside $\bar{\mathrm{\Omega }}$. (Zero extension of W_0^{1,p} has no boundary derivative, Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms)

Translation estimate. $\mathrm{\parallel }{\tau }_h{\tilde{u}}_j-{\tilde{u}}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }D{\tilde{u}}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$, under the Axiom of Choice. (The translation estimate for $W^{1,p}$ functions on ${\mathbb{R}}^n$)

Automatic tails. A family in $L^p({\mathbb{R}}^n)$ whose members all vanish almost everywhere outside the fixed bounded set $\bar{\mathrm{\Omega }}$ satisfies the tightness condition of the Fr'echet--Kolmogorov criterion. (Uniformly supported families have vanishing tails, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space)

The Fr'echet--Kolmogorov criterion. A bounded family in $L^p({\mathbb{R}}^n)$ with vanishing tails and uniform translation control is totally bounded, its closure is compact, and every sequence in it has an $L^p({\mathbb{R}}^n)$-convergent subsequence. (The Fr'echet--Kolmogorov compactness criterion in $L^p({\mathbb{R}}^n)$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Restriction is contractive. $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{W_0^{1,p}(\mathrm{\Omega })}$, and the $L^p(\mathrm{\Omega })$ norm of the restriction never exceeds the $L^p({\mathbb{R}}^n)$ norm of an extension. (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions)

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

Extension operator. There is a bounded linear $E:W^{1,p}(\mathrm{\Omega })\to W^{1,p}({\mathbb{R}}^n)$ with $(Eu){\mathrm{\mid }}_{\mathrm{\Omega }}=u$ and $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}\le \mathrm{\parallel }E\mathrm{\parallel }\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$. (Sobolev extension domains and extension operators)

Cutoff. Since $\bar{\mathrm{\Omega }}$ is compact and contained in the open set $V:=B(0,R)$, with $R>0$ large enough to contain $\bar{\mathrm{\Omega }}$, there is $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $\eta =1$ on $\bar{\mathrm{\Omega }}$ and $\mathrm{supp}\eta \subseteq V$, a fixed bounded set. This construction also works for the empty domain, choosing any ball $V$. (A Euclidean bump for a compact set inside an open set, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space)

Products with a smooth cutoff. If $v\in W^{1,p}({\mathbb{R}}^n)$ and $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, then $\eta v\in W^{1,p}({\mathbb{R}}^n)$ with $\mathrm{\parallel }\eta v{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}\le C_{\eta }\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}$ for a constant depending only on $\eta$ and $p$. (Weak Leibniz rule with a smooth factor)

Translation estimate and automatic tails. Under the Axiom of Choice, $\mathrm{\parallel }{\tau }_{h}v-v{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }\mathrm{\mid }Dv\mathrm{\mid }{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$; a family supported in one fixed bounded set has vanishing tails. (The translation estimate for $W^{1,p}$ functions on ${\mathbb{R}}^n$, Uniformly supported families have vanishing tails)

The Fr'echet--Kolmogorov criterion. A bounded family in $L^p({\mathbb{R}}^n)$ with vanishing tails and uniform translation control has an $L^p({\mathbb{R}}^n)$-convergent subsequence. (The Fr'echet--Kolmogorov compactness criterion in $L^p({\mathbb{R}}^n)$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Restriction and norms. $\mathrm{\parallel }g{\mathrm{\mid }}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }g{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$; and $(E{u}_j){\mathrm{\mid }}_{\mathrm{\Omega }}=u_j$ almost everywhere. (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions)

Bounded $C^1$ domains are extension domains. (Bounded C^k domains admit integer-order Sobolev extension)

Given: the Axiom of Choice, a nonempty bounded connected extension domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ of finite positive measure, $1\le p<\mathrm{\infty }$, and the mean $u_{\mathrm{\Omega }}$ of $u$. Nonempty openness supplies a ball inside $\mathrm{\Omega }$, and boundedness supplies a containing ball; thus $0<\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }<\mathrm{\infty }$ by Euclidean balls have positive finite Lebesgue measure.

Rellich compactness. Every sequence bounded in $W^{1,p}(\mathrm{\Omega })$ has a subsequence converging in $L^p(\mathrm{\Omega })$. (Compactness of $W^{1,p}(\mathrm{\Omega })\hookrightarrow L^p(\mathrm{\Omega })$ on bounded extension domains, Sobolev extension domains and extension operators)

The mean is continuous for the $L^p$ norm. $\mathrm{\mid }{u}_{\mathrm{\Omega }}\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{-1\mathrm{/}p}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p}$ and hence $\mathrm{\mid }{\int }_{\mathrm{\Omega }}(u_j-u)\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1-1\mathrm{/}p}\mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$, by H"older's inequality on the finite-measure set $\mathrm{\Omega }$. (Holder's inequality for integrals, including the endpoint cases, The space $L^p(\mu )$ as the quotient by null functions)

Weak gradients vanish when tested against convergent subsequences. If $u_j\to u$ in $L^p(\mathrm{\Omega })$ and $\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\to 0$, then for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and every coordinate $i$, ${\int }_{\mathrm{\Omega }}u{\mathrm{\partial }}_i\phi ={\lim }_j{\int }_{\mathrm{\Omega }}{u}_j{\mathrm{\partial }}_i\phi =-{\lim }_j{\int }_{\mathrm{\Omega }}{D}_i{u}_j\phi =0$. (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions)

Zero gradient implies constancy on components. If $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ and $D_{i}u=0$ almost everywhere for all $i$, then $u$ is almost everywhere constant on each connected component of $\mathrm{\Omega }$. (Zero weak gradient gives componentwise constants)