Integer-order Sobolev extension from a half-space

Given: the Axiom of Choice; the half-space $H={\mathbb{R}}^{n-1}\times (0,\mathrm{\infty })$; $k\in {\mathbb{N}}_0$; $1\le p\le \mathrm{\infty }$; $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$; a class $u\in W^{k,p}(H;\mathbb{K})$; and a test function $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$.

Sobolev classes on an open set: $u\in W^{k,p}(H;\mathbb{K})$ means that for every multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ there is a class $D^{\alpha }u\in L^p(H;\mathbb{K})$ satisfying the weak identity ${\int }_{H}u{D}^{\alpha }\psi =(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_H(D^{\alpha }u)\psi$ for all $\psi \in C_c^{\mathrm{\infty }}(H;\mathbb{K})$, with $D^{0}u=u$; the norm is the ${\mathrm{\ell }}^p$ sum over $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, and the maximum of essential bounds for $p=\mathrm{\infty }$ (Integer-order Sobolev spaces and their norms).

Weak differentiation is local and linear: for open $V\subseteq {\mathbb{R}}^n$, $v=D^{\alpha }u$ weakly on ${\mathbb{R}}^n$ implies $v{\mathrm{\mid }}_V=D^{\alpha }(u{\mathrm{\mid }}_V)$ weakly on $V$, and the sum of two weakly differentiable classes is weakly differentiable with the sum of the derivatives (Linearity, locality, and commutation of weak derivatives).

Linear changes of variables: for the invertible linear map $T_j(x^{\mathrm{\prime }},t)=(x^{\mathrm{\prime }},-jt)$, whose determinant has absolute value $j$, and every nonnegative measurable $f$, ${\int }_{{\mathbb{R}}^n}fd{\lambda }_n=j{\int }_{{\mathbb{R}}^n}f\circ T_{j}d{\lambda }_n$; in particular ${\int }_{\{t<0\}}\mathrm{\mid }w(x^{\mathrm{\prime }},-jt){\mathrm{\mid }}^{p}d{x}^{\mathrm{\prime }}dt=j^{-1}{\int }_H\mathrm{\mid }w(x^{\mathrm{\prime }},\tau ){\mathrm{\mid }}^{p}d{x}^{\mathrm{\prime }}d\tau$ for every measurable $w$ (A linear map $T$ of ${\mathbb{R}}^n$ sends Lebesgue measurable sets to Lebesgue measurable sets, with ${\lambda }_n(T[E])=\mathrm{\mid }\det T\mathrm{\mid }{\lambda }_n(E)$ when $T$ is invertible and $T[E]$ Lebesgue null when it is not).

Traces of normal sections. Let $u\in W^{k,p}(H;\mathbb{K})$ with $k\ge 1$ and $1\le p\le \mathrm{\infty }$. For every multi-index $\gamma$ with $\mathrm{\mid }\gamma \mathrm{\mid }\le k-1$ and almost every $x^{\mathrm{\prime }}\in {\mathbb{R}}^{n-1}$ the section $s\mapsto D^{\gamma }u(x^{\mathrm{\prime }},s)$ has a representative that is absolutely continuous on every compact interval $[0,T]$, extends continuously to $s=0$, and satisfies $\frac{d}{ds}{D}^{\gamma }u(x^{\mathrm{\prime }},s)=D^{\gamma +e_n}u(x^{\mathrm{\prime }},s)$ for almost every $s$, where $e_n$ is the last unit vector; at $p=\mathrm{\infty }$ one first restricts to a finite exponent on compact sets, since $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\mathrm{\infty }}\subseteq W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,q}$. The trace ${\mathrm{tr}}_{\gamma }(x^{\mathrm{\prime }}):={\lim }_{s\to 0^{+}}{D}^{\gamma }u(x^{\mathrm{\prime }},s)$ exists for almost every $x^{\mathrm{\prime }}$, is measurable, and satisfies the estimate $\mathrm{\mid }{\mathrm{tr}}_{\gamma }(x^{\mathrm{\prime }})\mathrm{\mid }\le \frac{2}{T}{\int }_0^T\mathrm{\mid }{D}^{\gamma }u(x^{\mathrm{\prime }},s)\mathrm{\mid }ds+{\int }_0^T\mathrm{\mid }{D}^{\gamma +e_n}u(x^{\mathrm{\prime }},s)\mathrm{\mid }ds$ for every $0<T<\mathrm{\infty }$, whose right-hand side is a.e. finite and integrable in $x^{\mathrm{\prime }}$ over compact sets (The ACL characterisation of $W^{1,p}$, One-dimensional $W^{1,p}$ functions have unique absolutely continuous representatives, Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures, Holder's inequality for integrals, including the endpoint cases).

Half-space integration by parts with traces. Let $u\in W^{k,p}(H;\mathbb{K})$, let $\alpha =(\beta ,m)$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, and let $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$. Then ${\int }_{H}u{D}^{\alpha }\phi =(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_H(D^{\alpha }u)\phi +{\sum }_{r=0}^{m-1}(-1{)}^{r+1}{\int }_{{\mathbb{R}}^{n-1}}{\mathrm{tr}}_{r{e}_n}(x^{\mathrm{\prime }}){\mathrm{\partial }}_t^{m-1-r}{D}^{\beta }\phi (x^{\mathrm{\prime }},0)d{x}^{\mathrm{\prime }}\mathrm{.}$ For the reflected function $f(x^{\mathrm{\prime }},t)=u(x^{\mathrm{\prime }},-jt)$ on the lower half-space, whose $\alpha$-derivative is $D^{\alpha }f(x^{\mathrm{\prime }},t)=(-j{)}^m{D}^{\alpha }u(x^{\mathrm{\prime }},-jt)$ and whose traces of the normal derivatives at $t=0^{-}$ are $(-j{)}^r{\mathrm{tr}}_{r{e}_n}(x^{\mathrm{\prime }})$, and the same formula holds with the boundary signs $(-1{)}^r$ in place of $(-1{)}^{r+1}$. In both formulas the boundary term keeps the tangential derivatives on the test function; they are not also applied to the trace. This follows by integrating in the normal variable first, then moving the tangential derivatives in the interior term onto $u$. The one-dimensional integrations are justified by the absolutely continuous representatives of [F4] and assembled with Fubini's theorem (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included).

Vandermonde systems. For $k\ge 1$ the matrix $((-j{)}^m{)}_{0\le m\le k-1,1\le j\le k}$ is invertible, because its determinant is the Vandermonde product ${\prod }_{i<i^{\mathrm{\prime }}}((-i^{\mathrm{\prime }})-(-i))\mathrm{\ne }0$ over the distinct nodes $-1,-2,\dots ,-k$; hence the moment system ${\sum }_{j=1}^k{a}_j(-j{)}^m=1$, $m=0,\dots ,k-1$, has exactly one solution. This is finite linear algebra over $\mathbb{R}$ and uses no choice.

Extension domain and operator: $\mathrm{\Omega }$ is a $W^{k,p}$-extension domain when there is a bounded linear $E:W^{k,p}(\mathrm{\Omega };\mathbb{K})\to W^{k,p}({\mathbb{R}}^n;\mathbb{K})$ with $(Eu){\mathrm{\mid }}_{\mathrm{\Omega }}=u$ almost everywhere for every class $u$ (Sobolev extension domains and extension operators).

Choice use. The Axiom of Choice is invoked only through the ACL, one-dimensional absolutely-continuous and Fubini interfaces cited in [F4]–[F5]; the Vandermonde coefficients of [F6] and the reflection formula are explicit.