Approximate Extension in Sobolev Space

Abstract

Let $L^{m,p}({\mathbb{R}}^n)$ be the homogeneous Sobolev space for $p\in (n,\infty )$, $\mu$ be a Borel regular measure on ${\mathbb{R}}^n$, and $L^{m,p}({\mathbb{R}}^n)+L^p(d\mu )$ be the space of Borel measurable functions with finite seminorm $\Vert f{\Vert }_{L^{m,p}({\mathbb{R}}^n)+L^p(d\mu )}:={\text{inf}}_{f_1+f_2=f}\{\Vert f_1{\Vert }_{L^{m,p}({\mathbb{R}}^n)}^p+{\int }_{{\mathbb{R}}^n}|f_2{|}^{p}d\mu {\}}^{1/p}$. We construct a linear operator $T:L^{m,p}({\mathbb{R}}^n)+L^p(d\mu )\to L^{m,p}({\mathbb{R}}^n)$, that nearly optimally decomposes every function in the sum space: $\Vert Tf{\Vert }_{L^{m,p}({\mathbb{R}}^n)}^p+{\int }_{{\mathbb{R}}^n}|Tf-f{|}^{p}d\mu \le C\Vert f{\Vert }_{L^{m,p}({\mathbb{R}}^n)+L^p(d\mu )}^p$ with $C$ dependent on $m$, $n$, and $p$ only. For $E\subset {\mathbb{R}}^n$, let $L^{m,p}(E)$ denote the space of all restrictions to $E$ of functions $F\in L^{m,p}({\mathbb{R}}^n)$, equipped with the standard trace seminorm. For $p\in (n,\infty )$, we construct a linear extension operator $T:L^{m,p}(E)\to L^{m,p}({\mathbb{R}}^n)$ satisfying $Tf{|}_E=f{|}_E$ and $\Vert Tf{\Vert }_{L^{m,p}({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{L^{m,p}(E)}$, where $C$ depends only on $n$, $m$, and $p$. We show these operators can be expressed through a collection of linear functionals whose supports have bounded overlap.

arXiv commentarXiv admin note: text overlap with arXiv:1205.2525 by other authors