Approximate Extension in Sobolev Space

TL;DR

This paper develops linear extension and decomposition operators for Sobolev spaces with measures, nearly optimally extending functions from subsets and decomposing functions in sum spaces with controlled norms.

Contribution

It constructs explicit linear extension and decomposition operators for Sobolev spaces with measures, with bounds depending only on fundamental parameters.

Findings

Constructed a linear operator T for sum spaces with near-optimal decomposition.

Developed a linear extension operator T for Sobolev spaces restricted to subsets.

Operators are representable via linear functionals with bounded overlap.

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 $\|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(d\mu)} := \text{inf}_{f_1 +f_2 = f} \{ \|f_1\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |f_2|^p d\mu \}^{1/p}$. 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: $\|Tf\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |Tf-f|^p d\mu \leq C \|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(d\mu)}^p$ 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 $\|Tf\|_{L^{m,p}(\mathbb{R}^n)} \leq C \|f\|_{L^{m,p}(E)}$, 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.