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 $|f|{L^{m,p}(\mathbb{R}^n) + L^p(d\mu)} := \text{inf}{f_1 +f_2 = f} { |f_1|{L^{m,p}(\mathbb{R}^n)}^p + \int{\mathbb{R}^n} |f_2|^p d\mu }^{1/p}$. 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.