Approximate Extension in Sobolev Space

Let $L^{m,p}({\mathbb{R}}^n)$ be the homogeneous Sobolev space for $p \in (n,\infty)$,$\mu$beaBorelregularmeasureon$$\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}$\mathrm{.}Weconstructalinearoperator$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$C$dependenton$m$,$n$, 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 $p\in (n,\mathrm{\infty })$, we construct a linear extension operator $T:L^{m,p}(E)\to L^{m,p}({\mathbb{R}}^n)$ satisfying $Tf{\mathrm{\mid }}_E=f{\mathrm{\mid }}_E$ and $\mathrm{\parallel }Tf{\mathrm{\parallel }}_{L^{m,p}({\mathbb{R}}^n)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{m,p}(E)}$, 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.