The existence of a bounded linear extension operator for $L^{s,p}({\mathbb{R}}^n)$ when $\frac{n}{p}<\{s\}$

Abstract

Let $L^{s,p}({\mathbb{R}}^n)$ denote the homogeneous Sobolev-Slobodeckij space. In this paper, we demonstrate the existence of a bounded linear extension operator from the jet space $J_E^{\lfloor s\rfloor }{L}^{s,p}({\mathbb{R}}^n)$ to $L^{s,p}({\mathbb{R}}^n)$ for any $E\subseteq {\mathbb{R}}^n$, $p\in [1,\infty )$, and $s\in (0,\infty )$ satisfying $\frac{n}{p}<\{s\}$, where $\{s\}$ represents the fractional part of $s$. Our approach builds upon the classical Whitney extension operator and uses the method of exponentially decreasing paths.