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

TL;DR

This paper proves the existence of a bounded linear extension operator for certain Sobolev-Slobodeckij spaces when the fractional part of the smoothness parameter exceeds a specific ratio involving the dimension and integrability exponent.

Contribution

It establishes a new bounded linear extension operator for homogeneous Sobolev-Slobodeckij spaces under the condition that the fractional part of s exceeds n/p, extending classical Whitney methods.

Findings

Existence of a bounded linear extension operator under specified conditions.

Extension operator constructed using Whitney and exponentially decreasing paths.

Applicable for all subsets E of R^n, p in [1, ∞), and s with 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 $J^{\lfloor s \rfloor}_E L^{s,p}(\mathbb{R}^n)$ 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.