An example related to Whitney's extension problem for $L^{2,p}(\mathbb{R}^2)$ when $1<p<2$

TL;DR

This paper establishes the existence of a bounded linear extension operator for a specific fractional Sobolev space over fractal sets in the plane, extending Whitney's problem to this context for 1<p<2.

Contribution

It proves the existence of a bounded linear extension operator for $L^{2,p}$ spaces on fractal sets in $R^2$, utilizing Fefferman-Klartag's theorem for radially symmetric trees.

Findings

Existence of a bounded linear extension operator for $L^{2,p}$ on fractal sets.

Application of Fefferman-Klartag theorem to fractal structures.

Extension operator constructed for $1<p<2$.

Abstract

In this paper, we prove the existence of a bounded linear extension operator $T: L^{2,p}(E) \rightarrow L^{2,p}(\mathbb{R}^2)$ when $1<p<2$, where $E \subset \mathbb{R}^2$ is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman-Klartag on the existence of linear extension operators for radially symmetric binary trees.