Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$

TL;DR

This paper studies the boundary behavior of weighted Sobolev spaces on Ahlfors regular sets in bf3rf3nf3s, establishing trace and extension theorems for Besov functions on fractal sets like the Sierpi
42ski carpet.

Contribution

It introduces new trace and extension operators for weighted Sobolev spaces on fractal sets with bf3rf3nf3s measures, expanding the understanding of function spaces on irregular sets.

Findings

Existence of bounded trace operators from weighted Sobolev spaces to Besov spaces on fractals.

Existence of bounded extension operators from Besov spaces to weighted Sobolev spaces.

Application of results to classical fractals like Sierpi
42ski carpet, gasket, and von Koch snowflake.

Abstract

The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<\alpha\le 0$, the weighted measure $\mu_{\alpha}$ given by integrating the density $\omega(x)=\text{dist}(x, E)^\alpha$ yields a Muckenhoupt $\mathcal{A}_p$-weight in a ball $B$ containing $E$. For such sets $E$ we show the existence of a bounded linear trace operator acting from $W^{1,p}(B,\mu_\alpha)$ to $B^\theta_{p,p}(E, \mathcal{H}^Q\vert_E)$ when $0<\theta<1-\tfrac{\alpha+n-Q}{p}$, and the existence of a bounded linear extension operator from $B^\theta_{p,p}(E, \mathcal{H}^Q\vert_E)$ to $W^{1,p}(B, \mu_\alpha)$ when $1-\tfrac{\alpha+n-Q}{p}\le \theta<1$. We illustrate these results with $E$ as the Sierpi\'nski carpet, the Sierpi\'nski gasket, and the von Koch snowflake.