Measure density and extension of Besov and Triebel-Lizorkin functions

TL;DR

This paper characterizes extension domains for Besov and Triebel-Lizorkin spaces in metric measure spaces using a measure density condition, introducing a median-based Whitney extension method applicable even for $0<p<1$.

Contribution

It establishes a measure density criterion for extension domains and develops a median-based Whitney extension, extending results to the case $0<p<1$ and providing new insights even in Euclidean spaces.

Findings

Characterization of extension domains via measure density condition.

Development of median-based Whitney extension method.

New interpolation theorem for Haj extasciitilde asz-Besov spaces.

Abstract

We show that a domain is an extension domain for a Haj\l asz-Besov or for a Haj\l asz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case $0<p<1$. The necessity of the measure density condition is derived from embedding theorems; in the case of Haj\l asz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj\l asz-Besov spaces are intermediate spaces between $L^p$ and Haj\l asz-Sobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces $B^s_{p,q}$, $0<s<1$, $0<p<\infty$, $0<q\le\infty$, defined via the $L^p$-modulus of smoothness of a function.