Measure Density and Extension of Besov and Triebel–Lizorkin Functions

Abstract

We show that a domain is an extension domain for a Hajłasz–Besov or for a Hajłasz–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 Hajłasz–Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Hajłasz–Besov spaces are intermediate spaces between $L^p$ and Hajłasz–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.