Measure Density and Extension of Besov and Triebel–Lizorkin Functions

Abstract

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