Measure Density and Extension of Besov and Triebel-Lizorkin Functions

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-p,q(s), 0 < s < 1, 0 < p < infinity, 0 < q <= infinity, defined via the L-p-modulus of smoothness of a function.