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-p,q(s), 0 <s <1, 0 <p <infinity, 0 <q
Original language | English |
---|---|
Pages (from-to) | 334-382 |
Number of pages | 49 |
Journal | Journal of Fourier Analysis and Applications |
Volume | 22 |
Issue number | 2 |
DOIs | |
Publication status | Published - Apr 2016 |
MoE publication type | A1 Journal article-refereed |
Keywords
- Besov space
- Triebel-Lizorkin space
- Extension domain
- Measure density
- Metric measure space
- HAJLASZ-SOBOLEV SPACES
- METRIC MEASURE-SPACES
- HOMOGENEOUS TYPE
- REGULAR SUBSETS
- INTERPOLATION
- INEQUALITIES
- EXTENDABILITY
- RESTRICTIONS
- DOMAINS
- RD