Measure Density and Extension of Besov and Triebel-Lizorkin Functions

Toni Heikkinen, Lizaveta Ihnatsyeva, Heli Tuominen*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

14 Citations (Scopus)

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 languageEnglish
Pages (from-to)334-382
Number of pages49
JournalJournal of Fourier Analysis and Applications
Volume22
Issue number2
DOIs
Publication statusPublished - Apr 2016
MoE publication typeA1 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

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