Measure Density and Extension of Besov and Triebel–Lizorkin Functions

By T. Heikkinen, L. Ihnatsyeva, Heli Tuominen. 2014

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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 $$0<p<1$$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 $$L^p$$Lp 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 $$B^s_{p,q}$$Bp,qs, $$0<s<1$$0<s<1, $$0<p<\infty $$0<p<∞, $$0<q\le \infty $$0<q≤∞, defined via the $$L^p$$Lp-modulus of smoothness of a function.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e676b94ac24b0411e35e93d8266a33395f4c262b