Traces for Besov spaces on fractal h-sets and dichotomy results

TL;DR

This paper investigates the conditions under which traces of Besov spaces exist on fractal h-sets, establishing criteria for existence and non-existence, and explores a dichotomy phenomenon related to the density of smooth functions outside these sets.

Contribution

It extends previous work by providing new necessary conditions for trace existence on fractal h-sets and introduces a dichotomy framework for function space behavior.

Findings

Criteria for non-existence of traces on fractal h-sets

Existence conditions for traces of Besov spaces

Dichotomy between trace existence and density of smooth functions

Abstract

We study the existence of traces of Besov spaces on fractal $h$-sets $\Gamma$ with the special focus laid on necessary assumptions implying this existence, or, in other words, present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of [Br4] and a continuation of the recent paper [Ca2]. Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that -- depending on the function space and the set $\Gamma$ -- there occurs an alternative: either the trace on $\Gamma$ exists, or smooth functions compactly supported outside $\Gamma$ are dense in the space. This notion was introduced by Triebel in [Tr7] for the special case of $d$-sets.