Traces for Besov spaces on fractal h-sets and dichotomy results
Ant\'onio Caetano, Dorothee Haroske

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 -sets 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 -- there occurs an alternative: either the trace on exists, or smooth functions compactly supported outside are dense in the space. This notion was introduced by Triebel in [Tr7] for the special case of -sets.
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