Traces of Besov spaces on fractal $h$-sets and dichotomy results

Abstract

We study the existence of traces of Besov spaces on fractal h-sets Γ 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 Γ -there occurs an alternative: either the trace on Γ exists, or smooth functions compactly supported outside Γ are dense in the space. This notion was introduced by Triebel in [Tr7] for the special case of d-sets. 2010 Mathematics Subject Classification: Primary 46E35; Secondary 28A80. Key words and phrases: fractal h-sets, traces, Besov spaces of generalised smoothness, density of test functions, dichotomy. c 2015. Licensed under the CC-BY-NC-ND 4.0 license

Traces of weighted function spaces: Dyadic norms and Whitney extensions

“…[22] and the references therein as well as [18]. Works focusing on trace theorems for fractals and related subsets of a Euclidean space include [25,41,42,50,44,23,24,5,17] (we also refer to [55] for a recent result concerning traces on non-regular subsets of R d ), while trace theorems in more general metric settings have been considered e.g. in [14,43,28,31,30].…”

Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

“…Then there exist bounded linear operators One should observe that Theorems 1.1, 1.4 and 1.5 are exactly of the same form as the classical results in the Euclidean setting, and that Theorems 1.4 and 1.5 (along with their non-homogeneous counterparts below) are completely new in this generality. A (very incomplete) list of previous results in the setting where F is a subset of an Euclidean space includes [22,32,33,35,19,21,4,12] -the trace spaces appearing in these papers are sometimes defined in a non-intrinsic manner, however e.g. [32,33,19] employ intrinsic characterizations in terms of optimal polynomial approximations.…”