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

Volume 231 / 2015

António M. Caetano, Dorothee D. Haroske Studia Mathematica 231 (2015), 117-147 MSC: Primary 46E35; Secondary 28A80. DOI: 10.4064/sm8171-1-2016 Published online: 18 February 2016


We study the existence of traces of Besov spaces on fractal $h$-sets $\varGamma $ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). 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 $\varGamma $—there occurs an alternative: either the trace on $\varGamma $ exists, or smooth functions compactly supported outside $\varGamma $ are dense in the space. This notion was introduced by Triebel (2008) for the special case of $d$-sets.


  • António M. CaetanoCenter for R&D in Mathematics and Applications
    Department of Mathematics
    University of Aveiro
    3810-193 Aveiro, Portugal
  • Dorothee D. HaroskeInstitute of Mathematics
    Friedrich Schiller University Jena
    07737 Jena, Germany

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