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

António M. Caetano ¹   ; Dorothee D. Haroske ²

¹ Center for R&D in Mathematics and Applications Department of Mathematics University of Aveiro 3810-193 Aveiro, Portugal
² Institute of Mathematics Friedrich Schiller University Jena 07737 Jena, Germany

Studia Mathematica, Tome 231 (2015) no. 2, pp. 117-147

Résumé

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.

Détail

DOI : 10.4064/sm8171-1-2016

Keywords: study existence traces besov spaces fractal h sets vargamma special focus assumptions necessary existence other words present criteria non existence traces sense paper regarded extension bricchi continuation caetano closely connected problem existence traces notion dichotomy function spaces prove depending function space set vargamma there occurs alternative either trace vargamma exists smooth functions compactly supported outside vargamma dense space notion introduced triebel special d sets