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