Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension”. Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces $W_p^1$ on the “zero” cusp with a Hölder singularity at the vertex.