A note on the trace theorem for Besov-type spaces of generalized smoothness on $d$-sets

Abstract

The main goal of this paper is to give a complete proof of the trace theorem for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions on $d$-sets $D\subset {\mathbb{R}}^n$, $d\le n$. The proof closely follows the classical approach by Jonsson and Wallin and the trace theorem for classical Besov spaces. Here, the trace space is defined by means of differences. When $d=n$, as an application of the trace theorem, we give a condition under which the test functions $C_c^{\infty }(D)$ are dense in the trace space on $D$.

Previously part of the appendix of publication arXiv:1608.01384