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

TL;DR

This paper provides a comprehensive proof of the trace theorem for Besov-type spaces with generalized smoothness on d-sets, extending classical results and exploring density of test functions in the trace space.

Contribution

It offers a complete proof of the trace theorem for Besov-type spaces of generalized smoothness on d-sets, adapting classical methods to this broader context.

Findings

Complete proof of the trace theorem for Besov-type spaces on d-sets.

Identification of conditions for density of $C_c^ abla(D)$ in the trace space when $d=n$.

Extension of classical trace results to spaces with generalized smoothness.

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 $D\subset\mathbb R^n$, $d\leq n$. 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 $D$.