Homogeneous Sobolev and Besov spaces on special Lipschitz domains and their traces

Anatole Gaudin

Published 2023-05-02Version 1

We propose here to garnish the folklore of function spaces on Lipschitz domains. We prove the boundedness of the trace operator for homogeneous Sobolev and Besov spaces on a special Lipschitz domain with sharp regularity. In order to obtain such a result, we also provide appropriate definitions and properties so that our construction of homogeneous of Sobolev and Besov spaces on special Lipschitz domains, and their boundary, that are suitable for the treatment of non-linear partial differential equations and boundary value problems. The trace theorem for homogeneous Sobolev and Besov spaces on special Lipschitz domains occurs in range $s\in(\frac{1}{p},1+\frac{1}{p})$. While the case of inhomogeneous Sobolev and Besov spaces is very common and well known, the case of homogeneous function spaces seems to be new. This paper uses and improves several arguments exposed by the author in a previous paper for function spaces on the whole and the half-space.