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

Abstract: We aim to contribute to 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. To achieve this, we provide appropriate definitions and properties, ensuring our construction of these spaces is suitable for non-linear partial differential equations and boundary value problems. The trace theorem holds with the sharp range $s \in (\frac{1}{p}, 1 + \frac{1}{p})$. While the case of inhomogeneous function spaces is well-known, the case of homogeneous function spaces appears to be new, even for a smooth half-space. We refine several arguments from a previous paper on function spaces on the half-space and include a treatment for the endpoint cases $p=1$ and $p=+\infty$.