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

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 $p=+\infty$.
Comment: The paper has been thoroughly revised. The main results are now valid even in the absence of completeness of the normed spaces, and there is also a study of the endpoint cases $p=1$ and $p=+\infty$. For the reader's convenience, an appendix has been added that briefly reviews a few known facts. Importantly, Part B of the Appendix contains few results on the interpolation of non-complete spaces. To conduct an in-depth study, the preliminary section contains refined and sharpened results for the homogeneous function spaces on $\mathbb{R}^n$.70 pages. 3 Figures. Comments are welcome.This work was partially supported by the ANR project RAGE ANR-18-CE40-0012