Boundary Traces from Weighted Sobolev Spaces into Besov Spaces

Abstract

AbstractThe starting point in this chapter is the consideration of boundary traces from a weighted Sobolev space defined in the entire Euclidean space in §8.1. Next, in §8.3, we consider traces from weighted Sobolev spaces defined in a given \((\varepsilon ,\delta )\)-domain \(\Omega \subseteq {\mathbb {R}}^n\) by relying on P. Jones’ extension theorem to reduce matters to the full Euclidean setting considered earlier. The next order of business is to construct extension operators from boundary Besov spaces into our weighted Sobolev spaces, something we accomplish in §8.4. Weighted Sobolev spaces of negative order, duality, and the conormal derivative operators are discussed in §8.5. Finally, in §8.6 we study boundary traces from weighted “maximal” Sobolev spaces, defined by requiring the membership of distributional derivatives to solid maximal Lebesgue spaces introduced in [133].