Embeddings on unbounded open sets

Determine results for embeddings between Besov spaces B_{p0,q0}^{s0}(U) -> B_{p1,q1}^{s1}(U) when U is an unbounded open subset of R^n, clarifying the behavior and classification of such embeddings in this setting in analogy with the bounded Lipschitz domain and whole-space cases.

Background

The paper provides a complete classification of the “quality” of non-compactness for embeddings between Besov spaces on bounded Lipschitz domains and on R^n, distinguishing cases that are finitely strictly singular, strictly singular, or not strictly singular.

In the concluding remarks, the authors explicitly pose the question of extending these types of results to unbounded open sets, indicating that the current classification does not address that geometrical setting.