A classification theorem on non-compact embeddings between Besov spaces

We analyze the embedding properties between Besov spaces, defined on the total space Rⁿ R n and on bounded domains. We give a complete classification on whether or not these embedding maps satisfy certain weaker forms of compactness characterized by the so-called strictly and finitely strictly singular conditions. The result extends the recent findings on Sobolev embeddings by offering a refined description of the quality of non-compactness in the setting of Besov spaces.