Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator

Abstract

In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, idτ:Bs1,τ1p1,q1(Ω)↪Bs2,τ2p2,q2(Ω) and idτ:Fs1,τ1p1,q1(Ω)↪Fs2,τ2p2,q2(Ω), where Ω⊂Rd is a bounded domain, obtaining necessary and sufficient conditions for the continuity of idτ . This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces.