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

In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, \(\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )\) and \(\text {id}_\tau : {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega )\), where \(\Omega \subset {{{\mathbb {R}}}^d}\) is a bounded domain, obtaining necessary and sufficient conditions for the continuity of \(\text {id}_\tau \). 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.