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, $i{d}_{\tau }:B_{p_1,q_1}^{s_1,{\tau }_1}(\Omega )\to B_{p_2,q_2}^{s_2,{\tau }_2}(\Omega )$ and $i{d}_{\tau }:F_{p_1,q_1}^{s_1,{\tau }_1}(\Omega )\to F_{p_2,q_2}^{s_2,{\tau }_2}(\Omega )$, where $\Omega \subset {\mathbb{R}}^d$ is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $i{d}_{\tau }$. 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.

30 pages, no figures