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

TL;DR

This paper characterizes the conditions under which certain Besov-type and Triebel-Lizorkin-type space embeddings are continuous on bounded domains and constructs a universal extension operator for these spaces.

Contribution

It provides necessary and sufficient conditions for limiting embeddings and develops a universal extension operator for Besov-type and Triebel-Lizorkin-type spaces.

Findings

Established criteria for embedding continuity in limiting cases.

Constructed a bounded universal extension operator.

Extended previous results on compactness to limiting embeddings.

Abstract

In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_\tau: B_{p_1,q_1}^{s_1,\tau_1}(\Omega) \rightarrow B_{p_2,q_2}^{s_2,\tau_2}(\Omega)$ and $id_\tau : F_{p_1,q_1}^{s_1,\tau_1}(\Omega) \rightarrow F_{p_2,q_2}^{s_2,\tau_2}(\Omega)$, where $\Omega \subset \mathbb{R}^d$ is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $id_\tau$. 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.