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, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ id τ : B p 1 , q 1 s 1 , τ 1 ( Ω ) ↪ B p 2 , q 2 s 2 , τ 2 ( Ω ) and $$\text {id}_\tau : {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ id τ : F p 1 , q 1 s 1 , τ 1 ( Ω ) ↪ F p 2 , q 2 s 2 , τ 2 ( Ω ) , where $$\Omega \subset {{{\mathbb {R}}}^d}$$ Ω ⊂ R d is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $$\text {id}_\tau $$ id τ . 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.

Nuclear Embeddings of Morrey Sequence Spaces and Smoothness Morrey Spaces

“…When p = u or τ = 0 we obtain the usual Besov and Triebel-Lizorkin spaces defined on domains. In [10] we studied the extension operator of spaces A s,τ p,q ( ) and studied limiting embeddings. We obtained, for instance, that -in addition to the monotonicity in the smoothness parameter s and the fine index q, as recalled in (2.3) and (2.4), respectively, -there is some monotonicity in τ , too: we proved that A s,τ 1 p,q ( ) → A s,τ 2 p,q ( ) when 0 ≤ τ 2 ≤ τ 1 , cf.…”