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Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator
Functional Analysis · Submitted 24 Sep 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $id_\tau$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $id_\tau$. | contribution 0.93 | 0.96 | 0.76 | asserted 1.00 |
| s1 | This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. | background 0.97 | 0.38 | 0.03 | asserted 0.69 |
| s2 | Moreover, we also construct Rychkov's linear, bounded universal extension operator for these spaces. | contribution 0.95 | 0.95 | 0.08 | asserted 1.00 |
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Original 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.
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3 arXiv papers cite this in 6 sentences
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In [ref] we studied the extension operator of spaces $(Ω)$ and studied limiting embeddings.
Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces · Spaces on domains · 0.82
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