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Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding

Fan Bu, Dachun Yang, Wen Yuan, Mingdong Zhang

TL;DR

This work develops a comprehensive real-variable theory for generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces endowed with matrix $\mathcal{A}_{\infty}$ weights and growth functions. It establishes a suite of characterizations including $\varphi$-transforms, Peetre-type maximal functions, and Littlewood–Paley theory, and then derives molecular and wavelet decompositions via almost diagonal operators. The authors prove the invariance of certain Triebel–Lizorkin-type spaces with respect to the integrable index $p$ and formulate Sobolev-type embeddings in the matrix-weighted setting, showing the growth condition is both necessary and sufficient for operator boundedness and space nontriviality. The framework unifies and extends known scalar and matrix-weighted BTL-type spaces, provides new averaging-space tools, and yields sharp embedding and invariance results with potential applications to harmonic analysis and PDEs under matrix weights.

Abstract

In this article, using growth functions we introduce generalized matrix-weighted Besov-Triebel-Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. We first characterize these spaces, respectively, in terms of the $\varphi$-transform, the Peetre-type maximal function, and the Littlewood-Paley functions. Furthermore, after establishing the boundedness of almost diagonal operators on the corresponding sequence spaces, we obtain the molecular and the wavelet characterizations of these spaces. As applications, we find the sufficient and necessary conditions for the invariance of those Triebel-Lizorkin-type spaces on the integrable index and also for the Sobolev-type embedding of all these spaces. The main novelty exists in that these results are of wide generality, the growth condition of growth functions is not only sufficient but also necessary for the boundedness of almost diagonal operators and hence this new framework of Besov-Triebel-Lizorkin-type is optimal, some results either are new or improve the known ones even for known matrix-weighted Besov-Triebel-Lizorkin spaces, and, furthermore, even in the scalar-valued setting, all the results are also new.

Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding

TL;DR

This work develops a comprehensive real-variable theory for generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces endowed with matrix weights and growth functions. It establishes a suite of characterizations including -transforms, Peetre-type maximal functions, and Littlewood–Paley theory, and then derives molecular and wavelet decompositions via almost diagonal operators. The authors prove the invariance of certain Triebel–Lizorkin-type spaces with respect to the integrable index and formulate Sobolev-type embeddings in the matrix-weighted setting, showing the growth condition is both necessary and sufficient for operator boundedness and space nontriviality. The framework unifies and extends known scalar and matrix-weighted BTL-type spaces, provides new averaging-space tools, and yields sharp embedding and invariance results with potential applications to harmonic analysis and PDEs under matrix weights.

Abstract

In this article, using growth functions we introduce generalized matrix-weighted Besov-Triebel-Lizorkin-type spaces with matrix weights. We first characterize these spaces, respectively, in terms of the -transform, the Peetre-type maximal function, and the Littlewood-Paley functions. Furthermore, after establishing the boundedness of almost diagonal operators on the corresponding sequence spaces, we obtain the molecular and the wavelet characterizations of these spaces. As applications, we find the sufficient and necessary conditions for the invariance of those Triebel-Lizorkin-type spaces on the integrable index and also for the Sobolev-type embedding of all these spaces. The main novelty exists in that these results are of wide generality, the growth condition of growth functions is not only sufficient but also necessary for the boundedness of almost diagonal operators and hence this new framework of Besov-Triebel-Lizorkin-type is optimal, some results either are new or improve the known ones even for known matrix-weighted Besov-Triebel-Lizorkin spaces, and, furthermore, even in the scalar-valued setting, all the results are also new.
Paper Structure (18 sections, 58 theorems, 276 equations)

This paper contains 18 sections, 58 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Generalized Matrix-Weighted Function Spaces $\dot{A}_{p,q}^{s, \upsilon}(W)$
  3. Definition of $\dot{A}_{p,q}^{s, \upsilon}(W)$
  4. Properties of Growth Functions $\upsilon$ and Nontriviality of $\dot{A}_{p,q}^{s,\upsilon}(W)$
  5. Relations of $\dot{A}_{p,q}^{s, \upsilon}(W)$ with Known Function Spaces
  6. Averaging Spaces and Proof of Theorem \ref{['thm-phitransMWBTL']}
  7. Coincidence of Matrix-Weighted Spaces and Averaging Spaces
  8. Proof of Theorem \ref{['thm-phitransMWBTL']}
  9. Peetre-Type Maximal Function and Littlewood--Paley Function Characterizations of $\dot{A}^{s,\upsilon}_{p,q}(W)$
  10. Peetre-Type Maximal Function Characterization
  11. Littlewood--Paley Function Characterization
  12. Molecular and Wavelet Characterizations of $\dot{A}_{p,q}^{s, \upsilon}(W)$
  13. Boundedness of Almost Diagonal Operators on $\dot{a}_{p,q}^{s, \upsilon}(W)$
  14. Molecular and Wavelet Characterizations of $\dot{A}_{p,q}^{s, \upsilon}(W)$
  15. Applications
  16. ...and 3 more sections

Key Result

Theorem 2.5

Let $(A, a)\in\{(B, b), (F, f)\}$, $s\in\mathbb{R}$, $p\in(0, \infty)$, $q\in(0,\infty]$, and $W\in\mathcal{A}_{p,\infty}$. Assume that $\upsilon\in\mathcal{G}(\delta_1, \delta_2; \omega)$ with and $\varphi, \psi\in\mathcal{S}$ both satisfy cond1. Then the following statements hold.

Theorems & Definitions (143)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • Lemma 2.9
  • ...and 133 more