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Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calderón--Zygmund Operators

Dachun Yang, Wen Yuan, Mingdong Zhang

TL;DR

The paper develops a unified operator theory for generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces $\dot{A}^{s,\upsilon}_{p,q}(W)$ with matrix $\mathcal{A}_{\infty}$ weights by leveraging molecular and wavelet characterizations. It reduces operator boundedness to almost diagonal sequence operators and proves boundedness results for pseudo-differential operators, trace/extension, and Calderón–Zygmund operators under precise index-finiteness conditions and weight-dimension constraints, all within the framework $W\in\mathcal{A}_{p,\infty}$ and growth functions $\upsilon$. The main contributions include (i) boundedness of homogeneous pseudo-differential operators with symbols in $\dot{S}_{1,1}^{\eta}$ between shifted and unshifted spaces, (ii) trace and extension theorems for Besov- and Triebel–Lizorkin-type spaces with matrix weights, and (iii) boundedness of Calderón–Zygmund operators via atom-to-molecule mappings, including vector- and $L^2$-extension considerations. These results generalize and unify many scalar and matrix-weighted settings, improving existing scalar-case bounds and providing a versatile framework for harmonic analysis and PDEs in the matrix-weighted context.

Abstract

This article is a continuation of our work on generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. In this article, we establish the boundedness of pseudo-differential, trace, and Calderón--Zygmund operators on these spaces. The main tools involved in this article are the molecular and the wavelet characterizations of these spaces. Since generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces include many classical function spaces such as matrix-weighted Besov--Triebel--Lizorkin spaces, all the results in this article are of wide generality.

Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calderón--Zygmund Operators

TL;DR

The paper develops a unified operator theory for generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces with matrix weights by leveraging molecular and wavelet characterizations. It reduces operator boundedness to almost diagonal sequence operators and proves boundedness results for pseudo-differential operators, trace/extension, and Calderón–Zygmund operators under precise index-finiteness conditions and weight-dimension constraints, all within the framework and growth functions . The main contributions include (i) boundedness of homogeneous pseudo-differential operators with symbols in between shifted and unshifted spaces, (ii) trace and extension theorems for Besov- and Triebel–Lizorkin-type spaces with matrix weights, and (iii) boundedness of Calderón–Zygmund operators via atom-to-molecule mappings, including vector- and -extension considerations. These results generalize and unify many scalar and matrix-weighted settings, improving existing scalar-case bounds and providing a versatile framework for harmonic analysis and PDEs in the matrix-weighted context.

Abstract

This article is a continuation of our work on generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces with matrix weights. In this article, we establish the boundedness of pseudo-differential, trace, and Calderón--Zygmund operators on these spaces. The main tools involved in this article are the molecular and the wavelet characterizations of these spaces. Since generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces include many classical function spaces such as matrix-weighted Besov--Triebel--Lizorkin spaces, all the results in this article are of wide generality.
Paper Structure (6 sections, 28 theorems, 136 equations)

This paper contains 6 sections, 28 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Generalized Matrix-Weighted Besov-Type and Triebel--Lizorkin-type Spaces
  3. Boundedness of Pseudo-Differential, Trace, and Calderón--Zygmund Operators on $\dot{A}_{p, q}^{s, \upsilon}(W)$
  4. Pseudo-Differential Operators
  5. Trace and Extension Operators
  6. Calderón--Zygmund Operators

Key Result

Lemma 3.4

Let $p\in(0,\infty), W\in\mathcal{A}_{p,\infty}$, and $\{A_Q\}_{Q\in\mathcal{D}}$ be a sequence of reducing operators of order $p$ for $W$. If $\beta_1 \in \llbracket d_{p, \infty}^{\mathrm{lower}}(W),\infty)$ and $\beta_2\in \llbracket d_{p, \infty}^{\mathrm{upper}}(W),\infty)$, then there exists a

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Definition 3.5
  • ...and 57 more