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Maximal Function and Atomic Characterizations of Matrix-Weighted Hardy Spaces with Their Applications to Boundedness of Calderón--Zygmund Operators

Fan Bu, Yiqun Chen, Dachun Yang, Wen Yuan

TL;DR

This work develops a real-variable framework for matrix-weighted Hardy spaces $H^p_W$ with $p\in(0,1]$ and matrix weights $W$ in $A_p$, defining $H^p_W$ via the matrix-weighted grand maximal function and establishing maximal-function and atomic characterizations through reducing operators. It introduces $\mathbb{A}$-matrix-weighted maximal functions and proving their equivalence to the matrix-weighted Hardy norm, yielding a robust set of tools for matrix-weighted analysis. The authors prove a finite atomic characterization and a Calderón–Zygmund decomposition in the matrix-weight setting, derive a sublinear-operator boundedness criterion, and show Calderón–Zygmund operators are bounded on $H^p_W$, extending classical scalar results to the matrix-weighted context. The results rely on new matrix-weighted maximal functions and the interplay between weights, reducing operators, and atomic decompositions, providing foundational techniques for PDEs and operator theory with matrix weights.

Abstract

Let $p\in(0,1]$ and $W$ be an $A_p$-matrix weight, which in scalar case is exactly a Muckenhoupt $A_1$ weight. In this article, we introduce matrix-weighted Hardy spaces $H^p_W$ via the matrix-weighted grand non-tangential maximal function and characterize them, respectively, in terms of various other maximal functions and atoms, both of which are closely related to matrix weights under consideration and their corresponding reducing operators. As applications, we first establish the finite atomic characterization of $H^p_W$, then using it we give a criterion on the boundedness of sublinear operators from $H^p_W$ to any $γ$-quasi-Banach space, and finally applying this criterion we further obtain the boundedness of Calderón--Zygmund operators on $H^p_W$. The main novelty of these results lies in that the aforementioned maximal functions related to reducing operators are new even in the scalar weight case and we characterize these matrix-weighted Hardy spaces by a fresh and natural variant of classical weighted atoms via first establishing a Calderón--Zygmund decomposition which is also new even in the scalar weight case.

Maximal Function and Atomic Characterizations of Matrix-Weighted Hardy Spaces with Their Applications to Boundedness of Calderón--Zygmund Operators

TL;DR

This work develops a real-variable framework for matrix-weighted Hardy spaces with and matrix weights in , defining via the matrix-weighted grand maximal function and establishing maximal-function and atomic characterizations through reducing operators. It introduces -matrix-weighted maximal functions and proving their equivalence to the matrix-weighted Hardy norm, yielding a robust set of tools for matrix-weighted analysis. The authors prove a finite atomic characterization and a Calderón–Zygmund decomposition in the matrix-weight setting, derive a sublinear-operator boundedness criterion, and show Calderón–Zygmund operators are bounded on , extending classical scalar results to the matrix-weighted context. The results rely on new matrix-weighted maximal functions and the interplay between weights, reducing operators, and atomic decompositions, providing foundational techniques for PDEs and operator theory with matrix weights.

Abstract

Let and be an -matrix weight, which in scalar case is exactly a Muckenhoupt weight. In this article, we introduce matrix-weighted Hardy spaces via the matrix-weighted grand non-tangential maximal function and characterize them, respectively, in terms of various other maximal functions and atoms, both of which are closely related to matrix weights under consideration and their corresponding reducing operators. As applications, we first establish the finite atomic characterization of , then using it we give a criterion on the boundedness of sublinear operators from to any -quasi-Banach space, and finally applying this criterion we further obtain the boundedness of Calderón--Zygmund operators on . The main novelty of these results lies in that the aforementioned maximal functions related to reducing operators are new even in the scalar weight case and we characterize these matrix-weighted Hardy spaces by a fresh and natural variant of classical weighted atoms via first establishing a Calderón--Zygmund decomposition which is also new even in the scalar weight case.

Paper Structure

This paper contains 7 sections, 30 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Maximal Function Characterizations of $H_W^p$
  3. $\mathbb A$-Matrix-Weighted Maximal Functions
  4. Relations Between Matrix-Weighted Hardy Spaces and $\mathbb A$-Matrix-Weighted Hardy Spaces
  5. Atomic Characterization
  6. Finite Atomic Characterization and Its Applications
  7. Boundedness of Calderón--Zygmund Operators on $H^p_W$

Key Result

Theorem 2.10

Let $p\in(0,\infty)$ and $W\in A_{p,\infty}$. Assume that $\psi\in\mathcal{S}$ satisfies $\int_{\mathbb R^n}\psi(x)\,dx\neq0$. Let $a\in(0,\infty)$, $l\in(n/p+[d_{p,\infty}^{\mathrm{lower}}(W)+d_{p,\infty}^{\mathrm{upper}}(W)]/p,\infty)$, and $N\in\mathbb N$ satisfy $N> l+ d_{p,\infty}^{\mathrm{uppe where the positive equivalence constants are independent of $\vec{f}$.

Theorems & Definitions (82)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 72 more