Matrix-Weighted Campanato Spaces: Duality and Calder\'on--Zygmund Operators

TL;DR

This paper introduces matrix-weighted Campanato spaces and establishes their duality with Hardy spaces, providing conditions for the boundedness of Calderón--Zygmund operators in this context.

Contribution

It defines matrix-weighted Campanato spaces using reducing operators and characterizes their duality with Hardy spaces, extending the boundedness criteria for Calderón--Zygmund operators.

Findings

Duality between $H^p_W$ and $ ext{Campanato}$ spaces established.

Necessary and sufficient conditions for Calderón--Zygmund operators boundedness derived.

Characterizations of matrix-weighted Campanato spaces provided.

Abstract

Let $p\in(0,\infty)$, $q\in[1,\infty)$, $s\in\mathbb Z_+$, and $W$ be an $A_p$-matrix weight, which in the scalar case is exactly a Muckenhoupt $A_{\max\{1,p\}}$ weight. In this article, by using the reducing operators of $W$, we introduce matrix-weighted Campanato spaces $\mathcal L_{p,q,s,W}$. When $p\in(0,1]$, applying the atomic and the finite atomic characterizations of the matrix-weighted Hardy space $H^p_W$, we prove that the dual space of $H^p_W$ is precisely $\mathcal L_{p,q,s,W}$, which further induces several equivalent characterizations of $\mathcal L_{p,q,s,W}$. In addition, we obtain a necessary and sufficient condition for the boundedness of modified Calder\'on--Zygmund operators on $\mathcal L_{p,q,s,W}$ with $p\in(0,\infty)$, which, combined with the duality, further gives a necessary and sufficient condition for the boundedness of Calder\'on--Zygmund operators on $H^p_W$ with $p\in(0,1]$.