Matrix-Weighted Campanato Spaces: Duality and Calderón--Zygmund Operators

Let $p\in (0,\mathrm{\infty })$, $q\in [1,\mathrm{\infty })$, $s\in {\mathbb{Z}}_{+}$, 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 $H_W^p$, we prove that the dual space of $H_W^p$ 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 Calderón–Zygmund operators on ${\mathcal{L}}_{p,q,s,W}$ with $p\in (0,\mathrm{\infty })$, which, combined with the duality, further gives a necessary and sufficient condition for the boundedness of Calderón–Zygmund operators on $H_W^p$ with $p\in (0,1]$.