Matrix-weighted $Z$-spaces and tent spaces:characterizations, duality and interpolation

Abstract

Let $α\in\mathbb R$, $0<p<\infty$, $0<q,r\le\infty$, and $W$ be a matrix $\mathcal A_p$ weight on ${\mathbb R}^n$. We introduce the matrix-weighted $Z$-spaces $Z^{p,q,r}_α(W)$ and tent spaces $T^{p,q,r}_α(W)$ on the upper half-space and establish their characterizations using reducing operators. We then characterize homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces in terms of the corresponding $Z$-spaces and tent spaces, respectively, using Schwartz kernels satisfying Tauberian and finite moment conditions. Furthermore, we establish duality for the matrix-weighted $Z$-spaces and tent spaces and obtain real and complex interpolation results for these spaces and the associated homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces.