Vector valued estimates for matrix weighted maximal operators and product $\mathrm{BMO}$

TL;DR

This paper establishes vector valued maximal operator estimates with matrix weights, proves an $ ext{H}^1$-$ ext{BMO}$ duality for matrix functions, and applies these results to biparameter paraproduct bounds.

Contribution

It introduces new vector valued estimates for matrix weighted maximal operators and extends $ ext{H}^1$-$ ext{BMO}$ duality to matrix valued functions, with applications to biparameter paraproducts.

Findings

Proved Fefferman--Stein type inequalities for matrix weighted vector valued maximal operators.

Established an $ ext{H}^1$-$ ext{BMO}$ duality for matrix valued functions.

Derived upper bounds for biparameter paraproducts using the new estimates.

Abstract

We consider maximal operators acting on vector valued functions, that is, functions taking values on $\mathbb{C}^d,$ that incorporate matrix weights in their definitions. We show vector valued estimates, in the sense of Fefferman--Stein inequalities, for such operators. These are proven using an extrapolation result for convex body valued functions due to Bownik and Cruz-Uribe. Finally, we show an $\mathrm{H}^1$-$\mathrm{BMO}$ duality for matrix valued functions and we apply the previous vector valued estimates to show upper bounds for biparameter paraproducts. For the reader's convenience, we include an appendix explaining how to adapt the extrapolation for real convex body valued functions of Bownik and Cruz-Uribe to the setting of complex convex body valued functions that we treat.