Boundedness of Journ\'{e} operators with matrix weights

TL;DR

This paper develops a biparameter matrix-weighted theory for Journé operators, establishing boundedness results on matrix-weighted L^p spaces under the product matrix Muckenhoupt condition, and discusses open problems in the area.

Contribution

It introduces a comprehensive biparameter matrix-weighted framework for Journé operators, extending bounds to L^p spaces and highlighting open problems in matrix-weighted inequalities.

Findings

Established biparameter matrix-weighted bounds for Journé operators on L^2 spaces.

Extended bounds to L^p spaces for 1 < p < ∞ for paraproduct-free Journé operators.

Identified an open problem involving matrix-weighted Fefferman--Stein inequality.

Abstract

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journ\'e operators as well as other central operators under the assumption of the product matrix Muckenhoupt condition. In particular, we provide a complete theory for biparameter Journ\'e operator bounds on matrix-weighted $L^2$ spaces. We also achieve bounds in the general case of matrix-weighted $L^p$ spaces, for $1 < p < \infty$ for paraproduct-free Journ\'e operators. Finally, we expose an open problem involving a matrix-weighted Fefferman--Stein inequality, on which our methods rely in the general setting of matrix-weighted bounds for arbitrary Journ\'e operators and $p \neq 2.$