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

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 $p \neq 2.$