Boundedness of Journé operators with matrix weights

Abstract

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journé 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 Journé 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 Journé 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 Journé operators and $p\ne 2.$

67 pages, no figures. Several parts of the paper have been deleted as they follow from other works, other parts have been modified to improve clarity of exposition. To be published in Journal of Mathematical Analysis and Applications