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 L2 spaces. We also achieve bounds in the general case of matrix-weighted Lp spaces, for 1<p<∞ 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≠2.