Boundedness of Journe operators with matrix weights

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journe 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 Journe operator bounds on matrix -weighted L2 spaces. We also achieve bounds in the general case of matrix-weighted Lp spaces, for 1 < p < infinity for parapro duct-free Journe 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 Journe operators and p not equal 2. (c) 2023 Elsevier Inc. All rights reserved.