Matrix Weighted Lᵖ Estimates in the Nonhomogeneous Setting

We establish a modified pointwise convex body domination for vector-valued Haar shifts in the nonhomogeneous setting, strengthening and extending the scalar case developed in arXiv: 2309. 13943. Moreover, we identify a subclass of shifts, called L¹-normalized, for which the standard convex body domination holds without requiring any regularity assumption on the measure. Finally, we extend the best-known matrix weighted Lᵖ estimates for sparse forms to the nonhomogeneous setting. The key difficulty here is the lack of a reverse-Hölder inequality for scalar weights, which was used in arXiv: 1710. 03397 to establish Lᵖ matrix weighted estimates and only works in the doubling setting. Our approach relies instead on a generalization of the weighted Carleson embedding theorem which allows to control not only a fixed weight, but also collections of weights localized on different dyadic cubes that satisfy a certain compatibility condition.