A lattice approach to matrix weights

TL;DR

This paper extends matrix weight theory into Banach lattices, introducing new function spaces, proving an extrapolation theorem, and applying results to variable Lebesgue and Morrey spaces, advancing the understanding of matrix weights.

Contribution

It generalizes matrix weighted Lebesgue spaces within Banach lattices and establishes new extrapolation results and characterizations for these spaces.

Findings

Boundedness of convex-set valued maximal operator characterized by matrix Muckenhoupt condition

Established equivalences for weak-type boundedness in matrix weighted spaces

Extended scalar results to matrix-weighted variable Lebesgue and Morrey spaces

Abstract

In this paper we recontextualize the theory of matrix weights within the setting of Banach lattices. We define an intrinsic notion of directional Banach function spaces, generalizing matrix weighted Lebesgue spaces. Moreover, we prove an extrapolation theorem for these spaces based on the boundedness of the convex-set valued maximal operator. We also provide bounds and equivalences related to the convex body sparse operator. Furthermore, we introduce a weak-type analogue of directional Banach function spaces. In particular, we show that the weak-type boundedness of the set valued maximal operator on matrix weighted Lebesgue spaces is equivalent to the matrix Muckenhoupt condition, with equivalent constants. Finally, we apply our main results to matrix-weighted variable Lebesgue and Morrey spaces, obtaining new extrapolation results and characterizations extending the known ones of the scalar setting.