A lattice approach to matrix weights

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.