Matrix-weighted estimates beyond Calder\'on-Zygmund theory

We investigate matrix-weighted bounds for the sublinear non-kernel operators considered by F. Bernicot, D. Frey, and S. Petermichl. We extend their result to sublinear operators acting upon vector-valued functions. First, we dominate these operators by bilinear convex body sparse forms, adapting a recent general principle due to T. Hyt\"onen. Then we use this domination to derive matrix-weighted bounds, adapting arguments of F. Nazarov, S. Petermichl, S. Treil, and A. Volberg. Our requirements on the weight are formulated in terms of two-exponent matrix Muckenhoupt conditions, which surprisingly exhibit a rich structure that is absent in the scalar case. Consequently, we deduce that our matrix-weighted bounds improve the ones that were recently obtained by A. Laukkarinen. The methods we use are flexible, which allows us to complement our results with a limited range extrapolation theorem for matrix weights, extending the results of P. Auscher and J. M. Martell, as well as M. Bownik and D. Cruz-Uribe.