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

We investigate matrix-weighted bounds for the linear non-kernel operators considered by F. Bernicot, D. Frey, and S. Petermichl. First, we dominate these operators by bilinear convex body sparse forms, applying 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 obtained recently by A. Laukkarinen.