Matrix-weighted estimates beyond Calderón–Zygmund theory

Abstract We study matrix-weighted bounds for a class of sublinear non-kernel operators considered by F. Bernicot, D. Frey, and S. Petermichl, whose work did not include vector-valued functions. The main feature of our approach is the use of sparse bilinear convex body domination, adapted from a recent general principle due to T. Hytönen, which does not apply to the sublinear case. This sparse domination framework allows us to adapt and extend techniques from F. Nazarov, S. Petermichl, S. Treil, and A. Volberg to the matrix-weighted setting. This approach leads to sharper matrix-weighted norm estimates than the ones previously known. Our weight assumptions are formulated in terms of two-exponent matrix Muckenhoupt conditions, which reveal a surprisingly rich structure absent in the scalar setting. Moreover, we establish a limited range extrapolation theorem for matrix weights, which is more general than the one in the earlier work by P. Auscher and J. M. Martell, as well as M. Bownik and D. Cruz-Uribe. This highlights the versatility and potential of our approach for broader applications in weighted harmonic analysis.