Convex body domination for rough singular integrals

Abstract: Convex body domination is a technique, where operators acting on vector-valued functions are estimated via certain convex body averages of the input functions. This domination lets one deduce various matrix weighted bounds for these operators and their commutators. In this paper, we extend the sparse domination results for rough singular integrals due to Conde-Alonso, Culiuc, Di Plinio and Ou to the convex body setting. In particular, our methods apply to homogeneous rough singular integrals with unbounded angular part. We also note that convex body domination implies new two weight commutator bounds even in the scalar case.