Convex body domination for rough singular integrals

TL;DR

This paper extends convex body domination techniques to rough singular integrals, enabling new matrix weighted bounds and commutator estimates, even for operators with unbounded angular parts.

Contribution

It generalizes sparse domination results to the convex body setting for rough singular integrals, including unbounded angular parts, and derives new two-weight commutator bounds.

Findings

Extended convex body domination to rough singular integrals.

Derived new matrix weighted bounds for these operators.

Established novel two-weight commutator estimates in the scalar case.

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.