Quantitative weighted estimates for rough homogeneous singular integrals

TL;DR

This paper establishes a quadratic bound in the $A_2$ constant for the weighted norm of rough homogeneous singular integrals, advancing understanding of their behavior without kernel continuity.

Contribution

It provides the first known quadratic weighted bound for rough homogeneous singular integrals, using a novel decomposition and quantitative analysis of kernel regularity.

Findings

Bound is quadratic in the $A_2$ constant $[w]_{A_2}$

Decomposition into smooth pieces is effective for rough kernels

Results have implications for weighted bounds of powers of the Beurling transform

Abstract

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound that is quadratic in the $A_2$ constant $[w]_{A_2}$. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.