New bounds for bilinear Calder\'on-Zygmund operators and applications

TL;DR

This paper extends domination theorems to bilinear Calderón-Zygmund operators, providing new bounds and applications in weighted inequalities, with implications for multilinear operators and commutators.

Contribution

It introduces pointwise control of bilinear Calderón-Zygmund operators via sparse operators and derives new weighted estimates with broad applications.

Findings

Pointwise domination of bilinear Calderón-Zygmund operators established.

New mixed weighted estimates for bilinear dyadic positive operators derived.

Applications include bounds for multilinear operators, commutators, and Fourier multipliers.

Abstract

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calder\'on--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hyt\"onen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{\infty}$ constants inspired in the Fujii-Wilson and Hrus\v{c}\v{e}v classical constants. These estimates have many new applications including mixed bounds for multilinear Calder\'on--Zygmund operators and their commutators with $BMO$ functions, square functions and multilinear Fourier multipliers.