Weighted estimates for bilinear fractional integral operators and their commutators

TL;DR

This paper establishes new weighted norm inequalities for bilinear fractional integral operators and their commutators, including maximal function control, advancing understanding of their boundedness in weighted Lebesgue spaces.

Contribution

It introduces novel weighted estimates and maximal function control theorems for bilinear fractional integral operators and their commutators with BMO functions.

Findings

Weighted $L^p$ bounds for bilinear fractional integrals

Control of these operators via maximal functions for $A_ abla$ weights

New weighted estimates for bilinear maximal functions associated with the bilinear Hilbert transform

Abstract

In this paper we prove several weighted estimates for bilinear fractional integral operators and their commutators with BMO functions. We also prove maximal function control theorem for these operators, that is, we prove the weighted $L^p$ norm is bounded by the weighted $L^p$ norm of a natural maximal operator when the weight belongs to $A_\infty$. As a corollary we are able to obtain new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform.