Weighted estimates for bilinear fractional integral operators and their commutators on Morrey spaces

TL;DR

This paper establishes weighted estimates and maximal function control theorems for bilinear fractional integral operators and their commutators on Morrey spaces, extending classical inequalities and providing new bounds.

Contribution

It introduces new weighted estimates and maximal control results for bilinear fractional integrals and commutators on Morrey spaces, including a bilinear Stein-Weiss inequality.

Findings

Weighted Morrey space estimates for bilinear fractional integrals

Maximal function control theorems for these operators

New bounds for bilinear maximal functions related to the bilinear Hilbert transform

Abstract

This paper mainly dedicates to prove a plethora of weighted estimates on Morrey spaces for bilinear fractional integral operators and their general commutators with BMO functions of the form $$B_{\alpha}(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha}}dy,\qquad 0<\alpha<n.$$ We also prove some maximal function control theorems for these operators, that is, the weighted Morrey norm is bounded by the weighted Morrey norm of a natural maximal operator when the weight belongs to $A_{\infty}$. As a corollary, some new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform are obtained. Furthermore, we formulate a bilinear version of Stein-Weiss inequality on Morrey spaces for fractional integrals.