Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces

TL;DR

This paper extends weighted norm inequalities to bilinear maximal operators on variable Lebesgue spaces, introducing a bilinear $ ext{A}_ extbf{p}$ condition that characterizes boundedness, and applies it to bilinear singular integrals.

Contribution

It introduces a bilinear $ ext{A}_ extbf{p}$ condition for variable Lebesgue spaces, generalizing linear and classical bilinear results, and establishes its necessity and sufficiency.

Findings

Bilinear maximal operator satisfies weighted norm inequalities under the new $ ext{A}_ extbf{p}$ condition.

The bilinear $ ext{A}_ extbf{p}$ condition is necessary and sufficient for boundedness.

Weighted inequalities are proved for bilinear singular integral operators in variable Lebesgue spaces.

Abstract

We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable $\A_\pp$ condition, and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza and Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.