Weighted Multilinear Square Function Bounds

TL;DR

This paper establishes new weighted bounds for multilinear Littlewood-Paley-Stein square functions, relaxing regularity conditions and extending results to Lebesgue spaces with indices less than one, advancing harmonic analysis theory.

Contribution

It introduces weight-independent bounds for multilinear square functions and characterizes necessary and sufficient conditions for weighted bounds in convolution operators.

Findings

Weighted bounds hold under relaxed regularity conditions.

Necessary and sufficient conditions are identified for convolution operators.

Extended bounds to Lebesgue spaces with indices less than one.

Abstract

In this work we study boundedness of Littlewood-Paley-Stein square func- tions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are independent of weights, which is a new result even in the linear case. For a class of multilinear convolu- tion operators, we prove necessary and sufficient conditions for weighted Lebesgue space bounds. Using extrapolation theory, we extend weighted bounds in the multilinear setting for Lebesgue spaces with index smaller than one.