Weak and strong type estimates for the multilinear Littlewood-Paley operators

TL;DR

This paper establishes sharp weighted norm inequalities and various two-weight estimates for multilinear Littlewood-Paley operators, extending results to the linear case and addressing conjectures in harmonic analysis.

Contribution

It provides new sharp weighted bounds, two-weight inequalities, and local decay estimates for multilinear Littlewood-Paley functions, including results applicable to the linear case.

Findings

Sharp weighted estimate in terms of aperture and weights

Two-weight inequalities with bump and entropy bump estimates

Coifman-Fefferman inequality with $A_{}$ norm

Abstract

Let $S_{\alpha}$ be the multilinear square function defined on the cone with aperture $\alpha \geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_{\alpha}$. We first obtain a sharp weighted estimate in terms of aperture $\alpha$ and $\vec{w} \in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise $A_{\infty}$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g^*_{\lambda}$ function. Some results are new even in the linear case.