Weighted bounds for multilinear square functions

TL;DR

This paper establishes weighted bounds for multilinear square functions with dependence on aperture, extending linear case results and utilizing local mean oscillation techniques to avoid intrinsic square functions.

Contribution

It provides new weighted bounds for multilinear square functions with explicit aperture dependence, extending prior linear results and employing a novel proof technique.

Findings

Weighted bounds depend on aperture parameter α

Extension of linear case results to multilinear setting

Avoidance of intrinsic square functions in proofs

Abstract

Let $\vec{P}=(p_1,\dotsc,p_m)$ with $1<p_1,\dotsc,p_m<\infty$, $1/p_1+\dotsb+1/p_m=1/p$ and $\vec{w}=(w_1,\dotsc,w_m)\in A_{\vec{P}}$. In this paper, we investigate the weighted bounds with dependence on aperture $\alpha$ for multilinear square functions $S_{\alpha,\psi}(\vec{f})$. We show that $$ \|S_{\alpha,\psi}(\vec{f})\|_{L^p(\nu_{\vec{w}})} \leq C_{n,m,\psi,\vec{P}}~ \alpha^{mn}[\vec{w}]_{A_{\vec{P}}}^{\max(\frac{1}{2},\tfrac{p_1'}{p},\dotsc,\tfrac{p_m'}{p})} \prod_{i=1}^m \|f_i\|_{L^{p_i}(w_i)}. $$ This result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Calder\'on--Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.