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Weighted Bounds for Multilinear Square Functions

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Abstract

Let \(\vec {P}=(p_{1},\dotsc ,p_{m})\) with 1 < p 1, …, p m < ∞, 1/p 1+⋯+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 α for multilinear square functions \(S_{\alpha ,\psi }(\vec {f})\). We show that

$$\|S_{\alpha,\psi}(\vec{f})\|_{L^{p}(\nu_{\vec{w}})} \leq C_{n,m,\psi,\vec{P}} \alpha^{mn}[\vec{w}]_{A_{\vec{P}}}^{\max(\frac{1}{2},\tfrac{p_{1}^{\prime}}{p},\dotsc,\tfrac{p_{m}^{\prime}}{p})} \prod\limits_{i=1}^{m} \|f_{i}\|_{L^{p_{i}}(w_{i})}. $$

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 Calderón–Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.

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Authors and Affiliations

  1. Department of Mathematics, Macquarie University, Ryde, 2109, NSW, Australia

    The Anh Bui

  2. Department of Mathematical Sciences, Division of Mathematics, University of Gothenburg, Gothenburg, 41296, Sweden

    Mahdi Hormozi

  3. Iran’s National Elites Foundation, No. 17 Haghighat Talab Street, Southern Nejatollahi Avenue, Tehran, Iran

    Mahdi Hormozi

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  1. The Anh Bui
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  2. Mahdi Hormozi
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Correspondence to Mahdi Hormozi.

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Bui, T.A., Hormozi, M. Weighted Bounds for Multilinear Square Functions. Potential Anal 46, 135–148 (2017). https://doi.org/10.1007/s11118-016-9575-9

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  • DOI: https://doi.org/10.1007/s11118-016-9575-9

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