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Weak and Strong Type Estimates for the Multilinear Littlewood–Paley Operators

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Abstract

Let \(S_{\alpha }\) be the multilinear square function defined on the cone with aperture \(\alpha \ge 1\). 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.

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Acknowledgements

We would like to thank the anonymous referee for his/her careful reading that helped improving the quality of the paper.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, 28049, Madrid, Spain

    Mingming Cao

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

    Mahdi Hormozi

  3. FAMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, Córdoba, Argentina

    Gonzalo Ibañez-Firnkorn

  4. Departamento de Matematica, Universidad Nacional del Sur (UNS), Bahiá Blanca, Argentina

    Israel P. Rivera-Ríos

  5. INMABB, Universidad Nacional del Sur (UNS)-CONICET, Alem 1253, Bahiá Blanca, Argentina

    Israel P. Rivera-Ríos

  6. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, People’s Republic of China

    Zengyan Si

  7. Research Center for Mathematics and Data Science, Kwansei Gakuin University, Gakuen 2-1, Sanda, 669-1337, Japan

    Kôzô Yabuta

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  1. Mingming Cao
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  2. Mahdi Hormozi
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  4. Israel P. Rivera-Ríos
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  5. Zengyan Si
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  6. Kôzô Yabuta
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Correspondence to Zengyan Si.

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Communicated by Loukas Grafakos.

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M. C. acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). M. H. is supported by a grant from IPM. G. I.-F. is partially supported by CONICET and SECYT-UNC. I. P. R.-R. is partially supported by CONICET PIP 11220130100329CO and Agencia I+D+i PICT 2018-02501 and PICT 2019-00018. Z. S. is supported partly by Natural Science Foundation of Henan(No. 202300410184), the Key Research Project for Higher Education in Henan Province(No. 19A110017) and the Fundamental Research Funds for the Universities of Henan Province(No. NSFRF200329).

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Cao, M., Hormozi, M., Ibañez-Firnkorn, G. et al. Weak and Strong Type Estimates for the Multilinear Littlewood–Paley Operators. J Fourier Anal Appl 27, 62 (2021). https://doi.org/10.1007/s00041-021-09870-x

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  • DOI: https://doi.org/10.1007/s00041-021-09870-x

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