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A class of multilinear bounded oscillation operators on measure spaces and applications

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Abstract

In recent years, dyadic analysis has attracted a lot of attention due to the \(A_2\) conjecture. It has been well understood that in the Euclidean setting, Calderón–Zygmund operators can be pointwise controlled by a finite number of dyadic operators with a very simple structure, which leads to some significant weak and strong type inequalities. Similar results hold for Hardy–Littlewood maximal operators and Littlewood–Paley square operators. These owe to good dyadic structure of Euclidean spaces. Therefore, it is natural to wonder whether we could work in general measure spaces and find a universal framework to include these operators. In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calderón–Zygmund operators with a quantity of operators beyond the multilinear Calderón–Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get weighted estimates in the full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fréchet–Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy–Littlewood maximal operators on measure spaces, multilinear \(\omega \)–Calderón–Zygmund operators on spaces of homogeneous type, multilinear Littlewood–Paley square operators, multilinear Fourier integral operators, higher order Calderón commutators, maximally modulated multilinear singular integrals, and q-variation of \(\omega \)-Calderón–Zygmund operators.

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Acknowledgements

Mingming Cao acknowledges financial support from Spanish Ministry of Science and Innovation through the Ramón y Cajal 2021 (RYC2021-032600-I), through the “Severo Ochoa Programme for Centres of Excellence in R &D” (CEX2019-000904-S), and through PID2019-107914GB-I00, and from the Spanish National Research Council through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). Gonzalo Ibañez-Firnkorn was partially supported by CONICET and SECYT-UNC. Israel P. Rivera-Ríos was partially supported by FONCyT PICT 2018-02501 and PICT 2019-00018 and by Junta de Andalucía UMA18FEDERJA002 and FQM 354. Qingying Xue was partly supported by the National Key R &D Program of China (No. 2020YFA0712900) and NNSF of China (No. 12271041). Finally, the authors would like to thank the referee for careful reading and valuable comments, which lead to the improvement of this 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. INMABB, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, Argentina

    Gonzalo Ibañez-Firnkorn

  3. Departamento de Matemática, Universidad Nacional del Sur (UNS), Bahía Blanca, Argentina

    Gonzalo Ibañez-Firnkorn & Israel P. Rivera-Ríos

  4. Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain

    Israel P. Rivera-Ríos

  5. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Qingying Xue

  6. 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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Cao, M., Ibañez-Firnkorn, G., Rivera-Ríos, I.P. et al. A class of multilinear bounded oscillation operators on measure spaces and applications. Math. Ann. 388, 3627–3755 (2024). https://doi.org/10.1007/s00208-023-02619-5

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