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\(L^{p}\)-Improving Bounds and Weighted Estimates for Maximal Functions Associated with Curvature

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Abstract

In this paper, we establish weighted estimates for a wide class of maximal functions along some finite type curves and hypersurfaces. In particular, various impacts of non-isotropic dilations are considered. Via the methodology of sparse domination, the weighted estimates for the global maximal functions can be reduced to the \(L^{p}\)-improving properties of the corresponding localized maximal functions. We mainly focus on proving the \(L^{p}\rightarrow L^{q}\) bounds \((q >p)\) for localized maximal functions with non-isotropic dilations of curves and hypersurfaces whose curvatures vanish to finite order at some points. As a corollary, we obtain the weighted inequalities for the corresponding global maximal functions, which generalize the known unweighted estimates.

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Notes

  1. We will omit the dependence on the dimension n and dilation \(\delta _t\) for the sake of brevity.

  2. We set \({\mathcal {L}}_n'\) to be the dual range of exponents to \({\mathcal {L}}_n\), which is defined as \({\mathcal {L}}_n' := \{(\frac{1}{p}, \frac{1}{q'}):(\frac{1}{p},1-\frac{1}{q'}) \in {\mathcal {L}}_n\}\).

  3. This also avoids the extra technicality discussed in [5] Remark 1.4.

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Acknowledgements

We would like to express our gratitude to Prof. Joris Roos for his insightful and helpful suggestions and comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

    Wenjuan Li

  2. School of Mathematics and Statistics, Henan University, Kaifeng, 475000, China

    Huiju Wang

  3. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA

    Yujia Zhai

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  1. Wenjuan Li
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Correspondence to Huiju Wang.

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Communicated by Alex Iosevich.

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This work is supported by Natural Science Foundation of China (No. 12271435); China Postdoctoral Science Foundation (No. 2021M693139); ERC Project FAnFArE No. 637510, the region Pays de la Loire and CNRS.

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Li, W., Wang, H. & Zhai, Y. \(L^{p}\)-Improving Bounds and Weighted Estimates for Maximal Functions Associated with Curvature. J Fourier Anal Appl 29, 10 (2023). https://doi.org/10.1007/s00041-023-09993-3

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