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On the L2-bounds for hyper-parabolic maximal function*

NIE xudong1, YAN Dunyan2,†, YANG Yaqing2   

  1. 1 Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
    2 School of Mathematics Science, University of Chinese Academy of Scinces, Beijing 10049, China
  • Received:2026-02-04 Revised:2026-04-10 Online:2026-04-21
  • Contact: E-mail:ydunyan@mails.ucas.ac.cn
  • Supported by:
    *National Key R&D Program of China (2023YFC3007303)and National Natural Science Foundation of China(12271501)

Abstract: We estimate the $ L^{2}$-bounds for the maximal function over hyper-surface
$ \sup _{t>0} \frac{1}{\left|S^{n}\right|}\left|\int_{S^{n}} f(\bar{x}-t \bar{y}) d \sigma(\bar{y})\right|.$
We prove the $ L^{2}$-bounds of the maximal function can be controlled by $ Cn^{\frac{3}{4}}$, where C is independent of n.

Key words: maximal function, dimension-free estimate, Fourier multiplier

CLC Number: