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2021 Maximal Functions Along Convex Curves with Lacunary Directions
Haixia Yu
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Taiwanese J. Math. Advance Publication 1-26 (2021). DOI: 10.11650/tjm/211208

Abstract

In this paper, we obtain the $L^p(\mathbb{R}^2)$-boundedness of the maximal functions \[ M_{I,\gamma} f(x_1,x_2) := \sup_{j \in \mathbb{Z}} \sup_{\varepsilon > 0} \frac{1}{2\varepsilon} \int_{-\varepsilon}^{\varepsilon} |f(x_1-t, x_2 - 2^j \gamma(t))| \, \mathrm{d}t \] and \[ M_{II,\gamma} f(x_1,x_2) := \sup_{i,j \in \mathbb{Z}} \sup_{\varepsilon > 0} \frac{1}{2\varepsilon} \int_{-\varepsilon}^{\varepsilon} |f(x_1 - 2^i t, x_2 - 2^j \gamma(t))| \, \mathrm{d}t, \] where $p \in (1,\infty]$ and $\gamma$ is a convex curve satisfying some suitable curvature conditions.

Funding Statement

The author is supported by STU Scientific Research Foundation for Talents (No. NTF21038) and Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515110241).

Acknowledgments

The author would like to thank the referee for carefully reading the manuscript and for offering valuable comments, and also Lixin Yan for the many helpful suggestions.

Citation

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Haixia Yu. "Maximal Functions Along Convex Curves with Lacunary Directions." Taiwanese J. Math. Advance Publication 1 - 26, 2021. https://doi.org/10.11650/tjm/211208

Information

Published: 2021
First available in Project Euclid: 29 December 2021

Digital Object Identifier: 10.11650/tjm/211208

Subjects:
Primary: 42B25

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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