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June, 2022 Maximal Functions Along Convex Curves with Lacunary Directions
Haixia Yu
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Taiwanese J. Math. 26(3): 545-570 (June, 2022). 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

Download Citation

Haixia Yu. "Maximal Functions Along Convex Curves with Lacunary Directions." Taiwanese J. Math. 26 (3) 545 - 570, June, 2022. https://doi.org/10.11650/tjm/211208

Information

Received: 27 January 2021; Revised: 16 December 2021; Accepted: 22 December 2021; Published: June, 2022
First available in Project Euclid: 29 December 2021

Digital Object Identifier: 10.11650/tjm/211208

Subjects:
Primary: 42B25

Keywords: Littlewood–Paley operator , maximal function along curve , Oscillatory integral

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

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Vol.26 • No. 3 • June, 2022
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