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
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
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