Maximal Functions Along Convex Curves with Lacunary Directions

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.