Maximal functions associated to a family of flat curves in lacunary directions

Consider the maximal operator defined by \(\begin{aligned} \mathcal {M}^{\mathfrak {a}}_{\gamma }f(x_1,x_2)=\sup _{k\in \mathbb {Z}}\sup _{r>0}\frac{1}{2r}\int _{-r}^{r}|f(x_1-t,x_2-a_k\gamma (t))|dt, \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{M}}_{\gamma }^{\mathfrak{a}}f(x_1,x_2)=\underset{k\in \mathbb{Z}}{sup}\underset{r>0}{sup}\frac{1}{2r}{\int }_{-r}^r|f(x_1-t,x_2-a_k\gamma (t))|dt,\end{array}$$ where \(\{(t,\gamma (t))\}\) $\{(t,\gamma (t))\}$ is a convex curve and \(\mathfrak {a}=(a_k)\) $\mathfrak{a}=(a_k)$ is a lacunary sequence. We observe that \(\gamma '\) ${\gamma }^{\prime }$ doubling assumption does not imply to the \(L^p\) $L^p$ boundedness of \( \mathcal {M}^{\mathfrak {a}}_{\gamma }\) ${\mathcal{M}}_{\gamma }^{\mathfrak{a}}$ . However, under the infinitesimally doubling condition on \(\gamma '\) ${\gamma }^{\prime }$ , we obtain the \(L^p\) $L^p$ boundedness of \( \mathcal {M}^{\mathfrak {a}}_{\gamma }\) ${\mathcal{M}}_{\gamma }^{\mathfrak{a}}$ for all \(1<p<\infty \) $1<p<\infty$ .