Maximal operators along flat curves with one variable vector field

Abstract We study a maximal average along a family of curves $\{(t,m(x_1)\gamma(t)):t\in [-r,r]\}$ , where $\gamma|_{[0,\infty)}$ is a convex function and m is a measurable function. Under the assumption of the doubling property of $\gamma'$ and $1\leqslant m(x_1)\leqslant 2$ , we prove the $L^p(\mathbb{R}^2)$ boundedness of the maximal average. As a corollary, we obtain the pointwise convergence of the average in r > 0 without any size assumption for a measurable m .