Maximal operators along flat curves with one variable vector field

We study a maximal average along a family of curves {(t,m(x(1))gamma(t)):t is an element of[-r,r]}, where gamma|([0,infinity) )is a convex function and m is a measurable function. Under the assumption of the doubling property of gamma ' and 1 <= m(x(1))<= 2, we prove the L-p(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.