On $L^p$-improving bounds for maximal operators associated with curves of finite type in the plane

In this paper, we study the $L^p$-improving property for the maximal operators along a large class of curves of finite type in the plane with dilation set $E\subset [1,2]$. The $L^p$-improving region depends on the order of finite type and the fractal dimension of $E$. In particular, various impacts of non-isotropic dilations are also considered.