Sharp variational inequalities for average operators over finite type curves in the plane

The aim of this article is to establish the $L^p(\mathbb{R}^2)$-boundedness of the variational operator associated with averaging operators defined over finite type curves in the plane. Additionally, we present the necessary conditions for the boundedness of these operators in $L^p$. Furthermore, to prove one of these results, we establish a mixed-norm local smoothing estimate from $L^4$ to $L^4(L^2)$ corresponding to a family of Fourier integral operators that do not uniformly satisfy the cinematic curvature condition.