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

TL;DR

This paper proves the boundedness of variational operators over finite type curves in the plane and establishes necessary conditions, using advanced Fourier analysis techniques.

Contribution

It introduces new boundedness results for variational operators over finite type curves and develops mixed-norm local smoothing estimates without cinematic curvature.

Findings

Proved $L^p$-boundedness of variational operators

Established necessary conditions for boundedness

Developed mixed-norm local smoothing estimates

Abstract

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.