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

TL;DR

This paper establishes sharp $L^p$ bounds for variation operators associated with averaging over finite-type planar curves, distinguishing isotropic and non-isotropic dilations. The authors combine long/short variation decompositions, stationary-phase analysis, and local smoothing (MSS) to prove sufficiency: in the isotropic case, $q>\max\{m,p/2\}$ with $p>m$ (and $q>\max\{2,p/2\}$ with $p>1$ in the $a_0=0$ scenario), with non-isotropic dilations demanding $p>2$ and $q>\max\{2,p/2\}$. They also prove necessary conditions $q\ge p/2$ via carefully constructed test functions, showing the bounds are sharp. A crucial technical ingredient is a mixed-norm $L^4_x(L^2_t)$ estimate for Fourier integral operators not satisfying the cinematic curvature condition uniformly, established via angle-based decompositions and square-function techniques. Together, these results extend planar variational inequalities to finite-type curves with vanishing curvature and connect to local smoothing and Fourier-integral-operator theory in a nontrivial way.

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.