A lower bound for a variation norm operator associated with circular means

TL;DR

This paper demonstrates the failure of a local variation norm estimate for circular means in two dimensions and provides quantitative lower bounds related to Fourier multipliers on annuli.

Contribution

It establishes lower bounds for variation norm estimates associated with circular means, revealing fundamental limitations in two-dimensional harmonic analysis.

Findings

Local $L^p(V_2)$ variation norm estimate fails for circular means in 2D

Provides lower bounds for functions of exponential type

Connects to lower bounds for Fourier multipliers on annuli

Abstract

We prove that a local $L^p(V_2)$ variation norm estimate fails for circular means in two dimensions, and quantify this failure by proving lower bounds for functions of exponential type. This is related to lower bounds for Fourier multipliers supported on annuli, of the type considered by C\'ordoba.