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

TL;DR

This paper shows that the local $L^p$ variation norm bound for circular means in $\mathbb{R}^2$ fails, by deriving sharp lower bounds for the variation of the circular averages on exponential-type functions whose Fourier support lies in a ball of radius $\lambda$. The authors connect the problem to radial Fourier multipliers supported on annuli and invoke Córdoba's bounds for such multipliers, constructing Besicovitch-type sets to produce Fefferman-style lower bounds that scale like $\big(\log \lambda\big)^{\frac12-\frac1p}$ for $p>2$. The main contribution is a quantitative lower bound for $B_p(\lambda)$, which implies the unboundedness of the local variation operator $V_2^I A$ in the range $2<p\le4$, and provides deep insight into the interplay between variation-norm estimates and annular Fourier multipliers. Overall, the results illuminate the sharp limitations of local variation bounds for circular means and connect geometric Kakeya-type constructions with multiplier theory.

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 Córdoba.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof : Proof of Lemma \ref{['lem:aux2']}
• proof : Proof of Theorem \ref{['thm:kakeya']}, conclusion
• Remark 2.4