A remark on circular means of Fourier transforms of measures

TL;DR

This paper confirms that the existing decay estimates for circular means of measures are optimal for all p >= 2, and no improvements are possible for p < 2, impacting related geometric measure theory problems.

Contribution

It proves that the decay estimates established for p >= 2 cannot be improved for p < 2, settling a question raised in prior work.

Findings

Decay estimates are optimal for p >= 2

No improvements are possible for p < 2

Implications for Falconer's distance problem

Abstract

In a recent paper of Wolff the optimal decay of circular L^p means of compactly supported measures of finite energy was given for p>=2, with application to Falconer's distance problem. The question was then raised in that paper as to whether any non-trivial improvement to these estimates were available for p < 2. We answer this question in the negative.