On the sweeping out property for convolution operators of discrete measures

TL;DR

This paper proves that convolution operators generated by certain discrete measures on the unit circle exhibit strong sweeping out behavior, meaning they oscillate between 0 and 1 almost everywhere.

Contribution

It establishes the strong sweeping out property for convolution operators of discrete measures on the unit circle, a result not previously known.

Findings

Convolution operators of discrete measures can oscillate between 0 and 1 almost everywhere.

The sequence of measures satisfies the condition $ o 1$ on shrinking intervals around zero.

The result applies to measures with zero mass at zero and increasing concentration near zero.

Abstract

Let $\mu_n$ be a sequence of discrete measures on the unit $\ZT=\ZR/\ZZ$ with $\mu_n(0)=0$, and $\mu_n((-\delta,\delta))\to 1$, as $n\to\infty$. We prove that the sequence of convolution operators $(f\ast\mu_n)(x)$ is strong sweeping out, i.e. there exists a set $E\subset\ZT$ such that \md0 \lim\sup_{n\to\infty}(\ZI_E\ast\mu_n)(x)= 1,\quad \lim\inf_{n\to\infty}(\ZI_E\ast\mu_n)(x)= 0, \emd almost everywhere on $\ZT$.