On the sweeping out property for convolution operators of discrete measures

Abstract:

Let $\mu _n$ be a sequence of discrete measures on the unit circle $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ 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 $E\subset \mathbb {T}$ such that \begin{equation*} \limsup \limits _{n\to \infty } (\mathbb {I}_E\ast \mu _n)(x)= 1,\quad \liminf \limits _{n\to \infty }(\mathbb {I}_E\ast \mu _n)(x)= 0, \end{equation*} almost everywhere on $\mathbb {T}$.