Compactness of families of convolution operators with respect to convergence almost everywhere.

Abstract

For a given sequence of measures $\mu_n$ on the circle $\mathbb{T}$ weakly convergent to the Dirac measure, we ask, is it possible to extract a subsequence $n(j)$ such that for any $f$ in the space $L^1 (L^2 ,L^{\infty })$ the convolutions $f\ast\mu_{n(j)}$ converge to $f$ almost everywhere. We show that it is crucial whether the measures are absolutely continuous, discrete or singular (non-atomic).