Almost everywhere convergence of convolution powers on compact Abelian groups

作者：Conze Jean Pierre^*; Lin Michael

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2013, 49(2): 550-568.

DOI：10.1214/11-AIHP468

• Convolution powers
• Almost everywhere convergence
• Sweeping out
• Strictly-aperiodic probabilities

It is well-known that a probability measure mu on the circle T satisfies parallel to mu(n) * f - integral fdm parallel to(p) -%26gt; 0 for every f is an element of L-p, every (some) p is an element of [1, infinity), if and only if vertical bar(mu) over cap (n)vertical bar %26lt; 1 for every non-zero n is an element of Z (mu, is strictly aperiodic). In this paper we study the a.e. convergence of mu(n) * f for every f is an element of L-p whenever p %26gt; 1. We prove a necessary and sufficient condition, in terms of the Fourier-Stieltjes coefficients of mu, for the strong sweeping out property (existence of a Borel set B with lim sup mu(n) * 1(B) = 1 a.e. and lim inf mu(n) * 1(B) = 0 a.e.). The results are extended to general compact Abelian groups G with Haar measure in, and as a corollary we obtain the dichotomy: for mu strictly aperiodic, either mu(n) * f -%26gt; integral f dm a.e. for every p %26gt; 1 and every f is an element of L-p (G, m), or mu, has the strong sweeping out property.

• 出版日期2013-5