Almost everywhere convergence of convolution powers on compact abelian groups

Conze, Jean-Pierre; Lin, Michael

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 550-568.

Il est connu qu’une mesure de probabilité $\mu$ sur le cercle $𝕋$ satisfait $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ pour toute fonction $f\in L_p$ et pour tout $p\in [1,\infty )$ (ou pour un $p\in [1,\infty )$), si et seulement si $\mu$ est strictement apériodique (i.e. $|\widehat{\mu }\left(n\right)|<1$ pour tout $n$ non nul dans $\mathbb{Z}$). Nous étudions ici la convergence presque partout de ${\mu }^n*f$ pour $f\in L_p$, $p>1$. Nous montrons une condition nécessaire et suffisante portant sur les coefficients de Fourier-Stieltjes de $\mu$ pour la propriété de “balayage fort” (existence d’un borélien $B$ tel que $lim\; sup{\mu }^n*1_B=1$ p.p. et $lim\; inf{\mu }^n*1_B=0$ p.p.). Les résultats sont étendus aux groupes abéliens compacts généraux $G$ de mesure de Haar $m$. Comme corollaire nous obtenons la dichotomie suivante : pour $\mu$ strictement apériodique, soit ${\mu }^n*f\to \int f\mathrm{d}m$ p.p. pour tout $p>1$ et toute fonction $f\in L_p(G,m)$, soit $\mu$ vérifie la propriété de balayage fort.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|<1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $p>1$. 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 $m$, and as a corollary we obtain the dichotomy: for $\mu$ strictly aperiodic, either ${\mu }^n*f\to \int f\mathrm{d}m$ a.e. for every $p>1$ and every $f\in L_p(G,m)$, or $\mu$ has the strong sweeping out property.

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