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), p. 550-568 / Harvested from Numdam

Résumé

Il est connu qu’une mesure de probabilité $\mu $ sur le cercle $\mathbb {T}$ satisfait $\Vert \mu ^{n}*f-\int f\,\mathrm {d}m\Vert _{p}\rightarrow 0$ 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. $|\hat{\mu }(n)|<1$ 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 $\limsup \mu ^{n}*1_{B}=1$ p.p. et $\liminf \mu ^{n}*1_{B}=0$ 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 $\mu ^{n}*f\rightarrow \int f\,\mathrm {d}m$ p.p. pour tout $p>1$ et toute fonction $f\in L_{p}(G,m)$, soit $\mu $ vérifie la propriété de balayage fort.

Abstract

It is well-known that a probability measure $\mu $ on the circle $\mathbb {T}$ satisfies $\Vert \mu ^{n}*f-\int f\,\mathrm {d}m\Vert _{p}\rightarrow 0$ for every $f\in L_{p}$, every (some) $p\in [1,\infty )$, if and only if $|\hat{\mu }(n)|<1$ 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 $\limsup \mu ^{n}*1_{B}=1$ a.e. and $\liminf \mu ^{n}*1_{B}=0$ 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 $\mu ^{n}*f\rightarrow \int f\,\mathrm {d}m$ a.e. for every $p>1$ and every $f\in L_{p}(G,m)$, or $\mu $ has the strong sweeping out property.

Détail

Publié le : 2013-01-01

MR 3088381 | Zbl 1281.37005

DOI : https://doi.org/10.1214/11-AIHP468
Classification:  37A30,  28D05,  47A35,  60G50,  42A38

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