Almost everywhere convergence of convolution powers on compact abelian groups
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, p. 550-568
Il est connu qu’une mesure de probabilité μ sur le cercle 𝕋 satisfait μ n *f-fdm p 0 pour toute fonction fL p et pour tout p[1,) (ou pour un p[1,)), si et seulement si μ est strictement apériodique (i.e. |μ ^(n)|<1 pour tout n non nul dans ). Nous étudions ici la convergence presque partout de μ n *f pour fL p , p>1. Nous montrons une condition nécessaire et suffisante portant sur les coefficients de Fourier-Stieltjes de μ pour la propriété de “balayage fort” (existence d’un borélien B tel que lim supμ n *1 B =1 p.p. et lim infμ 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 μ strictement apériodique, soit μ n *ffdm p.p. pour tout p>1 et toute fonction fL p (G,m), soit μ vérifie la propriété de balayage fort.
It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n *f-fdm p 0 for every fL p , every (some) p[1,), if and only if |μ ^(n)|<1 for every non-zero n (μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n *f for every fL p whenever p>1. We prove a necessary and sufficient condition, in terms of the Fourier-Stieltjes coefficients of μ, for the strong sweeping out property (existence of a Borel set B with lim supμ n *1 B =1 a.e. and lim infμ 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 μ strictly aperiodic, either μ n *ffdm a.e. for every p>1 and every fL p (G,m), or μ has the strong sweeping out property.
DOI : https://doi.org/10.1214/11-AIHP468
Classification:  37A30,  28D05,  47A35,  60G50,  42A38
@article{AIHPB_2013__49_2_550_0,
     author = {Conze, Jean-Pierre and Lin, Michael},
     title = {Almost everywhere convergence of convolution powers on compact abelian groups},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     pages = {550-568},
     doi = {10.1214/11-AIHP468},
     zbl = {1281.37005},
     mrnumber = {3088381},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_2_550_0}
}
Conze, Jean-Pierre; Lin, Michael. Almost everywhere convergence of convolution powers on compact abelian groups. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 550-568. doi : 10.1214/11-AIHP468. http://www.numdam.org/item/AIHPB_2013__49_2_550_0/

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