Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze; Michael Lin

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 550-568
  • ISSN: 0246-0203

Abstract

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It is well-known that a probability measure μ on the circle 𝕋 satisfies ∥ μ n * f - ∫ f d m ∥ p → 0 for every f ∈ L p , every (some) p ∈ [ 1 , ∞ ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ∈ ℤ ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f ∈ L p whenever p g t ; 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 sup μ n * 1 B = 1 a.e. and lim inf μ 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 * f → ∫ f d m a.e. for every p g t ; 1 and every f ∈ L p ( G , m ) , or μ has the strong sweeping out property.

How to cite

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Conze, Jean-Pierre, and Lin, Michael. "Almost everywhere convergence of convolution powers on compact abelian groups." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 550-568. <http://eudml.org/doc/272032>.

@article{Conze2013,
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 $|\hat\{\mu \}(n)|&lt;1$ 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 $p&gt;1$. 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 $p&gt;1$ and every $f\in L_\{p\}(G,m)$, or $\mu $ has the strong sweeping out property.},
author = {Conze, Jean-Pierre, Lin, Michael},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {convolution powers; almost everywhere convergence; sweeping out; strictly aperiodic probabilities; Abelian groups},
language = {eng},
number = {2},
pages = {550-568},
publisher = {Gauthier-Villars},
title = {Almost everywhere convergence of convolution powers on compact abelian groups},
url = {http://eudml.org/doc/272032},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Conze, Jean-Pierre
AU - Lin, Michael
TI - Almost everywhere convergence of convolution powers on compact abelian groups
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 550
EP - 568
AB - 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 $|\hat{\mu }(n)|&lt;1$ 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 $p&gt;1$. 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 $p&gt;1$ and every $f\in L_{p}(G,m)$, or $\mu $ has the strong sweeping out property.
LA - eng
KW - convolution powers; almost everywhere convergence; sweeping out; strictly aperiodic probabilities; Abelian groups
UR - http://eudml.org/doc/272032
ER -

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