Strong $q$-variation inequalities for analytic semigroups  [ Inégalités de $q$-variation forte pour les semi-groupes analytiques ]
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, p. 2069-2097
Soit $T:{L}^{p}\left(\Omega \right)\to {L}^{p}\left(\Omega \right)$ une contraction positive, avec $1. Supposons $T$ analytique, au sens où il existe une constante $K\ge 0$ telle que $\parallel {T}^{n}-{T}^{n-1}\parallel \le K/n$ pour tout entier $n\ge 1$. Soit $2 et soit ${v}^{q}$ l’espace des suites complexes à $q$-variation forte bornée. On montre que pour tout $x\in {L}^{p}\left(\Omega \right)$, la suite ${\left(\left[{T}^{n}\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ appartient à ${v}^{q}$ pour presque tout $\lambda \in \Omega$, avec la majoration $\parallel {\left({T}^{n}\left(x\right)\right)}_{n\ge 0}{\parallel }_{{L}^{p}\left({v}^{q}\right)}\le C{\parallel x\parallel }_{p}$. Si l’on supprime l’hypothèse d’analyticité, on obtient une majoration $\parallel {\left({M}_{n}\left(T\right)x\right)}_{n\ge 0}{\parallel }_{{L}^{p}\left({v}^{q}\right)}\le C{\parallel x\parallel }_{p}$, où ${M}_{n}\left(T\right)={\left(n+1\right)}^{-1}{\sum }_{k=0}^{n}{T}^{k}\phantom{\rule{0.166667em}{0ex}}$ désigne la moyenne ergodique de $T$. On obtient également des résultats similaires pour les semi-groupes fortement continus ${\left({T}_{t}\right)}_{t\ge 0}$ de contractions positives sur ${L}^{p}$.
Let $T:{L}^{p}\left(\Omega \right)\to {L}^{p}\left(\Omega \right)$ be a positive contraction, with $1. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel {T}^{n}-{T}^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2 and let ${v}^{q}$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in {L}^{p}\left(\Omega \right)$, the sequence ${\left(\left[{T}^{n}\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to ${v}^{q}$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left({T}^{n}\left(x\right)\right)}_{n\ge 0}{\parallel }_{{L}^{p}\left({v}^{q}\right)}\le C{\parallel x\parallel }_{p}$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left({M}_{n}\left(T\right)x\right)}_{n\ge 0}{\parallel }_{{L}^{p}\left({v}^{q}\right)}\le C{\parallel x\parallel }_{p}$, where ${M}_{n}\left(T\right)={\left(n+1\right)}^{-1}{\sum }_{k=0}^{n}{T}^{k}\phantom{\rule{0.166667em}{0ex}}$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups ${\left({T}_{t}\right)}_{t\ge 0}$ of positive contractions on ${L}^{p}$-spaces.
DOI : https://doi.org/10.5802/aif.2743
Classification:  47A35,  37A99,  47B38
Mots clés: Théorie ergodique, opérateurs sur ${L}^{p}$, $q$-variation forte, semi-groupes analytiques.
@article{AIF_2012__62_6_2069_0,
author = {Le Merdy, Christian and Xu, Quanhua},
title = {Strong $q$-variation inequalities for analytic semigroups},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {62},
number = {6},
year = {2012},
pages = {2069-2097},
doi = {10.5802/aif.2743},
mrnumber = {3060752},
zbl = {1269.47011},
language = {en},
url = {http://www.numdam.org/item/AIF_2012__62_6_2069_0}
}

Le Merdy, Christian; Xu, Quanhua. Strong $q$-variation inequalities for analytic semigroups. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2069-2097. doi : 10.5802/aif.2743. http://www.numdam.org/item/AIF_2012__62_6_2069_0/

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