Strong $q$-variation inequalities for analytic semigroups

Le Merdy, Christian; Xu, Quanhua

doi:10.5802/aif.2743

Strong

q

-variation inequalities for analytic semigroups
[Inégalités de

q

-variation forte pour les semi-groupes analytiques]

Le Merdy, Christian ¹ ; Xu, Quanhua ²

¹ Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex France
² School of Mathematics and Statistics Wuhan University Wuhan 430072 Hubei China and Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex France

Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2069-2097.

Résumé
Abstract

Soit $T : L^{p} (Ω) \to L^{p} (Ω)$ une contraction positive, avec $1 < p < \infty$ . Supposons $T$ analytique, au sens où il existe une constante $K \geq 0$ telle que $∥ T^{n} - T^{n - 1} ∥ \leq K / n$ pour tout entier $n \geq 1$ . Soit $2 < q < \infty$ et soit $v^{q}$ l’espace des suites complexes à $q$ -variation forte bornée. On montre que pour tout $x \in L^{p} (Ω)$ , la suite ${([T^{n} (x)] (λ))}_{n \geq 0}$ appartient à $v^{q}$ pour presque tout $λ \in Ω$ , avec la majoration $∥ {(T^{n} (x))}_{n \geq 0} ∥_{L^{p} (v^{q})} \leq C {∥ x ∥}_{p}$ . Si l’on supprime l’hypothèse d’analyticité, on obtient une majoration $∥ {(M_{n} (T) x)}_{n \geq 0} ∥_{L^{p} (v^{q})} \leq C {∥ x ∥}_{p}$ , où $M_{n} (T) = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ désigne la moyenne ergodique de $T$ . On obtient également des résultats similaires pour les semi-groupes fortement continus ${(T_{t})}_{t \geq 0}$ de contractions positives sur $L^{p}$ .

Let $T : L^{p} (Ω) \to L^{p} (Ω)$ be a positive contraction, with $1 < p < \infty$ . Assume that $T$ is analytic, that is, there exists a constant $K \geq 0$ such that $∥ T^{n} - T^{n - 1} ∥ \leq K / n$ for any integer $n \geq 1$ . Let $2 < q < \infty$ and let $v^{q}$ be the space of all complex sequences with a finite strong $q$ -variation. We show that for any $x \in L^{p} (Ω)$ , the sequence ${([T^{n} (x)] (λ))}_{n \geq 0}$ belongs to $v^{q}$ for almost every $λ \in Ω$ , with an estimate $∥ {(T^{n} (x))}_{n \geq 0} ∥_{L^{p} (v^{q})} \leq C {∥ x ∥}_{p}$ . If we remove the analyticity assumption, we obtain an estimate $∥ {(M_{n} (T) x)}_{n \geq 0} ∥_{L^{p} (v^{q})} \leq C {∥ x ∥}_{p}$ , where $M_{n} (T) = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ denotes the ergodic average of $T$ . We also obtain similar results for strongly continuous semigroups ${(T_{t})}_{t \geq 0}$ of positive contractions on $L^{p}$ -spaces.

MR Zbl

DOI : 10.5802/aif.2743

Classification : 47A35, 37A99, 47B38
Keywords: Ergodic theory, operators on $L^p$, strong $q$-variation, analytic semigroups.
Mot clés : Théorie ergodique, opérateurs sur $L^p$, $q$-variation forte, semi-groupes analytiques.

Affiliations des auteurs :

Le Merdy, Christian ¹ ; Xu, Quanhua ²

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     title = {Strong $q$-variation inequalities for analytic semigroups},
     journal = {Annales de l'Institut Fourier},
     pages = {2069--2097},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
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     year = {2012},
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     zbl = {1269.47011},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2743/

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