Strong $q$-variation inequalities for analytic semigroups

Le Merdy, Christian; Xu, Quanhua

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

Soit $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ une contraction positive, avec $1<p<\infty$. 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 $2<q<\infty$ et soit $v^q$ l’espace des suites complexes à $q$-variation forte bornée. On montre que pour tout $x\in L^p\left(\Omega \right)$, 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 $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ désigne la moyenne ergodique de $T$. On obtient également des résultats similaires pour les semi-groupes fortement continus ${\left(T_t\right)}_{t\ge 0}$ de contractions positives sur $L^p$.

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1<p<\infty$. 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 $2<q<\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, 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 $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ 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.

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.

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