Strong q-variation inequalities for analytic semigroups
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2069-2097
Let T:L p (Ω)L p (Ω) be a positive contraction, with 1<p<. Assume that T is analytic, that is, there exists a constant K0 such that T n -T n-1 K/n for any integer n1. Let 2<q< and let v q be the space of all complex sequences with a finite strong q-variation. We show that for any xL p (Ω), the sequence [ T n (x) ] (λ) n0 belongs to v q for almost every λΩ, with an estimate (T n (x)) n0 L p (v q ) Cx p . If we remove the analyticity assumption, we obtain an estimate (M n (T)x) n0 L p (v q ) Cx p , where M n (T)=(n+1) -1 k=0 n T k denotes the ergodic average of T. We also obtain similar results for strongly continuous semigroups (T t ) t0 of positive contractions on L p -spaces.
Soit T:L p (Ω)L p (Ω) une contraction positive, avec 1<p<. Supposons T analytique, au sens où il existe une constante K0 telle que T n -T n-1 K/n pour tout entier n1. Soit 2<q< et soit v q l’espace des suites complexes à q-variation forte bornée. On montre que pour tout xL p (Ω), la suite [ T n (x) ] (λ) n0 appartient à v q pour presque tout λΩ, avec la majoration (T n (x)) n0 L p (v q ) Cx p . Si l’on supprime l’hypothèse d’analyticité, on obtient une majoration (M n (T)x) n0 L p (v q ) Cx p , où M n (T)=(n+1) -1 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 ) t0 de contractions positives sur L p .
DOI : https://doi.org/10.5802/aif.2743
Classification:  47A35,  37A99,  47B38
Keywords: Ergodic theory, operators on L p , strong q-variation, analytic semigroups.
@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},
     zbl = {1269.47011},
     mrnumber = {3060752},
     language = {en},
     url = {https://aif.centre-mersenne.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, Volume 62 (2012) no. 6, pp. 2069-2097. doi : 10.5802/aif.2743. https://aif.centre-mersenne.org/item/AIF_2012__62_6_2069_0/

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