Strong $q$-variation inequalities for analytic semigroups

[Inégalités de $q$-variation forte pour les semi-groupes analytiques]

Le Merdy, Christian ; Xu, Quanhua

Annales de l'Institut Fourier, Tome 62 (2012), p. 2069-2097 / Harvested from Numdam

Résumé

Soit $T\colon L^p(\Omega )\rightarrow L^p(\Omega )$ une contraction positive, avec $1<p<\infty $. Supposons $T$ analytique, au sens où il existe une constante $K\ge 0$ telle que $\Vert T^n-T^{n-1}\Vert \le K/n$ 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 $x\in L^p(\Omega )$, la suite $\bigl ([T^n(x)](\lambda )\bigr )_{n\ge 0}$ appartient à $v^q$ pour presque tout $\lambda \in \Omega $, avec la majoration $\Vert (T^n(x))_{n\ge 0}\Vert _{L^p(v^q)}\le C\Vert x\Vert _p$. Si l’on supprime l’hypothèse d’analyticité, on obtient une majoration $\Vert (M_n(T)x)_{n\ge 0}\Vert _{L^p(v^q)}\le C\Vert x\Vert _p$, 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 $(T_t)_{t\ge 0}$ de contractions positives sur $L^p$.

Abstract

Let $T\colon L^p(\Omega )\rightarrow L^p(\Omega )$ be a positive contraction, with $1<p<\infty $. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\Vert T^n-T^{n-1}\Vert \le K/n$ 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 $x\in L^p(\Omega )$, the sequence $\bigl ([T^n(x)](\lambda )\bigr )_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega $, with an estimate $\Vert (T^n(x))_{n\ge 0}\Vert _{L^p(v^q)}\le C\Vert x\Vert _p$. If we remove the analyticity assumption, we obtain an estimate $\Vert (M_n(T)x)_{n\ge 0}\Vert _{L^p(v^q)}\le C\Vert x\Vert _p$, 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 $(T_t)_{t\ge 0}$ of positive contractions on $L^p$-spaces.

Détail

Publié le : 2012-01-01

MR 3060752 | Zbl 1269.47011

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.

Bibliographie

[1] Akcoglu, M.; Sucheston, L. Dilations of positive contractions on $L_p$ spaces, Canad. Math. Bull., Tome 20 (1977), pp. 285-292 | MR 458230 | Zbl 0381.47004

[2] Asmar, N.; Berkson, E.; Gillespie, T. A. Transference of strong type maximal inequalities by separation-preserving representations, Amer. J. Math., Tome 113 (1991), pp. 47-74 | MR 1087801 | Zbl 0729.43003

[3] Blunck, S. Analyticity and discrete maximal regularity on $L_p$-spaces, J. Funct. Anal., Tome 183 (2001), pp. 211-230 | MR 1837537 | Zbl 0987.47027

[4] Bourgain, J. Pointwise ergodic theorems for arithmetic sets, Publ. Math. IHES, Tome 69 (1989), pp. 5-41 | Numdam | MR 1019960 | Zbl 0705.28008

[5] Campbell, J. T.; Jones, R. L.; Reinhold, K.; Wierdl, M. Oscillation and variation for the Hilbert transform, Duke Math. J., Tome 105 (2000), pp. 59-83 | MR 1788042 | Zbl 1013.42008

[6] Coifman, R.; Rochberg, R.; Weiss, G. Applications of transference: the $L_p$ version of von Neumann’s inequality and the Littlewood-Paley-Stein theory, Linear spaces and Approximation, Birkhäuser, Basel (1978), pp. 53-67 | MR 500219 | Zbl 0398.47005

[7] Coifman, R. R.; Weiss, G. Transference methods in analysis, Amer. Math. Soc., CBMS 31 (1977) | MR 481928 | Zbl 0377.43001

[8] Coulhon, T.; Saloff-Coste, L. Puissances d’un opérateur régularisant, Ann. Inst. H. Poincaré Probab. Statist., Tome 26 (1990) no. 3, pp. 419-436 | Numdam | MR 1066086 | Zbl 0709.47042

[9] Crescimbeni, R.; Macías, R. A.; Menárguez, T.; Torrea, J. L.; Viviani, B. The $\rho $-variation as an operator between maximal operators and singular integrals, J. Evol. Equ., Tome 9 (2009), pp. 81-102 | MR 2501353 | Zbl 1239.42012

[10] Delyon, B.; Delyon, F. Generalization of von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. France, Tome 127 (1999), pp. 25-41 | Numdam | MR 1700467 | Zbl 0937.47004

[11] Dunford, N.; Schwartz, J. T. Linear operators, Part 1, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, Pure and Applied Mathematics, Tome 7 (1958) | MR 117523 | Zbl 0084.10402

[12] Fendler, G. Dilations of one parameter semigroups of positive contractions on $L_p$-spaces, Canad. J. Math., Tome 48 (1997), pp. 726-748 | MR 1471054 | Zbl 0907.47039

[13] Gaposhkin, V. F. Ergodic theorem for functions of normal operators (Russian), Funktsional. Anal. i Prilozhen., Tome 18 (1984), pp. 1-6 | MR 739083 | Zbl 0557.47005

[14] Goldstein, J. A. Semigroups of linear operators and applications, Oxford University Press, New York (1985) | MR 790497 | Zbl 0592.47034

[15] Jones, R. L.; Kaufman, R.; Rosenblatt, J. M.; Wierdl, M. Oscillation in ergodic theory, Ergodic Theory Dynam. Systems, Tome 18 (1998), pp. 889-935 | MR 1645330 | Zbl 0924.28009

[16] Jones, R. L.; Reinhold, K. Oscillation and variation inequalities for convolution powers, Ergodic Theory Dynam. Systems, Tome 21 (2001) no. 6, pp. 1809-1829 | MR 1869071 | Zbl 1060.28013

[17] Jones, R. L.; Seeger, A.; Wright, J. Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc., Tome 360 (2008), pp. 6711-6742 | MR 2434308 | Zbl 1159.42013

[18] Jones, R. L.; Wang, G. Variation inequalities for the Fejér and Poisson kernels, Trans. Amer. Math. Soc., Tome 356 (2004), pp. 4493-4518 | MR 2067131 | Zbl 1065.42006

[19] Le Merdy, C.; Xu, Q. Maximal theorems and square functions for analytic operators on $L^p$-spaces (to appear in J. London Math. Soc. (see also arXiv:1011.1360v1))

[20] Lyubich, Yu. Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition, Studia Math., Tome 134 (1999), pp. 153-167 | MR 1688223 | Zbl 0945.47005

[21] Lépingle, D. La variation d’ordre $p$ des semi-martingales (French), Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Tome 36 (1976), pp. 295-316 | MR 420837 | Zbl 0325.60047

[22] Meyer-Nieberg, P. Banach lattices, Springer, Berlin-Heidelberg-NewYork (1991) | MR 1128093 | Zbl 0743.46015

[23] Nagy, B.; Zemanek, J. A resolvent condition implying power boundedness, Studia Math., Tome 134 (1999), pp. 143-151 | MR 1688222 | Zbl 0934.47002

[24] Nevanlinna, O. Convergence of iterations for linear equations, Birkhaüser, Basel (1993) | MR 1217705 | Zbl 0846.47008

[25] Oberlin, R.; Seeger, A.; Tao, T.; Thiele, C.; Wright, J. A variation norm Carleson theorem (Preprint 2010, arXiv:1011.1360v1)

[26] Pazy, A. Semigroups of linear operators and applications to partial differential equations, Springer (1983) | MR 710486 | Zbl 0516.47023

[27] Peller, V. An analogue of J. von Neumann’s inequality for the space $L^{p}$ (Russian), Dokl. Akad. Nauk SSSR, Tome 231 (1976) no. 3, pp. 359-542 | MR 435925 | Zbl 0394.47006

[28] Pisier, G. Complex interpolation and regular operators between Banach lattices, Arch. Math. (Basel), Tome 62 (1994) no. 3, pp. 261-269 | MR 1259842 | Zbl 0991.46007

[29] Pisier, G.; Xu, Q. The strong $p$-variation of martingale and orthogonal series, Probab. Th. Rel. Fields, Tome 77 (1988) no. 3, pp. 497-514 | MR 933985 | Zbl 0632.60004

[30] Stein, E. M. Topics in harmonic analysis related to the Littlewood-Paley theory, Princeton University Press, Ann. Math. Studies (1970) | MR 252961 | Zbl 0193.10502

[31] De La Torre, A. A simple proof of the maximal ergodic theorem, Canad. J. Math., Tome 28 (1976), pp. 1073-1075 | MR 417819 | Zbl 0336.47006

[32] Yosida, K. Functional Analysis, Springer Verlag (1968)