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Analogs of Wiener’s ergodic theorems for semisimple Lie groups. II. (English) Zbl 0978.22006
[Part I by A. Nevo and E. M. Stein in Ann. Math. (2) 145, 565-595 (1997; Zbl 0884.43004).]
Let \((X,{\mathcal B},m)\) be a standard Borel space with a Borel measurable ergodic \(G\)-action preserving the probability measure \(m\). If \(\nu_t\), \(t\in \mathbb{R}_+\), is a one-parameter family of probability measures on \(G\) such that the map \(t\to\nu_t\) is continuous when the space of probability measures is equipped with the weak-\(*\)-topology, then \(\nu_t\) is a pointwise ergodic family in \(L^p(X)\) if, for every \(f\in L^p(X)\), \[ \lim_{t\to\infty}\pi(\nu_t)f(x)=\int_Xf dm, \] where \(\pi(\nu_t)f(x):=\int_G f(g^{-1}x) d\nu_t(g)\) and the convergence is pointwise almost everywhere and in the \(L^p\) norm. The authors prove that if \(G\) is a connected semisimple group with finite center and no compact factors and if \(\nu_t\) is the family of ball averages on \(G\) defined by \[ \nu_t={1\over{m_G(B_t)}}\int_{B_t}\delta_g dm_G(g), \] where \(m_G\) is the Haar measure on \(G\), \(B_t\) is the ball of radius \(t\) with respect to the \(G\)-invariant Riemannian metric on \(G/K\), and \(\delta_g\) is the Dirac measure at \(g\), then \(\nu_t\) is a pointwise ergodic family in \(L^p(X)\) for every ergodic \(G\)-space \((X,{\mathcal B},m)\) as above. Moreover, again for every ergodic \((X,{\mathcal B},m)\), the family of ball averages satisfies the strong maximal inequality in \(L^p(X)\), \(1<p\leq\infty\), namely, \[ \left\|\sup_{t\geq 0}|\pi(\nu_t)f|\right\|_p\leq C_p\|f\|_p. \] The authors also show that for semisimple Lie groups satisfying property T, the convergence of the ball averages to the ergodic mean is at an exponential rate, namely there exists a positive constant \(b(G)\) such that for every \(\theta<b(G)\) and for all \(r\) and \(u\) such that \(0<u<1\) and \(2p=2r+pr\), then for almost every \(x\in X\) \[ \left|\pi(\nu_t)f(x)-\int_Gf dm\right|\leq B(x,f)\exp\left(-{{u\theta t}\over{4}}\right), \] where \(\|B(\cdot,f)\|_r\leq B\|f\|_p\) for all \(f\in L^p(X)\). The authors also generalize the above results to general radial averages and to actions with a spectral gap (where the group acting does not necessarily have property T).

MSC:
22D40 Ergodic theory on groups
22E30 Analysis on real and complex Lie groups
28D10 One-parameter continuous families of measure-preserving transformations
43A10 Measure algebras on groups, semigroups, etc.
43A90 Harmonic analysis and spherical functions
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[1] G. Besson, G. Courtois, and S. Gallot, Volume et entropie minimale des espaces localement symétriques , Invent. Math. 103 (1991), 417–445. · Zbl 0723.53029 · doi:10.1007/BF01239520 · eudml:143864
[2] J. L. Clerc and E. M. Stein, \(L^p\)- multipliers for non-compact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911–3912. · Zbl 0296.43004 · doi:10.1073/pnas.71.10.3911
[3] M. Cowling, The Kunze-Stein phenomenon , Ann. of Math. (2) 107 (1978), 209–234. JSTOR: · Zbl 0363.22007 · doi:10.2307/1971142 · links.jstor.org
[4] –. –. –. –., “Sur les coefficients des représentations unitaires des groupes de Lie simples” in Analyse harmonique sur les groupes de Lie (Sém. Nancy-Strasbourg, 1976–1978.), II, Lecture Notes in Math. 739 , Springer, Berlin, 1979, 132–178. · Zbl 0417.22010 · doi:10.1007/BFb0062491
[5] M. Cowling and A. Nevo, Uniform estimates for spherical functions on complex semisimple Lie groups , · Zbl 1007.22016 · doi:10.1007/s00039-001-8220-x
[6] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups . Duke Math. J. 71 (1993), 181–209. · Zbl 0798.11025 · doi:10.1215/S0012-7094-93-07108-6
[7] A. Eskin, G. A. Margulis, and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture , Ann. of Math. (2) 147 (1998), 93–141. JSTOR: · Zbl 0906.11035 · doi:10.2307/120984 · links.jstor.org
[8] R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups , Ergeb. Math. Grenzgeb. 101 , Springer, Berlin, 1988. · Zbl 0675.43004
[9] Harish-Chandra, Spherical functions on a semi-simple Lie group, I, Amer. J. Math. 80 (1958), 241–310. JSTOR: · Zbl 0093.12801 · doi:10.2307/2372786 · links.jstor.org
[10] –. –. –. –., Spherical functions on a semi-simple Lie group, II, Amer. J. Math. 80 (1958), 553–613. JSTOR: · Zbl 0093.12801 · doi:10.2307/2372772 · links.jstor.org
[11] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces , Pure Appl. Math. 80 , Academic Press, New York, 1978. · Zbl 0451.53038
[12] ——–, Groups and Geometric Analysis , Pure Appl. Math. 113 , Academic Press, Orlando, Fla., 1984.
[13] R. Howe, “On a notion of rank for unitary representations of the classical groups” in Harmonic Analysis and Group Representations , Centro Internazionale Matematico Estivo, Liguori, Naples, 1982, 223–331.
[14] R. Howe and C. C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), 72–96. · Zbl 0404.22015 · doi:10.1016/0022-1236(79)90078-8
[15] R. Howe and E. C. Tan, Non-Abelian Harmonic Analysis , Universitext, Springer, New York, 1992. · Zbl 0768.43001
[16] A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity , Inst. Hautes Etudes Sci. Publ. Math. 79 (1994), 131–156. · Zbl 0819.58027 · doi:10.1007/BF02698888 · numdam:PMIHES_1994__79__131_0 · eudml:104094
[17] D. A. Kazhdan, On a connection between the dual space of a group and the structure of its closed subgroups , Funct. Anal. Appl. 1 (1967), 63–65. · Zbl 0168.27602 · doi:10.1007/BF01075866
[18] D. Y. Kleinbock and G. A. Margulis, “Bounded orbits of nonquasiunipotent flows on homogeneous spaces” in Sinai’s Moscow Center on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2 171 (1996), 141–172. · Zbl 0843.22027
[19] A. W. Knapp, Lie Groups Beyond an Introduction , Progr. Math. 140 , Birkhäuser, Boston, 1996. · Zbl 0862.22006
[20] J-S. Li, “The minimal decay of matrix coefficients for classical groups” in Harmonic Analysis in China , Math. Appl. 327 , Kluwer Acad. Publ., Dordrecht, 1995, 146–169. · Zbl 0844.22021
[21] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups , Ergeb. Math. Grenzgeb. 17 , Springer, Berlin, 1991. · Zbl 0732.22008
[22] G. A. Margulis, A. Nevo, and E. M. Stein, in preparation.
[23] A. Nevo, Harmonic analysis and pointwise ergodic theorems for noncommuting transformations , J. Amer. Math. Soc. 7 (1994), 875–902. JSTOR: · Zbl 0812.22005 · doi:10.2307/2152735 · links.jstor.org
[24] –. –. –. –., Pointwise ergodic theorems for radial averages on simple Lie groups, I , Duke Math. J. 76 (1994), 113–140. · Zbl 0838.43013 · doi:10.1215/S0012-7094-94-07605-9
[25] –. –. –. –., Pointwise ergodic theorems for radial averages on simple Lie groups, II , Duke Math. J., 86 (1997), 239–259. · Zbl 0869.43005 · doi:10.1215/S0012-7094-97-08607-5
[26] –. –. –. –., Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups , Math. Res. Lett. 5 (1998), 305–325. · Zbl 0942.22007 · doi:10.4310/MRL.1998.v5.n3.a5
[27] A. Nevo and E. M. Stein, A generalization of Birkhoff’s pointwise ergodic theorem , Acta Math. 173 (1994), 135–154. · Zbl 0837.22003 · doi:10.1007/BF02392571
[28] –. –. –. –., Analogs of Wiener’s ergodic theorems for semi-simple Lie groups, I , Ann. of Math. (2) 145 (1997), 565–595. JSTOR: · Zbl 0884.43004 · doi:10.2307/2951845 · links.jstor.org
[29] H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients , Bull. Soc. Math. France 126 (1998), 355–380. · Zbl 0917.22008 · smf.emath.fr · numdam:BSMF_1998__126_3_355_0 · eudml:87787
[30] K. Petersen, Ergodic Theory , Cambridge Stud. Adv. Math. 2 , Cambridge Univ. Press, New York, 1983.
[31] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Ann. of Math. Stud. 63 , Princeton Univ. Press, Princeton, 1970. · Zbl 0193.10502
[32] –. –. –. –., Analytic continuation of group representations , Adv. Math. 4 (1972), 172–207. · Zbl 0194.16201 · doi:10.1016/0001-8708(70)90022-8
[33] ——–, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Princeton Math. Ser. 45 , Princeton Univ. Press, Princeton, 1993. · Zbl 0821.42001
[34] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239–1295. · Zbl 0393.42010 · doi:10.1090/S0002-9904-1978-14554-6
[35] J-O. Stromberg, Weak type \(L^1\) estimates for maximal functions on non-compact symmetric spaces , Ann. of Math. (2) 114 (1981), 115–126. JSTOR: · Zbl 0472.43010 · doi:10.2307/1971380 · links.jstor.org
[36] N. Wiener, The ergodic theorem , Duke Math. J. 5 (1939), 1–18. · Zbl 0021.23501 · doi:10.1215/S0012-7094-39-00501-6
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