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Quantitative ergodic theorems for actions of groups of polynomial growth
Part of:
Special aspects of infinite or finite groups
General theory of linear operators
Harmonic analysis in several variables
Ergodic theory
Published online by Cambridge University Press: 04 September 2025
Abstract
We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.
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References
Aldaz, J. M.. Boundedness of averaging operators on geometrically doubling metric spaces. Ann. Acad. Sci. Fenn. Math. 44(1) (2019), 497–503.10.5186/aasfm.2019.4430CrossRefGoogle Scholar
Anantharaman, C. et al. Théorèmes ergodiques pour les actions de groupes (Monographies de L’Enseignement Mathématique, 41). L’Enseignement Mathématique, Geneva, 2010.Google Scholar
Arroyo, Á. and Llorente, J. G.. On a class of singular measures satisfying a strong annular decay condition. Proc. Amer. Math. Soc. 147(10) (2019), 4409–4423.10.1090/proc/14576CrossRefGoogle Scholar
Auscher, P. and Routin, E.. Local Tb theorems and Hardy inequalities. J. Geom. Anal. 23(1) (2013), 303–374.Google Scholar
Avigad, J., Gerhardy, P. and Towsner, H.. Local stability of ergodic averages. Trans. Amer. Math. Soc. 362(1) (2010), 261–288.10.1090/S0002-9947-09-04814-4CrossRefGoogle Scholar
Bewley, T.. Extension of the Birkhoff and von Neumann ergodic theorems to semigroup actions. Ann. Inst. Henri Poincaré B (N.S.) 7 (1971), 283–291.Google Scholar
Birkhoff, G. D.. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17 (1931), 656–660.10.1073/pnas.17.2.656CrossRefGoogle ScholarPubMed
Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 5–41.10.1007/BF02698838CrossRefGoogle Scholar
Breuillard, E.. Geometry of locally compact groups of polynomial growth and shape of large balls. Groups Geom. Dyn. 8(3) (2014), 669–732.10.4171/ggd/244CrossRefGoogle Scholar
Buckley, S. M.. Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24(2) (1999), 519–528.Google Scholar
Calderón, A. P.. A general ergodic theorem. Ann. of Math. (2) 58 (1953), 182–191.10.2307/1969828CrossRefGoogle Scholar
Calderón, A. P.. Ergodic theory and translation-invariant operators. Proc. Natl. Acad. Sci. USA 59 (1968), 349–353.10.1073/pnas.59.2.349CrossRefGoogle ScholarPubMed
Chatard, J.. Applications des propriétés de moyenne
${\mathrm{d}}^{\prime }$
un groupe localement compact à la théorie ergodique. Ann. Inst. Henri Poincaré (N.S.) 6 (1970), 307–326.Google Scholar
${\mathrm{d}}^{\prime }$
un groupe localement compact à la théorie ergodique. Ann. Inst. Henri Poincaré (N.S.) 6 (1970), 307–326.Google ScholarChrist, M.. A
$T(b)$
theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2) (1990), 601–628.10.4064/cm-60-61-2-601-628CrossRefGoogle Scholar
$T(b)$
theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2) (1990), 601–628.10.4064/cm-60-61-2-601-628CrossRefGoogle ScholarCoifman, R. and Weiss, G.. Analyse harmonique non-commutative sur certains espaces homogènes. Étude de certaines intégrales singulières (Lecture Notes in Mathematics, 242). Springer-Verlag, Berlin–New York, 1971.10.1007/BFb0058946CrossRefGoogle Scholar
Coifman, R. and Weiss, G.. Transference Methods in Analysis (CBMS Regional Conferences in Mathematics, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 31). American Mathematical Society, Providence, RI, 1976.Google Scholar
Colding, T. H. and Minicozzi, W. P. II. Liouville theorems for harmonic sections and applications. Comm. Pure Appl. Math. 51(2) (1998), 113–138.10.1002/(SICI)1097-0312(199802)51:2<113::AID-CPA1>3.0.CO;2-E3.0.CO;2-E>CrossRefGoogle Scholar
David, G., Journé, J.-L. and Semmes, S.. Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoam. 1 (1985), 1–56.10.4171/rmi/17CrossRefGoogle Scholar
Doob, J. L.. Stochastic Processes. John Wiley & Sons, Inc., New York; Chapman & Hall Limited, London, 1953.Google Scholar
Dubins, L. E.. Some upcrossing inequalities for uniformly bounded martingales. Symposia Mathematica, Volume IX (Convegno di Calcolo delle Probabilità and Convegno di Teoria della Turbolenza, INDAM, Rome, 1971). Academic Press, London–New York, 1972, pp. 169–177.Google Scholar
Emerson, W. R.. The pointwise ergodic theorem for amenable groups. Amer. J. Math. 96 (1974), 472–487.10.2307/2373555CrossRefGoogle Scholar
Gromov, M.. Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53–73.10.1007/BF02698687CrossRefGoogle Scholar
Gromov, M.. Metric Structures for Riemannian and non-Riemannian Spaces. Birkhäuser, Boston, MA, 1999.Google Scholar
Guivarc’h, Y.. Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 353–379.Google Scholar
Herz, C.. The theory of
$p$
-spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154 (1971), 69–82.Google Scholar
$p$
-spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154 (1971), 69–82.Google ScholarHochman, M.. Upcrossing inequalities for stationary sequences and applications. Ann. Probab. 37(6) (2009), 2135–2149.10.1214/09-AOP460CrossRefGoogle Scholar
Hong, G.. The behaviour of square functions from ergodic theory in
${L}^{\infty }$
. Proc. Amer. Math. Soc. 143(11) (2015), 4797–4802.10.1090/proc12737CrossRefGoogle Scholar
${L}^{\infty }$
. Proc. Amer. Math. Soc. 143(11) (2015), 4797–4802.10.1090/proc12737CrossRefGoogle ScholarHong, G., Liao, B. and Wang, S.. Noncommutative maximal ergodic inequalities associated with doubling conditions. Duke Math. J. 170(2) (2021), 205–246.10.1215/00127094-2020-0034CrossRefGoogle Scholar
Hong, G. and Ma, T.. Vector valued
$q$
-variation for differential operators and semigroups I. Math. Z. 286(1–2) (2017), 89–120.10.1007/s00209-016-1756-0CrossRefGoogle Scholar
$q$
-variation for differential operators and semigroups I. Math. Z. 286(1–2) (2017), 89–120.10.1007/s00209-016-1756-0CrossRefGoogle ScholarHytönen, T. and Kairema, A.. Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126(1) (2012), 1–33.10.4064/cm126-1-1CrossRefGoogle Scholar
Jones, R. L., Kaufman, R., Rosenblatt, J. M. and Wierdl, M.. Oscillation in ergodic theory. Ergod. Th. & Dynam. Sys. 18(4) (1998), 889–935.10.1017/S0143385798108349CrossRefGoogle Scholar
Jones, R. L., Rosenblatt, J. M. and Wierdl, M.. Oscillation in ergodic theory: higher dimensional results. Israel J. Math. 135 (2003), 1–27.10.1007/BF02776048CrossRefGoogle Scholar
Jones, R. L., Seeger, A. and Wright, J.. Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc. 360 (2008), 6711–6742.10.1090/S0002-9947-08-04538-8CrossRefGoogle Scholar
Jones, R. L. and Wang, G.. Variation inequalities for the Fejér and Poisson kernels. Trans. Amer. Math. Soc. 356(11) (2004), 4493–4518.10.1090/S0002-9947-04-03397-5CrossRefGoogle Scholar
Kalikow, S. and Weiss, B.. Fluctuations of ergodic averages. Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997). Illinois J. Math. 43(3) (1999), 480–488.Google Scholar
Kinnunen, J. and Shukla, P.. The structure of reverse Hölder classes on metric measure spaces. Nonlinear Anal. 95 (2014), 666–675.Google Scholar
Kinnunen, J. and Shukla, P.. The distance of
${L}^{\infty }$
from BMO on metric measure spaces. Adv. Pure Appl. Math. 5(2) (2014), 117–129.CrossRefGoogle Scholar
${L}^{\infty }$
from BMO on metric measure spaces. Adv. Pure Appl. Math. 5(2) (2014), 117–129.CrossRefGoogle ScholarKohlenbach, U.. Proof-theoretic methods in nonlinear analysis. Proceedings of the International Congress of Mathematicians, Volume II. Ed. B. Sirakov, P. Ney de Souza and M. Viana. World Scientific, Hackensack, NJ, 2018, pp. 61–82; Invited Lectures.Google Scholar
Krause, B.. Polynomial ergodic averages converge rapidly: variations on a theorem of Bourgain. Preprint, 2014, arXiv:1402.1803.Google Scholar
Krause, B., Mariusz, M. and Trojan, B.. Two-parameter version of Bourgain’s inequality: rational frequencies. Adv. Math. 323 (2018), 720–744.10.1016/j.aim.2017.10.033CrossRefGoogle Scholar
Krengel, U.. On the speed of convergence in the ergodic theorem. Monatsh. Math. 86(1) (1978/79), 3–6.10.1007/BF01300052CrossRefGoogle Scholar
Lépingle, D.. La variation
${\mathrm{d}}^{\prime }$
ordre
$\mathrm{p}$
des semi-martingales. Z. Wahrsch. Verw. Gebiete 36(4) (1976), 295–316.10.1007/BF00532696CrossRefGoogle Scholar
${\mathrm{d}}^{\prime }$
ordre
$\mathrm{p}$
des semi-martingales. Z. Wahrsch. Verw. Gebiete 36(4) (1976), 295–316.10.1007/BF00532696CrossRefGoogle ScholarLewko, A. and Lewko, M.. Estimates for the square variation of partial sums of Fourier series and their rearrangements. J. Funct. Anal. 262(6) (2012), 2561–2607.10.1016/j.jfa.2011.12.007CrossRefGoogle Scholar
Lin, H., Nakai, E. and Yang, D.. Boundedness of Lusin-area and
${\textit{g}}_{\unicode{x3bb}}^{\ast }$
functions on localized BMO spaces over doubling metric measure spaces. Bull. Sci. Math. 135(1) (2011), 59–88.10.1016/j.bulsci.2010.03.004CrossRefGoogle Scholar
${\textit{g}}_{\unicode{x3bb}}^{\ast }$
functions on localized BMO spaces over doubling metric measure spaces. Bull. Sci. Math. 135(1) (2011), 59–88.10.1016/j.bulsci.2010.03.004CrossRefGoogle ScholarLindenstrauss, E.. Pointwise theorems for amenable groups. Invent. Math. 146(2) (2001), 259–295.10.1007/s002220100162CrossRefGoogle Scholar
López-Sánchez, L. D., Martell, J. M. and Parcet, J.. Dyadic harmonic analysis beyond doubling measures. Adv. Math. 267 (2014), 44–93.10.1016/j.aim.2014.08.001CrossRefGoogle Scholar
Losert, V.. On the structure of groups with polynomial growth. Math. Z. 195 (1986), 109–118.10.1007/BF01161604CrossRefGoogle Scholar
Mirek, M., Stein, E. M. and Trojan, B..
${\ell}^{\mathrm{p}}\left(\mathbb{Z}\right)$
-estimates for discrete operators of Radon type: variational estimates. Invent. Math. 209(3) (2017), 665–748.10.1007/s00222-017-0718-4CrossRefGoogle Scholar
${\ell}^{\mathrm{p}}\left(\mathbb{Z}\right)$
-estimates for discrete operators of Radon type: variational estimates. Invent. Math. 209(3) (2017), 665–748.10.1007/s00222-017-0718-4CrossRefGoogle ScholarMirek, M., Stein, E. M. and Zorin-Kranich, P.. A bootstrapping approach to jump inequalities and their applications. Anal. PDE 13(2) (2020), 527–558.10.2140/apde.2020.13.527CrossRefGoogle Scholar
Mirek, M., Stein, E. M. and Zorin-Kranich, P.. Jump inequalities via real interpolation. Math. Ann. 376(1–2) (2020), 797–819.10.1007/s00208-019-01889-2CrossRefGoogle Scholar
Moriakov, N.. Fluctuations of ergodic averages for actions of groups of polynomial growth. Studia Math. 240(3) (2018), 255–273.10.4064/sm8692-5-2017CrossRefGoogle Scholar
Nevo, A.. Pointwise ergodic theorems for actions of groups. Handbook of Dynamical Systems. Volume 1B. Ed. B. Hasselblatt and A. Katok. Elsevier B. V., Amsterdam, 2006, pp. 871–982.10.1016/S1874-575X(06)80038-XCrossRefGoogle Scholar
Paterson, A. L. T.. Amenability (Mathematical Surveys and Monographs, 29). American Mathematical Society, Providence, RI, 1988.10.1090/surv/029CrossRefGoogle Scholar
Pisier, G.. Complex interpolation and regular operators between Banach spaces. Arch. Math. (Basel) 62(3) (1994), 261–269.10.1007/BF01261367CrossRefGoogle Scholar
Pisier, G. and Xu, Q.. The strong
$\textit{p}$
-variation of martingales and orthogonal series. Probab. Theory Related fields 77(4) (1988), 497–514.10.1007/BF00959613CrossRefGoogle Scholar
$\textit{p}$
-variation of martingales and orthogonal series. Probab. Theory Related fields 77(4) (1988), 497–514.10.1007/BF00959613CrossRefGoogle ScholarTao, T.. Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys. 28(2) (2008), 657–688.10.1017/S0143385708000011CrossRefGoogle Scholar
Tao, T.. Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence, RI, 2008.10.1090/mbk/059CrossRefGoogle Scholar
Tempelman, A.. Ergodic theorems for general dynamical systems. Dokl. Akad. Nauk SSSR 176 (1967), 790–793 (in Russian).Google Scholar
Tempelman, A.. Ergodic theorems for general dynamical systems. Tr. Moskov. Mat. Obšč. 26 (1972), 95–132 (in Russian).Google Scholar
Tempelman, A.. Pointwise ergodic theorems for bounded Lamperti representations of amenable groups. Proc. Amer. Math. Soc. 143(11) (2015), 4989–5004.10.1090/proc/12616CrossRefGoogle Scholar
Tessera, R.. Volume of spheres in doubling metric measured spaces and in groups of polynomial growth. Bull. Soc. Math. France 135(1) (2007), 47–64.10.24033/bsmf.2525CrossRefGoogle Scholar
Wiener, N.. The ergodic theorem. Duke Math. J. 5(1) (1939), 1–18.10.1215/S0012-7094-39-00501-6CrossRefGoogle Scholar
Zorin-Kranich, P.. Variation estimates for averages along primes and polynomials. J. Funct. Anal. 268(1) (2015), 210–238.10.1016/j.jfa.2014.10.018CrossRefGoogle Scholar
Zorin-Kranich, P.. Variational truncations of singular integrals on spaces of homogeneous type. Preprint, 2020, arXiv:2009.04541.10.1016/j.aim.2021.107832CrossRefGoogle Scholar