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Pointwise convergence of certain continuous-time double ergodic averages

Part of: Ergodic theory

Published online by Cambridge University Press:  04 May 2021

MICHAEL CHRIST ,
POLONA DURCIK ,
VJEKOSLAV KOVAČ  and
JORIS ROOS
MICHAEL CHRIST
Affiliation:
Department of Mathematics, University of California, Berkeley, CA94720, USA (e-mail: [email protected])
POLONA DURCIK
Affiliation:
Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA92866, USA (e-mail: [email protected])
VJEKOSLAV KOVAČ
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, 10000Zagreb, Croatia (e-mail: [email protected])
JORIS ROOS*
Affiliation:
Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, MA01854, USA School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, UK
*
e-mail: [email protected]
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Abstract

We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

Keywords

multiple ergodic averageconvergence almost everywhereCalderón transference principlemultilinear estimate

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2270 - 2280
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Assani, I.. Pointwise convergence of ergodic averages along cubes. J. Anal. Math. 110 (2010), 241–269.CrossRefGoogle Scholar
Austin, T.. Norm convergence of continuous-time polynomial multiple ergodic averages. Ergod. Th. & Dynam. 32(2) (2012), 361–382.CrossRefGoogle Scholar
Bergelson, V. and Leibman, A.. A nilpotent Roth theorem. Invent. Math. 147(2) (2002), 429–470.CrossRefGoogle Scholar
Bergelson, V. and Leibman, A.. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J.Amer. Math. Soc. 9(3) (1996), 725–753.CrossRefGoogle Scholar
Bergelson, V., Leibman, A. and Moreira, C. G.. From discrete- to continuous-time ergodic theorems. Ergod. Th. & Dynam. 32(2) (2012), 383–426.CrossRefGoogle Scholar
Bourgain, J.. Double recurrence and almost sure convergence. J. Reine Angew. Math. 404 (1990), 140–161.Google Scholar
Bourgain, J.. On the pointwise ergodic theorem on Lp for arithmetic sets. Israel J. Math. 61(1) (1988), 73–84.CrossRefGoogle Scholar
Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 5–45, with an appendix by the authors Furstenberg, H., Katznelson, Y. and Ornstein, D. S..CrossRefGoogle Scholar
Buczolich, Z. and Mauldin, R. D.. Divergent square averages. Ann. of Math. (2) 171(3) (2010), 1479–1530.CrossRefGoogle Scholar
Calderón, A.-P.. Ergodic theory and translation-invariant operators. Proc. Natl Acad. Sci. USA 59 (1968), 349–353.CrossRefGoogle ScholarPubMed
Christ, M.. On trilinear oscillatory integral inequalities and related topics. Preprint, 2020, arXiv:2007.12753.Google Scholar
Christ, M., Durcik, P. and Roos, J.. Trilinear smoothing inequalities and a variant of the triangular Hilbert transform. Preprint, 2020, arXiv:2008.10140.CrossRefGoogle Scholar
Chu, Q. and Frantzikinakis, N.. Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations. Ergod. Th. & Dynam. 32(3) (2012), 877–897.CrossRefGoogle Scholar
Donoso, S. and Sun, W.. A pointwise cubic average for two commuting transformations. Israel J. Math. 216(2) (2016), 657–678.CrossRefGoogle Scholar
Donoso, S. and Sun, W.. Pointwise convergence of some multiple ergodic averages. Adv. Math. 330 (2018), 946–996.CrossRefGoogle Scholar
Donoso, S. and Sun, W.. Pointwise multiple averages for systems with two commuting transformations. Ergod. Th. & Dynam. 38(6) (2018), 2132–2157.CrossRefGoogle Scholar
Frantzikinakis, N.. Joint ergodicity of sequences. Preprint, 2021, arXiv:2102.09967.Google Scholar
Frantzikinakis, N.. Some open problems on multiple ergodic averages. Bull. Hellenic Math. Soc. 60 (2016), 41–90.Google Scholar
Frantzikinakis, N., Lesigne, E. and Wierdl, M.. Random sequences and pointwise convergence of multiple ergodic averages. Indiana Univ. Math. J. 61(2) (2012), 585–617.CrossRefGoogle Scholar
Huang, W., Shao, S. and Ye, X.. Pointwise convergence of multiple ergodic averages and strictly ergodic models. J. Anal. Math. 139(1) (2019), 265–305.CrossRefGoogle Scholar
Jones, R. L., Seeger, A. and Wright, J.. Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc. 360(12) (2008), 6711–6742.CrossRefGoogle Scholar
Krause, B., Mirek, M. and Tao, T.. Pointwise ergodic theorems for non-conventional bilinear polynomial averages. Preprint, 2020, arXiv:2008.00857.Google Scholar
LaVictoire, P.. Universally L 1-bad arithmetic sequences. J. Anal. Math. 113 (2011), 241–263.CrossRefGoogle Scholar
O’Neil, R.. Convolution operators and L(p,q) spaces. Duke Math. J. 30 (1963), 129–142.CrossRefGoogle Scholar
Varadarajan, V. S.. Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191–220.CrossRefGoogle Scholar
Walsh, M. N.. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) 175(3) (2012), 1667–1688.CrossRefGoogle Scholar