Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T04:04:50.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Norm variation of ergodic averages with respect to two commuting transformations

Published online by Cambridge University Press:  17 August 2017

POLONA DURCIK ,
VJEKOSLAV KOVAČ ,
KRISTINA ANA ŠKREB  and
CHRISTOPH THIELE
POLONA DURCIK
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany email [email protected], [email protected]
VJEKOSLAV KOVAČ
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia email [email protected]
KRISTINA ANA ŠKREB
Affiliation:
Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića Miošića 26, 10000 Zagreb, Croatia email [email protected]
CHRISTOPH THIELE
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany email [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 658 - 688
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I. A. (Eds). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1992.Google Scholar
Austin, T.. On the norm convergence of non-conventional ergodic averages. Ergod. Th. & Dynam. Sys. 30(2) (2010), 321–338.Google Scholar
Avigad, J. and Rute, J.. Oscillation and the mean ergodic theorem for uniformly convex Banach spaces. Ergod. Th. & Dynam. Sys. 35(4) (2015), 1009–1027.Google Scholar
Bernicot, F.. L p estimates for non smooth bilinear Littlewood–Paley square functions on ℝ. Math. Ann. 351(1) (2011), 1–49.Google Scholar
Bernicot, F. and Shrivastava, S.. Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators. Indiana Univ. Math. J. 60(1) (2011), 233–268.Google Scholar
Birkhoff, G. D.. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17(12) (1931), 656–660.Google Scholar
Bourgain, J.. Almost sure convergence and bounded entropy. Israel J. Math. 63(1) (1988), 79–97.Google Scholar
Bourgain, J.. Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson, and D. S. Ornstein. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 5–45.Google Scholar
Bourgain, J.. Double recurrence and almost sure convergence. J. Reine Angew. Math. 404 (1990), 140–161.Google Scholar
Conze, J.-P. and Lesigne, E.. Théorèmes ergodiques pour des mesures diagonales. Bull. Soc. Math. France 112(2) (1984), 143–175.Google Scholar
Demeter, C. and Thiele, C.. On the two-dimensional bilinear Hilbert transform. Amer. J. Math. 132(1) (2010), 201–256.Google Scholar
Do, Y., Oberlin, R. and Palsson, E. A.. Variation-norm and fluctuation estimates for ergodic bilinear averages. Indiana Univ. Math. J. 66 (2017), 55–99.Google Scholar
Donoso, S. and Sun, W.. Pointwise multiple averages for systems with two commuting transformations. Ergod. Th. & Dynam. Sys. 1–26. doi:10.1017/etds.2016.127. Published online 14 March 2017.Google Scholar
Durcik, P.. An L 4 estimate for a singular entangled quadrilinear form. Math. Res. Lett. 22(5) (2015), 1317–1332.Google Scholar
Durcik, P.. $L^{p}$ estimates for a singular entangled quadrilinear form.  Trans. Amer. Math. Soc.   doi:10.1090/tran/6850. Published online 30 March 2017.Google Scholar
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.Google Scholar
Furstenberg, H. and Katznelson, Y.. An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math. 38(1) (1978), 275–291.Google Scholar
Furstenberg, H., Katznelson, Y. and Ornstein, D.. The ergodic theoretical proof of Szemerédi’s theorem. Bull. Amer. Math. Soc. (N.S.) 7(3) (1982), 527–552.Google 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.Google Scholar
Jones, R. L., Ostrovskii, I. V. and Rosenblatt, J. M.. Square functions in ergodic theory. Ergod. Th. & Dynam. Sys. 16(2) (1996), 267–305.Google 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.Google Scholar
Kovač, V.. Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(3) (2011), 813–846.Google Scholar
Kovač, V.. Boundedness of the twisted paraproduct. Rev. Mat. Iberoam. 28(4) (2012), 1143–1164.Google Scholar
Kovač, V.. Quantitative norm convergence of double ergodic averages associated with two commuting group actions. Ergod. Th. & Dynam. Sys. 36(3) (2016), 860–874.Google Scholar
Kovač, V. and Škreb, K. A.. One modification of the martingale transform and its applications to paraproducts and stochastic integrals. J. Math. Anal. Appl. 426(2) (2015), 1143–1163.Google Scholar
Kovač, V. and Thiele, C.. A T (1) theorem for entangled multilinear dyadic Calderón–Zygmund operators. Illinois J. Math. 57(3) (2013), 775–799.Google Scholar
Kovač, V., Thiele, C. and Zorin-Kranich, P.. Dyadic triangular Hilbert transform of two general functions and one not too general function. Forum Math. Sigma 3 (2015) (e25), 27 pages. doi:10.1017/fms.2015.25.Google Scholar
Lacey, M.. On bilinear Littlewood–Paley square functions. Publ. Mat. 40(2) (1996), 387–396.Google Scholar
Lacey, M. and Thiele, C.. L p estimates on the bilinear Hilbert transform for 2 < p < ∞ . Ann. of Math. (2) 146(3) (1997), 693–724.Google Scholar
Lacey, M. and Thiele, C.. On Calderón’s conjecture. Ann. of Math. (2) 149(2) (1999), 475–496.Google Scholar
Mirek, M., Stein, E. and Trojan, B.. $\ell ^{p}(\mathbb{Z}^{d})$ -estimates for discrete operators of Radon type: variational estimates. Invent. Math. doi:10.1007/s00222-017-0718-4. Published online 31 January 2017.Google Scholar
Mohanty, P. and Shrivastava, S.. A note on the bilinear Littlewood–Paley square function. Proc. Amer. Math. Soc. 138(6) (2010), 2095–2098.Google Scholar
von Neumann, J.. Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. USA 18(1) (1932), 70–82.Google Scholar
Ratnakumar, P. K. and Shrivastava, S.. On bilinear Littlewood–Paley square functions. Proc. Amer. Math. Soc. 140(12) (2012), 4285–4293.Google Scholar
Rubio de Francia, J. L.. A Littlewood–Paley inequality for arbitrary intervals. Rev. Mat. Iberoam. 1(2) (1985), 1–14.Google Scholar
Tao, T.. Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys. 28(2) (2008), 657–688.Google Scholar
Tao, T.. Cancellation for the multilinear Hilbert transform. Collect. Math. 67(2) (2016), 191–206.Google Scholar
Thiele, C.. Wave Packet Analysis (CBMS Regional Conference Series in Mathematics, 105) . American Mathematical Society, Providence, RI, 2006.Google Scholar
Walsh, M. N.. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) 175(3) (2012), 1667–1688.Google Scholar
Wolfram Research, Inc., Mathematica, ver. 9.0, Champaign, IL, 2012.Google Scholar
Zorin-Kranich, P.. Cancellation for the simplex Hilbert transform. Math. Res. Lett., to appear, Preprint, 2015, arXiv:1507.02436.Google Scholar