Skip to main content Accessibility help
×
Home

Norm variation of ergodic averages with respect to two commuting transformations

  • POLONA DURCIK (a1), VJEKOSLAV KOVAČ (a2), KRISTINA ANA ŠKREB (a3) and CHRISTOPH THIELE (a1)

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.

Copyright

References

Hide All
[1] Abramowitz, M. and Stegun, I. A. (Eds). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1992.
[2] Austin, T.. On the norm convergence of non-conventional ergodic averages. Ergod. Th. & Dynam. Sys. 30(2) (2010), 321338.
[3] Avigad, J. and Rute, J.. Oscillation and the mean ergodic theorem for uniformly convex Banach spaces. Ergod. Th. & Dynam. Sys. 35(4) (2015), 10091027.
[4] Bernicot, F.. L p estimates for non smooth bilinear Littlewood–Paley square functions on ℝ. Math. Ann. 351(1) (2011), 149.
[5] 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), 233268.
[6] Birkhoff, G. D.. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17(12) (1931), 656660.
[7] Bourgain, J.. Almost sure convergence and bounded entropy. Israel J. Math. 63(1) (1988), 7997.
[8] 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), 545.
[9] Bourgain, J.. Double recurrence and almost sure convergence. J. Reine Angew. Math. 404 (1990), 140161.
[10] Conze, J.-P. and Lesigne, E.. Théorèmes ergodiques pour des mesures diagonales. Bull. Soc. Math. France 112(2) (1984), 143175.
[11] Demeter, C. and Thiele, C.. On the two-dimensional bilinear Hilbert transform. Amer. J. Math. 132(1) (2010), 201256.
[12] Do, Y., Oberlin, R. and Palsson, E. A.. Variation-norm and fluctuation estimates for ergodic bilinear averages. Indiana Univ. Math. J. 66 (2017), 5599.
[13] Donoso, S. and Sun, W.. Pointwise multiple averages for systems with two commuting transformations. Ergod. Th. & Dynam. Sys. 126. doi:10.1017/etds.2016.127. Published online 14 March 2017.
[14] Durcik, P.. An L 4 estimate for a singular entangled quadrilinear form. Math. Res. Lett. 22(5) (2015), 13171332.
[15] 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.
[16] Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204256.
[17] Furstenberg, H. and Katznelson, Y.. An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math. 38(1) (1978), 275291.
[18] Furstenberg, H., Katznelson, Y. and Ornstein, D.. The ergodic theoretical proof of Szemerédi’s theorem. Bull. Amer. Math. Soc. (N.S.) 7(3) (1982), 527552.
[19] Jones, R. L., Kaufman, R., Rosenblatt, J. M. and Wierdl, M.. Oscillation in ergodic theory. Ergod. Th. & Dynam. Sys. 18(4) (1998), 889935.
[20] Jones, R. L., Ostrovskii, I. V. and Rosenblatt, J. M.. Square functions in ergodic theory. Ergod. Th. & Dynam. Sys. 16(2) (1996), 267305.
[21] Jones, R. L., Seeger, A. and Wright, J.. Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc. 360(12) (2008), 67116742.
[22] Kovač, V.. Bellman function technique for multilinear estimates and an application to generalized paraproducts. Indiana Univ. Math. J. 60(3) (2011), 813846.
[23] Kovač, V.. Boundedness of the twisted paraproduct. Rev. Mat. Iberoam. 28(4) (2012), 11431164.
[24] Kovač, V.. Quantitative norm convergence of double ergodic averages associated with two commuting group actions. Ergod. Th. & Dynam. Sys. 36(3) (2016), 860874.
[25] 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), 11431163.
[26] Kovač, V. and Thiele, C.. A T (1) theorem for entangled multilinear dyadic Calderón–Zygmund operators. Illinois J. Math. 57(3) (2013), 775799.
[27] 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.
[28] Lacey, M.. On bilinear Littlewood–Paley square functions. Publ. Mat. 40(2) (1996), 387396.
[29] Lacey, M. and Thiele, C.. L p estimates on the bilinear Hilbert transform for 2 < p < . Ann. of Math. (2) 146(3) (1997), 693724.
[30] Lacey, M. and Thiele, C.. On Calderón’s conjecture. Ann. of Math. (2) 149(2) (1999), 475496.
[31] 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.
[32] Mohanty, P. and Shrivastava, S.. A note on the bilinear Littlewood–Paley square function. Proc. Amer. Math. Soc. 138(6) (2010), 20952098.
[33] von Neumann, J.. Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. USA 18(1) (1932), 7082.
[34] Ratnakumar, P. K. and Shrivastava, S.. On bilinear Littlewood–Paley square functions. Proc. Amer. Math. Soc. 140(12) (2012), 42854293.
[35] Rubio de Francia, J. L.. A Littlewood–Paley inequality for arbitrary intervals. Rev. Mat. Iberoam. 1(2) (1985), 114.
[36] Tao, T.. Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys. 28(2) (2008), 657688.
[37] Tao, T.. Cancellation for the multilinear Hilbert transform. Collect. Math. 67(2) (2016), 191206.
[38] Thiele, C.. Wave Packet Analysis (CBMS Regional Conference Series in Mathematics, 105) . American Mathematical Society, Providence, RI, 2006.
[39] Walsh, M. N.. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) 175(3) (2012), 16671688.
[40]Wolfram Research, Inc., Mathematica, ver. 9.0, Champaign, IL, 2012.
[41] Zorin-Kranich, P.. Cancellation for the simplex Hilbert transform. Math. Res. Lett., to appear, Preprint, 2015, arXiv:1507.02436.

Norm variation of ergodic averages with respect to two commuting transformations

  • POLONA DURCIK (a1), VJEKOSLAV KOVAČ (a2), KRISTINA ANA ŠKREB (a3) and CHRISTOPH THIELE (a1)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed