Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:59:29.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-commutative ergodic averages of balls and spheres over Euclidean spaces

Published online by Cambridge University Press:  14 June 2018

GUIXIANG HONG
GUIXIANG HONG*
Affiliation:
School of Mathematics and Statistics, Wuhan University and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China email guixiang.hong@whu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we establish a non-commutative analogue of Calderón’s transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case—operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein–Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener’s pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones’ pointwise ergodic theorem, we use again the transference principle together with the Littlewood–Paley method, which is different from Jones’ original variational method that is still unavailable in the non-commutative setting.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 2 , February 2020 , pp. 418 - 436
Copyright
© Cambridge University Press, 2018 

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

Anantharaman-Delaroche, C.. On ergodic theorems for free group actions on noncommutative spaces. Probab. Theory Related Fields 135(4) (2006), 520–546.Google Scholar
Bekjan, T.. Noncommutative maximal ergodic theorems for positive contractions. J. Funct. Anal. 254(9) (2008), 2401–2418.Google Scholar
Bekjan, T., Chen, Z. and Osekowski, A.. Noncommutative maximal inequalities associated with convex functions. Trans. Amer. Math. Soc. 369 (2017), 409–427.Google Scholar
Benson, C., Jenkins, J. and Ratcliff, G.. Bounded K-spherical functions on Heisenberg groups. J. Funct. Anal. 105 (1992), 409–443.Google Scholar
Bergh, J. and Löfström, J.. Interpolation Spaces. An Introduction. Springer, New York, 1976.Google Scholar
Chen, Y., Ding, Y., Hong, G. and Liu, H.. Weighted jump and variational inequalities for rough operators. J. Funct. Anal. 274 (2018), 2446–2475.Google Scholar
Chen, Z., Yin, Z. and Xu, Q.. Harmonic analysis on quantum tori. Comm. Math. Phys. 322(3) (2013), 755–805.Google Scholar
Defant, A. and Junge, M.. Maximal theorems of Menchoff–Rademacher type in noncommutative L q-spaces. J. Funct. Anal. 206 (2004), 322–355.Google Scholar
Dirksen, S.. Weak-type interpolation for noncommutative maximal operators. J. Operator Theory 73(2) (2015), 515–532.Google Scholar
Faraut, J. and Harzallah, K.. Deux cours d’analyse harmonique. Birkhäuser, Basel, 1987.Google Scholar
Grafakos, L.. Classical Fourier Analysis (Graduate Texts in Mathematics) , 2nd edn. Springer, New York, 2008.Google Scholar
Hong, G.. The behavior of the bounds of matrix-valued maximal inequality in ℝ n for large n . Illinois J. Math. 57(3) (2013), 855–869.Google Scholar
Hong, G.. Noncommutative maximal ergodic theorems for spherical means on the Heisenberg group. Preprint, 2016, arXiv:1611.01651.Google Scholar
Hong, G., Liao, B. and Wang, S.. Noncommutative maximal ergodic inequalities associated with doubling conditions. Preprint, 2018, arXiv:1705.04851.Google Scholar
Hong, G. and Ma, T.. Vector-valued q-variation for ergodic averages and analytics semigroups. J. Math. Anal. Appl. 437 (2016), 1084–1100.Google Scholar
Hong, G. and Ma, T.. Vector-valued q-variation for differential operators and semigroups I. Math. Z. 286(1) (2017), 89–120.Google Scholar
Hong, G. and Sun, M.. Noncommutative multi-parameter Wiener–Wintner type ergodic theorem. Preprint, 2016, arXiv:1602.00927.Google Scholar
Hu, Y.. Maximal ergodic theorems for some group actions. J. Funct. Anal. 254 (2008), 1282–1306.Google Scholar
Jaming, P.. The spherical ergodic theorem revisited. Expo. Math. 27(3) (2009), 257–269.Google Scholar
Jones, R.. Ergodic averages on spheres. J. Anal. Math. 61 (1993), 29–45.Google Scholar
Junge, M.. Doob’s inequality for non-commutative martingales. J. reine angew. Math. 549 (2002), 149–190.Google Scholar
Junge, M. and Parcet, J.. Mixed-norm Inequalities and Operator Space Lp Embedding Theory (Memoirs of the American Mathematical Society, 952) . American Mathematical Society, Providence, RI, 2010.Google Scholar
Junge, M. and Xu, Q.. Noncommutative maximal ergodic theorems. J. Amer. Math. Soc. 20 (2006), 385–439.Google Scholar
Junge, M. and Xu, Q.. Noncommutative Burkholder/Rosenthal inequalities II: applications. Israel J. Math. 167 (2008), 227–282.Google Scholar
Lacey, M. T.. Ergodic averages on circles. J. Anal. Math. 67 (1995), 199–206.Google Scholar
Lance, E. C.. Ergodic theorems for convex sets and operator algebras. Invent. Math. 37(3) (1976), 201–214.Google Scholar
Litvinov, S.. A non-commutative Wiener–Wintner theorem. Preprint, 2014, arXiv:1405.4427v1 [math.OA].Google Scholar
Mei, T.. Operator Valued Hardy Spaces (Memoirs of the American Mathematical Society, 188) . American Mathematical Society, Providence, RI, 2007.Google Scholar
Nevo, A.. Pointwise ergodic theorems for actions of groups. Handbook of Dynamical Systems. Vol. 1, Part B. North-Holland, Amsterdam, 2006, Ch. 13, pp. 871–982.Google Scholar
Pisier, G.. Non-commutative Vector-valued L p-space and Completely p-summing Maps (Astérisque, 247) . Société Mathématique de France, Paris, 1998.Google Scholar
Pisier, G.. Introduction to Operator Space Theory (London Mathematical Society Lecture Note Series, 294) . Cambridge University Press, Cambridge, 2003.Google Scholar
Pisier, G. and Xu, Q.. Non-commutative L p-spaces. Handbook of the Geometry of Banach Spaces. Vol. II. Eds. Johnson, W. B. and Lindenstrauss, J.. Elsevier, Amsterdam, 2003, pp. 1459–1517.Google Scholar
Rubio de Francia, J. L.. Maximal functions and Fourier transforms. Duke Math. J. 53 (1986), 395–404.Google Scholar
Stein, E. M. and Strömberg, J. O.. Behavior of maximal functions in ℝ n for large n . Ark. Math. 21 (1983), 259–269.Google Scholar
Strichartz, R.. L p harmonic analysis and Radon transform on the Heisenberg group. J. Funct. Anal. 96 (1991), 350–406.Google Scholar
Wiener, N.. The ergodic theorem. Duke Math. J. 5 (1939), 1–18.Google Scholar