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Weighted and vector-valued variational estimates for ergodic averages

Published online by Cambridge University Press:  14 July 2016

BEN KRAUSE  and
PAVEL ZORIN-KRANICH Open the ORCID record for PAVEL ZORIN-KRANICH [Opens in a new window]
BEN KRAUSE
Affiliation:
UCLA Math Sciences Building, Los Angeles, CA 90095-1555, USA email benkrause23@math.ucla.edu
PAVEL ZORIN-KRANICH
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany email pzorin@uni-bonn.de
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Abstract

We prove weighted and vector-valued variational estimates for ergodic averages on $\mathbb{R}^{d}$. The weighted square function estimate relating ergodic averages to the dyadic martingale is obtained using an $\ell ^{r}$ version of a reverse Hölder inequality for variation seminorms.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 244 - 256
Copyright
© Cambridge University Press, 2016 

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References

Bergh, J. and Löfström, J.. Interpolation Spaces. An Introduction (Grundlehren der Mathematischen Wissenschaften, 223) . Springer, Berlin, 1976.Google Scholar
Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 5–45. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein.Google Scholar
Bergh, J. and Peetre, J.. On the spaces V p  (0 < p ≤∞). Boll. Unione. Mat. Ital. (4) 10 (1974), 632–648.Google Scholar
Buckley, S. M.. Summation conditions on weights. Michigan Math. J. 40(1) (1993), 153–170.Google Scholar
Duoandikoetxea, J.. Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260(6) (2011), 1886–1901.CrossRefGoogle Scholar
Fefferman, C. and Stein, E. M.. Some maximal inequalities. Amer. J. Math. 93 (1971), 107–115.Google Scholar
García-Cuerva, J. and Rubio de Francia, J. L.. Weighted Norm Inequalities and Related Topics, Vol. 116 (North-Holland Mathematics Studies. Notas de Matemática [Mathematical Notes], 104) . North-Holland, Amsterdam, 1985.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., Rosenblatt, J. M. and Wierdl, M.. Oscillation in ergodic theory: higher dimensional results. Israel J. Math. 135 (2003), 1–27.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.CrossRefGoogle Scholar
Krause, B.. On higher-dimensional oscillation in ergodic theory. Preprint, 2013. New York J. Math. to appear.Google Scholar
Lépingle, D.. La variation d’ordre p des semi-martingales. Z. Wahrscheinlichkeitstheorie verw. Geb. 36(4) (1976), 295–316.CrossRefGoogle Scholar
Lacey, M. T., Petermichl, S. and Reguera, M. C.. Sharp A 2 inequality for Haar shift operators. Math. Ann. 348(1) (2010), 127–141.Google Scholar
Ma, T., Torrea, J. L. and Xu, Q.. Weighted variation inequalities for differential operators and singular integrals. J. Funct. Anal. 268(2) (2015), 376–416.CrossRefGoogle Scholar
Ma, T., Torrea, J. L. and Xu, Q.. Weighted variation inequalities for differential operators and singular integrals in higher dimensions. Preprint, 2015.Google Scholar
Pisier, G. and Xu, Q. H.. The strong p-variation of martingales and orthogonal series. Probab. Theory Related Fields 77(4) (1988), 497–514.Google Scholar
Wittwer, J.. A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7(1) (2000), 1–12.Google Scholar