Skip to main content
SpringerLink
Log in

Vector valued q-variation for differential operators and semigroups I

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper, we establish \(\mathcal B\)-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space \(\mathcal B\) has martingale cotype property. These results generalize, on the one hand Pisier and Xu’s result on the variational inequalities for \(\mathcal B\)-valued martingales, on the other hand many classical variational inequalities in harmonic analysis and ergodic theory. Moreover, we show that Rademacher cotype q is necessary for the \(\mathcal B\)-valued q-variational inequalities. As applications of the variational inequalities, we deduce the jump estimates and obtain quantitative information on the rate of convergence. It turns out the rate of convergence depends upon the geometry of the Banach space under consideration, while Cowling and Leinert have shown that convergence holds for all separable Banach spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bourgain, J.: Pointwise ergodic theorems for arithmetic sets. Publ. Math. IHES 69, 5–41 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Campbell, J., Jones, R., Reinhold, K., Wierdl, M.: Oscillation and variation for the Hilbert transform. Duke Math. J. 105, 59–83 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Campbell, J.T., Jones, R.L., Reinhold, K., Wierdl, M.: Oscillation and variation for singular integrals in higher dimensions. Trans. Am. Math. Soc. 355, 2115–2137 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cowling, M., Leinert, M.: Pointwise convergence and semigroups acting on vector-valued functions. Bull. Aust. Math. Soc. 84, 44–48 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Do, Y., Muscalu, C., Thiele, C.: Variational estimates for paraproducts. Rev. Mat. Iberoam. 28, 857–878 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fendler, G.: Dilations of one parameter semigroups of positive contractions on \(L_p\)-spaces. Can. J. Math. 49, 736–748 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Publishing Co, Amsterdam (1985)

    MATH  Google Scholar 

  8. Gillespie, T.A., Torrea, J.L.: Dimension free estimates for the oscillation of Riesz transforms. Israel J. Math. 141, 125–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hong, G.: The behaviour of square functions from ergodic theory in \(L^\infty \). Proc. Am. Math. Soc. 143(11), 4797–4802 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hong, G., Ma, T.: Vector valued q-variation for ergodic averages and analytic semigroups. J. Math. Anal. Appl. 437(2), 1084–1100 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hytönen, T.P., Lacey, M.T., Parissis, I.: A variation norm Carleson theorem for vector-valued Walsh-Fourier series. Rev. Mat. Iberoam. 30(3), 979–1014 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jones, R., Kaufman, R., Rosenblatt, J., Wierdl, M.: Oscillation in ergodic theory. Ergod. Theory Dyn. Syst. 18, 889–935 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jones, R., Rosenblatt, J.M., Wierdl, M.: Oscillation in ergodic theory: higher dimensional results. Israel J. Math. 135, 1–27 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jones, R.L., Seeger, A., Wright, J.: Strong variational and jump inequalities in harmonic analysis. Trans. Am. Math. Soc. 360, 6711–6742 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jones, R.L., Wang, G.: Variation inequalities for the Fejér and Poisson kernels. Trans. Am. Math. Soc. 356, 4493–4518 (2004)

    Article  MATH  Google Scholar 

  16. Junge, M., Xu, Q.: Noncommutative maximal ergodic theorems. J. Am. Math. Soc. 20, 385–439 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Krause, B., Zorin-Kranich, P.: Weighted and vector-valued variational estimates for ergodic averages. arXiv:1409.7120

  18. Le Merdy, C., Xu, Q.: Strong q-variation inequalities for analytic semigroups. Ann. Inst. Fourier 62, 2069–2097 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lépingle, D.: La variation d’ordre \(p\) des semi-martingales. Z. Wahrscheinlichkeitstheorie Verwandte Geb. 36, 295–316 (1976)

    Article  MATH  Google Scholar 

  20. Ma, T., Torrea, J.L., Xu, Q.: Weighted variation inequalities for differential operators and Singular integrals. J. Funct. Anal. 268(2), 376–416 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martínez, T., Torrea, J.L., Xu, Q.: Vector-valued Littlewood-Paley-Stein theory for semigroups. Adv. Math. 203(2), 430–475 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mas, A.: Variation for singular integrals on Lipschitz graphs: \(L^p\) and endpoint estimates. Trans. Am. Math. Soc. 365(11), 5759–5781 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mas, A., Tolsa, X.: Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs. Proc. Lond. Math. Soc. 105, 49–86 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mas, A., Tolsa, X.: Variation for the Riesz transform and uniform rectifiability. J. Eur. Math. Soc. (to appear)

  25. Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  26. Oberlin, R., Seeger, A., Tao, T., Thiele, C., Wright, J.: A variation norm Carleson theorem. J. Eur. Math. Soc. 14, 421–464 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Peller, V.: An analogue of J. von Neumann’s inequality for the space \(L^p\). Dokl. Akad. Nauk SSSR 231(3), 539–542 (1976) (Russian)

  28. Pisier, G.: Factorization of operator valued analytic functions. Adv. Math. 93(1), 61–125 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pisier, G.: Complex interpolation and regular operators between Banach spaces. Arch. Math. (Basel) 62(3), 261–269 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pisier, G., Xu, Q.: The strong \(p\)-variation of martingales and orthogonal series. Probab. Theory Relat. Fields 77, 497–514 (1988)

    Article  MATH  Google Scholar 

  31. Qian, J.: The \(p\)-variation of partial sum processes and the empirical process. Ann. Probab. 26, 1370–1383 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press and the University of Tokio Press, Princeton, NJ (1970)

  33. Taggart, R.J.: Pointwise convergence for semigroups in vector-valued \(L_p\) spaces. Math. Z. 261, 933–949 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions and corrections, which help improve the present paper. Guixiang Hong is partially supported by NSFC No. 11601396, and 1000 Young Talent Researcher Programm of China-429900018-101150 (2016). Tao Ma is partially supported by NSFC No. 11271292 and 988 NSFC No. 11431011.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Guixiang Hong & Tao Ma

Authors
  1. Guixiang Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guixiang Hong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hong, G., Ma, T. Vector valued q-variation for differential operators and semigroups I. Math. Z. 286, 89–120 (2017). https://doi.org/10.1007/s00209-016-1756-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1756-0

Keywords

Mathematics Subject Classification

Search

Navigation