Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T21:42:50.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal Inequalities of Noncommutative Martingale Transforms

Part of: Stochastic processes Selfadjoint operator algebras

Published online by Cambridge University Press:  22 November 2019

Yong Jiao ,
Fedor Sukochev  and
Dejian Zhou
Yong Jiao
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha410083, China Email: [email protected]
Fedor Sukochev
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington2052, Australia Email: [email protected]
Dejian Zhou*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha410083, China Email: [email protected]
*
* Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decomposition and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities.

Keywords

noncommutative martingalemaximal inequalitiesmartingale transformfractional integral

MSC classification

Primary: 46L53: Noncommutative probability and statistics
Secondary: 60G42: Martingales with discrete parameter 46L52: Noncommutative function spaces 60G46: Martingales and classical analysis
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 1 , February 2021 , pp. 221 - 248
Check for updates
Copyright
© Canadian Mathematical Society 2019

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.)

Footnotes

Yong Jiao is supported by NSFC(11471337, 11722114).

References

Bañuelos, R., The foundational inequalities of D. L. Burkholder and some of their ramifications. Illinois J. Math. 54(2012), no. 3, 789–868.CrossRefGoogle Scholar
Bekjan, T., Chen, Z., Perrin, M., and Yin, Z., Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales. J. Funct. Anal. 258(2010), no. 7, 2483–2505. https://doi.org/10.1016/j.jfa.2009.12.006CrossRefGoogle Scholar
Bekjan, T., Chen, Z., and Osȩkowski, A., Noncommutative maximal inequalities associated with convex functions. Trans. Amer. Math. Soc. 357(2017), no. 1, 409–427. https://doi.org/10.1090/tran/6663Google Scholar
Bergh, J. and Löfström, J., Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.CrossRefGoogle Scholar
Burkholder, D., Martingale transforms. Ann. Math. Statist. 37(1966), 1494–1504. https://doi.org/10.1214/aoms/1177699141CrossRefGoogle Scholar
Chao, J.-A. and Ombe, H., Commutators on dyadic martingales. Proc. Japan Acad. Ser. A Math. Sci. 61(1985), no. 2, 35–38.CrossRefGoogle Scholar
Cuculescu, I., Martingales on von Neumann algebras. J. Multivariate Anal. 1(1971), no. 1, 17–27. https://doi.org/10.1016/0047-259X(71)90027-3CrossRefGoogle Scholar
Defant, A. and Junge, M., Maximal theorems of Menchoff-Rademacher type in non-commutative L q-spaces. J. Funct. Anal. 206(2004), no. 2, 322–355. https://doi.org/10.1016/j.jfa.2002.07.001CrossRefGoogle Scholar
Dirksen, S., de Pagter, B., Potapov, D., and Sukochev, F., Rosenthal inequalities in noncommutative symmetric spaces. J. Funct. Anal. 261(2011), no. 10, 2890–2925. https://doi.org/10.1016/j.jfa.2011.07.015CrossRefGoogle Scholar
Dodds, P. G., Dodds, T. K., de Pagter, B., and Sukochev, F. A., Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces. J. Funct. Anal. 148(1997), no. 1, 28–69. https://doi.org/10.1006/jfan.1996.3055CrossRefGoogle Scholar
Dodds, P. G., Dodds, T. K., de Pagter, B., and Sukochev, F. A., Lipschitz continuity of the absolute value in preduals of semifinite factors. Integral Equations Operator Theory 34(1999), no. 1, 28–44. https://doi.org/10.1007/BF01332490CrossRefGoogle Scholar
Hong, G., Junge, M., and Parcet, J., Algebraic Davis decomposition and asymmetric Doob inequalities. Comm. Math. Phys. 346(2016), no. 3, 995–1019. https://doi.org/10.1007/s00220-016-2581-3CrossRefGoogle Scholar
Hong, G., Junge, M., and Parcet, J., Asymmetric Doob inequalities in continuous time. J. Funct. Anal. 273(2017), no. 4, 1479–1503. https://doi.org/10.1016/j.jfa.2017.05.001CrossRefGoogle Scholar
Hong, G., López-Sánchez, L., Martell, J., and Parcet, J., Calderón-Zygmund operators associated to matrix-valued kernels. Int. Math. Res. Not. IMRN(2014), no. 5, 1221–1252. https://doi.org/10.1093/imrn/rns250CrossRefGoogle Scholar
Hong, G. and Mei, T., John-Nirenberg inequality and atomic decomposition for noncommutative martingales. J. Funct. Anal. 263(2012), no. 4, 1064–1097. https://doi.org/10.1016/j.jfa.2012.05.013CrossRefGoogle Scholar
Jiao, Y., Martingale inequalities in noncommutative symmetric spaces. Arch. Math. (Basel) 98(2012), no. 1, 87–97. https://doi.org/10.1007/s00013-011-0343-1CrossRefGoogle Scholar
Jiao, Y., Osȩkowski, A., and Wu, L., Inequalities for noncommutative differentially subordinate martingales. Adv. Math. 337(2018), 216–259. https://doi.org/10.1016/j.aim.2018.08.016CrossRefGoogle Scholar
Jiao, Y., Osȩkowski, A., and Wu, L., Noncommutative good-𝜆 inequalities. arxiv:1805.07057.Google Scholar
Jiao, Y., Sukochev, F., Zanin, D., and Zhou, D., Johnson-Schechtman inequalities for noncommutative martingales. J. Funct. Anal. 272(2017), no. 3, 976–1016. https://doi.org/10.1016/j.jfa.2016.11.002CrossRefGoogle Scholar
Jiao, Y., Zhou, D., Wu, L., and Zanin, D., Noncommutative dyadic martingales and Walsh-Fourier series. J. Lond. Math. Soc. (2) 97(2018), no. 3, 550–574. https://doi.org/10.1112/jlms.12121CrossRefGoogle Scholar
Junge, M., Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549(2002), 149–190. https://doi.org/10.1515/crll.2002.061Google Scholar
Junge, M. and Perrin, M., Theory of 𝓗p-spaces for continuous filtrations in von Neumann algebras. Astérisque(2014), No. 362.Google Scholar
Junge, M. and Xu, Q., Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31(2003), no. 2, 948–995. https://doi.org/10.1214/aop/1048516542Google Scholar
Junge, M. and Xu, Q., Noncommutative maximal ergodic theorems. J. Amer. Math. Soc. 20(2007), no. 2, 385–439. https://doi.org/10.1090/S0894-0347-06-00533-9CrossRefGoogle Scholar
Junge, M. and Xu, Q., Noncommutative Burkholder/Rosenthal inequalities. II. Applications. Israel J. Math. 167(2008), 227–282. https://doi.org/10.1007/s11856-008-1048-4CrossRefGoogle Scholar
Lord, S., Sukochev, F., and Zanin, D., Singular traces. Theory and applications. De Gruyter Studies in Mathematics, 46, De Gruyter, Berlin, 2013.Google Scholar
Mei, T., Operator valued Hardy spaces. Mem. Amer. Math. Soc. 188(2007), no. 881. https://doi.org/10.1090/memo/0881Google Scholar
Mei, T. and Parcet, J., Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities. Int. Math. Res. Not.(2009), no. 8, 1433–1487. https://doi.org/10.1093/imrn/rnn165CrossRefGoogle Scholar
Osȩkowski, A., Weak type inequality for noncommutative differentially subordinated martingales. Probab. Theory Related Fields 140(2008), no. 3-4, 553–568. https://doi.org/10.1007/s00440-007-0075-0CrossRefGoogle Scholar
Parcet, J., Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory. J. Funct. Anal. 256(2009), no. 2, 509–593. https://doi.org/10.1016/j.jfa.2008.04.007CrossRefGoogle Scholar
Parcet, J. and Randrianantoanina, N., Gundy’s decomposition for non-commutative martingales and applications. Proc. London Math. Soc. (3) 93(2006), no. 1, 227–252. https://doi.org/10.1017/S0024611506015863CrossRefGoogle Scholar
Perrin, M., A noncommutative Davis’ decomposition for martingales. J. Lond. Math. Soc. (2) 80(2009), no. 3, 627–648. https://doi.org/10.1112/jlms/jdp045CrossRefGoogle Scholar
Pisier, G., Non-commutative vector valued L p-spaces and completely p-summing maps. Astérisque(1998), no. 247.Google Scholar
Pisier, G. and Xu, Q., Non-commutative martingale inequalities. Comm. Math. Phys. 189(1997), no. 3, 667–698. https://doi.org/10.1007/s002200050224CrossRefGoogle Scholar
Pisier, G. and Xu, Q., Non-commutative L p-spaces. Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. https://doi.org/10.1016/S1874-5849(03)80041-4Google Scholar
Randrianantoanina, N., Non-commutative martingale transforms. J. Funct. Anal. 194(2002), no. 1, 181–212.Google Scholar
Randrianantoanina, N., Square function inequalities for non-commutative martingales. Israel J. Math. 140(2004), 333–365. https://doi.org/10.1007/BF02786639CrossRefGoogle Scholar
Randrianantoanina, N., Conditioned square functions for noncommutative martingales. Ann. Probab. 35(2007), no. 3, 1039–1070. https://doi.org/10.1214/009117906000000656CrossRefGoogle Scholar
Randrianantoanina, N. and Wu, L., Noncommutative fractional integrals. Studia Math. 229(2015), no. 2, 113–139. https://doi.org/10.4064/sm7989-1-2016Google Scholar
Rubio de Francia, J., Martingale and integral transforms of Banach space valued functions. In: Probability and Banach spaces (Zaragoza, 1985). Lecture Notes in Math., 1221, Springer, Berlin, 1986, pp. 195–222. https://doi.org/10.1007/BFb0099115CrossRefGoogle Scholar
Scheckter, T. and Sukochev, F., Weak type estimates for the noncommutative Vilenkin–Fourier series. Integral Equations Operator Theory 90(2018), no. 6, Art. 64. https://doi.org/10.1007/s00020-018-2489-8CrossRefGoogle Scholar
Stein, E., Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.Google Scholar
Sukochev, F. and Ferleger, S., Harmonic analysis in UMD-spaces: applications to basis theory. Russian Mat. Zametki 58(1995), no. 6, 890–905. https://doi.org/10.1007/BF02304891Google Scholar
Takesaki, M., Theory of operator algebras. I. Springer-Verlag, New York-Heidelberg, 1979.CrossRefGoogle Scholar
Wu, L., Multipliers for noncommutative Walsh–Fourier series. Proc. Amer. Math. Soc. 144(2016), no. 3, 1073–1085. https://doi.org/10.1090/proc/12831CrossRefGoogle Scholar