Summary
Let (f n ) be a martingale. We establish a relationship between exponential bounds for the probabilities of the typeP(|f n |>λ·‖T(f n )‖∞) and the size of the constantC p appearing in the inequality ‖f *‖ p ≦C p ‖T *(f)‖ p , for some quasi-linear operators acting on martingales.
Article PDF
Similar content being viewed by others
References
Bañuelos, R.: A sharp good-λ inequality with an application to Riesz transforms. Mich. Math. J.35, 117–125 (1988)
Bañuelos, R., Moore, C.N.: Sharp estimates for the nontangential maximal function and the Lusin area function. Trans. Am. Math. Soc.312, 641–662 (1989)
Bourgain, J.: On the behaviour of the constant in the Littlewood-Paley inequality. In: Lindenstrauss, J., Milman, V.D. (eds.) Israel Siminar GAFA 1987/88 (Lect. Notes Math., vol. 1376, pp. 202–208) Berlin Heidelberg New York: Springer 1989
Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab.1, 19–42 (1973)
Burkholder, D.L.: Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Adv. Math.26, 182–205 (1977)
Burkholder, D.L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab.12, 647–702 (1984)
Burkholder, D.L.: Sharp inequalities for martingales and stochastic integrals. Actes du Colloque Paul Lévy. Astérisque157–158, 75–94 (1988)
Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math.124, 250–304 (1970)
Burkholder, D.L., Davis, B., Gundy, R.F.: Integral inequalities for convex functions of operators on martingales. In: LeCam, L.M., Neyman, J., Scott, E.L. (eds.) Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, II. pp. 223–240. Berkeley: University of California Press, 1972
Chang, S., Wilson, M., Wolff, J.: Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv.60, 217–246 (1985)
Davis, B.: On the integrability of the martingale square function. Isr. J. Math.8, 187–190 (1970)
Davis, B.: On theL p-norms of stochastic integrals and other martingales. Duke Math. J.43, 697–704 (1976)
Garsia, A.M.: Martingale inequalities. Seminar notes on recent progress. Reading: Benjamin 1973
Hall, P., Marron, J.S.: Choice of kernel order in density estimation. Ann. Stat.16, 161–172 (1988)
Herz, C.: Bounded mean oscillation and regulated martingales. Trans. Am. Math. Soc.193, 199–215 (1974)
Herz, C.: An interpolation principle for martingale inequalities. J. Funct. Anal.22, 1–7 (1976)
Hitczenko, P.: Best constants in the decoupling inequality for non-negative random variables. Stat. Probab. Lett.9, 327–329 (1990)
Hitczenko, P.: Best constants in martingale version of Rosenthal's inequality (preprint 1989)
Johnson, W.B., Schechtman, G., Zinn, J.: Best constants in moment inequalities for linear combination of independent and exchangeable random variables. Ann. Probab.13, 234–253 (1985)
Kazamaki, N., Kikuchi, M.: Some remarks on ratio inequalities for continuous martingales. Stud. Math.94, 97–102 (1989)
McConnell, T.R.: Decoupling and stochastic integration in UMD Banach space (preprint 1988)
Meyer, P.A.: Martingales and stochastic integrals. Berlin Heidelberg New York: Springer 1972
Murai, T., Uchiyama, A.: Good λ inequalities for the area integral and the nontangential maximal function. Stud. Math.83, 251–262, 1986
Neveu, J.: Discrete parameter Martingales, Amsterdam Oxford New York: North-Holland 1975
Wang, G.: Some sharp inequalities for conditionally symmetric martingales. Ph.D. dissertation, University of Illinois at Urbana-Champaign (1989)
Author information
Authors and Affiliations
Additional information
This research was supported in part by NSF Grant, no. DMS-8902418
On leave from Academy of Physical Education, Warsaw, Poland
Rights and permissions
About this article
Cite this article
Hitczenko, P. Upper bounds for theL p -norms of martingales. Probab. Th. Rel. Fields 86, 225–238 (1990). https://doi.org/10.1007/BF01474643
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01474643