On Strongly Orthogonal Martingales in Umd Banach Spaces

• Autorzy: Yaroslavtsev Ivan S.

Identyfikatory

DOI

10.37190/0208-4147.41.1.10

• Języki publikacji: EN

Abstrakty

EN

In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space X and for any X-valued strongly orthogonal martingales M and N such that N is weakly differentially subordinate to M, one has, for all 1 < p < 1, [formula] with the sharp constant _χp;X being the norm of a decoupling-type martingale transform and lying in the range, [formula], where β_p;X is the UMD_p constant of X, h_p;X is the norm of the Hilbert transform on L^p(R; X), [formula] are the Gaussian decoupling constants.

Słowa kluczowe

EN

strongly orthogonal martingales; weak differential; subordination; UMD; sharp estimates; decoupling constant; martingale transform; Hilbert transform; diagonally plurisubharmonic function

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2021
• Tom: Vol. 41, Fasc. 1
• Strony: 153--171
• Opis fizyczny: Bibliogr. 37 poz.

Twórcy

autor

Yaroslavtsev Ivan S.

• Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

Bibliografia

• [1] R. Bañuelos, A. Bielaszewski, and K. Bogdan, Fourier multipliers for non-symmetric Lévy processes, in: Marcinkiewicz Centenary Volume, Banach Center Publ. 95, Inst. Math. Polish Acad. Sci., Warszawa, 2011, 9-25.
• [2] R. Bañuelos and K. Bogdan, Lévy processes and Fourier multipliers, J. Funct. Anal. 250 (2007), 197-213.
• [3] R. Bañuelos and A. Osękowski, Martingales and sharp bounds for Fourier multipliers, Ann. Acad. Sci. Fenn. Math. 37 (2012), 251-263.
• [4] R. Bañuelos and G. Wang, Orthogonal martingales under differential subordination and applications to Riesz transforms, Illinois J. Math. 40 (1996), 678-691.
• [5] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163-168.
• [6] J. Bourgain, On martingales transforms in finite-dimensional lattices with an appendix on the K-convexity constant, Math. Nachr. 119 (1984), 41-53.
• [7] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in: Conference on Harmonic Analysis in Honor of Antoni Zygmund (Chicago, IL, 1981), Wadsworth, Belmont, CA, 1983, 270-286.
• [8] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702.
• [9] D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, in: Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206, Springer, Berlin, 1986, 61-108.
• [10] D. L. Burkholder, Martingales and singular integrals in Banach spaces, in: Handbook of the Geometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, 233-269.
• [11] S. G. Cox and S. Geiss, On decoupling in Banach spaces, arXiv:1805.12377 (2018).
• [12] C. Dellacherie and P.-A. Meyer, Probabilities and Potential. B, North-Holland Math. Stud. 72, North-Holland, Amsterdam, 1982.
• [13] D. J. H. Garling, Brownian motion and UMD-spaces, in: Probability and Banach Spaces (Zaragoza, 1985), Lecture Notes in Math. 1221, Springer, Berlin, 1986, 36-49.
• [14] S. Geiss, A counterexample concerning the relation between decoupling constants and UMD-constants, Trans. Amer. Math. Soc. 351 (1999), 1355-1375.
• [15] S. Geiss, S. Montgomery-Smith, and E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 362 (2010), 553-575.
• [16] S. Geiss and I. S. Yaroslavtsev, Dyadic and stochastic shifts and Volterra-type operators, in preparation.
• [17] B. Hollenbeck, N. J. Kalton, and I. E. Verbitsky, Best constants for some operators associated with the Fourier and Hilbert transforms, Studia Math. 157 (2003), 237-278.
• [18] T. P. Hytönen, J. M. A. M. van Neerven, M. C. Veraar, and L. Weis, Analysis in Banach Spaces, Vol. I. Martingales and Littlewood-Paley theory, Ergeb. Math. Grenzgeb. 63, Springer, 2016.
• [19] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren Math. Wiss. 288, Springer, Berlin, 2003.
• [20] O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, New York, 2002.
• [21] T. R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10 (1989), 283-295.
• [22] J. M. A. M. van Neerven, M. C. Veraar, and L. W. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab. 35 (2007), 1438-1478.
• [23] A. Osękowski, Strong differential subordination and sharp inequalities for orthogonal processes, J. Theoret. Probab. 22 (2009), 837-855.
• [24] A. Osękowski and I. S. Yaroslavtsev, The Hilbert transform and orthogonal martingales in Banach spaces, Int. Math. Res. Notices (online, 2019).
• [25] G. Pisier, Martingales in Banach Spaces, Cambridge Stud. Adv. Math. 155, Cambridge Univ. Press, 2016.
• [26] P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed., Stochastic Modelling Appl. Probab. 21, Springer, Berlin, 2005.
• [27] J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, in: Probability and Banach Spaces (Zaragoza, 1985), Lecture Notes in Math. 1221, Springer, Berlin, 1986, 195-222.
• [28] M. C. Veraar, Stochastic integration in Banach spaces and applications to parabolic evolution equations, PhD thesis, TU Delft, 2006.
• [29] M. C. Veraar, Continuous local martingales and stochastic integration in UMD Banach spaces, Stochastics 79 (2007), 601-618.
• [30] M. C. Veraar, Randomized UMD Banach spaces and decoupling inequalities for stochastic integrals, Proc. Amer. Math. Soc. 135 (2007), 1477-1486.
• [31] M. C. Veraar and I. S. Yaroslavtsev, Cylindrical continuous martingales and stochastic integration in infinite dimensions, Electron. J. Probab. 21 (2016), art. 59, 53 pp.
• [32] G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522-551.
• [33] I. S. Yaroslavtsev, On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions, Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), 1988-2018.
• [34] I. S. Yaroslavtsev, Burkholder-Davis-Gundy inequalities in UMD Banach spaces, arXiv:1807.05573 (2018).
• [35] I. S. Yaroslavtsev, Even Fourier multipliers and martingale transforms in infinite dimensions, Indag. Math. (N.S.) 29 (2018), 1290-1309.
• [36] I. S. Yaroslavtsev, Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces, Studia Math. 243 (2018), 269-301.
• [37] I. S. Yaroslavtsev, Martingale decompositions and weak differential subordination in UMD Banach spaces, Bernoulli 25 (2019), 1659-1689.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).