UNIVERSITY
OF WROC£AW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
44.1 43.2 43.1 42.2 42.1 41.2 41.1
40.2 40.1 39.2 39.1 38.2 38.1 37.2
37.1 36.2 36.1 35.2 35.1 34.2 34.1
33.2 33.1 32.2 32.1 31.2 31.1 30.2
30.1 29.2 29.1 28.2 28.1 27.2 27.1
26.2 26.1 25.2 25.1 24.2 24.1 23.2
23.1 22.2 22.1 21.2 21.1 20.2 20.1
19.2 19.1 18.2 18.1 17.2 17.1 16.2
16.1 15 14.2 14.1 13.2 13.1 12.2
12.1 11.2 11.1 10.2 10.1 9.2 9.1
8 7.2 7.1 6.2 6.1 5.2 5.1
4.2 4.1 3.2 3.1 2.2 2.1 1.2
1.1
 
 
WROC£AW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 41, Fasc. 1,
pages 153 - 171
DOI: 10.37190/0208-4147.41.1.10
Published online 22.4.2021
 

On strongly orthogonal martingales in UMD Banach spaces

Ivan S. Yaroslavtsev

Abstract: 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 \(1<p<\infty\), \[\mathbb E \|N_t\|^p \leq \chi_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0,\] with the sharp constant \(\chi_{p, X}\) being the norm of a decoupling-type martingale transform and lying in the range \[\begin{aligned} \textstyle\max\Bigl\{\sqrt{\beta_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} &\leq \max\{\beta_{p, X}^{\gamma,+}, \beta_{p, X}^{\gamma, -}\}\\& \leq \chi_{p, X} \leq \min\{\beta_{p, X}, \hbar_{p,X}\},\end{aligned}\] where \(\beta_{p, X}\) is the UMD\(_p\) constant of \(X\), \(\hbar_{p, X}\) is the norm of the Hilbert transform on \(L^p(\mathbb R; X)\), and \(\beta_{p, X}^{\gamma,+}\) and \(\beta_{p, X}^{\gamma, -}\) are the Gaussian decoupling constants.

2010 AMS Mathematics Subject Classification: Primary 60G44, 60H05; Secondary 60B11, 32U05.

Keywords and phrases: strongly orthogonal martingales, weak differential subordination, UMD, sharp estimates, decoupling constant, martingale transform, Hilbert transform, diagonally plurisubharmonic function.

R. Bañuelos, A. Bielaszewski, and K. Bogdan, Fourier multipliers for non-symmetric Levy processes, in: Marcinkiewicz Centenary Volume, Banach Center Publ. 95, Inst. Math. Polish Acad. Sci., Warszawa, 2011, 9-25.

R. Bañuelos and K. Bogdan, Levy processes and Fourier multipliers, J. Funct. Anal., 250(1):197-213, 2007.

R. Bañuelos and A. Osêkowski, Martingales and sharp bounds for Fourier multipliers, Ann. Acad. Sci. Fenn. Math., 37(1):251-263, 2012.

R. Bañuelos and G. Wang, Orthogonal martingales under differential subordination and applications to Riesz transforms, Illinois J. Math., 40(4):678-691, 1996.

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21(2):163-168, 1983.

J. Bourgain, On martingales transforms in finite-dimensional lattices with an appendix on the K-convexity constant, Math. Nachr., 119():41-53, 1984.

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.

D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab., 12(3):647-702, 1984.

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.

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.

S. G. Cox and S. Geiss, On decoupling in Banach spaces, arXiv:1805.12377 (2018).

C. Dellacherie and P.-A. Meyer, Probabilities and Potential. B, North-Holland Math. Stud. 72, North-Holland, Amsterdam, 1982.

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.

S. Geiss, A counterexample concerning the relation between decoupling constants and UMD-constants, Trans. Amer. Math. Soc., 351(4):1355-1375, 1999.

S. Geiss, S. Montgomery-Smith, and E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc., 362(2):553-575, 2010.

S. Geiss and I. S. Yaroslavtsev, Dyadic and stochastic shifts and Volterra-type operators, in preparation.

B. Hollenbeck, N. J. Kalton, and I. E. Verbitsky, Best constants for some operators associated with the Fourier and Hilbert transforms, Studia Math., 157(3):237-278, 2003.

T. P. Hytonen, 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.

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren Math. Wiss. 288, Springer, Berlin, 2003.

O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, New York, 2002.

T. R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist., 10(2):283-295, 1989.

J. M. A. M. van Neerven, M. C. Veraar, and L. W. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab., 35(4):1438-1478, 2007.

A. Osêkowski, Strong differential subordination and sharp inequalities for orthogonal processes, J. Theoret. Probab., 22(4):837-855, 2009.

A. Osêkowski and I. S. Yaroslavtsev, The Hilbert transform and orthogonal martingales in Banach spaces, Int. Math. Res. Notices (online, 2019).

G. Pisier, Martingales in Banach Spaces, Cambridge Stud. Adv. Math. 155, Cambridge Univ. Press, 2016.

P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed., Stochastic Modelling Appl. Probab. 21, Springer, Berlin, 2005.

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.

M. C. Veraar, Stochastic integration in Banach spaces and applications to parabolic evolution equations, PhD thesis, TU Delft, 2006.

M. C. Veraar, Continuous local martingales and stochastic integration in UMD Banach spaces, Stochastics, 79(6):601-618, 2007.

M. C. Veraar, Randomized UMD Banach spaces and decoupling inequalities for stochastic integrals, Proc. Amer. Math. Soc., 135(5):1477-1486, 2007.

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.

G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab., 23(2):522-551, 1995.

I. S. Yaroslavtsev, On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions, Ann. Inst. H. Poincare Probab. Statist., 55(4): 1988-2018, 2019.

I. S. Yaroslavtsev, Burkholder–Davis–Gundy inequalities in UMD Banach spaces, arXiv:1807.05573 (2018).

I. S. Yaroslavtsev, Even Fourier multipliers and martingale transforms in infinite dimensions, Indag. Math. (N.S.), 29(5):1290-1309, 2018.

I. S. Yaroslavtsev, Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces, Studia Math., 243(3):269-301, 2018.

I. S. Yaroslavtsev, Martingale decompositions and weak differential subordination in UMD Banach spaces, Bernoulli, 25(3):1659-689, 2019.

Download:    Abstract    Full text