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On Strongly Orthogonal Martingales in Umd Banach Spaces

Yaroslavtsev Ivan S.

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

153--171

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 βp;X is the UMDp constant of X, hp;X is the norm of the Hilbert transform on Lp(R; X), [formula] are the Gaussian decoupling constants.

Parabolic martingales and non-symmetric Fourier multipliers

Bogdan K., Wojciechowski Ł.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 2

241--253

We give a class of Fourier multipliers with non-symmetric symbols and explicit norm bounds on Lp spaces by using the stochastic calculus of Lévy processes and Burkholder–Wang estimates for differentially subordinate martingales.

Vector-Valued Singular Integrals Revisited-with Random Dyadic Cubes

Hytönen T. P.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 3

269--283

The vector-valued T(1) theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a simplified approach to these results is presented, exploiting Nazarov, Treil and Volberg's method of random dyadic cubes, which allows one to circumvent the most subtle parts of the original arguments.

Maximal inequalities for stochastic integrals

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 3

273-287

We find the optimal universal constant Cp (1 < p ≤ ∞) in the following inequality. If X = (Xt)t>o is a martingale and Y = [wzór] for some predictable process H taking values in [-1,1], then E[wzór].