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Parabolic martingales and non-symmetric Fourier multipliers

100%

Bogdan K. , Wojciechowski Ł.

Probability and Mathematical Statistics

2012

tom 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

70%

Hytönen T. P.

tom 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.

On Strongly Orthogonal Martingales in Umd Banach Spaces

51%

Yaroslavtsev I. S.

Probability and Mathematical Statistics

2021

tom 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.