scholarly journals Inequality for Burkholder’s martingale transform

2015 ◽  
Vol 8 (4) ◽  
pp. 765-806 ◽  
Author(s):  
Paata Ivanisvili
Keyword(s):  
Martingale Transform

scholarly journals Projective Stochastic Equations and Nonlinear Long Memory

2014 ◽  
Vol 46 (4) ◽  
pp. 1084-1105
Author(s):  
Ieva Grublytė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Volterra Series ◽  
Bernoulli Shift ◽  
Moving Average ◽  
Stochastic Equations ◽  
Partial Sums ◽  
Martingale Transform ◽  
New Class ◽  
Moving Average Processes ◽  
Nonlinear Stochastic Equations

A projective moving average {Xt, t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on ‘intermediate’ lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Sharp $L^p$-bounds for a small perturbation of Burkholder's martingale transform

2012 ◽  
Vol 61 (2) ◽  
pp. 751-773 ◽  
Author(s):  
Nicholas Boros ◽  
Alexander Volberg ◽  
Prabhu Janakiraman
Keyword(s):  
Small Perturbation ◽  
Martingale Transform

scholarly journals Perturbation of Burkholder’s martingale transform and Monge–Ampère equation

2012 ◽  
Vol 230 (4-6) ◽  
pp. 2198-2234 ◽  
Author(s):  
Nicholas Boros ◽  
Prabhu Janakiraman ◽  
Alexander Volberg
Keyword(s):  
Martingale Transform ◽  
Monge Ampere

scholarly journals On the Local Tb Theorem: A Direct Proof under the Duality Assumption

2015 ◽  
Vol 59 (1) ◽  
pp. 193-222 ◽  
Author(s):  
Michael T. Lacey ◽  
Antti V. Vähäkangas
Keyword(s):  
Direct Proof ◽  
Principal Point ◽  
Random Grids ◽  
Martingale Transform ◽  
Zygmund Operator ◽  
Point Of Interest ◽  
Flexible Framework ◽  
Euclidean Setting ◽  
Dual Case

AbstractWe give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderón–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1/p + 1/q ⩽ 1, the ‘dual case’ 1/p + 1/q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.

LOGARITHMIC GROWTH FOR MATRIX MARTINGALE TRANSFORMS

2001 ◽  
Vol 64 (3) ◽  
pp. 624-636 ◽  
Author(s):  
T. A. GILLESPIE ◽  
S. POTT ◽  
S. TREIL ◽  
A. VOLBERG
Keyword(s):  
Hankel Operators ◽  
Martingale Transform ◽  
Martingale Transforms ◽  
Logarithmic Growth

An example is given of an operator weight W that satisfies the dyadic operator Hunt–Muckenhoupt–Wheeden condition [Aopf ]d2 for which there exists a dyadic martingale transform on L2 (W) that is unbounded. The construction relates weighted boundedness to the boundedness of dyadic vector Hankel operators.

scholarly journals Boundedness of vector-valued martingale transforms on extreme points and applciations

2004 ◽  
Vol 76 (2) ◽  
pp. 207-222 ◽  
Author(s):  
Teresa Martínez ◽  
José L. Torrea
Keyword(s):  
Hardy Space ◽  
Extreme Points ◽  
Atomic Decompositions ◽  
Martingale Transform ◽  
Nikodym Property ◽  
Dense Sets ◽  
Martingale Transforms ◽  
General Filtration ◽  
Vector Valued ◽  
Martingale Cotype

AbstractLet Β1, Β2be a pair of Banach spaces andTbe a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent:Tis bounded fromintofor somep(or equivalently for everyp) in the range 1 <p< ∞;Tis bounded fromintoBMOB2;Tis bounded fromBMOB1intoBMOB2;Tis bounded frominto. Applications toUMDand martingale cotype properties are given. We also prove that the Hardy spacedefined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.

scholarly journals Martingale Transform and Random Abel-Dini Series

1980 ◽  
Vol 8 (3) ◽  
pp. 475-482 ◽  
Author(s):  
Louis H. Y. Chen
Keyword(s):  
Martingale Transform

scholarly journals Martingale transform and L\'evy processes on Lie groups

2014 ◽  
Vol 63 (4) ◽  
pp. 1109-1138 ◽  
Author(s):  
Rodrigo Banuelos ◽  
David Applebaum
Keyword(s):  
Lie Groups ◽  
Martingale Transform
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