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A martingale approach to general Franklin systems

Anna Kamont, Paul F. X. Müller (2006)

Studia Mathematica

We prove unconditionality of general Franklin systems in L p ( X ) , where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

A sharp maximal inequality for continuous martingales and their differential subordinates

Adam Osękowski (2013)

Czechoslovak Mathematical Journal

Assume that X , Y are continuous-path martingales taking values in ν , ν 1 , such that Y is differentially subordinate to X . The paper contains the proof of the maximal inequality sup t 0 | Y t | 1 2 sup t 0 | X t | 1 . The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

Abel means of operator-valued processes

G. Blower (1995)

Studia Mathematica

Let ( X j ) be a sequence of independent identically distributed random operators on a Banach space. We obtain necessary and sufficient conditions for the Abel means of X n . . . X 2 X 1 to belong to Hardy and Lipschitz spaces a.s. We also obtain necessary and sufficient conditions on the Fourier coefficients of random Taylor series with bounded martingale coefficients to belong to Lipschitz and Bergman spaces.

An extension of a boundedness result for singular integral operators

Deniz Karlı (2016)

Colloquium Mathematicae

We study some operators originating from classical Littlewood-Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one-dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are the G* and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on L p . Moreover, we generalize a classical multiplier theorem by weakening its...

Atomic decomposition of predictable martingale Hardy space with variable exponents

Zhiwei Hao (2015)

Czechoslovak Mathematical Journal

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let ( Ω , , ) be a probability space and p ( · ) : Ω ( 0 , ) be a -measurable function such that 0 < inf x Ω p ( x ) sup x Ω p ( x ) < . It is proved that a predictable martingale Hardy space 𝒫 p ( · ) has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with...

BMO and commutators of martingale transforms

Svante Janson (1981)

Annales de l'institut Fourier

The commutator of multiplication by a function and a martingale transform of a certain type is a bounded operator on L p , 1 &lt; p &lt; , if and only if the function belongs to BMO. This is a martingale version of a result by Coifman, Rochberg and Weiss.

Carthaginian enlargement of filtrations

Giorgia Callegaro, Monique Jeanblanc, Behnaz Zargari (2013)

ESAIM: Probability and Statistics

This work is concerned with the theory of initial and progressive enlargements of a reference filtration 𝔽 F with a random timeτ. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an 𝔽 F -martingale in the enlarged filtrations. Also, we address martingales’ characterization in the enlarged filtrations in terms of martingales in the reference filtration, as well as predictable...

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