We give a short summary of Varopoulos’ generalised Hardy-LittlewoodSobolev inequality for self-adjoint C0 semigroups and give a new probabilistic representation of the classical fractional integral operators on Rn as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-LittlewoodSobolev inequality based on Burkholder-Gundy and Doob’s inequalities...

During the Sassanid Empire in Persia (226-652 AD), there was a renaissance of humanistic sciences, including medicine, in the city of Gondi-Shapur. When the Islamic center of power moved to Baghdad in about 750 AD, physicians of Gondi-Shapur, including the dean of the medical school (a Nestorian Christian), gradually moved to Baghdad constructing hospitals and medical schools. Aided by the Pers...

We deene stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of It^ o's predictable representation theorem, as well as product form and functional form of It^ o's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener s...

In this paper, after generalizing the pathwise Burkholder-DavisGundy (BDG) inequalities from discrete time to cadlag semimartingales, we present several applications of the pathwise inequalities. In particular we show that they allow to extend the classical BDG inequalities (1) to the Bessel process of order α ≥ 1 (2) to the case of a random exponent p (3) to martingales stopped at a time τ whi...

This note deals with the question: what remains of the Burkholder-Davis-Gundy inequalities when stopping times T are replaced by arbitrary random times ρ? We prove that these inequalities still hold when T is a pseudo-stopping time and never holds for ends of predictable sets.

The history of Arabian medicine can only be studied satisfactorily in conjunction with the general history of Islam, which began to assume political importance in AD 622. In that year, the prophet Mohammed united the warring tribes of Arabia through a common religious and social ideal. This vision enabled the Arabs to conquer half of the known world and to build an empire destined to replace th...