Related papers: Noncommutative Burkholder/Rosenthal inequalities a…
This paper is devoted to the study of $\Phi$-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative $\Phi$-moment analogues of martingale transformations, Stein's inequalities, Khintchine's…
We report recent advances on noncommutative martingale inequalities associated with convex functions. These include noncommutative Burkholder-Gundy inequalities associated with convex functions due to the present authors and Dirksen and…
This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. In particular, we…
In this paper we study Johnson-Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space…
We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\leq 1$. A new feature of our Davis decomposition is a simultaneous control of $\H_1^c$ and $\H_q^c$ norms for any noncommutative martingale in…
We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, noncommutative Dunford-Schwartz and Stein…
We provide generalizations of Burkholder's inequalities involving conditioned square functions of martingales to the general context of martingales in noncommutative symmetric spaces. More precisely, we prove that Burkholder's inequalities…
We establish the weighted fractional Orlicz-Hardy inequalities for various Orlicz functions. Further, we identify the critical cases for each Orlicz function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic…
We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish…
We prove a weak-type (1, 1) inequality involving conditioned versions of square functions for martingales in noncommutative $L^p$-spaces associated with finite von Neumann algebras. As application, we determine the optimal orders for the…
We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb R}^n)$. In all the cases, the conditions for the boundedness are given…
We study the boundedness of intrinsic square functions and their commutators on generalized Orlicz-Morrey spaces $M^{\Phi,\varphi}(\mathbb{R}^n)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type…
We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds:…
We propose a novel approach in noncommutative probability, which can be regarded as an analogue of good-$\lambda$ inequalities from the classical case due to Burkholder and Gundy (Acta Math {\bf124}: 249-304,1970). This resolves a…
We investigate Sobolev spaces $W^{1,\Phi}$ associated to Musielak-Orlicz spaces $L^\Phi$. We first present conditions for the boundedness of the Voltera operator in $L^\Phi$. Employing this, we provide necessary and sufficient conditions…
This paper is devoted to the study of $\Phi$-moments of sums of independent/freely independent random variables. More precisely, let $(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let…
In this paper, we obtain several inequalities of Ostrowski type that the absolute values of n-time differntiable functions are convex.
For displacement convex functionals in the probability space equip\-ped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type \L oja\-sie\-wicz inequalities. \chg{We also discuss the more…
We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic…
We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-\sigma)}(x_{0})}\,\varPhi(x, |\nabla(u-k)_{\pm}|)\,dx \leqslant…