Related papers: Sharp weak-type inequalities for differentially su…

We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if $x$ is a self-adjoint noncommutative martingale and $y$ is weakly differentially subordinate to $x$ then $y$ admits a…

Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and $\mathfrak{c}_p$ such that $$ \,\,\,\,\sup_{t\geq 0}\left|\left|X_t\right|\right|_p\leq…

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:…

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…

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0,1,2,...,\] almost surely, then…

For any two real-valued continuous-path martingales $X=\{X_t\}_{t\geq 0}$ and $Y=\{Y_t\}_{t\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of…

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…

For $p\in\lbrack2,\infty]$ a mixed Littlewood-type inequality asserts that there is a constant $C_{(m),p}\geq1$ such that \[ \left( \sum_{i_{1}=1}^{\infty}\left( \sum_{i_{2},...,i_{m}=1}^{\infty }|T(e_{i_{1}},...,e_{i_{m}})|^{2}\right)…

We prove that non-commutative martingale transforms are of weak type $(1,1)$. More precisely, there is an absolute constant $C$ such that if $\M$ is a semi-finite von Neumann algebra and $(\M_n)_{n=1}^\infty$ is an increasing filtration of…

The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0<p<\infty$, what is the optimal…

We are given two martingales on the filtration of the two dimensional Brownian motion. One is subordinated to another. We want to give an estimate of $L^p$-norm of a subordinated one via the same norm of a dominating one. In this setting…

\small{In 2004, Del Pino and Dolbeault \cite{DPDo} and Gentil \cite{G} investigated, independently, best constants and extremals associated to sharp Euclidean $L^p$-entropy inequalities. In this work, we present some important advances in…

In 2003, Del Pino and Dolbeault [14] and Gentil [19] investigated, independently, best constants and extremals associated to Euclidean Lp-entropy inequalities for p > 1. In this work, we present some contributions in the Riemannian context.…

In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$…

For each 1 < p < infinity, there exists a positive constant c_p, depending only on p, such that the following holds. Let (d_k), (e_k) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences…

In this paper we address the question of finding the best $L^p$-norm constant for martingale transforms with one-sided orthogonality. We consider two martingales on a probability space with filtration $\mathcal{B}$ generated by a…

Let $\{Z_t, t\geq 0\}$ be a strictly stable process on $\R$ with index $\alpha\in (0,2]$. We prove that for every $p > \alpha$, there exists $\gamma = \gamma (\alpha, p)$ and $\k = \k (\alpha, p)\in (0, +\infty)$ such that…

Let $(x_k)_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras…

We find the sharp constant $C=C(\tau,p, \mathbb{E}G, \mathbb{E}F)$ of the following inequality $\|(G^{2}+ \tau^{2} F^{2})^{1/2} \|_{p} \leq C \|F\|_{p},$ where $G$ is the transform of a martingale $F$ under a predictable sequence…

We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p<2$, while it…