Related papers: Some remarks on Davis inequality for biparameter f…

The classical Davis inequality $\mathbb{E} Mf\simeq \mathbb{E} Sf$, where $(Sf)^2=\sum_{k}\left|f_{k}-f_{k-1}\right|^2$ is the square function and $Mf= \sup_n \left|f_n\right|$ is the maximal function, is true with a universal constant for…

We extend an inequality of Merryfield, valid in the continuous setting, to discrete multiparameter martingales. As a consequence, we obtain the $L^p$ comparison of the maximal function with the square function: \begin{align*} E[(Sf)^p]…

For a Hilbert space valued martingale $(f_n)$ and an adapted sequence of positive random variables $(w_n)$, we show the weighted Davis type inequality \[ \mathbb{E} \Bigl( |f_0| w_0 + \frac{1}{4} \sum_{n=1}^{N} \frac{|df_n|^2}{f^*_n} w_n…

In this note we prove the following good-$\lambda$ inequality, for $r>2$, all $\lambda > 0$, $\delta \in \big(0, \frac{1}{2} \big)$ \[ \nu\big\{ V_r(f) > 3 \lambda ; \mathcal{M}(f) \leq \delta \lambda\big\} \leq 4 \nu\{s(f) > \delta…

We prove that for any martingale with respect to a biparameter atomic filtration satisfying $(F_4)$ condition there is a martingale having the same joint distribution but with respect to the canonical $(F_4)$ filtration. Even in one…

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…

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann…

A version of Littlewood-Paley-Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $I_k = I_k^1 \times I_k^2$ in ${\mathbb{Z}_+ \times \mathbb{Z}_+}$ and a family of functions $f_k$…

Let $f$ be a martingale with values in a uniformly $p$-smooth Banach space and $w$ any positive weight. We show that $\mathbb{E} (f^* \cdot w) \lesssim \mathbb{E}(S_p f \cdot w^*)$, where $\cdot^*$ is the martingale maximal operator and…

We consider a Markov chain X_1, X_2, ..., X_n belonging to a class of iterated random functions, which is "one-step contracting" with respect to some distance d. If f is any separately Lipschitz function with respect to d, we use a well…

The optimal function $f$ satisfying $$ \mathbb{E} |\sum_{1}^n X_i | \ge f(\mathrbb{E}|X_1|,...,\mathbb{E}|X_n|) $$ for every martingale $(X_1,X_1+X_2, ...,\sum_{i=1}^n X_i)$ is shown to be given by $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1}…

This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincar\'e inequality (PI), which is shown to yield filter stability. The proof is based upon a…

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-point…

Relative to the Gaussian measure on $\mathbb{R}^d$, entropy and Fisher information are famously related via Gross' logarithmic Sobolev inequality (LSI). These same functionals also separately satisfy convolution inequalities, as proved by…

Let $\alpha \in (0,2)$ and consider the operator $\sL$ given by \[ \sL f(x)=\int[ f(x+h)-f(x)-1_{(|h|\leq 1)}h\cdot \grad f(x)]\frac{n(x,h)}{|h|^{d+\alpha}} \d h, \] where the term $1_{(|h|\leq 1)}h\cdot \grad f(x)$ is not present when…

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 show that D. L\'{e}pingle's $L_1(\ell_2)$-inequality \begin{equation*} \Big\| \big( \sum_n \mathbb E[f_n | \mathscr F_{n-1}]^2 \big)^{1/2}\Big\|_1 \leq 2\cdot \Big\| \big( \sum_n f_n^2 \big)^{1/2} \Big\|_1, \qquad f_n\in\mathscr F_n,…

The main objective consists in generalizing a well-known It{\^o} formula of J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C,…

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:…

This paper is about elliptic and parabolic partial differential operators with discontinuities in the gradient which are compatible with a Finsler norm in a sense to be made precise. Examples of this type of problems arise in a number of…