Related papers: Martingale Type, the Gamlen-Gaudet Construction an…
In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak type and strong type inequalities on Doob's maximal operator and…
Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $\Delta^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that,…
Let $G$ be a locally compact, compactly generated group of polynomial growth and let $\omega$ be a weight on $G$. Under proper assumptions on the weight $\omega$, the Banach space $L^p(G,\omega)$ is a Banach \ast-algebra. In this paper we…
The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then…
This work is concerned with the theory of initial and progressive enlargements of a reference filtration F with a random time {\tau}. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of…
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the…
The purpose of this paper is to introduce $\omega$-Chebyshev-greedy and $\omega$-partially greedy approximation classes and to study their relation with $\omega$-approximation spaces, where the latter are a generalization of the classical…
We introduce the mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}})(\mathbb{R}^n)$ and $(L^{\vec{p}},L^{\vec{s}})^{\alpha}(\mathbb{R}^n)$, and show their some basic properties. In addition, we find the predual…
This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $\alpha\in (0,1/2)$. Suppose that $|X\pmod p|\leq (\alpha+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the…
We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators $\Gamma$ on a Hilbert space. Our assumption on $\Gamma$ is expressed in terms of…
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $\mathcal{F}_{\infty}$ (the $\sigma$-algebra generated by…
A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space…
Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied…
We deal with a real valued integral operator L of Laplace transformation type acting between Lebesgue spaces on the semi-axis. Sufficient conditions for belonging L to Schatten type classes are obtained. Some upper asymptotic estimates for…
In this chapter, we study Information Geometry from a particular non-parametric or functional point of view. The basic model is a probabilities subset usually specified by regularity conditions. For example, probability measures mutually…
A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration. We study and implement a particular type of…
The notion of a \textbf{$\boldsymbol{\mathcal{C}}$-filtered} object, where $\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture…
For a given continuous function $g:~\Omega\rightarrow\mathbb{C}$, we establish some Schwarz type Lemmas for mappings $f$ in $\Omega$ satisfying the {\rm PDE}: $\Delta f=g$, where $\Omega$ is a subset of the complex plane $\mathbb{C}$. Then…
In [Y.~K.~Hu, K.~A.~Kopotun, X.~M.~Yu, Constr. Approx. 2000], the authors have obtained a characterization of best $n$-term piecewise polynomial approximation spaces as real interpolation spaces between $L^p$ and some spaces of bounded…
We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, $L^p$ data and generators satisfying weak integrability conditions. We deal with equations…