We determine the optimal constants $C_{p,q}$ in the moment inequalities ${\left|\right|g\left|\right|}_p\le C_{p,q}{\left|\right|f\left|\right|}_q$, 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

We extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in $L_p$, p > -2.

Let f be a conditionally symmetric martingale and let S(f) denote its square function. (i) For p,q > 0, we determine the best constants $C_{p,q}$ such that $su{p}_n{\left(\right|fₙ|}^p{)/(1+Sₙ²\left(f\right))}^q\le C_{p,q}$. Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants $C_{N,q}^{\text{'}}$ such that $su{p}_n\left(f{ₙ}^{2N-1}\right){(1+Sₙ²\left(f\right))}^q\le C_{N,q}^{\text{'}}$. These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if...

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${\left|\right|X\left|\right|}_{BMO}\equiv su{p}_T\left|\right|E\left[\right|X_{\infty }-X_T\left|\right|{ℱ}_T{\left]\right||}_{\infty }<\infty$, where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b\left(X\right)=b>0:su{p}_T\left|\right|E\left[exp\left(b²(⟨X{⟩}_{\infty }-⟨X{⟩}_T)\right)\right|{ℱ}_T]{\left|\right|}_{\infty }<\infty$, where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q\left(X\right)ₜ=E\left[⟨X{⟩}_{\infty }\right|ℱₜ]-E\left[⟨X{⟩}_{\infty }\right|ℱ₀]$. We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty }$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1/4d₁(q\left(X\right),L^{\infty })\le b\left(X\right)\le 4/d₁(q\left(X\right),L^{\infty })$ hold for every continuous BMO-martingale X. ...

Consider the sequence ${\left(Cₙ\right)}_{n\ge 1}$ of positive numbers defined by C₁ = 1 and $C_{n+1}=1+Cₙ²/4$, n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For $f={\left(fₙ\right)}_{n\in \mathbb{Z}₊}\in ℳ$, let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and ${\left|\right|Mg\left|\right|}_X{\le \left|\right|Mf\left|\right|}_X$, then ${\left|\right|Sg\left|\right|}_X{\le C\left|\right|Sf\left|\right|}_X$, where C is a constant independent of f and g.

We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than $-1$. Without symmetry assumption, it holds if and only if the parameter is in the interval $(-1,0]$. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of...

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^2$-approximation of a probability density function $f$. If $f$ itself is a step-function the number of jumps may be unknown.

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(\Omega ,ℱ,\mathbb{P})$ be a probability space and $p(·):\Omega \to (0,\infty )$ be a $ℱ$-measurable function such that $0<{inf}_{x\in \Omega }p\left(x\right)\le {sup}_{x\in \Omega }p\left(x\right)<\infty$. It is proved that a predictable martingale Hardy space ${𝒫}_{p(·)}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy...

For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf\left(x\right)=1/\left|B\right(0,\left|x\right|\left)\right|{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$, $Tf\left(x\right)=1/\left|B\right(x,\left|x\right|\left)\right|{\int }_{B(x,|x\left|\right)}f\left(t\right)dt$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

The purpose of this work is a study of the following insurance reserve model: $R\left(t\right)=\eta +{\int }_0^{t}p(s,R\left(s\right))ds+{\int }_0^t\sigma (s,R\left(s\right))d{W}_s-Z\left(t\right)$, t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: $in{f}_{0\le t\le T}PR\left(t\right)\ge c\ge \gamma$ is considered.

We consider the operator $-d^2/d{r}^2-V$ in $L_2\left({ℝ}_{+}\right)$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $\mathrm{tr}{(-d^2/d{r}^2-V)}_{-}^{\gamma }\le C_{\gamma ,\alpha }{\int }_{{ℝ}_{+}}{(V\left(r\right)-1/\left(4r^2\right))}_{+}^{\gamma +(1+\alpha )/2}{r}^{\alpha }dr$ for any $\alpha \in [0,1)$ and $\gamma \ge (1-\alpha )/2$. This includes a Lieb-Thirring inequality in the critical endpoint case.

We establish the following sharp local estimate for the family ${R_j}_{j=1}^d$ of Riesz transforms on ${ℝ}^d$. For any Borel subset A of ${ℝ}^d$ and any function $f:{ℝ}^d\to ℝ$, ${\int }_A|R_{j}f\left(x\right)|dx\le C_p{\left|\right|f\left|\right|}_{L^p\left({ℝ}^d\right)}{\left|A\right|}^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_p={[2^{q+2}\Gamma (q+1)/{\pi }^{q+1}{\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^{q+1}]}^{1/q}$, 1 < p < 2, and $C_p={[4\Gamma (q+1)/{\pi }^q{\sum }_{k=0}^{\infty }1/{(2k+1)}^q]}^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $\parallel {\sum }_{k=0}^{\infty }d{f}_k{\parallel }_p\le C_p\parallel ({\sum }_{k=0}^{\infty }\left(\right|d{f}_k\left|²\right|{ℱ}_{k-1}{\left)\right)}^{1/2}{\parallel }_p+\parallel ({\sum }_{k=0}^{\infty }|d{f}_k{{|}^p)}^{1/p}{\parallel }_p,$ with $C_p=O(p/lnp)$ as p → ∞.

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.

Let $B_p$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^p$-norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p\left(x\right)}$-norm inequalities of the Hardy operator.

Let $u(x,y)$ be a harmonic function in the half-plane ${\mathbf{R}}_{+}^{n+1}$, $n\ge 2$. We define a family of functionals $D(u;r),-\infty \&gt;r\&gt;\infty$, that are analogs of the family of local times associated to the process $u(x_t,y_t)$ where $(x_t,y_t)$ is Brownian motion in ${\mathbf{R}}_{+}^{n+1}$. We show that $D\left(u\right)={sup}_{r}D(u;r)$ is bounded in $L^p$ if and only if $u(x,y)$ belongs to $H^p$, an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.

We present new inequalities of $L_p$ norms for sums of positive functions. These inequalities are useful for investigation of convergence of simple partial fractions in $L_p\left(ℝ\right)$.

Let $X={\sum }_{i=1}^k{a}_i{U}_i$, $Y={\sum }_{i=1}^k{b}_i{U}_i$, where the $U_i$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_i,b_i$ are real constants. We prove that if $b{²}_i$ is majorized by $a{²}_i$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^p$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $\alpha$ with drift and diffusion coefficients $b$, $\sigma$. When $\alpha \in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha \in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha \in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our...