Related papers: Extremal problems in BMO and VMO involving the Gar…

Let $\mathcal{N}\mathcal{F}$ be the class of smooth non-flat curves near the origin and near infinity previously introduced by the second author and let $\gamma\in\mathcal{N}\mathcal{F}$. We show - via a unifying approach relative to the…

One of the purposes of this paper is to prove that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L^{2,1}(G)\ast L^{2,1}({G})\subseteq L^{2,\infty}({G}). Let {K} be a maximal compact subgroup…

For an open set $\Om \subset \R^2$ let $\lambda(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown…

We prove that all extreme points of the unit ball of a Lipschitz-free space over a compact metric space have finite support. Combined with previous results, this completely characterizes extreme points and implies that all of them are also…

We unify several Bellman function problems into one setting. For that purpose we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in $\mathbb{R}^2$). We show how the unit ball…

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on…

In this paper, we will study the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_\lambda$-function on the generalized Morrey spaces $L^{p,\Phi}$ for…

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the…

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…

Let $f = f(z,t)$ be a function holomorphic in $z \in O \subseteq {\mathbb C}^d$ for fixed $t\in \Omega$ and measurable in $t$ for fixed $z$ and such that$z \mapsto f(z,\cdot)$ is bounded with values in$E := L_{p}(\Omega)$, $1\le p \le…

For an ergodic Brownian diffusion with invariant measure $\nu$, we consider a sequence of empirical distributions ($\nu$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($\gamma$n) n$\ge$1 along an adapted…

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-\alpha/2}$ be the fractional integrals of $L$ for $0<\alpha<n.$ For any locally integrable function $b$, The…

Consider a finite dimensional complex Hilbert space $\cH$, with $dim(\cH) \geq 3$, define $\bS(\cH):= \{x\in \cH \:|\: ||x||=1\}$, and let $\nu_\cH$ be the unique regular Borel positive measure invariant under the action of the unitary…

We first study a $d$-dimensional branching Brownian motion (BBM) among mild Poissonian obstacles, where a random trap field in $\mathbb{R}^d$ is created via a Poisson point process. The trap field consists of balls of fixed radius centered…

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such…

Here we advance the study of boundary the value problem for extremal functions of mean distortion and the associated Teichm\"uller spaces interpolating between the classical examples of extremal quasiconformal mappings, and the more recent…

Given $\alpha \in (0, \infty)$ and $r \in (0, \infty)$, let ${\cal D}_{r, \alpha}$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha^2$. Consider the Poisson point process having uniform intensity density on ${\cal…

We compute the exact $L^2$ operator norm of the Cauchy transform \[ (C_\Omega f)(z)=\frac1\pi\int_\Omega \frac{f(w)}{z-w}\,dA(w) \] on a circular annulus $\Omega=A(r,R)=\{r<|z|<R\}$. Exploiting rotational symmetry and a Fourier mode…

In this note, in particular, we establish the following result: Let $X$ be a real Banach space, $\varphi\in X^*\setminus \{0\}$ and $\psi:X\to {\bf R}$ a Lipschitzian functional with Lipschitz constant equal to $\varphi\|_X^{*}$. Then, we…

We show that the fermion, in the context of a system that comprises many such entities - which, by virtue of the Pauli exclusion principle, possesses a Fermi surface at zero temperature - may itself be thought of as a collection of…