Related papers: Lusin approximation for functions of bounded varia…

Let $\Omega$ be a smooth bounded domain in $\mathbb R^n$ and u be a measurable function on $\Omega$ such that $|u(x)|=1$ almost everywhere in $\Omega$. Assume that u belongs to the $B^s_{p,q}(\Omega)$ Besov space. We investigate whether…

We study Lusin-measurable functions with values in locally convex spaces. In particular, the behavior of pointwise limits of sequences of Lusin-measurable functions and exhibit pathological phenomena arising in the nonmetrizable setting.…

Let $U\subseteq\mathbb{R}^d$ be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. We also show…

It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space $\mathcal{X}$, a continuous map $\gamma\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ…

Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, $E$ a separable real Banach space and $p\geq 1$. Given a sequence of functions $f, f_1, f_2,...$ from $T\times E$ to ${\bf R}$, under general assumptions, we prove that, for each…

We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $\Omega \subset \mathbb{R}^n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz…

An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space $\mathbb{R}^d$ of specified Lebesgue measures, balls centered at the origin are maximizers of certain functionals defined by…

The best uniform polynomial approximation of the checkmark function $f(x)=|x-\alpha |$ is considered, as $\alpha$ varies in $(-1,1)$. For each fixed degree $n$, the minimax error $E_n (\alpha)$ is shown to be piecewise analytic in $\alpha$.…

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…

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators.…

A topological space is $Suslin$ ($Lusin$) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space $X$ let $C_p(X)$, $C_k(X)$ and $C_{{\downarrow}F}(X)$ be the space of continuous real-valued functions on $X$,…

Let $\mathrm{Lip}_0(X)$ be the space of all Lipschitz scalar-valued functions on a pointed metric space $X$. We characterize the approximation property for $\mathrm{Lip}_0(X)$ with the bounded weak* topology using as tools the tensor…

Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the…

We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a…

We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…

Suppose $\{X_{t}:t\ge 0\}$ is a supercritical superprocess on a Luzin space $E$, with a non-local branching mechanism and probabilities $\mathbb{P}_{\delta_{x}}$, when initiated from a unit mass at $x\in E$. By ``supercritical", we mean…

We prove that if $\Omega\subseteq\mathbb{R}^N$ is a set with finite perimeter with $\mathscr{H}^{N-1}(\partial \Omega\setminus\partial^* \Omega)=0$, then any set of finite perimeter $E\subseteq\mathbb{R}^N$ can be approximated by a…

We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form \begin{gather*} \min \biggl\{ \displaystyle\int_{\Omega} F(x,Dw)dx \ : \ w \in…

We study generalized Poincar\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we…

We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$…