Related papers: On a problem of Talagrand concerning separately co…

Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.

Let $X$ be a Polish space. We prove that the generic compact set $K\subseteq X$ (in the sense of Baire category) is either finite or there is a continuous gauge function $h$ such that $0<\mathcal{H}^{h}(K)<\infty$, where $\mathcal{H}^h$…

Given a closed (and non necessarily compact) basic semi-algebraic set $K\subseteq R^n$, we solve the $K$-moment problem for continuous linear functionals. Namely, we introduce a weighted $\ell_1$-norm $\ell_w$ on $R[x]$, and show that the…

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$.…

A set $E$ in a Banach space $X$ is compactivorous if for every compact set $K$ in $X$ there is a nonempty, (relatively) open subset of $K$ which can be translated into $E$. In a separable Banach space, this is a sufficient condition which…

Let $K$ be a quadratic number field of discriminant $\Delta_K$, let $E$ be a $\mathbb Q$-curve without CM completely defined over $K$ and let $\omega_E$ be an invariant differential on $E$. Let $L(E,s)$ be the $L$-function of $E$. In this…

Consider a finite collection $\{T_1, \ldots, T_J\}$ of differential operators with constant coefficients on $\mathbb{T}^2$ and the space of smooth functions generated by this collection, namely, the space of functions $f$ such that $T_j f…

In this note we establish the weak unique continuation theorem for caloric functions on compact $RCD(K,2)$ spaces and show that there exists an $RCD(K,4)$ space on which there exist non-trivial eigenfunctions of the Laplacian and…

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This…

This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to Talagrand (\cite{TalL1L2}). As an application, the main result of this note is a…

In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($\delta$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at…

We consider the following nonlinear singular elliptic equation $$-{div} (|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\la g(x) \quad{in} \RR^N,$$ where $g$ belongs to an appropriate weighted Sobolev space, and $p$ denotes the…

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$, and every continuous function on $K$ is locally a…

To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $\lambda_K: \hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if…

Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a…

We construct a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ which is a solution to $\Div (A \nabla u)=0$ in the sense of distributions, where $A$ is continuous and $u \not \in W^{1,p}_{\mathrm{loc}} (B(0,1))$ for $p > 1$. We also give a…

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space $B_1(X)$ of all Baire-one…

We answer a question of Banakh, Jab\l{}o\'nska and Jab\l{}o\'nski by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is…

Let A and B be $C^*$-algebras, A separable, and B $\sigma$-unital and stable. It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy…

This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The…