Related papers: Removable Sets for H\"older Continuous p(x)-Harmon…

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it…

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic measure which is…

It has been recently understood that the harmonic measure on the boundary $E = \partial \Omega$ of a domain $\Omega$ in $\mathbb{R}^n$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^{n - 1}$ on $E$ if and only…

We prove that bounded harmonic functions of anisotropic fractional Laplacians are H\"older continuous under mild regularity assumptions on the corresponding L\'evy measure. Under some stronger assumptions the Green function, Poisson kernel…

A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a simply connected domain $\Omega\subseteq\mathbb{C}$ is \textit{p-harmonic} if $f$ satisfies the $p$-harmonic equation $\Delta ^pf=0.$ In this paper, we…

Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that $I_H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$ and continuous on $\bar\DD$, we show that $f(T)$ is compact if and only if $f$…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

Let k be an algebraically closed field of characteristic 0. The question of irreducibility of the punctual Hilbert scheme Hilb_d P^n and its Gorenstein locus for various d was studied in [CEVV8, CN9, CN10, CN11]. In this short paper we…

It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these…

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with…

We introduce the concept of F-decomposable systems, well-ordered inverse systems of Hausdorff compacta with fully closed bonding mappings. A continuous mapping between Hausdorff compacta is called fully closed if the intersection of the…

Let $\Omega\subset\mathbb R^{n+1}$ be open and let $E\subset \partial\Omega$ with $0<H^s(E)<\infty$, for some $s\in(n,n+1)$, satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually…

We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where…

We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set. That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in…

In this paper, we study meromorphic functions on a domain $\Omega \subset \mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a…

We study the question of constructive approximation of the harmonic measure $\omega_x^\Omega$ of a connected bounded domain $\Omega$ with respect to a point $x\in\Omega$. In particular, using a new notion of computable harmonic…

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann…

For higher order integral functionals with $p(x)$ growth with respect to the highest order derivative $D^m u$, we prove that $D^m u$ is H\"older continuous on an open subset $\Omega_0 \subset \Omega$ of full Lebesgue- measure, provided that…

Suppose that $p \in (1,\infty]$, $\nu \in [1/2,\infty)$, $\mathcal{S}_\nu = \left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |\phi| < \frac{\pi}{2\nu}\right\}$, where $\phi$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and…

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincare type estimates, Helmholtz type decompositions, regular…