Related papers: Absolute continuity between the surface measure an…

We consider averages $\kappa$ of spectral measures of rank one perturbations with respect to a $\sigma$-finite measure $\nu$. It is examined how various degrees of continuity of $\nu$ with respect to $\alpha$-dimensional Hausdorff measures…

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 prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$,…

Let $\mathcal{S}_H^0$ denote the class of all functions $f(z)=h(z)+\overline{g(z)}=z+\sum^\infty_{n=2} a_nz^n +\overline{\sum^\infty_{n=2} b_nz^n}$ that are sense-preserving, harmonic and univalent in the open unit disk $|z|<1$. The…

Fix $c\in (1,23/22)$. Let $\alpha$ and $\beta$ be two distinct non-zero real numbers with $|\alpha|\neq |\beta|$. It is shown that for any measure preserving system $(X,\mathcal{X},\mu,T)$ and any $f,g\in L^{\infty}(\mu)$, the limit…

Let $\Omega\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary…

We investigate the set of (real Dolbeault classes of) balanced metrics $\Theta$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of all such $[\Theta]…

In this note we prove that a regular continuous open image of the Sorgenfrey line with an uncountable weight has a closed subspace that is homeomorphic to the Sorgenfrey line. As a corollary we deduce the theorem in the title.

We show that there are infinitely many pairwise nonhomothetic, complete, periodic metrics with constant scalar curvature that are conformal to the round metric on $S^n\setminus S^k$, where $k < \frac{n-2}{2}$. These metrics are obtained by…

Let the measure algebra of a topological group be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly…

We study continuity properties of Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\Omega, \mathbb{R}^n)$, $n \ge 2$, that satisfy the following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq K(x) J_f(x) + \Sigma (x)\]…

We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally…

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. In this article we present results for harmonic functions on…

A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal…

We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular the Green function in $R^2 \backslash K$…

For a metrizable space $X$ and a finite measure space $(\Omega,\mathfrak{M},\mu)$ let $M_{\mu}(X)$ and $M^f_{\mu}(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $\mu$) of…

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a…

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system $$|qx-p|< \, \psi(t) \ \…

Suppose $ F $ is an integrand associated with a uniformly convex $ \mathscr{C}^{3} $-norm, and $ V $ is a $ n $-dimensional varifold in an open subset of $ \mathbf{R}^{n+1} $ such that $ \mathscr{H}^n \llcorner \operatorname{spt} \| V \| $…

We prove that any continuous mapping $f:E\to Y$ on a completely metrizable subspace $E$ of a perfect paracompact space $X$ can be extended to a Lebesgue class one mapping $g:X\to Y$ (i.e. for every open set $V$ in $Y$ the preimage…