Related papers: Quasiconformal maps, analytic capacity, and non li…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior.…

This paper deals with $(K_1, K_2)$-quasiregular mappings. It is shown, by Morrey's Lemma and isoperimetric inequality, that every $(K_1, K_2)$-quasiregular mapping satisfies a H\"older condition with exponent $\alpha$ on compact subsets of…

It is shown that for any sufficiently regular even Minkowski valuation $\Phi$ which is homogeneous and intertwines rigid motions, and for any convex body $K$ in a smooth neighborhood of the unit ball, there exists a sequence of positive…

For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of…

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is $k^{th}$-stratum…

A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is…

We show that for any dimension t>2(1+alpha K)/(1+K) there exists a compact set E of dimension t and a function alpha-Holder continuous on the plane, which is K-quasiregular only outside of E. To do this, we construct an explicit…

We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincar\'e inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincar\'e…

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of…

We show that if $K$ is a compact metrizable space with finitely many accumulation points, then the closed unit ball of $C(K)$ is a plastic metric space, which means that any non-expansive bijection from $B_{C(K)}$ onto itself is in fact an…

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the…

In this paper, we first prove that a compact K\"ahler manifold is projective if it satisfies certain quasi-positive curvature conditions, including quasi-positive $S_2^\perp,\, S_2^+,\,\mbox{Ric}_3^\perp, \,\mbox{Ric}_3^+$ or…

Let $L:[0,1]\setminus\{d\}\rightarrow [0,1]$ be a one-dimensional Lorenz like expanding map ($d$ is the point of discontinuity), $\mathcal{P}=\{ (0,d),(d,1) \}$ be a partition of $[0,1]$ and $C^{\alpha}([0,1],\mathcal{P})$ the set of…

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a…

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and…

Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to…

Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Gr\"unbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$ where $H^+$ is a half-space of boundary $H$.…

We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space. This approach is applicable even if the space does not support any Poincar\'e inequalities. As an application of the…

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n, p\geq2$, $\Gamma$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is…