Related papers: Smooth approximation of bi-Lipschitz orientation-p…

We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved. This answers a question raised by Daneri and Pratelli.…

Linearly repetitive Delone sets are shown to be rectifiable by a bi-Lipschitz homeomorphisms of the Euclidean space that sends the Delone set to the set of points with integer coordinates.

We show that homeomorphisms $f$ in ${\Bbb R}^n$, $n\geqslant3$, of finite distortion in the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with a condition on $\varphi$ of the Calderon type and, in particular, in the Sobolev classes…

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular…

We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\to\Omega$ between Lyapunov domains (equivalently, bounded $C^{1,\alpha}$ domains) in $\mathbb{R}^n$, $\alpha\in(0,1]$, is globally Lipschitz on…

We characterize maps between $n$-dimensional N\"obeling manifolds that can be approximated by homeomorphisms.

We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded…

We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for…

This article considers the Lipschitz space with mixed logarithmic smoothness of $2\pi$ periodic functions of several variables. We obtain equivalent descriptions of the norm of the Lipschitz space and prove embedding theorems between Besov…

For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$, the properties are studied of fields of `good directions', that is the directions with respect to which $\partial\Omega$ can be locally represented as the graph of a…

We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of…

We establish that every $K$-quasiconformal mapping of $w$ of the unit disk $\ID$ onto a $C^2$-Jordan domain $\Omega$ is Lipschitz provided that $\Delta w\in L^p(\ID)$ for some $p>2$. We also prove that if in this situation $K\to 1$ with…

For some smooth special case of generalized $\varphi-$divergences as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) $\ell_{1}-$distance and (weighted) $\ell_{1}-$norm.

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces $W^r_p$ of mixed smoothness $r$, established estimates of the approximation error…

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the…

Let $(\Omega, \mathfrak{A}, \mu)$ and $(\Gamma, \mathfrak{B}, \nu)$ be two arbitrary measure spaces, and $p\in [1,\infty]$. Set $$L^p(\mu)_+^\mathrm{sp}:= \{f\in L^p(\mu): \|f\|_p =1; f\geq 0\ \mu\text{-a.e.} \}$$ i.e., the positive part of…

We introduce a generalized version of the local Lipschitz number $\textrm{lip}\,u$, and show that it can be used to characterize Sobolev functions $u\in W_{\textrm{loc}}^{1,p}(\mathbb R^n)$, $1\le p\le \infty$, as well as functions of…

First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$…

In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable…

We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is…