Related papers: Sobolev homeomorphic extensions from two to three …

We show that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if its (un)stable lamination is a projective limit of meridians. The proof is through…

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for…

Let $\Omega\subseteq\mathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(\Omega,\mathcal{R}^2)$ be a homeomorphism between $\Omega$ and $f(\Omega)$. Then there exists a sequence of diffeomorphisms…

Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $\varphi \colon \overline{X} \to \overline{Y}$. We consider the class of Sobolev mappings $f \in W^{1,n} (X, \mathbb{R}^n)$ with a strictly positive…

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak…

We show that there exists a planar Jordan domains $\Omega$ with boundary of Hausdorff dimension $1$ such that, for any conformal maps $\varphi \colon \mathbb D \to \Omega$, any homeomorphic extension of $\varphi$ or $\varphi^{-1}$ to the…

Given $f:\partial (-1,1)^n\to{\mathbb R}$, consider its radial extension $Tf(X):=f(X/\|X\|_{\infty})$, $\forall\, X\in [-1,1]^n\setminus\{0\}$. In "On some questions of topology for $S^1$-valued fractional Sobolev spaces" (RACSAM 2001), the…

We construct whole-space extensions of functions in a fractional Sobolev space of order $s\in (0,1)$ and integrability $p\in (0,\infty)$ on an open set $O$ which vanish in a suitable sense on a portion $D$ of the boundary $\partial O$ of…

Let $\Omega,\Omega'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:\Omega\to\Omega'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_{\Omega}(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak…

Let $L^{m,p}(\R^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p(\R^n)$. Assume that $n< p < \infty$. For $E \subset \R^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in…

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $\Omega$ is a $(1,p)$-extension…

We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in…

Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $\Omega\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(\Omega,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(\Omega)$ are…

Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{\rho_\nu\}_{\nu\in(0,\nu_0)}$ with…

Inspired by questions of convergence in continued fraction theory, Erd\H{o}s, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of M\"obius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with…

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control…

In this article, we study Sobolev homeomorphisms and composition operators on homogeneous Lie groups. We prove that a measurable homeomorphism $\varphi: \Omega \to\widetilde{\Omega}$ belongs to the Sobolev space $L^{1}_{q}(\Omega;…

Every homeomorphism h : X -> Y between planar open sets that belongs to the Sobolev class W^{1,p}(X,Y), 1<p<\infty, can be approximated in the Sobolev norm by diffeomorphisms.

Under a condition of the Calderon type on $\varphi$, we show that a homeomorphism $f$ of finite distortion in $W^{1,\varphi}_{\rm loc}$ and, in particular, $f\in W^{1,q}_{\rm loc}$ for $q>n-1$ in ${\Bbb R}^n$, $n\ge 3$, is a lower…

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to…