Related papers: How to Recognise Extension domains

For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $\Omega\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a…

Let $\Omega \subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $\Omega$ maps $L^{p}(\Omega) \to W^{1,p}(\Omega)$ for all $p > 1$, when the smoothness…

We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that…

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) $. It is shown that if the outward unit normal vector $N$ of the…

We show that a bounded planar simply connected domain $\Omega$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $\Omega^c$ there exists a curve $\gamma \subset \Omega^c$ connecting $x$ and $y$ with $$…

For each $p>n$ we use local oscillations to give intrinsic characterizations of the trace of the Sobolev space $W^1_p(\Omega)$ to the boundary of an arbitrary domain $\Omega\subset R^n$.

Let $\Omega\subset R^n$ be a bounded convex domain with $n\ge2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\infty$, we build up a global second order…

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…

This paper is devoted to the study of fractional (q,p)-Sobolev-Poincare inequalities in irregular domains. In particular, we establish (essentially) sharp fractional (q,p)-Sobolev-Poincare inequality in s-John domains and in domains…

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right), \end{align*} where $N\geq1$, $0<s<1$, $p=1,2$, $1\leq…

Let $\Omega\subset\mathbb{R}^n$ be an $(\epsilon,\delta,D)$-domain, with $\epsilon\in(0,1]$, $\delta\in(0,\infty]$, and $D\subset \partial \Omega$ being a closed part of $\partial \Omega$, which is a general open connected set when…

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…

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ \lambda_{s,p}(\Omega):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(\Omega),\,\, \|u\|_{L^p(\Omega)}=1 \right\}, $$…

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, be an open set with $s$-Ahlfors regular boundary $\partial \Omega$, for some $s \in(0,n]$, such that either $s=n$ and $\Omega$ is a corkscrew domain with the pointwise John condition, or…

Let $\Omega \subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(\Omega)^2$ for every $f\in L^p_0(\Omega)$, if…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in…

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p\colon\overline{\Omega}\times \overline{\Omega}\to (1,\infty)$ and $q:\partial \Omega \rightarrow (1,\infty)$ are…

Let $\Omega\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $\Omega$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partial\Omega$ has a holomorphic extension to…