Related papers: On Sobolev extension domains in $R^n$

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},\omega)$, where $\omega$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $\alpha\to1^-$ of Marchaud fractional derivatives of order $\alpha$.…

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $\Omega\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(\Omega, V)$ with the so-called…

Estimates for Kolmogorov and linear widths of some weighted Sobolev classes on a John domain are obtained.

A boundary behavior of closed open discrete mappings of Sobolev and Orlicz--Sobolev classes in ${\Bbb R}^n,$ $n\ge 3,$ is studied. It is proved that, mappings mentioned above have a continuous extension to boundary point $x_0$ of a domain…

In analogy with the definition of ``extended Sobolev scale" on $\mathbb{R}^n$ by Mikhailets and Murach, working in the setting of the lattice $\mathbb{Z}^n$, we define the ``extended Sobolev scale" $H^{\varphi}(\mathbb{Z}^n)$, where…

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…

We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on the geometry near their boundaries. We use this quantitative…

Let $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(\Omega,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For…

We construct and investigate the properties of tempered ultradistribution spaces in Sobolev spaces. A new Sobolev space preserving the original properties and condition whose derivatives are linear continuous operators embedding in $L^p$…

We consider integrals in the sense of Choquet with respect to the $\delta$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<\delta\le n$. In…

In this paper order estimates for the Kolmogorov widths of weighted Sobolev classes on a multi-dimensional domain are obtained. The classes are defined by conditions on the highest-order derivatives and the derivative of order zero.

We prove the solvability in Sobolev spaces $W^{1,2}_p$, $p>d+1$, of the terminal-boundary value problem for a class of fully nonlinear parabolic equations, including parabolic Bellman's equations, in bounded cylindrical domains with VMO…

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincar\'e inequality. For each $\theta \in [0,p)$, we…

We collect and extend results on the limit of $\sigma^{1-k}(1-\sigma)^k |v|_{l+\sigma,p,\Omega}^p$ as $\sigma$ tends to $0^+$ or $1^-$, where $\Omega$ is $\mathbb{R}^n$ or a smooth bounded domain, $k$ is 0 or 1, $l$ is a nonnegative…

We study extensions of sets and functions in general metric measure spaces. We show that an open set has the strong BV extension property if and only if it has the strong extension property for sets of finite perimeter. We also prove…

Let $\Lambda \subset R$ be a strictly increasing sequence. For $r = 1,2$, we give a simple explicit expression for an equivalent norm on the trace spaces $W_p^r(R)|_\Lambda$, $L_p^r(R)|_\Lambda$ of the non-homogeneous and homogeneous…

We extend the definition of weak and strong convergence to sequences of Sobolev-functions whose underlying domains themselves are converging. In contrast to previous works, we do so without ever assuming any sort of reference configuration.…

We introduce the non-homogeneous analogs of Van Schaftingen's classes. We show that these classes refine the embedding $W^{1,n}\subset bmo$. The analogous results established on bounded Lipschitz domains and Riemannian manifolds with…

The paper deals with the operator $u\rightarrow gu$ defined in the Sobolev space $W^{r,p}(\Omega)$ and which takes values in $L^p(\Omega)$ when $\Omega$ is an unbounded open subset in $R^n$. The functions $g$ belong to wider spaces of $L^p$…

There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one…