Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(mathbb{R}^{n})$-spaces to arbitrary compact subsets of $mathbb{R}^{n}$. The case $p in (1,n]$

Let $S subset mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $mathcal{H}^{d}_{infty}(S) > 0$ for some $d in (0,n]$. For each $p in (max{1,n-d},n]$ an almost sharp intrinsic description of the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(mathbb{R}^{n})$ is given. Furthermore, for each $p in (max{1,n-d},n]$ and $varepsilon in (0, min{p-(n-d),p-1})$ new bounded linear extension operators from the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ into the space $W_{p-varepsilon}^{1}(mathbb{R}^{n})$ are constructed.

On some geometric properties of sets related to the $d$-Hausorff content and new Whitney-type coverings

Let $S subset mathbb{R}^{n}$ be a~$d$-thick set for some $d in (0,n)$, i.e., there exists a number $lambda in (0,1]$ such that the~$d$-Hausdorff content $mathcal{H}^{d}_{infty}(Q cap S) geq lambda (l(Q))^{d}$ for all cubes~$Q subset mathbb{R}^{n}$ centered at~$x in S$ with side lengths $l(Q) in (0,1]$. Given a cube $Q=Q(x,l)$ with $x in mathbb{R}^{n}$ and $l in (0,1]$, we show that if the $d$-Hausdorff content $mathcal{H}^{d}_{infty}(Q cap S)$ is sufficiently small then the cube $Q$ is $S$-porous, i.e., there is a cube $Q subset Q setminus S$ with $l(Q) approx l(Q)$. In the case of an arbitrary set $E subset mathbb{R}^{n}$, given a cube $Q=Q(x,l)$ with $x in mathbb{R}^{n}$ and $l in (0,1]$ we show that if the $d$-Hausdorff content $mathcal{H}^{d}_{infty}(Q cap E)$ is sufficiently small then the cube $Q$ is $S$-hollow i.e., there exists a Borel set $U subset Q setminus E$ with $mathcal{H}^{n}(U) approx mathcal{H}^{n}(Q)$. Furthermore, we introduce the special $(d,lambda)$-thick distance $rho_{E,d,lambda}$ and show that $U$ can be chosen to lie in the complement of the $delta l$-neighborhood of $E$ for some $delta=delta(n,d,lambda) in (0,1)$. As an application we obtain new Whitney-type coverings. The sharpness of the results is illustrated by several examples.

Restrictions of Sobolev $W_{p}^{1}(mathbb{R}^{2})$-spaces to planar rectifiable curves

We construct explicit examples of Frostman-type measures concentrated on arbitrary planar rectifiable curves of positive length. Based on such constructions we obtain for each $p in (1,infty)$ an exact description of the trace space of the first-order Sobolev space $W^{1}_{p}(mathbb{R}^{2})$ to an arbitrary planar rectifiable curve $Gamma subset mathbb{R}^{2}$ of positive length.