Speaker
Description
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 $p \in (\max\{1,n-d\},n]$ we give an almost sharp intrinsic description of the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$. Furthermore, for each$\varepsilon \in (0, \min\{p-(n-d),p-1\})$ we construct a new bounded linear extension operator $\mathrm{Ext}_{S,d,\varepsilon}$ mapping the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ to the space $W_{p-\varepsilon}^{1}(\mathbb{R}^{n})$ such that $\mathrm{Ext}_{S,d,\varepsilon}$ is a right inverse operator for the corresponding trace operator. The construction of the operator $\mathrm{Ext}_{S,d,\varepsilon}$ does not depend on $p$ and based on new delicate combinatorial methods.
