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

Abstract

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 $W_p^1({\mathbb{R}}^2)$ to an arbitrary planar rectifiable curve $\Gamma \subset {\mathbb{R}}^2$ of positive length.