Restrictions of Sobolev W_p^1(R^2)-spaces to planar rectifiable curves

Abstract

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