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 \(\Gamma\subset\mathbb{R}^{2}\) of positive length. Based on such constructions we obtain for each \(p \in (1,\infty)\) an exact description of the trace space \(W^{1}_{p}(\mathbb{R}^{2})|_{\Gamma}\) of the first-order Sobolev space \(W^{1}_{p}(\mathbb{R}^{2})\) to an arbitrary simple rectifiable curve \(\Gamma\) of positive length.