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

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