*preprint*

**Inserted:** 28 jul 2017

**Last Updated:** 20 feb 2018

**Year:** 2017

v1, 42 pages; v2, 34 pages: introduction on pages 1-3 expanded and clarified, sections 3,4 and 5 shortened, result in subsection 4.3 improved (see Theorem 4.3), proof of Proposition 7.3 expanded and clarified; v3, 28 pages: introduction expanded, sections 2-5 shortened, statement and proof of Theorem 7.1 (i) improved, proof of Proposition 7.3 clarified

**Abstract:**

This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R}}^n,$ improving another previous result of ours. We show that this constant is also the value of the $n,1$ global capacity of any point from ${\mathbf{R}}^n,$ $n \ge 2.$ We also prove the embedding $H_{0}^{1,(n,1)}(\Omega) \hookrightarrow C(\bar{\Omega}) \cap L^{\infty}(\Omega),$ where $\Omega \subset {\mathbf{R}}^n$ is open and $n \ge 2$ is an integer. In the last section of the paper we show that the relative and the global $(p,1)$ and $p,1$ capacities are Choquet whenever $1 \le n<p<\infty$ or $1<n=p<\infty.$

**Keywords:**
Sobolev spaces, Lorentz spaces, capacity