*Published Paper*

**Inserted:** 27 sep 2012

**Last Updated:** 25 jan 2014

**Journal:** Journal of Mathematical Sciences

**Volume:** 196

**Number:** 2

**Pages:** 152-164

**Year:** 2014

**Doi:** 10.1007/s10958-013-1647-4

**Notes:**

Proceedings of the conference "Monge-Kantorovich optimal transportation problem, transport metrics and their applications" (EIMI, St-Petersburg, June 2012)

**Abstract:**

We recall some known and present several new results about Sobolev spaces defined with respect to a measure $\mu$, in particular a precise pointwise description of the tangent space to $\mu$ in dimension 1. This allows to obtain an interesting, original compactness result which stays open in $\mathbb{R}^d$, $d>1$, and can be applied to a new transport problem, with gradient penalization.

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