Characterization of Sobolev-Slobodeckij spaces using curvature energies

Abstract

We give a new characterization of Sobolev-Slobodeckij spaces W^{1+s,p} for n/p<1+s, where n is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in L^p and the appropriate energy is finite.

arXiv comment11 pages; the definition of curvature energies was slightly changed, which allowed to extend the result; some proofs got simplified, mainly thanks to [Dor85]