A new characterization of Sobolev spaces on ℝ ⁿ

In this paper we present a new characterization of Sobolev spaces on ℝ n. Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of ℝ n and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space. © 2011 Springer-Verlag.