A new characterization of Sobolev spaces on $${\mathbb{R}^{n}}$$

Abstract

In this paper we present a new characterization of Sobolev spaces on $${\mathbb{R}^n}$$ . 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 $${\mathbb{R}^n}$$ and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.