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

TL;DR

This paper introduces a novel quadratic multiscale characterization of Sobolev spaces on Euclidean spaces that relies solely on metric and measure, enabling broader definitions on metric measure spaces.

Contribution

It provides a new, metric-dependent characterization of Sobolev spaces that can be extended to general metric measure spaces, broadening their applicability.

Findings

Characterization depends only on metric and Lebesgue measure

Applicable to Sobolev spaces of any order of smoothness

Potential for defining Sobolev spaces on general metric measure spaces

Abstract

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). 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.