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.
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Alabern, R., Mateu, J. & Verdera, J. A new characterization of Sobolev spaces on \({\mathbb{R}^{n}}\) . Math. Ann. 354, 589–626 (2012). https://doi.org/10.1007/s00208-011-0738-0
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DOI: https://doi.org/10.1007/s00208-011-0738-0