Calculus of Variations and Geometric Measure Theory

S. Di Marino - D. Lučić - E. Pasqualetto

A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space

created by pasqualetto on 06 May 2020
modified by lučić on 26 Jun 2020



Inserted: 6 may 2020
Last Updated: 26 jun 2020

Pages: 8
Year: 2020

ArXiv: 2005.02924 PDF


We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.