Abstract
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.
Highlights
In recent years, the theory of weakly differentiable functions over an abstract metric measure space (X, d, μ) has been extensively studied
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
As a consequence of our arguments, we prove that if the Sobolev norm is closable on compactly-supported smooth functions, the reference measure is absolutely continuous with respect to the Lebesgue measure
Summary
The theory of weakly differentiable functions over an abstract metric measure space (X, d, μ) has been extensively studied. W 1,2(X, d, μ) is a Banach space, but it might be non-Hilbert: for instance, consider the Euclidean space endowed with the ∞-norm and the Lebesgue measure (cf [4, Remark 4.7]) Those metric measure spaces whose associated Sobolev space is Hilbert (which are said to be infinitesimally Hilbertian, cf [9]) play a very important role. We refer to the introduction of [11] for an account of the main advantages and features of this class of spaces The aim of this manuscript is to provide a quick proof of the following result (cf Theorem 11):.
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