# Weighted Sobolev spaces on metric measure spaces

• Luigi Ambrosio , Andrea Pinamonti and Gareth Speight

## Abstract

We investigate weighted Sobolev spaces on metric measure spaces (X,d,𝔪). Denoting by ρ the weight function, we compare the space W1,p(X,d,ρ𝔪) (which always coincides with the closure H1,p(X,d,ρ𝔪) of Lipschitz functions) with the weighted Sobolev spaces Wρ1,p(X,d,𝔪) and Hρ1,p(X,d,𝔪) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W1,p(X,d,ρ𝔪)=Hρ1,p(X,d,𝔪). We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality Wρ1,p(X,d,𝔪)=Hρ1,p(X,d,𝔪).

Funding statement: The authors acknowledge the support of the grant ERC ADG GeMeThNES. The second author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

## Acknowledgements

We would like to thank M. D. Surnachev for kindly explaining the change of weight in [27] (which we adapted in Proposition 4.4) and for informing us of his generalization of Zhikov’s result to more general exponents [26].

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