[BibTeX]

*Accepted paper*

**Inserted:** 11 jun 2014

**Last Updated:** 14 feb 2016

**Journal:** J. Reine Angew. Math.

**Year:** 2015

**Abstract:**

We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $\rho$ the weight function, we compare the space $W^{1,p}(X,d,\rho m)$ (which always concides with the closure $H^{1,p}(X,d,\rho m)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_\rho(X,d,m)$ and $H^{1,p}_\rho(X,d,m)$ 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 $W^{1,p}(X, d,\rho m)=H^{1,p}_\rho(X,d,m)$. We also adapt the results in 21 and in the recent paper 25 to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.

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