# Weighted Sobolev spaces on metric measure spaces

**submitted**
**year:** 2014

**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
$W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.

The paper is available on the
cvgmt preprint server.