Weighted Sobolev spaces on metric measure spaces

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 al- ways concides with the closure H1,p(X,d,ρm) of Lipschitz functions) with the weighted Sobolev spaces W1,p(X,d,m) and H1,p(X,d,m) defined as in the Euclidean theory of ρρ weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W1,p(X,d,ρm) = H1,p(X,d,m). We also adapt the results proved by Muckenhoupt and the ones proved by Zhikov to the metric measure set- ting, considering appropriate conditions on ρ that ensure the equality W1,p(X,d,m) = H1,p(X,d,m)