Weighted Sobolev spaces on metric measure spaces

We investigate weighted Sobolev spaces on metric measure spaces (X,d,m). Denoting by p the weight function, we compare the space W1,p(X,d, pm) (which always coincides with the closure H1,p(X, d, pm) of Lipschitz functions) with the weighted Sobolev spaces W1,pp (X, d,m) and H1,pp (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, pm) D H1,pp (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 p that ensure the equality W1,pp (X, d,m) D H1,pp (X, d,m).