Weighted Sobolev Spaces on Metric Measure Spaces

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 $W_{\rho }^{1,p}(X,d,m)$ and $H_{\rho }^{1,p}(X,d,m)$ 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 $W^{1,p}(X,d,\rho m)=H_{\rho }^{1,p}(X,d,m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W_{\rho }^{1,p}(X,d,m)=H_{\rho }^{1,p}(X,d,m)$.

26 pages. Removed Proposition 5.2 which was incorrect in previous versions of the paper; this does not affect any other part of the paper