Weighted Sobolev Spaces on Weighted Metric Measure Spaces

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Date(s) - 07/10/2014
11 h 00 min - 12 h 00 min


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 the results proved by Muckenhoupt and the ones proved by Zhikov to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W_{\rho}^{1,p} (X, d, m) =H_{\rho}^{1,p} (X, d, m)$. This is a joint work with Luigi Ambrosio and Gareth Speight.

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