Weighted Sobolev spaces on metric measure spaces

Abstract

Abstract We investigate weighted Sobolev spaces on metric measure spaces {(X,\mathrm{d},\mathfrak{m})} . Denoting by ρ the weight function, we compare the space {W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} (which always coincides with the closure {H^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} of Lipschitz functions) with the weighted Sobolev spaces {W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} and {H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{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,\mathrm{d},\rho\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{% m})} . We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality {W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},% \mathfrak{m})} .