Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - A. Pinamonti - G. Speight

Weighted Sobolev Spaces on Metric Measure Spaces

created by pinamonti on 11 Jun 2014
modified by speight on 16 May 2017


Accepted paper

Inserted: 11 jun 2014
Last Updated: 16 may 2017

Journal: J. Reine Angew. Math.
Year: 2015

ArXiv: 1406.3000 PDF


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 $W^{1,p}_\rho(X,d,m)$ and $H^{1,p}_\rho(X,d,m)$ 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 $W^{1,p}(X,d,\rho m)=H^{1,p}_\rho(X,d, m)$. 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 $W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.


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