Weighted Sobolev Spaces on Metric Measure Spaces

TL;DR

This paper studies weighted Sobolev spaces on metric measure spaces, establishing conditions under which different definitions of these spaces coincide, extending classical Euclidean results to more general metric settings.

Contribution

It generalizes the theory of weighted Sobolev spaces to metric measure spaces, proving equivalences between various definitions under mild assumptions.

Findings

Proves $W^{1,p}(X,d, ho m)=H^{1,p}_ ho(X,d,m)$ under certain conditions

Adapts Muckenhoupt and Zhikov results to metric measure spaces

Identifies conditions on weights ensuring space equivalences

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