Weighted Sobolev Spaces on Metric Measure Spaces

2014/06/11 by Ambrosio, Luigi, Pinamonti, Andrea, Speight, Gareth · 2 citations
#46E35 #49J52 #58J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1406.3000

Abstract

We investigate weighted Sobolev spaces on metric measure spaces (X,d,m). Denoting by ρ the weight function, we compare the space W^1,p(X,d,ρm) (which always concides with the closure H^1,p(X,d,ρm) of Lipschitz functions) with the weighted Sobolev spaces W^1,p_ρ(X,d,m) and H^1,p_ρ(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,ρm)=H^1,p_ρ(X,d, m). We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on ρ that ensure the equality W^1,p_ρ(X,d,m)=H^1,p_ρ(X,d,m).

Cited by

• Density problem for Sobolev spaces on Gehring Hayman domains with the ball separation condition in metric measure spaces
• Minimization of Degenerate Nonlinear Functionals under Radial Symmetry

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