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manuscripta mathematica

, Volume 99, Issue 3, pp 311–328 | Cite as

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

  • Yong Hah Lee

Abstract:

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively.

Mathematics Subject Classification (1991): 31C05, 31C20, 53C21, 58G03, 58G20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Yong Hah Lee
    • 1
  1. 1.Department of Mathematics, Seoul National University, Seoul 151-742, Korea. e-mail: yhlee@math.snu.ac.krKR

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