Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds |
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Authors: | Yong Hah Lee |
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Institution: | (1) Department of Mathematics, Seoul National University, Seoul 151-742, Korea. e-mail: yhlee@math.snu.ac.kr, KR |
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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.
Received: 23 July 1998 / Revised version: 10 February 1999 |
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Keywords: | Mathematics Subject Classification (1991): 31C05 31C20 53C21 58G03 58G20 |
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