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A discrete analogue of the harmonic morphism and Green kernel comparison theorems. (English) Zbl 1002.05049
Summary: We give a discrete analogue of the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping between two graphs preserving local harmonic functions. We characterize harmonic morphisms in terms of horizontal conformality. Many examples including coverings, non-complete extended $$p$$-sums and collapsings ar given. Introducing the horizontal and vertical Laplacians, the Green kernel estimates are obtained for the harmonic morphism. As applications, a general and sharp estimate of the Green kernel for an infinite tree is obtained.

MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 58E20 Harmonic maps, etc. 31C20 Discrete potential theory
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