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Spaces of harmonic functions. (English) Zbl 0963.31004
The paper is mainly concerned with the dimensions of certain spaces of harmonic functions on a complete Riemannian manifold $$M$$ of dimension $$n$$. It is shown that the study of harmonic functions on $$M$$ can be reduced to the study of harmonic functions on each end of $$M$$. In particular, precise dimension relations are established for various spaces associated to bounded harmonic functions and positive harmonic functions on $$M$$ and on its ends. A similar result is obtained for spaces of harmonic functions with polynomial growth. This leads, under certain hypotheses on $$M$$, to finite dimensionality results for the space of all harmonic functions on $$M$$ with polynomial growth of at most some prescribed degree.
Some existence and uniqueness results for bounded harmonic maps on $$M$$ are also proved.

##### MSC:
 31C12 Potential theory on Riemannian manifolds and other spaces 31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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