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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.

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