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Proceedings of the American Mathematical Society

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Bounded in the mean solutions of $ \triangle u=Pu$ on Riemannian manifolds

Authors: Kwang-nan Chow and Moses Glasner
Journal: Proc. Amer. Math. Soc. 26 (1970), 261-265
MSC: Primary 53.72; Secondary 30.00
MathSciNet review: 0271871
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Abstract: Let $ \Phi $ be a convex positive increasing function and $ d = {\lim _{t \to \infty }}\Phi (t)/t$. A harmonic function $ u$ on a Riemann surface $ R$ is called $ \Phi $-bounded if $ \Phi (\vert u\vert)$ is majorized by a harmonic function on R. M. Parreau has shown that if $ d < \infty $ ( $ d = \infty $, resp.), then every positive (bounded, resp.) harmonic function on $ R$ reduces to a constant if and only if every $ \Phi $-bounded harmonic function does. In this paper analogues of these results are given for the equation $ \Delta u = Pu(P \geqq 0)$ on a Riemannian manifold.

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Keywords: Riemannian manifold, Riemann surface, harmonic space, harmonic function, solution of $ \Delta u = Pu$, bounded in the mean, Dirichlet problem, $ Q$-compactification, relatively hyperbolic, Harnack principle
Article copyright: © Copyright 1970 American Mathematical Society