Spaces of Harmonic Functions |
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Authors: | Sung Chiung-Jue; Tam Luen-Fai; Wang Jiaping |
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Institution: | Department of Mathematics, National Chung Cheng University Chiayi, Taiwan 62117, cjsung{at}math.ccu.edu.tw
Department of Mathematics, Chinese University of Hong Kong Shatin, Hong Kong, lftam{at}math.cuhk.edu.hk
School of Mathematics, University of Minnesota Minneapolis, MN 55455, USA, jiaping{at}math.umn.edu |
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Abstract: | It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam 21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mn of dimension n,denoted by Hd(Mn), it was proved by Li and Tam 20] that thedimension of the space H1(M) always satisfies dimH1(M) dimH1(Rn)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau 32] whether in generaldim Hd(Mn) dimHd(Rn)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers511] by showing among other things that dimHd(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper 16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and ,
(1.1) for all x M and r R, where Vx(r) is the volume of the geodesicball Bx(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M,
(1.2) for all p M and r > 0. |
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