© 2000 by London Mathematical Society
© The London Mathematical Society
Spaces of Harmonic Functions
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
Received 7 September 1998. Revision received 9 March 1999.
It is important and interesting to study harmonic functions on a Riemannian manifold. In an earlier work of Li and Tam [21] it was demonstrated that the dimensions of various spaces of bounded and positive harmonic functions are closely related to the number of ends of a manifold. For the linear space consisting of all harmonic functions of polynomial growth of degree at most d on a complete Riemannian manifold Mn of dimension n, denoted by Hd(Mn), it was proved by Li and Tam [20] that the dimension of the space H1(M) always satisfies dimH1(M) 
dimH1(Rn) when M has non-negative Ricci curvature. They went on to ask as a refinement of a conjecture of Yau [32] whether in general dim Hd(Mn) dimHd(Rn)for all d. Colding and Minicozzi made an important contribution to this question in a sequence of papers [5–11] by showing among other things that dimHd(M) is finite when M has non-negative Ricci curvature. On the other hand, in a very remarkable paper [16], Li produced an elegant and powerful argument to prove the following. Recall that M satisfies a weak volume growth condition if, for some constant A and 
,
 |
for all x 
M and r R, where Vx(r) is the volume of the geodesic ball Bx(r) in M; M has mean value property if there exists a constant B such that, for any non-negative subharmonic function f on M,
 |
for all p M and r > 0.