By Leo Sario, Mitsuru Nakai, Cecilia Wang, Lung Ock Chung (auth.)

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**Example text**

Then I. i. i0 34 on A . We denote by normalized by g(~,z) f~jdg(~,z) the Green's function on = -i, Ci is the circle r = ri, z, and consider the annulus B If with pole at A = [r2~ ~ r ~ r2~+l ]. I where 8/~n and It follows that, for { c ~Z lhll ~ M and on B , z c BW. of the sectors v Tln, 2 l+k Consider the harmonic function on C2~ U C2~+l. for which v hI ~ O on s(z) v on Tln and B and the maximmm Ihll ~ M lhll ~ 2 -~ on with h and we conclude X. C2~+l , s = 0 on tends to zero. v ~v+l Rln U ~In ' and the smallest positive integer = (r = ~~ r 2~ + r2~+l)]" D on c2~UC2~+l, D .

A dx I A ... du = - / for u,v ~ A 0. dv-v**d~<~u=-jO u*&v- v*Au as m~intained. CHAPTER I HARMC~IC FUNCTIC~S We designate the family of harmonic functions by functions in a given class X = P, B, D, C X by HX. of functions which are positive, bounded, Dirichlet finite, or R, N > 2, for which the class of parabolic Green's functions ~G < 0 ~ and the subfamily of We are interested in the classes bounded and Dirichlet finite, respectively. N-manifolds H, < ~HB" g° For HX(R) Let ~ be the class of Riemannian N consists of constant, and denote by 0G N-manifolds, characterized by the nonexisteflce of harmonic In §l, we establish the strict inchsion relations N = 2, we base the proof of the strictness on the original counterexamples establishing these relations for Rien~nn surfaces.

N-ball is ideally suited for testing the dependence, in a concrete setup, of the existence of functions with various boundedness properties on the metric of a fixed base manifold. Using a Riemannian manifold, rather than a Riemann surface, as the carrier, has the additional advantage that the important class of meaningful. In §3, we give a systematic discussion of The chapter closes with a brief the harmonic null c l a s s e s account, Lp functions becomes 0Nu~p. in §4, of the relationships of n0Nx and completeness of a manifold.