Bôcher's Theorem

The usual proofs of Bocher's Theorem rely either on the theory of superharmonic functions ([4], Theorem 5.4) or series expansions using spherical harmonics ([5], Chapter X, Theorem XII). (The referee has called our attention to the proof given by G. E. Raynor [7]. Raynor points out that the original proof of Maxime Bocher [2] implicitly uses some non-obvious properties of the level surfaces of a harmonic function.) In this, paper we offer a different and simpler approach to this theorem. The only results about harmonic functions needed are the minimum principle, Harnack's Inequality, and the solvability of the Dirichlet problem in Bn. We will investigate a harmonic function by studying its dilates. For u a function defined on Bn \ {0} and r E (0, 1), the dilate ur is the function defined on (l/r)Bn \ {0} by