On the converse of the maximum principle

Abstract

A version of the converse of the maximum principle due to W. Rudin is as follows: If A is a linear space of continuous functions on the closed unit disc which contains all polynomials and if every function in A satisfies the maximum principle then every function in A is harmonic in the unit disc. The corresponding versions for the n-harmonic functions on the polydisc of $￠^n$, for the pluriharmonic functions and m-harmonic functions on the unit ball of $￠^n$, and for the ordinary harmonic functions on the unit ball of $R^N$ are proved. The series expansions of the corresponding Poission kernels are essential in the proofs.