Nonreflexivity of Banach spaces of bounded harmonic functions on Riemann surfaces

Abstract

We give a simple, short, and easy proof to the Masaoka theorem that if Dirichlet finiteness and boundedness for harmonic functions on a Riemann surface coincide with each other, then the dimension of the linear space of Dirichlet finite harmonic functions on the Riemann surface and the dimension of the linear space of bounded harmonic functions on the Riemann surface are finite and identical. The essence of our proof lies in the observation that the former of the above two Banach spaces is reflexive while the latter is not unless it is of finite dimension.