Dirichlet Finite Solutions of Δu = Pu

Abstract

The purpose of this paper is to give a necessary and also a sufficient condition for a Dirichlet finite harmonic function on a Riemann surface to be represented as a difference of a Dirichlet finite solution of Au=Pu (P_O) and a Dirichlet finite potential of signed measure. 1. Let P=P(z) dx dy (z=x+iy) be a nonnegative not identically zero o-Holder continuous (0<<?1) second order differential on a Riemann surface R and PD(R) be the Hilbert space of all Dirichlet finite solutions of (1) AU(Z) = P(Z)U(Z), A = 4a2./I aZa, on R with the scalar product given by mixed Dirichlet integral, i.e. (u, v)=D11(u, v)=fS11 duA*dv, not the energy integral. The study of PD(R) was begun by Royden [6]. We will use the fact shown by Nakai [2] that PD(R) forms a vector lattice under the natural order in PD(R). We also use the Glasner-Katz maximum principle [1] that the modulus of every function in PD(R) takes its maximum on the Royden harmonic boundary. The recent result of Nakai [3] that PBD(R) is dense in PD(R) will not be made use of. Let A(R) be the Royden harmonic boundary and HD(R) be the class of Dirichlet finite harmonic functions on R. (For the basic materials from the Royden compactification and the class HD(R) we refer to the monograph of Sario and Nakai [7].) One of the important problems in the theory of PD(R) which is not fully developed yet is to describe the distribution of PD(R)|A(R) in HD(R)|A(R). We will prove a theorem which contributes to this question. 2. If R is parabolic, then PD(R)= {0} (cf. Royden [6]), which case offers no interest. Therefore we assume throughout the paper that R is hyperbolic. Let M(R) be the class of all Dirichlet finite Tonelli functions on R and MA(R) the subclass of M(R) consisting of functions f with f JA(R)=O (cf. [7]). We then have the orthogonal decomposition M(R) = HD(R) + MA(R), Received by the editors May 26, 1971. AMS 1970 subject classifications. Primary 31A05, 58G99. 1 The work was sponsored by the U.S. Army Research Office-Durham, Grant DA-ARO-D-31-124-71-G20, UCLA. (e, American Mathematical Society 1972