Vol. 108, No. 2, 1983

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Integral comparison theorems for relative Hardy spaces of solutions of the equations Δu = Pu on a Riemann surface

Takeyoshi Satō

Vol. 108 (1983), No. 2, 407–430

We consider two partial differential equations of elliptic type Δu = Pu and Δu = Qu, which are invariantly defined on a Riemann surface R. M. Nakai showed that the Banach spaces PB, QB of bounded solutions on R of these equations are isometrically isomorphic under the condition R |P Q| < +, where it is assumed that R is of hyperbolic type. Let PHep and QHep, 1 < p +, be the relative Hardy spaces of quotients of solutions of the preceding equations by elliptic measures of R. In this paper we shall prove that the above condition is also sufficient for PHep and QHep to be isometrically isomorphic. For this purpose we shall introduce a mapping between the P-Martin and Q-Martin boundaries of R, and give some properties of this mapping.

Mathematical Subject Classification 2000
Primary: 30F25
Secondary: 31A35, 58G30
Received: 11 November 1981
Published: 1 October 1983
Takeyoshi Satō