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.
