Vol. 92, No. 1, 1981

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ISSN: 0030-8730
The equations Δu = Pu (P 0) on Riemann surfaces and isomorphisms between relative Hardy spaces

Takeyoshi Satō

Vol. 92 (1981), No. 1, 173–194
Abstract

It has been demonstrated by M. Nakai that the Banach spaces PB (the space of bounded solutions on R of the equation Δu = Pu, P 0) and HB (the space of bounded harmonic functions on R) are isometrically isomorphic whenever the condition

∫
P(z)G(z,w )dx dy < + ∞
R         0

is valid for some point w0 in R (z = x + iy). Here, G(z,w) is the harmonic Green’s function on R. In this paper we shall show, under the preceding condition that the Hardy space Hp, 1 < p +, of harmonic functions on a hyperbolic Riemann surface R is isometrically isomorphic to the relative Hardy space PHwp of quotients of solutions of Δu = Pu by the P-elliptic measure w of R.

Mathematical Subject Classification 2000
Primary: 30F15
Secondary: 30F20, 31A35
Milestones
Received: 3 January 1979
Published: 1 January 1981
Authors
Takeyoshi Satō