Hostname: page-component-7bb8b95d7b-495rp Total loading time: 0 Render date: 2024-09-12T23:33:10.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Canonically Isomorphic Spaces of Bounded Solutions of △u = Pu

Published online by Cambridge University Press:  20 November 2018

Moses Glasner
Moses Glasner*
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a hyperbolic Riemann surface and P, Q nonnegative C1 2-forms on R. The space of bounded solutions of △u = Pu (△u = Qu, respectively) on R is denoted by PB(R) (QB(R), respectively). A vector space isomorphism S between PB(R) and QB(R) is called canonical if for each u ϵ PB(R), there is a potential pu on R with \u — Su\ ≦ pu. The canonical isomorphism theme in the study of the equation △u = Pu was introduced in H. Royden's paper [9] on the order comparison condition.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 2 , 01 April 1976 , pp. 446 - 448
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Glasner, M., Comparison theorems for bounded solutions of Au = Pu, Trans. Amer. Math. Soc. 202 (1975), 173–179.Google Scholar
2. Lahtinen, A., On the equation Au = Pu and the classification of acceptable densities on Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. M533 (1972).Google Scholar
3. Loeb, P., An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier (Grenoble) 16 (1966), 167–208.Google Scholar
4. Loeb, P. and Walsh, B., A maximal regular boundary for solutions of elliptic differential equations, Ann. Inst. Fourier (Grenoble) 18 (1968), 283–308.Google Scholar
5. Maeda, F.-Y., Boundary value problems for the equation Au — qu = 0 with respect to an ideal boundary, J. Sci. Hiroshima Univ. 32 (1968), 85–146.Google Scholar
6. Nakai, M., The space of bounded solutions of Au = Pu on a Riemann surface, Proc. Japan Acad. 36 (1960), 267–272.Google Scholar
7. Nakai, M., Order comparisons on canonical isomorphisms, Nagoya Math. J. 50 (1973), 67–87.Google Scholar
8. Nakai, M., Banach spaces of bounded solutions of Au = Pu on hyperbolic Riemann surfaces, Nagoya Math. J. 53 (1974), 141–155.Google Scholar
9. Royden, H., The equation Au = Pu and the classification of open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. h\27 (1959).Google Scholar