Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-16T16:43:39.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Order Comparisons on Canonical Isomorphisms

Published online by Cambridge University Press:  22 January 2016

Mitsuru Nakai
Mitsuru Nakai*
Affiliation:
Nagoya University
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.

Consider a nonnegative Hölder continuous 2-form P(z)dxdy (z = x + iy) on a connected Riemann surface R. We denote by P(R) the linear space of solutions u of the equation Δu = Pu on R and by PX(R) the subspace of P(R) consisting of those u with a certain boundedness property X. We also use the standard notations H(R) and HX(R) for P(R) and PX(R) with P ≡ 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 50 , June 1973 , pp. 67 - 87
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

[1] Constantinescu, C.-Cornea, A.: Ideale Ränder Riemannscher Flachen, Springer, 1963.CrossRefGoogle Scholar
[2] Glasner, M.-Katz, R.: On the behavior of solutions of Δu=Pu at the Royden boundary, J. d’Analyse Math., 22 (1969), 345–354.CrossRefGoogle Scholar
[3] Glasner, M.-Nakai, M.: Riemannian manifolds with discontinuous metrics and the Dirichlet integral, Nagoya Math., J., 46 (1972), 1–48.CrossRefGoogle Scholar
[4] Loeb, P.: An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier, 16 (1966), 167–208.CrossRefGoogle Scholar
[5] Maeda, F.-Y.: Boundary value problems for the equation Δu—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 the equation Δu=Pu on a Riemann surface, Proc. Japan Acad., 36 (1960), 267–272.Google Scholar
[7] Nakai, M.: The space of Dirichlet-finite solutions of the equation Δu—Pu on a Riemann surface, Nagoya Math. J., 18 (1961), 111–131.CrossRefGoogle Scholar
[8] Nakai, M.: The equation Δu=Pu on E™> with almost rotation free P≧O, Tôhoku Math. J., 23 (1971), 413–431.CrossRefGoogle Scholar
[9] Nakai, M.: Dirichlet finite solutions of Δu=Pu on open Riemann surfaces, Kôdai Math. Sem. Rep., 23 (1971), 385–397.Google Scholar
[10] Nakai, M.: The equation Δu=Pu on the unit disk with almost rotation free P>0, J. Diff. Eq., 11 (1972), 307–320.CrossRefGoogle Scholar
[11] Nakai, M.: Canonical isomorphisms of energy finite solutions of Δu—Pu on open Riemann surfaces, Proc. Japan Acad, (to appear).Google Scholar
[12] Nakai, M.: Uniform densities on hyperbolic Riemann surfaces, Nagoya Math. J., 51 (1973) (to appear).CrossRefGoogle Scholar
[13] Nakai, M.: Banach spaces of bounded solutions of Δu—Pu (P>0) on hyperbolic Riemann surfaces (to appear).0)+on+hyperbolic+Riemann+surfaces+(to+appear).>Google Scholar
[14] Royden, H.: The equation Δu=Pu, and the classification of open Riemann surfaces, Ann. Acad. Sci. Fenn., 271 (1959), 1–27.Google Scholar
[15] Sario, L.-Nakai, M.: Classification Theory of Riemann Surfaces, Springer, 1970.CrossRefGoogle Scholar
[16] Singer, I.: Dirichlet finite solutions of Δu—Pu, Proc. Amer. Math. Soc, 32 (1972), 464–468.Google Scholar
[17] Singer, I.: Boundary isomorphism between Dirichlet finite solutions of Δu=Pu and harmonic functions, Nagoya Math. J., 50 (1973), 7–20.CrossRefGoogle Scholar