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Nonreflexivity of Banach spaces of bounded harmonic functions on Riemann surfaces. (English) Zbl 1218.30117

Summary: We give a simple, short, and easy proof for the Masaoka theorem that if Dirichlet finiteness and boundedness for harmonic functions on a Riemann surface coincide with each other, then the dimension of the linear space of Dirichlet finite harmonic functions on the Riemann surface and the dimension of the linear space of bounded harmonic functions on the Riemann surface are finite and identical. The essence of our proof lies in the observation that the former of the above two Banach spaces is reflexive while the latter is not, unless it is of finite dimension.

MSC:

30F20 Classification theory of Riemann surfaces
30F25 Ideal boundary theory for Riemann surfaces
30F15 Harmonic functions on Riemann surfaces
46A25 Reflexivity and semi-reflexivity
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[1] L. V. Ahlfors and L. Sario, Riemann surfaces , Princeton Mathematical Series, No. 26, Princeton Univ. Press, Princeton, NJ, 1960. · Zbl 0196.33801
[2] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen , Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 32, Springer, Berlin, 1963. · Zbl 0112.30801
[3] J. L. Doob, Boundary properties for functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573-621. · Zbl 0121.08604
[4] J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263-321. · Zbl 0001.14102
[5] N. Dunford and J. T. Schwartz, Linear operators (Part I: General Theory) , Pure and Applied Mathematics, Vol. 7, Interscience Publishers, 1967. · Zbl 0084.10402
[6] F.-Y. Maeda, Dirichlet integrals on harmonic spaces , Lecture Notes in Mathematics, 803, Springer, Berlin, 1980. · Zbl 0426.31001
[7] H. Masaoka, The classes of bounded harmonic functions and harmonic functions with finite Dirichlet integrals on hyperbolic Riemann surfaces, RIMS Kôkyûroku (Lecture Notes Series at Research Inst. Kyoto Univ.), 1553 (2007), 132-136.
[8] H. Masaoka, The class of harmonic functions with finite Dirichlet integrals and the harmonic Hardy spaces on a hyperbolic Riemann surface, RIMS Kôkyûroku (Lecture Notes Series at Research Inst. Kyoto Univ.), 1669 (2009), 81-90.
[9] H. Masaoka, The classes of bounded harmonic functions and harmonic functions with finite Dirichlet integrals on hyperbolic Riemann surfaces, Kodai Math. J. 33 (2010), no. 2, 233-239. · Zbl 1195.30058
[10] M. Nakai, Extremal functions for capacities, in the Workshop on Potential Theory 2007 (Hiroshima, 2007) , 83-102.
[11] M. Nakai, Extremal functions for capacities, J. Math. Soc. Japan 61 (2009), no. 2, 345-361. · Zbl 1185.31002
[12] M. Nakai and S. Segawa, Toki covering surfaces and their applications, J. Math. Soc. Japan 30 (1978), no. 2, 359-373. · Zbl 0371.30016
[13] M. Nakai, S. Segawa and T. Tada, Surfaces carrying no singular functions, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 10, 163-166. · Zbl 1193.30058
[14] K. Noshiro, Cluster sets , Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 28 Springer, Berlin, 1960. · Zbl 0102.32903
[15] M. Ohtsuka, Extremal length and precise functions , GAKUTO International Series. Mathematical Sciences and Applications, 19, Gakkōtosho, Tokyo, 2003.
[16] L. Sario and M. Nakai, Classification theory of Riemann surfaces , Grundlehren der mathematischen Wissenschaften in Einzelldarstellungen, Bd. 164, Springer-Verlag, 1970. · Zbl 0199.40603
[17] K. Yosida, Functional Analysis , Grundlehren der mathematischen Wissenschaften in Einzelldarstellungen, Bd. 123, Springer-Verlag, 1965.
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