We give a simple, short, and easy proof to 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.
References
C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 32, Springer, Berlin, 1963. MR159935C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 32, Springer, Berlin, 1963. MR159935
J. L. Doob, Boundary properties for functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573–621. MR173783 10.5802/aif.126J. L. Doob, Boundary properties for functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573–621. MR173783 10.5802/aif.126
J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321. MR1501590 10.1090/S0002-9947-1931-1501590-9J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321. MR1501590 10.1090/S0002-9947-1931-1501590-9
N. Dunford and J. T. Schwartz, Linear operators (Part I: General Theory), Pure and Applied Mathematics, Vol. 7, Interscience Publishers, 1967. MR1009162N. Dunford and J. T. Schwartz, Linear operators (Part I: General Theory), Pure and Applied Mathematics, Vol. 7, Interscience Publishers, 1967. MR1009162
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.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.
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.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.
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. MR2681537 10.2996/kmj/1278076339 euclid.kmj/1278076339
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. MR2681537 10.2996/kmj/1278076339 euclid.kmj/1278076339
M. Nakai, Extremal functions for capacities, J. Math. Soc. Japan 61 (2009), no. 2, 345–361. MR2532892 10.2969/jmsj/06120345 euclid.jmsj/1242220713
M. Nakai, Extremal functions for capacities, J. Math. Soc. Japan 61 (2009), no. 2, 345–361. MR2532892 10.2969/jmsj/06120345 euclid.jmsj/1242220713
M. Nakai and S. Segawa, Toki covering surfaces and their applications, J. Math. Soc. Japan 30 (1978), no. 2, 359–373. MR477031 10.2969/jmsj/03020359 euclid.jmsj/1240432561
M. Nakai and S. Segawa, Toki covering surfaces and their applications, J. Math. Soc. Japan 30 (1978), no. 2, 359–373. MR477031 10.2969/jmsj/03020359 euclid.jmsj/1240432561
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. MR2591361 10.3792/pjaa.85.163 euclid.pja/1259763077
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. MR2591361 10.3792/pjaa.85.163 euclid.pja/1259763077
M. Ohtsuka, Extremal length and precise functions, GAKUTO International Series. Mathematical Sciences and Applications, 19, Gakkōtosho, Tokyo, 2003. MR2068961M. Ohtsuka, Extremal length and precise functions, GAKUTO International Series. Mathematical Sciences and Applications, 19, Gakkōtosho, Tokyo, 2003. MR2068961
L. Sario and M. Nakai, Classification theory of Riemann surfaces, Grundlehren der mathematischen Wissenschaften in Einzelldarstellungen, Bd. 164, Springer-Verlag, 1970. MR264064L. Sario and M. Nakai, Classification theory of Riemann surfaces, Grundlehren der mathematischen Wissenschaften in Einzelldarstellungen, Bd. 164, Springer-Verlag, 1970. MR264064
