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Transactions of the American Mathematical Society

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The Dirichlet norm and the norm of Szegő type

Author: Saburou Saitoh
Journal: Trans. Amer. Math. Soc. 254 (1979), 355-364
MSC: Primary 30F30; Secondary 30C40
MathSciNet review: 539923
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Abstract: Let S be a smoothly bounded region in the complex plane. Let $ g(z,t)$ denote the Green's function of S with pole at t. We show that

$\displaystyle \iint_S {\vert f'(z){\vert^2}\,dx\,dy\, \leqslant \,\frac{1}{2}\i... ... {\frac{{\partial g(z,t)}} {{\partial {n_z}}}} \right)}^{ - 1}}\vert dz\vert} }$

holds for any analytic function $ f(z)$ on $ S\, \cup \,\partial S$. This curious inequality is obtained as a special case of a much more general result.

References [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437 (14:479c)
  • [2] S. Saitoh, The kernel functions of Szegö type on Riemann surfaces, Kōdai Math. Sem. Rep. 24 (1972), 410-421. MR 0322161 (48:525)
  • [3] -, The Rudin kernel and the extremal functions in Hardy classes, Kōdai Math. Sem. Rep. 25 (1973), 37-47. MR 0322162 (48:526)
  • [4] -, On some completenesses of the Bergman kernel and the Rudin kernel, Pacific J. Math. 56 (1975), 581-596. MR 0422598 (54:10584)
  • [5] -, The exact Bergman kernel and the kernels of Szegö type, Pacific J. Math. 71 (1977), 545-557. MR 0454007 (56:12258)
  • [6] -, The Bergman norm and the Szegö norm, Trans. Amer. Math. Soc. 249 (1979), 261-279. MR 525673 (80h:30011)
  • [7] -, Integral transform by the product of two Szegö kernel functions (submitted).
  • [8] M. Schiffer and D. C. Spencer, Functionals of finite Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1954. MR 0065652 (16:461g)

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Keywords: Bergman kernel, kernel of Szegö type, compact bordered Riemann surface, critical points of the Green's function, direct product of two spaces of Szegö type, Dirichlet integral of analytic function
Article copyright: © Copyright 1979 American Mathematical Society

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