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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0539923-5
PII: S 0002-9947(1979)0539923-5
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