Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30F30, 30C40

Retrieve articles in all journals with MSC: 30F30, 30C40

Additional Information

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

American Mathematical Society