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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 Bergman norm and the Szegő norm


Author: Saburou Saitoh
Journal: Trans. Amer. Math. Soc. 249 (1979), 261-279
MSC: Primary 30C40
MathSciNet review: 525673
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Abstract: Let G denote an arbitrary bounded regular region in the plane and $ {H_2}\left( G \right)$ the analytic Hardy class on G with index 2. We show that the generalized isoperimetric inequality

\begin{multline}\frac{1}{\pi }\,\iint\limits_G {{{\left\vert {\varphi \left( z \... ...t}^{2}}\,\left\vert dz \right\vert}\,\,\,\,\,\,\,(z\,=\,x\,+\,iy) \end{multline}

holds for any $ \varphi $ and $ \psi \, \in \,{H_2}\left( G \right)$. We also determine necessary and sufficient conditions for equality.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0525673-8
PII: S 0002-9947(1979)0525673-8
Keywords: Generalizations of isoperimetric inequality, Bergman, Szegö, and weighted Szegö kernels, the general theory of reproducing kernels, products of kernels, Rudin kernels, Szegö kernel with characteristic, multiplicative functions
Article copyright: © Copyright 1979 American Mathematical Society