The Dirichlet norm and the norm of Szegő type

Saitoh, Saburou

doi:10.1090/S0002-9947-1979-0539923-5

The Dirichlet norm and the norm of Szegő type
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Trans. Amer. Math. Soc. 254 (1979), 355-364 Request permission

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 \[ \iint _S {|f’(z){|^2} dx dy \leqslant \frac {1}{2}\int _{\partial S} {|f’(z){|^2}{{\left ( {\frac {{\partial g(z,t)}} {{\partial {n_z}}}} \right )}^{ - 1}}|dz|} }\] 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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Integral transform by the product of two Szegö kernel functions

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