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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some sharp weak-type inequalities for holomorphic functions on the unit ball of $ {\bf C}\sp n$


Author: Bogusław Tomaszewski
Journal: Proc. Amer. Math. Soc. 95 (1985), 271-274
MSC: Primary 32A35; Secondary 32A40
MathSciNet review: 801337
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Abstract: Let $ {B^n} = \{ z \in {{\mathbf{C}}^n}:\vert z\vert < 1\} $, $ {S^n} = \partial {B^n}$ and let $ {\sigma _n}$ be the Haar measure on $ {S^n}$. Then for all $ f \in {H^p}(1 \leqslant p < \infty )$ such that $ \operatorname{Im} (f(0)) = 0$ and $ t > 0$,

$\displaystyle {\sigma _n}(\{ z \in {S^n}:\vert f(z)\vert \geqslant t\} ) \leqslant {C_p} \cdot \frac{{\vert\vert\operatorname{Re} \,f\vert\vert _p^p}} {{{t^p}}}$

for some constant $ {C_p}$ depending only on $ p$. The best constant $ {C_p}$ is found for $ 1 \leqslant p \leqslant 2$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0801337-6
PII: S 0002-9939(1985)0801337-6
Article copyright: © Copyright 1985 American Mathematical Society



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