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

 

Derivatives of Hardy functions


Author: Boo Rim Choe
Journal: Proc. Amer. Math. Soc. 110 (1990), 781-787
MSC: Primary 32A35
MathSciNet review: 1028041
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Abstract: Let $ B$ be the open unit ball of $ {C^n}$, and set $ S = \partial B$. It is shown that if $ \varphi \in {L^p}\left( S \right),\varphi > 0$, is a lower semicontinuous function on $ S$ and $ 1/q > 1 + 1/p$, then, for a given $ \varepsilon > 0$, there exists a function $ f \in {H^p}\left( B \right)$ with $ f\left( 0 \right) = 0$ such that $ \left\vert {{f^ * }} \right\vert = \varphi $ almost everywhere on $ S$ and $ \int_B {{{\left\vert {\nabla f} \right\vert}^q}dV < \varepsilon } $ where $ V$ denotes the normalized volume measure on $ B$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1028041-8
PII: S 0002-9939(1990)1028041-8
Keywords: Derivatives, $ {H^p}$-functions
Article copyright: © Copyright 1990 American Mathematical Society