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Homogeneous polynomials on strictly convex domains

Author: Piotr Kot
Journal: Proc. Amer. Math. Soc. 135 (2007), 3895-3903
MSC (2000): Primary 32A05, 32A40
Published electronically: September 10, 2007
MathSciNet review: 2341939
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Abstract: We consider a circular, bounded, strictly convex domain $ \Omega\subset\mathbb{C}^{d}$ with boundary of class $ C^{2}$. For any compact subset $ K$ of $ \partial\Omega$ we construct a sequence of homogeneous polynomials on $ \Omega$ which are big at each point of $ K$. As an application for any $ E\subset\partial\Omega$ circular subset of type $ G_{\delta}$ we construct a holomorphic function $ f$ which is square integrable on $ \Omega\setminus\mathbb{D}E$ and such that $ E=E_{\Omega}^{2}(f):=\left\{z\in\partial\Omega: \int_{\mathbb{D}z}\left\vert f\right\vert^{2}d\mathfrak{L}_{\mathbb{D}z}^{2} =\infty\right\} $ where $ \mathbb{D}$ denotes unit disc in $ \mathbb{C}$.

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Piotr Kot
Affiliation: Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland

Keywords: homogeneous polynomials, exceptional sets, highly nonintegrable holomorphic function
Received by editor(s): September 8, 2005
Received by editor(s) in revised form: September 20, 2006
Published electronically: September 10, 2007
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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