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Homogeneous polynomials on strictly convex domains

Authors:	Piotr Kot

Institution:	Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland

Abstract:	We consider a circular, bounded, strictly convex domain $\Omega\subset\mathbb{C}^{d}$ with boundary of class $C^{2}$ . For any compact subset of $\partial\Omega$ we construct a sequence of homogeneous polynomials on $\Omega$ which are big at each point of . As an application for any $E\subset\partial\Omega$ circular subset of type $G_{\delta}$ we construct a holomorphic function 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 $\mathbb{C}$ .

Keywords:	homogeneous polynomials exceptional sets highly nonintegrable holomorphic function

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