Homogeneous polynomials on strictly convex domains

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 $\Omega \setminus \mathbb DE$ and such that $E=E_{\Omega }^{2}(f):=\left \{z\in \partial \Omega : \int _{\mathbb Dz}\left |f\right |^{2}d\mathfrak {L}_{\mathbb Dz}^{2} =\infty \right \}$ where $\mathbb D$ denotes unit disc in $\mathbb C$.