Homogeneous polynomials on strictly convex domains

Abstract

We consider a circular, bounded, strictly convex domain Ω ⊂ C d \Omega \subset \mathbb C^{d} with boundary of class C 2 C^{2} . For any compact subset K K of ∂ Ω \partial \Omega we construct a sequence of homogeneous polynomials on Ω \Omega which are big at each point of K K . As an application for any E ⊂ ∂ Ω E\subset \partial \Omega circular subset of type G δ G_{\delta } we construct a holomorphic function f f which is square integrable on Ω ∖ D E \Omega \setminus \mathbb DE and such that E = E Ω 2 ( f ) := { z ∈ ∂ Ω : ∫ D z | f | 2 d L D z 2 = ∞ } 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 D \mathbb D denotes unit disc in C \mathbb C .