The exceptional sets for functions of the Bergman space in the unit ball

Jakóbczak, Piotr

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 4 (1993) no. 2, pp. 79-85

Résumé

Let \( D \) be a domain in \( C^{2} \). Given \( w \in C \), set \( D_{w} = \{ z \in C \mid (z,w) \in D\} \). If \( f \) is a holomorphic and square-integrable function in \( D \), then the set \( E(D,f) \) of all \( w \) such that \( f ( \cdot, w) \) is not square-integrable in \( D_{w} \) has measure zero. We call this set the exceptional set for \( f \). In this Note we prove that whenever \( 0 r 1 \) there exists a holomorphic square-integrable function \( f \) in the unit ball \( B \) in \( C^{2} \) such that \( E(B,f) \) is the circle \( C(0, r) = \{z \in C \mid |z| = r\} \).