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Summary:
Let $D$ be a domain in $\mathbb{C}^2$. For $w \in \mathbb{C} $, let $D_w = \lbrace z \in \mathbb{C} \mid (z,w) \in D \rbrace $. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E(D,f)$ of all $w$ such that $f(.,w)$ is not square-integrable in $D_w$ is of measure zero. We call this set the exceptional set for $f$. In this note we prove that for every $0<r<1$,and every $G_\delta $-subset $E$ of the circle $C(0,r) = \lbrace z \in \mathbb{C} \mid | z | =r \rbrace $,there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $\mathbb{C}^2$ such that $E(B,f) = E.$
References:
[1] P. Jakóbczak: The exceptional sets for functions from the Bergman space. Portugaliae Mathematica 50, No 1 (1993), 115–128. MR 1300590
[2] P.Jakóbczak: The exceptional sets for functions of the Bergman space in the unit ball. Rend. Mat. Acc. Lincei s.9, 4 (1993), 79–85. MR 1233394 | Zbl 0788.46061
[3] J.Janas: On a theorem of Lebow and Mlak for several commuting operators. Studia Math. 76 (1983), 249–253. DOI 10.4064/sm-76-3-249-253 | MR 0729105
[4] B.W.Šabat: Introduction to Complex Analysis. Nauka, Moskva, 1969. (Russian) MR 0584932 | Zbl 0169.09001
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