The exceptional sets for functions of the Bergman space in the unit ball (Q1261198)

Let \(L^ 2 H(D)\) be the space of all \(L^ 2\) functions (with respect to Lebesgue measure) which are holomorphic on the domain \(D\subset C^ 2\). For \(w\in C\), let \(D_ w= D\cap(C\times \{w\})\), and let \(p(D_ w)=\{z\in C: (z,w)\in D\}\). Finally, for \(f\in L^ 2 H(D)\), let \(E(D,f)\) be the ``exceptional'' set consisting of all \(w\in C\) such that \(p(D_ w)\neq\emptyset\) and \(f|_{D_ w}\not\in L^ 2(p(D_ w))\). The author continues his study of these exceptional sets [cf. Port. Math., 50, No. 1, 115-128 (1993)]. The main result is the following theorem, which answers a question posed in the earlier paper. Theorem: Let \(B\) denote the unit ball in \(C^ 2\). For any \(r\in (0,1)\), there exists a function \(f\in L^ 2 H(B)\) such that \(E(B,f)=\{z\in C: | z|=r\}\).