The exceptional sets for functions of the Bergman space in the unit ball. (English) Zbl 0788.46061

Jakóbczak, Piotr

Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 4, No. 2, 79-85 (1993).

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 \(E(B,f)=\{z\in C: | z|=r\}\).

Reviewer: R.M.Aron (Kent)

MSC:

46J15	Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46E20	Hilbert spaces of continuous, differentiable or analytic functions

Keywords:

Bergman space in the unit ball; exceptional sets