Descriptions of exceptional sets in the circles for functions from the Bergman space. (English) Zbl 0901.32006

Jakóbczak, Piotr

Czech. Math. J. 47, No. 4, 633-649 (1997).

For a domain \(D\subset \mathbb{C}^2\) and \(w\in \mathbb{C}\), let \(D_w= \{z\in \mathbb{C}; (z, w) \in D\}\). For \(f\) holomorphic and square-integrable in \(D\), the set \(E(D,f)= \{w; f(\cdot, w) \notin L_2 (D_w)\}\) is of measure zero. It is proved that for each \(0<r<1\) and each \(G_\delta\)-subset \(E\) of \(\{z\in \mathbb{C}; | z| =r\}\) there exists \(f\in L^2H\) in the unit ball \(B\) in \(\mathbb{C}^2\) such that \(E(B,f) =E\).

Reviewer: K.Karták (Praha)

MSC:

32A37

Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))

Keywords:

holomorphic and square-integrable function; Bergman space

References:

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