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Traces of functions from \(H^{\infty}(B^ n)\) on some sets of hyperplanes. (Russian. English summary) Zbl 0591.32007
Let \({\mathfrak B}\) be the open unit ball in the complex n-dimensional space \({\mathbb{C}}^ n\). For each \(a\in {\mathfrak B}\) set \(T_ a=\{z\in {\mathfrak B}: (z,a)=| a|^ 2\},\) where (z,a) denotes the usual inner product. Consider an at most countable subset A of \({\mathfrak B}\) and set \(T_ A=\cap_{a\in A}(T_ a\cap {\mathfrak B}).\) The author posed in his earlier note in Lect. Notes Math. 1043, 577-578 (1984) the following problem: 1) Find a necessary and sufficient condition on A in order to have \(f\in H^{\infty}({\mathfrak B})\) with \(f^{-1}(0)=T_ A\). 2) Find a necessary and sufficient condition on A such that (*) for each family \(\{f_ a\}_{a\in A}\) of functions \(f_ a\in H^{\infty}(T_ a\cap {\mathfrak B})\) with \(\| f_ a\|_{\infty}\leq 1\) there exists an \(f\in H^{\infty}({\mathfrak B})\) with \(f| T_ a\cap {\mathfrak B}=f_ a\) for any \(a\in A.\)
In the paper under review he studied some simpler cases. For any given \(\xi\in \partial {\mathfrak B}\), \(0<q<1\), \(\delta >0\) and \(c>0\), we define a (q,\(\delta\),c)-wedge \(E_{q,\delta,c}(\xi)\) by setting \(E_{q,\delta,c}(\xi)=\{z\in {\mathfrak B}: | Im(1-(z,\xi_ 0)| \leq c\cdot Re(1-(z,\xi \quad_ 0);\quad | z|^ 2-| (z,\xi_ 0)|^ 2\leq q(1-| (z,\xi_ 0)|^ 2);\quad 0<Re(1-(z,\xi_ 0))<\delta \}.\)
Theorem 1: Suppose that \(f\in H^{\infty}({\mathfrak B})\) with \(f\not\equiv 0\), \(A\subset E_{q,\delta,c}(\xi)\) and \(T_ A\subset f^{-1}(0)\). Then \(\sum_{a\in A}(1-| a|)^ k<c,\) where \(k=1\) if \(0<q<\); \(k=1+\epsilon\) if \(q=\); \(k=\pi /\{2 \arctan ((q^ 2-(2q-1)^ 2)^{1/2}/(2q-1)\}+\epsilon,\) and the constant c depends on f,q,c,\(\delta\),\(\epsilon\).
Theorem 2: Let \(A\subset E_{q,\delta,c}(\xi)\) with \(0<q<1\), \(\xi\in \partial {\mathfrak B}\) and suppose that there exists a \(\delta_ 0>0\) such that \(\inf \{| [(z,a/| a|)-| a|]/[1-(z,a)]|: z\in T_ b\cap {\mathfrak B}\}\geq \delta_ 0\) for any a,b\(\in A\) with \(a\neq b\). Then the property (*) holds for A.
Reviewer: M.Hasumi

MSC:
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32T99 Pseudoconvex domains
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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