# zbMATH — the first resource for mathematics

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
Full Text: