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Generalization of a theorem of Bohr for bases in spaces of holomorphic functions of several complex variables. (English) Zbl 0988.32005
Bohr’s theorem [H. Bohr, Proc. Lond. Math. Soc. (2) 13, 1-5 (1914; JFM 44.0289.01)] states that given an analytic function in the unit disk, $$f(z) = \sum_k c_k z^k$$, such that $$|f(z)|< 1$$ for any $$z$$ in the disk, then $$\sum_k |c_k z^k|<1$$ for $$|z|<1$$. This was generalized to the polydisk by H. P. Boas and D. Khavinson [Proc. Am. Math. Soc. 125, No. 10, 2975-2979 (1997; Zbl 0888.32001)], and to more general domains in $$\mathbb C^n$$ by the first-named author [L. Aizenberg, Proc. Am. Math. Soc. 128, No. 4, 1147-1155 (2000; Zbl 0948.32001), see also L. Aizenberg, A. Aytuna and P. Djakov, Proc. Am. Math. Soc. 128, No. 9, 2611-2619 (2000; Zbl 0958.46015)].
In this work, the authors bring to light general features of Bohr-type phenomena. Denote by $$M$$ a complex manifold, by $$H$$ a space of holomorphic functions on $$M$$, and for any $$f \in H$$ and $$E \subset \subset M$$, $$|f|_E := \sup_E |f|$$. Letting $$\{ \varphi_n, n \geq 0\}$$ be a basis of $$H$$, if $$f = \sum_n f_n \varphi_n$$, write $$\|f\|_E := \sum_n |f_n||\varphi_n|_E$$. Say that $$H$$ has the Bohr Property iff there exist subsets $$U \subset K \subset M$$, where $$U$$ is open and $$K$$ is compact, such that for any $$f \in H$$, $$\|f\|_U \leq |f|_K$$. Note that for the space $$H(M)$$ of all holomorphic functions on $$M$$, general theorems of A. Dynin and B. S. Mityagin [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8, 535-540 (1960; Zbl 0104.08504)] show that for any $$K_1 \subset \subset M$$, there exist $$K_2 \subset \subset M$$ and $$C >0$$ such that $$\|f\|_{K_1} \leq |f|_{K_2}$$.
The main abstract result of the paper is that if $$H(M)$$ has a basis $$\{ \varphi_n, n \geq 0\}$$ such that $$\varphi_0 =1$$ (which is a necessary condition for the Bohr property) and that there exists $$z_0 \in M$$ such that $$\varphi_n (z_0) =0$$, $$n\geq 1$$, then $$H(M)$$ has the Bohr property.
In another section, the authors consider the case where the space $$H$$ is a Hilbert space of analytic functions on a bounded domain $$D \subset \mathbb C^n$$, and $$\{ \varphi_n$$, $$n \geq 0\}$$ is an orthogonal basis. Furthermore, the Hilbert norm is an $$L^2$$ norm with respect to a Borel measure $$\mu$$, and point evaluations are continuous. Suppose further that $$\mu$$ is representing for a point $$z_0 \in D$$. Then a Bohr property takes place iff there exist an open set $$U \ni z_0$$, a constant $$C>0$$, and a compact set $$K$$ such that $$\|f\|_U \leq C |f|_K$$, for all bounded holomorphic $$f$$. An application of this is given to show that certain doubly orthogonal bases enjoy the Bohr property.

##### MSC:
 32A70 Functional analysis techniques applied to functions of several complex variables 32A10 Holomorphic functions of several complex variables 46E10 Topological linear spaces of continuous, differentiable or analytic functions 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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