Generalization of a theorem of Bohr for bases in spaces of holomorphic functions of several complex variables.

*(English)*Zbl 0988.32005Bohr’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.

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.

Reviewer: Pascal J.Thomas (Toulouse)

##### 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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\textit{L. Aizenberg} et al., J. Math. Anal. Appl. 258, No. 2, 429--447 (2001; Zbl 0988.32005)

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##### References:

[1] | Aizenberg, L.A., The spaces of functions analytic in (p,q)-circular regions, Soviet math. dokl., 2, 79-82, (1960) |

[2] | Aizenberg, L.A.; Dautov, Sh.A., Holomorphic functions of several complex variables with nonnegative real part: traces of holomorphic and plurisubharmonic functions on the Shilov boundary, Math. USSR sb., 28, 301-313, (1976) |

[3] | Aizenberg, L., Multidimensional analogues of Bohr’s theorem on power series, Proc. amer. math. soc., 128, 1147-1155, (2000) · Zbl 0948.32001 |

[4] | Aytuna, A., Stein spaces M for which O(M) is isomorphic to a power series space, Advances in the theory of Fréchet spaces, (1988), Kluwer Academic Dordrecht/Boston/London, p. 115-154 · Zbl 0743.46017 |

[5] | A. Aytuna, Common bases for some pairs of analytic function spaces, preprint, Tagungsbericht 34-1995/Mathematisches Forschungsinstitut Oberwolfach. |

[6] | Boas, H.P.; Khavinson, D., Bohr’s power series theorem in several variables, Proc. amer. math. soc., 125, 2975-2979, (1997) · Zbl 0888.32001 |

[7] | Bohr, H., A theorem concerning power series, Proc. London math. soc. (2), 13, 1-5, (1914) · JFM 44.0289.01 |

[8] | Djakov, P.B.; Mityagin, B.S., Modified construction of a nuclear Fréchet space without basis, J. funct. anal., 23, 415-433, (1976) · Zbl 0339.46004 |

[9] | Dynin, A.; Mityagin, B., Criterion for nuclearity in terms of approximate dimension, Bull. Polish acad. sci. math., 8, 535-540, (1960) · Zbl 0104.08504 |

[10] | Dragilev, M.M., Canonical form of a bases in a space of analytic functions, Uspekhi math. nauk, 15, 181-188, (1960) · Zbl 0096.27301 |

[11] | Fornaess, J.E.; Stout, E.L., Spreading polydiscs on complex manifolds, Amer. J. math., 99, 933-960, (1977) · Zbl 0384.32004 |

[12] | Meise, R.; Vogt, D., Introduction to functional analysis, (1997), Clarendon Oxford |

[13] | Mityagin, B.S., Approximate dimension and bases in nuclear spaces, Uspekhi mat. nauk, 16, 63-132, (1961) · Zbl 0104.08601 |

[14] | Shapiro, H.S., Stefan Bergman’s theory of doubly-orthogonal functions: an operator-theoretic approach, Proc. roy. irish acad. sect. A, 79, 49-58, (1979) · Zbl 0364.30012 |

[15] | Sidon, S., Über einen satz von herrn Bohr, Math. Z., 26, 731-732, (1927) · JFM 53.0281.04 |

[16] | Stein, E.M., Boundary behavior of holomorphic functions of several complex variables, (1972), Princeton Univ. Press and Univ. of Tokyo Press Princeton · Zbl 0242.32005 |

[17] | Titchmarsh, E.C., The theory of functions, (1939), Oxford Univ. Press London · Zbl 0022.14602 |

[18] | Tomić, M., Sur un théorème de H. Bohr, Math. scand., 11, 103-106, (1962) · Zbl 0109.30202 |

[19] | Zahariuta, V.P.; Kadampatta, S.N., On the existence of extendable bases in spaces of functions, analytic on compacta, Math. notes, 27, 334-340, (1980) · Zbl 0458.46006 |

[20] | Zahariuta, V.P., Spaces of analytic functions and complex potential theory, Linear topol. spaces complex anal., 1, 74-146, (1994) · Zbl 0859.30041 |

[21] | Zobin, M.M.; Mityagin, B.S., Examples of nuclear Fréchet spaces without basis, Funktsional. anal. i prilozhen., 84, 35-47, (1984) |

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