-domains of holomorphy

ANNALES

POLONICI MATHEMATICI LXIII.1 (1996)

On balanced L

-domains of holomorphy

by Marek Jarnicki (Krak´ow) and Peter Pflug (Oldenburg)

Abstract. We show that any bounded balanced domain of holomorphy is an L

- domain of holomorphy.

It is known [Pfl] that any bounded domain of holomorphy G ⊂ C

with G = int G is an L

(G)-domain of holomorphy, where L

(G) denotes the space of all Lebesgue square integrable holomorphic functions on G. Ob- serve that L

(E) = L

(E \ {0}), where E := {λ ∈ C : |λ| < 1}. For Reinhardt domains there is a complete description of the L

-domains of holomorphy (cf. [Jar-Pfl]). In [Sic] J. Siciak formulated the following prob- lem: Which balanced domains G ⊂ C

are L

(G)-domains of holomorphy?

Here “balanced” means that E · G = G.

The aim of this note is to give a partial solution of this problem (using a deep result due to T. Ohsawa and K. Takegoshi [Ohs-Tak]).

Theorem. Any bounded balanced domain of holomorphy G ⊂ C

is an L

(G)-domain of holomorphy. Consequently, the L

(G)-envelope of holo- morphy of any bounded balanced domain G ⊂ C

coincides with the standard envelope of holomorphy.

P r o o f. We proceed by induction on n. Obviously, the result is true for n = 1. Assume that the assertion holds for n − 1.

Let G ⊂ C

be a bounded balanced domain of holomorphy. Suppose that G is not an L

(G)-domain of holomorphy. Since G is balanced, we may assume that there are domains ∅ 6= V ⊂ U ∩ G U such that:

(a) for any f ∈ L

(G) there exists an e f ∈ O(U ) with e f |

= f |

, where O(U ) denotes the space of all holomorphic functions on U ;

(b) there exists a continuous curve γ : [0, 1] → U with γ(0) ∈ V , γ(1) =:

b ∈ ∂G, γ([0, 1)) ⊂ G, and [0, 1)b ⊂ G;

(c) b = (b

, 0, . . . , 0).

1991 Mathematics Subject Classification: 32D05, 32D10.

Key words and phrases: balanced domain of holomorphy, L

-holomorphic function.

[101]

102 M. J a r n i c k i and P. P f l u g

To see that (b) is always possible we can proceed as follows. Let U , V be as in (a) and let γ : [0, 1] → U be an arbitrary curve with γ(0) ∈ V , γ(1) =: b ∈ ∂G, and γ([0, 1)) ⊂ G. Observe that there is τ

∈ (0, 1] such that [0, τ

)b ⊂ G and τ

b ∈ ∂G. It suffices to show that we can replace U , V , γ by τ

U , τ

V , τ

γ. Take an arbitrary function f ∈ L

(G) and put f

(z) := f (τ

z), z ∈ G. Then f

∈ L

(G) and therefore there exists f e

∈ O(U ) with e f

|

= f

|

. Define e f (z) := e f

(z/τ

), z ∈ τ

U . Then f ∈ O(τ e

U ) with e f |

= f |

.

We choose τ ∈ (0, 1) and a domain W with τ γ([0, 1]) ⊂ W ⊂ G ∩ U , τ γ(0) ∈ V , and [τ, 1]b ⊂ U . Take (n − 1)-dimensional domains e U and e V and an ε > 0 such that [τ, 1]b ⊂ e U ×(εE) ⊂ U and τ b ∈ e V ×{0} ⊂ ( e U ×{0})∩W . We define e G := G ∩ (C

× {0}) and consider e G as a bounded balanced domain of holomorphy in C

.

Let f ∈ L

( e G). Using [Ohs-Tak] we find a function F ∈ L

(G) with F |

G

e = f . By our construction there exists e F ∈ O(U ) such that e F |

= F |

. Then the identity theorem implies that e F |

= F |

; in particular, F (·, 0)| e

V

e = f |

e

. Thus f |

e

extends as a holomorphic function to e U with U 6⊂ e e G, which contradicts our assumption for the (n−1)-dimensional case.

R e m a r k. We do not know any characterization of unbounded balanced L

-domains of holomorphy. Even more, there is no description of unbounded balanced domains G with L

(G) 6= {0}. For Reinhardt domains see [Jar-Pfl].

References

[Jar-Pfl] M. J a r n i c k i and P. P f l u g, Existence domains of holomorphic functions of restricted growth, Trans. Amer. Math. Soc. 304 (1987), 385–404.

[Ohs-Tak] T. O h s a w a and K. T a k e g o s h i, On the extension of L

-holomorphic func- tions, Math. Z. 195 (1987), 197–204.

[Pfl] P. P f l u g, Quadratintegrable holomorphe Funktionen und die Serre Vermu- tung , Math. Ann. 216 (1975), 285–288.

[Sic] J. S i c i a k, Balanced domains of holomorphy of type H

, Mat. Vesnik 37 (1985), 134–144.

INSTITUTE OF MATHEMATICS CARL VON OSSIETZKY UNIVERSIT ¨AT OLDENBURG

JAGIELLONIAN UNIVERSITY FACHBEREICH MATHEMATIK

REYMONTA 4 POSTFACH 2503

30-059 KRAK ´OW, POLAND D-26111 OLDENBURG, GERMANY E-mail: JARNICKI@IM.UJ.EDU.PL E-mail: PFLUGVEC@DOSUNI1.RZ.UNI-OSNABRUECK.DE

Current address of Peter Pflug:

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Re¸ cu par la R´ edaction le 1.6.1995