HOLOMORPHIC FUNCTIONS WITH

SINGULARITIES ON ALGEBRAIC SETS

Józef Siciak

IMUJ PREPRINT 2000/21

Abstract. The aim of the paper is to prove the following Theorem: Let P be a non-zero polynomial of two complex variables. Put A := {(z_{1}, z_{2}); P (z_{1}, z_{2}) = 0}, A_{z_{2}}^{1} := {z_{1}; P (z_{1}, z_{2}) = 0}, A_{z_{1}}^{2} := {z_{2}; P (z_{1}, z_{2}) = 0}. Let E_{1}, E_{2} be two closed subsets of ℂ with positive logarithmic capacities. Put X := (E_{1} × ℂ) ∪ (ℂ × E_{2}). Let f: X \ A ∋ (z_{1}, z_{2}) ↦ f(z_{1}, z_{2}) ∈ ℂ be a function separately holomorphic on X \ A, i.e. f(z_{1}, ·) ∈ O(ℂ \ A_{z_{1}}^{2}) for every z_{1} ∈ E_{1}, and f(·, z_{2}) ∈ O(ℂ \ A_{z_{2}}^{1}) for every z_{2} ∈ E_{2}.

Then there exists a unique function f̃ ∈ O(ℂ^{2} \ A) with f̃ = f on X \ A.

If E_{1} = E_{2} = ℝ and P (z_{1}, z_{2}) = z_{1} − z_{2}, we get the result due to O. Öktem [6].

1. Introduction

The aim of this paper is to prove the following theorem.

Theorem 1.1. Given n ≥ 2, let E_{j} (j = 1, …, n) be a closed subset of the complex plane ℂ of the positive logarithmic capacity. Put

(*) X := (ℂ×E_{2} ×⋯×E_{n})∪(E_{1} ×ℂ×E_{3} ×⋯×E_{n})∪⋯∪(E_{1} ×⋯×E_{n−1} ×ℂ).

Let P be a non-zero polynomial of n complex variables. Put

(**) A := {z ∈ ℂ^{n}; P (z) = 0}, A_{z_{1},...,z_{j−1},z_{j+1},...,z_{n}}^{j} := {z_{j} ∈ ℂ; z ∈ A}

1980 Mathematics Subject Classification (1985 Revision). 32A10,32A99, 32D15, 32D10. Key words and phrases. algebraic set, separately holomomorphic function with singularities

on algebraic set, analytic continuation, envelopes of holomorphy, plurisubharmonic functions, pluripotential theory.

Research supported by KBN grant 2 PO3A 04514 .

for (z_{1}, …, z_{j−1}, z_{j+1}, …, z_{n}) ∈ ℂ^{n−1}, j = 1, …, n. Let f : X \ A ↦ ℂ be a function separately holomorphic on X \ A in the sense that

f(z_{1}, …, z_{j−1}, ·, z_{j+1}, …, z_{n}) ∈ O(ℂ \ A_{z_{1},...,z_{j−1},z_{j+1},...,z_{n}}^{j}),

if z_{k} ∈ E_{k}(k ≠ j), j = 1, …, n.

Then f ∈ O(ℂ^{n} \ A), i.e. there exists a unique function f̃ ∈ O(ℂ^{n} \ A) with f̃ = f on X \ A.

If n = 2, E_{1} = E_{2} = ℝ and P (z_{1}, z_{2}) = z_{1}−z_{2}, we get the result due to O. Öktem [6]. Properties of separately holomorphic functions of the above type were used by O. Öktem ([6,6a]) to characterize the range of the exponential Radon transform (which in turn is of interest for mathematical tomography). Theorem 1.1 shows that the Main conjecture of paper [6a] is true at least for a class of special cases interesting for applications in mathematical tomography.

Let D_{1} and D_{2} be two domains in ℂ^{n} with D_{1} ⊂ D_{2}. In the sequel we shall say that a function f defined and holomorphic on D_{1} is holomorphic on D_{2}, if there exists a unique function f̃ holomorphic on D_{2} such that f̃ = f on D_{1}.

We shall need the following three known theorems.

Theorem 1.2. Let F_{j} be a nonpolar relatively closed subset of a domain D_{j} on the complex z_{j}-plane, j = 1, …, n. Let f : X ↦ ℂ be a function of n complex variables separately holomorphic on the set X := D_{1}×F_{2}×⋯×F_{n} ∪ … ∪F_{1}×⋯×F_{n−1}×D_{n}.

Then the function f is holomorphic on a neighborhood of the set

D_{1} × (F_{2})_{reg} × ⋯ × (F_{n})_{reg} ∪ … ∪ (F_{1})_{reg} × ⋯ × (F_{n−1})_{reg} × D_{n},

where (F_{j})_{reg} is the set of points a of F_{j} such that F_{j} is locally regular (in the sense of the logarithmic potential theory) at a.

Theorem 1.3. Let D ⊂ ℂ^{m} (resp. G ⊂ ℂ^{n}) be a domain with a pluripolar boundary. Let E (resp. F) be a non-pluripolar relatively closed subset of D (resp. G).

Then every function f : X ↦ ℂ separately holomorphic on the set X := E × G ∪ D × F is holomorphic on D × G.

Theorems 1.2 and 1.3 are direct consequences of (e.g.) the main result of [4].

Theorem 1.4 [2]. Let A be an analytic subset (of pure codimension 1) of the envelope of holomorphy D̂ of a domain D ⊂ ℂ^{n}.

Then D̂ \ A is the envelope of holomorphy of D \ A.

2. Proof of Theorem 1.1

We shall consider three cases.

Case 1^{o}. A = ∅.

In this case Theorem 1.1 follows from Theorem 1.3.

Case 2^{o}. P depends on all variables, i.e. for every k with 1 ≤ k ≤ n the polynomial P can be written in the form

(#)  P (z) = ∑_{j=0}^{d_{k}} p_{kj}(z_{1}, …, z_{k−1}, z_{k+1}, …, z_{n})z_{k}^{j},

where d_{k} ≥ 1 and p_{kd_{k}} ≠ 0, k = 1, …, n.

We shall show that in the case 2^{o} our theorem follows from the following Lemma.

Lemma 2.1. If the polynomial P depends on all variables, then there exists a function g holomorphic on the domain ℂ^{n} \ A such that g = f on F_{1} × ⋯ × F_{n}, where F_{1} × ⋯ × F_{n} ⊂ ℂ^{n} \ A and F_{j} is a non-polar subset of E_{j} (j = 1, …, n).

In order to prove Theorem 1.1 it is sufficient to show that g = f on X \ A.

First we shall consider the case of n = 2. Fix (a_{1}, a_{2}) ∈ X \ A. We need to show that g(a_{1}, a_{2}) = f(a_{1}, a_{2}). Without loss of generality we may assume that a_{1} ∈ E_{1}. It is clear that A_{a_{1}}^{2} ≠ ℂ (because P (a_{1}, a_{2}) ≠ 0). Therefore the set F_{2} \ A_{a_{1}}^{2} is nonpolar. Hence there exist z_{2}^{o} ∈ F_{2} \ A_{a_{1}}^{2} and r > 0 such that B(z_{2}^{o}, r) ∩ F_{2} is non-polar, and ({a_{1}} × B(z_{2}^{o}, r)) ∩ A = ∅ (equivalently, P (a_{1}, z_{2}) ≠ 0 for all z_{2} ∈ B(z_{2}^{o}, r)). We know that g(z_{1}, z_{2}) = f(z_{1}, z_{2}) on F_{1}×F_{2}. Fix z_{2} ∈ B(z_{2}^{o}, r)∩F_{2}. The functions g(·, z_{2}) and f(·, z_{2}) are holomorphic on the domain ℂ \ A_{z_{2}}^{1} and identical on the nonpolar subset F_{1} of the domain. Hence g(·, z_{2}) = f(·, z_{2}) on ℂ \ A_{z_{2}}^{1}. It is clear that a_{1} ∈ ℂ \ A_{z_{2}}^{1}. By the arbitrary character of z_{2} it follows that g(a_{1}, z_{2}) = f(a_{1}, z_{2}) for all z_{2} ∈ B(z_{2}^{o}, r) ∩ F_{2}. The functions g(a_{1}, ·) and f(a_{1}, ·) are holomorphic on the domain ℂ \ A_{a_{1}}^{1}, and identical on the nonpolar subset B(z_{2}^{o}, r) ∩ F_{2} of the domain. Therefore g(a_{1}, ·) = f(a_{1}, ·) on ℂ \ A_{a_{1}}^{2}; in particular g(a_{1}, a_{2}) = f(a_{1}, a_{2}).

Now consider the case of n > 2 and assume that Theorem 1.1 is true in ℂ^{k} with 2 ≤ k ≤ n − 1. Fix a = (a_{1}, …, a_{n}) ∈ X \ A. Without loss of generality we may assume a_{1} ∈ E_{1}. Put a = (a_{1}, a′) with a′ = (a_{2}, …, a_{n}). It is clear that A_{a_{1}}^{(2,...,n)} := {z′ ∈ ℂ^{n−1}; P (a_{1}, z′) = 0} ≠ ℂ^{n−1}. Hence A_{a_{1}}^{(2,...,n)} is pluripolar. Therefore there exist ż′ = (ż_{2}, …, ż_{n}) ∈ F_{2} × ⋯ × F_{n} and r > 0 such that P (ż′, r) ∩ F′ is non-pluripolar, and ({a_{1}} × P (ż′, r)) ∩ A = ∅, where P (ż′, r) := (z′ ∈ ℂ^{n−1}; |z_{j} − ż_{j}| < r(j = 2, …, n)} and F′ := F_{2} × ⋯ × F_{n}. We know that g(z_{1}, z′) = f(z_{1}, z′) for all (z_{1}, z′) ∈ F_{1} × F′.

Fix z′ ∈ F′ ∩ P (ż′, r). The functions g(·, z′) and f(·, z′) are holomorphic on the domain ℂ\A_{z′}^{1} and g(·, z′) = f(·, z′) on the nonpolar subset F_{1} of the domain. Hence g(·, z′) = f(·, z′) on ℂ\A_{z′}^{1}. It is clear that a_{1} ∈ ℂ\A_{z′}^{1}. By the arbitrary character of z′ ∈ F′ ∩ P (ż′, r) it follows that g(a_{1}, z′) = f(a_{1}, z′) for all z′ ∈ F′ ∩ P (ż′, r) = (F_{2} ∩ B(ż_{2}, r)) × ⋯ × (F_{n} ∩ B(ż_{n}, r)), and F_{j} ∩ B(ż_{j}, r) is a nonpolar subset of E_{j} (j = 2, …, n). The function f(a_{1}, ·) is separately analytic on X′ \ A_{a_{1}}^{(2,...,n)}, where

X′ := ℂ × E_{3} × ⋯ × E_{n} ∪ ⋯ ∪ E_{2} × ⋯ × E_{n−1} × ℂ.

The function g(a_{1}, ·) is holomorphic on ℂ \ A_{a_{1}}^{(2,...,n)}, and g(a_{1}, z′) = f(a_{1}, z′) for all z′ ∈ F′ ∩ P (ż′, r). By the induction assumption g(a_{1}, z′) = f(a_{1}, z′) for all z′ ∈ ℂ^{n−1} \ A_{a_{1}}^{(2,...,n)}. In particular g(a_{1}, a′) = f(a_{1}, a′).

Case 3^{o}. The polynomial P depends on k variables, where 1 ≤ k ≤ n − 1. Without loss of generality we may assume that P (z_{1}, …, z_{n}) ≡ p(z_{1}, …, z_{k}), where p is a polynomial of k variables z_{1}, …, z_{k}. Put A_{k} := {z ∈ ℂ^{k}; p(z) = 0}. Then A = A_{k}×ℂ^{n−k}. The set A_{k} is pluripolar in ℂ^{k}, the set A_{k}×ℂ^{n−k} is pluripolar in ℂ^{n}. Therefore there exists a point a ∈ ((E_{1})_{reg} ×⋯×(E_{n})_{reg})\(A_{k} ×ℂ^{n−k}). Let r > 0 be so small that A_{k} × ℂ^{n−k} is disjoined with the polydisk B̄(a_{1}, r) × ⋯ × B̄(a_{n}, r).

The set F_{j} := E_{j}∩B̄(a_{j}, r) has positive capacity (j = 1, …, n). The set F_{1}×⋯×F_{n} is non-pluripolar.

By 2^{o} for each (z_{k+1}, …, z_{n}) ∈ E_{k+1} × ⋯ × E_{n} the function f(·, …, ·, z_{k+1}, …, z_{n}) is holomorphic on ℂ^{k} \ A_{k}, and for each (z_{1}, …, z_{k}) ∈ F_{1} × ⋯ × F_{k} the function f(z_{1}, …, z_{k}, ·, …, ·) is holomorphic on O(ℂ^{n−k}). Hence by Theorem 1.3 the function f is holomorphic on (ℂ^{k} \ A_{k}) × ℂ^{n−k} = ℂ^{n} \ A. This ends the proof of Theorem 1.1.

3. Proof of Lemma 2.1

Put

A^{k} := {z ∈ ℂ^{n}; p_{kd_{k}}(z_{1}, … z_{k−1}, z_{k+1}, …, z_{n}) = 0}, k = 1, …, n.

Then the set

B := A ∪ A^{1} ∪ ⋯ ∪ A^{n}

is pluripolar. We know that the set (E_{j})_{reg} is not polar. Therefore the cartesian product (E_{1})_{reg} × ⋯ × (E_{n})_{reg} is not pluripolar, and hence

(E_{1})_{reg} × ⋯ × (E_{n})_{reg} \ B ≠ ∅.

Fix

z^{o} = (z_{1}^{o}, …, z_{n}^{o}) ∈ (E_{1})_{reg} × ⋯ × (E_{n})_{reg} \ (A ∪ A^{1} ∪ ⋯ ∪ A^{n}).

Then there exists r_{o} > 0 such that

(##)  (B̄(z_{1}^{o}, 2r_{o}) × ⋯ × B̄(z_{n}^{o}, 2r_{o})) ∩ (A ∪ A^{1} ∪ ⋯ ∪ A^{n}) = ∅,

where B(z_{j}^{o}, 2r_{o}) := {z_{j} ∈ ℂ; |z_{j} −z_{j}^{o}| < 2r_{o}}. In particular, p_{kd_{k}}(z_{1}, …, z_{k−1}, z_{k+1}, …, z_{n}) ≠ 0 on B̄(z_{1}^{o}, 2r_{o}) × ⋯ × B̄(z_{k−1}^{o}, 2r_{o}) × B(z_{k+1}^{o}, 2r_{o}) × ⋯ × B̄(z_{n}^{o}, 2r_{o}).

We shall show that Lemma 2.1 follows from the following Proposition.

Proposition 3.1. If 0 < δ < min{1, r_{o}} is sufficiently small then for each k = 1, …, n there exists a sequence {c_{kj}}_{j∈ℤ} of functions holomorphic on the open set

Ω_{k} := B(z_{1}^{o}, δ) × ⋯ × B(z_{k−1}^{o}, δ) × ℂ × B(z_{k+1}^{o}, δ) × ⋯ × B(z_{n}^{o}.δ)

such that (i) the series

f_{k}(z) := ∑_{j=−∞}^{∞} c_{kj}(z)P (z)^{j}

is locally uniformly convergent on Ω_{k} \ A, so that the function f_{k} is holomorphic on Ω_{k} \ A;

(ii) f_{k}(z) = f(z) on the set F_{1} × ⋯ × F_{n}, where

F_{j} := E_{j} ∩ B(z_{j}^{o}, δ), j = 1, …, n.

In order to prove Lemma 2.1 let us observe that by (##) and by (ii) f_{j} = f_{k} = f on the non-pluripolar subset F_{1} ×⋯×F_{n} of the domain (Ω_{j} ∩Ω_{k})\A. Therefore the function

f_{o} := f_{1} ∪ ⋯ ∪ f_{n}

is well defined and holomorphic on Ω \ A with Ω := Ω_{1} ∪ ⋯ ∪ Ω_{n}. Moreover f_{o} = f on F_{1} × ⋯ × F_{n}. The set Ω is a Reinhardt domain with centre z^{o} whose envelope of holomorphy is ℂ^{n}. Therefore by Grauert-Remmert Theorem 1.4 there exists a function g holomorphic on ℂ^{n} \ A such that g = f_{o} on Ω \ A; in particular g = f on F_{1} × ⋯ × F_{n}. The proof of Lemma 2.1 is finished.

4. Proof of Proposition 3.1

Fix integer k with 1 ≤ k ≤ n. Without loss of generality we may assume that k = 1. Let {a_{1}, …, a_{s}} := {z_{1} ∈ ℂ; P (z_{1}, z_{2}^{o}, …, z_{n}^{o}) = 0} be the zero set of the polynomial P (·, z_{2}^{o}, …, z_{n}^{o}). By (##) the number m given by

2m := min{|p_{1d_{1}}(z′)|; |z_{j} − z_{j}^{o}| ≤ r_{o} (j = 2, …, n)}.

is positive.

Let R_{o} > max{1, r_{o}} be so large that B(a_{j}, 2) ⊂ B(0, R_{o}) (j = 1, …, s) and

(4.0)  |P (z)| ≥ m|z_{1}|^{d_{1}} for all |z_{1}| ≥ R_{o}, |z_{j} − z_{j}^{o}| ≤ r_{o}(j = 2, …, n).

We claim that for every R > R_{o} and for each ϵ, 0 < ϵ < 1, so small that

B̄(z_{1}^{o}, r_{o}) ∩ (∪_{j=1}^{s}B̄(a_{j}, ϵ)) = ∅, B̄(a_{j}, ϵ) ∩ B̄(a_{l}, ϵ) = ∅(j ≠ l)

there exists δ = δ(ϵ, R) such that 0 < 2δ < r_{o} and f is bounded and holomorphic on the set

\{z ∈ ℂ^{n}; ϵ < |z_{1} − a_{j}| < \frac{3}{2}R (j = 1, …, s), |z_{l} − z_{l}^{o}| < δ(l = 2, …, n)\}.

Indeed, f is separately holomorphic on the set

D_{1} × F_{2} × ⋯ × F_{n} ∪ ⋯ ∪ F_{1} × ⋯ × F_{n−1} × D_{n}

with F_{1} := E_{1} ∩ B̄(z_{1}^{o}, r_{o}), F_{j} := E_{j} ∩ B(z_{j}^{o}, r_{o}) (j = 2, …, n), D_{1} := ℂ \ (B̄(a_{1}, ϵ) ∪ ⋯ ∪ B̄(a_{s}, ϵ)), D_{j} := B(z_{j}^{o}, r_{o}) (j = 2, …, n). For each j the set F_{j} is locally regular at z_{j}^{o}. Hence by Theorem 1.2 there exists δ = δ(ϵ, R) such that O < 2δ < r_{o} and f is holomorphic on the domain

(†) \{z ∈ ℂ^{n}; \frac{ϵ}{2} < |z_{1} − a_{j}| < 2R(j = 1, …, s), |z_{ℓ} − z_{ℓ}^{o}| < 2δ (ℓ = 2, …, n)\},

where δ depends on ϵ and R. Moreover, we may assume that P (z) ≠ 0 for all z in the domain (†).

Observe that by (#) the function

W (ω, z) := \frac{P (ω, z′) − P (z_{1}, z′)}{ω − z_{1}} ≡ ∑_{l=1}^{d_{1}} p_{1l}(z′)[ω^{l−1} + ω^{l−2}z_{1} + ⋯ + z_{1}^{l−1}]

is a polynomial of n + 1 variables ω, z_{1}, …, z_{n}.

It is clear that for every j ∈ ℤ the function

(&)  Φ_{j}(ω, z) := W (ω, z)\frac{f(ω, z′)}{P (ω, z′)^{j+1}}

is holomorphic on the set \{(ω, z) ∈ ℂ^{n+1}; \frac{ϵ}{2} < |ω − a_{j}| < 2R (j = 1, …, s), z_{1} ∈ ℂ, z′ ∈ B(z_{2}^{o}, 2δ) × ⋯ × B(z_{n}^{o}, 2δ)}.

Therefore the function

(4.1)  c_{1j}(z) := \frac{1}{2πi} ∫_{C(0,R)} Φ_{j}(ω, z)dω

is holomorphic on the set ℂ × B(z_{2}^{o}, 2δ) × ⋯ × B(z_{n}^{o}, 2δ); here C(0, R) denotes the positively oriented circle of centre 0 and radius R. Moreover, by (4.0) for every compact subset K of ℂ there exist positive constants M_{1} = M_{1}(K, R, ϵ) and M_{2} = M_{2}(K, R, ϵ) such that

(4.2)  |c_{1j}(z)| ≤ M_{1}(M_{2}R^{d_{1}})^{j}

for all j ∈ ℤ and z ∈ K × B(z_{2}^{o}, δ) × ⋯ × B(z_{n}^{o}, δ).

For a fixed z′ ∈ F_{2} × ⋯ × F_{n} with F_{j} := E_{j} ∩ B(z_{j}^{o}, δ) the function Φ_{j}(·, ·, z′) is holomorphic on {ω ∈ ℂ; P (ω, z′) ≠ 0} × ℂ. Hence, by the Cauchy residue theorem,

(4.3)  c_{1j}(z) = \frac{1}{2πi} ∫_{∂D_{+}(z′,ρ)} Φ_{j}(ω, z)dω, z ∈ ℂ × (F_{2} × ⋯ × F_{n}),

where ρ is any positive real number and

D_{+}(z′, ρ) := {z_{1} ∈ ℂ; |P (z_{1}, z′)| < ρ.}

In the formula (4.3) the integration is taken over the positively oriented boundary of the open set D_{+}(z′, ρ) (the interior of the lemniscate on the z_{1}-plane).

We claim that the functions c_{1j} defined by (4.1) have the required properties. We already know that they are holomorphic on Ω_{1} := ℂ × B(z_{2}^{o}, δ) × ⋯ × B(z_{n}^{o}, δ). In order to prove that they have all the required properties let us observe that, given z′ ∈ F_{2} × ⋯ × F_{n} and 0 < r < 1, we have

f(z) = \frac{1}{2πi} ∫_{∂D(z′,r)} \frac{f(ω, z′)}{ω − z_{1}}dω, z_{1} ∈ D(z′, r) := \{z_{1} ∈ ℂ; r < |P (z_{1}, z′)| < \frac{1}{r}\}.

Hence

f(z) = \frac{1}{2πi} ∫_{∂D_{+}(z′, \frac{1}{r})} \frac{f(ω, z′)}{ω − z_{1}}dω − \frac{1}{2πi} ∫_{∂D_{−}(z′,r)} \frac{f(ω, z′)}{ω − z_{1}}dω

for all z_{1} ∈ D(z′, r), where D_{+}(z′, \frac{1}{r}) := \{z_{1} ∈ ℂ; |P (z_{1}, z′)| < \frac{1}{r}\}, D_{−}(z′, r) := \{z_{1} ∈ ℂ; |P (z_{1}, z′)| > r\}, D(z′, r) = D_{+}(z′, \frac{1}{r}) ∩ D_{−}(z′, r).

Observe that

\frac{f(ω, z′)}{ω − z_{1}} = \frac{P (ω, z′) − P (z_{1}, z′)}{ω − z_{1}} · \frac{f(ω, z′)}{P (ω, z′) − P (z_{1}, z′)} = ∑_{j=0}^{∞} Φ_{j}(ω, z)P (z)^{j}

for all ω ∈ ℂ with |P (ω, z′)| = \frac{1}{r} and all z_{1} ∈ D_{+}(z′, \frac{1}{r}) , the series being uniformly convergent with respect to ω ∈ ∂D_{+}(z′, \frac{1}{r}).

Similarly,

\frac{f(ω, z′)}{ω − z_{1}} = − ∑_{j=1}^{∞} Φ_{j}(ω, z)P (z)^{−j}

for all ω ∈ ∂D_{−}(z′, r) and all z_{1} ∈ D_{−}(z′, r), the series being uniformly convergent with respect to ω ∈ ∂D_{−}(z′, r).

By (4.3) it follows that

(4.4)  f(z) = ∑_{j=−∞}^{∞} c_{1j}(z)P (z)^{j}, z_{1} ∈ D(z′, 0), z′ ∈ F_{2} × ⋯ × F_{n}.

Moreover, for every ρ > 0, for every z′ ∈ F_{2} × ⋯ × F_{n}, and for every compact subset K of ℂ there exists M = M(ρ, z′, K) > 0 such that

|c_{1j}(z)| ≤ Mρ^{−j}, j ∈ ℤ, z_{1} ∈ K, z′ ∈ F_{2} × ⋯ × F_{n}.

Hence for all r > 0, z_{1} ∈ ℂ, z′ ∈ F_{2} × ⋯ × F_{n} one gets the inequalities

|c_{1j}(z_{1}, z′)| ≤ M(\frac{1}{r}, z′, \{z_{1}\})r^{j}, j ≥ 0,

|c_{1j}(z_{1}, z′)| ≤ M(r, z′, {z_{1}})r^{|j|}, j ≤ 1.

By the arbitrary character of r > 0 it follows that

lim sup_{|j|→∞} \frac{1}{|j|} log |c_{1j}(z)| = −∞, z_{1} ∈ C, z′ ∈ F_{2} × ⋯ × F_{n}.

By (4.2) the sequence \{\frac{1}{|j|} log |c_{1j}|\} is locally uniformly upper bounded on Ω_{1}. Put u(z) := lim sup \frac{1}{|j|} log |c_{1j}(z)|, z ∈ Ω_{1}. Then the upper semicontinuous regularization u^{∗} of u is plurisubharmonic in Ω_{1}, and by Bedford-Taylor theorem [3] on negligible sets the set {z ∈ F_{1} × ⋯ × F_{n}; −∞ = u(z) = u^{∗}(z)} is non-pluripolar. Therefore u^{∗} ≡ −∞ in Ω_{1}.

Given a compact subset K of Ω_{1} \ A, there exists r = r(K) with 0 < r < 1 such that r < |P (z)| < \frac{1}{r} for all z ∈ K. Fix k > 0 so large that \frac{1}{r}e^{−k} < \frac{1}{2}. By Hartogs Lemma there exists j_{o} = j_{o}(k, K) such that

\frac{1}{|j|} log |c_{1j}(z)P (z)^{j}| ≤ −k + log \frac{1}{r}, z ∈ K, |j| > j_{o},

i.e.

|c_{1j}(z)|P (z)^{j}| ≤ 2^{−|j|}, z ∈ K, |j| > j_{o}.

It follows that the series ∑_{j=−∞}^{∞} c_{1j}(z)P (z)^{j} is uniformly convergent on every compact subset of Ω_{1} \ A. Its sum f_{1} is holomorphic on Ω_{1} \ A. By (4.4) f_{1} = f on F_{1} × ⋯ × F_{n}. The proof of Proposition 3.1 is completed.

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Jagiellonian University Institute of Mathematicics Reymonta 4 30-059 Kraków Poland e-mail: siciak@im.uj.edu.pl