Uniform Extendibility of the Bergman Kernel for Generalized Minimal Balls

where f ∈ L2(Ω) and w ∈ Ω. A bounded domain Ω is said to satisfy the condition Q if the Bergman projection maps C∞ 0 (Ω) into the space O(Ω) of all holomorphic functions onΩ that can be extended holomorphically to a neighborhood ofΩ. It was also known that the condition Q holds on a domain Ω whenever the ∂-Neumann problem is globally real analytic hypoelliptic on Ω [1–3]. When a domain is a smoothly bounded pseudoconvex domain with real analytic boundary, there are many well-known examples on which the globally real analytic hypoellipticity of the ∂-Neumann problem holds:


Introduction
Let Ω be a bounded domain in C  .The Bergman space  2 (Ω) is the space of holomorphic functions on Ω which are square integrable.The Bergman space has the Bergman kernel (, ), which is uniquely determined by the properties that it is an element of  2 (Ω) in , is conjugate symmetric, and reproduces  2 (Ω).The Bergman projection  of  2 (Ω) onto  2 (Ω) is given by where  ∈  2 (Ω) and  ∈ Ω.
A bounded domain Ω is said to satisfy the condition Q if the Bergman projection maps  ∞ 0 (Ω) into the space O(Ω) of all holomorphic functions on Ω that can be extended holomorphically to a neighborhood of Ω.
It was also known that the condition Q holds on a domain Ω whenever the -Neumann problem is globally real analytic hypoelliptic on Ω [1][2][3].When a domain is a smoothly bounded pseudoconvex domain with real analytic boundary, there are many well-known examples on which the globally real analytic hypoellipticity of the -Neumann problem holds: (i) strictly pseudoconvex domains [4][5][6], (ii) smooth bounded pseudoconvex circular domains with ∑  =1   (/  ) ̸ = 0 near the boundary, where () is the defining function for domains [7], (iii) Reinhardt domains [8].
In this paper, we consider a class of convex circular domains which neither are Reinhardt nor have smooth boundaries and prove that the condition Q holds on each domain in the class defined above.Recently, there are many results of computing the Bergman kernels for various domains explicitly.See [9][10][11][12][13][14].
Let us restrict our attention to circular domains which are not Reinhardt.For  = ( 1 , . . .,   ) ∈ C  , consider the norm where  •  = ∑  =1     and || 2 =  • .The norm  * is the smallest norm in C  that coincides with the Euclidean norm in R  , under certain restriction [15].The minimal ball B * := { ∈ C  :  * () < 1} is the first known bounded domain in C  which is neither Reinhardt nor homogeneous and for which the Bergman kernel can be computed explicitly as the closed form [16].
(ii) Note also that, for  = 1 and  =  = (1), the domain Ω ,, is the minimal ball with radius 1/ √ 2, which is not Reinhardt but satisfies the condition Q [16].If at least one of   's is greater than 1, then Ω ,, is neither smooth nor Reinhardt.
In 2002, Youssfi [17] computed the Bergman kernel for the generalized minimal ball Ω ,, using the transformation formula of the Bergman kernel under proper holomorphic mappings [18][19][20].See the Bergman kernel for Ω ,, in Theorem 4. The Bergman kernel for Ω ,, contains an infinite series H ,, () defined by It is the crucial part to investigate the infinite series H ,, (), when we study the Bergman kernel for Ω ,, .
In [16], it was proved that the minimal ball satisfies the condition Q.In this paper, we generalize this result to the generalized minimal balls Ω ,, as follows.
Theorem 2. The condition Q holds on each domain where  ,, () is defined as in ( 4) and all   's are positive integers.
Similarly as in [16], we obtain the properties of the proper holomorphic mappings on Ω ,, .Corollary 3. Let  ⊂ C  1  1 +⋅⋅⋅+    be any bounded circular domain which contains the origin.
(ii) If  is smooth then there is no proper holomorphic mapping from Ω ,, into .
In Section 2, we review the notation and the Bergman kernel for Ω ,, following [17].In Section 3, we prove the theorem using the characterization theorem (Lemma 6).In the final section, the explicit formulas of H ,, () have been obtained in some cases and we show that the minimal ball satisfies the condition Q using Theorem 2.

Journal of Function Spaces
Now we prove the main theorem of this paper.At first we need to represent H ,, () in terms of Appell hypergeometric functions  ()   when   's are positive integers.If we write then we have where  0 () =  0 ( 1 , . . .,   ) is a polynomial such that there exists a  with 1 ≤  ≤  satisfying that the degree in   is strictly less than ℎ  − 1 and Thus we have If   's are positive integers, then there are positive integers   and   such that   =     +  and 0 ≤   ≤   −1.It follows that where Here we used the notation From (19), there exists an analytic function  3 () such that where The following lemma is useful when we show that a bounded domain satisfies the condition Q. Lemma 6 ([22, 23]).For a bounded domain Ω, the condition Q holds on Ω if and only if, for each compact subset  of Ω, there exists an open set   containing Ω such that (i) for each fixed  0 ∈ (,  0 ) is holomorphic in  on   , (ii) (, ) is continuous on   × .
The above property for smoothly bounded domains was proved by Chen [22] and duplicated for general bounded domains by Thomas [23].
At first we need to study the singular points of f(, ) defined as in (16) where  ± , is one of either  + , or  − , .Let  0 be the number of  − , 's.For fixed  and  with  0 +1 ≤  ≤ , 1 ≤  ≤   , we have where which means that  satisfies condition (ii).

Examples
In fact, it is difficult to write H ,, () explicitly.In the final section, we give some explicit formulas for H ,, ().
Proof.Consider the Taylor expansion of the function in the right hand side of (46).See the details in [11] or [13].