Uniform Extendibility of the Bergman Kernel for Generalized Minimal Balls
We consider a class of convex domains which contains non-Reinhardt domains with nonsmooth boundary. We show that the domains of this class satisfy the condition Q.
Let be a bounded domain in . The Bergman space is the space of holomorphic functions on which are square integrable. The Bergman space has the Bergman kernel , which is uniquely determined by the properties that it is an element of in , is conjugate symmetric, and reproduces . The Bergman projection of onto is given by where and .
A bounded domain is said to satisfy the condition if the Bergman projection maps into the space of all holomorphic functions on that can be extended holomorphically to a neighborhood of .
It was also known that the condition holds on a domain whenever the -Neumann problem is globally real analytic hypoelliptic on [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:(i)strictly pseudoconvex domains [4–6],(ii)smooth bounded pseudoconvex circular domains with near the boundary, where is the defining function for domains ,(iii)Reinhardt domains .
In this paper, we consider a class of convex circular domains which neither are Reinhardt nor have smooth boundaries and prove that the condition holds on each domain in the class defined above. Recently, there are many results of computing the Bergman kernels for various domains explicitly. See [9–14].
Let us restrict our attention to circular domains which are not Reinhardt. For , consider the norm where and . The norm is the smallest norm in that coincides with the Euclidean norm in , under certain restriction . The minimal ball is the first known bounded domain in which is neither Reinhardt nor homogeneous and for which the Bergman kernel can be computed explicitly as the closed form .
For any positive integers and , we denote by the -dimensional complex vector space of all -matrices with complex coefficients. If is an element of , we set Let be a positive integer and let and be multi-indices with and for all . We denote by the number of those ’s that are equal to . If , we may assume without loss of generality that and for . Let with , and consider the function defined for by and define the generalized minimal balls , where Note that where .
Remark 1. (i) Note that, for , the domain is a complex ellipsoid, which is a smoothly bounded Reinhardt domain and clearly satisfies the condition .
(ii) Note also that, for and , the domain is the minimal ball with radius , which is not Reinhardt but satisfies the condition . If at least one of ’s is greater than 1, then is neither smooth nor Reinhardt.
In 2002, Youssfi  computed the Bergman kernel for the generalized minimal ball using the transformation formula of the Bergman kernel under proper holomorphic mappings [18–20]. See the Bergman kernel for in Theorem 4. The Bergman kernel for contains an infinite series defined by It is the crucial part to investigate the infinite series , when we study the Bergman kernel for .
In , it was proved that the minimal ball satisfies the condition . In this paper, we generalize this result to the generalized minimal balls as follows.
Theorem 2. The condition holds on each domain where is defined as in (4) and all ’s are positive integers.
Similarly as in , we obtain the properties of the proper holomorphic mappings on .
Corollary 3. Let be any bounded circular domain which contains the origin.(i)If is a proper holomorphic mapping, then extends holomorphically to a neighborhood of .(ii)If is smooth then there is no proper holomorphic mapping from into .
In Section 2, we review the notation and the Bergman kernel for following . In Section 3, we prove the theorem using the characterization theorem (Lemma 6). In the final section, the explicit formulas of have been obtained in some cases and we show that the minimal ball satisfies the condition using Theorem 2.
2. Bergman Kernel Function for
In this section we review the Bergman kernel for following the notations which were introduced in . For with , we define
Consider the differential operator acting on functions for by where
We also consider the differential operator acting on functions for defined by where
Note that depends on . Since for , then we have .
We also consider the product group and extend each element , , , of to an element of by
If is a smooth function defined on the region of with , we consider the function on the region of with in by where . Then define . For example, we consider and . Then,
Consider the series defined for by where is defined in (7). The series converges for . Finally, we set
We extend the operation to a -valued bilinear mapping defined for by if , , and
In 2002, Youssfi  computed the Bergman kernel for as follows.
Theorem 4 (, Theorem ). The Bergman kernel function for the domain is given by the formula where and . The latter product is understood in the sense of (9). More precisely, one has and , where
Remark 5. In fact, the weighted Bergman kernel and Szegö kernel have been obtained in .
3. Proof of Theorem 2
For the study of , we need to introduce Appell’s hypergeometric functions. Recall that Appell’s hypergeometric function  is defined by where .
Now we prove the main theorem of this paper. At first we need to represent in terms of Appell hypergeometric functions when ’s are positive integers. If we write then we have where is a polynomial such that there exists a with satisfying that the degree in is strictly less than and
Thus we have
If ’s are positive integers, then there are positive integers and such that and . It follows that where Here we used the notation and .
Note that Appell’s hypergeometric functions converge for and diverge for . Thus there exist a nonnegative constant and an analytic function such that
From (28), there exist a nonnegative constant and an analytic function such that From (19), there exists an analytic function such that where
The following lemma is useful when we show that a bounded domain satisfies the condition .
Lemma 6 ([22, 23]). For a bounded domain , the condition holds on if and only if, for each compact subset of , there exists an open set containing such that(i)for each fixed is holomorphic in on ,(ii) is continuous on .
The above property for smoothly bounded domains was proved by Chen  and duplicated for general bounded domains by Thomas .
At first we need to study the singular points of defined as in (16).
Lemma 7. Let be a smooth function on a region of with . Denote If is defined as in (16), then there exists a function such that(i) equal to ,(ii) for .
Proof. If we define and , then is equal to one of
where is one of either or .
Let be the number of ’s. For fixed and with , , we have where (, resp.) means summation running over which contains (, resp.).
From the above formula, we see that a numerator has a factor . Repeating this process for all and , we obtain that is equal to certain function times which means that satisfies condition (ii).
Let and be defined as in (23). By Lemma 7 the Bergman kernel of has poles at with for some . Write and as where and . Then it follows that
The following lemma is needed when we estimate the function .
Lemma 8. For and with , let Then one has
Proof. By the Schwartz inequality, we have
Then, by using Lemma 8, the identity (41), and the inequality for , we have Thus by Lemma 6, each domain satisfies the condition .
In fact, it is difficult to write explicitly. In the final section, we give some explicit formulas for .
Lemma 9. Let with and . For any and with , one has
Proof. Consider the Taylor expansion of the function in the right hand side of (46). See the details in  or .
Lemma 10. Let be a polynomial of degree with variable . Then there exist unique constants for such that
Proof. For a given polynomial , there exist unique constants for such that Note that, for , If we write , then we obtain (47).
Example 11. The complex -dimensional ball with radius : if and , then using Lemma 9, we have
Example 12. The Thullen domain with a positive integer : if , , and , then is linearly biholomorphic to the Thullen domain . In this case, we have If we apply Lemma 9 to a variable , then we obtain By Lemma 10, there exist constants such that where is a polynomial for a positive integer . Note that .
Example 13. A complex ellipsoid : if , , and , then the following formula was proved in : where and .
Example 14. The minimal ball with radius : if , , and , the infinite series is equal to Since , we have which is analytic in and this shows that the minimal ball satisfies the condition .
Example 15. Let , , and with and . Then the infinite series is equal to Next, for nonnegative integers and , we need to compute the infinite series It is interesting to compare this series (58) with the left hand side in (46). We can express the infinite series (58) in terms of finite series using Lemma 9. Precisely, the series (58) is equal to
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was supported by NRF-2010-0011841 from the National Research Foundation of Korea.
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