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Abstract and Applied Analysis
Volume 2018, Article ID 5360857, 13 pages
https://doi.org/10.1155/2018/5360857
Research Article

Estimates on the Bergman Kernels in a Tangential Direction on Pseudoconvex Domains in

Department of Mathematics, Sogang University, Seoul 04107, Republic of Korea

Correspondence should be addressed to Sanghyun Cho; rk.ca.gnagos@ohchs

Received 2 March 2018; Accepted 5 July 2018; Published 18 July 2018

Academic Editor: Milan Pokorny

Copyright © 2018 Sanghyun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a smoothly bounded pseudoconvex domain in and assume that where , the boundary of . Then we get optimal estimates of the Bergman kernel function along some “almost tangential curve” .

1. Introduction

Let be a bounded domain in . A natural operator on is the orthogonal projection where denotes the holomorphic functions on . There is a corresponding kernel function , called the Bergman kernel function on . The nature of the singularity of tells us much about the holomorphic function theory of the domain in question and has been studied extensively since Bergman’s original inquiries [1].

One of the methods for the estimates of the Bergman kernel is to construct maximal size of polydiscs in where we have a plurisubharmonic function with maximal Hessian. For strongly pseudoconvex domains in , these polydiscs are of size in normal direction and of size in tangential directions. For weakly pseudoconvex domains, the size of the polydisc in tangential directions depends on the boundary geometry of near , and hence we need complete analysis of the boundary geometry near .

However these analyses and hence the optimal estimates on the Bergman kernels are done only for special type of pseudoconvex domains of finite type in . These domains are, for example, pseudoconvex domains of finite type in [24], decoupled, convex, or uniformly extendable domains of finite type in [57], or pseudoconvex domains in with positive eigenvalues [8, 9]. For the estimates for weighted Bergman projections, one can also refer to [1012]. Nevertheless, the optimal estimates for general pseudoconvex domains of finite type in , , are not known, even for case.

Assume that is a smoothly bounded domain in with smooth defining function with smooth boundary, . Regular finite 1-type at , denoted by , is the maximum order of vanishing of for all one complex dimensional regular curve , , and . Thus satisfieswhere , , denotes finite -type in the sense of D’Angelo [13]. Note that where is the type in the sense of Bloom-Graham.

Remark 1. Consider the domain [13] in defined by Then and while as the complex analytic curve lies in the boundary. Note that is not regular curve.

In the sequel, we let be a smoothly bounded pseudoconvex domain in , and assume that where . Let be the “almost tangential curve" connecting a point and as defined in (20). Note that for each . Set , where is defined in (51).

Theorem 2. Let be a smoothly bounded pseudoconvex domain in and assume that where . Then , the Bergman kernel function of at , satisfies

Theorem 3. Let and be as in Theorem 2. For each , there is a constant , independent of , such thatfor and .

Remark 4. (1) In Theorems 2 and 3, we do not assume that , but we assume only that (see Remark 1). With this weaker condition, we get optimal estimates for Bergman kernel function along special “almost tangential” direction, , but not normal or arbitrary direction.
In [14], Herbort gives an example of a domain where the Bergman kernel grows logarithmically when approaches to in normal direction. Setand for each small , set . Thus approaches to in normal direction as . In this case, Herbort shows that ; that is, the kernel grows logarithmically. For the same domain in (6), we note that and hence , in (4). Set . Then approaches to in “almost tangential direction”. In the Appendix of this paper, we will show that

In Section 2, we will construct special coordinates which reflect the regular finite type condition, , and then show that vanishes to order in -direction. We then consider the slices of by fixing . Then the domains become domains in , and hence we can handle them. Also, the condition acts like the condition on these slices.

For the estimates of , Catlin [2, 15] constructed plurisubharmonic functions with maximal Hessian near each thin -strip of (Section of [2]). In this paper, however, we will construct these functions only on nonisotropic polydiscs for each (Proposition 23). This avoids complicated technical parts in Section of [2]. To get estimates of , , , we consider dilated domains for each . Then the polydisc becomes , independent of , where is a polydisc of radius one with center at the origin. Therefore the uniform -subelliptic estimates for -equation hold on , and the estimates for follow.

Remark 5. Let be a smoothly bounded pseudoconvex domain in , and assume that , where . Then the conditions of Theorems 2 and 3 are satisfied. Near future, using the results of Theorems 2 and 3, we hope we can prove some function theories on , for example, the existence of peak function for that peaks at or necessary conditions for the Hölder estimates for -equation.

2. Special Coordinates

In the sequel, we let be a smoothly bounded pseudoconvex domain in and assume that , . Note that and are positive integers. Without loss of generality, we may assume that . In the sequel, we let and be multi-indices and set and , etc. In Theorem 3.1 in [16], You constructed special coordinates which represent the local geometry of near .

Theorem 6. Let be a smoothly bounded pseudoconvex domain in with smooth defining function and assume , . Then there is a holomorphic coordinate system about such thatwhere

(Idea of the proof) by the standard holomorphic coordinate changes, has the Taylor series expansion as in (8). Since , there is a regular curve which we may assume that satisfying for all sufficiently small . Set . Then, in coordinates, has representation satisfying (8). Also (9) follows from the condition that .

Remark 7. (1) The second condition in (8) and property (9) say that vanishes to order along axis and order along axis.
(2) There are much more terms (mixed with , and their conjugates), compared to the -extensible domain cases, in the summation part of (8).

In conjunction with multitype , we need to consider the dominating terms (in size) among the mixed terms in and variables in the summation part of (8). Using the notations of Section in [16], set Then there are for and for , such that

Remark 8. (1) Here, ’s and ’s are the exponents of and , respectively, in the dominating terms in the summation part of (8).
(2) If , then the expression in (8) will be similar to that of case in [2], and hence we need not consider the above complicated pairs.

Set . If , then for some . In this case, setThen , , and are colinear points in the first quadrant of the plane, and (resp., ) is the intercept of -axis (resp., -axis) of this line. Let be the line segment from to for , set , , and set As in Corollary 3.8 and Remark 3.9 in [16], we can rewrite (8) so thatwhere is a nontrivial homogeneous polynomial of degree given by and there are a small constant , and such thatfor all with all . Property (16) means that there is with terms mixed in and variables for . Let be the constant (direction) in (16) and we will fix in the rest of this paper. In the sequel, we set equal to or , .

Remark 9. (a) defined in (12) is strictly decreasing on .
(b) Each of the summation parts of (14) contains the terms of the form where ’s are the pairs, defined in (12), on the polyline .
(c) Each term of the first summation part in (14) is pure in or variables.
(d) Each term of the second summation part in (14) has terms mixed in and , and it corresponds to the pair of integers , the vertices of the polyline .

Lemma 10. Let be the term containing only or variables in the first sum of (14). Then

Proof. From (8) and (14), we see that . On the other hand, since the regular 1-type at is equal to , there is such that .

In the sequel, we let be a small neighborhood of where has expression as in (14). Since , we may assume that for all for a uniform constant by shrinking if necessary. For each fixed and for each satisfying , for a sufficiently small to be chosen, we set , where is the composition of the projection onto plane and then the projection onto along the direction. Using the Taylor series method in variable about , we see that Since and on , it follows from (17) that for near .

Now for each small , set and set . For a small constant to be chosen, set , and for a fixed small satisfying , set connecting and .

Following the same arguments as in the proof of Proposition 1.2 in [2], for each fixed , we can construct special coordinates about so that, in terms of new coordinates, there is no pure terms in or variables in the first summation part of in (14). We will fix variable and consider the coordinate changes only on variables.

Proposition 11. For each fixed , there is a holomorphic coordinate system such that in the new coordinates defined by where and where , , depends smoothly on , the function given by satisfies

Proof. For , define where . Then we have Assume that (22) and (23) hold for . That is, we have defined so that can be written as If we define , where then satisfies (26) for replaced by . If we proceed up to and set , then by setting , we see that (22) and (23) hold.

In the sequel, we will use the coordinate changes in Proposition 11 only at , (in particular at in Section 3). We want to study the dependence of about . For each , set , and we note that where is defined in the inductive step of the proof of Proposition 11. Set and set . Then and

To study the dependence of and hence dependence of about in (23), we thus need to study the dependence of on variable. For a convenience, set , i.e., remove tilde’s, and assume that satisfies for a sufficiently small to be chosen. In view of (19), we see that satisfies (31). In following we let be the given coordinates, and we let be the coordinates obtained from holomorphic coordinates changes of , as in -th step of coordinate changes in the proof of Proposition 11. Also we let (resp., ), , denote any partial derivative operator of order with respect to and (resp., and ) variables. According to the coordinate changes in Proposition 11, we note that .

Proposition 12. Assume that satisfies (31). Then for each , we haveand for each with , for some in (14), we have

Proof. We will prove by induction on . From (14), (17), and (31) one obtainsand hence (32) follows for . Since , it follows, from (31) and chain rule, that because we are evaluating at . This proves (32) for .
By induction, assume that (32) holds for . For the defined in (29), it follows, from (34) and induction hypothesis, thatfor . Since we are evaluating at , it follows, for , thatBy (30), (36), and (37) and by induction, (32) holds for because if .
Now we prove (33). Assume with , where are the pairs corresponding to the second summation part of (14). Note that the first summation part of (14) will be annihilated by because it contains the pure terms of or mixed with . Thus it follows from (14), (16), and (31) thatSince , it follows from (31) and (38) that Similarly, since , it follows from (36) that because for . This proves (33) for .
By induction assume that (33) holds for . If with and , that is, if has mixed derivatives of and , we note that (37) becomesIf , (33) follows from (41) and induction hypothesis of (33). If , it follows, from (36), (41), and induction hypothesis of (33), that because for . Therefore (33) is proved for .

Recall the expression of and coefficient functions in (23).

Corollary 13. Assume that satisfies (31). Thenand if for some in (14), then

Proof. From (23) we see that where and . Hence it follows from (32) that Assume for some . Thus and it follows from (23), (33), and (41) that because .

Remark 14. Suppose that and . Then , , and are colinear points. From the standard interpolation method, we havefor all sufficiently small . Assume that and for any of . Therefore it follows from (43) and (48) that Therefore the terms of the form , with for some , in the summation part in (23), are the major terms which bounds the other summation terms from above.

In the sequel, we assume that satisfies (31). As in Section in [2], for each , setIn view of Remark 14, we will consider only for , . From (9) and (44) we note thatbecause . For each sufficiently small , setand setFrom (51) and (52), we see that if , then

Lemma 15. For each , .

Proof. Set and . Then Therefore because .

Proposition 16. Assume satisfies (31). Thenwhere .

Proof. By (31), we note that . Assume that . Then by (51). Therefore it follows, from (50) and (52), that and hence it follows from (52) and (53) that Thus follows. Similarly, one can show that .

Let be a small constant to be determined (in Remark 22). By Lemma 15, for each , independent of . Therefore there is a smallest integer , , such thatThen for some by (53). In following, for the fixed integer in (59), set , , and as usual. If we define , where is defined in (22), we may regard that is a biholomorphism. For each , set . For each small , defineand set and when .

Proposition 17. The function satisfies

Proof. Recall that , and in (32). When , it follows from (12) () and (32) that Assume . Then by (12), for some , and hence it follows that . Therefore one obtains, from (48)–(52), thatFrom (32) and (63), it follows that