Abstract and Applied Analysis
Volume 2018 (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

Sanghyun Cho

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

Correspondence should be addressed to Sanghyun Cho; shcho@sogang.ac.kr

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 [2–4], decoupled, convex, or uniformly extendable domains of finite type in [5–7], or pseudoconvex domains in with positive eigenvalues [8, 9]. For the estimates for weighted Bergman projections, one can also refer to [10–12]. 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 satisfies

where , , 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 that

for 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. Set

and 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 that

where

(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, set

Then , , 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 that

where is a nontrivial homogeneous polynomial of degree given by

and there are a small constant , and such that

for 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 have

and for each with , for some in (14), we have

Proof. We will prove by induction on . From (14), (17), and (31) one obtains

and 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, that

for . Since we are evaluating at , it follows, for , that

By (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) that

Since , 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) becomes

If , (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). Then

and 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 have

for 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 , set

In view of Remark 14, we will consider only for , . From (9) and (44) we note that

because . For each sufficiently small , set

and set

From (51) and (52), we see that if , then

Lemma 15. For each , .

Proof. Set and . Then

Therefore because .

Proposition 16. Assume satisfies (31). Then

where .

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 that

Then 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 , define

and 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), that

From (32) and (63), it follows that

Using the coordinates defined in (14), set

Then , , are tangential holomorphic vector fields and on for a uniform constant . For any with , define

In -coordinates defined by , set , and set , . If we define

then by functoriality,

Lemma 18. There is a small constant such that

provided is sufficiently small.

Proof. Since the level sets of are pseudoconvex, it follows from (61) that

Recall that is the term which contains only or variables in the first summation part of (14). Therefore it follows, from (17), (19), and (23), that

because . If , it follows from (61) and (71) and by using the Taylor series method that

provided is sufficiently small. Thus (69) follows from (68), (70), and (72).

In the sequel, we let and be the constants which may different from time to time but depend only on the derivatives of or up to order . Recall that , . By using (61), and by using Taylor series method, one obtains that

provided is sufficiently small, where and . Note that we can write

where . By applying or successively to , we obtain that

where, by (73) and by using induction method, satisfies

Combining the estimate in (61), (75), and (76), one obtains that

Assume that (59) holds. Thus for some , and hence it follows from (53) that . Therefore it follows from (23) and (50) that there exist integers with , such that

For these , it follows from (61), (75), (76), and (78) and by using the Taylor series method that there are constants such that

provided is sufficiently small.

Lemma 19. There is such that

Proof. By functoriality, we have

From (23), we see that

and it follows from (61) that

Therefore (80) follows from (73), (81), and (83) and by using Taylor series method.

Note that , for some , and hence there exist , with . In view of (79), we may assume that

is valid (when , we replace by ). Set

By using the estimates (73)–(76), one obtains that

and similarly,

for , where and .

Lemma 20. Assume that (59) holds. Then

Proof. Suppose . In view of (51)–(53), (56), and (59), we see that

and hence it follows that

This together with (73)–(78) implies the estimate (88).

In the sequel, we write

Lemma 21. There is a positive number , independent of and , such that if and if (59) holds, then there are constants and , independent of , and , such that

Proof. Suppose . From (87) and (88), we note that

for where and . Using (84)–(88) and (93) and by using small (large) constant method, one obtains that

for some and provided is sufficiently small.

Remark 22. From now on, we fix constants and , which depend only on the derivatives of or of order up to on , satisfying (69), (73), (80), and (86)–(92), and set for a convenience. Now we choose and fix and then fix so that

3. Estimates on the Bergman Kernels

Recall that where and where is the projection defined before (19). Also note that where and where is the holomorphic coordinate function defined in Proposition 11 about . Also recall defined in (20). In this section we estimate the Bergman kernel function , for and .

To get optimal estimates of the Bergman kernel, we need to construct a plurisubharmonic function which has maximal Hessian near each thin neighborhood of as in [2, 15]. It contains complicated estimates depending on the type conditions of each boundary points. In this paper, however, we will construct such functions only at . This will make the estimates much simpler than those in [2, 15] but still contain many complicated estimates.

Note that and are fixed in Remark 22 and hence the type and the integer defined in (59) depend only on . Recall that , , , and . From (54) we have

Let us write .

Proposition 23. There exist a smooth plurisubharmonic function on that satisfies the following:
(i) , for , and is supported in .
(ii) There exist a small constant such that if , then

(iii) If where is defined in (22), then

holds for all where .

Proof. For each fixed , we note that the integers and , defined in (59), will be fixed. Set and . Note that provided is sufficiently small. Since , it follows from (80) that

for . From now on, we fix and set .
We may assume that the level sets of are pseudoconvex on and on , where we may assume that . Also by (95). Therefore it follows from (69) and (99) that

for .
Let be defined by

where is a smooth function such that for and for , satisfying . Set . Note that has similar expression as in (22). Thus it follows, from (22), (29), (30), and chain rule, that

Here and . Since , one obtains

where and .
Suppose that satisfies . Using the fact that , , and the fact that , it follows from (100)–(103) that

We note that the negative part in (104) contains instead of .
Let be a smooth convex function such that for and for and . Set and set . Suppose . Then , and hence (79) holds for with ; that is,

For those with , it follows from (104) (with ) and (105) that

If , then and hence . Hence is a smooth plurisubharmonic function supported on , and .
Now assume and assume that (59) holds. Then (79) holds for some positive integers with . Let be the function defined in (85). From (88) and (95), we see that

, because and . Set

where is a smooth function that satisfies , for , for , and for all . Thus and because . By (107) we note that on , and we also note that if , in particular, outside . From (92), we obtain that

for , because .
Assuming that , we note that the negative coefficient part of of the Hessian of in (104) is controlled by the first term in the third line of (109), and the error terms of the coefficients of and in the third line of (109) are controlled by the corresponding coefficients of the Hessian of in (104). In either or cases, it follows from (104), (106), and (109) that

for .
Note that parameters, , , , , and , are fixed in Remark 22, independent of . Therefore the upper bound of follows from (84)–(88), (96), (99), (102), and (103). Note that , if , and this property holds on if we take sufficiently small; say, . Also note that on . This fact together with (96) and (110) proves properties (i) and (ii). Property (iii) follows from (22), (30), (32), and (96).

For each , set .

Proposition 24. There is a small constant such that for all sufficiently small .

Proof. From (22)–(29), we obtain that

for all sufficiently small . Assume and write

From (61), and by using Taylor series method, one obtains that

for a uniform constant . Similarly, we obtain . Combining these estimates and (111) and if we set , then we obtain that

Remark 25. (1) Set . Then, by functoriality, Proposition 23 holds, where is replaced by , and is replaced by .

For each fixed , and for each fixed , set . Note that on . Thus it follows, from transformation formula, that

In view of Propositions 23 and 24, there is a smooth plurisubharmonic weight function which has maximal Hessian on . We also note that by (56). If we use these properties and (115), we get the following estimates for the Bergman kernel function at as in Theorem 6.1 in [2]:

This proves Theorem 2.

Now we want to get derivative estimates of for and . In view of (115), we will estimate where and . We will follow the methods in [3, 9] which use dilated coordinates. For each fixed , we recall that , and . Define a dilation map given by

set

and set

where and where is defined in Proposition 23.

Set

and write . The properties of , which follow from Propositions 23 and 24 and Remark 25, are summarized in the following proposition.

Proposition 26. For each there is , defined on , such that
(1) is smooth plurisubharmonic in , and ;
(2) , for some ;
(3) if ;
(4) .

The weight function with the properties in Proposition 26 is the key ingredient for the derivative estimates of the Bergman kernel function off the diagonal. Set and let be the Neumann operator on . Then we have the following estimates of (Proposition 3.14 in [3]).

Proposition 27. Let be a form and . Then there is , independent of , so that

Note that . Set

From (117) and Proposition 24, we note that

independent of . Let with in a neighborhood of and on . From (123), we see that , independent of . Therefore we have the following elliptic estimates:

where is the complex Laplacian on .

Remark 28. The estimates in (124) are on the polydisc , strictly inside of , independent of . Therefore we gain two derivatives in (124) and it is stable; that is, is independent of . Also we note that we do not require that . Since where , we can also apply the estimate (121) on .

Let , , and be polyradial. In terms of -coordinates in (117), we have the following well known representation of Bergman kernel function on .

Let with outside and on . Combining (121)–(125), we can prove the following lemma as in the proof of Theorem 4.2 in [3].

Lemma 29. For each there is such that

Now, if we use the estimate (126) with , we can prove Theorem 3 as in the proof of Theorem 4.2 in [3].

Appendix

We recall Herbort’s example in (6). Therefore and hence , , and in our notations. For each fixed , set . Then and approaches to in “almost tangential direction” as the points do along . In this case, we will show that

which is exactly same result as Theorem 2.

Proof of (A.1). Set

Then the polydisc about is contained in . Therefore the upper bound follows. Let us show lower bounds.
Let be the unit polydisc in . Since the localization lemma is valid for , we will estimate . Set

and set

Then and . Set . Then . Note that

where we have used . Set and . Then

where we have used the polar coordinates: , in the second line.
Set and . Then

Write . Set . Then

Also,

Combining (A.5)–(A.9), we obtain that . Therefore .

Remark 30. Set . Then is a peak function that peaks at for the domain .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author was partially supported by the Sogang University Research Fund.

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