Let Ω be a smoothly bounded pseudoconvex domain in and assume that where z₀ ∈ bΩ, the boundary of Ω. Then we get optimal estimates of the Bergman kernel function along some “almost tangential curve” C_b(z₀, δ₀) ⊂ Ω ∪ {z₀}.

1. Introduction

Let Ω be a bounded domain in

. A natural operator on Ω is the orthogonal projection

(1)

where H(Ω) denotes the holomorphic functions on Ω. There is a corresponding kernel function K_Ω(z, w), called the Bergman kernel function on Ω. The nature of the singularity of K_Ω(z, w) 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 Ω where we have a plurisubharmonic function with maximal Hessian. For strongly pseudoconvex domains in , these polydiscs are of size δ > 0 in normal direction and of size δ^1/2 in tangential directions. For weakly pseudoconvex domains, the size of the polydisc in tangential directions depends on the boundary geometry of Ω near z₀ ∈ bΩ, and hence we need complete analysis of the boundary geometry near z₀.

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 (n − 2) 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 , n > 2, are not known, even for n = 3 case.

Assume that Ω is a smoothly bounded domain in

with smooth defining function r with smooth boundary, bΩ. Regular finite 1-type at z₀ ∈ bΩ, denoted by

, is the maximum order of vanishing of r∘γ for all one complex dimensional regular curve γ, γ(0) = z₀, and γ^′(0) ≠ 0. Thus

satisfies

(2)

where Δ_q(z₀), 1 ≤ q ≤ n − 1, denotes finite q-type in the sense of D’Angelo [13]. Note that Δ_n−1(z₀) = T_BG(z₀) where T_BG(z₀) is the type in the sense of Bloom-Graham.

Remark 1. Consider the domain Ω [13] in defined by

(3)

Then

and Δ₂(0) = 4 while Δ₁(0) = ∞ as the complex analytic curve γ(t) = (t³, t², 0) lies in the boundary. Note that γ(t) is not regular curve.

In the sequel, we let Ω be a smoothly bounded pseudoconvex domain in , and assume that where z₀ ∈ bΩ. Let C_b(z₀, δ₀) ⊂ Ω ∪ {z₀} be the “almost tangential curve" connecting a point and z₀ ∈ bΩ as defined in (20). Note that dist(z^δ, bΩ) ≈ δ for each z^δ ∈ C_b(z₀, δ₀). Set τ₁ = δ^1/η, τ₂ = τ(z^δ, δ) where τ(z^δ, δ) is defined in (51).

Theorem 2. Let Ω be a smoothly bounded pseudoconvex domain in and assume that where z₀ ∈ bΩ. Then K_Ω(z^δ, z^δ), the Bergman kernel function of Ω at z^δ ∈ C_b(z₀, δ₀), satisfies

(4)

Theorem 3. Let Ω and z₀ ∈ bΩ be as in Theorem 2. For each α = (α₁, α₂, α₃), there is a constant C_α > 0, independent of δ > 0, such that

(5)

for z ∈ Ω and z^δ ∈ C_b(z₀, δ₀).

Remark 4. (1) In Theorems 2 and 3, we do not assume that Δ₁(z₀) < ∞, 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, C_b(z₀, δ₀), but not normal or arbitrary direction.

(2) In [14], Herbort gives an example of a domain where the Bergman kernel grows logarithmically when z ∈ Ω approaches to z₀ ∈ bΩ in normal direction. Set

(6)

and for each small δ > 0, set z^δ = (0,0, −δ). Thus z^δ approaches to 0 ∈ bΩ_H in normal direction as δ → 0. In this case, Herbort shows that K_Ω(z^δ, z^δ) ≈ δ⁻³(−log⁡δ) ⁻¹; that is, the kernel grows logarithmically. For the same domain Ω_H in (6), we note that

and hence τ₁ = δ^1/6, τ₂ = δ^1/3 in (4). Set C_b(z₀, δ₀)≔{(δ^1/6/2, 0, −δ) : 0 ≤ δ ≤ δ₀}. Then z^δ≔(δ^1/6/2, 0, −δ) ∈ C_b(z₀, δ₀) approaches to 0 ∈ bΩ in “almost tangential direction”. In the Appendix of this paper, we will show that

(7)

In Section 2, we will construct special coordinates which reflect the regular finite type condition, , and then show that r(z) vanishes to order η in z₁-direction. We then consider the slices of Ω by fixing z₁. Then the domains become domains in , and hence we can handle them. Also, the condition Δ₂(z₀) < ∞ acts like the condition Δ₁(z₀) < ∞ on these slices.

For the estimates of K(z, w), Catlin [2, 15] constructed plurisubharmonic functions with maximal Hessian near each thin δ-strip of bΩ (Section 3 of [2]). In this paper, however, we will construct these functions only on nonisotropic polydiscs Q_aδ(z^δ)⊂⊂Ω for each z^δ ∈ C_b(z₀, δ₀) (Proposition 23). This avoids complicated technical parts in Section 3 of [2]. To get estimates of K_Ω(z, z^δ), z ∈ Ω, z^δ ∈ C_b(z₀, δ₀), we consider dilated domains D_δ for each δ > 0. Then the polydisc Q_aδ(z^δ)⊂⊂Ω becomes P(0,1)⊂⊂D_δ, independent of δ > 0, where P(0,1) is a polydisc of radius one with center at the origin. Therefore the uniform 1/2-subelliptic estimates for -equation hold on P(0,1), and the estimates for K_Ω(z, z^δ) follow.

Remark 5. Let Ω be a smoothly bounded pseudoconvex domain in , and assume that Δ₁(z₀) < ∞, where z₀ ∈ bΩ. 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 Ω, for example, the existence of peak function for Ω that peaks at z₀ ∈ bΩ or necessary conditions for the Hölder estimates for -equation.

2. Special Coordinates

In the sequel, we let Ω be a smoothly bounded pseudoconvex domain in and assume that , z₀ ∈ bΩ. Note that m and η are positive integers. Without loss of generality, we may assume that z₀ = 0. In the sequel, we let α = (α₁, α₂, α₃) and β = (β₁, β₂, β₃) be multi-indices and set α^′ = (α₁, α₂) and z^′ = (z₁, z₂), etc. In Theorem 3.1 in [16], You constructed special coordinates which represent the local geometry of bΩ near z₀.

Theorem 6. Let Ω be a smoothly bounded pseudoconvex domain in with smooth defining function r and assume , 0 ∈ bΩ. Then there is a holomorphic coordinate system z = (z₁, z₂, z₃) about 0 such that

(8)

where

(9)

(Idea of the proof) by the standard holomorphic coordinate changes, r(w) has the Taylor series expansion as in (8). Since , there is a regular curve which we may assume that satisfying |r(γ(t))| ≈ |t|^η for all sufficiently small . Set z = (w₁, w₂ + γ₂(w₁), w₃). Then, in z coordinates, r(z) has representation satisfying (8). Also (9) follows from the condition that m = Δ₂(0).

Remark 7. (1) The second condition in (8) and property (9) say that r(z) vanishes to order η along z₁ axis and order m along z₂ axis.

(2) There are much more terms (mixed with z₁, z₂ and their conjugates), compared to the h-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 z₁ and z₂ variables in the summation part of (8). Using the notations of Section 3.2 in [16], set

(10)

Then there are (p_ν, q_ν) ∈ S for ν = 0,1, …, N and η_ν, λ_ν > 0 for ν = 1, …, N, such that

(11)

Remark 8. (1) Here, p_ν’s and q_ν’s are the exponents of z₁ and z₂, respectively, in the dominating terms in the summation part of (8).

(2) If Δ₁(z₀) < ∞, 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 t₀ = η. If 1 ≤ k ≤ m, then q_ν−1 < k ≤ q_ν for some ν = 1, …, N. In this case, set

(12)

Then (p_ν−1, q_ν−1), (t_l, l), and (p_ν, q_ν) are colinear points in the first quadrant of the plane, and λ_ν (resp., η_ν) is the intercept of q-axis (resp., p-axis) of this line. Let L_ν be the line segment from (p_ν−1, q_ν−1) to (p_ν, q_ν) for ν = 1, …, N, set L = L₁ ∪ L₂ ∪ …∪L_N, Γ_L = {(α^′, β^′) ∈ Γ; (α₁ + β₁, α₂ + β₂) ∈ L}, and set

(13)

As in Corollary 3.8 and Remark 3.9 in [16], we can rewrite (8) so that

(14)

where

is a nontrivial homogeneous polynomial of degree p_ν given by

(15)

and there are a small constant a₀ > 0, and

such that

(16)

for all α₂ + β₂ = q_ν with all ν = 1, …, N − 1. Property (16) means that there is (α^′, β^′) with terms mixed in z₂ and

variables for |z₁ − d| < a₀. Let |d| = 1 be the constant (direction) in (16) and we will fix d in the rest of this paper. In the sequel, we set

equal to z_l or

, l = 1,2, 3.

Remark 9. (a) {t_k} defined in (12) is strictly decreasing on k.

(b) Each of the summation parts of (14) contains the terms of the form where (t_k, k)’s are the pairs, defined in (12), on the polyline L.

(d) Each term of the second summation part in (14) has terms mixed in z₂ and , and it corresponds to the pair of integers (p_ν, q_ν), the vertices of the polyline L.

Lemma 10. Let d₀(z₁)≔r(z₁, 0,0) be the term containing only z₁ or variables in the first sum of (14). Then

(17)

Proof. From (8) and (14), we see that . On the other hand, since the regular 1-type at 0 ∈ bΩ is equal to , there is such that .

In the sequel, we let V be a small neighborhood of z₀ = 0 ∈ bΩ where r(z) has expression as in (14). Since (∂r/∂z₃)(0) ≠ 0, we may assume that |(∂r/∂z₃)(z)| ≥ c₀ for all z ∈ V for a uniform constant c₀ > 0 by shrinking V if necessary. For each fixed δ > 0 and for each z = (z₁, z₂, z₃) ∈ V satisfying |z₁ − dδ^1/η| < γδ^1/η, for a sufficiently small γ > 0 to be chosen, we set

, where π(z) is the composition of the projection onto z₁z₃ plane and then the projection onto bΩ along the Re⁡z₃ direction. Using the Taylor series method in z₃ variable about e_δ, we see that

(18)

Since |e_δ| ≪ 1 and

on V, it follows from (17) that

(19)

for z₁ near 0.

Now for each small δ > 0, set z(δ)≔(dδ^1/η, 0,0) and set

. For a small constant b > 0 to be chosen, set z^δ = (dδ^1/η, 0, e_δ − bδ) ∈ Ω, and for a fixed small δ₀ > 0 satisfying

, set

(20)

connecting

and z₀ = 0 ∈ bΩ.

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 z₂ or variables in the first summation part of r(z) in (14). We will fix z₁ variable and consider the coordinate changes only on z^′′ = (z₂, z₃) variables.

Proposition 11. For each fixed , there is a holomorphic coordinate system such that in the new coordinates ζ^′′ defined by

(21)

where

(22)

and where

, l = 2,3, …, m, depends smoothly on

, the function given by

satisfies

(23)

Proof. For , define

(24)

where w^′′ = (w₂, w₃). Then we have

(25)

Assume that (22) and (23) hold for l ≥ 2. That is, we have defined so that can be written as

(26)

If we define Φ^l = Φ^l−1∘ϕ^l, where

(27)

then

satisfies (26) for l replaced by l + 1. If we proceed up to l = m 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

(28)

where ρ_l is defined in the inductive step of the proof of Proposition 11. Set

(29)

and set ρ₀ = r. Then

and

(30)

To study the dependence of

and hence dependence of

about

in (23), we thus need to study the dependence of

variable. For a convenience, set

, i.e., remove tilde’s, and assume that

satisfies

(31)

for a sufficiently small γ > 0 to be chosen. In view of (19), we see that

satisfies (31). In following we let z be the given coordinates, and we let ζ be the coordinates obtained from holomorphic coordinates changes of z, as in l-th step of coordinate changes in the proof of Proposition 11. Also we let

(resp.,

), k = 1,2, 3, denote any partial derivative operator of order s with respect to z_k and

(resp., ζ_k and

) variables. According to the coordinate changes in Proposition 11, we note that

Proposition 12. Assume that satisfies (31). Then for each i = 0,1, …, m + 1, we have

(32)

and for each α₂, β₂ > 0 with α₂ + β₂ = q_ν, for some q_ν in (14), we have

(33)

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

(34)

and hence (32) follows for i = 0. Since ρ₁(z₁, ζ^′′) = r(z₁, ζ₂, z₃ + e₀ζ₃), it follows, from (31) and chain rule, that

(35)

because we are evaluating at

. This proves (32) for i = 1.

By induction, assume that (32) holds for i = 0,1, …, s. For the defined in (29), it follows, from (34) and induction hypothesis, that

(36)

for i = 1, …, s. Since we are evaluating at ζ₂ = 0, it follows, for s ≥ 1, that

(37)

By (30), (36), and (37) and by induction, (32) holds for i = s + 1 because t_k ≤ t_s if k ≥ s.

Now we prove (33). Assume α₂ + β₂ = q_ν with α₂ > 0, β₂ > 0 where (p_ν, q_ν) 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 z₂ or mixed with . Thus it follows from (14), (16), and (31) that

(38)

Since ρ₁(z₁, ζ^′′) = r(z₁, ζ₂, z₃ + e₀ζ₃), it follows from (31) and (38) that

(39)

Similarly, since ρ₂(z₁, ζ^′′) = ρ₁(z₁, ζ₂, ζ₃ + e₁ζ₂), it follows from (36) that

(40)

because

for q_ν ≥ 2. This proves (33) for i = 2.

By induction assume that (33) holds for i = 2, …, s. If k = q_ν = α_ν + β_ν with α_ν > 0 and β_ν > 0, that is, if has mixed derivatives of ∂/∂ζ₂ and , we note that (37) becomes

(41)

If k = q_ν ≤ s, (33) follows from (41) and induction hypothesis of (33). If q_ν > s, it follows, from (36), (41), and induction hypothesis of (33), that

(42)

because

for q_ν > s. Therefore (33) is proved for i = s + 1.

Recall the expression of ρ = ρ_m+1 and coefficient functions in (23).

Corollary 13. Assume that satisfies (31). Then

(43)

and if j + k = q_ν for some q_ν in (14), then

(44)

Proof. From (23) we see that

(45)

where

and j, k > 0. Hence it follows from (32) that

(46)

Assume q_ν = j + k ≤ m for some q_ν. Thus j, k > 0 and it follows from (23), (33), and (41) that

(47)

because

Remark 14. Suppose that q_ν−1 < l ≤ q_ν and p_ν ≤ t_l < p_ν−1. Then (p_ν, q_ν), (t_l, l), and (p_ν−1, q_ν−1) are colinear points. From the standard interpolation method, we have

(48)

for all sufficiently small a, b ≥ 0. Assume that j, k > 0 and l = j + k ≠ q_ν for any of ν = 1,2, …, N. Therefore it follows from (43) and (48) that

(49)

Therefore the terms of the form

, with j + k = q_ν for some q_ν, 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 1 in [2], for each

, set

(50)

In view of Remark 14, we will consider A_l(z₁) only for l = q_ν, 0 ≤ ν ≤ N − 1. From (9) and (44) we note that

(51)

because q₀ = m. For each sufficiently small δ > 0, set

(52)

and set

(53)

From (51) and (52), we see that if δ^′ < δ, then

(54)

Lemma 15. For each 0 < ϵ ≤ 1, .

Proof. Set and . Then

(55)

Therefore

because 0 < ϵ ≤ 1.

Proposition 16. Assume satisfies (31). Then

(56)

where

Proof. By (31), we note that |z₁| ≈ δ^1/η. Assume that . Then by (51). Therefore it follows, from (50) and (52), that

(57)

and hence it follows from (52) and (53) that

(58)

Thus

follows. Similarly, one can show that

Let 0 < σ < 1 be a small constant to be determined (in Remark 22). By Lemma 15,

for each 0 < σ < 1, independent of δ > 0. Therefore there is a smallest integer

, 0 ≤ s ≤ m − 1, such that

(59)

Then t_s = q_ν(s) for some q_ν(s) by (53). In following, for the fixed integer

in (59), set δ_s = σ^sδ,

, and τ₁ = δ^1/η as usual. If we define

, where Φ₃(ζ^′′) is defined in (22), we may regard that

is a biholomorphism. For each

, set

. For each small γ > 0, define

(60)

and set

and

when s = 0.

Proposition 17. The function satisfies

(61)

Proof. Recall that ρ = ρ_m+1, and |z₁| = δ^1/η in (32). When k = 0, it follows from (12) (t₀ = η) and (32) that

(62)

Assume 1 ≤ k ≤ m. Then by (12), q_ν−1 < k ≤ q_ν for some ν, and hence it follows that p_ν ≤ t_k ≤ p_ν−1. Therefore one obtains, from (48)–(52), that

(63)

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

(64)

Using the z coordinates defined in (14), set

(65)

Then L_k, k = 1,2, are tangential holomorphic vector fields and |L₃r| ≥ c₀ > 0 on V∩Ω for a uniform constant c₀ > 0. For any j, k with j, k > 0, define

(66)

In ζ-coordinates defined by

, set

, k = 1,2, 3 and set

, k = 1,2. If we define

(67)

then by functoriality,

(68)

Lemma 18. There is a small constant c₂ > 0 such that

(69)

provided γ > 0 is sufficiently small.

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

(70)

Recall that

is the term which contains only z₁ or

variables in the first summation part of (14). Therefore it follows, from (17), (19), and (23), that

(71)

because |z₁| = |dδ^1/η| = τ₁ = δ^1/η. If

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

(72)

provided γ > 0 is sufficiently small. Thus (69) follows from (68), (70), and (72).

In the sequel, we let c₂ and C₂ be the constants which may different from time to time but depend only on the derivatives of r or ρ up to order η. Recall that

, k = 1,2. By using (61), and by using Taylor series method, one obtains that

(73)

provided γ > 0 is sufficiently small, where

and

. Note that we can write

(74)

where

. By applying

successively to

, we obtain that

(75)

where, by (73) and by using induction method, R_j+k−1 satisfies

(76)

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

(77)

Assume that (59) holds. Thus t_s = q_ν(s) for some q_ν(s), and hence it follows from (53) that

. Therefore it follows from (23) and (50) that there exist integers j, k > 0 with j + k = t_s = q_ν(s), such that

(78)

For these j, k > 0, it follows from (61), (75), (76), and (78) and by using the Taylor series method that there are constants c₂, C₂ > 0 such that

(79)

provided γ > 0 is sufficiently small.

Lemma 19. There is C₂ > 0 such that

(80)

Proof. By functoriality, we have

(81)

From (23), we see that

(82)

and it follows from (61) that

(83)

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

Note that

, for some q_ν(s), and hence there exist j > 0, k > 0 with j + k = t_s. In view of (79), we may assume that

(84)

is valid (when j = 1, we replace

). Set

(85)

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

(86)

and similarly,

(87)

for

, where

and

Lemma 20. Assume that (59) holds. Then

(88)

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

(89)

and hence it follows that

(90)

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

In the sequel, we write

(91)

Lemma 21. There is a positive number σ > 0, independent of and δ, such that if and if (59) holds, then there are constants c₂ > 0 and C₂ > 0, independent of , δ and σ > 0, such that

(92)

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

(93)

for j, k = 1,2 where

and

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

(94)

for some c₂ > 0 and C₂ > 0 provided σ > 0 is sufficiently small.

Remark 22. From now on, we fix constants c₂ > 0 and C₂ > 0, which depend only on the derivatives of r or ρ of order up to η on V, satisfying (69), (73), (80), and (86)–(92), and set for a convenience. Now we choose and fix γ > 0 and then fix σ > 0 so that

(95)

3. Estimates on the Bergman Kernels

Recall that where z(δ) = (dδ^1/η, 0, −δ) and where π is the projection defined before (19). Also note that where and where is the holomorphic coordinate function defined in Proposition 11 about . Also recall C_b(z₀, δ₀) defined in (20). In this section we estimate the Bergman kernel function K_Ω(z, z^δ), for z ∈ Ω and z^δ ∈ C_b(z₀, δ₀).

To get optimal estimates of the Bergman kernel, we need to construct a plurisubharmonic function which has maximal Hessian near each thin neighborhood of bΩ 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 σ > 0 and γ > 0 are fixed in Remark 22 and hence the type t_s and the integer s defined in (59) depend only on

. Recall that δ_s = σ^sδ, τ₁ = δ^1/η,

, and

. From (54) we have

(96)

Let us write L = a₁L₁ + a₂L₂ + a₃L₃.

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 b > 0 such that if , then

(97)

(iii) If where Φ₃ is defined in (22), then

(98)

holds for all

where

Proof. For each fixed , we note that the integers and t_s, defined in (59), will be fixed. Set and . Note that provided δ > 0 is sufficiently small. Since δ_s = σ^sδ, it follows from (80) that

(99)

for

. From now on, we fix

and set λ_s = σ^−2sλ.

We may assume that the level sets of r are pseudoconvex on V and on V∩Ω, where we may assume that . Also 4C₂γ^1/2 ≤ c₂/10 by (95). Therefore it follows from (69) and (99) that

(100)

for

Let ψ(ζ) be defined by

(101)

where χ is a smooth function such that χ(t) = 1 for t < γ²/9 and χ(t) = 0 for t ≥ γ², satisfying |D^kχ| ≤ C_kγ^−2k. Set

. Note that

has similar expression as in (22). Thus it follows, from (22), (29), (30), and chain rule, that

(102)

Here α = (α₁, α₂, α₃) and |α| = α₁ + α₂ + α₃. Since

, one obtains

(103)

where

and

Suppose that z satisfies Ψ(z) ≥ 1/4. Using the fact that L_kr = 0, k = 1,2, and the fact that 84C₂γ^−9/2 = (c₂/5)λ ≤ (c₂/5)λ_s, it follows from (100)–(103) that

(104)

We note that the negative part in (104) contains γ^1/2λ instead of γ^1/2λ_s.

Let h be a smooth convex function such that h(t) = 0 for t ≤ 1/2 and h(t) > 0 for t > 1/2 and h(9/8) ≤ 1. Set and set . Suppose . Then s = 0, and hence (79) holds for δ_s = δ with j = k = 1; that is,

(105)

For those z with Ψ(z) ≥ 1/4, it follows from (104) (with λ_s = λ) and (105) that

(106)

If Ψ(z) ≤ 1/4, 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 j, k with j + k = t_s. Let G(z) be the function defined in (85). From (88) and (95), we see that

(107)

, because

and t_s ≤ m. Set

(108)

where ϕ(t) is a smooth function that satisfies ϕ(t) = t, for t ≤ 1/16, ϕ(t) = 0 for t ≥ 1, and ϕ(t) ≤ 1/8 for all t. Thus

and

because h(9/8) ≤ 1. By (107) we note that ϕ(z) = z on

, and we also note that

, in particular,

outside

. From (92), we obtain that

(109)

for

, because γ^1/2C₂ ≤ c₂/40.

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

(110)

for

Note that parameters, c₂, C₂, γ, σ, and λ, are fixed in Remark 22, independent of δ > 0. Therefore the upper bound of follows from (84)–(88), (96), (99), (102), and (103). Note that , if r(z)>−δ/4λ_s = −δσ^2s/4λ, and this property holds on if we take b > 0 sufficiently small; say, 0 < 2b < σ^2m/λ². Also note that Ψ = 1 on . This fact together with (96) and (110) proves properties (i) and (ii). Property (iii) follows from (22), (30), (32), and (96).

For each z^δ = (dδ^1/η, 0, e_δ − bδ) ∈ C_b(z₀, δ₀), set .

Proposition 24. There is a small constant a > 0 such that R_2aδ(ζ^δ)⊂⊂Ω for all sufficiently small δ > 0.

Proof. From (22)–(29), we obtain that

(111)

for all sufficiently small δ > 0. Assume ζ ∈ R_2aδ(ζ^δ) and write

(112)

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

(113)

for a uniform constant C₂ > 0. Similarly, we obtain |E₂| ≤ 4aC₂δ. Combining these estimates and (111) and if we set a = b/24C₂, then we obtain that

(114)

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

For each fixed δ > 0, and for each fixed

, set

. Note that

on V. Thus it follows, from transformation formula, that

(115)

In view of Propositions 23 and 24, there is a smooth plurisubharmonic weight function

which has maximal Hessian on Q_aδ(z^δ)⊂⊂Ω. We also note that

by (56). If we use these properties and (115), we get the following estimates for the Bergman kernel function K_Ω(z^δ, z^δ) at z^δ ∈ C_b(z₀, δ₀) as in Theorem 6.1 in [2]:

(116)

This proves Theorem 2.

Now we want to get derivative estimates of K(z, z^δ) for z ∈ Ω and z^δ ∈ C_b(z₀, δ₀). In view of (115), we will estimate

where

and

. We will follow the methods in [3, 9] which use dilated coordinates. For each fixed δ > 0, we recall that τ₁ = δ^1/η, τ₂ = τ(z^δ, δ) and τ₃ = δ. Define a dilation map D_δ given by

(117)

set

(118)

and set

(119)

where

and where

is defined in Proposition 23.

Set

(120)

and write

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

Proposition 26. For each δ > 0 there is λ_δ(w), defined on Ω_δ, such that

(1) λ_δ(w) is smooth plurisubharmonic in Ω_δ, and |λ_δ| ≤ 1;

(2) , for some ;

(3) if w ∈ P(0,1);

(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 N_δ be the Neumann operator on Ω_δ. Then we have the following L² estimates of N_δ (Proposition 3.14 in [3]).

Proposition 27. Let h ∈ L² be a (0,1) form and supp⁡h ⊂ P. Then there is C > 0, independent of δ > 0, so that

(121)

Note that D_δ(ζ^δ) = 0. Set

(122)

From (117) and Proposition 24, we note that

(123)

independent of δ > 0. Let

with ξ₁ = 1 in a neighborhood of 0 and ξ₂ = 1 on supp⁡ξ₁. From (123), we see that supp⁡ξ₂ ⊂ P(0,1)⊂⊂P(0,2)⊂⊂Ω_δ, independent of δ > 0. Therefore we have the following elliptic estimates:

(124)

where □_δ is the complex Laplacian on Ω_δ.

Remark 28. The estimates in (124) are on the polydisc P(0,1)⊂⊂P(0,2)⊂⊂Ω_δ, strictly inside of Ω_δ, independent of δ > 0. Therefore we gain two derivatives in (124) and it is stable; that is, C_s is independent of δ > 0. Also we note that we do not require that Δ₁(z₀) < ∞. Since where , we can also apply the estimate (121) on P(0,2).

Let

, ∫ϕ = 1, and ϕ be polyradial. In terms of w-coordinates in (117), we have the following well known representation of Bergman kernel function on Ω_δ.

(125)

Let χ ∈ C^∞(Ω_δ) with χ = 1 outside P(0,1) and χ = 0 on supp⁡ϕ. Combining (121)–(125), we can prove the following lemma as in the proof of Theorem 4.2 in [3].

Lemma 29. For each s ≥ 0 there is C_s > 0 such that

(126)

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

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.

Appendix

We recall Herbort’s example Ω_H in (6). Therefore η = 6 = Δ₁(0) and hence τ₁ = δ^1/6, τ₂ = δ^1/3, and τ₃ = δ in our notations. For each fixed δ > 0, set z^δ = (δ^1/6/2, 0, −δ). Then z^δ ∈ Ω_H and approaches to 0 ∈ bΩ_H in “almost tangential direction” as the points do along C_b(z₀, δ₀). In this case, we will show that

(A.1)

which is exactly same result as Theorem 2.

Proof of (A.1). Set

(A.2)

Then the polydisc P_δ(z^δ) about z^δ is contained in Ω_H. Therefore the upper bound

follows. Let us show lower bounds.

Let Δ₃ be the unit polydisc in . Since the localization lemma is valid for Ω_H, we will estimate . Set

(A.3)

and set

(A.4)

Then Ω_H = G₁ ⊂ G₂ and z^δ ∈ G₁∩G₂. Set f_δ = 8δ^11/6z₁/(z₃ − δ) ². Then f_δ(z^δ) = 1. Note that

(A.5)

where we have used

. Set

and

. Then

(A.6)

where we have used the polar coordinates:

, k = 1,2 in the second line.

Set and . Then

(A.7)

Write

. Set δ^−2/3x^1/2 = x^′. Then

(A.8)

Also,

(A.9)

Combining (A.5)–(A.9), we obtain that

. Therefore

Remark 30. Set f(z) = exp⁡(z₃/(1 − z₃)). Then f is a peak function that peaks at 0 ∈ bΩ_H for the domain Ω_H.

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

Abstract

1. Introduction

2. Special Coordinates

3. Estimates on the Bergman Kernels

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Appendix

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

Abstract

1. Introduction

2. Special Coordinates

3. Estimates on the Bergman Kernels

Conflicts of Interest

Acknowledgments

Appendix

Open Research

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References

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