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and Applied Analysis 3 In conjunction with multitype M(0) = (1,m,m3), 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 Γ = {(α, β) ; aα ,β ̸ = 0, m ≤ α + β ≤ η, and α , β > 0} S = {(p, q) ; α1 + β1 = p, α2 + β2 = q for some (α, β) ∈ Γ} ∪ {(η, 0)} . (10) Then there are (p], q]) ∈ S for ] = 0, 1, . . . , N and η], λ] > 0 for ] = 1, . . . , N, such that (1) (p0, q0) = (η, 0) , (pN, qN) = (0,m) , λN = m, η1 = η (2) p0 > p1 > . . . > pN and q0 < q1 < . . . < qN (3) λ1 < λ2 < . . . < λN and η1 > η2 > . . . > ηN (4) p]−1 η] + q]−1 λ] = 1 and p] η] + q] λ] = 1 (5) aα ,β = 0 if α1 + β1 η] + α2 + β2 λ] < 1 for each ] = 1, . . . , N. (11) Remark 8. (1) Here, p]’s and q]’s are the exponents of z1 and z2, respectively, in the dominating terms in the summation part of (8). (2) If Δ 1(z0) < ∞, then the expression in (8) will be similar to that of C case in [2], and hence we need not consider the above complicated pairs. Set t0 = η. If 1 ≤ k ≤ m, then q]−1 < k ≤ q] for some ] = 1, . . . , N. In this case, set tk = η] (1 − k λ]) , i.e., tk η] + k λ] = 1. (12) Then (p]−1, q]−1), (tl, l), and (p], q]) are colinear points in the first quadrant of the plane, and λ] (resp., η]) is the intercept of q-axis (resp., p-axis) of this line. Let L] be the line segment from (p]−1, q]−1) to (p], q]) for ] = 1, . . . , N, set L = L1∪L2 ∪ . . . ∪ LN, ΓL = {(α, β) ∈ Γ; (α1 + β1, α2 + β2) ∈ L}, and set Λ = {(α, β) ∈ ΓL; α + β = (p], q]) , α2 > 0, β2 > 0, ] = 1, . . . , N} . (13) As in Corollary 3.8 and Remark 3.9 in [16], we can rewrite (8) so that r (z) = Re z3 + ∑ ΓL−Λ aα,βzαzβ + N ∑ ]=1 ∑ α2+β2=q] α2>0,β2>0 M] α2,β2 (z1) zα2 2 zβ2 2 + O(z3 |z| + N ∑ ]=1 q] ∑ l=q]−1 z1[tl]+1 z2l + z2m+1) , (14) whereM] α2,β2(z1) is a nontrivial homogeneous polynomial of degree p] given by M] α2,β2 (z1) = ∑ α1+β1=p] aα1,β1zα1 1 zβ1 1 , (15) and there are a small constant a0 > 0, and d ∈ {z1 ∈ C; |z1| = 1} such that M] α2,β2 (z1) ̸ = 0 for z1 − d < a0, and M] α2,β2 (dδ1/η) ≈ δp]/η, (16) for all α2 + β2 = q] with all ] = 1, . . . ,N − 1. Property (16) means that there is (α, β) with terms mixed in z2 and z2 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 ?̂?l equal to zl or zl, l = 1, 2, 3. Remark 9. (a) {tk} defined in (12) is strictly decreasing on k. (b) Each of the summationparts of (14) contains the terms of the form ?̂?tk 1 ?̂?k 2 where (tk, k)’s are the pairs, defined in (12), on the polyline L. (c) Each term of the first summation part in (14) is pure in z2 or z2 variables. (d) Each term of the second summation part in (14) has terms mixed in z2 and z2, and it corresponds to the pair of integers (p], q]), the vertices of the polyline L. Lemma 10. Let d0(z1) fl r(z1, 0, 0) be the term containing only z1 or z1 variables in the first sum of (14). Then d0 (z1) ≈ z1η . (17) Proof. From (8) and (14), we see that |r(z1, 0, 0)| ≲ |z1|η. On the other hand, since the regular 1-type at 0 ∈ bΩ is equal to η = Treg Ω (0), there is c1 > 0 such that |r(z1, 0, 0)| ≥ c1|z1|η. 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 − dδ1/η| < γδ1/η, for a sufficiently small γ > 0 to be chosen, we set π(z) = (z1, 0, eδ) fl ?̃? ∈ bΩ, where π(z) 4 Abstract and Applied Analysis 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 eδ, we see that r (z1, 0, 0) = 2Re [∂r (?̃?) ∂z3 (−eδ)] + O (e2δ) . (18) Since |eδ| ≪ 1 and 2Re(∂r/∂z3) = 1 + O(|z|) ≥ 1/2 on V, it follows from (17) that eδ ≈ z1η , (19) for z1 near 0. Now for each small δ > 0, set z(δ) fl (dδ1/η, 0, 0) and set ?̃?δ = π(z(δ)) = (dδ1/η, 0, eδ) ∈ bΩ. For a small constant b > 0 to be chosen, set zδ = (dδ1/η, 0, eδ − bδ) ∈ Ω, and for a fixed small δ0 > 0 satisfying r(zδ0) < 0, set Cb (z0, δ0) fl {zδ: zδ = (dδ1/η, 0, eδ − bδ) , 0 ≤ δ ≤ δ0} ⊂ Ω ∪ {z0} , (20) connecting zδ0 ∈ V ∩ Ω and z0 = 0 ∈ bΩ. Following the same arguments as in the proof of Proposition 1.2 in [2], for each fixed ?̃? ∈ V, we can construct special coordinates about z so that, in terms of new coordinates, there is no pure terms in z2 or z2 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 ?̃? = (?̃?1, ?̃?2, ?̃?3) ∈ V, there is a holomorphic coordinate system z = Φz(ζ) = (ζ2, Φ3(ζ)) such that in the new coordinates ζ defined by Φz (ζ) = (?̃?2 + ζ2, Φ3 (ζ)) , (21) where Φ3 (ζ) = ?̃?3 + ( ∂r ∂z3 (?̃?)) −1 (ζ3 2 − m ∑ l=1 cl (?̃?) ζl 2) , (22) and where cl(?̃?), l = 2, 3, . . . , m, depends smoothly on ?̃?, the function given by ρ(?̃?1, ζ) fl r(?̃?1, Φz(ζ)) satisfies ρ (?̃?1, ζ) = r (?̃?) + Re ζ3 + m ∑ j+k=2 j,k>0 aj,k (?̃?1) ζj 2ζ2 + O (ζ3 ζ + ζ2m+1) . (23) Proof. For ?̃? ∈ V, define Φ1 (w) = (?̃?2 + w2, ?̃?3 + ( ∂r ∂z3 (?̃?)) −1 (w3 2 − ∂r ∂z2 (?̃?) w2)) , (24) wherew = (w2, w3). Then we have ρ2 (?̃?1, w) fl r (?̃?1, Φ1 (w)) = r (?̃?) + Rew3 + O (w2) . (25) Assume that (22) and (23) hold for l ≥ 2. That is, we have definedΦl−1 : C → C so that ρl(?̃?1, w) fl r(?̃?1, Φl−1(w)) can be written as ρl (?̃?1, w) = Rew3 + l−1 ∑ j+k=2 j,k>0 al−1 j,k (?̃?) wj 2wk2 + O (w3 w + w2l) . (26) If we defineΦl = Φl−1 ∘ φl, where φl (ζ) = (ζ2, ζ3 − 2 l! ∂ lρl ∂wl 2 (?̃?1, 0, 0) ζl 2) , (27) then ρl+1(?̃?1, ζ) fl r(?̃?1, Φl(ζ)) satisfies (26) for l replaced by l + 1. If we proceed up to l = m and set Φz = Φm = Φ1 ∘ φ2 ∘ ⋅ ⋅ ⋅ ∘ φm, then by setting ρ = ρm+1 = r(?̃?, Φz(⋅)), we see that (22) and (23) hold. In the sequel, we will use the coordinate changes in Proposition 11 only at ?̃? = (?̃?1, 0, ?̃?3) ∈ V, (in particular at ?̃?δ = (dδ1/η, 0, eδ) ∈ bΩ in Section 3). We want to study the dependence of Φz about ?̃?. For each ?̃? = (?̃?1, 0, ?̃?3) ∈ V, set c0(?̃?) fl (∂r/∂z3)(?̃?) = 1 + O(|?̃?|), and we note that cl (?̃?) = 1 l! ∂ lρl ∂wl 2 (?̃?1, 0, 0) , l = 1, 2, . . . , m, (28) where ρl is defined in the inductive step of the proof of Proposition 11. Set e0 (?̃?) = 12c0 (?̃?)−1 , e1 (?̃?) = −c0 (?̃?)−1 ∂r ∂z2 (?̃?) and el (?̃?) = −c0 (?̃?)−1 cl (?̃?) l = 2, . . . , m, (29) and set ρ0 = r. Then ρ1(?̃?1, ζ) = r(?̃?1, ζ2, ?̃?3 + e0ζ3) and ρi+1 = ρi (?̃?1, ζ2, ζ3 + ei (?̃?) ζi 2) , i = 1, 2, . . . , m. (30) To study the dependence of Φz and hence dependence of aj,k(?̃?1) about ?̃?1 in (23), we thus need to study the dependence of el(?̃?) on ?̃? variable. For a convenience, set ?̃? = (z1, 0, z3), i.e., remove tilde’s, and assume that ?̃? satisfies z1 − dδ1/η < γδ1/η, and z3 ≲ z1η , (31) Abstract and Applied Analysis 5and Applied Analysis 5 for a sufficiently small γ > 0 to be chosen. In view of (19), we see that ?̃?δ = (dδ1/η, 0, eδ) 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 Dsk (resp., Dsk), k = 1, 2, 3, denote any partial derivative operator of order swith respect to zk and zk (resp., ζk and ζk) variables. According to the coordinate changes in Proposition 11, we note that D1 = D1. Proposition 12. Assume that ?̃? = (z1, 0, z3) ∈ V satisfies (31). Then for each i = 0, 1, . . . , m + 1, we have Dl1 1Dk2ρi (?̃?) ≲ z1tk−l1 , 0 ≤ k ≤ m, 0 ≤ l1 ≤ η, (32) and for each α2, β2 > 0 with α2 + β2 = q], for some q] in (14), we have Dl1 1Dq] 2 ρi (?̃?) ≈ Dl1 1Dq] 2 r (?̃?) ≈ z1p]−l1 , l1 ≤ p], 2 ≤ i ≤ m + 1. (33) Proof. We will prove by induction on i. From (14), (17), and (31) one obtains Dl1 1Dk2r (?̃?) ≲ z1η−l1 + z1tk−l1 + z3 + z1[tk]+1−l1 ≲ z1tk−l1 , (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 Dl1 1Dk2ρ1 (?̃?) ≲ Dl1 1Dk2r (?̃?) + z3 ≲ z1tk−l1 , (35) because we are evaluating at ζ = (z1, 0, 0).This proves (32) for i = 1. By induction, assume that (32) holds for i = 0, 1, . . . , s. For the ei(?̃?) defined in (29), it follows, from (34) and induction hypothesis, that Dl1 1 ei (?̃?) ≲ l1 ∑ j=0 Dj1Di2ρi (?̃?) ≲ l1 ∑ j=0 z1ti−j ≲ z1ti−l1 , (36) for i = 1, . . . , s. Since we are evaluating at ζ2 = 0, it follows, for s ≥ 1, that Dl1 1Dk2ρs+1 (?̃?) = Dl1 1Dk2ρs (?̃?) , if k < s, and = Dl1 1Dk2ρs (?̃?) + O( l1 ∑ j=0 Dj1es (?̃?)) , if k ≥ s. (37) By (30), (36), and (37) and by induction, (32) holds for i = s+1 because tk ≤ ts if k ≥ s. Now we prove (33). Assume α2 + β2 = q] with α2 > 0, β2 > 0 where (p], q]) are the pairs corresponding to the second summation part of (14). Note that the first summation part of (14) will be annihilated byDq] 2 because it contains the pure terms of z2 or z2 mixed with ?̂?k 1 . Thus it follows from (14), (16), and (31) that D

abstract and applied analysis

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Estimates on the Bergman Kernels in a Tangential Direction on Pseudoconvex Domains in C3