AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2015/731068 731068 Research Article On Sharp Hölder Estimates of the Cauchy-Riemann Equation on Pseudoconvex Domains in Cn with One Degenerate Eigenvalue http://orcid.org/0000 0001 5441 4397 Cho Sanghyun 1 You Young Hwan 2 Zhu Chun-Gang 1 Department of Mathematics Sogang University Seoul 121-742 Republic of Korea sogang.ac.kr 2 Department of Mathematics Indiana University East Richmond IN 47374 USA iue.edu 2015 26112015 2015 08 10 2015 05 11 2015 2015 Copyright © 2015 Sanghyun Cho and Young Hwan You. 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.

Let Ω be a smoothly bounded pseudoconvex domain in Cn with one degenerate eigenvalue and assume that there is a smooth holomorphic curve V whose order of contact with bΩ at z0bΩ is larger than or equal to η. We show that the maximal gain in Hölder regularity for solutions of the ¯-equation is at most 1/η.

1. Introduction

For any open set UCn, we let Λδ(U) denote the space of functions in Hölder class δ0 on U. Let Ω be a smoothly bounded pseudoconvex domain in Cn and z0bΩ. Suppose that there exists a neighborhood U of z0 such that, for all ¯-closed forms α, with αΛδ(Ω), we can solve ¯u=α in Ω with a gain of regularity of the solution u; that is,(1)uΛδ+ϵUΩCαΛδΩ,for some ϵ>0. In this event, we want to find a necessary condition and determine how large ϵ can be. When z0Ω, it is well known that ϵ=1. However, when z0bΩ, ϵ>0 depends on the boundary geometry of Ω near z0.

Note that the Hölder estimates of ¯-equation are well known when Ω is bounded strongly pseudoconvex domain in Cn. However, for weakly pseudoconvex domains in Cn, Hölder estimates are known only for special pseudoconvex domains, that is, pseudoconvex domains of finite type in C2, convex finite type domains in Cn, and pseudoconvex domains of finite type with diagonal Levi-form in Cn, and so forth. Proving Hölder estimates for general pseudoconvex domains in Cn is one of big questions in several complex variables. Meanwhile, it is of great interest to find a necessary condition or optimal possible gain of the Hölder estimates for ¯.

Several authors have obtained necessary conditions for Hölder regularity of ¯ on restricted classes of domains . Let TBG(z0), the “Bloom-Graham” type, be the maximum order of contact of bΩ with any (n-1)-dimensional complex analytic manifold at z0. If TBG(z0)=N, then Krantz  showed that ϵ1/N. Krantz’s result is sharp for ΩC2 and when α is a (0,n-1)-form. Also McNeal  proved sharp Hölder estimates for (0,1)-form α under the condition that Ω has a holomorphic support function at z0Ω. Note that the existence of holomorphic support function is satisfied for restricted domains and it is often the first step to prove the Hölder estimates for ¯-equation .

Straube  proved necessary condition for Hölder regularity gain of Neumann operator N. More specifically, if Neumann operator N has Hölder regularity gain of 2ϵ, then ϵ1/η, where η is larger than or equal to order of contact of an analytic variety (possibly singular) V at z0. However, it should be emphasized that there is no natural machinery to pass between necessary conditions for Hölder regularity of ¯-Neumann operator and that of ¯, in contrast to the case of L2-Sobolev topology.

Let Ω={z:r(z)<0}, where r is a smooth defining function of Ω, and let V be a smooth 1-dimensional analytic variety passing through z0bΩ. We say V has order of contact larger than or equal to η with bΩ at z0bΩ if there is a positive constant C>0 such that(2)rzCz-z0η,for all zV sufficiently close to z0. Here smooth means that γ(0)0 if γ(t) represents a parametrization of V. Recently, the second author, You , proved a necessary condition for Hölder estimates for bounded pseudoconvex domains of finite type in C3. That is, if there is a 1-dimensional smooth analytic variety V passing through z0bΩ and the order of contact of V with bΩ is larger than or equal to η>0, then the gain of the regularity in Hölder norm should be less than or equal to 1/η. To get a necessary condition for Hölder estimates, we first need a complete analysis of boundary geometry near z0bΩ of finite type.

In this paper we prove a necessary condition for the sharp Hölder estimates of ¯-equation near z0bΩ when Ω is a smoothly bounded pseudoconvex domain in Cn and the Levi-form of bΩ at z0bΩ has (n-2)-positive eigenvalues. Our method used to prove the following main theorem will be useful for a study of necessary conditions of Hölder estimates of ¯-equation for other kinds of finite type domains.

Theorem 1.

Let Ω be a smoothly bounded pseudoconvex domain in Cn and assume that the Levi-form of bΩ at z0bΩ has (n-2)-positive eigenvalues. Assume that there is a smooth holomorphic curve V whose order of contact with bΩ at z0bΩ is larger than or equal to η. If there exists a neighborhood U of z0 and a constant C>0 so that, for each αL0,1(Ω) with ¯α=0, there is a uΛϵ(UΩ¯) such that ¯u=α and(3)uΛϵUΩ¯CαLΩ,then ϵ1/η.

To prove Theorem 1 we use the analysis of the local geometry near z0bΩ in  and use the method developed in . In particular Proposition 4 is a key coordinate change which shows that z1 which represents the smooth variety V and the terms mixed with z1 and strongly pseudoconvex directions vanishes up to order m:=[η+1/2], where [x] denotes the largest integer less than or equal to x.

Remark 2.

In general, we note that N:=TBG(z0)η. Thus we have ϵ1/η1/N in (3). We also note that η is a positive integer.

2. Special Coordinates

Let (Ω,z0,η) be as in the statement of Theorem 1 and let r be a smooth defining function of Ω near z0. We may assume that there is a coordinate system z~=(z~1,,z~n) about z0 such that z0=0 and |r/z~n|c>0, for some constant c>0, in a small neighborhood U of z0. In this section, we construct special coordinates z=(z1,,zn) near z0bΩ which change the given smooth holomorphic curve V into the z1-axis. We will exclude the trivial case, η=2, and hence we assume that η3 is a positive integer. Set m:=[η+1/2].

As in the proof of Proposition  2.2 in , after a linear change of coordinates followed by standard holomorphic changes of coordinates, we can remove inductively the pure terms such as z~1j, z~¯1k terms as well as z~1jz~α, z~¯1jz~¯α terms, 2αn-1, in the Taylor series expansion of r(z~) so that r(z~) can be written as(4)rz~=Rez~n+j+kη,j,k>0a~j,kz~1jz~¯1k+α=2n-1z~α2+α=2n-1j+km,k>0Rea~j,kαz~1jz~¯1kzα+Oz~nz~+z~2z~+z~z~1m+1+z~1η+1,where z~=(z~2,,z~n-1). Let V be the smooth 1-dimensional variety satisfying (2). Without loss of generality, we may assume that (2) is satisfied in z~-coordinates defined in (4). Let γ:CV, γ(t)=(γ1(t),,γn(t)), be a local parametrization of V. We may assume that γ1(0)0, and, hence, after reparametrization, we can write γ(t)=(t,γ2(t),,γn(t)) and it satisfies(5)rγtCtη.

Lemma 3.

γ n ( t ) vanishes to order at least η.

Proof.

The proof is similar to the proof of Lemma  2.3 in . Since γ(0)=0, γn(t) vanishes to order s>0. Suppose that s<η; that is, γn(t)=asts+O(ts+1) for s<η. In terms of z coordinates in (4), we can write (6)rγt=as2ts+a¯s2t¯s+j+kη+1,j,k>0cj,ktjt¯k+Ots+1. Since r(γ(t)) vanishes to order at least η, there must be some cancelation between the parenthesis part and summation part. However, this is impossible because parenthesis part consists only of pure terms while summation part consists of mixed power terms.

Proposition 4.

There is a holomorphic coordinate system z with Φ(z)=z~ such that, in terms of z coordinates, r~(z):=rΦ(z) can be written as(7)r~z=Rezn+j+k=η,j,k>0aj,kz1jz¯1k+α=2n-1zα2+α=2n-1j+k=m,k>0Reaj,kαz1jz¯1kzα+Oznz+z2z+zz1m+1+z1η+1,and it satisfies(8)r~t,0,,0,0tη.

Proof.

With z~-coordinates defined in (4), define Φ:CnCn, z~=Φ(z), by(9)Φz=z1,z2+γ2z1,,zn-1+γn-1z1,zn,and set r~(z)=rΦ(z). In terms of z coordinates, r~(z) can be written as(10)r~z=Rezn+j+kη,j,k>0aj,kz1jz¯1k+α=2n-1zα2+α=2n-12j+km,k>0Reaj,kαz1jz¯1kzα+Oznz+z2z+zz1m+1+z1η+1.Since γn(t) vanishes to order η, it follows from (5), (9), and (10) that (11)r~t,0,,0=rt,γ2t,,γn-1t,0tη, and hence (8) is proved. Also we note that (12)r~t,0,,0=j+kη,j,k>0aj,ktjt¯k+Otη+1, and hence aj,k=0, for j+k<η, because of (8). This fact together with (10) proves that the first summation part in (7) is homogeneous polynomial of order η.

Now we want to show that aj,kα=0, for j+k<m, in the third summation part in (7). On the contrary, let 0<s<m be the least integer such that aj,kα0 for some j+k=s and α. In order to show that this is a contradiction, we use variants of the methods in Lemma 4.1 and Proposition  4.4 in . For t with 0<t<1, define a scaling map(13)z=Htszt1/2sw1,t1/2w2,,t1/2wn-1,twn,and set ρst=t-1((Hts)r~) and then set ρ~=limt0+ρst. Note that 2s<η, and hence the first summation part in (7) will be disappeared in this limiting process. Also note that ρ~ is the limit in the C-topology of ρst which, for each t>0, is a defining function of a pseudoconvex domain Ωt, and hence ρ~ is a defining function of a pseudoconvex domain Ω~ given by(14)ρ~w=Rewn+α=2n-1wα2+Reα=2n-1Pαw1,w¯1wα,where Pα(w1,w¯1) is a plurisubharmonic, nonholomorphic, polynomial of order s provided it is nontrivial. Therefore the Hessian matrix A:=2ρ~/wjw¯k1j,kn-1 is semidefinite Hermitian matrix and hence detA0. Note that(15)detA=2Reα=2n-12Pαw1w¯1wα-α=2n-1Pαw¯120.Assume Pα is nontrivial for some α; say, α=2. For each |w1|<1, take an appropriate argument of w2 satisfying Re2P2/w1w¯1w20. By (15), it follows that Pα/w¯1=0 at w=(w1,w2,0,,0), and hence Pα is holomorphic function of w1 at w for each 2αn-1. This is a contradiction proving our proposition.

3. A Construction of Special Functions

Let us take the coordinates z=(z1,,zn) defined in Proposition 4 near z0bΩ. In this section, we construct a family of uniformly bounded holomorphic functions fδδ>0 with large derivatives in zn-direction along some curve ΓΩ defined in (39).

In the sequel, we set z=(z2,,zn) and z=(z2,,zn-1). We will consider slices of Ω in z1-direction. From (7), rδ(z):=r~(dδ1/η,z2,,zn) can be written as(16)rδz=Rezn+bηδ+k=2n-1zk2+α=2n-1Rebαδm/ηzα+Oznδ1/η+znz+z3+zδm+1/η+z2δ1/η+δ1+1/η,where bη=dηj+k=η,j,k>0aj,k and where aj,k’s are fixed constants in (7). Note that bηR1. Define (17)w=z,wn=zn+bηδ, and write w=z for a convenience. Then bηδ term is absorbed in the expression of (16).

Let π be the projection onto bΩ along zn-direction. Set zδ=(dδ1/η,0,,0) and set z~δ=π(zδ):=(dδ1/η,0,,0,z~n). Note that |z~n|δ. Define a biholomorphism Φδ:Cn-1Cn-1, Φδ(ζ)=z=(ζ,Φn(ζ)), by(18)ζ=z,Φnζ=ζn+z~n-α=2n-1bαδm/ηζα,and set ρδ(ζ):=rδΦδ(ζ). Then ρδ(0)=0, and, in terms of ζ coordinates, ρδ(ζ) can be written as (19)ρδζ=Reζn+k=2n-1ζk2+Oζnζ+ζ3+Oζnδ1/η+ζδm+1/η+ζ2δ1/η+δ1+1/η.

Set Ω~δ:=Ω{(dδ1/η,z2,,zn)}, the z1 slice of Ω, and set U~δ=U{(dδ1/η,z2,,zn)}. Also set Ωδ=Φδ-1(Ω~δ), and set Uδ={(dδ1/η,ζ);Φ(dδ1/η,ζ)U~δ}. Then Ωδ is pseudoconvex domain in Cn-1 and bΩδUδ is uniformly strongly pseudoconvex, independent of δ>0, provided U is sufficiently small. In the same manner as in Proposition  4.1 in  or Proposition  2.5 in  (our case is much simpler because bΩδUδ is uniformly strongly pseudoconvex independent of δ), we can push out bΩδ near z~δbΩδUδ uniformly independent of δ>0: For each small γ>0, set Bγ={ζ:|ζ|<γ}. Set (20)Jδζ=δ2+ζn2+k=2n-1ζk41/2, and for each small σ>0 we set (21)Wδ,a,σ=ζ:ρδζ<σJδζBa, where a>0 is chosen so that B2aUδ. Then Wδ,a,σ is the maximally pushed out domain of Ωδ near z~δ reflecting strong pseudoconvexity.

To connect the pushed out part Wδ,a,σ and Ωδ, we use a bumping family Ωδt0tτCn-1 with front Ba as in Theorem  2.3 in  or Theorem  2.6 in  (again the construction of a bumping family is much simpler because Ωδ is uniformly strongly pseudoconvex). Set (22)Dδ,σt=ΩδtBaWδ,a,σΩδt. Then Dδ,σt becomes a pseudoconvex domain in Cn-1 which is pushed out near the origin provided t>0 and σ>0 are sufficiently small. In the sequel, we fix these t0 and σ0 and we note that these choices of t0 and σ0>0 are independent of δ>0. Set Dδ:=Dδ,σ0t0Cn-1.

According to Section 3 of , or by a method similar to dimension two case of , there exists L2(Dδ) holomorphic function fδ satisfying(23)fδζn0,,0,-bδ21δ,for some bR independent of δ where b is taken so that (0,,0,-bδ/2)ΩδCn-1. Note that fδ is independent of z1.

Recall that the domains Ωδ or Dδ are the domains in Cn-1 obtained by fixing ζ1=dδ1/η. Define a biholomorphism Ψ:CnCn by(24)Ψζ1,ζ=ζ1,Φδζ,and set ρ(ζ)=r~Ψ(ζ). For a small constant 0<c<d to be determined, set (25)Pδ,cζ:ζ1-dδ1/η<cδ1/η,ζk<a1,  k=2,,n, where a1=a/2n. In terms of ζ coordinates, for each 0<σσ0, and for each 0<c<d, set (26)Ωδ,cσ=Pδ,cζ1,ζ:ρdδ1/η,ζ<σJδζ, which is obtained by moving Wδ,a,σ along ζ1 direction, and set (27)Ωδ,c=Pδ,cζ:ρζ<0. Note that Ωδ,cσ and Ωδ,c are small neighborhoods of zδ including ζ1 direction.

Lemma 5.

For sufficiently small c>0, we have Ωδ,cΩδ,cσ/2, or, equivalently,(28)ρdδ1/η,ζ-ρζ<σ2Jδζ,for  ζ=ζ1,ζPδ,c.

Proof.

Assume ζΩδ,c. Then(29)ρζ-ρdδ1/η,ζcδ1/ηmaxζ~1-dδ1/η<cδ1/ηD1ρζ~1,ζ.Note that Φδ is independent of ζ1. Since ρ(ζ)=r~Ψ(ζ), it follows from (7) and (24) that(30)D1ρζ~1,ζδ1-1/η+ζn+ζ2+δm-1/ηζδ-1/ηJδζ,because δ(m-1)/η|ζ|δ-1/η(δ2m/η+ζ2) and 2mη. Combining (29) and (30), we obtain (28) provided c>0 is sufficiently small.

For each σ>0 and a2>0, set Uδ,a2σ:=Ω¯δ,a2σ. Since fδ is independent of ζ1, we see that fδ is holomorphic on Ωδ,cσ. We will show that fδ is bounded uniformly on Uδ,a2σ/8 for some 0<a2<ca1 to be determined. For each q=(q1,q)Uδ,a2σ/8, set τ1=a2δ1/η, τk=a2Jδ(q)1/2, 2kn-1, and τn=a2Jδ(q), and define a nonisotropic polydisc Qa2(q) by (31)Qa2qζ:ζk-qk<τk,  1kn. In order to proceed as in Section 7 of , we first show the following lemma which is similar to Lemma  4.3 in .

Lemma 6.

There is an independent constant 0<a2<c such that(32)Qa2qUδ,a23σ/4Ωδ,cσ,for  q=q1,qUδ,a2σ/8.

Proof.

Assume ζ=(ζ1,ζ)Qa2(q). Then we have (33)Jδq2δ2+2ζn2+2ζn-qn2+8k=2n-1ζk4+ζk-qk48Jδζ2+2a2Jδq2+8k=2n-1a2Jδq2=8Jδζ2+8n-14a22Jδq2. If we take a2>0 so that (8n-14)a221/2, we obtain that (1/4)Jδ(q)Jδ(ζ). This shows that qQa~2(ζ), where a~2=4a2. By the same argument, we have Jδ(ζ)4Jδ(q) provided (8n-14)a~221/2. Therefore, if 0<a21/8·1/4n-7, we obtain that(34)14JδqJδζ4Jδq,for  ζQa2q.Since δm/ηδ1/2Jδ(ζ)1/2, it follows from (7) that(35)DkρζJδζ1/2,ζ=ζ1,ζQa2q,2kn-1.Combining (34) and (35), one obtains(36)ρdδ1/η,ζ~·ζ-qC2a21/2Jδq,for each ζ,ζ~Qa2(q), for some C2>0, where denotes the gradient of ζ=(ζ2,,ζn) variables.

Now we prove (32). Assume qUδ,a2σ/8 and ζQa2(q). Since ρ(dδ1/η,q)σ/8Jδ(q), we can write(37)ρdδ1/η,ζσ8Jδq+ρdδ1/η,ζ~ζ-q,for some ζ~=(ζ~1,ζ~)Qa2(q). Combining (34), (36), and (37), we obtain that (38)ρdδ1/η,ζσ2Jδζ+C2a21/2Jδq<3σ4Jδζ, provided 16C2a21/2<σ. This proves (32).

Let bη=dηj+k=η,j,k>0aj,k be the number in (16), and define(39)Γ=z:z=dδ1/η,0,,0,-bδ2-bηδ,δ>0.Then Γ=Ψ(Γ~), where Γ~={ζ:ζ=(dδ1/η,0,,0,-bδ/2)} and where Ψ is defined in (24), and b>0 is the number in (23). Note that ΓΩ for all sufficiently small δ>0 provided d>0 is sufficiently small.

Remark 7.

In the above discussion, σ>0 is any number such that 0<σσ0. Thus, in particular, we can fix σ=σ0.

Theorem 8.

f δ is bounded holomorphic function in Ωδ,a2σ/8 and, along Γ, fδ satisfies(40)fδζndδ1/η,0,,0,-bδ21δ,for some bR independent of δ.

Proof.

By (23) and (24), we already know that there is a L2 holomorphic function fδ on Ωδ,cσ satisfying estimate (40). We only need to show that fδ is bounded in Ωδ,a2σ/8. Assume qUδ,a2σ/8=Ω¯δ,a2σ/8Ωδ,cσ. Then Qa2(q)Ωδ,cσ by Lemma 6. Now if we use the mean value theorem on polydisc Qa2(q)Ωδ,cσ and the fact that fδL2(Ωδ,cσ) is holomorphic we will get the boundedness of fδ on Ω¯δ,a2σ/8.

4. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

Without loss of generality, we may assume that Ω={ζCn;ρ(ζ)<0}, where ρ(ζ)=ρΨ(ζ) and where Ψ is given in (24). Let f=fδ be the bounded holomorphic function in Ωδ,a2σ/8 defined in Theorem 8, and set α=¯gδ, where (41)gδ=ϕζ1-dδ1/ηcδ1/ηϕζ2a2ϕζ3a2ϕζna2f0,ζ2,,ζn and where (42) ϕ t = 1 , t 1 2 , 0 , t 3 4 . Note that(43)αLδ-1/η.Now set (44)hζ1,,ζn=uζ1,ζ2,,ζn-gδ, where uΛϵ(UΩ) solves ¯u=α as in the statement of Theorem 1, and hence h is holomorphic. Set q1δ(θ)=(dδ1/η+4/5cδ1/ηeiθ,0,,0,-bδ/2) and q2δ(θ)=(dδ1/η+4/5cδ1/ηeiθ,0,,0,-bδ), where θR. Let us estimate the lower and upper bounds of the integral (45)Hδ=12π02πhq1δθ-hq2δθdθ. From the definition of ϕ we have gδ(q1δ(θ))=gδ(q2δ(θ))=0, and it follows from (3) and (43) that(46)Hδ=12π02πuq1δθ-uq2δθdθδϵαLδϵ-1/η.

For the lower bound estimate, we start with an estimate of the holomorphic function f=fδ with a large nontangential derivative constructed in Theorem 8. For each sufficiently small δ>0, set ζδ=(0,,0,-bδ/2) and ζ~δ=(0,,0,-bδ), and set ζδ=(dδ1/η,ζδ) and ζ~δ=(dδ1/η,ζ~δ). Then Taylor’s theorem of f in ζn variable shows that (47)f0,,0,ζn=fζδ+fζnζδζn+bδ2+Oζn+bδ22. Now we take ζn=-bδ. Since f/ζnζδ1/δ, it follows that(48)fζ~δ-fζδ=fζnζδ-bδ2+Oδ21,for all sufficiently small δ>0. Returning to the lower bound estimate of Hδ, the mean value property, (3), (43), and (48) give us(49)Hδ=12π02πhq1δθ-hq2δθdθ=hζδ-hζ~δfζ~δ-fζδ-uζ~δ-uζδ1-δϵ-1/η,because gδ(ζδ)=f(ζδ) and gδ(ζ~δ)=f(ζ~δ). If we combine (46) and (49), we obtain that(50)1δϵ-1/η.If we assume ϵ>1/η and δ0, (50) will be a contradiction. Therefore, ϵ1/η.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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