On Sharp Hölder Estimates of the Cauchy-Riemann Equation on Pseudoconvex Domains in C n with One Degenerate Eigenvalue

and Applied Analysis 3


Introduction
For any open set  ⊂ C  , we let Λ  () denote the space of functions in Hölder class  ≥ 0 on .Let Ω be a smoothly bounded pseudoconvex domain in C  and  0 ∈ Ω.Suppose that there exists a neighborhood  of  0 such that, for all closed forms , with  ∈ Λ  (Ω), we can solve  =  in Ω with a gain of regularity of the solution ; that is, for some  > 0. In this event, we want to find a necessary condition and determine how large  can be.When  0 ∈ Ω, it is well known that  = 1.However, when  0 ∈ Ω,  > 0 depends on the boundary geometry of Ω near  0 .Note that the Hölder estimates of -equation are well known when Ω is bounded strongly pseudoconvex domain in C  .However, for weakly pseudoconvex domains in C  , Hölder estimates are known only for special pseudoconvex domains, that is, pseudoconvex domains of finite type in C 2 , convex finite type domains in C  , and pseudoconvex domains of finite type with diagonal Levi-form in C  , and so forth.Proving Hölder estimates for general pseudoconvex domains in C  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 [1][2][3][4].Let  BG ( 0 ), the "Bloom-Graham" type, be the maximum order of contact of Ω with any ( − 1)-dimensional complex analytic manifold at  0 .If  BG ( 0 ) = , then Krantz [2] showed that  ≤ 1/.Krantz's result is sharp for Ω ⊂ C 2 and when  is a (0,  − 1)-form.Also McNeal [3] proved sharp Hölder estimates for (0, 1)-form  under the condition that Ω has a holomorphic support function at  0 ∈ Ω.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 [4].
Straube [5] proved necessary condition for Hölder regularity gain of Neumann operator .More specifically, if Neumann operator  has Hölder regularity gain of 2, then  ≤ 1/, where  is larger than or equal to order of contact of an analytic variety (possibly singular)  at  0 .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  2 -Sobolev topology.
Let Ω = { : () < 0}, where  is a smooth defining function of Ω, and let  be a smooth 1-dimensional analytic variety passing through  0 ∈ Ω.We say  has order of contact larger than or equal to  with Ω at  0 ∈ Ω if there is a positive constant  > 0 such that for all  ∈  sufficiently close to  0 .Here smooth means that   (0) ̸ = 0 if () represents a parametrization of .Recently, the second author, You [6], proved a necessary condition for Hölder estimates for bounded pseudoconvex domains of finite type in C 3 .That is, if there is a 1-dimensional smooth analytic variety  passing through  0 ∈ Ω and the order of contact of  with Ω is larger than or equal to  > 0, then the gain of the regularity in Hölder norm should be less than or equal to 1/.To get a necessary condition for Hölder estimates, we first need a complete analysis of boundary geometry near  0 ∈ Ω of finite type.
In this paper we prove a necessary condition for the sharp Hölder estimates of -equation near  0 ∈ Ω when Ω is a smoothly bounded pseudoconvex domain in C  and the Leviform of Ω at  0 ∈ Ω has ( − 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 C  and assume that the Levi-form of Ω at  0 ∈ Ω has ( − 2)-positive eigenvalues.Assume that there is a smooth holomorphic curve  whose order of contact with Ω at  0 ∈ Ω is larger than or equal to .If there exists a neighborhood  of  0 and a constant  > 0 so that, for each  ∈  0,1 ∞ (Ω) with  = 0, there is a  ∈ Λ  ( ∩ Ω) such that  =  and then  ≤ 1/.
To prove Theorem 1 we use the analysis of the local geometry near  0 ∈ Ω in [7] and use the method developed in [6].In particular Proposition 4 is a key coordinate change which shows that  1 which represents the smooth variety  and the terms mixed with  1 and strongly pseudoconvex directions vanishes up to order  := [( + 1)/2], where [] denotes the largest integer less than or equal to .

Special Coordinates
Let (Ω,  0 , ) be as in the statement of Theorem 1 and let  be a smooth defining function of Ω near  0 .We may assume that there is a coordinate system z = (z 1 , . . ., z ) about  0 such that  0 = 0 and |/z  | ≥  > 0, for some constant  > 0, in a small neighborhood  of  0 .In this section, we construct special coordinates  = ( 1 , . . .,   ) near  0 ∈ Ω which change the given smooth holomorphic curve  into the  1 -axis.We will exclude the trivial case,  = 2, and hence we assume that  ≥ 3 is a positive integer.Set  := [(+1)/2].
As in the proof of Proposition 2.2 in [7], after a linear change of coordinates followed by standard holomorphic changes of coordinates, we can remove inductively the pure terms such as z 1 , z 1 terms as well as z 1 z , z 1 z terms, 2 ≤  ≤  − 1, in the Taylor series expansion of (z) so that (z) can be written as ã, z Re (ã where z = (z 2 , . . ., z−1 ).Let  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).
Proof.The proof is similar to the proof of Lemma 2.3 in [6].
Since (0) = 0,   () vanishes to order  > 0. Suppose that  < ; that is,   () =     + O( +1 ) for  < .In terms of  coordinates in (4), we can write Since (()) vanishes to order at least , 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.

A Construction of Special Functions
Let us take the coordinates  = ( 1 , . . .,   ) defined in Proposition 4 near  0 ∈ Ω.In this section, we construct a family of uniformly bounded holomorphic functions {  } >0 with large derivatives in   -direction along some curve Γ ⊂ Ω defined in (39).
To connect the pushed out part  ,, and Ω  , we use a bumping family {Ω   } 0≤≤ ⊂ C −1 with front   as in Theorem 2.3 in [11] or Theorem 2.6 in [10] (again the construction of a bumping family is much simpler because Ω  is uniformly strongly pseudoconvex).Set Then   , becomes a pseudoconvex domain in C −1 which is pushed out near the origin provided  > 0 and  > 0 are sufficiently small.In the sequel, we fix these  0 and  0 and we note that these choices of  0 and  0 > 0 are independent of  > 0. Set   := According to Section 3 of [10], or by a method similar to dimension two case of [9], there exists for some  ∈ R independent of  where  is taken so that (0, . . ., 0, −/2) ∈ Ω  ⊂ C −1 .Note that   is independent of  1 .