Abstract and Applied Analysis
Volume 2015 (2015), Article ID 731068, 6 pages
http://dx.doi.org/10.1155/2015/731068

Research Article

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

Sanghyun Cho¹ and Young Hwan You²

¹Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
²Department of Mathematics, Indiana University East, Richmond, IN 47374, USA

Received 8 October 2015; Accepted 5 November 2015

Academic Editor: Chun-Gang  Zhu

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.

Abstract.

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

1. Introduction

For any open set , we let denote the space of functions in Hölder class on . Let be a smoothly bounded pseudoconvex domain in and . Suppose that there exists a neighborhood of such that, for all -closed forms , with , we can solve in with a gain of regularity of the solution ; that is,

for some . In this event, we want to find a necessary condition and determine how large can be. When , it is well known that . However, when , depends on the boundary geometry of near .

Note that the Hölder estimates of -equation are well known when is bounded strongly pseudoconvex domain in . However, for weakly pseudoconvex domains in , Hölder estimates are known only for special pseudoconvex domains, that is, pseudoconvex domains of finite type in , convex finite type domains in , and pseudoconvex domains of finite type with diagonal Levi-form in , and so forth. Proving Hölder estimates for general pseudoconvex domains in 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 [1–4]. Let , the “Bloom-Graham” type, be the maximum order of contact of with any -dimensional complex analytic manifold at . If , then Krantz [2] showed that . Krantz’s result is sharp for and when is a -form. Also McNeal [3] proved sharp Hölder estimates for -form under the condition that has a holomorphic support function at . 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 , then , where is larger than or equal to order of contact of an analytic variety (possibly singular) at . 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 -Sobolev topology.

Let , where is a smooth defining function of , and let be a smooth 1-dimensional analytic variety passing through . We say has order of contact larger than or equal to with at if there is a positive constant such that

for all sufficiently close to . Here smooth means that 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 . That is, if there is a 1-dimensional smooth analytic variety passing through and the order of contact of with is larger than or equal to , then the gain of the regularity in Hölder norm should be less than or equal to . To get a necessary condition for Hölder estimates, we first need a complete analysis of boundary geometry near of finite type.

In this paper we prove a necessary condition for the sharp Hölder estimates of -equation near when is a smoothly bounded pseudoconvex domain in and the Levi-form of at has -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 be a smoothly bounded pseudoconvex domain in and assume that the Levi-form of at has -positive eigenvalues. Assume that there is a smooth holomorphic curve whose order of contact with at is larger than or equal to . If there exists a neighborhood of and a constant so that, for each with , there is a such that and

then .

To prove Theorem 1 we use the analysis of the local geometry near in [7] and use the method developed in [6]. In particular Proposition 4 is a key coordinate change which shows that which represents the smooth variety and the terms mixed with and strongly pseudoconvex directions vanishes up to order , where denotes the largest integer less than or equal to .

Remark 2. In general, we note that . Thus we have in (3). We also note that is a positive integer.

2. Special Coordinates

Let be as in the statement of Theorem 1 and let be a smooth defining function of near . We may assume that there is a coordinate system about such that and , for some constant , in a small neighborhood of . In this section, we construct special coordinates near which change the given smooth holomorphic curve into the -axis. We will exclude the trivial case, , and hence we assume that is a positive integer. Set .

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 , terms as well as , terms, , in the Taylor series expansion of so that can be written as

where . Let be the smooth 1-dimensional variety satisfying (2). Without loss of generality, we may assume that (2) is satisfied in -coordinates defined in (4). Let , , be a local parametrization of . We may assume that , and, hence, after reparametrization, we can write and it satisfies

Lemma 3. vanishes to order at least .

Proof. The proof is similar to the proof of Lemma  2.3 in [6]. Since , vanishes to order . Suppose that ; that is, 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.

Proposition 4. There is a holomorphic coordinate system with such that, in terms of coordinates, can be written as

and it satisfies

Proof. With -coordinates defined in (4), define , , by

and set . In terms of coordinates, can be written as

Since vanishes to order , it follows from (5), (9), and (10) that

and hence (8) is proved. Also we note that

and hence , for , because of (8). This fact together with (10) proves that the first summation part in (7) is homogeneous polynomial of order .
Now we want to show that , for , in the third summation part in (7). On the contrary, let be the least integer such that for some and . In order to show that this is a contradiction, we use variants of the methods in Lemma 4.1 and Proposition  4.4 in [8]. For with , define a scaling map

and set and then set . Note that , and hence the first summation part in (7) will be disappeared in this limiting process. Also note that is the limit in the -topology of which, for each , is a defining function of a pseudoconvex domain , and hence is a defining function of a pseudoconvex domain given by

where is a plurisubharmonic, nonholomorphic, polynomial of order provided it is nontrivial. Therefore the Hessian matrix is semidefinite Hermitian matrix and hence . Note that

Assume is nontrivial for some ; say, . For each , take an appropriate argument of satisfying . By (15), it follows that at , and hence is holomorphic function of at for each . This is a contradiction proving our proposition.

3. A Construction of Special Functions

Let us take the coordinates defined in Proposition 4 near . In this section, we construct a family of uniformly bounded holomorphic functions with large derivatives in -direction along some curve defined in (39).

In the sequel, we set and . We will consider slices of in -direction. From (7), can be written as

where and where ’s are fixed constants in (7). Note that . Define

and write for a convenience. Then term is absorbed in the expression of (16).

Let be the projection onto along -direction. Set and set . Note that . Define a biholomorphism , , by

and set . Then , and, in terms of coordinates, can be written as

Set , the slice of , and set . Also set , and set . Then is pseudoconvex domain in and is uniformly strongly pseudoconvex, independent of , provided is sufficiently small. In the same manner as in Proposition  4.1 in [9] or Proposition  2.5 in [10] (our case is much simpler because is uniformly strongly pseudoconvex independent of ), we can push out near uniformly independent of : For each small , set . Set

and for each small we set

where is chosen so that . Then is the maximally pushed out domain of near reflecting strong pseudoconvexity.

To connect the pushed out part and , we use a bumping family 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 is uniformly strongly pseudoconvex). Set

Then becomes a pseudoconvex domain in which is pushed out near the origin provided and are sufficiently small. In the sequel, we fix these and and we note that these choices of and are independent of . Set .

According to Section 3 of [10], or by a method similar to dimension two case of [9], there exists holomorphic function satisfying

for some independent of where is taken so that . Note that is independent of .

Recall that the domains or are the domains in obtained by fixing . Define a biholomorphism by

and set . For a small constant to be determined, set

where . In terms of coordinates, for each , and for each , set

which is obtained by moving along direction, and set

Note that and are small neighborhoods of including direction.

Lemma 5. For sufficiently small , we have , or, equivalently,

Proof. Assume . Then

Note that is independent of . Since , it follows from (7) and (24) that

because and . Combining (29) and (30), we obtain (28) provided is sufficiently small.

For each and , set . Since is independent of , we see that is holomorphic on . We will show that is bounded uniformly on for some to be determined. For each , set , , , and , and define a nonisotropic polydisc by

In order to proceed as in Section 7 of [9], we first show the following lemma which is similar to Lemma  4.3 in [9].

Lemma 6. There is an independent constant such that

Proof. Assume . Then we have

If we take so that , we obtain that . This shows that , where . By the same argument, we have provided . Therefore, if , we obtain that

Since , it follows from (7) that

Combining (34) and (35), one obtains

for each , for some , where denotes the gradient of variables.
Now we prove (32). Assume and . Since , we can write

for some . Combining (34), (36), and (37), we obtain that

provided . This proves (32).

Let be the number in (16), and define

Then , where and where is defined in (24), and is the number in (23). Note that for all sufficiently small provided is sufficiently small.

Remark 7. In the above discussion, is any number such that . Thus, in particular, we can fix .

Theorem 8. is bounded holomorphic function in and, along , satisfies

for some independent of .

Proof. By (23) and (24), we already know that there is a holomorphic function on satisfying estimate (40). We only need to show that is bounded in . Assume . Then by Lemma 6. Now if we use the mean value theorem on polydisc and the fact that is holomorphic we will get the boundedness of on .

4. Proof of Theorem 1

Without loss of generality, we may assume that , where and where is given in (24). Let be the bounded holomorphic function in defined in Theorem 8, and set , where

and where

Note that

Now set

where solves as in the statement of Theorem 1, and hence is holomorphic. Set and , where . Let us estimate the lower and upper bounds of the integral

From the definition of we have , and it follows from (3) and (43) that

For the lower bound estimate, we start with an estimate of the holomorphic function with a large nontangential derivative constructed in Theorem 8. For each sufficiently small , set and , and set and . Then Taylor’s theorem of in variable shows that

Now we take . Since , it follows that

for all sufficiently small . Returning to the lower bound estimate of , the mean value property, (3), (43), and (48) give us

because and . If we combine (46) and (49), we obtain that

If we assume and , (50) will be a contradiction. Therefore,

Conflict of Interests

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

References

N. Kerzman, “Hölder and estimates on the solutions of in a strongly pseudoconvex domains,” Communications on Pure and Applied Mathematic, vol. 24, pp. 301–379, 1971.

S. G. Krantz, “Characterizations of various domains of holomorphy via estimates and applications to a problem of Kohn,” Illinois Journal of Mathematics, vol. 23, no. 2, pp. 267–285, 1979.

J. McNeal, “On sharp Hölder estimates for the solutions of the equations, Several complex variables and complex geometry,” in Proceedings of Symposia in Pure Mathematics, vol. 52, part 3, pp. 277–285, American Mathematical Society, Providence, RI, USA, 1989.

R. M. Range, “On Hölder estimates for on weakly pseudoconvex domains,” in Proceedings of International Conference on Onseveral Complex Variables, pp. 247–267, Cortona, Italy, 1976-1977.

E. J. Straube, “A Remark on hölder smoothing and subellipticity of the ∂-neumann operator,” Communications in Partial Differential Equations, vol. 20, no. 1-2, pp. 267–275, 1995.

Y. H. You, “Necessary conditions for Hölder regularity gain of equation in ,” http://arxiv.org/abs/1504.05432

S. Cho, “Boundary behavior of the Bergman kernel function on some pseudo convex domains in ,” Transactions of the American Mathematical Society, vol. 345, no. 2, pp. 803–817, 1994.

D. W. Catlin, “Boundary invariants of pseudoconvex domains,” Annals of Mathematics, vol. 120, no. 3, pp. 529–586, 1984.

D. W. Catlin, “Estimates of invariant metrics on pseudoconvex domains of dimension two,” Mathematische Zeitschrift, vol. 200, no. 3, pp. 429–466, 1989.

S. Cho, “Estimates of invariant metrics on some pseudoconvex domains in ,” Journal of the Korean Mathematical Society, vol. 32, pp. 661–678, 1995.

S. Cho, “Extension of complex structures on weakly pseudoconvex compact complex manifolds with boundary,” Mathematische Zeitschrift, vol. 211, no. 1, pp. 105–120, 1992.