Journal of Function Spaces
Volume 2015 (2015), Article ID 295759, 10 pages
http://dx.doi.org/10.1155/2015/295759

Research Article

Gain of Regularity in Extension Problem on Convex Domains

M. Jasiczak

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Ulica Umultowska 87, 61-614 Poznan, Poland

Received 19 March 2015; Accepted 18 June 2015

Academic Editor: Alberto Fiorenza

Copyright © 2015 M. Jasiczak. 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.

We investigate the extension problem from higher codimensional linear subvarieties on convex domains of finite type. We prove that there exists a constant such that on Bergman spaces with there appears the so-called “gain regularity.” The constant depends on the minimum of the dimension and the codimension of the subvariety. This means that the space of functions which admit an extension to a function in the Bergman space is strictly larger than , where is a subvariety.

1. Introduction

Let be a bounded pseudoconvex domain, a plurisubharmonic function, and a complex linear hyperplane. In [1], Ohsawa and Takegoshi proved that there is a constant depending only on the diameter of such that for any function holomorphic in with

there is a function satisfying and

The symbol stands for the measure induced on by the volume element of , that is, the -dimensional Lebesgue measure, , is the volume measure in . This result turned out to be extremely useful in both complex analysis and complex geometry (cf. references in [2] and a recent article by Demailly et al. [3]). Not surprisingly, it stimulated a lot of research. There are at least three natural ways of generalizing the Ohsawa-Takegoshi Theorem. The first one is to study the extension problem on different function spaces, for instance other spaces, with the prominent case of bounded extensions of bounded holomorphic functions (for strictly pseudoconvex domains, this problem was solved by Henkin [4] and Amar [5], for convex finite type domains and linear affine subvarieties by Diederich and Mazzilli in [6], and the case of varieties which are not necessarily linear was studied by Alexandre in [7]). The second one is to investigate dependence of the problem on the geometry of or . Lastly, it is also natural to ask when the extension can be realized by a linear operator.

Cumenge in [8] considered the extension problem in case of strictly pseudoconvex domains and subvarieties of codimension which are nonsingular and cut transversally. She proved that in this case any holomorphic function in , admits an extension to a holomorphic function which belongs to . The symbol stands for a function which defines the domain . Such a function is uniformly comparable with , when is smooth.

In order to motivate our study, let us restrict our attention to the case. Comparing with the Ohsawa-Takegoshi Theorem for , the result of Cumenge says that there is a strictly larger class of functions than which admit an extension to a holomorphic function in . There appears the so-called “gain of regularity.” Observe also that each complex codimension counts as one in the exponent of the weight . We say that the problem is isotropic in the case of strictly pseudoconvex domains. Such phenomena have been central in the whole -problem theory and PDE’s in general (a natural example here is subelliptic estimates [9–12] on finite type domains).

It came therefore as a surprise when Diederich and Mazzilli proved in [2] that there are elementary pseudoconvex domains and subvarieties with no “regularity gain.” Specifically, they proved that for arbitrary integers , and for any there is a bounded pseudoconvex domain with smooth polynomial boundary and a function , where

such that does not have a holomorphic extension in (resp. for ). This result implies in particular that for each there is a positive integer and a bounded pseudoconvex domain with smooth polynomial boundary, such that intersects transversally at all points and such that there is a holomorphic function , not admitting an extension in . It seems important to notice that the minimum of the dimension and the codimension of the subvariety is maximally possible.

In this paper, we consider bounded convex domains of finite type, so in particular domains considered by Diederich and Mazzilli in [2], and complex linear subvarieties of higher codimension. We prove a result which goes in the opposite direction comparing with the results of Diederich and Mazzilli. Namely, on such domains and subvarieties there always is the “gain of regularity” in the extension problem. We will make clear this statement below.

In order to present the result, we need however some definitions. For consider linear affine mappings

where , , . Let be the joint zero set of . Assume that and . For a positive Borel measure supported on , we write to denote the space of all functions holomorphic in such that

We can now present the main result.

Theorem 1. Assume that is a bounded convex domain of finite type defined by a smooth function such that on . If and

then there exists a positive Borel measure supported on such that

and an operator

satisfying for any .

In other words, if for it holds that , then the class of functions which admit an extension in is strictly larger than . Also for any dimension of there is also a space with where the “gain of regularity” appears.

We also establish the following result.

Theorem 2. Assume that is a bounded convex domain of finite type defined by a smooth function such that on . There exists a positive Borel measure supported on such that

and an operator

satisfying for any .

Thus, the space of functions which admit an extension in is always larger than .

It is natural to ask what happens when condition (6) is not satisfied. We do not have any answer yet. What is clear however is that the estimates which we provide do not work anymore. Another natural and closely related question is whether the operator maps into for any between and . In particular, whether maps into without any gain. We are only able to prove the following fact.

Theorem 3. Assume that , and then

for any .

In view of Ohsawa-Takegoshi Theorem, it suggests that our method of extending holomorphic functions is not optimal.

Convex finite (d’Angelo 1-) type domains are important class of domains where geometric aspects of function theory are studied. The finite type conditions were discovered in connection with the -Neumann problem (see the fundamental works of Kohn [9, 10] and Catlin [11, 12], and see also [13] for more information on the type condition). By the results in [14, 15], the assumption that a convex domain is of finite d’Angelo 1-type is equivalent to the assumption that there exists a constant such that all complex lines have order of contact at most with (cf. also [16] for an important generalization of this property for the multitype).

There are a few cornerstones in the study of function theory on convex domains. The first step was made by Bruna et al. [17] and McNeal [15, 18] who introduced the correct notion of pseudoballs and pseudometric on such domains. This was used to describe boundary behaviour of the Bergman kernel by McNeal [18] and the Szegö kernel by McNeal and Stein [19]. The fundamental step was made by Bruna et al. in [20]. The authors showed op. cit. how to estimate kernel functions in terms of boundary distances. Another breakthrough came when Diederich and Fornæss [21] constructed support functions for convex finite type domains. This made it possible to answer many analytic questions such as the quantitative behaviour of the -equation on spaces [22–24] and Hölder spaces [25, 26]. This made it also possible to study extension problems by means of integral operators, for instance by means of operators constructed by Berndtsson and Andersson [27] and Berndtsson [28]. As we have already written, this was started by Diederich and Mazzilli in [6]. We continue this task in this paper. We mention also that other aspects of function theory on convex finite type domains such as duality problems were also studied [29]. We remark that recently Nikolov et al. [30] found a mistake in [15, 18]. This however has no influence on our work since crucial estimates, in particular formula (49) below, remain valid.

2. Proof

Let be a bounded convex domain with -boundary of finite type . We may assume that has been chosen to be convex on and smooth in . We assume that the domain is of type . As was stated in the Introduction section, this means that the maximal order of contact of with complex lines is equal at most .

In order to prove Theorems 1 and 2, we use an extension operator constructed by Berndtsson in [28]—the construction relies on previous results by Berndtsson and Andersson in [27]. For the machinery developed by Berndtsson and Andersson to work, we need appropriate holomorphic support functions. Such functions, depending smoothly on and of optimal contact behaviour, were constructed by Diederich and Fornæss in [21]. The paper [25] contains crucial estimates, which we will use in the proofs. Since such estimates were used by many authors before, most notably by Diederich et al. in [25] and Fischer in [22, 26], we are rather brief in this aspect of the proof. For the same reason, we do not include separate background on geometry of convex finite type domains. Such information can naturally be found in papers by McNeal [15, 18]. It was also given in many papers on convex finite type domains; we refer the reader for instance to [20] or [25].

The extension operator is an integral operator of the form

defined by a kernel function .

The proof of boundedness of on spaces in Theorem 1 is based on the following modification of Schur’s test.

Proposition 4. Let be positive Borel measures on and let be a positive weight function. If there exist nonnegative functions , such that

then the operator

is a bounded operator between and .

The proof of Proposition 4 is an easy modification of the standard case, which can be found for instance in [31] p. 52. Therefore, we omit it.

Sufficiently close to the boundary of the domain , and only this is the case of interest, the extension operator takes the form

where

Symbol stands for the -vector dual to the volume form in , while denotes the contraction between the exterior algebras and of the tangent and cotangent bundles.

Crucial in the whole construction is the function which is the support function constructed for convex domains of finite type in [21] by Diederich and Fornæss. Function is appropriately decomposed, like in Hefer’s Lemma, to yield the form

Details of the construction can be found in [25].

In order to apply Proposition 4, we need to choose measures , and functions , , and . Since we are interested in values of the operator in the space , we set and . It remains therefore to find appropriate functions , , and . To accomplish this task, we need to recall some information on convex domains of finite type. Let, as in [15, 18],

be a complex directional boundary distance. For a fixed point and fixed radius , we define the -extremal basis centered at as in [15]. Once the basis is chosen, we write to denote .

Functions , , and have to capture the nonisotropic nature of the problem. Following [20], we therefore define the following.

Definition 5. Assume that is a bounded convex domain of finite type in , defined by a function which is smooth in , convex in and such that on . Let be an -covector at . The nonisotropic norm of at is defined as

With this definition, we can formulate Lemma 6.

Lemma 6. Assume that is a bounded convex domain of finite type in defined by a function which is smooth in , convex in and such that on . For the variety , consider the operator . If is sufficiently large, then there exists a constant such that

Proof. We define polydiscs

and for the corresponding polyannuli

The constant is chosen in such a way that for all , (cf. Proposition  3.1 (ii) in [25]). For simplicity, we assume that .
Then, we have for fixed that

For fixed , we use this cover with and estimate

We show how to estimate the integral of a typical term of the kernel function over . We will keep denoting the typical term by the same symbol .
It is a consequence of Lemma 3.2 in [25] that we may assume that

in . Therefore,

for . In order to estimate the right-hand side in (26), we choose the -extremal basis at . Let be the corresponding coordinates of a point and let be a unitary transformation such that ; that is, . We use functorial properties of the contraction operation

Symbol denotes the pullback operation.
Lemma 3.4 in [25] gives us the following estimate in :

where the sum is over all permutations , of .
This gives

where the sum is again over all permutations of .
Since and , we obtain the following estimate:

The last estimate follows from the fact that

It remains now to estimate the integral over . We cannot simply say that the integral over is bounded since we claim that the integral behaves like the norm of the form . Thus, in particular we claim that the integral vanishes as tends to .
Naturally, if for a positive constant , then

with a constant which depends only on , since it can easily be proved that . Also in this case, since uniformly for unit vectors and points . This establishes (20) in provided .
Since in , we may assume that and obtain the estimate

Trivial estimates of the form give therefore under the assumption that that

Since uniformly for unit vectors , we also have

provided . This completes the proof.

We can now prove Theorem 2.

Proof of Theorem 2. It follows from Fubini’s Theorem and Lemma 6 that if is sufficiently large, then

where

Since , it follows immediately that . It remains to notice that . In other words, in order to complete the proof it suffices to show that there exists a holomorphic function satisfying

which does not belong to . We consider the convex domain in . We may assume that is a plane tangent to at and lies on the left-hand side of . Since is convex, it suffices to take for suitable .

We intend to complete the proof of Theorem 1. For this, we need the following lemma.

Lemma 7. Assume that is a bounded convex domain of finite type in , defined by a function which is smooth in , convex in and such that on . For the variety , consider the operator . Assume that

For any such that

there exists a constant such that

In the proof of Lemma 7, we will use the following two lemmas.

Lemma 8. There exists an open set and a constant such that if and with , then

Lemma 9. Let be a -covector at . There exists a constant such that if ,, then

where is the -extremal basis at .

We omit the proof of Lemma 8 since it can be proved in the same way as Lemma 4.2 in [25] or Lemma 3.3 in [32]. We concentrate on the proof of Lemma 9.

Proof of Lemma 9. Let be unit vectors. We can write

and also

If , then and, as a result, uniformly for any unit vector

when . If , then uniformly for any unit vector

This is Proposition  2.3 in [18].
Thus, if , then for any unit vector

It is proved in [18] (Proposition  2.2, cf. also [30]) that if , where is the -extremal basis at , then

Thus, we have if . For any nonzero vectors , we can therefore estimate

which completes the proof.

Proof of Lemma 7. Naturally,

with an independent constant . This follows from the fact that in (Lemma 3.2 [22]) and that

which holds true for every . Hence, we need to show that under our assumptions

is uniformly bounded for . This time we therefore fix and consider the cover , of the polydisc . Again we work with a typical term of keeping the same notation. It follows from Lemma 8 that if , then

Let be the -extremal basis at . Let be the corresponding coordinates of a point and let be a unitary map such that ; that is, . Using the map, we change coordinates in the integral

Functoriality of the contraction and Lemma 3.3 in [22] together with (54) yield the following estimate:

where denotes the pullback of the volume form and

The sum is over all permutations of the set . We use Lemma 9 and obtain the following estimate:

since . The last estimate is a consequence of the fact that is a convex function provided and by the assumption . It is important to realize that estimate (58) is made with respect to the same -extremal basis at as for instance in (56).
We have therefore the following estimate of the integral:

Since , we also have

Therefore, we are led to estimate the following expression:

In other words, we need to estimate on . Here, we use Wirtinger’s formula according to which

As a result,

In order to have control over (61), we need to get rid of variables in the form . Since , on , we have on the following equality:

Thus, whenever , we obtain the estimate

Under the assumption that , this gives

where is a permutation of the set .
From (60), we therefore obtain

where . Thus,

Since , we obtain for some that

Thus,

This completes the proof.

We can now prove Theorem 1.

Proof of Theorem 1. If , then there exists such that . Choose such a small positive number .
As has already been stated, we apply Proposition 4. Thus, we set , , , , and . Then, it follows from Lemmas 6 and 7 and Proposition 4 that

where , provided is sufficiently large.
Since , we have that . Since is convex, it is therefore easy to show that .

Lastly we provide sketch of Theorem 3.

Proof of Theorem 3. We again apply Schur criteria, that is, Proposition 4. This time however with . In view of Lemma 6, we set ,.
In order to complete the proof, we need to know that

For the estimates in the proof of Lemma 7 to work, we must have . Indeed, only under this assumption we can have control over the expression

which appears in (60). We also need to have to obtain estimate (66). The readers easily convince themselves that if and , then the estimates in the proof of Lemma 7 can be repeated.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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