Gain of Regularity in Extension Problem on Convex Domains

We investigate the extension problem from higher codimensional linear subvarieties on convex domains of finite type. We prove that there exists a constant d such that on Bergman spaces H(D) with 1 ≤ p < d there appears the so-called “gain regularity.” The constant d 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 H(D) is strictly larger than H(D ∩ A), where A is a subvariety.


Introduction
Let  be a bounded pseudoconvex domain,  :  → R ∪ {−∞} a plurisubharmonic function, and  ⊂ C  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 (2 − 2)dimensional Lebesgue measure, , is the volume measure in C  .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  ≥ 1 which are nonsingular and cut  transversally.She proved that in this case any holomorphic function in   ((−)   ∩ ),  ≥ , 1 ≤  < ∞, 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 dist(⋅, ), when  is smooth.
In order to motivate our study, let us restrict our attention to the  2 case.Comparing with the Ohsawa-Takegoshi Theorem for  = 0, the result of Cumenge says that there is a strictly larger class of functions than  2 ( ∩ ) ∩ () which admit an extension to a holomorphic function in  2 ().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][10][11][12] on finite type domains).
such that  does not have a holomorphic extension in  2(+2)/(+4)+ () (resp. 2(+2)/+ () for  = ∞).This result implies in particular that for each  > 0 there is a positive integer  and a bounded pseudoconvex domain  ⊂ C 2+1 with smooth polynomial boundary, such that   intersects  transversally at all points  ∩   and such that there is a holomorphic function  ∈  ∞ ( ∩   ), not admitting an extension in  2+ ().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.
Theorem 1. Assume that  ⊂ C  is a bounded convex domain of finite type defined by a smooth function  such that  ̸ = 0 on .If  > 1 and then there exists a positive Borel measure  supported on ∩ such that and an operator satisfying  ∩ | ∩ =  for any  ∈   ( ∩ , ).
In other words, if for  > 1 it holds that 1 + 1/( − 1) > min{ − , } > 1, 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  > 1 where the "gain of regularity" appears.
We also establish the following result.
Theorem 2. Assume that  ⊂ C  is a bounded convex domain of finite type defined by a smooth function  such that  ̸ = 0 on .There exists a positive Borel measure  supported on  ∩  such that and an operator satisfying  ∩ | ∩ =  for any  ∈  1 ( ∩ , ).
Thus, the space of functions which admit an extension in  1 () is always larger than  1 ( ∩ ).
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 1 and ∞.In particular, whether  ∩ maps  2 ( ∩ ) into  2 () without any gain.We are only able to prove the following fact.Theorem 3. Assume that min{,  − } ≤ 2, and then  ∩ :   ( ∩ ) →   () (11) for any 1 ≤  ≤ 2.
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 Fornaess [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][23][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.

Proof
Let  = { < 0} be a bounded convex domain with  ∞boundary of finite type .We may assume that  has been chosen to be convex on C  and smooth in C  \ {0}.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 Fornaess 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 ℎ 1 , ℎ 2 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 C  , while ⌋ denotes the contraction between the exterior algebras Λ and Λ * 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 Fornaess.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 , ℎ 1 , and ℎ 2 .Since we are interested in values of the operator   ∩ in the space   (), we set  =  ∩ and ] = .It remains therefore to find appropriate functions , ℎ 1 , and ℎ 2 .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 (V , 1 , . . ., V ,  ) centered at  as in [15].Once the basis is chosen, we write   (, ) to denote (, V ,  , ).Functions ℎ 1 , ℎ 2 , 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 C  ,  > 1 defined by a function  which is smooth in C  \ {0}, convex in C  and such that  ̸ = 0 on .Let Ω be an (, 0)-covector at  ∈ .The nonisotropic norm |Ω| N () of Ω 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 C  ,  > 1 defined by a function  which is smooth in C  \ {0}, convex in C  and such that  ̸ = 0 on .For the variety  = ( 1 , . . .,   ), consider the operator   ∩ .If  is sufficiently large, then there exists a constant  such that Proof.We define polydiscs and for  ∈ N 0 the corresponding polyannuli The constant  1 is chosen in such a way that  1  /2 () ⊃ (1/2)  () for all ,  (cf.Proposition 3.1 (ii) in [25]).For simplicity, we assume that  1 = 1.Then, we have for fixed  0 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] for  ∈   |()| ().In order to estimate the right-hand side in (26), we choose the 2 ) at .Let ( 1 , . . .,   ) be the corresponding coordinates of a point  and let Φ be a unitary transformation such that  = Φ() +  := (); that is,  = Φ ⋆ (−).We use functorial properties of the contraction operation Symbol * denotes the pullback operation.Lemma 3.4 in [25] The last estimate follows from the fact that det ( It remains now to estimate the integral over  \   0 ().We cannot simply say that the integral over  \   0 () is bounded since we claim that the integral behaves like the norm provided  ≥ 2.This completes the proof.
We can now prove Theorem 2.
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 C  ,  > 1 defined by a function  which is smooth in C  \ {0}, convex in C  and such that  ̸ = 0 on .For the variety  = ( 1 , . . .,   ), consider the operator   ∩ .Assume that For any  > 0 such that there exists a constant   such that In the proof of Lemma 7, we will use the following two lemmas. where 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.
Lastly we provide sketch of Theorem 3.