Research Article | Open Access
On Extremal Problems in Certain New Bergman Type Spaces in Some Bounded Domains in
Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains in , we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains.
The goal of this paper is to add several new results for distances in analytic Bergman type spaces of functions of several variables. It turns out that our distance theorem we proved before in case of unit disk, a sharp result under certain natural additional condition, is valid also in various domains and various Bergman type analytic spaces. Namely, we look at analytic Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. These analytic spaces act as direct extensions of well-known analytic Bergman spaces in the unit disk. These analytic spaces are relatively new and we will include some basic facts on them in our paper. They will also be needed for proofs of our assertions partially. We will start this paper with two sharp results on distances in Bergman type spaces in two domains: Siegel domains of second type and in minimal homogeneous domains in . Then one side estimates for distance function in Lie ball and bounded symmetric domains of tube type will be given based directly on recent advances related to boundedness of Bergman type projections in Bergman type spaces in these type domains.
Our intention in this paper is the same as in our previous papers on this topic. Namely, we collect some facts from earlier investigation concerning Bergman projection with positive Bergman kernel and Bergman kernel and use them for our purposes in estimates of function (distance function).
First we need an embedding of our quazinormed analytic space (in any domain) into another one (); this immediately poses a problem of for all . Then we need the Bergman reproducing formula for all function from space. Then, finally, we use the boundedness of Bergman type projections with positive kernel acting from to together with Forelli-Rudin type sharp estimates of Bergman kernel. These three tools were used in general Siegel domain of second type, polydisk, and unit ball in [1–4] (see also various references there). We continue to use these tools providing new sharp (and not sharp) results in various spaces of analytic functions in this paper.
Note that our theorem on Siegel domains was formulated in  without proof. We provide the complete proof here. We also note that various problems, related to Bergman type projections, are applied in many problems in function theory (see, e.g.,  and references there).
First we provide a known result in the unit disk with complete proof taken from our previous papers [1, 2]. In the unit disk case all arguments here are short and transparent and are based on several tools like Forelli-Rudin type estimate and estimates for Bergman type projections with positive Bergman kernel. Then we will see arguing similarly as in unit disk and we will easily complete the proof of more complicated cases. The complete formulations of our last theorems will be given, but sometimes sketches of proofs will be added and details of proofs of higher-dimensional cases will be left to readers.
Note that it is easy to see that our assertions may have various applications in approximation theory; for example see  and references there.
The base of our proofs is properties of Bergman projection in various domains given in [7–12]. The estimates of Bergman kernel from [7–12] are also playing an important role below in our proofs. Note that arguments we use below are very close to arguments which were used before in [1, 2, 13]. As a result we alert the reader that the exposition is sketchy sometimes.
Throughout the paper, we write (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables being discussed.
The notation means that there is a positive constant , such that . We will write for two expressions if there is a positive constant such that .
This paper can be viewed as continuation of  where various other cases (domains) were also considered. In  the base of all our proofs in complex domains in higher dimension was the Bergman reproducing formula, while here all our assertions are based on some recent results on boundedness of Bergman projections with positive Bergman kernel in Bergman spaces in such type domains.
2. Notations, Definitions, and Preliminaries
We will need various definitions and assertions for formulations of main results. These are assertions on various types of domains we consider in this paper and analytic functions on them.
We denote by the unit ball in . As usual, we denote by the space of all holomorphic functions in . For and , denote by or the space of all functions holomorphic in and satisfying the condition where is the Lebesgue measure in .
Further, for a complex number with , put
Let , .
The following theorem is well-known and it has many applications in various problems in function theory (see ).
Theorem A. Assume that , and that the complex number satisfies the condition Then each function admits the following integral representations: where is the Hermitian inner product in .
We will start this section with various known assertions on analytic function spaces on Siegel domains of second type. Next we will continue adding some lemmas on each Bergman type analytic space on each domain in higher dimension which we will consider in this paper. We alert the reader that some assertions below will serve as introductory material and will not be used during the proof to make the reading of paper more convenient.
Let be a usual homogeneous Siegel domain of second type. Let denote the Lebesgue measure on (for all other bounded domains in this paper we will also use the same notation below) and let as usual be the space of holomorphic functions on endowed as usual with the topology of uniform convergence on compact subsets of .
The Bergman projection of is as usual the orthogonal projection of onto its subspace consisting of holomorphic functions. Moreover it is known that is the integral operator defined on by the Bergman kernel which for was computed for example in [18, 19].
Let be a real number, for example. We fix it. Since is homogeneous, the function does not vanish on ; we can set Let be an arbitrary positive number. The weighted Bergman space is defined as usual by . We put .
The so-called weighted Bergman projection is the orthogonal projection of onto . This fact can be found in [8, 10]. It is proved in [8, 10] that there exists a real number such that if and that for . is the integral operator defined on by the weighted Bergman kernel . In all our work we will assume that .
We denote by Bergman kernel for the Siegel domain of the second type, which differs from Bergman kernel by constant. We will use it in text also.
Lemma A. Let . Take for large fixed . Then the function satisfies the estimate and .
Lemma B. For each sufficiently large and for each such that one has the reproducing formula
We will need for our theorems some basic facts for Siegel domains of second type. We denote by , , and parameters of a Siegel domains of second type (see [4, 5, 8, 10]). We will use usual operations between two vectors for such parameters below in our text.
Lemma C. Let and be in , . Then for and ,
Lemma D. Let be a vector of such that for all and is a real number such that . Then for all such that , , .
We list in Lemma E other properties of Bergman kernel. The last estimate in assertion below is an embedding theorem which connect so-called growth spaces with Bergman spaces. This allows to pose a distance problem (see also the complete analogue of this result in other simpler domains in [1, 17]).
Lemma E. Let , , . Then and for all , , , in . For all ,
The following result concerns the boundedness of Bergman type projection with positive Bergman kernel in weighted Bergman spaces. Note that this fact is classical in simpler domains and it has also many applications in analytic function theory.
Proposition A. Let and be in such that and , . Then is bounded from into if
The following assertion provides integral representation for a certain so-called analytic “growth space” on Siegel domains of the second type.
Proposition B. Let and be two vectors of such that , , . Let be in such that then .
The following result explains the structure of functions from Bergman spaces on Siegel domains of second type. It is an extension of a classical theorem on atomic decomposition of Bergman spaces in the unit disk on a complex plane.
Proposition C. Let be a symmetric Siegel domain of second type, , , . Then there are two constants and such that for every there exists an sequence such that where is a lattice in and the following estimate holds:
Let be a bounded domain in . We say that is a minimal domain with a center if the following condition is satisfied: for every biholomorphism with , (s is the complex Jacobian of the map ), we have
Let now us denote by the Bergman kernel of , that is, the reproducing kernel of . It is known that is a minimal domain with a center if and only if for any , (see  and references there).
From , Proposition 3.6, we see that is a minimal domain with a center if and only if for any .
Every bounded homogeneous domain is biholomorphic to a representative bounded homogeneous domain.
We fix a minimal bounded homogeneous domain with a center . For a bounded linear operator on (this is standard analytic part of standard space in ) the Berezin symbol of is defined by where is a normalized Bergman kernel in Bergman space in minimal bounded homogeneous domain . For a Borel measure on , we define a function on by which is called the Berezin symbol of the measure . Since is a bounded function on , is a continuous function if is finite.
We will provide some basic facts for a minimal bounded homogeneous domains.
Lemma F (see ). There exists a constant such that for all and .
Lemma G (see ). There exists a sequence satisfying the following conditions. (1). (2).(3)There exists a positive integer such that each point belongs to at most of the sets .
Lemma H (see ). There exists a constant such that for all , , and , where is a space of analytic functions in .
Theorem B (see ). Take any . Then, there exists such that for all with , where means the Bergman distance on .
Theorem C (see ). There exist integers such that for . Recall that the Bergman kernel of the Siegel disk is given by
We will denote the weighted reproducing Bergman kernel for weighted Bergman spaces in this type domains below simply as omitting index .
Note from lemmas above (see ) that we have
This allows putting distance problems for these domains which we solve in Theorem 8.
We need now some preliminaries for Bergman spaces in Lie ball.
Let denote each of the following domains in , :(1)the tube over the spherical cone (2)the Lie ball
Obviously, the first domain is unbounded while the second one is bounded. It is well- known that they are biholomorphically equivalent and, in Elie Cartans classification of bounded symmetric domains , they are representatives of class IV (according to Huas numbering ). We are interested in bounded Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains in this paper.
To make the exposition easier, we remind the readers about the basic definitions again.
Let denote the space of holomorphic functions in domain and let, as above, be Lebesgue measure in . For every , the Bergman space is defined by . For every , we set for ; this is a norm under which is a Banach space. The Bergman projection of is the orthogonal projection of the Hilbert space onto its closed subspace . Moreover, is the integral operator associated with a kernel called the Bergman kernel of . Finally, let denote the integral operator associated with the positive kernel .
The following results were proved in .
Theorem D (see ). For every , the Bergman projection is unbounded on .
Theorem E (see ). Let . The operator is bounded on if and only if . Furthermore, the Bergman projection is bounded from to when .
For the tube domain some of these results were announced in . The question whether is bounded on when belongs to remains open. The case of all homogeneous Siegel domains of second type has recently been considered by . Bekolle and A. Temgoua Kagou. They proved that there is a range of , around , where the Bergman projection is bounded in , while there is a range of , around and , where it is unbounded (see ). In all cases the critical result is not known.
We add some basic facts on Bergman kernel on these domains.
Proposition D (see ). The Bergman kernel of is given by where , .
Definition 1 (see ). Let denote the positive kernel defined on the cone , , , .
Proposition E (see ). For each , there exists a constant such that, for all and , Moreover, there exists a constant such that, for each such that and each such that , where denotes the interval .
Let be the linear fractional mapping from onto which is given in . In particular, we assume that , where and is holomorphic outside , where is a polynomial such that . In view of the change of variables formula, the Bergman kernel of has the following expression in terms of that of : where s is the complex Jacobian of the map . On the other hand, since is a circular domain, for each real number , and thus, there exists a constant such that for each . Hence, from (32), we get The following property of Bergman kernel is vital.
Lemma I (see ). For all and in ,
We add some basic facts on bounded symmetric domains of tube type from .
Let be an irreducible bounded symmetric domain of tube type in . That is, is conformally equivalent to a tube domain over a symmetric cone in . Irreducible symmetric cones are completely classified (see ), being either light-cones or cones of positive-definite symmetric or hermitian matrices, namely, We write for the rank of the cone (which in light-cones is ) and for the associated determinant function (which in light-cones is the Lorentz form ).
An important open question in these domains, and , concerns the boundedness of the associated Bergman projections, that is, the orthogonal projection mapping into the subspace of holomorphic functions . In contrast with Cauchy-Szego projections (which are not bounded in for any , if ), the -boundedness of Bergman projections has been conjectured in a small interval around , namely, At the moment, positive results are only known to hold in a proper subinterval with a small improvement over this range in the case of light-cones.
Also, we are interested in applying the transference principle to the family of weighted Bergman projections in and . Using the notation in the text from , Chapter XIII, these operators are defined for by where the Bergman kernels and their associated measures have the explicit expressions for certain constants , . Here denotes the unique polynomial (holomorphic in and antiholomorphic in ) such that is -invariant and , .
The Bergman kernels in these two domains are related by the formula
Corollary 2 (see ). Let and . Then the following are equivalent:(a) is bounded from ;(b) is bounded from .
Remark 3. The statements in the corollary can only hold if .
The transference principle also applies to the positive operators In this case we can even state a stronger result. We consider a new operator, acting on functions in by Here, with a slight abuse of notation, we still write for the measure in and for the closed unit ball in . We write , that is, the space with mixed norm given by where .
Proposition F (see ). Let and . The following are equivalent:(a) is bounded from ;(b) is bounded from ;(c) is bounded from for all ;(d) is bounded from .
The Bergman projection for a bounded domain in , along with the associated Bergman kernel function, has proved fundamental for the theory of boundary behavior of holomohic mappings (see [27, 28] and their references).
It is well-known that on smooth bounded strongly pseudoconvex domains the Bergman projection preserves each Sobolev space , . The same is true more generally for those smooth bounded weakly pseudoconvex domains which admit subelliptic estimates for the -Neumann problem (see [27, 28]).
By the mean value property for we have for any bounded domain where and is Lebesgues measure on and is Euclidean distance from to the boundary of .
This immediately pose a dist problem in all such very general domains. We give one side estimates based on recent results on Bergman projection in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains.
We follow also our sharp recent result in unit disk. Note if bounded domain has smooth boundary, then A bounded domain in , is called smooth if there is a defining function such that the boundary and the gradient of does not vanish in (see [7, 12, 14, 23, 29]).
3. On Distance Function in Bergman Type Spaces in Certain Domains in Higher Dimension
In this section we provide main results of the paper. Based fully on preliminaries of previous section, the plan of this section is the following: first we formulate a result in the unit disk and then repeat arguments we provided in proof of that theorem in various situations in Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. Since all proofs are short the repetitions of arguments which are needed in higher dimension will be omitted; sometimes sketches will be given.
Our results on Siegel domains and minimal homogeneous domains are sharp.
Let be, as usual, the unit disk on the complex plane, and let be the normalized Lebesgue measure on . Let be the space of all analytic functions on the unit disk . For and , define the fractional derivative of the function as usual in the following manner: We will write if . Obviously, for all , if .
For , , , the weighted analytic Besov space is the class of analytic functions satisfying where is circle and is the Lebesgue measure on the circle .
We denote by the analytic Besov spaces in the unit disk for all real numbers. Note also that for we have that these spaces are Bergman spaces according to definition above for unit ball and we will use this notation below for all negative numbers in Besov spaces.
It is well-known that , , , .
Let further , , , .
It is easy to note that, based on previous section results, the complete analogues of the embedding we just provided are valid also in Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. We leave this easy task to readers. This allows posing a dist problem in each space we consider in this paper.
In the following theorem (see ) we calculate distances from a weighted Bloch class to Bergman spaces for . We will see that almost each argument below is also valid not only in unit disk but based on preliminaries in previous section in Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains. So this theorem is very typical for us though it is known.
Theorem 4. Let , , , , and . Let . Then the following are equivalent: (a); (b).
Proof. First we show that . For , we have
where is a well-known Bergman representation constant (see ).
For , So .
For , , we have So we finally have It remains to prove that . Let us assume that . Then we can find two numbers such that , and a function , , and . Hence as above we easily get from that , and hence Since for , (see ) where , , , , and where , , . we get So, we arrive at a contradiction. The theorem is proved.
The following theorem is a version of Theorem 4 for the case .
Theorem 5. Let , , , , and . Let . Then the following are equivalent: (a); (b).
The proof of Theorem 5 is the same actually as the proof of Theorem 4. The only difference is the boundedness of Bergman type projection operator but with the positive Bergman kernel. This fact will be heavily used by us below.
Indeed the close inspection of the proof of Theorem 4 shows that the proof of Theorem 5 is the same as the proof of Theorem 4, but here we will use (58) (see below) instead of (55). For , , , , which follows immediately from Hölder’s inequality and (56).
Remark 6. Analytic spaces in unit disk are well-known in literature as so-called growth spaces (see, e.g., ). It can be shown easily that these spaces are Banach spaces. These spaces are playing a vital role in this paper and are embedded in Bergman spaces , for large enough (the same is valid in any bounded domain with boundary). Hence the representation (4) is valid also in the unit disk with large enough index depending on for all functions from such classes. The mentioned embedding is well-known and almost obvious and we leave the proof of it to interested readers. This fact is crucial for the proofs of Theorems 7 and 8 below and along with well-known Forelli-Rudin estimates, for to each domain we defined, (see ), actually serves as base of both proofs.
We formulate a general theorem for Siegel domains of second type in . Then we use the same ideas to formulate the same result in minimal bounded homogeneous domains in using very recent advances of Yamaji (see [20, 22] and references there).
We formulate finally also some new one side estimates for distances based on projection theorems for symmetric domain of tube type, Lie ball, without proofs, since arguments are similar to unit disk case. Our results for Siegel domains of second type and minimal homogeneous domains are sharp.
Theorem 7. Let be Siegel domain of second type. Let Let and where is a positive number. Then the following two quantities are equivalent , where for all and so that, , , , and , and for certain fixed vector , depending on and on parameters of the Siegel domains , , and .
Proof of Theorem 7. We will follow the proof of Theorem 4. First we show that . Let , , , , . Then , , , for the same large enough set depending on parameters of Siegel domain, for all .
Hence we have that Also, we have that Note then we used that for
We show the reverse implication now following again unit disk case arguments (see Theorems 4 and 5). We have the following. Let us assume that . Then we can find two numbers , such that and a function , and hence and hence We used that (see [5, 8]) These estimates give a lower estimate for .
Theorem 7 is proved.
For and , denote by the space of all functions holomorphic in and satisfying the condition where is the Lebesgue measure in domain.
Let denote reproducing weighted Bergman kernel for weighted Bergman space (see [11, 12]) where is a Bergman kernel for . Based on preliminaries of previous section on Bergman type spaces in minimal homogeneous domains and arguments of proof of Theorems 4 and 5 and comments related to them, we can formulate the following result.
Theorem 8. Assume that and let be bounded minimal homogeneous domains with boundary. Let . Then for we have that , where for all for some fixed .
We add only the full sketch of proof of this theorem since it is quite similar to proofs of previous theorems. We consider the operator defined by is operator on . The fact that Bergman type projection with positive Bergman kernel is a bounded operator in spaces (weighted Lebegue spaces) can be seen in recent paper . It is well-known also, since is a Hilbert space that the Bergman reproducing formula for all functions from these spaces is valid and since our domain is bounded, this space for large enough contains for any . So the Bergman representation formula for each function taken from is also valid with large enough . These two facts along with Forelly-Rudin type estimates for these domains (see ) complete the proof.
All other results of this paper are one side estimates for distance function and are fully based on results on boundedness of Bergman type projection with positive Bergman kernel as we have seen in unit disk case in Bergman type spaces, but over specific domain which we mentioned above in previous section as separate assertions.
Theorem 9. Let be Lie ball with boundary. Let , , . Let , . Then
We omit details of the proof referring to unit disk case.
Based on preliminaries of previous section on Bergman type spaces in bounded symmetric domains of tube type and arguments of proof of Theorems 4 and 5 and comments related to them, we can formulate the following result.
Theorem 10. Let be bounded symmetric domain of tube type with boundary. Let , , . Let , . Then