`Abstract and Applied AnalysisVolumeΒ 2010, Article IDΒ 354961, 23 pageshttp://dx.doi.org/10.1155/2010/354961`
Research Article

Weighted Anisotropic Integral Representations of Holomorphic Functions in the Unit Ball of

Institute of Mathematics, National Academy of Sciences of Armenia, Marshal Bagramian Avenue 24 B, Yerevan 375019, Armenia

Received 15 November 2010; Accepted 27 December 2010

Copyright Β© 2010 Arman Karapetyan. 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 obtain weighted integral representations for spaces of functions holomorphic in the unit ball and belonging to area-integrable weighted -classes with βanisotropicβ weight function of the type , . The corresponding kernels of these representations are estimated, written in an integral form, and even written out in an explicit form (for ).

1. Introduction

Denote by the unit ball in the complex -dimensional space . For and , denote by the space of all functions holomorphic in and satisfying the condition where is the Lebesgue measure in . Further, for a complex number with , put

We have the following theorem.

Theorem 1.1. Assume that , and that the complex number satisfies the condition Then each function admits the following integral representations: where is the Hermitean inner product in .

For , that is, for the case of the unit disc , this theorem was established in [1, 2], where the formulas (1.4) are important in the theory of factorization of meromorphic functions in the unit disc.

For , the theorem was proved in [3] (when ) and in [4, 5] (when ).

In monographs [6, 7], one can find numerous applications of the formulas (1.4) in the complex analysis.

In the present paper, we generalize Theorem 1.1 in the following way.

Assume that and satisfies the conditions Then we introduce the spaces of functions holomorphic in and satisfying the condition Section 3 contains detailed investigation of these spaces with βanisotropicβ weight function.

For these βanisotropicβ spaces, similarities of the integral representations (1.4) are obtained, but this time a special kernels (where , are associated with , and in a special way) appear instead of (Theorem 4.7). Theorem 4.5 gives the description (in a multiple series form) and the main properties of these kernels. Theorem 4.8 makes it possible to represent the kernels as integrals taken over . Finally, in the special case we write out these kernels in an explicit form (see Theorem 4.12).

2. Preliminaries

In this section, we present several well-known facts which will be used in what follows.

Fact 1. For and , put then Moreover, if and , then

Remark 2.1. Assume that and is a continuous positive (i.e., ) function in . If , then when the corresponding integral exists.

Fact 2. For ,

Fact 3. If , then

Fact 4. If , then

As a consequence of Stirlingβs Formula, we have the following fact.

Fact 5. For arbitrary and for In addition, if , and , then

Fact 6. Assume that , and ; then is a nondecreasing function of , that is,

Corollary 2.2. Assume that , and is a continuous positive (i.e., ) function in . Then if and . In particular,

3. Main Function Spaces

Suppose that . We put Further, we shall write (β) only if the following conditions are satisfied: Similarly, if , then we shall write (β) if only (β).

The following multidimensional analogue of Fact 1 is valid.

Lemma 3.1. For (β) and for arbitrary multi-index , put then
Moreover, if (β) and are arbitrary multi-indices such that , then

Proof. We intend to establish (3.3) by induction in . For we simply arrived at (2.2). Assume the validity of (3.3) for some and proceed to the case of . Note that arbitrary can be written as , where . Consequently, for (β) and for multi-index , we have A simple change of variable: in the inner integral gives the following recurrent relation: Note that (β) (for all ) due to the condition (β). Consequently, in (3.6) we can apply (3.3) to due to our inductive assumption. As a result, we obtain (3.3) but this time for . Thus, the inductive argument is completed and the formula (3.3) is established.
Now suppose that , and there exists such that . Then we can split arbitrary in and so that and . Hence In view of (2.4), the inner integral in (3.7) is equal to 0, so (3.4) is also proved.

Corollary 3.2. If (β), then

Remark 3.3. In the integrals (see (3.2)) instead of (β), arbitrary (β) can be considered and the formulas (3.3), (3.4), and (3.8) remain true after the replacement of by .

Definition 3.4. Assume that and (β). Denote by the space of all complex-valued functions in with We obviously have Correspondingly, denote by the subspace of functions holomorphic in . Note that .

Lemma 3.5. Assume that (β), and . Then where is the volume of the unit ball of .

Proof. Fix an arbitrary ; put and . Since is subharmonic in , we have Note that Hence for Combining (3.12) and (3.14), we obtain

Corollary 3.6. is a closed subspace in .

Definition 3.7. If and , then put Similarly, if and , then put In particular, if , note also that .

Lemma 3.8. Assume that (β), and . Then for all , In particular, .

Proof. Evidently, it suffices to fix and consider the case . In other words, it is sufficient to show that To this end, we proceed as follows: An application of Corollary 2.2 (see (2.12)) to the inner integral gives the desired inequality.

Corollary 3.9. Assume that (β), and . Then for all , In particular, .

Lemma 3.10. Assume that (β), and . Then for all and for arbitrary ,

Proof. Since the case coincides with (3.18), we can suppose that . Further, similarly to the proof of Lemma 3.8, it suffices to fix and consider the case where In view of Theorem 4.5(b), (4.26)-(4.27) and due to Lebesgueβs dominated convergence theorem, we have and the series in (4.28) and (4.29) converge absolutely.
Taking into account (3.4) and Remark 3.3, we conclude that all terms in the right-hand side of (4.29) (except of the case ) are equal to zero. Hence the left-hand side of (4.29) is equal to Thus, (4.23β) has been proved.
Now let us proceed to (4.22β). If , then in view of (3.4) and Remark 3.3
If , then in view of (3.4), (3.2), (3.3) and Remark 3.3 Consequently, (4.28), (4.31), and (4.32) together yield Thus, (4.22β) also has been proved.
Now we intend to make passage in (4.22β) and (4.23β). Evidently, the left-hand sides tend to and correspondingly. It remains to show that the right-hand sides of (4.22β) and (4.23β) tend to the right-hand sides of (4.22) and (4.23), respectively. In view of the estimate (4.19), it suffices to show that If , then due to the condition (4.20) and Corollary 3.13 If , then an application of HΓΆlder integral inequality to gives Here the integral over is finite due to Corollary 3.2 and the condition . Consequently, , in view of Corollary 3.13. Thus, the theorem is proved.

Theorem 4.8. For , and for ,

Proof. For according to (4.10) and (2.6), we have The summation of these relations over yields (see (2.5)) Thus, (4.37) is proved.

Remark 4.9. Under the conditions of the theorem, the formula (4.37) easily implies the assertions (c), (d), and (e) of Theorem 4.5.

Remark 4.10. If we take in (4.37), then the formal application of (2.7) gives the following formula: that is, we arrive at the kernel of the integral representations (1.4).

Remark 4.11. In fact, the kernel , defined for (β) by (4.9β)-(4.9ββ), can be considered as an analytic continuation in of the integral , defined for by (4.37).
Now we consider an interesting special case , when (in comparison with (4.9β)-(4.9ββ) or (4.37)) the kernel can be written out in an explicit form.

Theorem 4.12. Assume that (β), that is, . Then for ,

Proof. According to (4.9β)-(4.9ββ)-(4.10) and in view of the formula (2.5), Thus, (4.41) is established.

Remark 4.13. During the proof we regroup the double series: , which is legitimate in view of Theorem 4.5(a). In fact, we apply Fubiniβs theorem for double series.

Remark 4.14. The analysis of the proof shows that we have established (4.41) only for those and , which satisfy the condition . In fact, this is quite sufficiently since both sides of (4.41) are holomorphic in , antiholomorphic in , and continuous in (see Theorem 4.5(c), (d)).

Remark 4.15. If one takes in Theorem 4.12, then which coincides with (4.11) (or (4.40)) for the case .

Remark 4.16. Note that for the same case and under slightly restrictive conditions , the formula (4.37) gives ()

Remark 4.17. It can be shown (we omit the proof) that under the conditions the following interesting formula is valid ():

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