Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 725705, 14 pages

http://dx.doi.org/10.1155/2013/725705

Research Article

Holomorphic Spaces in the Unit Ball of

Augusto Guadalupe Miss Paredes,¹ Lino Feliciano Reséndis Ocampo,² and Luis Manuel Tovar Sánchez³

¹Departamento de Ciencias Básicas, Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas del IPN, Avenida Instituto Politécnico Nacional No. 2580, Colonia La Laguna Ticomán, 07340 Delegación Gustavo A. Madero, DF, Mexico

²Universidad Autónoma Metropolitana, Unidad Azcapotzalco, C.B.I. Área de Análisis Matemático y sus Aplicaciones, Avenida Pablo 180, 02200 México, DF, Mexico

³Escuela Superior de Física y Matemáticas del IPN, Edificio 9, Unidad ALM, 07300 Zacatenco, DF, Mexico

Received 5 June 2013; Accepted 24 August 2013

Academic Editor: Ruhan Zhao

Copyright © 2013 Augusto Guadalupe Miss Paredes et al. 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 introduce the and vector spaces of holomorphic functions defined in the unit ball of , generalizing previous work like Ouyang et al. (1998), Stroethoff (1989), and Choa et al. (1992). Likewise, we characterize those spaces in terms of harmonic majorants as a generalization of Arellano et al. (2000).

1. Introduction

1.1. Preliminaries in One Complex Variable

Let be the open unit disk in the complex plane . For , let be the Möbius transformation defined by

For , we denote Green's function of with logarithmic singularity at by

The Bloch space is defined as the set of analytic functions , such that

For , Aulaskari and Lappan [1] introduced in 1994 the spaces as the family of analytic functions satisfying

For , this definition was extended in [2].

Theorem 1 (see [2]). Let and be an analytic function. Then, if and only if

With the aim of generalizing and enclosing several weighted function spaces, Zhao in [3] introduced the spaces in the next way.

Let ,, , and be an analytic function. We will say that belongs to if satisfies the integral condition

So far, Theorem 1 is generalized for functions of by the following.

Theorem 2 (see [3]). Let , , , and be an analytic function. Then, if and only if

Zhao has shown that for certain intervals of , , and , makes as a Banach space.

In this paper, we present the spaces of holomorphic functions in the unit ball of , that generalize the spaces introduced by Zhao in [3] for analytic functions in the unit disk. At the same time, this work generalizes, mainly, several results of Ouyang et al for -holomorphic functions appearing in [4]. However, the techniques, the methods, and the structure of our work results are completely different to the quoted reference. At the beginning of Section 2, we introduce the spaces and , that is, when we use in the integral representation, the invariant Green function, or when we use the biholomorphism . We remark that the generalization to several complex variables of requires, for and , different intervals to those used in the one-dimensional case. In Theorems 12 and 13, we draw attention to the continuity of the integral expressions defining these spaces to conclude easily the inclusions of the little classes and in and , respectively.

In Section 2.2, we present what is the most natural Bloch space associated to and several characterizations that we give in Theorem 22. It is important to compare this statement with the results of [4, Proposition 3.6] and [5, Theorem 2.4]. In Section 3, we present in Corollary 30 the equivalence between and , generalizing Proposition 3.4 of [4] and Theorem 1 of [3]. Finally in Section 4, we obtain the characterizations of and in terms of harmonic majorants. Giving in this way, the holomorphic version of the results of Arellano et al. in [6].

1.2. Preliminaries in Several Variables

Consider the vector space with the Hermitian product defined as

and the Euclidean norm on defined as

The open ball with center and radius denoted by is defined as

and its closure is

We will denote , , and .

The function , of class , is harmonic in if

for all . Moreover, results harmonic in if and only if

Let . The Möbius function is defined as

with and , for , and define (see [7]).

This set of functions is a part of the family of automorphisms on denoted by .

Lemma 3 (see [7, 8]). The function satisfies the following properties:

(i);(ii);(iii)the real Jacobian for in is given by

Lemma 4. The real Jacobian of satisfies

Proof. It follows directly from .

Define, for and ,

1.3. Invariant Gradient and Measure

The collection of all holomorphic functions in is denoted by . Let denote the complex gradient of and denote the radial derivative of . Let , where is the normalized volume measure in . Then, is a Möbius invariant; that is

for each -integrable function and .

Let denote the invariant Laplacian of [8], and let denote the invariant gradient of [7]. It is well known [7] that, for ,

where . In [7], the invariant Green's function is defined as , where

As in [7], the -dimensional invariant Bloch space is the set of holomorphic functions , such that their Bloch invariant norm satisfies

this is equivalent [4, Lemma 3.2] to, say,

Proposition 5 (see [7]). Let be an integer; then there are positive constants and , such that, for all ,

Proposition 6 (see [9]). For , one has

Theorem 7 (see [7, Theorem 1.12]). Suppose that is real and . Then, the integral

has the following asymptotic properties.

(i)If , is bounded in .(ii)If , then

as .(iii)If , then

as .

Let ,  . We say that , , and satisfy condition (A) if

and satisfy condition (B) if

Proposition 8. Suppose that , , and satisfy condition (A). Then,

Proof. We have the following equalities:

The singularity at 0 is integrable if . Now, if we estimate

as if and , , then the result follows from Theorem 7 for any case of .

In a similar way, we have the following.

Proposition 9. Suppose that , , and satisfy condition (B). Then,

2. Function Spaces and

2.1. Main Definitions

In this section, we define the spaces , and their little associated spaces. We study some properties and relations between them.

Suppose that , and satisfy condition (A), and let . Define

Suppose that , , and satisfy condition (B), and let . Define

It is clear that

Associated to these functions, we define the following vector spaces of holomorphic functions.

(i)Suppose that , , and satisfy condition (A). belongs to if it satisfies

and belongs to the little space if it satisfies

(ii)Suppose that , , and satisfy condition (B). belongs to if it satisfies

and belongs to the little space if it satisfies

(iii)Let . belongs to the Dirichlet space if

As , then for , the sets and are normed vector function spaces, with norms given by

This is a direct consequence of the usual theory when we consider the measures ,, respectively. For and nonnegative , , we have ; as a consequence, the sets and result in metric vector spaces, with the metric given by

respectively (see [10]). We will prove their completeness in Theorem 26.

Example 10. Let denotes the class of holomorphic functions on a neighborhood of . Let . Suppose that , , and satisfy condition (A). By (19) we have, for example, the estimation

So, in the definitions, the parameters , , and were chosen to guarantee that the vector spaces and include at least this wide class of holomorphic functions, as Propositions 5, 8, and 9 show. Compare this example with Proposition 3.7 and Theorem 3.8 from [4].

Let , with power series development given by

such that . Then, belongs to each one of the previous spaces. By (37), we get the same result for and .

The spaces are Möbius invariant in the following sense.

Theorem 11. Suppose that , , and satisfy condition (A). If and is a Möbius transformation, then .

Proof. Let ,, and . By the change of variables formula and the Möbius invariance of , and , we have

where we have used in the last line the estimations of Lemmas 3 and 4.

Observe that if and , then

and these spaces are Möbius invariant in the classical way.

Theorem 12. Suppose that , , and satisfy condition (A). Suppose that

for all . Then, the function is continuous on .

Proof. Let be fixed and , such that . Consider a neighborhood with . Let be a sequence contained in and convergent to . Consider now and define . Then, is bounded in . Thus, by bounded convergence theorem,

converges to

Now, integrating in and using change of variables,

Consider

and define the measure by

Now, if we take , , then . Besides, on , results are bounded by a constant , because . Now,

where denotes the characteristic function of the set . As

then by Lebesgue’s dominated convergence theorem, we have that (55) converges to

Theorem 13. Suppose that , , and satisfy condition (B). Suppose that

for all . Then, the function is continuous on .

Proof. Let and , such that . If , the function defined by

results to be uniformly continuous on . As , then

If is nonconstant, given that , there exists , such that and ; then

Thus, if , then

Remark 14. Last two theorems are the versions in of Theorem 3.1 of [6].

Some relations between the little spaces , and the spaces , are given in the following results.

Proposition 15. Suppose that , , and satisfy condition (A). If , then the function

is continuous on .

Proof. By the Theorem 12, it suffices to prove that if , given , there exists such that if , then ; that is, . By hypothesis, . So, given , there exists , such that if , then , or implies . Moreover, as is compact, results uniformly continuous on .

Proposition 16. Suppose that , , and satisfy condition (A). Then, is contained in .

Proof. In fact if , as the function is uniformly continuous on , then is bounded and so

With the same arguments, we get .

2.2. Characterization as Bloch Spaces

Now, we are going to present several characterizations of Bloch spaces similar to the characterizations obtained by Stroethoff [11, Theorems 4 and 7] and Zhao [3, Theorem 1].

For , the -Bloch spaces, , are the families of holomorphic functions satisfying

It is easy to verify that is a norm on the vector space and contains at least the holomorphic functions on domains , with . The respective little -Bloch spaces satisfy

It is clear that .

Proposition 17. (i) Suppose that , , and satisfy condition (A). Then,

and there exists , such that

(ii) If . Then,

Proof. (i) Let ; then by Propositions 5 and 6 and (20)

where we have used that is a subharmonic function; (i) follows from this estimation.

(ii) Let and be fixed. By the absolute continuity of the integral and by Lemma 3, given , there exists , such that if we choose sufficiently close to 1 then

With the same kind of arguments used in (i), we obtain

for all , sufficiently close to 1, and we conclude the proof.

The following result says that Dirichlet spaces are the limit case of and , when .

Theorem 18. (i) Suppose that , , and satisfy condition (A). Then,

(ii) Suppose that , , and satisfy condition (B). Then,

Proof. We prove the first inclusion. The proof of the second one is similar. Let and . By Proposition 17, there exists , such that

Let , such that if ; then . Thus,

By Proposition 8, the first integral goes to zero as . For the second one, consider (75) as following:

Moreover, for the third, using the change of variable ,

and this concludes the proof.

Proposition 19. Suppose that , , and satisfy condition (A), with . Then,

Proof. Let . Then,

Using change of variables and by Proposition 5 we have

and this integral results convergent because

when . Finally, if .

Similar results to Propositions 17 and 19 can be obtained for .

Proposition 20. Suppose that , , and satisfy condition (B), with . Then,

and there exists , such that

Proof. Let ; then

and we finish the proof as in the Proposition 17.

Proposition 21. Suppose that , , and satisfy condition (A), with . Then,

Proof. Let . By hypothesis,

Thus,

and by Theorem 7, we obtain that if ,

and the result follows.

We can synthesize the last propositions in the following.

Theorem 22. Suppose that , , and satisfy condition (B). The following statements are equivalent.

(i);(ii) for each ;(iii) for some ;(iv) for each ;(v) for some .

Jointly with these properties, there are similar results for the little spaces , , and .

For these cases, the inclusions follow immediately from the previous estimations. For the other inclusions we have the following.

Proposition 23. Suppose that , , and satisfy condition (A) and . Then,

Proof. Since , then . Suppose that ; then

Given , there exists , such that

if . We split the integral

as

By hypothesis, the second integral satisfies

Now,

and the function

attains its maximum on the compact set . Therefore, if , the last estimation tends to be zero.

The proof of

is similar.

Lemma 24. Let and . Then, is a uniformly continuous function from the metric space equipped with the Bergman metric to the metric space equipped with the Euclidean metric.

Proof. Let be a piecewise smooth curve. Thus, the lenght of in the Bergman metric of is

where is the Bergman matrix of (see [7]). Let be a holomorphic function. Then,

Now, if is a geodesic in the Bergman metric, with , , and , we have

and the proof is complete.

Theorem 25. Let . The spaces and are complete metric spaces.

Proof. Let be a Cauchy sequence. Given , there exists , such that ,   for all   . In particular, is convergent. Let . We apply the estimation of Lemma 24 to ,, and. Thus,

Then, converges uniformly on compact subsets of to some holomorphic function, say . Therefore, converges uniformly on compact subsets of to , for . So, we can get converges to uniformly on compact sets of when . For , we have

and taking limit when , we get

that is, and hence .

Let be a Cauchy sequence. Then, converges to . Given , there exists , such that , for all . Moreover, there exists , such that

Thus,

for all ; therefore, is a complete space.

Theorem 26. Suppose that , , and satisfy condition (A). The spaces and are complete metric spaces.

Proof. We give the proof only in the case . Let be a Cauchy sequence. Since is too a Cauchy sequence in , then converges to some .

We will prove that . Given , there exists , such that

By Fatou's Lemma, as ,

Since converges to , if we take supremum on and roots in both sides of the last inequality, we obtain

Since , we get that .

If is a Cauchy sequence in , then converges to . Given , there exists , such that if ,

Thus,

therefore, .

In the case , the proof is similar.

As a consequence of the previous results and Proposition 17, we have the following.

Corollary 27. Suppose that , , and satisfy condition (A). The spaces , , , and are Banach spaces, and the inclusion operators and are bounded.

Similar results are valid for the spaces and .

3. The Equality

In this section, we prove the equality .

Theorem 28. Let , , , , and . Then,

where .

Proof. Let . Define

For , as is subharmonic on , we have

Thus,

By Proposition 5,

This integral results convergent because . Denote the last integral by . Then,

and the result follows.

Theorem 29. Suppose that , , and satisfy condition (A). If , then

where for some .

Proof. From Proposition 6 and (113),

Now, if ,

Finally,

with  .

By the Möbius invariance of the space , if we apply the last result to , we obtain the following.

Corollary 30. Suppose that , , and satisfy condition (A) and . Then, for ,

Theorem 31. Suppose that , , and satisfy condition (A) and . Then,

if and only if

Proof. We will abbreviate , then from the last corollary and (20)

using the change of variables , we obtain

Then,

and from here we obtain the result.

Corollary 32. Suppose that , , and satisfy condition (A). Then,

4. Harmonic Majorants in and

For and , the function defined by

is the real Poisson kernel for the ball . If is a real function, integrable on , we define the Poisson integral formula of as

where is the area measure of , for . This formula gives the classic solution to the Dirichlet problem for . Moreover, if , then is harmonic on and has nontangential limit a. e. on (see [12]).

Theorem 33 ((Harnack) see [13]). Consider . Let be a monotone sequence of real harmonic functions on . Suppose that is bounded for some . Then, the sequence converges uniformly on compact sets to a harmonic function on .

Proposition 34. Let be a continuous function on . For , let the function be defined by

Then, results in a uniformly continuous function on . Besides, if , then

for every .

Proof. The function  is uniformly continuous; that is, given that , there exists , such that , for all , with . Let . There exist , such that and . Thus, if , then and . In particular, and . Therefore, if , then .

It follows directly that if , then

so the result follows.

Observe that Poisson's integral formula defines a continuous harmonic function on , with boundary values given by .

Corollary 35. If is as in the previous proposition and for every , , define as

Then, one has besides that

Theorem 36. If is a continuous function and is the function defined in the previous corollary, let be defined by

for every . Then, results in a harmonic function, or on . In the first case, can be obtained through Poisson's integral formula applied to . Besides,

If , then the nontangential limits

exist almost everywhere on .

Proof. Let be an increasing sequence of positive numbers in , such that . For every , we define the harmonic continuous function

Then, if , it follows that

Then, the sequence is increasing, so

for every .

By Harnack's theorem, or is a harmonic function on , and the limit does not depend on the sequence. As

it follows that if and there exists almost everywhere.

For every fixed, let

this is a sequence of positive Lebesgue’s integrable functions, such that for ,

Then, by Lebesgue's monotone convergence theorem,

as .

Let be an open set and let . We say that a harmonic function is the least harmonic majorant of if

(a), for every ;(b) is any harmonic function on , such that , for every , then .

The least harmonic majorant of a continuous function, bounded and radially increasing is not necessarily the Poisson extension of its boundary values; see, for instance, Example 2.2 of [14].

In a similar way as [6, 14], consider a function . For , let us introduce the family of functions defined by

By Theorem 12 and Proposition 34, this function is well defined for every and results in a continuous function on . Moreover, as , for ,

Let be the harmonic function associated with , that is, the solution of Dirichlet's problem, with boundary values given by . It follows from Proposition 34 that

In the same way as we did for , we introduce the family of functions defined by

In this case we obtain Theorem 13 and Proposition 34.

Consider

So, we can summarize the previous results in the next theorems.

Theorem 37. Suppose that , , and satisfy condition (A). Then, belongs to the class if and only if the function defined by

is bounded. The function is then the Dirichlet solution, with boundary values given by . Further, is the least harmonic majorant of the family of harmonic functions , .

Analogous result is obtained for and .

Now, we can formulate an analogous statement to the Theorem 3.1 in [14] to characterize functions of and through their harmonic majorant, as in Theorem 3.9 of [6].

Theorem 38. Suppose that , , and satisfy condition (A) and . Then, the following conditions are equivalent:

(i);(ii) is bounded and is the classical Dirichlet solution with boundary values given by ;(iii) is bounded and is the classical Dirichlet solution with boundary values given by .

With this development, we have obtained a new version of Theorem 1.3 in [6], in terms of harmonic majorants and and their corresponding boundary values and , for the invariant measure.

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