#### Abstract

Composition operators from Bloch-type spaces to classes, from to , and from to are considered. The criteria for these operators to be bounded or compact are given. Our study also includes the corresponding hyperbolic spaces.

#### 1. Introduction

Let denote the algebra of all analytic functions in the unit disc , and let be the subset of consisting of those for which . Every induces the composition operator acting on , defined by . By Littlewood’s subordination principle any such composition operator maps every Hardy and Bergman space into itself. For the theory of composition operators in analytic function spaces see [1, 2]. Clearly every composition operator maps also into itself. Hyperbolic classes are subsets of and are defined by using the hyperbolic derivative of . The hyperbolic derivative of the composition satisfies the equality which can be understood as a kind of chain rule.

For , the -Bloch space consists of those for which The closure of polynomials in is the little space -Bloch which consists of those for which , as . The spaces and are the classical Bloch space and the little Bloch space , respectively. For the theory of Bloch spaces, see the classical reference [3] and also [4, 5].

The hyperbolic -Bloch classes and are the sets of those for which and , respectively. In the special case it is simply denoted and . Clearly and are not linear spaces since the sum of two functions in does not necessarily belong to . Moreover, the Schwarz-Pick lemma implies for all , and therefore the hyperbolic -Bloch classes are only considered when .

Let Green’s function of be defined as , where is the automorphism of which interchanges the points zero and . For any , the automorphism is its own inverse and satisfies the fundamental equalities which can be verified by straightforward calculations.

For , , and , the spaces and consist of those for which and respectively. Here denotes the element of the Lebesgue area measure on . The family of function spaces, introduced by Zhao in [6], is known as the general family of function spaces. For , the space is a Banach space with respect to the norm , and so is as a closed subspace of ; see [6, Section 2]. When , the space is a complete metric space with the (invariant) metric defined by . The metric is also -homogeneous; that is, for , and therefore the space is a quasi-Banach space for . If , then the space reduces to the space of constant functions by [6, Proposition 2.12]. Therefore from now on it is always assumed that the parameters , , and of the spaces or in question satisfy , , , and without mentioning it every time. Many classical function spaces can be found among the family by choosing the parameters appropriately. In order to connect the results of the present paper to the ones in the existing literature some information on this matter is gathered in Table 1. For example, stands for the classical Bergman space and denotes the weighted Dirichlet space. The interested reader is invited to see [6, 7] for more information and the definitions of the spaces.

The class is defined as the set of those for which Similarly, , if (5) with replaced by , is satisfied. It is sometimes convenient to set , where . By Schwarz-Pick lemma , if , and hence the classes and are considered only when .

A composition operator is said to be bounded if there exists a positive constant such that for all . Further, is said to be bounded if (7) is satisfied for all . On the other hand, is said to be bounded if (7) is satisfied for all and . Hereafter a bounded operator mapping from one hyperbolic class into another is understood in an analogous manner.

The purpose of this paper is, on one hand, to complete in part certain results in the existing literature and, on the other hand, to continue the line of research of [8] on composition operators in hyperbolic function classes. The spaces/classes of interest in this work are -Bloch spaces, Dirichlet-type spaces, and -spaces as well as their hyperbolic counterparts. The boundedness of the composition operator is discussed in several different cases, using the standard tools such as the change of variable formula by Stanton and different kind of characterizations of Carleson measures.

The remainder of this paper is organized as follows. In Section 2, the main results are presented together with necessary definitions. In Section 3 some auxiliary results on hyperbolic classes are given and the necessary background material involving Carleson measures and Nevanlinna counting function is introduced. Sections 4–11 contain the proofs of the main results in chronological order.

#### 2. Main Results

Bounded and compact composition operators mapping from or into or have been studied in many particular cases in [9, 10]; see also [11–15], the most general result being found in [16]. The first result in the present paper extends in part these results on bounded composition operators to the corresponding hyperbolic classes under certain conditions on the parameters.

Theorem 1. *Let , , , , and . Then the following statements are equivalent:*(1)* is bounded;*(2)* is bounded;*(3)*.** Moreover, if , then (1)–(3) are equivalent to*(4)* is bounded;*(5)* is bounded.*

Theorem 2. *Let , , , , and . Then the following statements are equivalent:*(1)* is bounded;*(2)* is compact;*(3)*.** In case the conditions (1)–(3) are equivalent to*(4)* is bounded.*

Theorem 3. *Let , , , , and . Then the following statements are equivalent: *(1)* is bounded;*(2)* and
**Moreover, if , then (1) and (2) are equivalent to *(3)* is bounded.*

It is now proceeded to study the case when the target space is or . The following result should be compared with [8, Theorem 1.3] and [10, Theorem (iii)].

Theorem 4. *Let , , , , and . Then, the following statements are equivalent: *(1)* is bounded;*(2)* is bounded;*(3)*.**Moreover, if , , then (1)–(3) are equivalent to *(4)* is bounded;*(5)* is bounded.*

Since the test functions used in the proof of Theorem 4 are included in the little versions of the domain spaces, the next corollary follows.

Corollary 5. *Let , , , , and . Then the following statements are equivalent: *(1)* is bounded;*(2)* is bounded;*(3)*.**Moreover, if , , then (1)–(3) are equivalent to *(4)* is bounded;*(5)* is bounded.*

Theorem 6. *Let , , , and . Then the following statements are equivalent: *(1)* is bounded;*(2)* is bounded;*(3)*.**Moreover, if , , then (1)–(3) are equivalent to *(4)* is bounded;*(5)* is bounded.*

Theorem 7. *Let , , , and . Then the following statements are equivalent: *(1)* is bounded;*(2)* is bounded;*(3)* and
**Moreover, if , , then (1)–(3) are equivalent to *(4)* is bounded;*(5)* is bounded.*

If the domain and the target class both are some classes with , then the situation seems to be more complicated. The following result characterizes bounded composition operators mapping from into when . Note that in the hyperbolic case the condition is indeed needed in the proof while the condition only guarantees that the target class is not the whole class .

Theorem 8. *Let , , , , and . Then the following statements are equivalent: *(1)* is bounded;*(2)*.** Moreover, if and , then (1) and (2) are equivalent to *(3)* is bounded.*

*Remark 9. *The condition (1) in Theorem 8 is a special case of (2) in Theorem 4. If , then by Lemma 12, and Theorem 8 implies that is bounded if and only if
However, a straightforward calculation shows that (10) is satisfied if and only if
Thus, by taking and , we see that [10, Theorem (iii)] remains valid also when the domain space is the classical Dirichlet space .

Theorem 10. *Let and . Then the following conditions are equivalent: *(1)* is bounded (compact);*(2)* is bounded (compact) and .*

Function belongs to if and only if for every and . In particular, this holds for . In this sense, Theorem 10 is related to [9, Theorem 5.2].

Theorem 11. *Let , , , and . Then the following conditions are equivalent: *(1)* is bounded;*(2)* is bounded and .*

For suitable choice of parameters the nonhyperbolic cases of Theorems 8–11 reduce to [9, Theorem 5.2]. The reasoning there, however, is slightly different.

#### 3. Auxiliary Results and Background Material

Some basic properties of the hyperbolic classes are gathered in the following lemma.

Lemma 12. *Let and . Then the following assertions hold: *(1)*, where is a positive constant independent of ;*(2)* if ;*(3)*.*

Lemma 12 can be proved in a similar manner as the corresponding results for the spaces ; see [6, 17] for details. Note that (1) implies the inclusion , .

A positive Borel measure on is said to be a* bounded **-Carleson measure*, if
where denotes the arc length of a subarc of ;
is the Carleson box based on , and the supremum is taken over all subarcs of such that . Moreover, if
then is said to be a* compact **-Carleson measure*. If , then a bounded (resp., compact) 1-Carleson measure is just a standard bounded (resp., compact) Carleson measure.

For and , let the pseudohyperbolic disc be defined by . The pseudohyperbolic disc is an Euclidean disc centered at with radius ; see [18, page 3].

In the following lemma we have gathered some well-known and useful characterizations of bounded -Carleson measures. For the proof, see [19, Theorem 13], [20, Lemma 2.1], [21, pp. 89–90], and [22, Proposition 2.1].

Lemma A. *Let be a positive Borel measure on , , and . Then the following statements are equivalent: *(1)*;*(2)*;*(3)*.** Moreover, the expressions , , and are comparable.*

Another auxiliary result needed is Luecking’s [21] characterization of Carleson measures in terms of functions in the weighted Bergman spaces.

Theorem B. *Let be a positive measure on , and let . Then is a bounded -Carleson measure if and only if there is a positive constant , depending only on and , such that
**
for all analytic functions in , in particular for all , the standard weighted Bergman space. Moreover, if is a bounded -Carleson measure, then , where depends only on , and , and
*

The following change of variables formula by Stanton, [23, 24], was apparently first used by Shapiro [25] in the study of composition operators. It also plays a key role in some of our proofs.

Lemma C. *Let and be positive measurable functions on , and let . Then
**
where
*

If , then is the generalized Nevanlinna counting function

For the study of compactness we need the following well-known result; see [1, Proposition 3.11] for a similar result. The following can be deduced by a result of Tjani; see [26].

Lemma D. *Let . Then is compact if and only if for any bounded sequence in with uniformly on compact subsets of as , as .*

#### 4. Proof of Theorem 1

It will be shown first that the conditions (3), (4), and (5) are equivalent by proving the implications . Since (4) clearly implies (5), (see Lemma 12), it suffices to prove the other two implications.

##### 4.1. Proof of

Let and ; that is, . Then and therefore is bounded if (3) is satisfied.

##### 4.2. Proof of

Let first . Suppose that is bounded, and define for . Then and therefore for all . Since is bounded, there is a positive constant such that Taking limit as , Fatou’s lemma yields (3) with .

If , by [10, Theorem ], there are functions and in , such that . Since Hence the functions are bounded, and , , satisfy , . Therefore in for and also . Applying the assumption that is bounded for the functions and , and using the asymptotic inequalities the condition (3) with follows.

##### 4.3. The Rest of the Assertions

It was shown in [16, Theorems 1.1 and 1.4] that (1), (2) and (3) are equivalent for . In fact, He and Jiang required this restriction just to see that (2) implies (3). Now, we prove that such implication holds for* any *. Suppose that is bounded. Let be a sequence in such that , and consider the functions

By [27, Theorem 1], each and there is a constant such that . Since is bounded, for any . Let now . Then for any and . An integration with respect to using Fubini theorem and a Zygmund’s result on gap series (see Theorem 8.20 on page 215 of Volume I of [28]) we get Since for any we have being a constant independent of neither nor . An application of Fatou’s lemma in the above inequality yields (3).

#### 5. Proof of Theorem 2

The equivalence of (1), (2), and (3) is proved in [16]. Hence it remains to show that these together are equivalent to (4).

##### 5.1. The Necessity of (4)

Let . It follows from (1) that is bounded and hence is bounded by Theorem 1. Furthermore, (3) implies that for every ; that is, .

##### 5.2. The Sufficiency of (4)

Let be bounded. Choosing and as in the proof of Theorem 1 one obtains Thus (3) holds, and the proof is complete.

#### 6. Proof of Theorem 3

Since (1) and (2) are equivalent by [16], it remains to prove the necessity and the sufficiency of (3).

##### 6.1. The Necessity of (3)

By (1) the operator is bounded, which implies by Theorem 1 that is bounded and For every and there exists such that when . For this fixed , By (2), the right hand side tends to zero as . Hence for every .

##### 6.2. The Sufficiency of (3)

It is enough to show that (3) implies (2). The condition (3) implies that is bounded, whence by Theorem 1 Furthermore, since the function belongs to , one obtains and (2) is satisfied.

#### 7. Proof of Theorem 4

It will be shown that the conditions (3), (4), and (5) are equivalent by proving the implications . Since (4) clearly implies (5) by Lemma 12, it suffices to prove the other two implications. The equivalence of (1), (2), and (3) is proved for example, in [29]; see Corollaries 2.10 and 2.12.

##### 7.1. Proof of

If , then and it follows that (3) implies (4).

##### 7.2. Proof of

Suppose that is bounded, and define , where and . By the assumption there exists a positive constant such that for all . By Lemma 12, and an application of [30, Lemma 2.5] shows that there exists a positive constant such that for all and if . But since the right hand side of (39) is a decreasing function of , it follows that for all , and . This together with (38) yields for all , and the condition (3) follows by choosing .

It is worth noticing that the implication in the case can also be proved by using the functions for which , , and , where is a positive constant, for all .

#### 8. Proof of Theorem 6

Since (1) clearly implies (2) and (2) implies (3) by Theorem 4 and the fact that identity function belongs to with the present parameters, the first thing to show is that (3) implies (1). By Theorem 4 it remains to prove the inclusion of to the space with every . But this follows from the (3) and the inequality

In a similar manner (3) implies (4) and (4) clearly implies (5). It remains to show that (5) implies (3). Let , where and . Given , for all , being a positive constant independent of and close enough to the unit circle. The condition (3) follows by choosing .

#### 9. Proof of Theorem 8

It will be shown that (1) and (2) are equivalent, and (2) and (3) are equivalent.

##### 9.1. Proof of

By Lemma C, where . By Theorem B and Lemmas A and C, and it follows that is bounded if (2) is satisfied.

##### 9.2. Proof of

This implication can be proved by using the test functions such that . These functions satisfy the inequality for all by Forelli-Rudin estimates [5, Lemma ].

##### 9.3. Proof of

This implication can be proved in a similar manner as the implication since the assertion in Theorem B holds also when is replaced with a subharmonic function , , and is replaced by . The proof of this fact will be presented next since some of the details will be needed later on. Since is subharmonic, there exists a positive constant , depending only on and , such that This together with Lemma C implies Then the symmetry of the characteristic functions of pseudohyperbolic discs and the Minkowski inequality yield The desired inequality follows by the fact that for . Thus (2) implies (3) by Lemma C.

##### 9.4. Proof of

To prove this implication, define