Abstract
Characterizing the hyperbolic Hardy classes, several -functions of hyperbolic type are introduced. Using this, necessary and sufficient conditions on the inducing self-maps are established for the boundedness of the composition operators from logarithmic Bloch spaces into Hardy spaces.
1. Introduction
This paper is to characterize the class of holomorphic self-map of the open complex unit disc for which the composition operator induced by maps logarithmic Bloch space boundedly into Hardy space . Main result of this paper is Theorem 2 whose primitive form is as follows.
Theorem 1. If is a holomorphic self-map of and if , , then the following are equivalent:(i) for all holomorphic on satisfying (ii);(iii)
When the above equivalence is known. We refer to [1–3], wherein the results are investigated in and in the ball of , respectively.
Also, the case was considered in [4], so the main case under consideration is . In the latter case, a different approach, based on duality, is used.
Note that Theorem 1 not only characterizes the composition operators mapping logarithmic Bloch functions into the Hardy space but also introduces a kind of -function. The result will be stated precisely and more extensively in Section 3.
The restriction of the range of is essential. If , then it reduces to a trivial result. If , then it corresponds to another space instead of Hardy space. These will be treated separately in the last section.
2. Preliminaries
2.1. Hardy Space and Hyperbolic Hardy Class
Let be the unit disc of the complex plane. For and for subharmonic in , we denote Then the right side limit is monotone increasing. And by definition, the Hardy space consists of holomorphic in for which , while the Yamashita hyperbolic Hardy class consists of holomorphic self-map of for which Here denotes the hyperbolic distance of and in , namely, Though is not a linear space, it has, as hyperbolic counterparts, many properties analogous to those of .
For holomorphic self-maps of , we let following the notation of Yamashita and let Then they have the following basic properties:for any , where Here and throughout, means , denotes the Laplacian: , means that is bounded by a positive uniform constant times , and means that either both sides are zero or the quotient lies between two positive uniform constants. We refer to [1, 5–7] for (6)~(11). For a general theory of and , we refer to [8–10] and [7, 11, 12], respectively.
2.2. Logarithmic Bloch Space
For , denotes the weighted Bloch space consisting of holomorphic functions in satisfying is a Banach space equipped with the norm . Though the invariance “ for all and ” is satisfied only when , it is not difficult to see that we still have once is fixed.
2.3. Bloch Pullback Problem
The composition operator induced by a holomorphic self-map of will be denoted as usual by , that is, . For two function spaces and , we denote by the set of self-maps for which . In this notation, (i)(ii) of Theorem 1 can be expressed as The problem of characterizing is a kind of Bloch pullback problem. Bloch pullback problem was initiated by P. Ahern and W. Rudin. See [13–20] for Bloch-BMOA pullbacks and [1–3, 17, 21] for Bloch-Hardy pullbacks.
2.4. New Hyperbolic -Functions
For holomorphic in , it is well known that -function of Littlewood-Paley defined bysatisfies andSee [10, 22, 23]. Analogously, for holomorphic self-map of , the hyperbolic version of -function defined bysatisfies andprovided . See [17].
We pay attention to the absence of the square root in the defining of in (17) when we compare it to that of in (15). The difference actually explains lots of known parallelism (see [11]) between and . Suggested by (11), we define for Note that . Of course, main objective of introducing (19) is to establish an equivalence as (18).
3. Equivalence between Norms
3.1. Main Result Revisited
Equipped with the notions introduced in Section 2, Theorem 1 can be stated as the first part of the following.
Theorem 2. Let , and . Let be a holomorphic self-map of . Then the following (i), (ii), and (iii) are mutually equivalent:(i)(ii)(iii). Moreover, if we assume thenandwhere denotes
It follows by (13) that the restriction is not an essential one because is bounded if and only if is with . Note that we used the notation also for the case when are not normed spaces.
3.2. More on -Function Equivalence
For (20), we in fact can prove more extensively the following: we define for and that Then
Theorem 3. Under the assumption of Theorem 2 with ,
Immediately after this with (21) is the following.
Corollary 4. Under the assumption of Theorem 2 with ,
To cover all of our results stated up to here, it is sufficient to prove (24) and (21). This will be done in Section 5 after stating preparatory lemmas in Section 4.
4. Preparatory Lemmas
We describe some lemmas, whose proof will be deferred to Section 6, that will be used in proving our main theorem.
Lemma 5. Let be a holomorphic self-map of and . Then, for any positive the function is subharmonic in . Furthermore, for , is an increasing function of andwhere is the Poisson kernel:
The subharmonicity of in Lemma 5 gives the following, where .
Lemma 6. Let and . Then for holomorphic self-maps of ,
We need the following inequalities which are not difficult to guess. See, for example, [24] or [25] for (29) and (30), and see [26] for (31).
Lemma 7. Let and . Then, for ,
5. Proof of Main Results
Let be a holomorphic self-map of with . We are sufficient to show (24) and (21). We assume is not constant because there’s nothing left to prove when is a constant. Let us denote for simplicity by the boundary of and the arc length measure on normalized to be .
5.1. Proof of (24)
For notational clearance we prove only for . But replacing and , respectively, by and , it is easy to check that the proof below works for general in the same way.
To show , we divide it into two cases: and .
If , then by Hölder’s inequality, (10) and Lemma 6, it follows that
Next, suppose . Set and let .
We make use of the representation where the supremum is taken with respect to all nonnegative trigonometric polynomials with , .
If we set the Poisson integral of on , then (34), a change of the order of the integration, and (33) givewhile straightforward calculation gives where and . Whence by (36) we have
Now, using Green’s theorem with limiting process and Hölder’s inequality, we obtain
On the other hand, direct differentiation gives and where denotes the holomorphic function in whose real part is with ; whence Since , we have, by Schwarz inequality, Applying Hölder’s inequality with triple and applying (11) to the last integral, we arrive at From (16), we know , and it follows from the theorem of M. Riesz ([8, 23]) that . Thus
Gathering estimates (38), (39), and (45) up, we arrive at for all positive trigonometric polynomials with . That is, Therefore we obtain via the arithmetic-geometric mean inequality that
We next show Set again. Take such that . Hölder’s inequality with paring gives where . Applying Hölder’s inequality one more time to the last quantity with paring , Therefore by applying the arithmetic-geometric mean inequality and (48), we obtain the desired inequality. This proof is completed.
5.2. Proof of (21)
If , then (16), the definition of , and (24) gives This with simple inequality gives whence this verifies .
We next show . For each nondyadic , let where is the Rademacher function (see [8, 23]) defined by Then is holomorphic in and by (30) there is a positive constant such that so that with Thus, the definition of gives
On the other hand, it follows by (29) thatwhile Khinchin’s inequality (see [23] Theorem 8.4) gives
Now (57), (58), and (59) together verify . This proof is completed.
6. Proof of Lemmas
For measurable functions in , let us denote as usual
6.1. Proof of Lemma 5
Let and . From we get where Thus, Noting that and , we have Thus, is subharmonic if .
For , subharmonicity of implies that is an increasing function of : Also, by subharmonicity, Letting , monotone convergence theorem guarantees (27) and (28) also for . This proof is completed.
6.2. Proof of Lemma 6
First let . Apply Green’s Theorem of the form valid for . Recalling (11), it follows by a limiting process after integrating with respect to that Thus, the first equivalence follows.
The second equivalence follows from Lemma 5: setting , the inequality and the increasing property of give
Next, letting we obtain (28) for also. This proof is completed.
7. Remarks and Acknowledgement
In view of our main result, Theorem 2, there arises a natural question: what is when ?
Let . If we denote the identity map of by , that is, , thensimply because which gives , equivalently , for all .
Let be holomorphic self-map of satisfying . Then is subordinate to . By (73) and Littlewood Subordination Theorem ([8]), we have for any . But by (13) we can remove the condition . Therefore we have for any .
The case was considered in [26] (partially) in connection with (31) and the concept of the hyperbolic Nevanlinna class (see [27]).
Data Availability
No data were used to support this study.
Disclosure
The present author noticed the publication of [4, 25, 26] after finishing this paper.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This work was supported by a Research Grant of Andong National University.