Journal of Function Spaces
Volume 2018 (2018), Article ID 8150908, 7 pages
https://doi.org/10.1155/2018/8150908

Research Article

Composition Operators Mapping Logarithmic Bloch Functions into Hardy Space

E. G. Kwon

Department of Mathematics-Education, Andong National University, Andong 36729, Republic of Korea

Correspondence should be addressed to E. G. Kwon; egkwon@andong.ac.kr

Received 29 March 2018; Accepted 29 May 2018; Published 4 July 2018

Academic Editor: Stanislav Hencl

Copyright © 2018 E. G. Kwon. 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.

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 by

satisfies and

See [10, 22, 23]. Analogously, for holomorphic self-map of , the hyperbolic version of -function defined by

satisfies and

provided . 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 then

and

where 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 and

where 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 .

Note, by (11) and (27),

and

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) give

while 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) that

while 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, , then

simply 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.

References

E. G. Kwon, “Composition of Blochs with bounded analytic functions,” Proceedings of the American Mathematical Society, vol. 124, no. 5, pp. 1473–1480, 1996.

E. Doubtsov and A. N. Petrov, “Bloch-to-Hardy composition operators,” Central European Journal of Mathematics, vol. 11, no. 6, pp. 985–1003, 2013.

E. G. Kwon, “Hyperbolic mean growth of bounded holomorphic functions in the ball,” Transactions of the American Mathematical Society, vol. 355, no. 3, pp. 1269–1294, 2003.

E. Doubtsov, “Hyperbolic Hardy classes and logarithmic Bloch spaces,” in Methods of Fourier analysis and approximation theory, Appl. Numer. Harmon. Anal., pp. 33–42, Springer, 2016.

E. G. Kwon, “Fractional integration and the hyperbolic derivative,” Bulletin of the Australian Mathematical Society, vol. 38, no. 3, pp. 357–364, 1988.

E. G. Kwon, “Mean growth of the hyperbolic Hardy class functions,” Mathematica Japonica, vol. 35, no. 3, pp. 451–460, 1990.

S. Yamashita, “Functions with Hp hyperbolic derivative,” Mathematica Scandinavica, vol. 53, no. 2, pp. 238–244, 1983.

P. L. Duren, The theory of Hp functions, Academic Press, New York, NY, USA, 1970.

J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, NY, USA, 1981.

M. Pavlović, Function classes on the unit disc (De Gruyter Studies in Mathematics 52), Berlin, Germany, De Gruyter, 2013.

S. Yamashita, “Hyperbolic Hardy classes and hyperbolically Dirichlet-finite functions,” Hokkaido Mathematical Journal, vol. 10, no. Special Issue, pp. 709–722, 1981.

S. Yamashita, “Holomorphic functions of hyperbolically bounded mean oscillation,” Bollettino dell Unione Matematica Italiana. Sezione B, vol. 5, no. 3, pp. 983–1000, 1986.

P. Ahern, “On the behavior near a torus of functions holomorphic in the ball,” Pacific Journal of Mathematics, vol. 107, no. 2, pp. 267–278, 1983.

P. Ahern and W. Rudin, “Bloch functions, BMO, and boundary zeros,” Indiana University Mathematics Journal, vol. 36, no. 1, pp. 131–148, 1987.

E. Doubtsov, “BLOCH-to-BMOA compositions on complex balls,” Proceedings of the American Mathematical Society, vol. 140, no. 12, pp. 4217–4225, 2012.

E. Doubtsov, “Composition Operators Between Bloch and BMOA Spaces on Polydisks,” Journal of Mathematical Sciences, vol. 229, pp. 518–527, 2018.

E. G. Kwon, “Hyperbolic g-function and Bloch pullback operators,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 626–637, 2005.

E. G. Kwon, “On analytic functions of Bergman BMO in the ball,” Canadian Mathematical Bulletin, vol. 42, no. 1, pp. 97–103, 1999.

E. G. Kwon and M. Pavlović, “Bibloch mappings and composition operators from Bloch type spaces to BMOA,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 303–313, 2011.

W. Ramey and D. Ullrich, “Bounded mean oscillation of Bloch pull-backs,” Mathematische Annalen, vol. 291, no. 4, pp. 591–606, 1991.

R. Zhao, “Composition operators from Bloch type spaces to Hardy and Besov spaces,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 749–766, 1999.

M. Pavlović, “On the Littlewood–Paley g-function and Calderóns area theorem,” Expositiones Mathematicae, vol. 32, pp. 169–195, 2013.

A. Zygmund, Trigonometric Series, Cambridge University Press, London, UK, 1959.

D. Girela, M. Pavlović, and J. Á. Peláez, “Spaces of analytic functions of Hardy-Bloch type,” Journal d'Analyse Mathématique, vol. 100, pp. 53–81, 2006.

A. N. Petrov, “Reverse estimates in logarithmic Bloch spaces,” Archiv der Mathematik, vol. 100, no. 6, pp. 551–560, 2013.

A. N. Petrov, “Integral inverse estimates for logarithmic Bloch spaces in the ball,” Journal of Mathematical Sciences, vol. 202, pp. 556–572, 2014.

E. Doubtsov, “Characterizations of the hyperbolic Nevanlinna class in the ball,” Complex Variables and Elliptic Equations. An International Journal, vol. 54, no. 2, pp. 119–124, 2009.