# Products of differentiation and composition from hardy spaces to Zygmund-type spaces of analytic functions on the upper half-plane.

1. Introduction

Let [[PI].sub.+] = {z [member of] C : Imz > 0} be the upper half-plane in the complex plane C and H([[PI].sub.+]) the space of all analytic functions on [[PI].sub.+]. For p [member of][1, [infinity]) the Hardy space [H.sup.p]([[PI].sub.+]) consists of all f [member of] H([[PI].sub.+]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Hardy space with the norm [parallel]*[parallel][H.sup.p]([[PI].sub.+]) becomes a Banach space. For some information of this space see  and .

The n-th weighted-type space consists of all f [member of] H([[PI].sub.+]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When n = 2, we call the space the Zygmund-type space and denote it by [Z.sub.[infinity]]([[PI].sub.+]). The quantity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a semi-norm on [Z.sub.[infinity]]([[PI].sub.+]) and a norm on [Z.sub.[infinity]]([[PI].sub.+])/[P.sub.1], where [P.sub.1] is the set of all polynomials whose degrees are less than or equal to 1. Under the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the Zygmund-type space is a Banach space. For some details of the weighted-type, Bloch-type and Zygmund-type spaces on the half-plane, see, e.g., [4, 6, 8] and the references therein.

Let [phi] be an analytic self-map of [[PI].sub.+]. Products of differentiation operator and composition operator on H([[PI].sub.+]) are defined by

[D.sub.[phi],R]f(z) = (f o [phi])'(z) = f'([phi](z))[phi]'(z), z [member of] [[PI].sub.+],

and

[D.sub.[phi],L]f(z) = f'([phi](z)), z [member of] [[PI].sub.+].

A typical problem is to provide function theoretic characterizations when [phi] induces a bounded or compact products of differentiation operator and composition operator between two given spaces of analytic functions. Hibschweiler and Portnoy  considered the behavior of the differentiation on the range of the composition operator on Hardy or weighted Bergman spaces on the unit disk. Ohno  recently has studied the products of differentiation operator and composition operator on Bloch and little Bloch spaces on the unit disc. Here we continue this line of research and investigate the products of differentiation operator and composition operator from Hardy spaces to spaces defined as above on the upper half-plane.

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a [??] b means that there is a positive constant C such that a/C [less than or equal to] b [less than or equal to] Ca.

2. Main Results

In this section we first quote two auxiliary lemmas. The first was proved in .

Lemma 2.1. Suppose p [greater than or equal to] 1, n [member of] N and w [member of] [[PI].sub.+], then the function

[f.sub.w,n](z) = [(Imw).sup.n-1/p]/[(z - [bar.w]).sup.n]

belongs to [H.sup.p]([[PI].sub.+]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The next lemma was essentially showed in the proof of Theorem 3.1 of .

Lemma 2.2. Suppose p [greater than or equal to] 1, then there exists a positive constant C independent of f such that

[absolute value of [f.sup.(n)](z)] [less than or equal to] C[parallel]f[parallel][H.sup.p]([[PI].sub.+])/[(Imz).sup.n+1/p].

We now formulate and prove the main results of this paper.

Theorem 2.3. Suppose p [greater than or equal to] 1 and [phi] is an analytic self-map of [[PI].sub.+], then the operator [D.sub.[phi],R]: [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Moreover, if the operator [D.sub.[phi],R] : [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+])/[P.sub.1] is bounded, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

where [P.sub.1] is the set of all polynomials whose degrees are less than or equal to 1.

Proof. Suppose the operator [D.sub.[phi],R]: [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded. Lemma 2.1 shows that the function

[f.sub.w,1](z) = [(Im[phi](w)).sup.1-1/p]/(z - [bar.[phi](w)])

belongs to [H.sup.p]([[PI].sub.+]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

By an easy calculation we have

[f'.sub.w,1](z) = -[(Im[phi](w)).sup.1-1/p]/[(z - [bar.[phi](w)]).sup.2], [f".sub.w,1] = 2[[(Im[phi](w)).sup.1-1/p]/[(z - [bar.[phi](w)]).sup.3]] (2.6)

and

[f'".sub.w,1](z) = -6[[(Im[phi](w)).sup.1-1/p]/[(z - [bar.[phi](w)]).sup.4]] (2.7)

Replacing (2.6) and (2.7) in (2.5), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Then by (2.8) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

So if we can show that conditions in (2.1) and (2.2) hold, then from the inequality (2.9) it follows that the condition in (2.3) holds.

For w [member of] [[PI].sub.+], set the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to check that

[g'.sub.w]([phi](w)) = 0, [g".sub.w]9[phi](w)) = 0 and [g'".sub.w]([phi](w)) = C/[(Im[phi](w)).sup.3+1/p].

By Lemma 2.1 we have [g.sub.w] [member of] [H.sup.p]([[PI].sub.+]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From this, and since [D.sub.[phi],R]: [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded, it implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

from which the condition in (2.1) holds.

For w [member of] [[PI].sub.+], taking the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[h'.sub.w]([phi](w)) = 0, [h".sub.w]([phi](w)) = C/[(Im[phi](w)).sup.2+1/p], [h'".sub.w]([phi](w)) = 0,

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From this and the boundedness of [D.sub.[phi],R]: [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

from which the condition in (2.2) holds.

From (2.9), (2.10) and (2.11), we prove that the condition in (2.3) holds. Moreover, from (2.9), (2.10) and (2.11) it implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

Conversely, suppose conditions in (2.1), (2.2) and (2.3) hold. By Lemma 2.2, for f [member of] [H.sup.p]([[PI].sub.+]) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

From conditions in(2.1), (2.2), (2.3) and (2.13), this shows that [D.sub.[phi],R] : [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded. By (2.12) and (2.13), we get the asymptotic relation in (2.4).

Theorem 2.4. Suppose p [greater than or equal to] 1 and [phi] is an analytic self-map of [[PI].sub.+], then the operator [D.sub.[phi],L]: [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)

Moreover, if the operator [D.sub.[phi],L] : [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+])/[P.sub.1] is bounded, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)

where [P.sub.1] is the set of all polynomials whose degrees are less than or equal to 1.

Proof. Suppose the operator [D.sub.[phi],L]: [H.sup.p] ([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded. Then the function [f.sub.w,1] [member of] [H.sup.p]([[PI].sub.+]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

Replacing (2.6) and (2.7) in (2.17), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.18)

By (2.18) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.19)

So if we can show that the condition in (2.15) holds, then from the inequality (2.19) it follows that the condition in (2.14) holds.

For w [member of] [[PI].sub.+], taking the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[g".sub.w]([phi](w)) = C/[(Im[phi](w)).sup.2+1/p], [g"'.sub.w]([phi](w)) = 0,

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From this and the boundedness of [D.sub.[phi],L]: [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

from which the condition in (2.15) holds.

Moreover, from (2.19) and (2.20) it implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)

Conversely, suppose conditions in (2.14) and (2.15) hold. By Lemma 2.2, for f [member of] [H.sup.p]([[PI].sub.+]) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.22)

From conditions in (2.14), (2.15) and (2.22), it follows that [D.sub.[phi],L] : [H.sup.p]([[PI].sub.+]) [right arrow] [Z.sub.[infinity]]([[PI].sub.+]) is bounded. From (2.21) and (2.22), we obtain the asymptotic relation in (2.16).

References

 P. Duren, Theory of [H.sup.p] spaces, Pure and Applied Mathematics, vol. 38 Academic Press, New York, 1970.

 R. A. Hibschweiler, N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math., 35(3):843-855, 2005.

 K. Hoffman, Banach spces of analytic functions, Prentice-Hall Series in Morden Analysis Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.

 S. Li and S. Stevic, Volterra-type operators on Zygmund spaces, Journal of Inequalities and Applications, vol. 2007, Article ID 32124, 10 pages, 2007.

 S. Ohno, Products of composition and differentiation on Bloch spaces, Bull. Korean Math. Soc., 46(6):1135-1140, 2009.

 S. D. Sharma, A. K. Sharma and S. Ahmed, Composition operators between Hardy and Bloch-type spaces of the upper half-plane, Bull. Korean Math. Soc., 43(3):475-482, 2007.

 S. Stevic, Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane, Abstract and Applied Analysis, vol 2009, Article ID 161528, 8 pages, 2009.

 Y. Yang, Z. J. Jiang, Products of differentiation and composition from weighted Bergman spaces to some spaces of analytic functions on the upper half-plane, Int. Journal of Math. Analysis, 4(22):1085-1094, 2010.

Zhi Jie Jiang

School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China.

E-mail: matjzj@126.com

(1) Supported by the Special Foundation for Young Scientists of Sichuan Province (No. 09ZC115).
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