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Norm of an integral operator on some analytic function spaces on the unit disk
Journal of Inequalities and Applications volume 2013, Article number: 342 (2013)
Abstract
If f is an analytic function in the unit disc , a class of integral operators is defined as follows:
The norm of on some analytic function spaces is computed in this paper.
MSC:47B38, 32A35.
1 Introduction
Let be the unit disk of a complex plane ℂ. Denote by the class of functions analytic in . Let dσ denote the normalized Lebesgue area measure in and the Green function with logarithmic singularity at a, i.e., , where is the Möbius transformation of .
For , the is the space of all functions , for which
We know that , the space of all analytic functions of bounded mean oscillation [1, 2]. For all , the space is the same and equal to the Bloch space , consisting of analytic functions f in such that
See [3, 4] for the theory of Bloch functions.
For , the α-Bloch space, denoted by , is the space of all functions f in , for which
Obviously, for .
For any , the next two integral operators on are induced as follows:
Let denote the multiplication operator, that is, .
Let . Then
If f is a constant, then all results about , or are trivial. In general, f is assumed to be non-constant. Both integral operators have been studied by many authors. See [5–21] and the references therein.
Norm of composition operator, weighted composition operator and some integral operators have been studied extensively by many authors, see [22–34] for example. Recently, Liu and Xiong discussed the norm of integral operators and on the Bloch space, Dirichlet space, BMOA space and so on in [35].
In this paper, we study the norm of integral operator . The norm of on several analytic function spaces is computed.
2 Main results
In this section, we state and prove our main results. In order to formulate our main results, we need an auxiliary result which is incorporated in the following lemma.
Lemma 2.1 Let . For any , the function
is analytic in and .
Proof By (1.1) and [[1], Proposition 1, p.109], we have
where . Taking , we have
Since
we have
A simple computation shows
Also, it is easy to see
Thus,
and
Then the proof is complete. □
First, we consider the norm of on , .
Theorem 2.2 Let . If , then is bounded on if and only if . Moreover,
Proof For any with , it is trivial that . To prove the converse, define . Given any , there exists such that . Let , where
It is easy to see that
Henceforth,
Taking and by the subharmonicity of , we obtain
By Lemma 2.1 we have
Since ϵ is arbitrary, we have and the proof is complete. □
Next, we consider the norm of from () to .
Theorem 2.3 Let . If , then is bounded from space to space if and only if . Moreover, we have
Proof If , then (1.2) gives
From a part of the proof of estimate (2.2) for , we see that
and so
This leads to
On the other hand, define . Given any , there exists such that . Let , where
This together with Lemma 2.1 gives the following:
Since ϵ is arbitrary, we have
The proof is complete. □
Finally, we consider the norm of the integral operator on , .
Theorem 2.4 Let and . Then the integral operator is bounded on if and only if . Moreover,
Proof For any with , by (1.3) we have
This implies .
Now we need to show the reverse inequality. Define . Given any , there exists such that . Put
where is any path in from 0 to z, and a single-valued analytic branch is specified. By Theorem 13.11 in [[36], p.274], we know h is an analytic function in and . Also, it is easy to check . In fact,
On the other hand, we have
Hence, the assertion follows by (2.4) and (2.5). Thus
Since the ϵ is arbitrary, the proof is complete. □
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Acknowledgements
The first author is supported by the National Natural Science Foundation of China (No. 11126284). The second author is supported by the project of Department of Education of Guangdong Province (No. 2012KJCX0096).
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Li, H., Li, S. Norm of an integral operator on some analytic function spaces on the unit disk. J Inequal Appl 2013, 342 (2013). https://doi.org/10.1186/1029-242X-2013-342
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DOI: https://doi.org/10.1186/1029-242X-2013-342