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Weighted composition followed and proceeded by differentiation operators from spaces to Bloch-type spaces
Journal of Inequalities and Applications volume 2012, Article number: 160 (2012)
Abstract
In this paper, we investigate boundedness and compactness of the weighted composition followed and proceeded by differentiation operators from spaces to Bloch-type spaces and little Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are obtained.
MSC:47B38, 30D45.
1 Introduction
Let Δ be an open unit disc in the complex plane, and let be the class of all analytic functions on Δ. The α-Bloch space () is, by definition, the set of all function f in such that
Under the above norm, is a Banach space. When , is the well-known Bloch space. Let denote the subspace of , for f
This space is called a little α-Bloch space.
Assume that μ is a positive continuous function on , having the property that there exist positive numbers s and t, , and , such that
Then μ is called a normal function (see [9]).
Denote (see, e.g., [2, 4, 10])
It is known that is a Banach space with the norm (see [4]).
Let denote the subspace of , i.e.,
This space is called a little Bloch-type space. When , the induced space becomes the α-Bloch space .
Throughout this paper, we assume that K is a right continuous and nonnegative nondecreasing function. For , , we say that a function belongs to the space (see, [11]), if
where dA denotes the normalized Lebesgue area measure on Δ, is the Green function with logarithmic singularity at a, that is, , where for . When , , the space equals to , which is introduced by Zhao in [13]. Moreover (see [13]), we have that and for , and for . When , is a Banach space with the norm
From [11], we know that , if and only if
Moreover, (see [11], Theorem 2.1]).
Throughout the paper, we assume that
otherwise consists only of constant functions (see [11]).
Let φ be a nonconstant analytic self-map of Δ, and let ϕ be an analytic function in Δ. We define the linear operators
They are called weighted composition followed and proceeded by differentiation operators respectively, where and D are composition and differentiation operators respectively. The boundedness and compactness of on the Hardy spaces were investigated by Hibschweiler and Portnoy in [3] and by Ohno in [8]. In [6], Li and Stević studied the boundedness and compactness of the operator on the α-Bloch spaces. In [7], Li and Stević studied the boundedness and compactness of the composition and differentiation operators between and α-Bloch spaces. In [12], Yang studied the boundedness and compactness of the operator (or ) from to the Bloch-type spaces.
In this paper, we investigate the operators and from spaces to Bloch-type spaces and little Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given. Our results also generalize some known results in [12].
Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant C such that .
2 Statement of the main results
In this paper, we shall prove the following results.
Theorem 2.1 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, , , and K is a nonnegative nondecreasing function on such that
where denote the characteristic function of the set A. Then is bounded if and only if
Theorem 2.2 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, , , and K is a nonnegative nondecreasing function on such that (2.1) hold. Then is compact if and only if is bounded, and
Theorem 2.3 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, , , and K is a nonnegative nondecreasing function on such that (2.1) hold. Then is compact if and only if
From the above three theorems, we get the following
Corollary 2.4 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Then the following statements hold.
-
(i)
is bounded if and only if
-
(ii)
is compact if and only if is bounded, and
-
(iii)
is compact if and only if
Theorem 2.5 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, , , and K is a nonnegative nondecreasing function on such that (2.1) hold. Then the following statements hold.
-
(i)
is bounded if and only if
-
(ii)
is compact if and only if is bounded, and
-
(iii)
is compact if and only if
From Theorem 2.5, we get the following
Corollary 2.6 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Then the following statements hold.
-
(i)
is bounded if and only if
-
(ii)
is compact if and only if is bounded, and
-
(iii)
is compact if and only if
3 Proofs of the main results
In this section, we will prove our main results. For this purpose, we need some auxiliary results.
Lemma 3.1 Let φ be an analytic self-map of Δ, ϕ be an analytic function in Δ. Suppose , . Then is compact if and only if is bounded and for any bounded sequence in which converges to zero uniformly on compact subsets of Δ as , and (or ) as .
Lemma 3.1 can be proved by standard way (see [1], Proposition 3.11]).
Lemma 3.2 A closed set of is compact if and only if it is bounded and satisfies
Proof First of all, we suppose that is compact and let . By the definition of , we can choose an -net which center at in respectively, and a positive number r () such that , for and . If , for some , so we have
for . This establishes (3.1). □
On the other hand, if is a closed bounded set which satisfies (3.1) and is a sequence in , then by the Montel’s theorem, there is a subsequence which converges uniformly on compact subsets of Δ to some analytic function f, and also converges uniformly to on compact subsets of Δ. According to (3.1), for every , there is an r, , such that for all , , if . It follows that , if . Since converges uniformly to f and converges uniformly to on , it follows that , i.e., , so that is compact.
Lemma 3.3 ([14])
Let and . Then we have
Proof of Theorem 2.1 First, suppose that the conditions in (2.2) hold. Then for any and , by use of the fact and Lemma 3.3, we have
Taking the supremum in (3.2) for , and employing (2.2), we deduce that
is bounded.
Conversely, suppose that is bounded. Then there exists a constant C such that for all . Taking the functions , and , which belong to , we get
and
From (3.3), (3.4), and the boundedness of the function , it follows that
For , let
by direct calculation, we get
From [5], we know that , for each . Moreover, there is a positive constant C such that . Hence, we have
for . Therefore, we obtain
Next, for , let
Then from [5], we see that and . Since
we have
Thus
Inequality (3.5) gives
Therefore, the first inequality in (2.2) follows from (3.9) and (3.10). From (3.7) and (3.8), we obtain
Inequalities (3.3) and (3.11) imply
and
Inequality (3.12) together with (3.13) implies the second inequality of (2.2). The proof of Theorem 2.1 is completed. □
Proof of Theorem 2.2 First, suppose that is bounded and (2.3) hold. Let be a sequence in such that , and converges to 0 uniformly on compact subsets of Δ as . By the assumption, for any , there exists a such that
and
hold for . Since is bounded, it follows from the proof of Theorem 2.1 that
Let . Then we have
From the fact that as on compact subsets of Δ, and Cauchy’s estimate, we conclude that and as on compact subsets of Δ. Letting in (3.14) and using the fact that ε is an arbitrary positive number, we obtain . Applying Lemma 3.1, the result follows.
Conversely, suppose that is compact. Then it is clear that is bounded. Let be a sequence in Δ such that as . For , let
Then and converges to 0 uniformly on compact subsets of Δ as . Since is compact, by Lemma 3.1, we have . On the other hand, from (3.6) we have
which implies that
if one of these two limits exists.
Next, for , set
Then is a sequence in . Notice that ,
And converges to 0 uniformly on compact subsets of Δ as . Since is compact, we have . On the other hand, since
we have
From (3.15) and (3.16), we get
The proof of Theorem 2.2 is completed. □
Proof of Theorem 2.3 First, let . By the proof of Theorem 2.1, we have
Taking the supremum in (3.18) over all such that , we can get
By Lemma 3.2, we see that the operator is compact.
Conversely, suppose that is compact. By taking and using the boundedness of , we get
From this, by taking the test function and using the boundedness of , it follows that
In the following, we distinguish two cases:
First, we assume that . From (3.19) and (3.20), we obtain
and
So the result follows in this case.
Secondly, we assume that . Let be a sequence such that . From the compactness of , we see that is compact. According to Theorem 2.2, we get
and
For any , from (3.19) and (3.22), there exists such that
for , and there exists such that
for . Therefore, when , and , we obtain
On the other hand, if , and , we have
From (3.23) and (3.24), we get the second equality of (2.4). Similarly to the above arguments, by (3.20) and (3.21), we can get the first equality of (2.4). The proof of Theorem 2.3 is completed. □
Similarly to the proofs of Theorems 2.1-2.3, we can get the proofs of Corollary 2.4, Theorem 2.5 and Corollary 2.6. We omit the proofs.
References
Cowen CC, Maccluer BD Studies in Advanced Mathematics. In Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton; 1995.
Fu X, Zhu X: Weighted composition operators on some weighted spaces in the unit ball. Abstr. Appl. Anal. 2008., 2008: Article ID 605807
Hibschweiler RA, Portnoy N: Composition followed by differentiation between Bergman and Hardy spaces. Rocky Mt. J. Math. 2005, 35(3):843–855. 10.1216/rmjm/1181069709
Hu ZJ, Wang SS: Composition operators on Bloch-type spaces. Proc. R. Soc. Edinb. A 2005, 135(6):1229–1239. 10.1017/S0308210500004340
Kotilainen M:On composition operators in type spaces. J. Funct. Spaces Appl. 2007, 5(2):103–122. 10.1155/2007/956392
Li SX, Stevič S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 2007, 9(2):195–205.
Li SX, Stevič S: Composition followed by differentiation betweenand α -Bloch spaces. Houst. J. Math. 2009, 35(1):327–340.
Ohno S: Products of composition and differentiation between Hardy spaces. Bull. Aust. Math. Soc. 2006, 73(2):235–243. 10.1017/S0004972700038818
Shields AL, Williams DL: Bonded projections, duality, and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 1971, 162: 287–302.
Stevič S:Norm of weighted composition operators from Bloch space to on the unit ball. Ars Comb. 2008, 88: 125–127.
Wulan HS, Zhou JZ: type spaces of analytic functions. J. Funct. Spaces Appl. 2006, 4(1):73–84. 10.1155/2006/910813
Yang WF:Products of composition and differentiation operators from to Bloch-type spaces. Abstr. Appl. Anal. 2009., 2009: Article ID 741920
Zhao RH: On a general family of function spaces. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 1996, 105: 1–56.
Zhu KH: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 1993, 23(3):1143–1177. 10.1216/rmjm/1181072549
Acknowledgements
The work is supported by the National Natural Science Foundation of China (Grant No. 11171080), and Foundation of Science and Technology Department of Guizhou Province (Grant No. [2010] 07).
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Long, J., Wu, P. Weighted composition followed and proceeded by differentiation operators from spaces to Bloch-type spaces. J Inequal Appl 2012, 160 (2012). https://doi.org/10.1186/1029-242X-2012-160
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DOI: https://doi.org/10.1186/1029-242X-2012-160