Weighted composition followed and proceeded by differentiation operators fromZygmund spaces to Bloch-type spaces
© Long et al.; licensee Springer. 2014
Received: 12 January 2014
Accepted: 8 April 2014
Published: 2 May 2014
The boundedness and compactness of the weighted composition followed andproceeded by differentiation operators from Zygmund spaces to Bloch-type spacesand little Bloch-type spaces are characterized.
MSC: 47B38, 30H30, 32C15.
KeywordsZygmund spaces Bloch-type spaces weighted composition followed and proceeded by differentiation operators boundedness compactness
This space is called the little α-Bloch space (see ).
Then μ is called a normal function (see ).
It is well known that is a Banach space with the norm (see ).
This space is called the little Bloch-type space. When , the induced space becomes the α-Bloch space.
They are called weighted composition followed and proceeded by differentiationoperators, respectively, where and D are composition and differentiationoperators respectively. Associated with φ is the composition operator and weighted composition operator for and . It is interesting to provide a function theoreticcharacterization for φ inducing a bounded or compact compositionoperator, weighted composition operator and related ones on various spaces (see,e.g., [10–19]). For example, it is well known that is bounded on the classical Hardy, Bloch and Bergmanspaces. Operators and as well as some other products of linear operatorswere studied, for example, in [20–29] (see also the references therein). There has been some considerablerecent interest in investigation various type of operators from or to Zygmund typespaces (see, [7, 11, 23, 30–37]).
In this paper, we investigate the operators and from Zygmund spaces to Bloch-type spaces and littleBloch-type spaces. Some sufficient and necessary conditions for the boundedness andcompactness of these operators are given.
Throughout this paper, constants are denoted by C, they are positive and maydiffer from one occurrence to the other. The notation means that there is a positive constant Csuch that .
2 Main results and proofs
In this section, we state and prove our main results. In order to formulate our mainresults, we quote several lemmas which will be used in the proofs of the mainresults in this paper. The following lemma can be proved in a standard way (see,e.g., Proposition 3.11 in ). Hence we omit the details.
Lemma 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. Thenis compact if and only ifis bounded and for any bounded sequencein (or) which converges to zero uniformly on compactsubsets of Δ as, and (or) as.
- (ii)⇒ (iii). Assume that is bounded, i.e., there exists a constant C such that
- (iii)⇒ (i). Assume that (2.1) and (2.2) hold. Then, for every , from (1.4), we have(2.15)
Taking the supremum in (2.15) for , and employing (2.1) and (2.2), we deduce that is bounded. The proof of Theorem 2.3 iscompleted. □
- (iii)is bounded,(2.16)
- (ii)⇒ (iii). Assume that is compact. Then it is clear that is bounded. By Theorem 2.3 we know that is bounded. Let be a sequence in Δ such that as and , (if such a sequence does not exist then (2.16) and (2.17) are vacuously satisfied). Set(2.18)
if one of these two limits exists.
- (iii)⇒ (i). Suppose that is bounded and conditions (2.16) and (2.17) hold. From Theorem 2.3, it follows that(2.22)
By combining this with Lemma 2.1 the result easily follows. The proof ofTheorem 2.4 is completed. □
Proof of Theorem 2.5 Assume that is bounded. Then, it is clear that is bounded. Taking the test functions and respectively, we obtain (2.25).
as . Since the operator is bounded, we have , which implies the boundedness of. □
- (ii)⇒ (iii). Assume that is compact. Then is bounded. From the proof of Theorem 2.5, we have(2.29)
from which the result follows in this case.
- (iii)⇒ (i). Let . Then we have
From Lemma 2.2 it follows that the operator is compact. □
Similarly to the proofs of Theorems 2.3-2.6, we can get the following results; weomit the proof.
- (iii)is bounded,
The authors would like to thank the referees for their valuable suggestions whichgreatly improve the present article. The work was supported by the UnitedTechnology Foundation of Science and Technology Department of Guizhou Provinceand Guizhou Normal University (Grant No. LKS12), and National NaturalScience Foundation of China (Grant No. 11171080, Grant No. 11171277).
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