Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces

and Applied Analysis 3


Introduction
Let D denote the open unit disk of the complex plane C and (D) the space of all analytic functions in D.
The logarithmic Bloch space is defined as follows: ( The space B log is a Banach space under the norm ‖‖ It is obvious that there are unbounded B log functions.For example, consider the function () = log log(/(1 − )).
There are also bounded functions that do not belong to B log .In fact, the interpolating Blaschke products do not belong to B log .It is easily proved that, for 0 <  < 1, B  ⫋ B log ⫋ B. B log first appeared in the study of boundedness of the Hankel operators on the Bergman space.Attele in [1] proved that, for  ∈  2  (D), the Hankel operator   :  1  (D) →  1 (D) is bounded if and only if ‖‖ B log < ∞, thus giving one reason, and not the only reason, why log-Bloch-type spaces are of interest.Ye in [2] proved that B log,0 is a closed subspace of B log .Galanopoulos in [3] characterized the boundedness and compactness of the composition operator   : B log →   log and the boundedness and compactness of the weighted composition operator   : B log → B log .Ye in [4] characterized the boundedness and compactness of the weighted composition operator   between the logarithmic Bloch space B log and the -Bloch spaces B  on the unit disk and the boundedness and compactness of the weighted composition operator   between the little logarithmic Bloch space B 0 log and the little -Bloch spaces B  0 on the unit disk.Li in [5] characterized the boundedness and compactness of the weighted composition operator   from Bergman spaces    into the logarithmic Bloch space B log on the unit disk.Ye in [6] characterized the boundedness and compactness of the weighted composition operator   from the general function space (, , ) into the logarithmic Bloch space B log on the unit disk.Colonna and Li in [7] studied the boundedness and compactness of the weighted composition operators from Hardy space into the logarithmic Bloch space and the little logarithmic Bloch space.Petrov in [8] obtains sharp reverse estimates for the D  ,  () =  ()  () ( ()) ,  ∈ (D) , where  ∈ (D) and  is a nonconstant holomorphic selfmap of D.
If  = 0, then D  , becomes the weighted composition operator   , defined by    () =  ()  ( ()) ,  ∈ D, which, for () ≡ 1, is reduced to the composition operator   for some recent articles on weighted composition operators on some  ∞ -type spaces, for example, [14][15][16] and references therein.If  = 1,() =   (), then D  , = D  , which was studied in [17][18][19][20][21].When  = 1, () ≡ 1, then D  , =   D, which was studied in [17,19].If  = 1, () = , then D  , =   D, that is, the product of differentiation operator and multiplication operator   defined by    = .Zhu in [13] completely characterized the boundedness and compactness of linear operators which are obtained by taking products of differentiation, composition, and multiplication operators from Bergman type spaces to Bers spaces.Stević in [12] studied the boundedness and compactness of the weighted differentiation composition operator D  , from mixed-norm spaces to weighted-type spaces or the little weighted-type space (see also [22][23][24]).Zhu in [25] studied the boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces.Yang in [21] studied the boundedness and compactness of the operator   D and D  from   (, ) to B  and B ,0 spaces.Liu and Yu in [18] studied the boundedness and compactness of the operator D  between  ∞ and Zygmund spaces.Ye and Zhou in [26] studied the boundedness and compactness of the weighted composition operators from Hardy to Zygmund type spaces.Stević in [27] studied the boundedness and compactness of the generalized composition operator from mixed-norm space to the Blochtype space, the little Bloch-type space, the Zygmund space, and the little Zygmund space.For other recently introduced products of operators on spaces of holomorphic functions see [13,16].Motivated by the results [12,18,23,24,27], we consider the boundedness and compactness of the operators D  , from the logarithmic Bloch spaces to the Zygmund-type spaces and the little Zygmund-type spaces.For the proof, we need different test functions and some complex calculations kills.
Throughout this paper, we will use the letter  to denote a positive constant that can change its value at each occurrence.

Auxiliary Results
Here we prove and quote some auxiliary results which will be used in the proofs of the main results in this paper.

Lemma 1.
Let  be a positive integer.Suppose  ∈ B log ; there exists a constant  such that       () Proof.We use induction on .Using the definition of the logarithmic Bloch spaces we have the case holds for  = 1.Assume the case  =  holds; since By the Cauchy integral formula we obtain Note that we have for every  ∈ D. Hence the case  =  + 1 holds.The desired result follows.The proof of this lemma is complete.
Lemma 2 (see [4,28]).Let The following criterion for the compactness is a useful tool and it follows from standard arguments (e.g., [ The proof is similar to that of Lemma 1 in [31]; hence we omit it.

Boundedness and
On the other hand, we have Applying conditions ( 20) and ( 21), we deduce that the operator for all  ∈ B log,0 .For () =   /! ∈ B log,0 , we have that Taking () =  +1 /( + 1)! ∈ B log,0 ; we have that By ( 23), (24), and the boundedness of the function (), we get In the same way, taking () =  +2 /(+2)!∈ B log,0 , we have that By ( 23), ( 25), (26), and the boundedness of the function (), we have that For a fixed  ∈ D, set We get that By Lemma 2 we have sup Thus combining (47) with (46) we get the condition (19), finishing the proof of the theorem.Theorem 6.Let  ∈ (D), and let n be a nonnegative integer,  a holomorphic self-map of D, and  a weight.Then the following statements are equivalent: (1) D  , : B log → Z  is compact; (2) D  , : B log,0 → Z  is compact; (3) D  , : B log → Z  is bounded and lim Proof.
29, Proposition 3.11] or [30, Lemma 2.10]).Let  ∈ (D), and let n be a nonnegative integer,  a holomorphic self-map of D, and  a weight.Then D  , : B log (B log,0 ) → Z  is compact if and only if D  , : B log (B log,0 ) → Z  is bounded and, for any bounded sequence {  } in B log (B log,0 ) which converges to zero uniformly on compact subsets of D as  → ∞, we have ‖D  ,   ‖ Z  → 0 as  → ∞.

Compactness of D
, from B log (B log,0 ) to Z  (Z ,0 ) SpacesIn this section, we study the boundedness and compactness of D  , : B log (B log,0 ) → Z  (Z ,0 ).