On Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space

Let D be the open unit disk in the complex plane C, φ an analytic self-map of D and H(D) the space of all analytic functions on D. In order to unify the products of composition, multiplication, and differentiation operators, Stević and Sharma introduced the following so-called Stević-Sharma operator: Tψ1,ψ2,φ f (z) =ψ1(z)f (φ(z)) +ψ2(z)f ′(φ(z)), f ∈ H(D), where ψ1,ψ2 ∈ H(D). Here we characterize the boundedness and compactness of the operator Tψ1,ψ2,φ from the Zygmund space to the Bloch-Orlicz space.


Introduction
If ψ ≡ , it becomes the composition operator, usually denoted by C ϕ . If ϕ(z) = z, it becomes the multiplication operator, usually denoted by M ψ . Hence, since W ϕ,ψ = M ψ C ϕ , it is a product-type operator. A standard problem is to provide function theoretic characterizations when ϕ and ψ induce a bounded or compact weighted composition operator (see, e.g., [-] and the references therein).
A systematic study of other product-type operators started by Stević et al. since the publication of papers [] and []. Before that there were a few papers in the topic, e.g., [].

The differentiation operator on H(D) is defined by
The next two product-type operators DC ϕ and C ϕ D, attracted some attention first (see, e.g., [-] and the references therein). The publication of [] attracted some attention in product-type operators involving integral-type ones (see, e.g., [-] and the references therein). Since that time there has been a great interest in various product-type operators on spaces of holomorphic functions. For example, the six product-type operators from Bergman spaces to Bloch type spaces were studied by Sharma in []. The next product-type operators W ϕ,ψ D and DW ϕ,ψ , which were considered in [] and [], are included in () as the first and sixth operators, respectively. For some other product-type operators, see, e.g., [, -] and the references therein.
In order to treat operators in () in a unified manner, Stević and Sharma introduced the following so-called Stević-Sharma operator: For example, in [] and [] the operator was studied on the weighted Bergman space. By using Stević-Sharma operator all six possible products of composition, multiplication, and differentiation operators can be obtained. More specifically we have etc., where ψ  , ψ  ∈ H(D). In this paper we characterize the boundedness and compactness of the Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space. As the applications of our main results, readers can obtain some characterizations for the boundedness and compactness for all six product-type operators in (), as well as above mentioned differences operators from the Zygmund space to the Bloch-Orlicz space. Now we present the needed spaces and some facts. For α > , the weighted Zygmund space Z α consists of all f ∈ H(D) such that It is a Banach space with the norm When α = , this space is the Zygmund space and is denoted by For some results on Zygmund-type spaces and some concrete operators on them, see, for example, [, , ] and the references therein.
Recently, the Bloch-Orlicz space was introduced in [] by Ramos Fernández. More precisely, let be a strictly increasing convex function such that () = . From these conditions it follows that lim t→+∞ (t) = +∞. The Bloch-Orlicz space associated with the function , denoted by B , is the class of all f ∈ H(D) such that Moreover, B is a Banach space with the norm Thus, for f ∈ B it follows that This equivalent norm is useful to us for the study of operator T ψ  ,ψ  ,ϕ from the Zygmund space to the Bloch-Orlicz space. It is obvious to see that if (t) = t p with p > , then the space B coincides with the weighted Bloch space B α , where α = /p. Also, if (t) = t log( + t), then B coincides with the Log-Bloch space (see []). For the generalization of the Log-Bloch spaces, see, for example, [, ].
Let X and Y be Banach spaces. It is said that a linear operator L : X → Y is bounded if there exists a positive constant K such that The operator L : X → Y is said to be compact if it maps bounded sets into relatively compact sets. It is well known that the norm of operator L : Z → B is defined by and written by L .
Throughout this paper, a positive constant C may differ from one occurrence to the other. The notation a b means that there exists a positive constant C such that a ≤ Cb. When a b and b a, we write a b.

Main results and proofs
In order to characterize the compactness, we need the following result, which is proved in a standard way []. So, the proof is omitted.
Lemma  Let ϕ be an analytic self-map of D and ψ  , ψ  ∈ H(D). Then the bounded operator T ψ  ,ψ  ,ϕ : Z → B is compact if and only if for every bounded sequence {f j } j∈N in Z such that f j →  uniformly on every compact subset of D as j → ∞, it follows that We state the following useful result whose first estimate was essentially proved in [], while the second essentially follows from the point evaluation estimate for the Bloch functions (see, e.g., []). See also [].
Lemma  For each f ∈ Z and z ∈ D, it follows that For w ∈ D and / < |w| < , we define the function By using this function, the test functions in the Zygmund space can be obtained as follows: From [] we have the next result on the functions g w and h w .
Lemma  Let w ∈ D and / < |w| < . Then Now we characterize the boundedness of the operator T ψ  ,ψ  ,ϕ : Z → B .
Theorem  Let ϕ be an analytic self-map of D and ψ  , ψ  ∈ H(D). Then the following statements are equivalent.
(ii) The functions ψ  , ψ  , and ϕ satisfy the following conditions: Moreover, if the operator T ψ  ,ψ  ,ϕ : Z → B is nonzero and bounded, then Proof (i) ⇒ (ii). Suppose that T ψ  ,ψ  ,ϕ : Z → B is bounded. For a fixed w ∈ D and |ϕ(w)| > /, let f (z) = h ϕ(w) (z)c  + c  , where Then by Lemma  By using the boundedness of T ψ  ,ψ  ,ϕ : Z → B to the function f , we have from which we get Then by the boundedness of T ψ  ,ψ  ,ϕ : Z → B , we obtain Considering h  (z) = z ∈ Z, by the boundedness of T ψ  ,ψ  ,ϕ : Z → B we have From (), (), the boundedness of ϕ, and the triangle inequality, we obtain From (), (), (), the boundedness of ϕ  , and the triangle inequality, we get Then from () we have From () and () we finally have M  < ∞.
Now we prove that M  < ∞. For a fixed w ∈ D and |ϕ(w)| > /, let g(z) = g ϕ(w) (z)c  . Then By using the boundedness of T ψ  ,ψ  ,ϕ : Z → B , we have From (), (), and the triangle inequality, it follows that and then Hence, from () and () we have M  < ∞.
(ii) ⇒ (i). By Lemma , for all f ∈ Z we have It is clear that Hence from () and () it follows that T ψ  ,ψ  ,ϕ : Z → B is bounded. Suppose that the operator T ψ  ,ψ  ,ϕ : Z → B is nonzero and bounded. Then from the proof of (i) ⇒ (ii) it is not hard to see that Since the operator T ψ  ,ψ  ,ϕ : Z → B is nonzero, we have T ψ  ,ψ  ,ϕ > . From this we can find a positive constant C such that  ≤ C T ψ  ,ψ  ,ϕ , which means that Then combing () and () gives It is clear from () and () that Hence from () and () the asymptotic expression of T ψ  ,ψ  ,ϕ follows. The proof is finished.
Next we characterize the compactness of operator T ψ  ,ψ  ,ϕ : Z → B .
Theorem  Let ϕ be an analytic self-map of D and ψ  , ψ  ∈ H(D). Then the following statements are equivalent.
The functions ψ  , ψ  , and ϕ satisfy the following conditions: Proof (i) ⇒ (ii). Suppose that (i) holds. Then it is clear that the operator T ψ  ,ψ  ,ϕ : Z → B is bounded. In the proof of Theorem , we have shown that M  < ∞, L  < ∞ and L  < ∞. Consider a sequence {ϕ(z i )} i∈N in D such that |ϕ(z i )| → as i → ∞. If such a sequence does not exist, then the last two conditions (ii) obviously hold. We may suppose, without loss of generality, that |ϕ(z i )| > / for all i ∈ N. Using this sequence, we define the function sequence Then from a calculation we see that sup i∈N f i Z ≤ C and f i →  uniformly on every compact subset of D as i → ∞. So by Lemma  Moreover, we have Hence we get From this, Lemmas  and , and since M  is finite, we obtain On the other hand, take the sequence g i (z) = g ϕ(z i ) (z)c i , i ∈ N, where c i = g ϕ(z i ) (ϕ(z i )). Then sup i∈N g i Z ≤ C, g i ϕ(z i ) = , g i ϕ(z i ) = log e  -|ϕ(z i )|  , g i (z i ) = ϕ(z i )  -|ϕ(z i )|  .