Product-Type Operators on the Space of Fractional Cauchy Transforms

Let S � S(D) be the class of all holomorphic self-maps of the unit diskD of the complex planeC, T be the boundary of D, N0 be the set of all nonnegative integers, and N be the set of all positive integers. Denote by H(D) the space of all holomorphic functions on D. We first recall the spaces we work on. Let α> 0 be a real number and M be the space of all complex Borel measures on T endowed with the total variation norm. ,e familyFα of fractional Cauchy transforms is the collection of holomorphic functions f in D for which


Introduction
Let S � S(D) be the class of all holomorphic self-maps of the unit disk D of the complex plane C, T be the boundary of D, N 0 be the set of all nonnegative integers, and N be the set of all positive integers. Denote by H(D) the space of all holomorphic functions on D.
We first recall the spaces we work on. Let α > 0 be a real number and M be the space of all complex Borel measures on T endowed with the total variation norm. e family F α of fractional Cauchy transforms is the collection of holomorphic functions f in D for which for some μ ∈ M. e space F α is a Banach space, with respect to the norm given by where the infimum extends over all measures μ. e fractional Cauchy transforms space F α plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory which has phenomenal development in connection with the Calderon-Zygmundtype singular integral theory. e space F α may be identified with M/H 1 0 , the quotient of the Banach space M by H 1 0 , and the subspace of L 1 consisting of functions with mean value 0 whose conjugate belongs to the Hardy space H 1 . Hence, F α is isometrically isomorphic to M/H 1 0 . Furthermore, M admits a decomposition M � L 1 ⊕M s , where M s is the space of Borel measures, which are singular with respect to Lebesgue measure, and H 1 0 ⊂ L 1 . According to the Lebesgue decomposition theorem, each μ ∈ M can be written as μ � μ a ⊕μ s , where μ a is absolutely continuous with respect to the Lebesgue measure and μ s is singular with respect to the Lebesgue measure (μ a ⊥μ s ). Consequently, F α is isometrically isomorphic to L 1 /H 1 0 ⊕M s . Hence, F α can be written as F α � F αa ⊕F αs , where F αa is isomorphic to L 1 /H 1 0 , the closed subspace of M of absolutely continuous measures, and F αs is isomorphic to M s , the subspace of M of singular measures. For further results about the space of fractional Cauchy transforms, we refer to [1][2][3][4][5][6][7][8][9][10][11][12] and references therein.
Let ϑ be a weight, that is, ϑ is a positive continuous function on D. A positive continuous function ] on the interval [0, 1) is said to be normal if there are δ ∈ [0, 1) and τ and η, 0 < τ < η such that (3) In this paper, we assume the normal weighted function With the norm For φ ∈ S and u ∈ H(D), the weighted composition operator, which plays an important role in the isometry theory of Banach spaces, induced by u and φ is given by We can regard this operator as a generalization for a multiplication operator M u induced by u and a compo- An extensive study concerning the theory of (weighted) composition operators has been established during the past four decades on various settings. We refer to standard references [13][14][15] for various aspects about the theory of composition operators acting on holomorphic function spaces, especially the problems of relating operator-theoretic properties of C φ to function theoretic properties of φ. e differentiation operator, on H(D), is defined by Note that D is typically unbounded on many familiar spaces of holomorphic functions. e differential operator plays an important role in various fields such as dynamical system theory and operator theory. e products of any two of C φ , M u , and D can be obtained in six ways, i.e., M u C φ , C φ M u , M u D, DM u , C φ D, and DC φ . Similarly, the products of all of C φ , M u , and D can also be obtained in six ways, i.e., In order to treat above product-type operators in a unified manner, Stević et al. [16], for the first time, introduced the so-called Stević-Sharma operator: for φ ∈ S, u 1 , u 2 ∈ H(D).
is operator is related to the various products of multiplication, composition, and differentiation operators. It is clear that all products of multiplication, composition, and differentiation operator in the following six ways can be obtained from the operator T u 1 ,u 2 ,φ by choosing different u 1 and u 2 . More specially, we have Recently, product-type operators on some spaces of holomorphic functions on the unit disk have become a subject of increasing interest (see [17][18][19] and references therein). Hibschweiler et al. [20] first characterized the boundedness and compactness of DC φ between Bergman spaces and Hardy spaces. Liu and Yu [21] investigated the boundedness and compactness of the operator DC φ from H ∞ and Bloch spaces to Zygmund spaces. Ohno [22] considered the boundedness and compactness of C φ D on Hardy space H 2 . Zhu [23] studied the boundedness and compactness of linear operators which are obtained by taking products of multiplication, composition, and differentiation operators from Bergman-type spaces to Berstype spaces. Quite recently, Zhang and Liu [24] presented the boundedness and compactness of the operator T u 1 ,u 2 ,φ from Hardy spaces to Zygmund-type spaces. Liu and Yu [25] gave the complete characterizations for the boundedness and compactness of the operator T u 1 ,u 2 ,φ from Hardy spaces to the logarithmic Bloch spaces. Liu et al. [26] investigated the compactness of the operator T u 1 ,u 2 ,φ on logarithmic Bloch spaces. Yu and Liu [27] characterized the boundedness and compactness of the operator T u 1 ,u 2 ,φ from H ∞ space to the logarithmic Bloch spaces. Jiang [28] considered the boundedness and compactness of the operator T u 1 ,u 2 ,φ from the Zygmund spaces to the Bloch-Orlicz spaces. Li and Guo [29] studied the boundedness and compactness of the operator T u 1 ,u 2 ,φ from Zygmund-type spaces to Bloch-Orlicz spaces.
Inspired by the above results, the purpose of the paper is devoted to the boundedness and compactness of the operator T u 1 ,u 2 ,φ from the fractional Cauchy transforms' spaces to the Zygmund-type spaces over the unit disk in terms of the function theoretic characterization of Julia-Carathéodory type. As the applications of our main results, readers easily can obtain the boundedness and compactness characterizations of all six product-type operators: from the space of fractional Cauchy transforms to the Zygmund-type spaces.

Preliminaries
In this section, we recall some basic facts and preliminary results to be used in the sequel.
Suppose X and Y are two Banach spaces with norms ‖.‖ X and ‖.‖ Y , respectively. Recall that a linear operator e bounded operator T: X ⟶ Y is said to be compact if the image of every bounded set of X is relatively compact in Y. Equivalently, T: X ⟶ Y is compact if and only if the image of every bounded sequence in X has a subsequence that converges in Y.
e following lemma gives a convenient compactness criterion for the Stević-Sharma operator A proof can be found in Proposition 3.11 of [13] for a single composition operator over the unit disk, and it can be easily modified for the operator T u 1 ,u 2 ,φ on F α . e following lemma is taken from [30] which is vital to construct the test functions on the space of fractional Cauchy transforms.
Based on Lemma 2, we can obtain the following lemma, see Lemma 2 of [31], for the detailed proof.
Furthermore, we need the following lemma to prove our main results.
Proof. For f ∈ F α , there is a μ ∈ M such that (1) holds. en, we have us, we have Taking infimum over all measures, μ ∈ M, for which (1) holds; the proof is complete.

Main Results and Proofs
In this section, we devote to investigating the boundedness and compactness of the operator T u 1 ,u 2 ,φ acting from the spaces of fractional Cauchy transforms to the Zygmund-type spaces in terms of the function theoretic characterization of Julia-Carathéodory type.
Conversely, assume that T u 1 ,u 2 ,φ : F α ⟶ Z ] is bounded. en, there exists a constant C such that for all f ∈ F α . It is elementary to deduce that z n ∈ F α , for n ∈ N 0 . First, take the function f(z) � 1, we obtain en, put f(z) � z, and we apply (20) to have Next, taking f(z) � (z 2 /2), (20) and (21) yield that Furthermore, putting f(z) � (z 3 /6), we deduce that
Fix w ∈ D and d, e, h ∈ R. Put It follows from Lemma 3 that g w ∈ F α and sup w∈D ‖g w ‖ F α ≤ C. In addition,
en, there exists K 0 ∈ N, for k > K 0 , such that