# Product-Type Operators on the Space of Fractional Cauchy Transforms

Authors: Zeng Fan; Xin Guo
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The space of fractional Cauchy transforms plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory. In this paper, we study the boundedness and compactness of product-type operators from the space of fractional Cauchy transforms to the Zygmund-type space in terms of the function theoretic characterization of Julia–Carathéodory type.


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## Body

## 1. Introduction

LetS=SD be the class of all holomorphic self-maps of the unit disk D of the complex plane C, 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 HD the space of all holomorphic functions on D.We first recall the spaces we work on. Letα>0 be a real number and ℳ be the space of all complex Borel measures on T endowed with the total variation norm. The family ℱα of fractional Cauchy transforms is the collection of holomorphic functions f in D for which(1)fz=∫Tdμξ1−ξ¯zα,z∈D,for some μ∈ℳ. The space ℱα is a Banach space, with respect to the norm given by(2)fℱα=infμ,where the infimum extends over all measures μ.The fractional Cauchy transforms spaceℱα plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory which has phenomenal development in connection with the Calderon–Zygmund-type singular integral theory. The space ℱα may be identified with ℳ/H01¯, the quotient of the Banach space ℳ by H01¯, and the subspace of L1 consisting of functions with mean value 0 whose conjugate belongs to the Hardy space H1. Hence, ℱα is isometrically isomorphic to ℳ/H01¯. Furthermore, ℳ admits a decomposition ℳ=L1⊕ℳs, where ℳs is the space of Borel measures, which are singular with respect to Lebesgue measure, and H01¯⊂L1. According to the Lebesgue decomposition theorem, each μ∈ℳ can be written as μ=μa⊕μs, where μa is absolutely continuous with respect to the Lebesgue measure and μs is singular with respect to the Lebesgue measure μa⊥μs. Consequently, ℱα is isometrically isomorphic to L1/H01¯⊕ℳs. Hence, ℱα can be written as ℱα=ℱαa⊕ℱαs, where ℱαa is isomorphic to L1/H01¯, the closed subspace of ℳ of absolutely continuous measures, and ℱαs is isomorphic to ℳs, the subspace of ℳ of singular measures. For further results about the space of fractional Cauchy transforms, we refer to [1–12] and references therein.Letϑ be a weight, that is, ϑ is a positive continuous function on D. A positive continuous function ν 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)νr1−rτis decreasing onδ,1andlimr⟶1νr1−rτ=0,νr1−rηis increasing onδ,1andlimr⟶1νr1−rη=∞.In this paper, we assume the normal weighted functionν:D⟶0,∞ is also radial, i.e., νz=νz,z∈D. Now, the Zygmund-type space Zν consists of all f∈HD such that(4)bνf=supz∈Dνzf″z<∞.With the normfZν=f0+f′0+bνf, the Zygmund-type space is a Banach space.Forφ∈S and u∈HD, the weighted composition operator, which plays an important role in the isometry theory of Banach spaces, induced by u and φ is given by(5)Wu,φf≔u⋅f∘φ,f∈HD.We can regard this operator as a generalization for a multiplication operatorMu induced by u and a composition operator Cφ induced by φ, where Muf=u⋅f and Cφf=f∘φ. 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 [13–15] for various aspects about the theory of composition operators acting on holomorphic function spaces, especially the problems of relating operator-theoretic properties of Cφ to function theoretic properties of φ. The differentiation operator, on HD, is defined by(6)Dfz=f′z,z∈D.Note thatD is typically unbounded on many familiar spaces of holomorphic functions. The differential operator plays an important role in various fields such as dynamical system theory and operator theory.The products of any two ofCφ, Mu, and D can be obtained in six ways, i.e., MuCφ, CφMu, MuD, DMu, CφD, and DCφ. Similarly, the products of all of Cφ, Mu, and D can also be obtained in six ways, i.e., MuCφD, CφMuD, MuDCφ, CφDMu, DMuCφ, and DCφMu. 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:(7)Tu1,u2,φfz=u1zfφz+u2zf′φz,f∈HD,for φ∈S, u1,u2∈HD. This 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 Tu1,u2,φ by choosing different u1 and u2. More specially, we have(8)MuDCφ=T0,u′,φ,CφMuD=T0,u∘φ,φ,MuCφD=T0,u,φ,DMuCφ=Tu′,uφ′,φ,CφDMu=Tu′∘φ,u∘φ,φ,DCφMu=Tu′∘φφ′,u∘φφ′,φ.Recently, product-type operators on some spaces of holomorphic functions on the unit disk have become a subject of increasing interest (see [17–19] and references therein). Hibschweiler et al. [20] first characterized the boundedness and compactness of DCφ between Bergman spaces and Hardy spaces. Liu and Yu [21] investigated the boundedness and compactness of the operator DCφ from H∞ and Bloch spaces to Zygmund spaces. Ohno [22] considered the boundedness and compactness of CφD on Hardy space H2. 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 Bers-type spaces. Quite recently, Zhang and Liu [24] presented the boundedness and compactness of the operator Tu1,u2,φ from Hardy spaces to Zygmund-type spaces. Liu and Yu [25] gave the complete characterizations for the boundedness and compactness of the operator Tu1,u2,φ from Hardy spaces to the logarithmic Bloch spaces. Liu et al. [26] investigated the compactness of the operator Tu1,u2,φ on logarithmic Bloch spaces. Yu and Liu [27] characterized the boundedness and compactness of the operator Tu1,u2,φ from H∞ space to the logarithmic Bloch spaces. Jiang [28] considered the boundedness and compactness of the operator Tu1,u2,φ from the Zygmund spaces to the Bloch–Orlicz spaces. Li and Guo [29] studied the boundedness and compactness of the operator Tu1,u2,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 operatorTu1,u2,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:(9)MuDCφ,CφMuD,MuCφD,DMuCφ,CφDMu,DCφMu,from the space of fractional Cauchy transforms to the Zygmund-type spaces.


## 2. Preliminaries

In this section, we recall some basic facts and preliminary results to be used in the sequel.SupposeX and Y are two Banach spaces with norms .X and .Y, respectively. Recall that a linear operator T from X to Y is bounded if there is a positive constant C such that TfY≤CfX, for all f in X. The 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.The following lemma gives a convenient compactness criterion for the Stević–Sharma operatorTu1,u2,φfz=u1zfφz+u2zf′φz acting from the space of fractional Cauchy transforms ℱα to the Zygmund-type spaces Zν.Lemma 1.
Supposeu1,u2∈HD,φ∈S. Then, the operator Tu1,u2,φ:ℱα⟶Zν is compact if and only if Tu1,u2,φ:ℱα⟶Zν is bounded and Tu1,u2,φfn⟶0 in Zν, for any bounded sequence fn in ℱα, such that fn⟶0 uniformly on compact subsets of D.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 Tu1,u2,φ on ℱα.The following lemma is taken from [30] which is vital to construct the test functions on the space of fractional Cauchy transforms.Lemma 2.
Letα,β>0,f∈HD(1)
Iff∈ℱα,z∈D, then there exists a C>0 such that fz≤Cfℱα/1−z2α(2)
Iff∈ℱα, then f′∈ℱα+1 and f′ℱα+1≤αfℱα(3)
Iff′∈ℱα+1, then f∈ℱα, and there exists a C>0 such that fℱα≤f0+Cf′ℱα+1(4)
Iff∈ℱα and g∈ℱβ, then fg∈ℱα+β and fgℱα+β≤fℱαgℱβBased on Lemma2, we can obtain the following lemma, see Lemma 2 of [31], for the detailed proof.Lemma 3.
Letα>0, s∈ℕ0, and z∈D. Put(10)τzsw=1−z2s1−z¯wα+s,w∈D.Then,τzs∈ℱα and supz∈Dτzsℱα<∞.Furthermore, we need the following lemma to prove our main results.Lemma 4.
Letα>0. Suppose that f∈ℱα and n∈ℕ0. Then, there is a positive constant C independent of f such that(11)fnz≤Cfℱα1−z2α+n.Proof.
Forf∈ℱα, there is a μ∈ℳ such that (1) holds. Then, we have(12)fnz=∏j=0n−1α+j∫Tξ¯ndμξ1−ξ¯zn+α.
Thus, we have(13)fnz≤C∫Tdμξ1−ξ¯zα+n≤C∫Tdμξ1−z2α+n=Cμ1−z2α+n.
Taking infimum over all measures,μ∈ℳ, for which (1) holds; the proof is complete.


## 3. Main Results and Proofs

In this section, we devote to investigating the boundedness and compactness of the operatorTu1,u2,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.Theorem 1.
Letα>0. Suppose u1,u2∈HD,φ∈S. Then, Tu1,u2,φ:ℱα⟶Zν is bounded if and only if the following conditions are satisfied:(14)supz∈Dνzu1″z1−φz2α<∞,(15)supz∈Dνz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1<∞,(16)supz∈Dνzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2<∞,(17)supz∈Dνzu2zφ′z21−φz2α+3<∞.Proof.
Suppose that (14)–(17) hold. Let f∈ℱα with fℱα≤1. Using Lemma 4, we have(18)Tu1,u2,φf0=u10fφ0+u20f′φ0≤Cu101−φ02α+u201−φ02α+1fℱα,Tu1,u2,φf′0=u1′0fφ0+u10φ′0+u2′0φ′0f′φ0+u20φ′0f″φ0≤Cu1′01−φ02α+u10φ′0+u2′01−φ02α+1+u20φ′01−φ02α+2fℱα,νzTu1,u2,φf″z=νzu1zfφz+u2zf′φz″≤νzu1″zfφz+νz2u1′zφ′z+u1zφ″z+u2″zf′φz+νzu1zφ′z2+2u2′zφ′z+u2zφ″zf″φz+νzu2zφ′z2f3φz≤Cνzu1″z1−φz2αfℱα+Cνz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1fℱα+Cνzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2fℱα+Cνzu2zφ′z21−φz2α+3fℱα.
Applying (14)–(17), it follows from the last above inequality that Tu1,u2,φ:ℱα⟶Zν is bounded.
Conversely, assume thatTu1,u2,φ:ℱα⟶Zν is bounded. Then, there exists a constant C such that(19)Tu1,u2,φfZν≤Cfℱα,for all f∈ℱα. It is elementary to deduce that zn∈ℱα, for n∈ℕ0. First, take the function fz=1, we obtain(20)supz∈Dνzu1″z<∞.
Then, putfz=z, and we apply (20) to have(21)supz∈Dνz2u1′zφ′z+u1zφ″z+u2″z<∞.
Next, takingfz=z2/2, (20) and (21) yield that(22)νzu1zφ′z2+2u2′zφ′z+u2zφ″z≤Tu1,u2,φfZν<∞.
Furthermore, puttingfz=z3/6, we deduce that(23)∞>Tu1,u2,φfZν≥νz|u2″zfφz+2u1′zφ′z+u1zφ″z+u1″zf′φz+u1zφ′z2+2u2′zφ′z+u2zφ″zf″φz+u2zφ′z2f3φz|=νz|u1″zφ3z6+2u1′zφ′z+u1zφ″z+u2″zφ2z2+u1zφ′z2+2u2′zφ′z+u2zφ″zφz+u2zφ′z2|.
Applying (20)–(22) gives that(24)supz∈Dνzu2zφ′z2<∞.
Fixw∈D and a,b,c∈ℝ. Consider the following test function:(25)fwz=a1−w21−w¯zα+1−1−w221−w¯zα+2+b1−w231−w¯zα+3+c1−w241−w¯zα+4.
Lemma3 gives that fw∈ℱα and supw∈Dfwℱα≤C. A straightforward calculation shows that(26)fw′z=aw¯α+11−w21−w¯zα+2−α+2w¯1−w221−w¯zα+3+α+3bw¯1−w231−w¯zα+4+α+4cw¯1−w241−w¯zα+5,fw″z=α+1α+2aw¯21−w21−w¯zα+3−α+2α+3w¯21−w221−w¯zα+4+α+3α+4bw¯21−w231−w¯zα+5+α+4α+5cw¯21−w241−w¯zα+6,fw3z=α+1α+2α+3aw¯31−w21−w¯zα+4−α+2α+3α+4w¯31−w221−w¯zα+5+α+3α+4α+5bw¯31−w231−w¯zα+6+α+4α+5α+6cw¯31−w241−w¯zα+7.
Puta=α+2/3α+3,b=α+2/α+3,andc=−α+2/3α+12 in (25) such that(27)fww=a+b+c−11−w2α,fw′w=fw″w=fw3w=0.
Thus, we have(28)C≥Tu1,u2,φfφwZν≥νwTu1,u2,φfφw″w=νwu1″wfφwφw=a+b+c−1νwu1″w1−φw2α.
Hence,(29)supz∈Dνzu1′′z1−φz2α<∞,namely, (14) holds.
Fixw∈D and d,e,h∈ℝ. Put(30)gwz=−1−w21−w¯zα+1+d1−w221−w¯zα+2+e1−w231−w¯zα+3+h1−w241−w¯zα+4.
It follows from Lemma3 that gw∈ℱα and supw∈Dgwℱα≤C. In addition,(31)gw′z=−α+1w¯1−w21−w¯zα+2+α+2dw¯1−w221−w¯zα+3+α+3ew¯1−w231−w¯zα+4+α+4hw¯1−w241−w¯zα+5,gw″z=−α+1α+2w¯21−w21−w¯zα+3+α+2α+3dw¯21−w221−w¯zα+4+α+3α+4ew¯21−w231−w¯zα+5+α+4α+5hw¯21−w241−w¯zα+6,gw3z=−α+1α+2α+3w¯31−w21−w¯zα+4+α+2α+3α+4dw¯31−w221−w¯zα+5+α+3α+4α+5ew¯31−w231−w¯zα+6+α+4α+5α+6hw¯31−w241−w¯zα+7.
Putd=3α+7/α+3,e=−α+23α+11/α+3α+4,andh=α+2/α+4 in (30) such that(32)gww=gw′′w=gw3w=0,gw′w=−α+1+dα+2+eα+3+hα+4w¯1−w2α+1.
Thus, we have(33)C≥Tu1,u2,φgφwZν≥νwTu1,u2,φgφw″w=νw2u1′zφ′w+u1wφ″w+u2″wgφw′φw=−α+1+dα+2+eα+3+hα+4⋅νw2u1′wφ′w+u1wφ″w+u2″wφw1−φw2α+1.
Hence,(34)supz∈Dνz2u1′zφ′z+u1zφ″z+u2″zφz1−φz2α+1<∞,which implies that(35)supφz>1/2νz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1≤supφz>1/22νz2u1′zφ′z+u1zφ″z+u2″zφz1−φz2α+1<∞.
Applying (21) yields that(36)supφz≤1/2νz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1≤Csupφz≤1/2νz2u1′zφ′z+u1zφ″z+u2″z<∞.
Combining (20) and (36), we can obtain(37)supz∈Dνz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1<∞,which means that (15) holds.
Fixw∈D and p,q,i∈ℝ. Consider the following test function:(38)hwz=p1−w21−w¯zα+1+i1−w221−w¯zα+2−1−|w|231−w¯zα+3+q1−w241−w¯zα+4.
Lemma3 yields that hw∈ℱα and supw∈Dhwℱα≤C. Furthermore, we obtain that(39)hw′z=α+1pw¯1−w21−w¯zα+2+α+2iw¯1−w221−w¯zα+3−α+3w¯1−w231−w¯zα+4+α+4qw¯1−w241−w¯zα+5,hw″z=α+1α+2pw¯21−w21−w¯zα+3+α+2α+3iw¯21−w221−w¯zα+4−α+3α+4w¯21−w231−w¯zα+5+α+4α+5qw¯21−w241−w¯zα+6,hw3z=α+1α+2α+3pw¯31−w21−w¯zα+4+α+2α+3α+4iw¯31−w221−w¯zα+5−α+3α+4α+5w¯31−w231−w¯zα+6+α+4α+5α+6qw¯31−w241−w¯zα+7.
Putp=−α−4/3α+10,i=3α+11/3α+10,andq=α+3/3α+10 in (38) such that(40)hww=hw′w=hw3w=0,hw′′w=pα+1α+2+iα+2α+3+qα+4α+5−α+3α+4w¯21−w23.
Thus, we have(41)C≥Tu1,u2,φhφwZν≥νwTu1,u2,φhφw″w=νwu1wφ′w2+2u2′wφ′w+u2zφ″w⋅hφwφw″⋅φw2=pα+1α+2+iα+2α+3+qα+4α+5−α+3α+4⋅νwu1wφ′w2+2u2′wφ′w+u2zφ″wφw21−φw2α+2.
Hence,(42)supz∈Dνzu1zφ′z2+2u2′zφ′z+u2zφ″zφz21−φz2α+2<∞,which implies that(43)supφz>1/2νzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2≤Csupφz>1/2νzu1zφ′z2+2u2′zφ′z+u2zφ″zφz21−φz2α+2<∞.
Applying (22) yields that(44)supφz≤1/2νzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2≤Csupφz≤1/2νzu1zφ′z2+2u2′zφ′z+u2zφ″zφz2<∞.
Together (43) with (44), we can obtain(45)supz∈Dνzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2<∞,which means that (16) holds.
Fixw∈D and l,m,n∈ℝ. Consider the following test function:(46)Jwz=n1−w21−w¯zα+1+l1−w221−w¯zα+2+m1−w231−w¯zα+3−1−w241−w¯zα+4.
Applying Lemma3 yields that Jw∈ℱα and supw∈DJwℱα≤C. A straightforward calculation shows that(47)Jw′z=α+1nw¯1−w21−w¯zα+2+α+2lw¯1−w221−w¯zα+3+α+3mw¯1−w231−w¯zα+4−α+4w¯1−w241−w¯zα+5,Jw″z=α+1α+2nw¯21−w21−w¯zα+3+α+2α+3lw¯21−w221−w¯zα+4+α+3α+4mw¯21−w231−w¯zα+5−α+4α+5w¯21−w241−w¯zα+6,Jw3z=α+1α+2α+3nw¯31−w21−w¯zα+4+α+2α+3α+4lw¯31−w221−w¯zα+5+α+3α+4α+5mw¯31−w231−w¯zα+6−α+4α+5α+6w¯31−w241−w¯zα+7.
Putl=−3,m=3,andn=1 in (46) such that(48)Jww=Jw′w=Jw″w=0,Jw3w=lα+2α+3α+4+mα+3α+4α+5+nα+1α+2α+3−α+4α+5α+6⋅w¯31−w2α+3.
Thus, we have(49)C≥Tu1,u2,φJφwZν≥νwTu1,u2,φJφw″w=lα+2α+3α+4+mα+3α+4α+5+nα+1α+2α+3−α+4α+5α+6⋅νwu2wφ′w2φw31−φw2α+3.
Hence,(50)supz∈Dνzu2zφ′z2φz31−φz2α+3<∞,which implies that(51)supφz>1/2νzu2zφ′z21−φz2α+3≤Csupφz>1/2νzu2zφ′z2φz31−φz2α+3<∞.
Applying (24) yields that(52)supφz≤1/2νzu2zφ′z21−φz2α+3≤Csupφz≤1/2νzu2zφ′z2<∞.
Together (51) with (52), we can obtain(53)supz∈Dνzu2zφ′z21−φz2α+3<∞,which means that (17) holds. The proof is complete.Theorem 2.
Letα>0. Suppose u1,u2∈HD,φ∈S. Then, Tu1,u2,φ:ℱα⟶Zν is compact if and only if Tu1,u2,φ:ℱα⟶Zν is bounded, and the following conditions are satisfied:(54)limφz⟶1νzu1″z1−φz2α=0,(55)limφz⟶1νz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1=0,(56)limφz⟶1νzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2=0,(57)limφz⟶1νzu2zφ′z21−φz2α+3=0.Proof.
Assume thatTu1,u2,φ:ℱα⟶Zν is bounded and (54)–(57) hold. Due to Lemma 1, in order to prove that Tu1,u2,φ:ℱα⟶Zν is compact, it suffices to show that, for any bounded sequence fk in ℱα with fk⟶0 uniformly on compact subsets of D, Tu1,u2,φfkZν⟶0ask⟶∞. We may assume that fkℱα≤1, for all k∈ℕ. By (54)–(57), we have that, for any ϵ>0, there exists r∈0,1 such that(58)νzu1″z1−φz2α<ϵ,(59)νz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1<ϵ,(60)νzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2<ϵ,(61)νzu2zφ′z21−φz2α+3<ϵ,for r<φz<1. From the boundedness of the operator Tu1,u2,φ:ℱα⟶Zν and the proof of Theorem 1, we obtain that (20)–(24) hold. Since fk⟶0 uniformly on compact subsets of D, Cauchy estimates show that fkn converges to 0 uniformly on compact subsets of D. Then, there exists K0∈ℕ, for k>K0, such that(62)supφz≤rνzTu1,u2,φfk′z≤supφz≤rνzu1″zfkφz+supφz≤rνz2u1′zφ′z+u1zφ″z+u2″zfk′φz+supφz≤rνzu1zφ′z2+2u2′zφ′z+u2zφ″zfk″φz+supφz≤rνzu1zφ′z2fk3φz≤ϵsupφz≤rνzu1″z+ϵsupφz≤rνz2u1′zφ′z+u1zφ″z+u2″z+ϵsupφz≤rνzu1zφ′z2+2u2′zφ′z+u2zφ″z+ϵsupφz≤rνzu1zφ′z2≤Cϵ.
From (58)–(61) and Lemma 4, we have(63)Tu1,u2,φfkZν=Tu1,u2,φfk0+Tu1,u2,φfk′0+supz∈DνzTu1,u2,φfk″z≤ν0u10fkφ0+u20fk′φ0+ν0u1′0fkφ0+u10φ′0+u2′0φ′0fk′φ0+u20φ′0fk″φ0+supz∈Dνzu1″zfkφz+supz∈Dνz2u1′zφ′z+u1zφ″z+u2″zfk′φz+supz∈Dνzu1zφ′z2+2u2′zφ′z+u2zφ″zfk″φz+supz∈Dνzu2zφ′z2fk″φ0fk3φz≤Cϵ+supr<φz<1νzu1″zfkφz+supr<φz<1νz2u1′zφ′z+u1zφ″z+u2″zfk′φz+supr<φz<1νzu1zφ′z2+2u2′zφ′z+u2zφ″zfk″φz+supr<φz<1νzu2zφ′z2fk″φ0fk3φz≤Cϵ+Csupr<φz<1νzu1″z1−φz2α+νz2u1′zφ′z+u1zφ″z+u2″z1−φz2α+1+νzu1zφ′z2+2u2′zφ′z+u2zφ″z1−φz2α+2+νzu2zφ′z21−φz2α+3≤Cϵ,which implies that Tu1,u2,φ:ℱα⟶Zν is compact.
Conversely, it is clear that the compactness ofTu1,u2,φ:ℱα⟶Zν implies that the boundedness of Tu1,u2,φ:ℱα⟶Zν. If φ∞<1, it is clear that (54)–(57) is vacuous and obviously hold. Hence, assume that φ∞=1. Let zk be a sequence in D such that φzk⟶1 as k⟶∞. Take the test functions(64)fkz=fφzkz,where fw is defined in (25). From the proof of Theorem 1, we have that supk∈ℕfkℱα≤C and(65)fkφzk=a+b+c−11−φzk2α,fk′φzk=fk′′φzk=fk3φzk=0.
It is obvious thatfk converges to 0 uniformly on compact subsets of D. From Lemma 1 and the compactness of Tu1,u2,φ:ℱα⟶Zν, we have(66)a+b+c−1νzku1″zk1−φzk2α≤Tu1,u2,φfkZν⟶0,as k⟶∞, which implies that (54) holds.
Take the test functions:(67)gkz=gφzkz,where gw is defined in (30). From the proof of Theorem 1, we have that supk∈ℕgkℱα≤C and(68)gkφzk=gk′′φzk=gk3φzk=0,gk′φzk=−α+1+dα+2+eα+3+hα+4φzk¯1−φzk2α+1.
It is obvious thatgk converges to 0 uniformly on compact subsets of D. From Lemma 1 and the compactness of Tu1,u2,φ:ℱα⟶Zν, we have(69)−α+1+dα+2+eα+3+hα+4⋅νzk2u1′zkφ′zk+u1zkφ″zk+u2″zkφzk1−φzk2α+1≤Tu1,u2,φgkZν⟶0,as k⟶∞. Thus, for φzk⟶1, we have(70)limk⟶∞νzk2u1′zkφ′zk+u1zkφ″zk+u2″zk1−φzk2α+1=0,which implies (55) holds.
Analogously, choosing the test functionshkz=hφzkz and Jkz=Jφzkz which is defined in (38) and (46), same to the above approach, we can deduce (56) and (57). Consequently, the proof is complete.Remark 1.
Due to Theorems1 and 2, we may easily obtain the characterizations for the boundedness and compactness of all six product-type operators:(71)MuDCφ,CφMuD,MuCφD,DMuCφ,CφDMu,DCφMu,from the space of fractional Cauchy transforms to the Zygmund-type spaces. We leave the details to the readers interested in this research area.



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