# Weighted Composition Operators fromH∞ to α,m-Bloch Space on Cartan-Hartogs Domain of the First Type

Authors: Jianbing Su; Ziyi Zhang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

LetYI be nonhomogeneous Cartan-Hartogs domain of the first type, ϕ a holomorphic self-map, and ψ a fixed holomorphic function on YI. We study the weighted composition operator ψCϕf=ψf∘ϕ for a function f holomorphic on YI. Our main results generalize both cases of the unit ploydisc and the unit ball obtained by Li and Stević (Li 2007 and Li 2008). Firstly, we obtain two crucial inequalities on YI; furthermore, the boundedness and compactness of operator ψCϕ from the space H∞ of all bounded holomorphic functions to the α,m-Bloch space Bα,m on YI are investigated.


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## Body

## 1. Introduction

LetΩ be a bounded domain in ℂn and HΩ be the set of all holomorphic functions on Ω. Let A,B be complex Banach spaces on Ω, let ψ be a fixed holomorphic function on Ω, and let ϕ be a holomorphic self-map of Ω. The weighted composition operator ψCϕ:A⟶B with the multiplication symbol ψ and the composition symbol ϕ is defined by
(1)ψCϕf=ψf∘ϕ,for a function f holomorphic on A. It should be mentioned that this operator can be regarded as a generalization of a multiplication operator and a composition operator on various Banach spaces; one can see [1] and reference within for more information on composition operators.Our primary objects of study in this article are bounded and compact weighted composition operators from the spaceH∞ of all bounded holomorphic functions to the α,m-Bloch space Bα,m on the Cartan-Hartogs domain of the first type, which is defined by Yin [2]. In the work of [3], Cartan first split the irreducible bounded symmetric domains into four types of Cartan domains and two exceptional domains whose complex dimensions are 16 and 27, respectively. Based on this pioneering work, Yin [2] constructed the Hua domains in the theory of several complex variables, which mainly contain the Cartan-Hartogs domains, Cartan-Egg domains, Hua domains, generalized Hua domains, and Hua construction. The Cartan-Hartogs domain of the first type is defined as follows:
(2)YIN,m,n;K≔W∈ℂN,Z∈RIm,n:W2K<detI−ZZ¯′,K>0,where
(3)RIm,n≔Z∈ℂm×n:I−ZZ¯′>0is the Cartan domain of the first type, Z¯′ denotes the conjugate transpose of Z, det denotes the determinant of a square matrix, N,m,n are some positive integers, and K is a positive real number. In particular, when m=1, W=0, and K=1, the Cartan-Hartogs domain of the first type turns to be the case of the unit ball; it is obvious that the unit ball is a specific case of the Hua domain. In [4], they verified that the Hua domain is not a homogeneous domain or a Reinhardt domain unless a ball. For simplicity, the Cartan-Hartogs domain of the first type is characterized as YI. Moreover, throughout this paper, we only consider the case of N=1 for convenience. However, we would like to mention that all of results obtained in this work can be extended to the case of N≥1 naturally.In [5], Ohno investigated the boundedness and compactness of weighted composition operators between H∞ and the Bloch space B in the open unit disc. In the setting of the unit ball, Du and Li [6] study the boundedness and compactness of weighted composition operators from H∞ to the Bloch space B, whose norm is defined by the radial derivative Rfz. Li and Stević [7] gave another characterization for the boundedness and compactness of weighted composition operators from H∞ to the α-Bloch space Bα, whose norm is defined by the gradient ∇fz. Actually, these two norms are equivalent (see [8] for details). In the setting of the unit polydisc, Li and Stević [9, 10] presented some necessary and sufficient conditions for the composition operators and weighted composition operators between H∞ and α-Bloch space Bα to be bounded and compact. Besides, there are various interesting works in the literature concerning the operators from the Bloch-type space with the normal weight μ or the logarithmic weight to H∞ in the unit disc, unit ball, or polydisc (cf. [6, 11–15]).Allen and Colonna [16] investigated the boundedness and compactness of the weighted composition operators from H∞ to the Bloch space B in the bounded homogeneous domain. In the case of the infinite dimensional bounded symmetric domains, Hamada in [17] studied the bounded weighted composition operators from H∞ to the Bloch space B on the infinite dimensional bounded symmetric domain, which is realized as the open unit ball of a JB∗-triple in [18].However, in the setting of the Hua domain, the related works only focus on the composition operators between the classic Bloch spaces, the Bloch-type space equipped with the special weightα or the normal weight μ (see, e.g., [19–23]).In the present paper, motivated by [7, 9], we characterize the boundedness and compactness of weighted composition operators from H∞ to α,m-Bloch space on the Cartan-Hartogs domain of the first type. The remainder of this article is organized as follows. In Section 2, we collect background materials necessary for the understanding of the statements of our main results. In Section 3, two important inequalities on the Cartan-Hartogs domain of the first type are derived. The first one, let K≥1 and α≥m, for a holomorphic function f in the Cartan-Hartogs domain of the first type, there exists a constant C>0 such that
(4)fBα,m≤fBα,m≤Cf∞.The second inequality is that, forZ,W,X,Y∈YI, we have
(5)2detI−ZX¯′1/m−W,YK/m≥detI−ZZ¯′1/m−W2K/m+detI−XX¯′1/m−∣Y2K/m,which is in a position to derive the Hua inequality (see [24]). Using these two inequalities and constructing some test functions on YI, Section 4 is devoted to studying the boundedness of the weighted composition operator ψCϕ:H∞YI⟶Bα,mYI, and in Section 5, the compactness of the weighted composition operator ψCϕ:H∞YI⟶Bα,mYI is also derived.Throughout the rest of the paper,C denotes some constants which may change from line to line.


## 2. Preliminaries

In this section, before we state the main results, we would like to collect some notations and crucial lemmas in order to prove the main results.Definition 1.
We useH∞=H∞YI to denote the space of all bounded holomorphic functions on YI. The space H∞ is a Banach algebra under the following supremum norm ∥·∥∞:
(6)f∞≔supZ,W∈YIfZ,W<+∞,forallf∈HYI.For a holomorphic functionf, the complex gradient of f at Z,W will be denoted by ∇fZ,W, that is
(7)∇fZ,W=∂fZ,W∂z11,∂fZ,W∂z12,⋯,∂fZ,W∂zmn,∂fZ,W∂W.Definition 2.
Letα>0. The α,m-Bloch space Bα,m=Bα,mYI consists of all holimorphic functions on HYI satisfying
(8)supZ,W∈YIdetI−ZZ¯′1/m−W2K/mα∇fZ,W<+∞.If we equip the norm(9)fBα,m≔f0,0+supZ,W∈YIdetI−ZZ¯′1/m−W2K/mα∇fZ,W,it is clear that the α,m-Bloch space Bα,m becomes a Banach space under the norm ∥·∥Bα,m which can be proved in a standard way.For more information onH∞ and the Bloch-type space, we refer to [25, 26] and references therein.Lemma 3 (see [27], Theorem 3.3.1) (Hadamard).
LetA=aij≥0 be an n×n Hermitian matrix. Then,
(10)detA≤∏i=1naii,and "=" holds if and only if A is a diagonal matrix.Lemma 4 (see [28]).
Ifxk≥−1, xk keep the same sign and n≥2, then,
(11)∏k=1n1+xk>1+∑k=1nxk.Remark 5.
Whenn=1 or xk=0k=2,⋯,n, we get ∏k=1n1+xk=1+∑k=1nxk.Lemma 6 (see [26], Proposition 5.1).
LetD be the unit disc on ℂn. H∞D⊂BD. Moreover,
(12)1−∣z2f′z≤f∞for all z∈D and f∈H∞.This lemma shows that any bounded analytic function onD is in the Bloch space. We will generalize this lemma to the Cartan-Hartogs domain of the first type in Section 3.Lemma 7 (see [29], Appendix: Theorem 1.1).
Let(13)Z=z11z12⋯z1nz21z22⋯z2n⋮⋮⋱⋮zm1zm2⋯zmnbe an m×n matrix m≤n; then, there exist an m×m unitary matrix U and an n×n unitary matrix V such that
(14)Z=Uλ10⋯00⋯00λ2⋯00⋯0⋮⋮⋮⋮⋮00⋯λm0⋯0Vλ1≥λ2≥⋯≥λm≥0,where λ12,⋯,λm2 are characteristic values of ZZ¯′.Lemma 8 (see [29], Theorem 3.1.1).
Let(15)Λ1=λ10⋯00λ2⋯0⋮⋮⋱000⋯λmλ1≥λ2≥⋯≥λm≥0,Λ2=μ10⋯00μ2⋯0⋮⋮⋱000⋯μmμ1≥μ2≥⋯≥μm≥0,satisfying
(16)λjμk<1j,k=1,⋯,m.
Then, there is an arranged square matrixP such that
(17)infUU¯′=I,VV¯′=IdetI−Λ1UΛ2U¯′V=detI−Λ1PΛ2P′,and the minimum value is obtained when U=P and V=I.Lemma 9 (see [28]).
Letak,bk≥0; then,
(18)∏k=1nak+bk1/n≥∏k=1nak1/n+∏k=1nbk1/n.Lemma 10.
LetK be a compact subset of YI and f be a holomorphic function on Bα,mYI. Then for every Z,W∈K, there exists a constant CK>0 such that
(19)fZ,W≤CKfBα,m.Proof.
This lemma is a special case of Lemma 2.4 in [23] by taking μZ,W=detI−ZZ¯′1/m−W2K/mα; here, we omit the details.Lemma 11.
Letϕ=ϕ11,ϕ12⋯ϕmn,ϕmn+1 be a holomorphic self-map of YI and ψ a holomorphic function on YI. The weighted composition operator ψCϕ:H∞YI⟶Bα,mYI is compact if and only if ψCϕ is bounded and for any bounded sequence fkk≥1inH∞YI converging to 0 uniformly on compact subsets of YI, limk⟶∞ψCϕfkBα,m=0.Proof.
Suppose thatψCϕ:H∞YI⟶Bα,mYI is compact. Let fkk≥1 be a bounded sequence in H∞YI with supk≥1fk∥∞≤M<∞, and fk⟶0 uniformly on compact subsets of YI as k⟶∞. By the definition of compactness of ψCϕ, the sequence ψCϕfkk≥1 has a subsequence ψCϕfkjj≥1 converging to f∈Bα,mYI. By (19), there exists a constant CK>0 such that
(20)ψCϕfkjZ,W−fZ,W≤CKψCϕfkj−fBα,m,for every Z,W∈YI. It follows that ψCϕfkjZ,W−fZ,W⟶0 uniformly on compact subsets of YI as j⟶∞. Therefore, fkj⟶0 uniformly on compact subsets of YI as j⟶∞. Owing to the definition of ψCϕ, we obtain f≡0; thus, limk⟶∞ψCϕfkBα,m=0.
Conversely, suppose thathkk≥1 is a sequence in the ball B0,M⊂H∞YI; then hk∞≤M<∞. It is obvious that hkk≥1 is uniformly bounded on compact subsets of YI. By Montel’s theorem, we know that hkk≥1 has a subsequence hkjj≥1 converges to h∈HYI uniformly on YI. Moveover, h∈H∞YI and h∞≤M. Hence the sequence hkj−hj≥1 is such that hkj−h∞≤2M and hkj−h⟶0 uniformly on compact subsets of YI. We Following from the hypothesis implies that
(21)limj⟶∞ψCϕhkj−hBα,m=limj⟶∞ψCϕhkj−ψCϕhBα,m=0,which yields that the set ψCϕB0,M is relatively compact.


## 3. Two Important Inequalities

In this section, we obtain two important inequalities onYI, which are essential in proving our main results. We remark that two inequalities below seem to be known in the unit ball, but we need to prove them correct on the Cartan-Hartogs domain of the first type.Theorem 12.
LetK≥1 and α≥m. There exists a positive constant C independent of f such that
(22)fBα,m≤fBm,m≤Cf∞,for all Z,W∈YI and f∈H∞.Proof.
Sinceα≥m and the definition of Bα,mYI, we have fBα,m≤fBm,m. For each Z,W∈YI, let
(23)Z=z11z12⋯z1nz21z22⋯z2n⋮⋮⋱⋮zm1zm2⋯zmn.
In view of (10) and (11), we have
(24)detI−ZZ¯′≤1−z112+z122+⋯+z1n2×1−z212+z222+⋯+z2n2×⋯×1−zm12+zm22+⋯+zmn2≤1−z112⋯1−z1n21−z212⋯1−zm12⋯1−∣zmn2.
Due toa2+b2≤a+ba≥0,b≥0, it leads to
(25)∇fZ,W=∂f∂z11Z,W2+∂f∂z12Z,W2+⋯+∂f∂zmnZ,W2+∂f∂WZ,W21/2≤∂f∂z11Z,W+∂f∂z12Z,W+⋯+∂f∂zmnZ,W+∂f∂WZ,W.
SinceW2K<detI−ZZ¯′, it is easy to see that W<1. Moreover, let a=detI−ZZ¯′, b=W2K; we have 0≤b<a<1. Making use of the following inequality a1/m−b1/mm≤a−b, it suffices to obtain
(26)detI−ZZ¯′1/m−W2K/mm≤detI−ZZ¯′−W2K.
In fact, to provea1/m−b1/mm≤a−b, we can consider a1/m≤a−b1/m+b1/m. Let c=a−b1/m,d=b1/m; it follows that we should prove cm+dm≤c+dm, which obviously holds. Moreover, according to Lemma 6, it leads to
(27)detI−ZZ¯′1/m−W2K/mm∇fZ,W≤detI−ZZ¯′−W2K∣∇fZ,W∣≤1−z1121−z122⋯1−zmn2−W2K×∣∂f∂z11Z,W∣+∣∂f∂z12Z,W∣+⋯+∣∂f∂zmnZ,W∣+∣∂f∂WZ,W∣≤1−z1121−z122⋯1−zmn2−1−z1121−z122⋯1−zmn2W2K×∣∂f∂z11Z,W∣+∣∂f∂z12Z,W∣+⋯+∣∂f∂zmnZ,W∣+∣∂f∂WZ,W∣≤1−z1121−z122⋯1−zmn21−WK2×∣∂f∂z11Z,W∣+∣∂f∂z12Z,W∣+⋯+∣∂f∂zmnZ,W∣+∣∂f∂WZ,W∣≤1−z112∣∂f∂z11Z,W∣+1−z122∣∂f∂z12Z,W∣+⋯+1−zmn2∣∂f∂zmnZ,W∣+1−WK2∣∂f∂WZ,W∣≤mn∥f∥∞+1−WK2∣∂fZ,W∂WK·∂WK∂W∣≤mn∥f∥∞+∥f∥∞·KWK−1≤mn+K∥f∥∞,which gives the desired estimate.Remark 13.
When the target is the unit ball inℂn, let m=1, W=0, and K=1; we have the inequality 1−Z2∣∇fZ∣≤n+1∥f∥∞, which arrives at the same conclusion in ([7], Lemma 3).Theorem 14.
LetZ,X∈ℂm×n, W,Y∈ℂN and K>0. If I−ZZ¯′>0, I−XX¯′>0, W2K<detI−ZZ¯′ and Y2K<detI−XX¯′. Then, the following inequality holds
(28)2∣detI−ZX¯′1/m−W,YK/m∣≥detI−ZZ¯′1/m−W2K/m+detI−XX¯′1/m−Y2K/m,and "=" holds if and only if Z,W=X,Y.Proof.
Whenm=n, since Z1,X1∈RIm,n, applying Lemma 7, there exist m×m unitary matrixes U1,U2 and n×n unitary matrixes V1,V2 such that
(29)Z1=U1λ10⋯00λ2⋯0⋮⋮⋱⋮00⋯λmV1=U1Λ1V11>λ1≥λ2≥⋯≥λm≥0,X1=U2μ10⋯00μ2⋯0⋮⋮⋱⋮00⋯μmV2=U2Λ2V21>μ1≥μ2≥⋯≥μm≥0.
Then, it turns out to(30)detI−Z1X1¯′=detI−U1Λ1V1V2¯′Λ2¯′U2¯′=detU1U1¯′−U1Λ1V1V2¯′Λ2¯′U2¯′=detU1detU1¯′−Λ1V1V2¯′Λ2¯′U2¯′=detI−Λ1V1V2¯′Λ2¯′U2¯′U1=detI−Λ1V1V2¯′Λ2¯′V2V1¯′U2¯′U1,and according to Lemma 8, there exists an arrange square matrix P such that
(31)∣detI−Z1X1¯′∣≥∣detI−Λ1PΛ2P′∣=∏i=1m1−λiμki.
Hence, using (18), we have
(32)2detI−Z1X1¯′1/m≥2∏i=1m1−λiμki1/m=2m∏i=1m1−λiμki1/m=∏i=1m2−2λiμki1/m≥∏i=1m1−λi2+1−μki21/m≥∏i=1m1−λi21/m+∏i=1m1−μki21/m=detI−Z1Z1¯′1/m+detI−X1X1¯′1/m,where ki is the rearrangement of i. Moreover, referring to the condition of equalities for (31) and (32), we obtain the inequality
(33)2detI−Z1X1¯′1/m≥detI−Z1Z1¯′1/m+detI−X1X1¯′1/m,which becomes an equality if and only if Z1=X1.
Whenm<n, there exists an unitary matrix Un such that
(34)Z=Z1m,0U,X=X1m,X2U.
By (32), we obtain
(35)2detI−ZX¯′1/m=2detI−Z1X1¯′1/m≥detI−Z1Z1¯′1/m+detI−X1X1¯′1/m≥detI−Z1Z1¯′1/m+detI−X1X1¯′−X2X2¯′1/m=detI−ZZ¯′1/m+detI−XX¯′1/m.
Thus, the inequality(36)2detI−ZX¯′1/m≥detI−ZZ¯′1/m+detI−XX¯′1/mholds when m≤n, and "=" holds if and only if Z=X. By the inequality of arithmetic and geometric means, we have
(37)2WK/mY¯′K/m≤W2K/m+Y2K/m,and the equality holds if and only if ∣W∣=∣Y∣. Therefore, combining (36) with (37) gives that
(38)2detI−ZX¯′1/m−W,YK/m≥2detI−ZX¯′1/m−2W,YK/m≥2detI−ZX¯′1/m−2WK/mY¯′K/m≥detI−ZZ¯′1/m+detI−XX¯′1/m−W2K/m−Y2K/m=detI−ZZ¯′1/m−W2K/m+detI−XX¯′1/m−Y2K/m.
The first inequality becomes an equality if and only ifdetI−ZX¯′1/m·W,YK/m¯′≥0, and the second inequality becomes an equality if and only if W=0, Y=0, or W=kYk>0, which implies "=" holds only when W=Y in (38). Hence, in this case, there is equality in (38) if and only if Z,W=X,Y.Corollary 15.
LetZ,X∈ℂm×n, W,Y∈ℂN, and K>0. If I−ZZ¯′>0, I−XX¯′>0, W2K<detI−ZZ¯′, and Y2K<detI−XX¯′. Then, the following inequality holds
(39)detI−ZX¯′1/m−W,YK/m2≥detI−ZZ¯′1/m−W2K/mdetI−XX¯′1/m−Y2K/m.Proof.
This proof only follows the elementary inequalitya+b/2≥aba≥0,b≥0; here, we omit the details.Corollary 16.
LetZ,X∈ℂm×n. If I−ZZ¯′>0 and I−XX¯′>0, then,
(40)2detI−ZX¯′1/m≥detI−ZZ¯′1/m+detI−XX¯′1/m.Proof.
SubstitutingW=0 and Y=0 into (28) leads to this inequality.Remark 17.
Sincea+b/2≥aba≥0,b≥0, we get
(41)detI−ZX¯′2=I−ZZ¯′1/m2m≥12detI−ZZ¯′1/m+12detI−XX¯′1/m2m≥detI−ZZ¯′1/mdetI−XX¯′1/mm=detI−ZZ¯′detI−XX¯′,which yields the Hua inequality discovered by Hua Loo-Keng in [24].


## 4. Boundedness ofψCϕ:H∞⟶Bα,m

In this section, we characterize the bounded weighted composition operator in the caseψCϕ:H∞YI⟶Bα,mYI. The following theorem describes such properties.We will begin by introducing some notations. Letϕ=ϕ11,ϕ12⋯ϕmn,ϕmn+1 be a holomorphic self-map of YI, denoting
(42)DϕZ,W=∂ϕ11Z,W∂z11⋯∂ϕ11Z,W∂zmn∂ϕ11Z,W∂W⋮⋱⋮⋮∂ϕmnZ,W∂z11⋯∂ϕmnZ,W∂zmn∂ϕmnZ,W∂W∂ϕmn+1Z,W∂z11⋯∂ϕmn+1Z,W∂zmn∂ϕmn+1Z,W∂W.Theorem 18.
ForK≥1 and α≥m, let ϕ=ϕ11,ϕ12,⋯,ϕmn,ϕmn+1 be a holomorphic self-map of YI, ψ a holomorphic function on YI, and Zϕ,Wϕ=ϕZ,W. If
(43)ψ∈Bα,m,supZ,W∈YIdetI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W<∞,
then the weighted composition operatorψCϕ:H∞YI⟶Bα,mYI is bounded.
Conversely, if the weighted composition operatorψCϕ:H∞YI⟶Bα,mYI is bounded, then,
(44)ψ∈Bα,m,supZ,W∈YIdetI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mψZ,WGZ,W<∞,where
(45)GZ,W=∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤n1mdetI−ZϕZϕ¯′1/mtrI−ZϕZϕ¯′−1IuvZϕ¯′×∂ϕuv∂zklZ,W+KmWϕ2K/m−2Wϕ¯′∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤n1mdetI−ZϕZϕ¯′1/mtrI−ZϕZϕ¯′−1IuvZϕ¯′×∂ϕuv∂WZ,W+KmWϕ2K/m−2Wϕ¯′∂ϕmn+1∂WZ,W21/2.Proof.
Assume that (43) holds. There exists a positive constant C1 such that
(46)detI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W≤C1,for all Z,W∈YI and Zϕ,Wϕ=ϕZ,W∈YI. Firstly, we know that
(47)∇CϕfZ,W=∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂z11Z,W+∂f∂Ymn+1ϕZ,W∂ϕmn+1∂z11Z,W,⋯,∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂zmnZ,W+∂f∂Ymn+1ϕZ,W∂ϕmn+1∂zmnZ,W,∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂WZ,W+∂f∂Ymn+1ϕZ,W∂ϕmn+1∂WZ,W.
Therefore, it leads to(48)∇CϕfZ,W2=∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂zklZ,W+∂f∂Ymn+1ϕZ,W∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂WZ,W+∂f∂Ymn+1ϕZ,W∂ϕmn+1∂WZ,W2≤2∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂zklZ,W2+2∑1≤k≤m1≤l≤n∂f∂Ymn+1ϕZ,W∂ϕmn+1∂zklZ,W2+2∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W∂ϕuv∂WZ,W2+2∂f∂Ymn+1ϕZ,W∂ϕmn+1∂WZ,W2≤2∑1≤u≤m1≤v≤n∂f∂YuvϕZ,W2+∂f∂Ymn+1ϕZ,W2×∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤n∂ϕuv∂zklZ,W2+∑1≤k≤m1≤l≤n∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤n∂ϕuv∂WZ,W2+∂ϕmn+1∂WZ,W2=2∇fϕZ,W2DϕZ,W2.
Namely,(49)∇CϕfZ,W≤2∇fϕZ,WDϕZ,W.
For a functionf∈H∞YI, we obtain the following estimate
(50)detI−ZZ¯′1/m−∣W2K/mα∇ψCϕfZ,W=detI−ZZ¯′1/m−W2K/mα∇ψZ,W·CϕfZ,W+ψZ,W·∇CϕfZ,W≤detI−ZZ¯′1/m−W2K/mα∇ψZ,WCϕfZ,W+detI−ZZ¯′1/m−W2K/mαψZ,W∇CϕfZ,W≤detI−ZZ¯′1/m−W2K/mα∇ψZ,WfϕZ,W+2detI−ZZ¯′1/m−W2K/mαψZ,W∇fϕZ,WDϕZ,W≤detI−ZZ¯′1/m−W2K/mα∇ψZ,WfϕZ,W+2detI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W×detI−ZϕZϕ¯′1/m−Wϕ2K/mm∇fϕZ,WψBα,mf∞+2C1fBm,m.
Sinceψ∈Bα,m and (22), it leads to
(51)ψCϕfBα,m=ψCϕf0,0+supZ,W∈YIdetI−ZZ¯′1/m−W2K/mα∇ψCϕfZ,W≤ψ0,0fϕ0,0+ψBα,mf∞+2C1fBm,m≤ψ0,0f∞+ψBα,mf∞+2CC1f∞≤C2f∞,which implies that ψCϕ:H∞YI⟶Bα,mYI is bounded.
Conversely, assume thatψCϕ:H∞YI⟶Bα,mYI is bounded. It follows that there exists a positive constant C such that
(52)ψCϕfBα,m≤Cf∞.
Letf≡1; we have ψBα,m≤C, which implies ψ∈Bα,m. For X,Y∈YI, define a test function fX,Y∈HYI by
(53)fX,YZ,W≔detI−XX¯′1/m−Y2K/mdetI−ZX¯′1/m−W,YK/m.
From (28), it follows that
(54)fX,YZ,W=detI−XX¯′1/m−Y2K/mdetI−ZX¯′1/m−W,YK/m≤detI−XX¯′1/m−Y2K/m1/2detI−ZZ¯′1/m−W2K/m+1/2detI−XX¯′1/m−Y2K/m≤2detI−XX¯′1/m−Y2K/mdetI−XX¯′1/m−Y2K/m=2,which implies fX,Y∈H∞YI and fX,Y∞≤2.
For the test functionf, we have
(55)∂fX,Y∂YuvϕZ,W=detI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2·FuvZϕ,X,∂fX,Y∂Ymn+1ϕZ,W=detI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,Y2K/m2·KmWϕ,YK/m−1Y¯′,where FuvZϕ,X=1/mdetI−ZϕX¯′1/mtrI−ZϕX¯′−1IuvX¯′. It leads to
(56)∇CϕfX,YZ,W=∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤n∂fX,Y∂YuvϕZ,W∂ϕuv∂zklZ,W+∂fX,Y∂Ymn+1ϕZ,W∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤n∂fX,Y∂YuvϕZ,W∂ϕuv∂WZ,W+∂fX,Y∂Ymn+1ϕZ,W∂ϕmn+1∂WZ,W21/2=∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤ndetI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2·FuvZϕ,X∂ϕuv∂zklZ,W+detI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2·KmWϕ,YK/m−1Y¯′∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤ndetI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2·FuvZϕ,X∂ϕuv∂WZ,W+detI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2·KmWϕ,YK/m−1Y¯′∂ϕmn+1∂WZ,W21/2=detI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2×∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤nFuvZϕ,X∂ϕuv∂zklZ,W+KmWϕ,YK/m−1Y¯′∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤nFuvZϕ,X∂ϕuv∂WZ,W+KmWϕ,YK/m−1Y¯′∂ϕmn+1∂WZ,W21/2.
Then, it follows that(57)detI−ZZ¯′1/m−W2K/mα∇CϕfX,YZ,W=detI−ZZ¯′1/m−W2K/mαdetI−XX¯′1/m−Y2K/mdetI−ZϕX¯′1/m−Wϕ,YK/m2×∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤nFuvZϕ,X∂ϕuv∂zklZ,W+KmWϕ,YK/m−1Y¯′∂ϕmn+1∂zklZ,W2+∑1≤u≤m1≤v≤nFuvZϕ,X∂ϕuv∂WZ,W+KmWϕ,YK/m−1Y¯′∂ϕmn+1∂WZ,W21/2.
Let(58)X,Y=Zϕ,Wϕ=ϕZ,W.
SincefϕZ,W=1, (52) and (57), we obtain
(59)2C≥ψCϕfβα,m≥detI−ZZ¯′1/m−W2K/mα∇ψCϕfZ,W=detI−ZZ¯′1/m−W2K/mαψZ,W·∇CϕfZ,W+∇ψZ,W·CϕfZ,W≥detI−ZZ¯′1/m−W2K/mαψZ,W·∇CϕfZ,W−detI−ZZ¯′1/m−W2K/mα∇ψZ,W·fϕZ,W≥detI−ZZ¯′1/m−W2K/mαdetI−ZϕZ¯ϕ′1/m−Wϕ2K/mψZ,W×∑1≤k<m1≤l≤n∑1≤u<m1≤v<nFuvZϕ,Zϕ∂ϕuv∂zklZ,W+KmWϕ2Km−2Wϕ′¯∂ϕmn+1∂zklZ,W2+∑1≤u<m1≤v<nFuvZϕ,Zϕ∂ϕuv∂WZ,W+KmWϕ2K/m−2Wϕ′¯∂ϕmn+1∂WZ,W21/2−detI−ZZ¯′1/m−W2K/mα∇ψZ,W≥detI−ZZ¯′1/m−W2K/mαdetI−ZϕZ¯ϕ′1/m−Wϕ2K/mψZ,WGZ,W−detI−ZZ¯′1/m−W2K/mα∇ψZ,W.
Sinceψ∈Bα,m, we obtain
(60)supZ,W∈YIdetI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/m∣ψZ,W∣GZ,W<∞.
The proof is completed.Remark 19.
Letm=1, W=0 and K=1, we obtain the following results in the case of the unit ball B=Z∈ℂn:Z2<1. Let α=1. If
(61)ψ∈B,supZ∈B1−Z21−ϕZ2ψZDϕZ<∞,then the weighted composition operator ψCϕ:H∞B⟶BB is bounded. Conversely, the weighted composition operator ψCϕ:H∞B⟶BB is bounded, then
(62)ψ∈BsupZ∈B∣ψZ∣1−Z21−ϕZ2∣DϕZ′ϕZ¯′∣<∞,where
(63)∣DϕZ∣=∑k,l=1n∂ϕlZ∂Zk21/2.Li and Stević investigated the boundedness of this weighted composition operator in [7], which is as the same as the above results; therefore, our main results cover and substantially improve the work of [7].


## 5. Compactness ofψCϕ:H∞⟶Bα,m

In this section, we characterize the compact weighted composition operatorψCϕ:H∞YI⟶Bα,mYI.Theorem 20.
ForK≥1 and α≥m, let ϕ=ϕ11,ϕ12⋯ϕmn,ϕmn+1 be a holomorphic self-map of YI, ψ a holomorphic function on YI, and Zϕ,Wϕ=ϕZ,W. If
(64)limϕZ,W⟶∂YIdetI−ZZ¯′1/m−W2K/mα∇ψZ,W=0,limϕZ,W⟶∂YIdetI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W=0,then the weighted composition operator ψCϕ:H∞YI⟶Bα,mYI is compact.
Conversely, if the weighted composition operatorψCϕ:H∞YI⟶Bα,mYI is compact, then
(65)limϕZ,W⟶∂YIdetI−ZZ¯′1/m−W2K/mα∣∇ψZ,W∣=0,limϕZ,W⟶∂YIdetI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/m∣ψZ,W∣GZ,W=0,where GZ,W is the same as (45).Proof.
Suppose that (64) holds. We have
(66)supZ,W∈YIdetI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W<∞.
Following from Theorem18, we obtain that ψCϕ:H∞YI⟶Bα,mYI is bounded. Let fkk≥1 be a bounded sequence, and fk converges to 0 uniformly on compact subsets of YI. Let M≔supk≥1∥fk∥∞. By the assumptions, for any ε>0, there is a constant δ∈0,1 such that
(67)detI−ZZ¯′1/m−W2K/mα∣∇ψZ,W∣<ε,(68)detI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W<ε,whenever distϕZ,W,∂YI<δ. Taking (49), (67), (68), and Theorem 12 into account, it turns out that
(69)detI−ZZ¯′1/m−W2K/mα∇ψCϕfkZ,W=detI−ZZ¯′1/m−W2K/mα∇ψZ,W·CϕfkZ,W+ψZ,W·∇CϕfkZ,W=detI−ZZ¯′1/m−W2K/mα∇ψZ,W·CϕfkZ,W+detI−ZZ¯′1/m−W2K/mαψZ,W·∇CϕfkZ,W≤detI−ZZ¯′1/m−W2K/mα∇ψZ,WfkϕZ,W+2detI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W×detI−ZϕZϕ¯′1/m−Wϕ2K/mm∇fkϕZ,W≤εfkϕZ,W+2εfkBm,m≤1+2Cεfk∞≤1+2CMε.
In addition, we set(70)Eδ≔distϕZ,W,∂YI≥δ.
Note thatEδ is a compact subset of YI. For ε defined in (67), it leads to fk⟶0 uniformly on Eδ as k⟶∞. Cauchy’s estimate gives that ∣∇fk∣⟶0 as k⟶∞ on compact subsets, in particular on ϕEδ. Hence, as k⟶∞, by (49) we obtain
(71)detI−ZZ¯′1/m−W2K/mα∇ψCϕfkZ,W≤detI−ZZ¯′1/m−W2K/mα∇ψZ,WfkϕZ,W+2detI−ZZ¯′1/m−W2K/mαdetI−ZϕZϕ¯′1/m−Wϕ2K/mmψZ,WDϕZ,W×detI−ZϕZϕ¯′1/m−Wϕ2K/mm∇fkϕZ,W⟶0.
According to the two inequalities (69) and (71), as k⟶∞, we have
(72)ψCϕfkBα,m=∣ψCϕfk0,0∣+supZ,W∈YIdetI−ZZ¯′1/m−W2K/mα∇ψCϕfkZ,W=ψ0,0·fkϕ0,0+supZ,W∈YIdetI−ZZ¯′1/m−W2K/mα∇ψCϕfkZ,W⟶0.
Consequently, making use of Lemma11, we get ψCϕ:H∞YI⟶Bα,mYI is compact.
Conversely, suppose thatψCϕ:H∞YI⟶Bα,mYI is compact. Let Xi,Yii≥1=ϕZi,Wii≥1 be a sequence on YI such that ϕZi,Wi⟶∂YI, as i⟶∞. If the sequence is nonexistent, conditions (c) and (d) obviously hold. Moreover, let us introduce a test function sequence fii≥1:
(73)fiZ,W≔detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m.
The proof of Theorem12 gives fi∈H∞ and ∥fi∥∞≤2. Due to (28), it gives that
(74)∣fiZ,W∣=detI−XiXi¯′1/m−Yi2K/m∣detI−ZXi¯′1/m−W,YiK/m∣≤detI−XiXi¯′1/m−Yi2K/m1/2detI−ZZ¯′1/m−W2K/m+1/2detI−XiXi¯′1/m−Yi2K/m≤2detI−XiXi¯′1/m−Yi2K/mdetI−ZZ¯′1/m−W2K/m.
Takingi⟶∞, we have Xi,Yi⟶∂YI. This implies detI−XiXi¯′1/m−Yi2K/m⟶0, as i⟶∞. Let E be a compact subset of YI. For Z,W∈E, it is easy to see that detI−ZZ¯′1/m−W2K/m has a positive lower bound. Hence, we obtain fi⟶0 uniformly on all compact subsets of YI, as i⟶∞.
SinceψCϕ:H∞YI⟶Bα,mYI is compact, according to Lemma 11, we have
(75)limk⟶∞∥ψCϕfi∥Bα,m=0.
For the test functionfi, we have
(76)∂fi∂YuvϕZ,W=detI−XiXi¯′1/m−Yi2K/mdetI−ZϕXi¯′1/m−Wϕ,YiK/m2·FuvZϕ,Xi,∂fi∂Ymn+1ϕZ,W=detI−XiXi¯′1/m−Yi2K/mdetI−ZϕXi¯′1/m−Wϕ,YiK/m2·KmWϕ,YiK/m−1Yi¯′,where FuvZϕ,Xi=1/mdetI−ZϕXi¯′1/mtrI−ZϕXi¯′−1IuvXi¯′. Thus,
(77)∇CϕfiZi,Wi=detI−XiXi¯′1/m−Yi2K/mdetI−ZϕiXi¯′−Wϕi,YiK2×∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤nFuvZϕi,Xi∂ϕuv∂zkliZi,Wi+KmWϕi,YiK/m−1Yi¯′∂ϕmn+1∂zkliZi,Wi2+∑1≤u≤m1≤v≤nFuvZϕi,Xi∂ϕuv∂WiZi,Wi+KmWϕi,YiK/m−1Yi¯′∂ϕmn+1∂WiZi,Wi21/2,and we have
(78)∣∇ψCϕfiZi,Wi∣=∣∇ψZi,Wi·CϕfiZi,Wi+ψZi,Wi·∇CϕfiZi,Wi∣=∇ψZi,Wi·fiϕZi,Wi+ψZi,Wi·∇CϕfiZi,Wi.
Let(79)Xi,Yi=Zϕi,Wϕi=ϕZi,Wi.
SincefiϕZi,Wi=1 and (78), we obtain that
(80)ψCϕfiBα,m≥detI−ZiZi¯′1/m−Wi2Kmα∇ψCϕfiZi,Wi=detI−ZiZi¯′1/m−Wi2Kmα∇ψZi,Wi+detI−ZiZi¯′1/m−Wi2KmαψZi,Wi·∇CϕfiZi,Wi≥detI−ZiZi¯′1/m−Wi2Kmα∇ψZi,Wi−detI−ZiZi¯′1/m−Wi2KmαψZi,Wi·∇CϕfiZi,Wi=detI−ZiZi¯′1/m−Wi2Kmα∇ψZi,Wi−detI−ZiZi¯′1/m−Wi2KmαψZi,Wi∇CϕfiZi,Wi=detI−ZiZi¯′1/m−Wi2Kmα∇ψZi,Wi−detI−ZiZi¯′1/m−Wi2K/mαdetI−ZϕiZϕi¯′1/m−Wϕi2K/mψZi,WiGZi,Wi,where
(81)GZi,Wi=∑1≤k≤m1≤l≤n∑1≤u≤m1≤v≤nFuvZϕi,Zϕi∂ϕuv∂zkliZi,Wi+KmWϕi2K/m−2Wϕi¯′∂ϕmn+1∂zkliZi,Wi2+∑1≤u≤m1≤v≤nFuvZϕi,Zϕi∂ϕuv∂WiZi,Wi+KmWϕi2K/m−2Wϕi¯′∂ϕmn+1∂WiZi,Wi21/2.
So we get(82)limϕZi,Wi⟶∂YIdetI−ZiZi¯′1/m−Wi2K/mαdetI−ZϕiZϕi¯′1/m−Wϕi2K/mψZi,WiGZi,Wi=limϕZi,Wi⟶∂YIdetI−ZiZi¯′1/m−Wi2K/mα∇ψZi,Wi,if one of these two limits exists.
Next, let(83)giZ,W≔detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m−detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m1/2.for a sequence Zi,Wii≥1 in YI such that ϕZi,Wi⟶∂YI, as i⟶∞. Then,
(84)∣giZ,W∣=∣detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m+detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m1/2∣≤∣detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m∣+detI−XiXi¯′1/m−Yi2K/mdetI−ZXi¯′1/m−W,YiK/m1/2=fiZ,W+fiZ,W1/2,
It is easy to obtaingii≥1 is a bounded sequence in H∞ and gi⟶0 uniformly on every compact subset of YI. Moreover, we notice that giϕZi,Wi=0 and
(85)∇giϕZi,Wi=GZi,Wi2detI−ZϕiZϕi¯′1/m−Wϕi2K/m.
By the similar method as above,(86)0⟵ψCϕgiBα,m≥detI−ZiZi¯′1/m−Wi2K/mα∇ψCϕgiZi,Wi=detI−ZiZi¯′1/m−Wi2K/mα∇ψZi,Wi·CϕgiZi,Wi+ψZi,Wi·∇CϕgiZi,Wi=detI−ZiZi¯′1/m−Wi2K/mα0+ψZi,Wi·GZi,Wi2detI−ZϕiZϕi¯′1/m−Wϕi2K/m=detI−ZiZi¯′1/m−Wi2K/mα2detI−ZϕiZϕi¯′1/m−Wϕi2K/mψZi,WiGZi,Wi.
And by (82),
(87)limϕZi,Wi⟶∂YIdetI−ZiZi¯′1/m−Wi2K/mα∣∇ψZi,Wi∣=0.
All of the proofs are complete.Remark 21.
Letm=1, W=0, and K=1; we get the following results in the case of the unit ball B=Z∈ℂn:Z2<1. Let α=1. If
(88)limZ⟶11−∣Z2∣∇ψZ∣=0,lim∣Z∣⟶1∣ψZ∣1−Z21−ϕZ2DϕZ=0,then, the weighted composition operator ψCϕ:H∞B⟶BB is compact. Conversely, the weighted composition operator ψCϕ:H∞B⟶BB is compact; then,
(89)limϕZ⟶11−Z2∣∇ψZ∣=0,limϕZ⟶1∣ψZ∣1−Z21−ϕZ2∣DϕZ′ϕZ¯′∣=0,where
(90)∣DϕZ∣=∑k,l=1n∂ϕlZ∂Zk21/2.
It turns out to be the same as the results obtained by Li and Stević in [7].



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*Source: 4732049-2022-04-15.xml*