# Composition Operators fromp-Bloch Space to q-Bloch Space on the Fourth Cartan-Hartogs Domains

Authors: Jianbing Su; Chao Zhang
Journal: Journal of Operators
 (2015)

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## Abstract

We obtain new generalized Hua’s inequality corresponding toYIV(N,n;K), where YIV(N,n;K) denotes the fourth Cartan-Hartogs domain in CN+n. Furthermore, we introduce the weighted Bloch spaces on YIV(N,n;K) and apply our inequality to study the boundedness and compactness of composition operator Cϕ from βp(YIV(N,n;K)) to βq(YIV(N,n;K)) for p≥0 and q≥0.


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## Body

## 1. Introduction

The study of composition operators on various Banach spaces of analytic functions has been a long active field in complex and functional analysis. The composition operators as well as related operators known as the weighted composition operators between the Bloch space and Lipschitz space were investigated in [1, 2] in the case of the unit disk. The study of the composition operators on the Bloch space was given in [3] for the polydisc, in [4, 5] for the unit ball, and in [6–9] for the bounded symmetric domains.In 1930s, all irreducible bounded symmetric domains were divided into six types by E. Cartan. The first four types of irreducible domains are called the classical bounded symmetric domains. The other two types, called exceptional domains, consist of one domain each (a16- and 27-dimensional domain). In 2000, Yin constructed four kinds of domains corresponding to the classical bounded symmetric domains, called the Cartan-Hartogs domains [10]. It is known that the Cartan-Hartogs domains are nonhomogeneous domains except the unit ball. So it is different from the bounded symmetric domains. The fourth Cartan-Hartogs domains, denoted by YIV(N,n;K), can be expressed as(1)YIVN,n;K≔w∈CN,z∈RIVn:w2K<1+zzT2-2z¯zT,K>0, where RIV(n)={z:z∈Cn,1+zzT2-2z¯zT>0,1-zzT2>0} is the fourth classical bounded symmetric domains [11] and zT is the transpose of z.For simplicity, we will writeYIV for YIV(N,n;K) if no ambiguity can arise.Letϕ=(ϕi)N+n be a holomorphic self-map of YIV. The class of all holomorphic functions on YIV is denoted by H(YIV). The composition operator Cϕ on H(YIV) is defined by (Cϕf)(z,w)=f(ϕ(z,w)) for all (z,w)∈YIV and f∈H(YIV).In 1955, Hua in [12] proved an inequality: if Z1, Z2 are n×n complex matrices and I-Z1Z1¯T, I-Z2Z2¯T are both Hermitian positive definite matrices, then (2)det⁡I-Z1Z1¯Tdet⁡I-Z2Z2¯T≤det⁡I-Z1Z2¯T2. Equality holds if and only if Z1=Z2.In 2015, Su et al. obtained generalized Hua’s inequality corresponding to the first Cartan-Hartogs domainYI (see Theorem  1 in [13]). From Theorem  1 in [13], it is easy to get more precise inequality (see Lemma 4). Furthermore, we obtain new generalized Hua’s inequality corresponding to YIV (see Lemma 5).In this paper, we define thep-Bloch space βp(YIV) as the space that consists of all f∈H(YIV) such that (3)fβp=f0,0+supz,w∈YIV⁡1-2z¯zT+zzT2-w2Kp∇fz,w<+∞, where(4)∇fz,w=∂fz,w∂z1,…,∂fz,w∂zn,∂fz,w∂w1,…,∂fz,w∂wN,∇fz,w2=∑1≤α≤n∂fz,w∂zα2+∑1≤β≤N∂fz,w∂wβ2.It is clear thatβp(YIV) is a set of constant functions when p<0, so we assume that p≥0.In this paper, we will obtain some results about the composition operators for the case of the weighted Bloch space on the fourth Cartan-Hartogs domain. In Section2, we state several auxiliary results most of which will be used in the proofs of the main results. In Sections 3 and 4, we establish the main results of the paper. We give the sufficient conditions and necessary conditions for the boundedness (in Section 3) and the compactness (in Section 4) of composition operator Cϕ from βp(YIV) to βq(YIV), where p≥0,q≥0.


## 2. Some Lemmas

In order to obtain our main results, we need the following lemmas.Lemma 1.
If(5)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w=O1,(6)z,w∈YIV,z2,w2=ϕz,w⟶∂YIV, then(7)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w<∞ for all (z,w)∈YIV and (z2,w2)=ϕ(z,w). Where ϕ=(ϕ1,ϕ2,…,ϕn,…,ϕN+n), (8)ϕ′z,w=∂ϕ1∂z1⋯∂ϕ1∂zn∂ϕ1∂w1⋯∂ϕ1∂wN⋮⋱⋮⋮⋮⋮∂ϕn∂z1⋯∂ϕn∂zn∂ϕn∂w1⋯∂ϕn∂wN∂ϕn+1∂z1⋯∂ϕn+1∂zn∂ϕn+1∂w1⋯∂ϕn+1∂wN⋮⋱⋮⋮⋮⋮∂ϕn+N∂z1⋯∂ϕn+N∂zn∂ϕn+N∂w1⋯∂ϕn+N∂wN,ϕ′z,w2=∑1≤α≤n∑1≤k≤N+n∂ϕk∂zα2+∑1≤β≤N∑1≤k≤N+n∂ϕk∂wβ2.Proof.
From (5), there exists a constant δ>0 such that(9)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w≤C1 whenever dist((z2,w2),∂YIV)<δ, where C1 is a positive number.
SetEδ={(z2,w2)∈YIV:dist((z2,w2),∂YIV)≥δ}. It is easy to know that Eδ is a compact subset of YIV. Thus, there exists a constant M∈(0,1) such that M≤1+z2z2T2-2z22-w22K≤1. So(10)11+z2z2T2-2z22-w22Kp≤1Mp<+∞.
Therefore, there exists a constantC such that(11)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w≤C for all (z,w)∈YIV and (z2,w2)=ϕ(z,w). The proof is completed.Lemma 2.
LetH be a compact subset of YIV. Then, there exists a constant C>0 such that (12)∫01z,w1+t4zzT2-2t2z2-tw2Kpdt<Cfor all (z,w)∈H.Proof.
Whenp=0, we have(13)∫01z,w1+t4zzT2-2t2z2-tw2Kpdt=∫01z,wdt. Since w2K<1+zzT2-2z2≤1 and z,w<2, ∫01z,wdt<2.
Whenp>0, denote Eδ≔{(z,w)∈YIV:1+zzT2-2z2-w2K≥δ},δ∈(0,1). For any compact subset H⊂YIV, there exists a constant δ∈(0,1) such that H⊂Eδ.
For any(z,w)∈H, t∈[0,1], we have(14)tw2K≤w2K≤1+zzT2-2z2≤1+t4zzT2-2t2z2. Furthermore,(15)1+t4zzT2-2t2z2-tw2K≥1+zzT2-2z2-w2K≥δ>0. Thus,(16)0<11+t4zzT2-2t2z2-tw2K≤11+zzT2-2z2-w2K≤1δ. So we have(17)∫01z,w1+t4zzT2-2t2z2-tw2Kpdt<2δp. Letting C=max{2,2/δp}, we can get(18)∫01z,w1+t4zzT2-2t2z2-tw2Kpdt<C for all (z,w)∈H. The proof is completed.Lemma 3.
Letf∈βp(YIV) and H be a compact subset of YIV. Then, there exists a constant C>0 such that(19)fz,w≤Cfβp∀z,w∈H.Proof.
By Lemma2, there exists a constant C>0 such that(20)fz,w=f0,0+∫01∇ftz,tw,z¯,w¯dt≤f0,0+∫01∇ftz,twz¯,w¯dt≤f0,0+fβp∫01z,w1+t4zzT2-2t2z2-tw2Kpdt≤Cfβp. The proof is completed.Lemma 4.
LetZ1,Z2∈Cm×n, W1,W2∈CN, and K>0. If Im-Z1Z1¯T>0, Im-Z2Z2¯T>0, W12K<det(Im-Z1Z1¯T), and W22K<det(Im-Z2Z2¯T), then(21)det⁡Im-Z1Z1¯T-W12Kdet⁡Im-Z2Z2¯T-W22K≤detIm-Z1Z2¯T-W1W2K2.Proof.
Setr1=W1, r2=W2. Obviously, r1,r2∈R. So we can get (21) by the process of the proof on Theorem  1 in [13].The following conclusion of Lemma5 is new generalized Hua’s inequality corresponding to YIV.Lemma 5.
Letz,u∈RIV(n), w,v∈CN, and K>0. If 1+zzT2-2z2-w2K>0 and 1+uuT2-2u2-v2K>0, then(22)1+zzT2-2z2-w2K1+uuT2-2u2-v2K≤1+zzTuuT¯-2zuT¯-wvK2.Proof.
Whenz,u∈RIV(n), there exists an orthogonal matrix Γ (see [11]) such that(23)z=z1∗,z2∗,z3∗,z4∗,0,…,0Γ,u=u1∗,u2∗,u3∗,u4∗,0,…,0Γ, where z1∗,z2∗,z3∗,z4∗,u1∗,u2∗,u3∗,u4∗∈C.
Set(24)Z=z1∗+iz2∗z1∗-iz2∗z3∗-iz4∗z3∗+iz4∗,U=u1∗+iu2∗u1∗-iu2∗u3∗-iu4∗u3∗+iu4∗. Then,(25)1+zzT2-2z2=det⁡I2-ZZT¯>0,1+uuT2-2u2=det⁡I2-UUT¯>0,1+zzTuuT¯-2zuT¯=det⁡I2-ZT¯U≠0. From Lemma 4, we can get (26)1+zzT2-2z2-w2K1+uuT2-2u2-v2K≤1+zzTuuT¯-2zuT¯-wvK2. The proof is completed.Lemma 6.
The composition operatorCϕ from βp(YIV) to βq(YIV) is compact if and only if Cϕfvβq→0 as v→∞ for every bounded sequence {fv} in βp(YIV) such that fv→0 uniformly on every compact subset of YIV.Proof.
Assume thatCϕ from βp(YIV) to βq(YIV) is compact. Let {fv} be a bounded sequence in βp(YIV) such that fv→0 uniformly on every compact subset of YIV. Suppose Cϕfvβq↛0 as v→∞. Then, there exists a subsequence {fvj} of {fv} such that infjCϕfvjβq>0. Since Cϕ is compact, there exists a subsequence of the bound subsequence {fvj}, still denoted as {fvj}, such that limj→∞f-Cϕfvjβq=0, f∈βq(YIV). For any compact set H⊂YIV, there exists a constant C depending only on H such that (27)f-Cϕfvjz,w≤Cf-Cϕfvjβq⟶0,j⟶∞. Thus, {f-Cϕfvj}→0 uniformly on compact subset H. For ∀ε>0, there exists a constant J1 such that(28)fz,w-fvjϕz,w<εwhenever j>J1 and (z,w)∈H.
Note that{fvj}→0 uniformly on compact subset H; then, for the above ε, there exists a constant J2 such that(29)fvjz,w<εwhenever j>J2 and (z,w)∈H. Let J=max{J1,J2}; from (28) and (29), we get (30)fz,w<fvjϕz,w+ε<2εwhenever j>J and (z,w)∈E:=H∩ϕ(H). So f(z,w)=0; furthermore, f≡0 on YIV. This is a contradiction and we have limj→∞Cϕfvjβq=0.
Conversely, let{fv} be a bounded sequence in βp(YIV) with fvβp≤C. Then, there exists a subsequence {fvj} of {fv} and {fvj}→f as j→+∞. Thus,(31)limj→∞⁡Cϕfvj-fβq=limj→∞⁡Cϕfvj-Cϕfβq=0. Therefore, Cϕ:βp(YIV)→βq(YIV) is compact. The proof is completed.


## 3. The Boundedness of Composition Operators

Theorem 7.
If(32)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w=O1,z2,w2⟶∂YIV,then Cϕ:βp(YIV)→βq(YIV) is bounded.
Conversely, ifCϕ:βp(YIV)→βq(YIV) is bounded, then(33)1+zzT2-2z2-w2KqGz,w1+z2z2T2-2z22-w22Kp=O1,z2,w2⟶∂YIV,where (z2,w2)=ϕ(z,w), (z,w)∈YIV, ϕ=(ϕ1,ϕ2,…,ϕn,ϕn+1,…,ϕN+n), and K>1.ϕ
′
(
z
,
w
) and ϕ′z,w2 see Lemma 1. G(z,w)=Az,wϕ′z,w(34)Az,w=2z21¯-2z21z2z2T¯,…,2z2n¯-2z2nz2z2T¯,Kw22K-2w21¯,…,Kw22K-2w2N¯.Proof.
Letf∈βp(YIV). Then,(35)∇f∘ϕz,w2=∑1≤α≤n∑1≤k≤N+n∂f∂Vkϕz,w∂ϕk∂zαz,w2+∑1≤β≤N∑1≤k≤N+n∂f∂Vkϕz,w∂ϕk∂wβz,w2≤n+N∑1≤α≤n∑1≤k≤N+n∂f∂Vkϕz,w2∂ϕk∂zαz,w2+n+N∑1≤β≤N∑1≤k≤N+n∂f∂Vkϕz,w2∂ϕk∂wβz,w2≤n+N∇fϕz,w2ϕ′z,w2. By Lemma 1 and condition (32), there exists a constant C>0 such that(36)1+zzT2-2z2-2w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w≤C for all (z,w)∈YIV. So(37)1+zzT2-2z2-2w2Kq∇Cϕfz,w≤n+N1+zzT2-2z2-2w2Kqϕ′z,w∇fϕz,w≤n+NCfβp=C′fβp. By Lemma 3, we have f(ϕ(0,0))≤C′′fβp. Thus,(38)Cϕfβq≤Cfβp. So we get that Cϕ:βp(YIV)→βq(YIV) is bounded.
For the conversion, assume thatCϕ:βp(YIV)→βq(YIV) is a bounded operator with (39)Cϕfβq≤Cfβpfor all f∈βp(YIV).
Ifp≠1/2, we use a family of test functions {f(u,v):(u,v)∈YIV} in βp(YIV) which is defined by(40)fu,vz,w=12p-111+zzTuuT¯-2zuT¯-w,vK2p-1-1. Then,(41)∂fu,v∂zα=12p-11-2p1+zzTuuT¯-2zuT¯-w,vK-2p2zαuuT¯-2uα¯=1+zzTuuT¯-2zuT¯-w,vK-2p2uα¯-2zαuuT¯,∂fu,v∂wβ=Kw,vK-1vβ¯1+zzTuuT¯-2zuT¯-w,vK2p.Thus,(42)∇fu,vz,w2=∑1≤α≤n2uαT¯-2zαuuT¯2+∑1≤β≤NKw,vK-1vβ¯21+zzTuuT¯-2zuT¯-w,vK4p.On one hand, we have(43)1+zzT2-2z2-w2Kp∇fu,vz,w≤1+zzT2-2z2-w2Kp1+uuT2-2u2-v2Kp1+zzTuuT¯-2zuT¯-wvK2p·∑1≤α≤n2uαT¯-2zαuuT¯2+∑1≤β≤NKw,vK-1vβ¯21/21+uuT2-2u2-v2Kp≤2u¯-2zuuT¯+KwK-1vK1+uuT2-2u2-v2Kp. It is easy to know that(44)2u¯-2zuuT¯≤2u+2zuuT¯≤4,KwK-1vK≤K. Obviously, f(u,v)(0,0)=0. So(45)fu,vβp=fu,v0,0+supz,w∈YIV⁡1+zzT2-2z2-w2Kp∇fz,w≤C11+uuT2-2u2-v2Kp,where C1=4+K.
On the other hand, we can get(46)1+zzT2-2z2-w2Kq∇Cϕfu,vz,w=1+zzT2-2z2-w2Kq∑1≤α≤n∑1≤k≤N+n∂fu,v∂Vkϕz,w∂ϕk∂zαz,w2+∑1≤β≤N∑1≤k≤N+n∂fu,v∂Vkϕz,w∂ϕk∂wβz,w21/2, where(47)∑1≤k≤n∂fu,v∂Vkϕz,w=∑1≤k≤n2ukT¯-2z2kuuT¯1+z2z2TuuT¯-2z2uT¯-w2,vK2p,∑n+1≤k≤N+n∂fu,v∂Vkϕz,w=∑n+1≤k≤N+nKw2,vK-1vk-n¯1+z2z2TuuT¯-2z2uT¯-w2,vK2p. Thus,(48)1+zzT2-2z2-w2Kq∇Cϕfu,vz,w=1+zzT2-2z2-w2Kq1+z2z2TuuT¯-2z2uT¯-w2,vK2p∑1≤α≤n∑1≤k≤n2ukT¯-2z2kuuT¯∂ϕk∂zαz,w+∑n+1≤k≤N+nKw2,vK-1vk-n¯∂ϕk∂zαz,w2+∑1≤β≤N∑1≤k≤n2ukT¯-2z2kuuT¯∂ϕk∂wβz,w+∑n+1≤k≤N+nKw2,vK-1vk-n¯∂ϕk∂wβz,w21/2. Set (u,v)=(z2,w2)=ϕ(z,w); by (39), we have (49)1+zzT2-2z2-w2Kq1+z2z2TuuT¯-2z2uT¯-w2,vK2p∑1≤α≤n∑1≤k≤n2ukT¯-2z2kuuT¯∂ϕk∂zαz,w+∑n+1≤k≤N+nKw2,vK-1vk-n¯∂ϕk∂zαz,w2+∑1≤β≤N∑1≤k≤n2ukT¯-2z2kuuT¯∂ϕk∂wβz,w+∑n+1≤k≤N+nKw2,vK-1vk-n¯∂ϕk∂wβz,w21/2≤CC11+z2z2T2-2z22-w22Kp. Furthermore, we have(50)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22KpGz,w≤C2. If p=1/2, set(51)fu,vz,w=ln⁡11+zzTuuT¯-2zuT¯-w,vK. Then,(52)∂fu,v∂zα=2uαT¯-2zαuuT¯1+zzTuuT¯-2zuT¯-w,vK,∂fu,v∂wβ=Kw,vK-1vβ¯1+zzTuuT¯-2zuT¯-w,vK. For the same reason, it can be proved that (33) holds. The details are omitted here. The proof is completed.


## 4. The Compactness of Composition Operators

Theorem 8.
If(53)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w=o1,z2,w2⟶∂YIV,then Cϕ:βp(YIV)→βq(YIV) is compact.
Conversely, ifCϕ:βp(YIV)→βq(YIV) is compact, then(54)1+zzT2-2z2-w2KqGz,w1+z2z2T2-2z22-w22Kp=o1,z2,w2⟶∂YIV,where (z2,w2)=ϕ(z,w), (z,w)∈YIV, ϕ=(ϕ1,ϕ2,…,ϕn,ϕn+1,…,ϕN+n), K>1, ϕ′z,w, and G(z,w); see Theorem 7.Proof.
Let{fv} be a bounded sequence in βp(YIV) with fvβp≤C and fv→0 uniformly on compact subsets of YIV. Furthermore, by Weierstrass Theorem, it is easy to show that {∇fv}→0 uniformly on compact subsets of YIV. Thus, for any ε>0, there exists δ>0 such that(55)1+zzT2-2z2-w2kq1+z2z2T2-2z22-w22kpϕ′z,w<ε whenever dist((z2,w2),∂YIV)<δ. Then,(56)1+zzT2-2z2-w2Kq∇Cϕfvz,w≤n+N1+zzT2-2z2-w2Kq∇fvϕz,wϕ′z,w≤n+Nεfvβp. Write Eδ={(z2,w2)∈YIV,dist((z2,w2),∂YIV)≥δ}, and then Eδ is a compact subset of YIV.
Let(z2,w2)∈Eδ; then, there exists a constant M∈(0,1) such that(57)11+z2z2T2-2z22-w22Kp≤1Mp<+∞. Therefore, there exists a constant C such that (58)1+zzT2-2z2-w2Kq1+z2z2T2-2z22-w22Kpϕ′z,w≤C. So(59)1+zzT2-2z2-w2Kq∇Cϕfvz,w≤Cn+N∇fvϕz,w.Since fv(ϕ(0,0))→0 as v→∞, from (56) and (59) we obtain(60)Cϕfvβq=fvϕ0,0+sup⁡1+zzT2-2z2-w2Kq∇fv∘ϕz,w⟶0,v⟶∞. By Lemma 6, we know that Cϕ:βp(YIV)→βq(YIV) is compact.
For the conversion, assume that (54) fails; then, there exists a sequence {(zj,wj)} in YIV with ϕ(zj,wj)→∂YIV as j→∞ and ε0>0 such that (61)1+zjzjT2-2zj2-wj2KqGzj,wj1+z2jz2jT2-2z2j2-w2j2Kp≥ε0for all j=1,2,….
We will construct a family of functions{fj(z,w)} satisfying the following three conditions:(I)
{fj(z,w)} is a bounded sequence in βp(YIV);(II)
{fj(z,w)} tends to zero uniformly on compact subsets of YIV;(III)
Cϕfjz,wβq↛0,j→∞.This contradicts with the compactness of Cϕ. Hence, we prove that (54) is necessary for that Cϕ:βp(YIV)→βq(YIV) being compact.
Ifp≠1/2, set(62)fjz,w=12p-11+z2jz2jT2-2z2j2-w2j2Kp1+zzTz2jz2jT¯-2zz2jT¯-w,w2jK2p-1.We have(63)1+zzT2-2z2-w2Kp∇fjz,w=1+zzT2-2z2-w2Kp1+z2jz2jT2-2z2j2-w2j2Kp·∑1≤α≤n2z2αjT¯-2zαz2jz2jT¯2+K2∑1≤β≤Nw,w2j2K-2w2βj¯21/21+zzTz2jz2jT¯-2zz2jT¯-w,w2jK2p≤1+zzT2-2z2-w2Kp1+z2jz2jT2-2z2j2-w2j2Kp1+zzTz2jz2jT¯-2zz2jT¯-ww2jK2p·2z2jT¯-zz2jz2jT¯+KwK-1w2jK. It is easy to know that(64)z2jT¯-zz2jz2jT¯≤z2jT¯+zz2jz2jT¯≤2,KwK-1w2jK≤K. Combining with Lemma 5, we can get 1+zzT2-2z2-w2Kp∇fjz,w≤C1, where C1=4+K. Thus, {fj(z,w)} is a bounded sequence in βp(YIV). This means that {fj(z,w)} satisfies condition (I).
By Lemma5,(65)fjz,w≤12p-11+z2jz2jT2-2z2j2-w2j2Kp1+zzT2-2z2-w2Kp-1/21+z2jz2jT2-2z2j2-w2j2Kp-1/2=12p-11+z2jz2jT2-2z2j2-w2j2K1/21+zzT2-2z2-w2Kp-1/2. When (z2j,w2j)→∂YIV as j→∞, then (1+z2jz2jT2-2z2j2-w2j2K)→0. Since (z,w)∈H and H is a compact subset of YIV, then inf(1+zzT2-2z2-w2K)>0. So fj(z,w)→0 as j→∞ uniformly on compact subsets of YIV. This means that {fj(z,w)} satisfies condition (II)(66)1+zjzjT2-2zj2-wj2Kq∇Cϕfjzj,wj=1+zjzjT2-2zj2-wj2Kq1+z2jz2jT2-2z2j2-w2j2Kp∑1≤α≤n∑1≤k≤n2z2kj¯-2z2kjz2jz2jT¯∂ϕk∂zαzj,wj+∑n+1≤k≤n+NKw2j2K-2w2k-nj¯∂ϕk∂zαzj,wj2+∑1≤β≤N∑1≤k≤n2z2kj¯-2z2kjz2jz2jT¯∂ϕk∂wβjzj,wj+∑n+1≤k≤n+NKw2j2K-2w2k-nj¯∂ϕk∂wβzj,wj21/2=1+zjzjT2-2zj2-wj2KqGzj,wj1+z2jz2jT2-2z2j2-w2j2Kp. Condition (61) shows that Cϕfj↛0 as j→+∞ and {fj(z,w)} satisfies condition (III).
Whenp=1/2, set(67)fjz,w=1+z2jz2jT2-2z2jT2-w2j2K1/2ln⁡1+zzTz2jz2jT¯-2zz2jT¯-w,w2jK, for all j=1,2,…. Then,(68)∂fj∂zα=1+z2jz2jT2-2z2j2-w2j2K1/22zαz2jz2jT¯-2z2αjT¯1+zzTz2jz2jT¯-2zz2jT¯-w,w2jK,∂fj∂wβ=1+z2jz2jT2-2z2j2-w2j2K1/2-Kw,w2jK-1w2βj¯1+zzTz2jz2jT¯-2zz2jT¯-w,w2jK.
It is not difficult to prove that{fj} is a bounded sequence in βp(YIV) and tends to zero uniformly on compact subsets of YIV. By (68),(69)1+zjzjT2-2zj2-wj2Kq∇Cϕfjzj,wj=1+zjzjT2-2zj2-wj2Kq1+z2jz2jT2-2z2j2-w2j2K1/2∑1≤α≤n∑1≤k≤n2z2kj¯-2z2kjz2jz2jT¯∂ϕk∂zαjzj,wj+∑n+1≤k≤n+NKw2j2K-2w2k-nj¯∂ϕk∂zαjzj,wj2+∑1≤β≤N∑1≤k≤n2z2kj¯-2z2kjz2jz2jT¯∂ϕk∂wβjzj,wj+∑n+1≤k≤n+NKw2j2K-2w2k-nj¯∂ϕk∂wβjzj,wj21/2=1+zjzjT2-2zj2-wj2KqGzj,wj1+z2jz2jT2-2z2j2-w2j2Kp. From the assumption, we get 1+zjzjT2-2zj2-wj2Kq∇Cϕfjzj,wj≥ε0. This means that {fj} satisfies condition (III). The proof is completed.



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*Source: 718257-2015-10-26.xml*