# On an Integral-Type Operator Acting between Bloch-Type Spaces on the Unit Ball

Authors: Stevo Stević; Sei-Ichiro Ueki
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Let𝔹 denote the open unit ball of ℂn. For a holomorphic self-map φ of 𝔹 and a holomorphic function g in 𝔹 with g(0)=0, we define the following integral-type operator: Iφgf(z)=∫01ℜf(φ(tz))g(tz)(dt/t), z∈𝔹. Here ℜf denotes the radial derivative of a holomorphic function f in 𝔹. We study the boundedness and compactness of the operator between Bloch-type spaces ℬω and ℬμ, where ω is a normal weight function and μ is a weight function. Also we consider the operator between the little Bloch-type spaces ℬω,0 and ℬμ,0.


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## Body

## 1. Introduction

Let𝔹 denote the open unit ball of the n-dimensional complex vector space ℂn and H(𝔹) the space of all holomorphic functions on 𝔹. For f∈H(𝔹) with the Taylor expansion f(z)=∑|γ|≥0aγzγ, let(1.1)ℜf(z)=∑|γ|≥0|γ|aγzγ
be the radial derivative of f, where γ=(γ1,…,γn) is a multi-index, |γ|=γ1+⋯+γn, and zγ=z1γ1…znγn. It is well known that(1.2)ℜf(z)=∑j=1nzj∂f∂zj(z)=〈∇f(z),z̅〉,
where ∇ is the usual gradient on ℂn.Letφ be a holomorphic self-map of 𝔹 and g∈H(𝔹) with g(0)=0. Then φ and g define an operator Iφg on H(𝔹) as follows:(1.3)Iφgf(z)=∫01ℜf(φ(tz))g(tz)dtt,f∈H(𝔹),z∈𝔹.The following important formula involvingℜ and Iφgf was proved, for example, in [1](1.4)ℜ[Iφgf](z)=ℜf(φ(z))g(z),z∈𝔹.Motivated by papers [2, 3], operators Iφg were introduced by the first author of the present paper and Zhu in [1, 4–6], where its boundedness and compactness from the α-Bloch space, the Zygmund space, the mixed-norm space, and the generalized weighted Bergman space into the Bloch-type space on the unit ball are studied. In our previous work [7], we studied the boundedness and compactness of Iφg acting between weighted-type spaces. For related operators on ℂn see, for example, [8–21] and the references therein.Letω be a strictly positive continuous function on 𝔹 (weight). If ω(z)=ω(|z|) for every z∈𝔹, we call it radial weight. A weight ω is called normal ([9, 22]) if it is radial and there are a and b, 0<a<b<∞ such that ω(r)/(1-r)a is decreasing on [0,1), ω(r)/(1-r)b is increasing on [0,1),(1.5)limr→1ω(r)(1-r)a=0,limr→1ω(r)(1-r)b=∞.A radial weightω is called typical if it is nonincreasing with respect to |z| and ω(z)→0 as |z|→1-. If ω is normal, then by the monotonicity of ω(r)/(1-r)a, for 0≤r1<r<1, we have that(1.6)ω(r)=(1-r)aω(r)(1-r)a<(1-r)aω(r1)(1-r1)a<ω(r1),
that is, ω is decreasing on [0,1). On the other hand, from the first equality in (1.5), we have that for any ε>0, there is a δ>0 such that(1.7)0<ω(r)<ε(1-r)a,(δ<r<1),
which implies limr→1-ω(r)=0. Hence every normal weight ω is also typical.For a weightω, the associated weightω̃ ([23]) is defined by(1.8)ω̃(z):=1sup{|f(z)|:f∈Hω∞,∥f∥Hω∞≤1},z∈𝔹.HereHω∞ denotes the weighted-type space consisting of all f∈H(𝔹) with (1.9)∥f∥Hω∞=supz∈𝔹ω(z)|f(z)|<∞
(see, e.g., [23, 24]). Associated weights assist us in studying of weighted-type spaces of holomorphic functions. It is known that associated weights are also continuous, 0<ω≤ω̃, and for each z∈𝔹, we can find an fz∈Hω∞,∥fz∥Hω∞≤1 such that fz(z)=1/ω̃(z). Let Hω,0∞ be the little  weighted-type  space, that is, the space of all f∈H(𝔹) such that ω(z)|f(z)|→0 as |z|→1-. If ω is typical, then the unit ball BHω∞ is the closure of BHω,0∞ for the compact open topology. Hence we have(1.10)ω̃(z)=1sup{|f(z)|:f∈Hω,0∞,∥f∥Hω∞≤1}
and so for each z∈𝔹, we can choose an fz∈BHω,0∞ such that fz(z)=1/ω̃(z). A weight ω is called essential if it satisfies that ω̃≤Cω for some positive constant C. By the arguments in [25], we see that a normal weight function is also essential. For some examples of essential weights, see, for example, [25]. Related results can also be found in [22, 26].TheBloch-type spaceℬω is the space of all holomorphic functions f on 𝔹 such that(1.11)bω(f)=supz∈𝔹ω(z)|ℜf(z)|<∞,
where ω is a weight (see, e.g., [20]). The little Bloch-type spaceℬω,0 consists of all f∈H(𝔹) such that (1.12)lim|z|→1-ω(z)|ℜf(z)|=0.
Both spaces ℬω and ℬω,0 are Banach spaces with the norm (1.13)∥f∥ℬω=|f(0)|+bω(f),
and ℬω,0 is a closed subspace of ℬω. When ω(r)=1-r2, the space ℬω is a classical Bloch space.The purpose of this paper is to characterize the boundedness and compactness of the operatorsIφg:ℬω→ℬμ and Iφg:ℬω,0→ℬμ,0.Throughout this paper, we assume thatφ is a holomorphic self-map of 𝔹 and g∈H(𝔹) with g(0)=0. Furthermore, some constants are denoted by C; they are positive and may differ from one occurrence to the other. The notation a⪯b means that there exists a positive constant C such that a≤Cb. Moreover, if both a⪯b and b⪯a hold, then one says that a≍b.


## 2. Auxiliary Results

Here we formulate and prove some auxiliary results which are used in the proofs of the main ones.The following lemma was proved in [20, Theorem  2.1].Lemma 2.1.
Letω be a normal weight function and f∈H(𝔹). Then f∈ℬω if and only if supz∈𝔹ω(z)|∇f(z)|<∞ and it holds that
(2.1)∥f∥ℬω≍|f(0)|+supz∈𝔹ω(z)|∇f(z)|.
Moreover, f∈ℬω,0 if and only if  lim|z|→1-ω(z)|∇f(z)|=0.As an application of Lemma2.1, we have the following result.Lemma 2.2.
Letω be a normal weight function and f∈ℬω. Then f∈ℬω,0 if and only if it holds that  limr→1-∥fr-f∥ℬω=0, where fr(z)=f(rz).Proof.
Take anf∈ℬω,0.  For a fixed ε>0, by Lemma 2.1, we can choose a δ0∈(0,1) such that
(2.2)ω(z)|∇f(z)|<ε2
for any z∈𝔹∖δ02𝔹̅. Since (∂fr/∂zj)(z)=r(∂f/∂zj)(rz) for j∈{1,…,n}, r∈(0,1), and z∈𝔹, we have
(2.3)∥fr-f∥ℬω≍supz∈𝔹ω(z)|r∇f(rz)-∇f(z)|≤supz∈𝔹∖δ0𝔹̅ω(z)|r∇f(rz)-∇f(z)|+supz∈δ0𝔹̅ω(z)|r∇f(rz)-∇f(z)|.
Since
(2.4)max|z|≤δ0|r∇f(rz)-∇f(z)|→0,asr→1-,
we see that the second term in (2.3) converges to 0 as r→1-.
Ifr∈(δ0,1) and z∈𝔹∖δ0𝔹̅, then by (2.2) we have
(2.5)ω(rz)|∇f(rz)|<ε2.
By (1.6) we have that ω(z)≤ω(rz) for r,|z|∈[0,1).
Hence we have(2.6)supz∈𝔹∖δ0𝔹̅ω(z)|r∇f(rz)-∇f(z)|≤supz∈𝔹∖δ0𝔹̅ω(rz)|∇f(rz)|+supz∈𝔹∖δ0𝔹̅ω(z)|∇f(z)|<ε,
for all r∈(δ0,1). This proves that limr→1-∥fr-f∥ℬω=0 whenever f∈ℬω,0.
Conversely, the normality ofω implies that for any ε>0 we have
(2.7)ω(z)|∇f(rz)|≤ε(1-|z|)asup|w|≤r|∇f(w)|→0,as|z|→1-,
so that fr∈ℬω,0 for any r∈(0,1). On the other hand, by the assumption limr→1-∥fr-f∥ℬω=0, we have that for every ε>0 there is an r1∈(0,1) such that
(2.8)∥fr-f∥ℬω<ε,
for r∈(r1,1).
By letting|z|→1- in the following inequality, which easily follows from Lemma 2.1:
(2.9)ω(z)|∇f(z)|≼ω(z)|∇f(rz)|+∥fr-f∥ℬω,
then using (2.7) and (2.8), we get f∈ℬω,0, as claimed.Corollary 2.3.
Letω be a normal weight function. Then the set of all holomorphic polynomials is dense in ℬω,0.Proof.
For the homogeneous expansionf=∑k=0∞Fk of an f∈ℬω,0, we set Pj=∑k=0jFk for each j∈ℕ. Since Pj→f uniformly on compact subsets of 𝔹 as j→∞, we see that ℜ[Prj]→ℜ[fr] uniformly on 𝔹 for any r∈(0,1). Moreover, we have
(2.10)∥Prj-f∥ℬω≤∥Prj-fr∥ℬω+∥fr-f∥ℬω≤supz∈r𝔹̅ω(z)supz∈𝔹|ℜ[Prj](z)-ℜ[fr](z)|+∥fr-f∥ℬω.
Combining this with Lemma 2.2, we get the desired result.The following lemma can be found in [1, Lemma  3]. Its proof is similar to the proof of the corresponding one-dimensional result in [27], for the case of the little Bloch space ℬ(1-r),0. Hence we omit the proof.Lemma 2.4.
A closed subsetK in ℬω,0 is compact if and only if it is bounded and
(2.11)lim|z|→1-supf∈Kω(z)|ℜf(z)|=0.The following lemma is very useful for estimating the norm of the Bloch-type space.Lemma 2.5.
Assume thatm is a positive integer and ω is normal. Then for every f∈H(𝔹),(2.12)supz∈𝔹ω(z)|f(z)|≍|f(0)|+supz∈𝔹(1-|z|)mω(z)|ℜmf(z)|.Proof.
For the details of the proof, we can refer [9] or [28].


## 3. The Boundedness of OperatorIφg

In this section we consider the boundedness of the operatorIφg:ℬω→ℬμ or Iφg:ℬω,0→ℬμ,0.Theorem 3.1.
Letω be a normal weight function and μ a weight function. Then the following conditions are equivalent:  (a)
Iφg:ℬω→ℬμ is bounded;(b)
Iφg:ℬω,0→ℬμ is bounded;(c)
φ and  g satisfy(3.1)supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z))<∞.
Moreover, if Iφg:ℬω→ℬμ is bounded, then
(3.2)∥Iφg∥ℬω→ℬμ≍supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z)).Proof.
The implication(a)⇒(b) is clear, so we only prove (b)⇒(c) and (c)⇒(a).
(b)⇒(c): assume that Iφg:ℬω,0→ℬμ is bounded and fix z∈𝔹. We may assume that φ(z)≠0. For w:=φ(z), there exists hw∈Hω,0∞ such that ∥hw∥Hω∞≤1 and hw(w)=1/ω̃(w). We define the function fw as follows:
(3.3)fw(v)=∫01hw(tv)〈tv,w〉|w|dtt,v∈𝔹.
Since ℜfw(v)=hw(v)(〈v,w〉/|w|), we see that fw∈ℬω,0 and ∥fw∥ℬω≤1. Hence, by (1.4), we have
(3.4)∥Iφg∥ℬω→ℬμ≥∥Iφgfw∥ℬμ≥μ(z)|g(z)||ℜfw(φ(z))|=μ(z)|g(z)||φ(z)|ω̃(φ(z)),
and so condition (3.1) is true.
(c)⇒(a): we assume (3.1) and take an f∈ℬω. Since ω is an essential weight (due to its normality), (1.4) gives
(3.5)μ(z)|ℜ[Iφgf](z)|=μ(z)|g(z)||ℜf(φ(z))|≤μ(z)|g(z)||φ(z)|ω̃(φ(z))|∇f(φ(z))|ω̃(φ(z))⪯μ(z)|g(z)||φ(z)|ω̃(φ(z))supw∈𝔹ω(w)|∇f(w)|,
for any z∈𝔹. By Lemma 2.1, we have supw∈𝔹ω(w)|∇f(w)|⪯∥f∥ℬω, and so we obtain
(3.6)∥Iφgf∥ℬμ⪯supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z))∥f∥Bω.
This implies that Iφg:ℬω→ℬμ is bounded. The relation (3.2) follows from (3.4) and (3.6). This completes the proof.Theorem 3.2.
Letω be a normal weight function and μ a weight function. Then the following conditions are equivalent:  (a)
Iφg : ℬω,0→ℬμ,0 is bounded;(b)
φ and g satisfy(3.7)lim|z|→1-μ(z)|g(z)||φ(z)|=0,supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z))<∞.Proof.
(a)⇒(b): as in the proof of Theorem 3.1, for fixed z∈𝔹 and w=φ(z), we see that φ and g satisfy the condition
(3.8)supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z))<∞.
On the other hand, since the normality ofω implies that the function πj(z):=zj(1≤j≤n) belongs to ℬω,0, we obtain that μ(z)|g(z)||φj(z)|→0 for each j, and so μ(z)|g(z)||φ(z)|→0 as |z|→1-.
(b)⇒(a): the assumption lim|z|→1-μ(z)|g(z)||φ(z)|=0 shows that Iφgp∈ℬμ,0 for any polynomial p. For each f∈ℬω,0, by Corollary 2.3, we can choose a sequence of polynomials {pj}j∈ℕ such that ∥f-pj∥ℬω→0 as j→∞. Furthermore, the assumption
(3.9)supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z))<∞
shows that Iφg: ℬω,0→ℬμ is bounded by Theorem 3.1. Thus we obtain
(3.10)0≤∥Iφgf-Iφgpj∥ℬμ≤∥Iφg∥ℬω→ℬμ∥f-pj∥ℬω→0(asj→∞).
Since Iφgf∈ℬμ,{Iφgpj}j∈ℕ⊂ℬμ,0, and ℬμ,0 is closed in ℬμ, we have Iφgf∈ℬμ,0 for any f∈ℬω,0. Hence Iφg(ℬω,0)⊆ℬμ,0 which means that Iφg:ℬω,0→ℬμ,0 is bounded. The proof is accomplished.The following corollary is an immediate consequence of Theorems3.1 and 3.2.Corollary 3.3.
Letω be a normal weight function and μ a weight function. Then Iφg:ℬω,0→ℬμ,0 is bounded if and only if lim|z|→1-μ(z)|g(z)||φ(z)|=0 and Iφg:ℬω→ℬμ is bounded.


## 4. The Compactness of OperatorIφg

In this section we characterize the compactness ofIφg:ℬω→ℬμ or Iφg:ℬω,0→ℬμ,0. To do this, we need the following standard lemma (see, e.g., [13, Lemma  3]).Lemma 4.1.
Letω and μ be weight functions. Suppose that the operator Iφg:ℬω(orℬω,0)→ℬμ is bounded. Then Iφg:ℬω(orℬω,0)→ℬμ is compact if and only if for every bounded sequence {fj}j∈ℕ in ℬω(orℬω,0) which converges to 0 uniformly on compact subsets of 𝔹, ∥Iφgfj∥ℬμ→0 as j→∞.Theorem 4.2.
Letω and μ be weight functions. Suppose that φ is a holomorphic self-map of 𝔹 such that ∥φ∥∞<1 and the operator Iφg:ℬω→ℬμ is bounded. Then Iφg:ℬω→ℬμ is compact. Here ∥φ∥∞ denotes the supremum supz∈𝔹|φ(z)|.Proof.
Since∥φ∥∞<1, we see that |φ(z)|≤r for some r∈(0,1) and any z∈𝔹. From the proof of Theorem 3.1, we see that the boundedness of Iφg:ℬω→ℬμ implies
(4.1)M:=supz∈𝔹μ(z)|g(z)||φ(z)|ω̃(φ(z))<∞.
Thus we obtain that
(4.2)supz∈𝔹μ(z)|g(z)||φ(z)|≤Msupw∈r𝔹̅ω̃(w)<∞.
Take a bounded sequence {fj}j∈ℕ in ℬω such that fj→0 uniformly on compact subsets of 𝔹 as j→∞. By (1.4), we have
(4.3)∥Iφgfj∥Bμ=supz∈𝔹μ(z)|g(z)||ℜfj(φ(z))|≤supz∈𝔹μ(z)|g(z)||φ(z)||∇fj(φ(z))|≤Msupw∈r𝔹̅ω̃(w)supw∈r𝔹̅|∇fj(w)|.
Since ∂fj/∂zk(1≤k≤n) also converges to 0 uniformly on r𝔹̅ as j→∞, (4.2) and (4.3) show that ∥Iφgfj∥ℬμ→0 as j→∞. From Lemma 4.1, it follows that Iφg:ℬω→ℬμ is compact, and so we get the assertions.Lemma 4.3.
Suppose thatω is a weight function. Then there exists a sequence {fk}k∈ℕ in the closed unit ball of ℬω such that fk→0 uniformly on compact subsets of 𝔹 as k→∞.Proof.
Let{wk}k∈ℕ⊂𝔹 with |wk|→1- as k→∞. For each wk, there exists hk:=hwk∈Hω∞ such that ∥hk∥Hω∞≤1 and hk(wk)=1/ω̃(wk). We define fk as follows:
(4.4)fk(z)=∫01hk(tz){〈tz,wk〉|wk|}1/(1-|wk|)dtt,z∈𝔹.
Since fk(0)=0 and |ℜfk(z)|≤|hk(z)|, we have {fk}k∈ℕ⊂ℬω and ∥fk∥ℬω≤1 for each k∈ℕ. For any compact subset 𝒦 of 𝔹, we can choose an r∈(0,1) such that 𝒦⊂r𝔹̅. Hence we obtain that for any z∈𝒦(4.5)|fk(z)|≤∫01|hk(tz)|t|wk|/(1-|wk|)dt≤∥hk∥Hω∞∫011ω(tz)t|wk|/(1-|wk|)dt≤maxw∈r𝔹̅1ω(w)(1-|wk|).
From the above inequality, it follows that fk converges to 0 uniformly on compact subsets of 𝔹 as k→∞. This completes the proof.Remark 4.4.
If we assume thatω is typical in Lemma 4.3, then we can choose hk∈Hω,0∞. In this case, hence, we see that fk belongs to ℬω,0 for each k∈ℕ.Theorem 4.5.
Letω be a normal weight function and μ a weight function. Suppose that the operator Iφg:ℬω→ℬμ is bounded and ∥φ∥∞=1. Then the following conditions are equivalent:  (a)
Iφg:ℬω→ℬμ is compact;(b)
Iφg:ℬω,0→ℬμ is compact;(c)
φ and g satisfy(4.6)lim|φ(z)|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z))=0.Proof.
(a)⇒(b): this implication is obvious.
(b)⇒(c): take a sequence {zk}k∈ℕ in 𝔹 with |φ(zk)|→1- as k→∞ and put wk=φ(zk) for each k. Then, by Remark 4.4 after Lemma 4.3, there exists a sequence {fk}k∈ℕ in ℬω,0 such that supk∈ℕ∥fk∥ℬω≤1 and fk→0 uniformly on compact subsets of 𝔹 as k→∞. By Lemma 4.1, the compactness of Iφg:ℬω,0→ℬμ implies that ∥Iφgfk∥ℬμ→0 as k→∞.
On the other hand, (1.4) gives ℜ[Iφgfk](z)=ℜfk(φ(z))g(z), and so we have
(4.7)∥Iφgfk∥ℬμ≥μ(zk)|ℜfk(φ(zk))||g(zk)|≥μ(zk)|ℜfk(φ(zk))||g(zk)||φ(zk)|.
From the construction (4.4) of fk, we obtain
(4.8)ℜfk(φ(zk))=|φ(zk)|1/(1-|φ(zk)|)ω̃(φ(zk)),
for each k∈ℕ. Combining this with (4.7), we have
(4.9)∥Iφgfk∥ℬμ≥μ(zk)|g(zk)||φ(zk)|ω̃(φ(zk))|φ(zk)|1/(1-|φ(zk)|).
Letting k→∞, we have
(4.10)limk→∞μ(zk)|g(zk)||φ(zk)|ω̃(φ(zk))=0,
for any sequence {zk}k∈ℕ with |φ(zk)|→1-. This proves that (4.6) is true.
(c)⇒(a): we will prove the following estimate:
(4.11)∥Iφg∥e⪯lim sup|φ(z)|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z)).
Here ∥Iφg∥e denotes the essential norm of Iφg:ℬω→ℬμ, namely,
(4.12)∥Iφg∥e=inf{∥Iφg+𝒦∥ℬω→ℬμ∣𝒦:ℬω→ℬμiscompact}.
Now we take a sequence {rl}l∈ℕ⊂(0,1) which increasingly converges to 1 and put
(4.13)Irlφgf(z)=∫01ℜf(rlφ(tz))g(tz)dtt.
Since ∥rlφ∥∞≤rl<1, Theorem 4.2 implies that Irlφg:ℬω→ℬμ is compact for each l∈ℕ. For any f∈ℬω with ∥f∥ℬω≤1, from (1.4) it follows that
(4.14)∥Iφgf-Irlφgf∥ℬω=supz∈𝔹μ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|≤supR<|φ(z)|<1μ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|+sup|φ(z)|≤Rμ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|,
for some fixed R∈(0,1). The essentiality of ω and Lemma 2.1 give
(4.15)μ(z)|g(z)||ℜf(φ(z))|≤μ(z)|g(z)||φ(z)|ω̃(φ(z))ω̃(φ(z))|∇f(φ(z))|⪯μ(z)|g(z)||φ(z)|ω̃(φ(z))supw∈𝔹ω(w)|∇f(w)|⪯μ(z)|g(z)||φ(z)|ω̃(φ(z)).
Similarly, we also have
(4.16)μ(z)|g(z)||ℜf(rlφ(z))|⪯μ(z)|g(z)||φ(z)|ω̃(rlφ(z)),
for each l∈ℕ. The normality of ω implies that
(4.17)ω(rlφ(z))(1-|rlφ(z)|)a≥ω(φ(z))(1-|φ(z)|)a,
for each l∈ℕ and some a>0, and so by the essentiality,
(4.18)ω̃(rlφ(z))(1-|rlφ(z)|)a≽ω̃(φ(z))(1-|φ(z)|)a.
Thus (4.16) and (4.18) give
(4.19)μ(z)|g(z)||ℜf(rlφ(z))|⪯μ(z)|g(z)||φ(z)|ω̃(φ(z)),
for each l∈ℕ. By (4.15) and (4.19), we obtain
(4.20)supR<|φ(z)|<1μ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|⪯supR<|φ(z)|<1μ(z)|g(z)||φ(z)|ω̃(φ(z)).
When|φ(z)|≤R, by using the mean value theorem, we have
(4.21)μ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|≤(1-rl)μ(z)|g(z)||φ(z)|sup|w|≤R|∇[ℜf](w)|≤1-rl1-Rmaxw∈R𝔹̅1ω(w)supz∈𝔹μ(z)|g(z)||φ(z)|×supw∈𝔹ω(w)(1-|w|)|∇[ℜf](w)|.
Since ω(w)(1-|w|) is also normal, by Lemmas 2.1 and 2.5, we have
(4.22)supw∈𝔹ω(w)(1-|w|)|∇[ℜf](w)|≍supw∈𝔹ω(w)(1-|w|)|ℜ2f(w)|≍supw∈𝔹ω(w)|ℜf(w)|.
Hence we obtain
(4.23)sup|φ(z)|≤Rμ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|⪯1-rl1-Rmaxw∈r𝔹̅1ω(w)supz∈𝔹μ(z)|g(z)||φ(z)|.
Since the boundedness of Iφg:ℬω→ℬμ implies supz∈𝔹μ(z)|g(z)||φ(z)|<∞, letting l→∞ in the above inequality, we have
(4.24)sup∥f∥ℬω≤1sup|φ(z)|≤Rμ(z)|g(z)||ℜf(φ(z))-ℜf(rlφ(z))|→0.
By using (4.14), (4.20), and (4.24) and letting R→1-, we obtain the desired estimate
(4.25)∥Iφg∥e⪯lim sup|φ(z)|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z)).
So if condition (4.6) is true, then ∥Iφg∥e=0, which means that Iφg:ℬω→ℬμ is compact. Our proof is accomplished.Theorem 4.6.
Letω be a normal weight function and μ a weight function. Suppose that the operator Iφg:ℬω,0→ℬμ,0 is bounded. Then Iφg:ℬω,0→ℬμ,0 is compact if and only if
(4.26)lim|z|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z))=0.Proof.
Suppose that (4.26) holds. For any f∈ℬω,0, by Lemma 2.1 and (1.4), we have
(4.27)μ(z)|ℜ[Iφgf](z)|=μ(z)|ℜf(φ(z))||g(z)|≤μ(z)|g(z)||φ(z)||∇f(φ(z))|⪯μ(z)|g(z)||φ(z)|ω̃(φ(z))ω(φ(z))|∇f(φ(z))|⪯μ(z)|g(z)||φ(z)|ω̃(φ(z))∥f∥ℬω.
Combining this with (4.26), we obtain
(4.28)lim|z|→1-sup∥f∥ℬω≤1μ(z)|ℜ[Iφgf](z)|=0.
Hence it follows from Lemma 2.4 that Iφg:ℬω,0→ℬμ,0 is compact.
Conversely, we assume thatIφg:ℬω,0→ℬμ,0 is compact. By Theorem 3.2, we see that(4.29)lim|z|→1-μ(z)|g(z)||φ(z)|=0.
Thus this implies (4.26) if ∥φ∥∞<1.
Now assume∥φ∥∞=1. We claim that
(4.30)lim sup|z|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z))=lim sup|φ(z)|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z)).
Further assume that {zk}k∈ℕ is a sequence in 𝔹 such that
(4.31)lim sup|z|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z))=limk→∞μ(zk)|g(zk)||φ(zk)|ω̃(φ(zk)).
If supk∈ℕ|φ(zk)|<1, then from this and (4.29) we have that both limits in (4.30) are equal to zero. If supk∈ℕ|φ(zk)|=1, then there is a subsequence {φ(zkl)}l∈ℕ such that |φ(zkl)|→1- as l→∞. Hence we have
(4.32)lim sup|z|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z))=liml→∞μ(zkl)|g(zkl)||φ(zkl)|ω̃(φ(zkl))≤lim sup|φ(z)|→1-μ(z)|g(z)||φ(z)|ω̃(φ(z)),
and so (4.30) holds.
SinceIφg:ℬω,0→ℬμ is also compact, by Theorem 4.5, we see that the second limit in (4.30) is equal to zero, so that (4.26) holds. This completes the proof.



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*Source: 214762-2010-06-27.xml*