# Products of Composition and Differentiation Operators from𝒬K(p,q) Spaces to Bloch-Type Spaces

Authors: Weifeng Yang
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We study the boundedness and compactness of the products of composition and differentiation operators from𝒬K(p,q) spaces to Bloch-type spaces and little Bloch-type spaces.


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## Body

## 1. Introduction

Let𝔻 be the open unit disk in the complex plane and H(𝔻) the class of all analytic functions on 𝔻. The α-Bloch space ℬα (α>0) on 𝔻 is the space of all analytic functions f on 𝔻 such that(1.1)∥f∥ℬα=|f(0)|+supz∈𝔻(1-|z|2)α|f′(z)|<∞.
Under the above norm,ℬα is a Banach space. When α=1, ℬ1=ℬ is the well-known Bloch space. Let  ℬ0α  denote the subspace of ℬα consisting of those f∈ℬα for which(1-|z|2)α|f′(z)|→0 as |z|→1. This space is called the little α-Bloch space.Assume thatμ is a positive continuous function on [0,1), and there exist positive numbers s and t,0<s<t, and δ∈[0,1) such that(1.2)μ(r)(1-r)sisdecreasingon[δ,1),limr→1μ(r)(1-r)s=0,μ(r)(1-r)tis  increasing  on[δ,1),limr→1μ(r)(1-r)t=∞,
then μ is called a normal function ([1]).Anf∈H(𝔻) is said to belong to the Bloch-type space ℬμ=ℬμ(𝔻), if (see, e.g, [2–5])(1.3)∥f∥ℬμ=|f(0)|+supz∈𝔻μ(|z|)|f′(z)|<∞.ℬμ is a Banach space with the norm ∥·∥ℬμ (see [3]). When  μ(r)=(1-r2)α, the induced space ℬμ becomes the α-Bloch space ℬα.Throughout this paper, we assume thatK:[0,∞)→[0,∞) is a nondecreasing continuous function. Assume that p>0, q>-2. A function f∈H(𝔻) is said to belong to 𝒬K(p,q) (see [6]) if(1.4)∥f∥=(supα∈𝔻∫𝔻|f′(z)|p(1-|z|2)qK(g(z,a))dA(z))1/p<∞,
where  dA denotes the normalized Lebesgue area measure in 𝔻 (i.e., A(𝔻)=1) and g(z,a) is the Green function with logarithmic singularity at a, that is, g(z,a)=log(1/|φa(z)|) (φa is a conformal automorphism defined by φa(z)=(a-z)/(1-a̅z) for a∈𝔻). If K(x)=xs, s≥0, the space 𝒬K(p,q) equals to F(p,q,s), which is introduced by Zhao in [7]. Moreover (see [7]) we have that, F(p,q,s)=ℬ(q+2)/p, and F0(p,q,s)=ℬ0(q+2)/p for s>1, F(p,q,s)⊆ℬ(q+2)/p, and F0(p,q,s)⊆ℬ0(q+2)/p for 0<s≤1, F(2,0,s)=Qs, and F0(2,0,s)=Qs,0, F(2,1,0)=H2, F(2,0,1)=BMOA, and F0(2,0,1)=VMOA. When p≥1, 𝒬K(p,q) is a Banach space under the norm(1.5)∥f∥𝒬K(p,q)=|f(0)|+∥f∥.
From [6], we know that 𝒬K(p,q)⊆ℬ(q+2)/p, 𝒬K(p,q)=ℬ(q+2)/p if and only if(1.6)∫01(1-r2)-2K(-logr)rdr<∞.
Moreover, ∥f∥ℬ(q+2)/p≤C∥f∥𝒬K(p,q) (see in [6, Theorem  2.1] or [8, Lemma  2.1]). Throughout the paper we assume that (see [6])(1.7)∫01(1-r2)qK(-logr)rdr<∞,
since otherwise 𝒬K(p,q) consists only of constant functions.Letφ denote a nonconstant analytic self-map of 𝔻. Associated with φ is the composition operator Cφ defined by Cφf=f∘φ for f∈H(𝔻). The problem of characterizing the boundedness and compactness of composition operators on many Banach spaces of analytic functions has attracted lots of attention recently, see, for example, [9, 10] and the reference therein.LetD be the differentiation operator on H(𝔻), that is, Df(z)=f′(z). For f∈H(𝔻), the products of composition and differentiation operators DCφ and CφD are defined, respectively, by(1.8)DCφ(f)=(f∘φ)′=f′(φ)φ′,CφD(f)=f′(φ),f∈H(𝔻).The boundedness and compactness ofDCφ on the Hardy space were investigated by Hibschweiler and Portnoy in [11] and by Ohno in [12]. The case of the Bergman spaces was studied in [11], while the case of the Hilbert-Bergman space was studied by Stević in [13]. In [14], Li and Stević studied the boundedness and compactness of the operator DCφ on α-Bloch spaces, while in [15] they studied these operators between H∞ and α-Bloch spaces. The boundedness and compactness of the operator DCφ from mixed-norm spaces to α-Bloch spaces was studied by Li and Stević in [16]. Norm and essential norm of the operator DCφ from α-Bloch spaces to weighted-type spaces were studied by Stević in [17]. Some related operators can be also found in [18–21]. For some other papers on products of linear operators on spaces of holomorphic functions, mostly integral-type and composition operators, see, for example, the following papers by Li and Stević: [5, 22–30].Motivated basically by papers [14, 15], in this paper, we study the operators DCφ and CφD from 𝒬K(p,q) space to ℬμ and ℬμ,0 spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given.Throughout this paper, constants are denoted byC, they are positive and may differ from one occurrence to the other. The notation A≍B means that there is a positive constant C such that B/C≤A≤CB.


## 2. Main Results and Proofs

In this section we give our main results and proofs. For this purpose, we need some auxiliary results. The following lemma can be proved in a standard way (see, e.g, in [9, Proposition  3.11]). A detailed proof, can be found, for example, in [31].Lemma 2.1.
Letφ be an analytic self-map of  𝔻. Suppose that  μ is normal,  p>0,  q>-2. Then  DCφ(orCφD):𝒬K(p,q)→ℬμ is compact if and only if  DCφ(orCφD):𝒬K(p,q)→ℬμ is bounded and for any bounded sequence  (fn)n∈ℕ in  𝒬K(p,q) which converges to zero uniformly on compact subsets of  𝔻, one has  ∥DCφfn∥ℬμ→0(or∥CφDfn∥ℬμ→0) as  n→∞.  The following lemma can be proved similarly as [32], one omits the details (see also [2, 4]).Lemma 2.2.
A closed set K inℬμ,0 is compact if and only if it is bounded and satisfies
(2.1)lim|z|→1-supf∈Kμ(|z|)|f′(z)|=0.Now one is in a position to state and prove the main results of this paper.Theorem 2.3.
Letφ be an analytic self-map of 𝔻. Suppose that μ is normal, p>0, q>-2, and K is a nonnegative nondecreasing function on [0,∞) such that
(2.2)∫01K(-logr)(1-r)min{-1,q}(log11-r)χ-1(q)rdr<∞,
where χO(x) denote the characteristic function of the set O. Then DCφ:𝒬K(p,q)→ℬμ is bounded if and only if
(2.3)supz∈𝔻μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p<∞,supz∈𝔻μ(|z|)|φ′′(z)|(1-|φ(z)|2)(2+q)/p<∞.Proof.
Suppose that the conditions in (2.3) hold. Then for any z∈𝔻 and f∈𝒬K(p,q),
(2.4)μ(|z|)|(DCφf)′(z)|=μ(|z|)|(f′(φ)φ′)′(z)|≤μ(|z|)|φ′(z)|2|f′′(φ(z))|+μ(|z|)|φ′′(z)||f′(φ(z))|≤μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p∥f∥ℬ(q+2)/p+μ(|z|)|φ′′(z)|(1-|φ(z)|2)(2+q)/p∥f∥ℬ(q+2)/p≤Cμ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p∥f∥𝒬K(p,q)+Cμ(|z|)|φ′′(z)|(1-|φ(z)|2)(2+q)/p∥f∥𝒬K(p,q),
where we have used the fact that ∥f∥ℬ(q+2)/p≤∥f∥𝒬K(p,q), as well as the following well-known characterization for α-Bloch functions (see, e.g., [33])
(2.5)supz∈𝔻(1-|z|2)β|φ′(z)|≍|φ′(0)|+supz∈𝔻(1-|z|2)(1+β)|φ″(z)|.
Taking the supremum in (2.4) for z∈𝔻, then employing (2.3) we obtain that DCφ:𝒬K(p,q)→ℬμ is bounded.
Conversely, suppose thatDCφ:𝒬K(p,q)→ℬμ is bounded, that is, there exists a constant C such that ∥DCφf∥ℬμ≤C∥f∥𝒬K(p,q) for all f∈𝒬K(p,q). Taking the functions f(z)≡z, and f(z)≡z2, which belong to 𝒬K(p,q), we get
(2.6)supz∈𝔻μ(|z|)|φ″(z)|<∞,(2.7)supz∈𝔻μ(|z|)|(φ′(z))2+φ″(z)φ(z)|<∞.
From (2.6), (2.7), and the boundedness of the function φ(z), it follows that
(2.8)supz∈𝔻μ(|z|)|φ′(z)|2<∞.
For w∈𝔻, let
(2.9)fw(z)=1-|w|2(1-zw¯)(q+2)/p.
By some direct calculation we have that
(2.10)fw′(w)=q+2pw¯(1-|w|2)((q+2)/p),fw″(w)=(q+2p)(q+2p+1)w¯2(1-|w|2)(q+2)/p+1.
From [8], we know that fw∈𝒬K(p,q), for each w∈𝔻, moreover there is a positive constant C such that supw∈𝔻∥fw∥𝒬K(p,q)≤C. Hence, we have
(2.11)C∥DCφ∥𝒬K(p,q)→ℬμ≥∥DCφfφ(λ)∥ℬμ≥-q+2pq+2+ppμ(|λ|)|φ′(λ)|2|φ(λ)|2(1-|φ(λ)|2)(2+q+p)/p+q+2pμ(|λ|)|φ″(λ)||φ(λ)|(1-|φ(λ)|2)(2+q)/p,
for λ∈𝔻. Therefore, we obtain
(2.12)μ(|λ|)|φ″(λ)||φ(λ)|(1-|φ(λ)|2)(2+q)/p≤C∥DCφ∥𝒬K(p,q)→ℬμ+q+2+ppμ(|λ|)|φ′(λ)|2|φ(λ)|2(1-|φ(λ)|2)(2+q+p)/p.
Next, for w∈𝔻, let
(2.13)gw(z)=(1-|w|2)2(1-zw¯)(q+2)/p+1-(q+2)/p+1(q+2)/p1-|w|2(1-zw¯)(q+2)/p.
Then from [8], we see that gw∈𝒬K(p,q) and supw∈𝔻∥gw∥𝒬K(p,q)<∞. Since
(2.14)gφ(λ)′(φ(λ))=0,|gφ(λ)′′(φ(λ))|=q+2+pp|φ(λ)|2(1-|φ(λ)|2)(2+q+p)/p,
we have
(2.15)∞>C∥DCφ∥𝒬K(p,q)→ℬμ≥∥DCφgφ(λ)∥ℬμ≥q+2+ppμ(|λ|)|φ′(λ)|2|φ(λ)|2(1-|φ(λ)|2)(2+q+p)/p.
Thus(2.16)sup|φ(λ)|>1/2μ(|λ|)|φ′(λ)|2(1-|φ(λ)|2)(2+q+p)/p≤sup|φ(λ)|>1/24μ(|λ|)|φ′(λ)|2|φ(λ)|2(1-|φ(λ)|2)(2+q+p)/p≤C∥DCφ∥𝒬K(p,q)→ℬμ<∞.
Inequality (2.8) gives
(2.17)sup|φ(λ)|≤1/2μ(|λ|)|φ′(λ)|2(1-|φ(λ)|2)(2+q+p)/p≤4(2+q+p)/p3(2+q+p)/psup|φ(λ)|≤1/2μ(|λ|)|φ′(λ)|2<∞.
Therefore, the first inequality in (2.3) follows from (2.16) and (2.17). From (2.12) and (2.15), we obtain
(2.18)supλ∈𝔻μ(|λ|)|φ″(λ)||φ(λ)|(1-|φ(λ)|2)(2+q)/p<∞.
Equations (2.6) and (2.18) imply
(2.19)sup|φ(λ)|>1/2μ(|λ|)|φ″(λ)|(1-|φ(λ)|2)(2+q)/p≤2sup|φ(λ)|>1/2μ(|λ|)|φ″(λ)||φ(λ)|(1-|φ(λ)|2)(2+q)/p<∞,(2.20)sup|φ(λ)|≤1/2μ(|λ|)|φ″(λ)|(1-|φ(λ)|2)(2+q)/p≤4(2+q)/p3(2+q)/psup|φ(λ)|≤1/2μ(|λ|)|φ′′(λ)|<∞.
Inequality (2.19) together with (2.20) implies the second inequality of (2.3). This completes the proof of Theorem 2.3.Theorem 2.4.
Letφ be an analytic self-map of 𝔻. Suppose that μ is normal, p>0, q>-2 and K is a nonnegative nondecreasing function on [0,∞) such that (2.2) holds. Then DCφ:𝒬K(p,q)→ℬμ is compact if and only if DCφ:𝒬K(p,q)→ℬμ is bounded,
(2.21)lim|φ(z)|→1μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p=0,lim|φ(z)|→1μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p=0.Proof.
Suppose thatDCφ:𝒬K(p,q)→ℬμ is bounded and (2.21) holds. Let (fk)k∈ℕ be a sequence in 𝒬K(p,q) such that supk∈ℕ∥fk∥𝒬K(p,q)<∞ and fk converges to 0 uniformly on compact subsets of 𝔻 as k→∞. By the assumption, for any ε>0, there exists a δ∈(0,1) such that
(2.22)μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+p+q)/p<ε,μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p<ε,
when δ<|φ(z)|<1. Since DCφ:𝒬K(p,q)→ℬμ is bounded, then from the proof of Theorem 2.3 we have
(2.23)M1:=supz∈𝔻μ(|z|)|φ″(z)|<∞,M2:=supz∈𝔻μ(|z|)|φ′(z)|2<∞.
Let K={z∈𝔻:|φ(z)|≤δ}. Then, we have(2.24)∥DCφfk∥ℬμ=supz∈𝔻μ(|z|)|(DCφfk)′(z)|+|fk′(φ(0))||φ′(0)|=supz∈𝔻μ(|z|)|(φ′fk′(φ))′(z)|+|fk′(φ(0))||φ′(0)|≤supz∈𝔻μ(|z|)|φ′(z)|2|fk′′(φ(z))|+supz∈𝔻μ(|z|)|φ″(z)||fk′(φ(z))|+|fk′(φ(0))||φ′(0)|≤supKμ(|z|)|φ′(z)|2|fk′′(φ(z))|+supKμ(|z|)|φ″(z)||fk′(φ(z))|+sup𝔻∖Kμ(|z|)|φ″(z)|2|fk′′(φ(z))|+sup𝔻∖Kμ(|z|)|φ″(z)||fk′(φ(z))|+|fk′(φ(0))||φ′(0)|≤supKμ(|z|)|φ′(z)|2|fk′′(φ(z))|+supKμ(|z|)|φ″(z)||fk′(φ(z))|+|fk′(φ(0))||φ′(0)|+Csup𝔻∖Kμ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+p+q)/p∥fk∥𝒬K(p,q)+sup𝔻∖Kμ(|z|)|φ″(z)|(1-|φ(z)|2)(2+p)/p∥fk∥𝒬K(p,q)≤M2supK|fk′′(φ(z))|+M1supK|fk′(φ(z))|+2Cε∥fk∥𝒬K(p,q)+|fk′(φ(0))||φ′(0)|.
The assumption that fk→0 as k→∞ on compact subsets of 𝔻 along with Cauchy's estimate give that fk′→0 and fk′′→0 as k→∞ on compact subsets of 𝔻. Letting k→∞ in (2.24) and using the fact that ε is an arbitrary positive number, we obtain limk→∞∥DCφfk∥ℬμ=0. Applying Lemma 2.1, the result follows.
Now, suppose thatDCφ:𝒬K(p,q)→ℬμ is compact. Then it is clear that DCφ:𝒬K(p,q)→ℬμ is bounded. Let (zk)k∈ℕ be a sequence in 𝔻 such that |φ(zk)|→1 as k→∞ (if such a sequence does not exist then condition (2.21) is vacuously satisfied). Let
(2.25)fk(z)=1-|φ(zk)|2(1-φ(zk)¯z)(q+2)/p.
Then, supk∈ℕ∥fk∥𝒬K(p,q)<∞ and fk converges to 0 uniformly on compact subsets of 𝔻 as k→∞. Since DCφ:𝒬K(p,q)→ℬμ is compact, by Lemma 2.1 we have limk→∞∥DCφfk∥ℬμ=0. On the other hand, from (2.11) we have
(2.26)C∥DCφfk∥ℬμ≥|-2+qp2+p+qpμ(|zk|)|φ′(zk)|2|φ(zk)|2(1-|φ(zk)|2)(2+q+p)/p+2+qpμ(|zk|)|φ′′(zk)||φ(zk)|(1-|φ(zk)|2)(2+q)/p|,
which implies that
(2.27)lim|φ(zk)|→12+q+ppμ(|zk|)|φ′(zk)|2|φ(zk)|2(1-|φ(zk)|2)(2+q+p)/p=lim|φ(zk)|→1μ(|zk|)|φ′′(zk)||φ(zk)|(1-|φ(zk)|2)(2+q)/p,
if one of these two limits exists.
Next, fork∈ℕ, set
(2.28)gk(z)=(1-|φ(zk)|2)2(1-φ(zk)¯z)(q+2)/p+1-q+2+pq+21-|φ(zk)|2(1-φ(zk)¯z)(q+2)/p.
Then (gk)k∈ℕ is a sequence in 𝒬K(p,q). Notice that gk′(φ(zk))=0,
(2.29)|gk′′(φ(zk))|=2+q+pp|φ(zk)|2(1-|φ(zk)|2)(2+q+p)/p,
andgk converges to 0 uniformly on compact subsets of 𝔻 as k→∞. Since DCφ:𝒬K(p,q)→ℬμ is compact, we have limk→∞∥DCφgk∥ℬμ=0. On the other hand, we have
(2.30)2+q+ppμ(|zk|)|φ′(zk)|2|φ(zk)|2(1-|φ(zk)|2)(2+q+p)/p≤∥DCφgk∥ℬμ.
Therefore
(2.31)lim|φ(zk)|→1μ(|zk|)|φ′(zk)|2(1-|φ(zk)|2)(2+q+p)/p=lim|φ(zk)|→1μ(|zk|)|φ′(zk)|2|φ(zk)|2(1-|φ(zk)|2)(2+q+p)/p=0.
This along with (2.27) implies
(2.32)lim|φ(zk)|→1μ(|zk|)|φ″(zk)|(1-|φ(zk)|2)(2+p)/p=0.
From the last two equalities, the desired result follows.Theorem 2.5.
Letφ be an analytic self-map of  𝔻. Suppose that  μ is normal,  p>0,  q>-2 and  K is a nonnegative nondecreasing function on  [0,∞) such that (2.2) holds. Then  DCφ:𝒬K(p,q)→ℬμ,0 is compact if and only if
(2.33)lim|z|→1μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p=0,lim|z|→1μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p=0.Proof.
Sufficiency. Letf∈𝒬K(p,q). By the proof of Theorem 2.3 we have
(2.34)μ(|z|)|(DCφf)′(z)|≤C(μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p+μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p)∥f∥𝒬K(p,q).
Taking the supremum in (2.34) over all f∈𝒬K(p,q) such that ∥f∥𝒬K(p,q)≤1, then letting |z|→1, we get
(2.35)lim|z|→1sup∥f∥𝒬K(p,q)≤1μ(|z|)|(DCφf)′(z)|=0.
From which by Lemma2.2 we see that the operator DCφ:𝒬K(p,q)→ℬμ,0 is compact.                   Necessity. Assume that DCφ:𝒬K(p,q)→ℬμ,0 is compact. By taking the function given by f(z)≡z and using the boundedness of DCφ:𝒬K(p,q)→ℬμ,0, we get
(2.36)lim|z|→1μ(|z|)|φ″(z)|=0.
From this, by taking the test functionf(z)≡z2 and using the boundedness of DCφ:𝒬K(p,q)→ℬμ,0 it follows that
(2.37)lim|z|→1μ(|z|)|φ′(z)|2=0.
If ∥φ∥∞<1, from (2.36) and (2.37), we obtain that
(2.38)lim|z|→1μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+q+p)/p≤1(1-∥φ∥∞2)(2+q+p)/plim|z|→1μ(|z|)|φ′(z)|2=0,lim|z|→1μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p≤1(1-∥φ∥∞2)(2+q)/plim|z|→1μ(|z|)|φ″(z)|=0,from which the result follows in this case.
Assume that∥φ∥∞=1. Let (φ(zk))k∈ℕ be a sequence such that limk→∞|φ(zk)|=1. From the compactness of DCφ:𝒬K(p,q)→ℬμ,0 we see that DCφ:𝒬K(p,q)→ℬμ is compact. From Theorem 2.4 we get
(2.39)lim|φ(z)|→1μ(|z|)|φ′(z)|2(1-|φ(z)|2)(2+p+q)/p=0,(2.40)lim|φ(z)|→1μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p=0.
From (2.36) and (2.40), we have that for every ε>0, there exists an r∈(0,1) such that
(2.41)μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p<ε,
when r<|φ(z)|<1, and there exists a σ∈(0,1) such that μ(|z|)|φ″(z)|≤ε(1-r2)(2+q)/p when σ<|z|<1. Therefore, when σ<|z|<1, and r<|φ(z)|<1, we have
(2.42)μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p<ε.
On the other hand, if σ<|z|<1, and |φ(z)|≤r, we obtain
(2.43)μ(|z|)|φ″(z)|(1-|φ(z)|2)(2+q)/p<1(1-r2)(2+q)/pμ(|z|)|φ″(z)|<ε.
Inequality (2.42) together with (2.43) gives the second equality of (2.33). Similarly to the above arguments, by (2.37) and (2.39) we get the first equality of (2.33). The proof is completed.From the above three theorems, we get the following corollary (see [14]).Corollary 2.6.
Letφ be an analytic self-map of 𝔻. Then the following statements hold. (i)
DCφ:ℬ→ℬ is bounded if and only if
(2.44)Supz∈𝔻(1-|z|2)|φ′(z)|2(1-|φ(z)|2)2<∞,supz∈𝔻(1-|z|2)|φ″(z)|1-|φ(z)|2<∞;(ii)
DCφ:ℬ→ℬ is compact if and only if DCφ:ℬ→ℬ is bounded,
(2.45)lim|φ(z)|→1(1-|z|2)|φ′(z)|2(1-|φ(z)|2)2=0,lim|φ(z)|→1(1-|z|2)|φ″(z)|1-|φ(z)|2=0;(iii)
DCφ:ℬ→ℬ0 is compact if and only if
(2.46)lim|z|→1(1-|z|2)|φ′(z)|2(1-|φ(z)|2)2=0,lim|z|→1(1-|z|2)|φ″(z)|1-|φ(z)|2=0.Similarly to the proofs of Theorems2.3–2.5, we can get the following result. We omit the proof.Theorem 2.7.
Letφ be an analytic self-map of  𝔻. Suppose that  μ is normal,  p>0,  q>-2 and  K is a nonnegative nondecreasing function on  [0,∞) such that  (2.2) holds. Then the following statements hold. (i)
CφD:𝒬K(p,q)→ℬμ is bounded if and only if
(2.47)supz∈𝔻μ(|z|)|φ′(z)|(1-|φ(z)|2)(2+p+q)/p<∞;(ii)
CφD:𝒬K(p,q)→ℬμ is compact if and only if  CφD:𝒬K(p,q)→ℬμ is bounded and
(2.48)lim|φ(z)|→1μ(|z|)|φ′(z)|(1-|φ(z)|2)(2+p+q)/p=0;(iii)
CφD:𝒬K(p,q)→ℬμ,0 is compact if and only if
(2.49)lim|z|→1μ(|z|)|φ′(z)|(1-|φ(z)|2)(2+p+q)/p=0.From Theorem2.7 we get the following corollary.Corollary 2.8.
Letφ be an analytic self-map of  𝔻. Then the following statements hold. (i)
CφD:ℬ→ℬ is bounded if and only if
(2.50)supz∈𝔻(1-|z|2)|φ′(z)|(1-|φ(z)|2)2<∞;(ii)
CφD:ℬ→ℬ is compact if and only if CφD:ℬ→ℬ is bounded and
(2.51)lim|φ(z)|→1(1-|z|2)|φ′(z)|(1-|φ(z)|2)2=0;(iii)
CφD:ℬ→ℬ0 is compact if and only if
(2.52)lim|z|→1(1-|(z)|2)|φ′(z)|(1-|φ(z)|2)2=0.



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*Source: 741920-2009-07-07.xml*