# Weighted Composition Operators and Integral-Type Operators between Weighted Hardy Spaces on the Unit Ball

Authors: Stevo Stević; Sei-Ichiro Ueki
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We study the boundedness and compactness of the weighted composition
operators as well as integral-type operators between weighted Hardy spaces on the unit ball.


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## Body

## 1. Introduction

Let𝔹 denote the open unit ball of the n-dimensional complex vector space ℂn, ∂𝔹 its boundary, and let H(𝔹) denote the space of all holomorphic functions on 𝔹. For 0<p<∞ and α≥0 we define the weighted Hardy space  Hαp(𝔹) as follows:
(1.1)Hαp(𝔹)={f∈H(𝔹):sup0<r<1(1-r)α∫∂𝔹|f(rζ)|pdσ(ζ)<∞},
where dσ is the normalized Lebesgue measure on ∂𝔹 (see, also [1], as well as [2], for an equivalent definition of the space). Note that for α=0 the weighted Hardy space becomes the Hardy space Hp(𝔹). We define the norm ∥·∥Hαp on this space as follows:
(1.2)∥f∥Hαpp=sup0<r<1(1-r)α∫∂𝔹|f(rζ)|pdσ(ζ).
With this norm Hαp(𝔹) is a Banach space when 1≤p<∞. For a related space on the unit polydisk; see [3]. In this paper, we investigate two types of operators acting between weighted Hardy spaces.Letφ be a holomorphic self-map of 𝔹 and u∈H(𝔹). Then φ and u induce a weighted composition operator  uCφ on H(𝔹) which is defined by uCφf=u(f∘φ). This type of operators has been studied on various spaces of holomorphic functions in ℂn, by many authors; see, for example, [4], recent papers [5–17], and the references therein.Letg∈H(𝔻) and φ be a holomorphic self-map of the open unit disk 𝔻 in the complex plane. Products of integral and composition operators on H(𝔻) were introduced by S. Li and S. Stević  in a private communication (see [18–21], as well as papers [22] and [23] for closely related operators) as follows:
(1.3)CφJgf(z)=∫0φ(z)f(ζ)g′(ζ)dζ,(1.4)JgCφf(z)=∫0zf(φ(ζ))g′(ζ)dζ.In [24] the first author of this paper has extended the operator in (1.4) in the unit ball settings as follows (see also [25, 26]). Assume g∈H(𝔹), g(0)=0, and φ is a holomorphic self-map of 𝔹, then we define an operator on the unit ball as follows:
(1.5)Pφg(f)(z)=∫01f(φ(tz))g(tz)dtt,f∈H(𝔹),z∈𝔹.
If n=1, then g∈H(𝔻) and g(0)=0, so that g(z)=zg0(z), for some g0∈H(𝔻). By the change of variable ζ=tz, it follows that
(1.6)Pφgf(z)=∫01f(φ(tz))tzg0(tz)dtt=∫0zf(φ(ζ))g0(ζ)dζ.
Thus the operator (1.5) is a natural extension of operator JgCφ in (1.4). For related operators see [27–33] as well as the references therein.In this paper we study the boundedness and compactness of the weighted composition operators as well as the integral-type operatorPφg, between different weighted Hardy spaces on the unit ball.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 a≤Cb. Moreover, if both a⪯b and b⪯a hold, then one says that a≍b.


## 2. Weighted Composition Operators

This section is devoted to studying weighted composition operators between weighted Hardy spaces. Weighted composition operators between different Hardy spaces on the unit ball were previously studied in [15, 34], while the composition operators on the unit ball were studied in [35, 36]. For the case of the unit disk see also [37].Before we formulate the main results in this section we quote several auxiliary results which will be used in the proofs of these ones.Lemma 2.1.
Let0<p<∞ and α≥0. Suppose that u∈H(𝔹) and φ is a holomorphic self-map of 𝔹. Then for each f∈H(𝔹)(2.1)∥uCφf∥Hαp≤liminfR→1-∥uCφfR∥Hαp,
where uCφfR(z)=u(z)f(Rφ(z)).Proof.
Fixr∈(0,1). Fatou's lemma shows that
(2.2)(1-r)α∫∂𝔹|u(rζ)f(φ(rζ))|pdσ(ζ)≤(1-r)αliminfR→1-∫∂𝔹|u(rζ)f(Rφ(rζ))|pdσ(ζ)=liminfR→1-(1-r)α∫∂𝔹|u(rζ)f(Rφ(rζ))|pdσ(ζ)≤liminfR→1-∥uCφfR∥Hαp.
Hence we have the desired inequality.Recall that anf∈H(𝔹) has the homogeneous expansion
(2.3)f(z)=∑k=0∞∑|γ|=kc(γ)zγ,
where γ=(γ1,…,γn) is a multi-index, |γ|=γ1+⋯+γn and zγ=z1γ1⋯znγn. For the homogeneous expansion of f and any integer j≥1, let
(2.4)Rjf(z)=∑k=j∞∑|γ|=kc(γ)zγ,
and Kj=I-Rj where If=f is the identity operator. Note that Kj is compact operator on Hαp(𝔹) for each j∈ℕ.Lemma 2.2.
If1<p<∞, then Rj converges to 0 pointwise in the Hardy space Hp(𝔹)as j→∞.Proof.
See [34, Corollary 3.4] .Lemma2.2 and the uniform boundedness principle show that {Rj} is an uniformly bounded sequence in Hp(𝔹).The following lemma is proved similar to [4, Lemma 3.16]. We omit its proof.Lemma 2.3.
IfuCφ is bounded from Hαp(𝔹) into Hβq(𝔹), then
(2.5)∥uCφ∥e,Hαp(𝔹)→Hβq(𝔹)≤liminfj→∞∥uCφRj∥Hαp(𝔹)→Hβq(𝔹),
where ∥·∥e,Hαp(𝔹)→Hβq(𝔹) and ∥·∥Hαp(𝔹)→Hβq(𝔹) denote the essential norm and the operator norm, respectively.Lemma 2.4.
Let0<p≤q<∞. Suppose that μ is a positive Borel measure on 𝔹 which satisfies
(2.6)μ(B(ζ,t))≤C1tqn/p(ζ∈∂𝔹,t>0),
for some positive constant C1. Then there exists a positive constant C2 which depends only on p,q, and the dimension n such that
(2.7)∫𝔹|f|qdμ≤C1C2∥f∥Hpq,
for any f∈Hp(𝔹). Here B(ζ,t)={z∈𝔹:|1-〈z,ζ〉|<t}.Proof.
See [38, page 13, Theorem] or [34, Lemma 2.1] .Let0<q<∞. For each r∈(0,1), a holomorphic self-map φ of 𝔹 and u∈H(𝔹), we define a positive Borel measure μu,φr on 𝔹 by
(2.8)μu,φr(E)=∫(φr)-1(E)|ur|qdσ,
for all Borel sets E of 𝔹. By the change of variables formula from measure theory, we can verify
(2.9)∫𝔹gdμu,φr=∫∂𝔹|u(rζ)|q(g∘φ)(rζ)dσ(ζ),
for each nonnegative measurable function g in 𝔹.Theorem 2.5.
Let0<p≤q<∞ and α,β≥0. Suppose that u∈H(𝔹) and φ is a holomorphic self-map of 𝔹. Then uCφ:Hαp(𝔹)→Hβq(𝔹) is bounded if and only if
(2.10)supw∈𝔹sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)<∞.Proof.
Forw∈𝔹 we put
(2.11)fw(z)={1-|w|2(1-〈z,w〉)2}α+n/p.
Then we see that fw∈Hαp(𝔹) and moreover supw∈𝔹∥fw∥Hαp≤C. By a straightforward calculation, we have
(2.12)∥uCφfw∥Hβqq=sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ),
for all w∈𝔹. Hence if uCφ:Hαp(𝔹)→Hβq(𝔹) is bounded, then u and φ satisfy the condition
(2.13)supw∈𝔹sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)≤C∥uCφ∥Hαp(𝔹)→Hβq(𝔹)q<∞.
Next we assume
(2.14)M:=supw∈𝔹sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)<∞.
Fixr∈(0,1) and R∈(0,1), respectively. For ζ∈∂𝔹 and t,0<t≤tR=1-R, we put w=(1-t)ζ and wR=(1-tR)ζ. Since the function fw(z), which is defined by (2.11) for this w, satisfies
(2.15)|fw(z)|q>4-q(α+n)/pt-qn/p(1-R)-qα/p
for all z∈B(ζ,t), we have
(2.16)μu,φr(B(ζ,t))tqn/p≤4q(α+n)/p(1-R)qα/p∫B(ζ,t)|fw(z)|qdμu,φr(z)≤4q(α+n)/p(1-R)qα/p∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)≤4q(α+n)/p(1-R)qα/p(1-r)βM.
By the same argument, the function fwR(z) gives the following estimate:
(2.17)μu,φr(B(ζ,2tR))≤9q(α+n)/p(1-R)qα/p(1-r)βMtRqn/p.
Now we need to prove that there exists a positive constant C such that
(2.18)μu,φr(B(ζ,t))≤C(1-R)qα/p(1-r)βMtqn/p,
for all ζ∈∂𝔹 and t>0. By the estimate (2.16), we see that the inequality (2.18) is true for all t∈(0,tR]. Thus we assume t>tR. By the same argument as in [36, pages 241-242, proof of Theorem 1.1] , we see that the inequality (2.17) shows that there exists a positive constant Cn which depends only on the dimension n such that
(2.19)μu,φr(B(ζ,t))≤Cn(ttR)n9q(α+n)/p(1-R)qα/p(1-r)βMtRqn/p=Cn9q(α+n)/p(1-R)qα/p(1-r)βMtntR(q/p-1)n≤Cn9q(α+n)/p(1-R)qα/p(1-r)βMtqn/p.
Hence for C=max{4q(α+n)/p,Cn9q(α+n)/p}, we have the inequality in (2.18).
Forf∈Hαp(𝔹) the dilate function fR belongs to the ball algebra, and so fR is in the Hardy space Hp(𝔹). Hence Lemma 2.4 gives
(2.20)∫𝔹|fR(z)|qdμu,φr(z)≤C′C(1-R)qα/p(1-r)βM∥fR∥Hpq,
for some positive constant C′ and all R∈(0,1). This implies that
(2.21)(1-r)β∫∂𝔹|uCφfR(rζ)|qdσ(ζ)≤C′CM[(1-R)α∫∂𝔹|f(Rζ)|pdσ(ζ)]q/p,
and so we have
(2.22)∥uCφfR∥Hβqq≤C′CM∥f∥Hαpq,
for all R∈(0,1). By Lemma 2.1 we have
(2.23)∥uCφf∥Hβqq≤C′C∥f∥Hαpq×supw∈𝔹sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ).
This completes the proof.The following proposition is proved in a standard way; see, for example, the proofs of the corresponding results in [4, 32, 33, 39]. Hence we omit its proof.Proposition 2.6.
Let0<p,q<∞ and α,β≥0. Suppose that u∈H(𝔹) and φ is a holomorphic self-map of 𝔹 which induce the bounded operator uCφ:Hαp(𝔹)→Hβq(𝔹). Then uCφ:Hαp(𝔹)→Hβq(𝔹) is compact if and only if for every bounded sequence {fj}j∈ℕ in Hαp(𝔹) which converges to 0 uniformly on compact subsets of 𝔹, {uCφfj}j∈ℕ converges to 0 in Hβq(𝔹).In the proof of Theorem2.8, we need the following lemma.Lemma 2.7.
Let1<p<∞, α≥0, and fw be the family of test functions defined in (2.11). Then fw→0 weakly in Hαp(𝔹) as |w|→1-.Proof.
The family{fw}w∈𝔹 is bounded in Hαp(𝔹) and fw→0 uniformly on compact subsets of 𝔹 as |w|→1-. By the definitions of the space Hαp(𝔹) and the norm ∥·∥Hαp, we see that Hαp(𝔹) is a subspace of the weighted Bergman space Aαp(𝔹) and
(2.24)∥f∥Aαp≤C(α,p,n)∥f∥Hαp(f∈Hαp(𝔹)),
for some positive constant C(α,p,n) which depends on α,p, and n. This inequality implies that the family {fw}w∈𝔹 is also bounded in Aαp(𝔹). Note also that the family converges to 0 uniformly on compact subsets of 𝔹 as |w|→1-. Hence fw→0 weakly in Aαp(𝔹) as |w|→1-.
In order to prove thatfw→0 weakly in Hαp(𝔹) as |w|→1-, we take an arbitrary bounded linear functional Λ on Hαp(𝔹). By the Hahn-Banach theorem, Λ can be extended to a bounded linear functional Λ̃ on Aαp(𝔹) so that Λ̃(fw)=Λ(fw) for all w∈𝔹. Since fw→0 weakly in Aαp(𝔹) as |w|→1-, we have Λ(fw)=Λ̃(fw)→0 as |w|→1-, and so fw→0 weakly in Hαp(𝔹) as |w|→1-.Theorem 2.8.
Let1<p≤q<∞ and α,β≥0. Suppose that u∈H(𝔹) and φ is a holomorphic self-map of 𝔹 such that uCφ:Hαp(𝔹)→Hβq(𝔹) is bounded. Then the qth power of the essential norm ∥uCφ∥e,Hαp(𝔹)→Hβq(𝔹) is comparable to
(2.25)lim sup|w|→1-sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ).
Hence uCφ:Hαp(𝔹)→Hβq(𝔹) is compact if and only if
(2.26)lim|w|→1-sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)=0.Proof.
To prove a lower estimate(2.27)∥uCφ∥e,Hαp(𝔹)→Hβq(𝔹)q≥limsup|w|→1-sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ),
we consider the test functions fw defined in (2.11). The family {fw}w∈𝔹 is bounded in Hαp(𝔹), say by L, and fw→0 uniformly on compact subsets of 𝔹 as |w|→1-. Thus by Lemma 2.7 we have that fw→0 weakly in Hαp(𝔹) as |w|→1-, so that ∥𝒦fw∥Hβq→0 as |w|→1- for every compact operator 𝒦:Hαp(𝔹)→Hβq(𝔹). Hence
(2.28)L∥uCφ-𝒦∥Hαp(𝔹)→Hβq(𝔹)≥limsup|w|→1-∥(uCφ-𝒦)fw∥Hβq≥limsup|w|→1-∥uCφfw∥Hβq.
This inequality and (2.12) give the lower estimate for ∥uCφ∥e,Hαp(𝔹)→Hβq(𝔹)q.
Next we prove an upper estimate. Takef∈Hαp(𝔹) with ∥f∥Hαp≤1. Fix ε>0 and put
(2.29)M1:=limsup|w|→1-sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ).
Then we can choose R0∈(0,1) such that
(2.30)sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)<M1+ε,
for w∈𝔹 with |w|>R0. Fix r∈(0,1) and R∈(R0,1). By the same argument as in the proof of inequality (2.20) in Theorem 2.5, we obtain that
(2.31)∫𝔹|(Rjf)R(z)|qdμu,φr(z)≤C(1-R)qα/p(1-r)β(M1+ε)∥(Rjf)R∥Hpq,
where the positive constant C is independent of r, R and a positive integer j. Since fR is in the ball algebra, Lemma 2.2 gives
(2.32)∥(Rjf)R∥Hpq=∥Rj(fR)∥Hpq≤supj≥1∥Rj∥Hp(𝔹)→Hp(𝔹)q∥fR∥Hpq.
Combining this with (2.31), we have
(2.33)(1-r)β∫∂𝔹|uCφ(RjfR)(rζ)|qdσ(ζ)≤C′(M1+ε)[(1-R)α∫∂𝔹|fR(ζ)|pdσ(ζ)]q/p≤C′(M1+ε)∥f∥Hαpq,
and so we have
(2.34)∥uCφ(RjfR)∥Hβqq≤C′(M1+ε)∥f∥Hαpq.
Letting R→1-, by Lemma 2.1, we obtain
(2.35)∥uCφRjf∥Hβqq≤C′(M1+ε)∥f∥Hαpq.
Since ε>0 is arbitrary, this estimate and Lemma 2.3 imply
(2.36)∥uCφ∥e,Hαp(𝔹)→Hβq(𝔹)q≤liminfj→∞∥uCφRj∥Hαp(𝔹)→Hβq(𝔹)q≤C′limsup|w|→1-sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ),
which completes the proof.Remark 2.9.
In the above proof, we used Lemma2.2. This lemma required the assumption 1<p<∞. Hence we cannot have an upper estimate for ∥uCφ∥e,Hαp(𝔹)→Hβq(𝔹) in the case 0<p≤1. However, Proposition 2.6 shows that the compactness of uCφ:Hαp(𝔹)→Hβq(𝔹)(0<p≤q<∞) is equivalent to
(2.37)lim|w|→1-sup0<r<1(1-r)β∫∂𝔹|u(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)=0.


## 3. Integral-Type Operators

Here we study the boundedness and compactness of the integral-type operatorsPφg between weighted Hardy spaces on the unit ball.Forf∈H(𝔹) with the Taylor expansion f(z)=∑|γ|≥0aγzγ, let ℜf(z)=∑|γ|≥0|γ|aγzγ be the radial derivative of f.The following lemma was proved in [24] (see also [25]).Lemma 3.1.
Assume thatφ is a holomorphic self-map of 𝔹, g∈H(𝔹) and g(0)=0. Then for every f∈H(𝔹)  one holds
(3.1)ℜ[Pφg(f)](z)=f(φ(z))g(z).
A positive continuous functionω on the interval [0,1) is called normal [40] if there is a δ∈[0,1) and a and b, 0<a<b such that
(3.2)ω(r)(1-r)aisdecreasingon[δ,1)andlimr→1ω(r)(1-r)a=0,ω(r)(1-r)bisincreasingon[δ,1)andlimr→1ω(r)(1-r)b=∞.
If one that a functionω:𝔹→[0,∞) is normal, one will also assume that it is radial.Lemma 3.2.
Assume that0<q≤∞, m is a positive integer and ω is normal. Then for every f∈H(𝔹)(3.3)sup0<r<1ω(r)Mq(f,r)≍|f(0)|+sup0<r<1(1-r)mω(r)Mq(ℜmf,r),
where
(3.4)Mq(f,r)=(∫∂𝔹|f(rζ)|qdσ(ζ))1/q,M∞(f,r)=supζ∈∂𝔹|f(rζ)|.Proof.
The proof of the lemma in the case1≤q≤∞ can be found in [27, Theorem 2]. However, due to an overlook, the proof for the case q∈(0,1) has a gap. Hence we will give a correct proof here in the case.
We may assume thatf(0)=0, otherwise we can consider the functions h(z)=f(z)-f(0). Also we may assume that δ=0, to avoid some minor technical difficulties.
By [27, Lemma 1], for each fixed q∈(0,1], there is a positive constant C depending only on q and the dimension n such that
(3.5)Mq(f,r)≤Cr(∫0r(r-t)q-1Mqq(ℜf,t)dt)1/q,
for every r∈(0,1) and f∈H(𝔹) such that f(0)=0.
From (3.5) and the fact that ω is normal, we have
(3.6)sup0≤r<1ω(r)Mq(f,r)≤Csup0≤r<1ω(r)1r(∫0r(r-t)q-1Mqq(ℜf,t)dt)1/q≤Csup0≤r<1(1-r)a1r(∫0r(r-t)q-1ωq(t)(1-t)aqMqq(ℜf,t)dt)1/q≤Csup0≤r<1(1-r)a1r(∫0r(r-t)q-1(1-t)aq+qdt)1/qsup0≤t<1(1-t)ω(t)Mq(ℜf,t)=Csup0≤r<1(1-r)a(∫01(1-u)q-1(1-ur)aq+qdu)1/qsup0≤t<1(1-t)ω(t)Mq(ℜf,t).
By [40, page 291, Lemma 6] there exists a positive constant C such that
(3.7)∫01(1-u)q-1(1-ur)aq+qdu≤C(1-r)aq,
for every r∈(0,1). Combining this with (3.6), we have
(3.8)sup0≤r<1ω(r)Mq(f,r)≤Csup0≤r<1(1-r)a(1(1-r)aq)1/qsup0≤t<1(1-t)ω(t)Mq(ℜf,t)=Csup0≤t<1(1-t)ω(t)Mq(ℜf,t).
The reverse inequality is proved by the following inequality:
(3.9)(1-r)Mq(ℜf,r)≤CMq(f,1+r2)
and the fact that ω(r)≍ω(1+r/2) for ω normal (see [27]). Hence, we obtain the result for the case m=1.
Form≥2 it should be only noticed that (1-r)mω(r) is still normal, that ℜmf(0)=0 for every integer m≥1, and use the method of induction.Theorem 3.3.
Let0<p≤q<∞ and α,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹. Then Pφg:Hαp(𝔹)→Hβq(𝔹) is bounded if and only if
(3.10)supw∈𝔹sup0<r<1(1-r)β+q∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)<∞.Proof.
Takef∈Hαp(𝔹) with ∥f∥Hαp≤1. Since the function (1-r)β/q for β>0 and 0<q<∞ is normal, Lemma 3.2 gives
(3.11)sup0<r<1(1-r)β/qMq(Pφgf,r)≍|Pφgf(0)|+sup0<r<1(1-r)(β/q)+1Mq(ℜ[Pφgf],r).
The assumption g(0)=0 implies Pφgf(0)=0, and Lemma 3.1 shows ℜ[Pφgf]=gCφf. Hence we obtain
(3.12)sup0<r<1(1-r)βMqq(Pφgf,r)≍sup0<r<1(1-r)β+qMqq(gCφf,r),
and so we obtain ∥Pφgf∥Hβqq≍∥gCφf∥Hβ+qqq. This implies that the boundedness of Pφg:Hαp(𝔹)→Hβq(𝔹) is equivalent to the boundedness of gCφ:Hαp(𝔹)→Hβ+qq(𝔹). So Theorem 2.5 shows that the condition
(3.13)supw∈𝔹sup0<r<1(1-r)β+q∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)<∞
is a necessary and sufficient condition for the boundedness of Pφg:Hαp(𝔹)→Hβq(𝔹). This completes the proof.The next proposition is proved similar to Proposition2.6.Proposition 3.4.
Let0<p,q<∞, and α,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹 which induce the bounded operator Pφg:Hαp(𝔹)→Hβq(𝔹). Then Pφg:Hαp(𝔹)→Hβq(𝔹) is compact if and only if for every bounded sequence {fj}j∈ℕ in Hαp(𝔹) which converges to 0 uniformly on compact subsets of 𝔹, {Pφgfj}j∈ℕ converges to 0 in Hβq(𝔹).Theorem 3.5.
Let0<p≤q<∞ and α,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹 which induce the bounded operator Pφg:Hαp(𝔹)→Hβq(𝔹). Then Pφg:Hαp(𝔹)→Hβq(𝔹) is compact if and only if
(3.14)lim|w|→1-sup0<r<1(1-r)β+q∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ)=0.Proof.
First we assume that condition (3.14) holds. Take a bounded sequence {fj}j∈ℕ⊂Hαp(𝔹) which converges to 0 uniformly on compact subsets of 𝔹. Theorem 2.8 and the remark in Section 2 show that gCφ:Hαp(𝔹)→Hβ+qq(𝔹) is compact. Thus Proposition 2.6 implies that
(3.15)limj→∞∥gCφfj∥Hβ+qq=0.
From (3.15) and since ∥Pφgfj∥Hβqq≍∥gCφfj∥Hβ+qqq, we have that ∥Pφgfj∥Hβqq→0 as j→∞. By Proposition 3.4, we see that Pφg:Hαp(𝔹)→Hβq(𝔹) is compact.
To prove the necessity of the condition in (3.14), we consider the family of test functions fw which is defined by (2.11). Hence we have
(3.16)∥Pφgfw∥Hβqq≍∥gCφfw∥Hβ+qqq=sup0<r<1(1-r)β+q∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ),
for all w∈𝔹. Since {fw}w∈𝔹 is a bounded sequence in Hαp(𝔹) and fw→0 uniformly on compact subsets of 𝔹 as |w|→1-, the compactness of Pφg and Proposition 3.4 show that ∥Pφgfw∥Hβqq→0 as |w|→1-. This fact along with (3.16) implies the condition in (3.14), finishing the proof of the theorem.Theorem 3.6.
Let1<p≤q<∞ and α,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹 which induce the bounded operator Pφg:Hαp(𝔹)→Hβq(𝔹). Then the qth power of the essential norm of Pφg is comparable to
(3.17)limsup|w|→1-sup0<r<1(1-r)β+q∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ).Proof.
To prove a lower estimate, we take an arbitrary compact operator𝒦:Hαp(𝔹)→Hβq(𝔹). Since Lemma 2.7 implies that the family of functions fw defined by (2.11) converges to 0 weakly in Hαp(𝔹) as |w|→1-, we obtain
(3.18)C∥Pφg-𝒦∥Hαp(𝔹)→Hβq(𝔹)≥limsup|w|→1-(∥Pφgfw∥Hβq-∥𝒦fw∥Hβq)≥limsup|w|→1-∥Pφgfw∥Hβq.
Combining this with (3.16), we have
(3.19)C∥Pφg∥e,Hαp(𝔹)→Hβq(𝔹)q≥limsup|w|→1-sup0<r<1(1-r)β+q∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ),
which is a lower estimate.
By some modification of Lemma2.3 and the application of Lemmas 3.1 and 3.2, we get
(3.20)∥Pφg∥e,Hαp(𝔹)→Hβq(𝔹)q≤liminfj→∞sup∥f∥Hαp≤1∥PφgRjf∥Hβqq≤Climinfj→∞sup∥f∥Hαp≤1∥gCφRjf∥Hβ+qqq.
As in the proof of Theorem 2.8, we obtain
(3.21)liminfj→∞sup∥f∥Hαp≤1∥gCφRjf∥Hβ+qqq≤Climsup|w|→1-sup0<r<1(1-r)β+q×∫∂𝔹|g(rζ)|q{1-|w|2|1-〈φ(rζ),w〉|2}q(α+n)/pdσ(ζ),
and so we have an upper estimate for ∥Pφg∥e,Hαp(𝔹)→Hβq(𝔹)q.


## 4. The CasePφg:Hα∞(𝔹)→Hβ∞(𝔹)

Whenp=∞ and α>0, we define the weighted-type space Hα∞(𝔹) as follows:
(4.1)Hα∞(𝔹)={f∈H(𝔹):sup0<r<1(1-r)αM∞(f,r)<∞}.
It is easy to see that f∈Hα∞(𝔹) if and only if supz∈𝔹(1-|z|)α|f(z)|<∞, so we define the norm ∥f∥Hα∞ on Hα∞(𝔹) by this supremum.Furthermore we consider the subspaceHα,0∞(𝔹) defined by
(4.2)Hα,0∞(𝔹)={f∈H(𝔹):limr→1-(1-r)αM∞(f,r)=0}.Theorem 4.1.
Letα,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹. Then Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is bounded if and only if
(4.3)supz∈𝔹(1-|z|)β+1|g(z)|(1-|φ(z)|)α<∞.
In this case, the operator norm ∥Pφg∥Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is comparable to the above supremum.Proof.
By the definition of the spaceHα∞(𝔹), f∈Hα∞(𝔹) satisfies the growth condition
(4.4)|f(w)|≤(1-|w|)-α∥f∥Hα∞(w∈𝔹),
so it follows from Lemma 3.1 and Lemma 3.2 that
(4.5)∥Pφgf∥Hβ∞≍supz∈𝔹(1-|z|)β+1|gCφf(z)|≤∥f∥Hα∞supz∈𝔹(1-|z|)β+1|g(z)|(1-|φ(z)|)α,
for every f∈Hα∞(𝔹).
Hence we obtain
(4.6)∥Pφg∥Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹)≤Csupz∈𝔹(1-|z|)β+1|g(z)|(1-|φ(z)|)α.
Now we prove the reverse inequality. Forw∈𝔹, we put
(4.7)fw(z)=1(1-〈z,w〉)α.
Note that fw∈Hα,0∞(𝔹) for each w∈𝔹 and moreover supw∈𝔹∥fw∥Hα∞≤1.
Whenφ(z)≠0, we have
(4.8)∥Pφg∥Hα,0∞(𝔹)→Hβ∞(𝔹)≥∥Pφgft(φ(z)/|φ(z)|)∥Hβ∞≍supw∈𝔹(1-|w|)β+1|gCφft(φ(z)/|φ(z)|)(w)|≥(1-|z|)β+1|g(z)||ft(φ(z)/|φ(z)|)(φ(z))|=(1-|z|)β+1|g(z)|(1-t|φ(z)|)α,
for all t∈(0,1). Letting t→1- in (4.8), we have
(4.9)∥Pφg∥Hα,0∞(𝔹)→Hβ∞(𝔹)≥C(1-|z|)β+1|g(z)|(1-|φ(z)|)α.
For the constant function1∈Hα,0∞(𝔹) we obtain
(4.10)∥Pφg1∥Hβq≍supw∈𝔹(1-|w|)β+1|gCφ1(w)|≥(1-|z|)β+1|g(z)|.
Inequality (4.10) shows that the estimate in (4.9) also holds when φ(z)=0.
Hence, from (4.9) we obtain
(4.11)∥Pφg∥Hα,0∞(𝔹)→Hβ∞(𝔹)≥Csupz∈𝔹(1-|z|)β+1|g(z)|(1-|φ(z)|)α,
which along with the obvious inequality
(4.12)∥Pφg∥Hα∞(𝔹)→Hβ∞(𝔹)≥∥Pφg∥Hα,0∞(𝔹)→Hβ∞(𝔹)
completes the proof of the theorem.For the compactness ofPφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹), we can also prove the following proposition which is similar to Proposition 2.6.Proposition 4.2.
Letα,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹 which induce the bounded operator Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹). Then Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is compact if and only if for every bounded sequence {fj}j∈ℕ in Hα∞(𝔹)(orHα,0∞(𝔹)) which converges to 0 uniformly on compact subsets of   𝔹, {Pφgfj}j∈ℕ converges to 0 in Hβ∞(𝔹).Theorem 4.3.
Letα,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹 such that Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is bounded. Then the essential norm ∥Pφg∥e,Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is comparable to
(4.13)limsup|φ(z)|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α.
In particular, Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is compact if and only if
(4.14)lim|φ(z)|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α=0.Proof.
First we consider the family{fw}w∈𝔹 where
(4.15)fw(z)=1-|w|(1-〈z,w〉)α+1.
We can easily check that fw∈Hα,0∞(𝔹), ∥fw∥Hα∞≤1 for all w∈𝔹 and fw→0 uniformly on compact subsets of 𝔹 as |w|→1-. Hence [40, page 296, Theorem 2] implies that fw→0 weakly in Hα,0∞(𝔹) as |w|→1-.
If∥φ∥∞<1, then as in the proof of [26, Theorem 3] it can be seen that the operator Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹) is compact, so that
(4.16)∥Pφg∥e,Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ∞(𝔹)=0.
On the other hand, the limit in (4.13) is vacuously equal to zero, from which the result follows in this case. If ∥φ∥∞=1, then take a sequence {φ(zj)}j∈ℕ in 𝔹 with |φ(zj)|→1 as j→∞ and put Fj(z)=fφ(zj)(z) for each j∈ℕ. Then {Fj}j∈ℕ is a bounded sequence in Hα,0∞(𝔹) and {Fj}j∈ℕ converges to 0 weakly in Hα,0∞(𝔹), as j→∞. Hence for every compact operator 𝒦:Hα,0∞(𝔹)→Hβ∞(𝔹) we have ∥𝒦Fj∥Hβ∞→0 as j→∞. So we have
(4.17)∥Pφg-𝒦∥Hα,0∞(𝔹)→Hβ∞(𝔹)≥limsupj→∞∥PφgFj-𝒦Fj∥Hβ∞≥limsupj→∞∥PφgFj∥Hβ∞≍limsupj→∞supw∈𝔹(1-|w|)β+1|g(w)||Fj(φ(w))|≥limsupj→∞(1-|zj|)β+1|g(zj)||Fj(φ(zj))|≥12α+1limsupj→∞(1-|zj|)β+1|g(zj)|(1-|φ(zj)|)α,
for all compact operators 𝒦:Hα,0∞(𝔹)→Hβ∞(𝔹). Taking the infimum over the set of all compact operators 𝒦:Hα,0∞(𝔹)→Hβ∞(𝔹), we obtain
(4.18)∥Pφg∥e,Hα,0∞(𝔹)→Hβ∞(𝔹)≥Climsupj→∞(1-|zj|)β+1|g(zj)|(1-|φ(zj)|)α.
Combining this with the estimate ∥Pφg∥e,Hα∞(𝔹)→Hβ∞(𝔹)≥∥Pφg∥e,Hα,0∞(𝔹)→Hβ∞(𝔹), we have
(4.19)∥Pφg∥e,Hα∞(𝔹)→Hβ∞(𝔹)≥Climsup|φ(z)|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α.
Next we prove an upper estimate. Assume that{rl}l∈ℕ⊂(0,1) is a sequence which increasingly converges to 1. For this {rl}l∈ℕ, we define the operators defined by
(4.20)Prlφgf(z)=∫01g(tz)f(rlφ(tz))dtt.
As in the proof of [26, Theorem 3], Proposition 4.2 shows that Prlφg:Hα∞(𝔹)→Hβ∞(𝔹) is compact for each l∈ℕ.
Put
(4.21)M2:=limsup|φ(z)|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α,
and fix ε>0. Then we can choose R∈(0,1) such that
(4.22)(1-|z|)β+1|g(z)|(1-|φ(z)|)α<M2+ε,
if R<|φ(z)|<1. Take f∈Hα∞(𝔹) with ∥f∥Hα∞≤1 and an integer l∈ℕ. By Lemma 3.1 and Lemma 3.2, we have
(4.23)∥Pφgf-Prlφgf∥Hβ∞≍supz∈𝔹(1-|z|)β+1|ℜ[Pφgf](z)-ℜ[Prlφgf](z)|=supz∈𝔹(1-|z|)β+1|g(z)||f(φ(z))-f(rlφ(z))|=sup|φ(z)|≤R(1-|z|)β+1|g(z)||f(φ(z))-f(rlφ(z))|+supR<|φ(z)|<1(1-|z|)β+1|g(z)||f(φ(z))-f(rlφ(z))|.
By using the mean value theorem and the asymptotic relation
(4.24)supz∈𝔹(1-|z|)α+1|∇f(z)|≍supz∈𝔹(1-|z|)α+1|ℜf(z)|,
we obtain
(4.25)sup|φ(z)|≤R|f(φ(z))-f(rlφ(z))|≤sup|φ(z)|≤R(1-rl)|φ(z)|sup|w|≤R|∇f(w)|≤(1-rl)R(1-R)α+1supw∈𝔹(1-|w|)α+1|∇f(w)|≍(1-rl)R(1-R)α+1supw∈𝔹(1-|w|)α+1|ℜf(w)|≍(1-rl)R(1-R)α+1∥f∥Hα∞.
Since the boundedness of Pφg:Hα∞(𝔹)→Hβ∞(𝔹) implies that Pφg1∈Hβ∞(𝔹), we see
(4.26)supz∈𝔹(1-|z|)β+1|g(z)|<∞,
and so we have
(4.27)sup∥f∥Hα∞≤1sup|φ(z)|≤R(1-|z|)β+1|g(z)||f(φ(z))-f(rlφ(z))|≤C(1-rl)R(1-R)α+1supz∈𝔹(1-|z|)β+1|g(z)|.
On the other hand, the monotonicity of M∞(f,r) shows
(4.28)|f(rlφ(z))|≤(1-|φ(z)|)-α∥frl∥Hα∞≤(1-|φ(z)|)-α∥f∥Hα∞.
Thus we have
(4.29)sup∥f∥Hα∞≤1supR<|φ(z)|<1(1-|z|)β+1|g(z)||f(φ(z))-f(rlφ(z))|≤2supR<|φ(z)|<1(1-|z|)β+1|g(z)|(1-|φ(z)|)α≤2(M2+ε).
From (4.23), (4.27), (4.29), and the compactness of Prlφg, we obtain
(4.30)∥Pφg∥e,Hα∞(𝔹)→Hβ∞(𝔹)≤∥Pφg-Prlφg∥Hα∞(𝔹)→Hβ∞(𝔹)≤C(1-rl)R(1-R)α+1supz∈𝔹(1-|z|)β+1|g(z)|+2(M2+ε).
Letting l→∞ and ε→0, we have
(4.31)∥Pφg∥e,Hα∞(𝔹)→Hβ∞(𝔹)≤2limsup|φ(z)|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α.
This completes the proof.WhenPφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ,0∞(𝔹) is bounded, we see that g∈Hβ+1,0∞(𝔹). By a standard argument as in the proof of [26, Corollary 3], we have
(4.32)limsup|z|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α=limsup|φ(z)|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α,
and so
(4.33)∥Pφg∥e,Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ,0∞(𝔹)≍limsup|z|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α.
Hence we obtain the following characterization for the compactness of the operator Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ,0∞(𝔹).Corollary 4.4.
Letα,β>0. Suppose that g∈H(𝔹) with g(0)=0 and φ is a holomorphic self-map of 𝔹 such that Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ,0∞(𝔹) is bounded. Then Pφg:Hα∞(𝔹)(orHα,0∞(𝔹))→Hβ,0∞(𝔹) is compact if and only if
(4.34)lim|z|→1-(1-|z|)β+1|g(z)|(1-|φ(z)|)α=0.



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*Source: 952831-2009-08-20.xml*