# Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space toF(p,q,s) Space on the Unit Ball

Authors: Yu-Xia Liang; Ze-Hua Zhou
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper characterizes the boundedness and compactness of the
product of extended Cesàro operator and composition operator from Lipschitz
space toF(p,q,s) space on the unit ball of ℂn.


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## Body

## 1. Introduction

Let𝔹 be the unit ball in the n-dimensional complex space ℂn, the closure of 𝔹 will be written as 𝔹¯. By dv we denote the Lebesgue measure on 𝔹 normalized so that v(𝔹)=1 and by dσ the normalized rotation invariant measure on the boundary S=∂𝔹 of 𝔹. Let H(𝔹) be the class of all holomorphic functions on 𝔹 and S(𝔹) the collection of all the holomorphic self-mappings of 𝔹. Denote by A(𝔹) the unit ball algebra of all continuous functions on 𝔹¯ that are holomorphic on 𝔹.Forf∈H(𝔹), let (1.1)Rf(z)=∑j=1nzj∂f∂zj(z)
be the radial derivative of f.We recall that theα-Bloch space ℬα for α≥0 consists of all f∈H(𝔹) such that (1.2)Bα(f)=sup⁡z∈B(1-|z|2)α|Rf(z)|<∞.
The expression ℬα(f) defines a seminorm while the natural norm is given by ∥f∥ℬα=|f(0)|+ℬα(f). This norm makes ℬα into a Banach space. When α=1, ℬ1=ℬ is the well known Bloch space.Forα∈(0,1), ℒα(𝔹) denotes the holomorphic Lipschitz space of order α which is the set of all f∈H(𝔹) such that, for some C>0,(1.3)|f(z)-f(w)|≤C|z-w|α
for every z,w∈𝔹. It is clear that each space ℒα(𝔹) contains the polynomials and is contained in the ball algebra A(𝔹). It is well known that ℒα(𝔹) is endowed with a complete norm ∥·∥ℒα that is given by(1.4)‖f‖Lα=|f(0)|+sup⁡z≠w;z,w∈B{|f(z)-f(w)||z-w|α}.
See [1, 2] for more information of the Lipschitz spaces on 𝔹.Fora∈𝔹, let g(z,a)=log|φa(z)|-1 be Green's function on 𝔹 with logarithmic singularity at a, where φa is the Möbius transformation of 𝔹 with φa(0)=a, φa(a)=0, and φa=φa-1.Let0<p,s<∞, -n-1<q<∞, a function f∈H(𝔹) is said to belong to F(p,q,s) if (see, e.g., [3–5])(1.5)‖f‖F(p,q,s)p=|f(0)|p+sup⁡a∈B∫B|Rf(z)|p(1-|z|2)qgs(z,a)dυ(z)<∞.IfX is a Banach space of holomorphic functions on a domain Ω and if φ is a (holomorphic) self-map of Ω, the composition operator of symbol φ is defined by Cφ(f)=f∘φ. The study of composition operators consists in the comparison of the properties of the operator Cφ with that of the function φ itself, which is called the symbol of Cφ. One can characterize boundedness and compactness of Cφ and many other properties. We refer to the books in [6, 7] and to some recent papers in [4, 5, 8] to learn much more on this subject.Leth∈H(𝔹), the following integral-type operator was first introduced in [9] (1.6)Thf(z)=∫01f(tz)Rh(tz)dtt,f∈H(B),z∈B.
This operator is called generalized Cesàro operator. It has been well studied in many papers, see, for example, [3, 9–24] as well as the related references therein.It is natural to discuss the product of extended Cesàro operator and composition operator. Forh∈H(𝔹) and φ∈S(𝔹), the product can be expressed as(1.7)ThCφf(z)=∫01f(φ(tz))Rh(tz)dtt,f∈H(B),z∈B.
It is interesting to characterize the boundedness and compactness of the product operator on all kinds of function spaces. Even on the disk of ℂ, some properties are not easily managed; see some recent papers in [18, 25–28].Building on those foundations, the present paper continues this line of research and discusses the operator in high dimension. The remainder is assembled as follows: in Section2, we state a couple of lemmas. In Section 3, we characterize the boundedness and compactness of the product ThCφ of extended Cesàro operator and composition operator from Lipschitz spaces to F(p,q,s) spaces on the unit ball of ℂn.Throughout the remainder of this paper,C will denote a positive constant, the exact value of which will vary from one appearance to the next. The notation A≍B means that there is a positive constant C such that B/C≤A≤CB.


## 2. Some Lemmas

To begin the discussion, let us state a couple of lemmas, which are used in the proofs of the main results.Lemma 2.1.
Suppose thatf,h∈H(𝔹). Then,
(2.1)R[ThCφ(f)](z)=f(φ(z))Rh(z).Proof.
The proof of this Lemma follows by standard arguments (see, e.g., [9, 29, 30]).Lemma 2.2 (see [2, 31]).
If0<α<1, then ℬ1-α=ℒα(𝔹); furthermore,
(2.2)‖f‖B1-α≍‖f‖Lα
as f varies through ℒα(𝔹).The following criterion for compactness follows from standard arguments similar to the corresponding lemma in [6]. Hence, we omit the details.Lemma 2.3.
Assume thath∈H(𝔹) and φ∈S(𝔹). Suppose that X or Y  is one of the following spaces ℒα(𝔹), F(p,q,s). Then, ThCφ:X→Y is compact if and only if ThCφ:X→Y is bounded, and for any bounded sequence {fk}k∈N in X which converges to zero uniformly on compact subsets of 𝔹 as k→∞, one has ∥ThCφfk∥Y→0 as k→∞.Lemma 2.4 (see [4, 5]).
Iff∈ℬα, then
(2.3)|f(z)|≤C‖f‖Ba,0<α<1,(2.3')|f(z)|≤C‖f‖Baln⁡⁡e1-|z|2,α=1,(2.3'')|f(z)|≤C‖f‖Ba(1-|z|2)α-1,α>1.The next lemma was obtained in [32].Lemma 2.5.
Ifa>0, b>0, then the elementary inequality holds
(2.4)(a+b)p≤{ap+bp,0<p<1,2p-1(ap+bp),p≥1.It is obvious that Lemma2.5 holds for the sum of finite number k, that is, (2.5)(a1+⋯+ak)p≤C(a1p+⋯+akp),
where a1,…,ak>0 and C is a positive constant.Lemma 2.6 (see [4, 5]).
For0<p, s<+∞, -n-1<q<+∞, q+s>-1, there exists C>0 such that
(2.6)sup⁡a∈B∫B(1-|w|2)p|1-〈z,w〉|n+1+q+p(1-|z|2)qgs(z,a)dν(z)≤C
for every ω∈𝔹.Lemma 2.7 (see [4]).
There is a constantC>0 so that, for all t>-1 and z∈𝔹, one has
(2.7)∫B|ln⁡11-〈z,w〉|2(1-|w|2)t|1-〈z,w〉|n+1+tdν(z)≤C(ln⁡11-|z|2)2.Lemma 2.8 (see [4, 5]).
Suppose that0<p, s<∞, -n-1<q<∞, and q+s>-1. If f∈F(p,q,s), then f∈ℬ(n+1+q)/p, and ∥f∥ℬ(n+1+q)/p≤C∥f∥F(p,q,s).Lemma 2.9.
Let{fk}k∈N be a bounded sequence in F(p,q,s) which converges to zero uniformly on compact subsets of the unit ball 𝔹, where (n+1+q)/p<1. Then, limk→∞supz∈𝔹|fk(z)|=0.Proof.
It follows from Lemma2.8 that F(p,q,s)⊆ℬ(n+1+q)/p and ∥f∥ℬ(n+1+q)/p≤C∥f∥F(p,q,s) for any f∈F(p,q,s). So, when (n+1+q)/p<1, the proof of this lemma is similar to that of Lemma 3.6 of [33], hence the proof is omitted.


## 3. The Boundedness and Compactness of the OperatorThCφ:ℒα(𝔹)→F(p,q,s)

Theorem 3.1.
Assume thatα∈(0,1), 0<p, s<∞,  -n-1<q<∞, q+s>-1, φ∈S(𝔹), and h∈H(𝔹). Then, ThCφ:ℒα→F(p,q,s) is bounded if and only if h∈F(p,q,s).Proof.
Assume thath∈F(p,q,s). Since 0<1-α<1, by Lemmas 2.2 and 2.4, for any f∈ℒα, we have
(3.1)|f(z)|≤C‖f‖B1-α≤C‖f‖Lα.
Since |ThCφf(0)|=0, by using Lemma 2.1 and relations (2.3) and (3.1), we have
(3.2)‖ThCφf‖F(p,q,s)p=sup⁡a∈B∫B|f(φ(z))Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤Csup⁡a∈B∫B|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)‖f‖B1-αp≤C‖h‖F(p,q,s)p‖f‖Lαp<∞.
Thus ThCφ:ℒα→F(p,q,s) is bounded.
Conversely, suppose thatThCφ:ℒα→F(p,q,s) is bounded. Taking the function f(z)=1∈ℒα, then
(3.3)‖ThCφf‖F(p,q,s)p=|ThCφf(0)|p+sup⁡a∈B∫B|R(ThCφf)(z)|p(1-|z|2)qgs(z,a)dν(z)=sup⁡a∈B∫B|f(φ(z))Rh(z)|p(1-|z|2)qgs(z,a)dν(z)=sup⁡a∈B∫B|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)=‖h‖F(p,q,s)p.
From which, the boundedness of ThCφ implies that h∈F(p,q,s). This completes the proof of this theorem.Next, we characterize the compactness ofThCφ:ℒα→F(p,q,s).Theorem 3.2.
Assume thatα∈(0,1), 0<p, s<∞, -n-1<q<∞, q+s>-1, φ∈S(𝔹), and h∈H(𝔹). Then, ThCφ:ℒα→F(p,q,s) is compact if and only if ThCφ:ℒα→F(p,q,s) is bounded, and
(3.4)lim⁡⁡r→1sup⁡a∈B∫{|φ(z)|>r}|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)=0.Proof.
Assume thatThCφ:ℒα→F(p,q,s) is bounded and (3.4) holds. It follows from Theorem 3.1 that h∈F(p,q,s).
Now, let{fj}j∈N be a bounded sequence of functions in ℒα such that fj→0 uniformly on the compact subsets of 𝔹 as j→∞. Suppose that supj∈N∥fj∥ℒα≤L. It follows from (3.4) that, for any ɛ>0, there exists r0∈(0,1) such that, for every r0<r<1,
(3.5)sup⁡a∈B∫{|φ(z)|>r}|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)<ε.
Set r0<r<1, then
(3.6)‖ThCφfj‖F(p,q,s)p=sup⁡a∈B∫B|fj(φ(z))|p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤sup⁡a∈B∫{|φ(z)|≤r}|fj(φ(z))|p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)+sup⁡a∈B∫{|φ(z)|>r}|fj(φ(z))|p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)=I1+I2,
where
(3.7)I1:=sup⁡a∈B∫{|φ(z)|≤r}|fj(φ(z))|p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z),I2:=sup⁡a∈B∫{|φ(z)|>r}|fj(φ(z))|p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z).
LetK={w:|w|≤r}, then K is a compact subset of 𝔹. Since fj→0 uniformly on compact subsets of 𝔹 as j→∞ and h∈F(p,q,s), we get
(3.8)I1≤sup⁡w∈K|fj(w)|psup⁡a∈B∫{|φ(z)|≤r}|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤‖h‖F(p,q,s)psup⁡w∈K|fj(w)|p≤Csup⁡w∈K|fj(w)|p⟶0,j⟶∞.
On the other hand, by (3.5) and Lemmas 2.2 and 2.4, it follows that
(3.9)I2≤C‖fj‖B1-αpsup⁡a∈B∫{|φ(z)|>r}|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤C∥fj∥Lαpɛ≤CLpε.
Since ɛ is arbitrary, from the above inequalities, we get
(3.10)lim⁡j→∞‖ThCφfj‖F(p,q,s)=0.
Hence, by (3.10) and Lemma 2.3, we conclude that ThCφ:ℒα→F(p,q,s) is compact.
For the converse direction, we suppose thatThCφ:ℒα→F(p,q,s) is compact. It is obvious that ThCφ:ℒα→F(p,q,s) is bounded.
Now, we prove (3.4). Setting the test functions fl(m)(z)=zlm for fixed l∈{1,…,n}, where z=(z1,…,zn) and m=1,2,…. It is easy to check that ∥fl(m)∥ℒα≤C, and fl(m)→0 uniformly on the compact subsets of 𝔹 as m→∞. Write φ=(φ1,…,φn), since ThCφ:ℒα→F(p,q,s) is compact, by Lemma 2.3, it follows that, as m→∞,
(3.11)‖ThCφfl(m)‖F(p,q,s)p=sup⁡a∈B∫B|φl(z)|mp|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)⟶0.
Note that |φ(z)|2=|φ1(z)|2+⋯+|φn(z)|2≤(|φ1(z)|+⋯+|φn(z)|)2; by the relation (3.11) and Lemma 2.5, we have
(3.12)sup⁡a∈B∫B|φ(z)|mp|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤sup⁡a∈B∫B(∑l=1n|φl(z)|)mp|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤Csup⁡a∈B∫B(∑l=1n|φl(z)|mp)|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)⟶0,m⟶∞.
This means that, for every ε>0, there is m0∈N such that, for every r∈(0,1),
(3.13)rm0psup⁡a∈B∫{|φ(z)|>r}|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)=sup⁡a∈B∫{|φ(z)|>r}rm0p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤sup⁡a∈B∫{|φ(z)|>r}|φ(z)|m0p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)≤sup⁡a∈B∫B|φ(z)|m0p|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)<ε.
Thus, when r>2-(1/m0p), by the above inequality, we obtain
(3.14)sup⁡a∈B∫{|φ(z)|>r}|Rh(z)|p(1-|z|2)qgs(z,a)dν(z)<2ε.
From which, the desired result (3.4) holds. This completes the proof of this theorem.Remark 3.3.
Whenφ(z)=z, the product of extended Cesàro operator ThCφ is the generalized extended Cesàro operator Th; thus, by Theorems 3.1 and 3.2, we have the following two corollaries.Corollary 3.4.
Assume thatα∈(0,1), 0<p, s<∞, -n-1<q<∞, q+s>-1, and h∈H(𝔹). Then, Th:ℒα→F(p,q,s) is bounded if and only if h∈F(p,q,s).Corollary 3.5.
Assume thatα∈(0,1), 0<p, s<∞, -n-1<q<∞, q+s>-1, and h∈H(𝔹). Then, Th:ℒα→F(p,q,s) is compact if and only if Th:ℒα→F(p,q,s) is bounded, and
(3.15)lim⁡r→1sup⁡a∈B∫|z|>r|Rh(z)|p(1-|z|2)qgs(z,a)=0.



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