# On an Integral-Type Operator from Zygmund-Type Spaces to Mixed-Norm Spaces on the Unit Ball

Authors: Stevo Stević
Journal: Abstract and Applied Analysis
 (2010)

---

## Abstract

The boundedness and compactness of an integral-type operator recently introduced by the author from Zygmund-type spaces to the mixed-norm space on the unit ball are characterized here.


---

## Body

## 1. Introduction

Let𝔹={z∈ℂn:|z|<1} be the open unit ball in ℂn, ∂𝔹 its boundary,dVN the normalized volume measure on 𝔹, and H(𝔹) the class of all holomorphic functions on 𝔹. Strictly positive, bounded, continuous functions on 𝔹 are called weights.For anf∈H(𝔹) with the Taylor expansion f(z)=∑|β|≥0aβzβ, let (1.1)Rf(z)=∑|β|≥0|β|aβzβ
be the radial derivative of f, where β=(β1,β2,…,βn) is a multi-index, |β|=β1+⋯+βn and zβ=z1β1⋯znβn.A positive, continuous functionν on the interval [0,1) is called normal [1] if there are δ∈[0,1) and a and b, 0<a<b such that(1.2)ν(r)(1-r)aisdecreasingon[δ,1),lim⁡r→1ν(r)(1-r)a=0,ν(r)(1-r)bisincreasingon[δ,1),lim⁡r→1ν(r)(1-r)b=∞.
If we say that a function ν:𝔹→[0,∞) is normal, we also assume that it is radial, that is, ν(z)=ν(|z|),z∈𝔹.Letμ be a weight. By 𝒵μ(𝔹)=𝒵μ, we denote the class of all f∈H(𝔹) such that
(1.3)z(f):=sup⁡z∈Bμ(z)|R2f(z)|<∞,
and call it the Zygmund-type class. The quantity z(f) is a seminorm. A norm on 𝒵μ can be introduced by ∥f∥𝒵=|f(0)|+z(f). Zygmund-type class with this norm will be called theZygmund-type space.Thelittle Zygmund-type space on 𝔹, denoted by 𝒵μ,0(𝔹)=𝒵μ,0, is the closed subspace of 𝒵μ consisting of functions f satisfying the following condition
(1.4)lim⁡|z|→1μ(z)|R2f(z)|=0.For0<p, q<∞, and ϕ normal, the mixed-norm space H(p,q,ϕ)(𝔹)=H(p,q,ϕ) consists of all functions f∈H(𝔹) such that
(1.5)‖f‖H(p,q,ϕ)=(∫01Mqp(f,r)ϕp(r)1-rdr)1/p<∞,
where
(1.6)Mq(f,r)=(∫∂B|f(rζ)|qdσ(ζ))1/q,
and dσ is the normalized surface measure on ∂𝔹. For p=q, ϕ(r)=(1-r2)(α+1)/p, and α>-1, the space is equivalent with the weighted Bergman space Aαp(𝔹).In [2], the present author has introduced products of integral and composition operators on H(𝔹) as follows (see also [3–5]). Assume g∈H(𝔹),g(0)=0, and φ is a holomorphic self-map of 𝔹, then we define an operator on H(𝔹) by  (1.7)Pφg(f)(z)=∫01f(φ(tz))g(tz)dtt,f∈H(B),z∈B.
The operator is an extension of the operator introduced in [6]. Here, we continue to study operator Pφg by characterizing the boundedness and compactness of the operator between Zygmund-type spaces and the mixed-norm space. For some results on related integral-type operators mostly in ℂn, see, for example, [3, 6–27] and the references therein.In 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. If both a⪯b and b⪯a hold, then one says that a≍b.


## 2. Auxiliary Results

In this section, we quote several lemmas which are used in the proofs of the main results.The first lemma was proved in [2].Lemma 2.1.
Assume thatφ is a holomorphic self-map of 𝔹,g∈H(𝔹), and g(0)=0. Then, for every f∈H(𝔹) it holds(2.1)R[Pφg(f)](z)=f(φ(z))g(z).The next Schwartz-type characterization of compactness [28] is proved in a standard way (see, e.g., the proof of the corresponding lemma in [11]), hence we omit its proof.Lemma 2.2.
Assumep,q>0,φ is a holomorphic self-map of 𝔹,g∈H(𝔹),g(0)=0,ϕ is normal, and μ is a weight. Then, the operator Pφg:𝒵μ(or𝒵μ,0)→H(p,q,ϕ) is compact if and only if for every bounded sequence (fk)k∈ℕ⊂𝒵μ(or𝒵μ,0) converging to 0 uniformly on compacts of 𝔹 we have limk→∞∥Pφgfk∥H(p,q,ϕ)=0.The next lemma is folklore and can be found, for example, in [6] (one-dimensional case for standard power weights is due to Flett [29, Theorems 6 and 7]).Lemma 2.3.
Assume that0<p, q<∞,ϕ is normal, and m∈ℕ. Then, the following asymptotic relationship holds for every f∈H(𝔹),
(2.2)∫01Mqp(f,r)ϕp(r)1-rdr≍|f(0)|p+∫01Mqp(Rmf,r)(1-r)mpϕp(r)1-rdr.Lemma 2.4.
Assume thatμ is normal and f∈𝒵μ.  Then,
(2.3)|f(z)|≤C‖f‖Zμ(1+∫0|z|∫0tdsμ(s)dt),z∈B.
Moreover, if
(2.4)∫01∫0tdsμ(s)dt<∞,
then
(2.5)|f(z)|≤C‖f‖Zμ,
for any z∈𝔹.Proof.
By Lemma 3.1 in [21] applied to ℜf we have that
(2.6)|Rf(z)|≤C‖f‖Zμ(1+∫0|z|dsμ(s)).
Hence, for |z|≥1/2, we have that
(2.7)|f(z)-f(z2)|≤∫1/21|Rf(tz)|dtt≤C‖f‖Zμ∫1/21(1+∫0t|z|dsμ(s))d(t|z|)|z|,
so that
(2.8)|f(z)|≤M∞(f,12)+C‖f‖Zμ(1+∫0|z|∫0tdsμ(s)dt),
where M∞(f,1/2)=max|z|≤1/2|f(z)|.
If|z|≤1/2, then by the mean value property of the function f(z)-f(0) (see [30]), Jensen’s inequality, and Parseval’s formula, we obtain
(2.9)max⁡|z|≤1/2|f(z)-f(0)|2≤4n∫|z|≤3/4|f(w)-f(0)|2dVN(w)≤4n∫|z|≤3/4|Rf(w)|2dVN(w)≤3nmax⁡|z|≤3/4|Rf(z)|2.
From (2.9) and (2.6), we obtain
(2.10)M∞(f,1/2)≤|f(0)|+(3)nmax⁡|z|≤3/4|Rf(z)|≤|f(0)|+(3)nC‖f‖Zμ(1+∫03/4dsμ(s))≤C‖f‖Zμ.
From (2.8) and (2.10), (2.3) follows, from which by (2.4) the second statement follows.Lemma 2.5.
Assumeμ is normal and (2.4) holds. Then, for every bounded sequence (fk)k∈ℕ⊂𝒵μ converging to 0 uniformly on compacts of 𝔹,  we have that
(2.11)lim⁡k→∞sup⁡z∈B|fk(z)|=0.Proof.
From (2.4), we have that for every ɛ>0, there is a δ∈(0,min{ɛ,1/2}) such that
(2.12)∫(1-δ)|z||z|∫0tdsμ(s)dt<ɛ,
for |z|>1-δ.
Hence, from (2.12) it follows that for each k∈ℕ and |z|≥1-δ(2.13)|fk(z)-fk((1-δ)z)|≤∫1-δ1|Rfk(tz)|dtt≤C‖fk‖Zμ∫1-δ1(1+∫0t|z|dsμ(s))dt≤C‖fk‖Zμ(ɛ+∫(1-δ)|z||z|∫0tdsμ(s)dt).
From (2.12) and (2.13), we obtain
(2.14)|fk(z)|≤sup⁡|w|≤1-δ|fk(w)|+2Cɛsup⁡k∈N‖fk‖Zμ.
Letting k→∞ in this inequality, using the assumption that fk converges to 0 on the compact |w|≤1-δ, and using the fact that ɛ is an arbitrary positive number, the lemma follows.


## 3. The Boundedness and Compactness ofPφg:𝒵μ(or𝒵μ,0)→H(p,q,ϕ)

The boundedness and compactness of the operatorPφg:𝒵μ(or𝒵μ,0)→H(p,q,ϕ) are characterized in this section.Theorem 3.1.
Assume thatp,q>0,φ is a holomorphic self-map of 𝔹,g∈H(𝔹),g(0)=0,ϕ and μ are normal, and  μ satisfies condition (2.4). Let
(3.1)G(z)=∫01g(tz)dtt.
Then, the following statements are equivalent:
(a)Pφg:𝒵μ,0→H(p,q,ϕ) is bounded;
(b)Pφg:𝒵μ→H(p,q,ϕ) is bounded;
(c)Pφg:𝒵μ,0→H(p,q,ϕ) is compact;
(d)Pφg:𝒵μ→H(p,q,ϕ) is compact;
(e)G∈H(p,q,ϕ).
Moreover, ifPφg:𝒵μ→H(p,q,ϕ) is bounded, then the following asymptotic relations hold:
(3.2)‖Pφg‖Zμ→H(p,q,ϕ)≍‖Pφg‖Zμ,0→H(p,q,ϕ)≍‖G‖H(p,q,ϕ).Proof.
The implications(d)⇒(b),(b)⇒(a),(d)⇒(c), and (c)⇒(a) are obvious.
(a)⇒(e) Since Pφg:𝒵μ,0→H(p,q,ϕ) is bounded and f(z)≡1∈𝒵μ,0, by Lemma 2.1 we have that G(z)=Pφg(1)(z)∈H(p,q,ϕ). Moreover,
(3.3)‖G‖H(p,q,ϕ)=‖Pφg(1)‖H(p,q,ϕ)≤‖Pφg‖Zμ,0→H(p,q,ϕ).
(e)⇒(d) Assume that (fk)k∈ℕ⊂𝒵μ is a bounded sequence converging to 0 uniformly on compacts of 𝔹. Then, by Lemmas 2.1, 2.3, and 2.5, we have
(3.4)‖Pφgfk‖H(p,q,ϕ)≍|Pφgfk(0)|+(∫01Mqp(gfk∘φ,r)ϕp(r)(1-r)1-pdr)1/p≤C‖G‖H(p,q,ϕ)sup⁡z∈B|fk(z)|⟶0,ask⟶∞,
which along with Lemma 2.2 implies the compactness of Pφg:𝒵μ→H(p,q,ϕ).
From (2.4) and by Lemmas 2.3 and 2.4, we have
(3.5)‖Pφgf‖H(p,q,ϕ)≤C(∫01Mqp(gf∘φ,r)ϕp(r)(1-r)1-pdr)1/p≤C‖f‖Zμ(∫01Mqp(g,r)ϕp(r)(1-r)1-pdr)1/p≤C‖f‖Zμ‖G‖H(p,q,ϕ).
This, together with (3.3) and the inequality
(3.6)‖Pφg‖Zμ,0→H(p,q,ϕ)≤‖Pφg‖Zμ→H(p,q,ϕ),
implies the asymptotic relations in (3.2), as desired.



---

*Source: 198608-2010-12-02.xml*