On an integral-type operator between bloch-type spaces

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Abstract

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cn. Let φ be a holomorphic self-map of B and gH(B), such that g(0)=0. We study the boundedness and compactness of the following integral-type operator recently introduced by StevićPφg(f)(z)=01f(φ(tz))g(tz)dtt,zB,between Bloch-type spaces. Our main results are natural extensions of some results in the following paper: S. Stević, On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball, J. Math. Anal. Appl. 354 (2009) 426–434.

Introduction

Let Bn=B be the unit ball of Cn, B1=D the open unit disk in C, H(B) the class of all holomorphic functions on B. For fH(B), letRf(z)=j=1nzjfzj(z)represent the radial derivative of f.

For α>0, recall that the α-Bloch space Bα=Bα(B), is the space consisting of all functions fH(B) such thatfBα=|f(0)|+supzB(1-|z|2)α|Rf(z)|<.Under the above norm, Bα is a Banach space. When α=1, we get the classical Bloch space B. For more information of the Bloch space and the α-Bloch space (see, e.g., [3], [43] and the references therein).

A positive continuous function μ on [0,1) is called normal, if there exist positive numbers s and t,0<s<t, and δ[0,1) such that [26]μ(r)(1-r)sis decreasing on[δ,1)andlimr1μ(r)(1-r)s=0;μ(r)(1-r)tis increasing on[δ,1)andlimr1μ(r)(1-r)t=.

For a normal function ω, an fH(B) is said to belong to the Bloch-type space Bω=Bω(B), ifBω(f)supzBω(|z|)|Rf(z)|<,(see, e.g., [39]). The Bloch-type space is a Banach space with the normfBω=|f(0)|+Bω(f).Let Bω,0 denote the subspace of Bω consisting of those fBω for whichlim|z|1ω(|z|)|Rf(z)|=0.This space is called the little Bloch-type space. When μ(r)=(1-r2)α, the induced spaces Bω and Bω,0 become the α-Bloch space Bα and the little α-Bloch space B0α.

Let φ be a holomorphic self-map of B. The composition operator Cφ is defined by(Cφf)(z)=(fφ)(z),fH(B).It is interesting to provide a function theoretic characterization of when φ induces a bounded or compact composition operator on various spaces. Recall that a linear operator is said to be bounded if the image of a bounded set is a bounded set, while a linear operator is compact if it takes bounded sets to sets with compact closure. The book [4] contains plenty of information on this topic. For some recent results in this research area, see, e.g., [3], [5], [6], [20], [30], [32], [40], [41], [42], [46] and the references therein.

Let gH(D) and φ be a holomorphic self-map of D. Products of integral and composition operators on H(D) were introduced by Li and Stević (see, [10], [16], [17], [23], as well as [18], [31] for a related operator) as follows:CφJgf(z)=0φ(z)f(ζ)g(ζ)dζandJgCφf(z)=0zf(φ(ζ))g(ζ)dζ.Operators in (2), (3) are extensions of the following classical integral operator:Tg(f)(z)=0zf(ζ)h(ζ)dζ,which was introduced in [25].

One of the interesting questions has been to extend operators in (2), (3) in the unit ball settings and to study their function theoretic properties between spaces of holomorphic functions on the unit ball in terms of inducing functions.

If gH(B) is such that g(0)=0 and φ is a holomorphic self-map of B, then in [35] Stević introduced the following operator on the unit ball:Pφg(f)(z)=01f(φ(tz))g(tz)dtt,fH(B),zB,and has shown that it is a natural extension of the operator in (3). In [34] Stević proposed a big research project regarding the operator. Some further results in this direction can be found in [36], [37]. A particular case of the operator in (4) was independently introduced in [45] and studied in [44].

For φ(z)=z and gRg the operator Pφg is reduced to operator Tg, so called, extended Cesàro operator or the Riemann–Stieltjes operator, which was studied, e.g., in [1], [2], [7], [8], [9], [11], [12], [13], [14], [15], [19], [21], [22], [27], [39] (see also the references therein). Some related integral-type operators can be found, e.g., in [2], [28], [29], [33], [38].

In this paper, we study the boundedness and compactness of the operator Pφg between Bloch-type spaces. Our main results are natural extensions of some results in [37]. As a corollary, we obtain some characterizations for the boundedness and compactness of the extended Cesàro operator between Bloch-type spaces.

Throughout this paper C will denote constants in this paper, they are positive and may differ from one occurrence to the other. ab means that there is a positive constant C such that aCb. Moreover, if both ab and ba hold, then one says that ab.

Section snippets

Auxiliary results

In order to prove our main results, we need some auxiliary results which are incorporated in the following lemmas. The following lemma can be found, for example, in [39], [42].

Lemma 1

Assume that ω is a normal function on [0,1). If fBω, then|f(z)|C1+0|z|dtω(t)fBωfor some C independent of f.

The following lemma can be proved similar to the proof of Lemma 1 in [24]. We omit the details.

Lemma 2

Assume that ω is a normal function on [0,1). A closed set K in Bω,0 is compact if and only if it is bounded and

Main results and their proofs

Now we are in a position to state and prove our main results in this paper.

Theorem 1

Assume that gH(B),g(0)=0,φ is a holomorphic self-map of B, μ and ω are normal functions on [0,1). Then the following statements are equivalent.

  • (i)

    Pφg:BωBμ is bounded;

  • (ii)

    Pφg:Bω,0Bμ is bounded;

  • (iii)

    supzBμ(|z|)|g(z)|1+0|φ(z)|dtω(t)<.

Moreover, when Pφg:BωBμ is bounded, the following relationship:PφgBωBμsupzBμ(|z|)|g(z)|1+0|φ(z)|dtω(t)holds.

Proof

(iii)(i). Assume that (7) holds. Note that Pφgf(0)=0 for every fH(B). For any fBω

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