Si XU Xuejun ZHANG Shenlian LI

Abstract Let n>1 and B be the unit ball in n dimensions complex space Cn.Suppose that ϕ is a holomorphic self-map of B and ψ ∈H(B) with ψ(0) = 0.A kind of integral operator,composition operator,is defined by

In this paper,the authors characterize the conditions that the composition operator Tϕ,ψ is bounded or compact on the normal weight Zygmund space Zµ(B).At the same time,the sufficient and necessary conditions for all cases are given.

Keywords Normal weight Zygmund space,Composition operator,Boundedness and compactness

1 Introduction

Let Cnbe the Euclidean space of complex dimensionn.Forz= (z1,···,zn) andw=(w1,···,wn) in Cn,the inner product ofzandwis denoted by

LetBdenote the unit ball in Cn.The class of all holomorphic functions onBis denoted byH(B).Forf∈H(B),the complex gradient ∇fand the radial derivativeRfare defined by

Definition 1.1A positive continuous functionµon [0,1) is called normal if there exist constants 0

Such as,and

(n=1,2,···,b>a>0) are the normal functions.

Without loss of generality,letr0=0 in this paper.

LetDbe the disc in complex plane C.Iff∈H(D) andis said to belong to the Zygmund space Z(D).In fact,the function 1-|z|2may be regarded as a kind of weight function.Later,the space is called as the Zygmund type space Zp(D) if the weight function 1-|z|2is generalized to (1-|z|2)p(p >0).In this paper,we generalize the weight function 1-|z|2to the normal functionµ(|z|),and generalize the variable from one complex variable to several complex variables.

Definition 1.2Letµbe a normal function on [0,1).A functionfis said to belong to the normal weight Zygmund space Zµ(B) iff∈H(B) and

It is easy to prove that Zµ(B) is a Banach space under the norm

In particular,it is just the Zygmund space Z(B) whenµ(r)=1-r2or the Zygmund type space Zp(B) whenµ(r)=(1-r2)p(0

Whenn >1,we gave several equivalent norms of Zµ(B) in [1].About various Zygmund type spaces,there have been a lot of work for examples see [1–27].

Definition 1.3Letµbe a normal function on [0,1).f∈H(B) is said to belong to the normal weight Bloch space Bµ(B) iff∈H(B) and

In particular,it is just the Bloch space B(B) whenµ(r)=1-r2.

It is known thatTherefore,theoperatorC(·) is extended to the weightedoperator as follows:

wheregis a given analytic function.

In several complex variables,the extendedoperator is defined by

wheregis a given holomorphic function onBwithg(0)=0.

No matter one complex variables or several complex variables,many mathematicians have done a lot of research on varioustype operators.For example,see [2–4,6–7,9–10,17,24–25,28–42].In practical applications,we often encounter the combination oftype operator and composition operator.In this paper,we consider the following compositiontype operator.

Definition 1.4Letϕ= (ϕ1,···,ϕn) be a holomorphic self-map ofBandψ∈H(B) withψ(0)=0.The compositiontype operator is defined by

Ifϕ(z)=z,thenTϕ,ψis just the extendedoperatorTψ.The purpose of this paper is to characterize the conditions that the compositiontype operatorTϕ,ψis bounded or compact on Zµ(B) whenn >1,and to give the sufficient and necessary conditions for all cases.Ultimately,this problem can be transformed into a kinds of weighted composition operator problem from the normal weight Zygmund space to the normal weight Bloch space in high dimensions.Many scholars have discussed similar problems (see [4,16,18,26–27]etc.).However,so far,for abstract normal weightµ,especially in high dimensions,the sufficient and necessary conditions forTϕ,ψto be bounded or compact on Zµ(B) have not been given.

In this paper,we use the symbolsc,c1,c2,c3,c4to denote positive constants independent of variablesz,w.But they may depend on some parameters or fixed values,with different values in different cases.We say that two quantitiesEandFare equivalent (denoted by “E≍F” in the following ) if there exist two positive constantsA1andA2such thatA1E≤F≤A2E.

2 Some Lemmas

Letµbe a normal function on [0,1) and

For anyWhen 0z∈B,let

Whenz0,we may decomposeutowith 〈z,ξ〉 = 0 andξ∈∂B.By computation,it is clear that

Therefore,there is a constantc>0 such thatfor all 0 ≤t<1.

Further,by (2.1),there exists

For more information on this metric,see [20–21,43–44].In order to prove the main results,we first give some lemmas.

Lemma 2.1Letµbe a normal function on [0,1) andf∈H(B).Then the following conditions are equivalent:

(1)f∈Zµ(B).

Further,I1≍I2≍I3≍||f||Zµ,and the controlling constants are independent off.In particular,I1≤||f||Zµ.

ProofThese results come from [1,Theorem 3.1]and [20,Lemma 2.1].

Lemma 2.2Letµbe a normal function on [0,1).Iff∈Zµ(B),then

ProofThese results comes from [2].

Lemma 2.3Letµbe a normal function on [0,1) and

Theng(r) is strictly increasing on [0,1) and

wherensis the integer part of (1-rs)−1,r0=0,µ(rs)=2−s(s=1,2,···).

ProofThese results come from [45,Theorem 1].

Lemma 2.4Letµbe a normal function on [0,1).Suppose thatkis a positive integer.Let 0

whenr0<|w|<1.

ProofThe first two results come from [19,Lemma 2.5].Notice that

This shows that the third result also holds.

Lemma 2.5Letµbe normal on [0,1).If the sequence {fj(z)} is bounded on Zµ(B) and converges to 0 uniformly on any compact subset ofB.

ProofThese results comes from [2].

Lemma 2.6Letµbe normal on [0,1) such that

For 0< r0<1 andf∈Zµ(B),if |∇f(z)| ≤mwhen |z| ≤r0,then there exists constantc>0 such that

for allr0<|z|<1,whereξ∈∂Bwith 〈z,ξ〉=0.

ProofBy a unitary transformation,we may letz= (|z|,0,···,0) with |z|<1 andξ=(0,1,0,···,0).For fixed 0 ≤ρ <1,we leth(η) =D1(Rf)(ρ,η,0,···,0).Iff∈Zµ(B),then by Lemma 2.1 we have

Therefore,for anyr0<|z|<1 and 0 ≤t≤|z|,we may obtain

Whenr0<|z|<1,we have

3 Boundedness of Tϕ,ψ

Theorem 3.1Letµbe a normal function on[0,1).Forn>1,suppose thatϕ=(ϕ1,···,ϕn)is a holomorphic self-map ofBandψ∈H(B) withψ(0)=0.ThenTϕ,ψis a bounded operator on Zµ(B) if and only if the following results hold:

whereRϕ(z)=(Rϕ1(z),···,Rϕn(z)).

ProofFirst,we prove sufficiency.

For anyf∈Zµ(B),we have

By Lemmas 2.1–2.2 and (2.2),we may obtain

If (3.1)–(3.3) hold,then by Lemma 2.2 and (3.4) we have

This shows thatTϕ,ψis bounded on Zµ(B) by Lemma 2.1.

Conversely,ifTϕ,ψis a bounded operator on Zµ(B),thenψ∈Zµ(B) by takingf0(z)=1 ∈Zµ(B).At the same time,we have

by takingf0,l(z)=zl∈Zµ(B) for anyl∈{1,2,···,n} and Lemma 2.1.

If there is always |ϕ(z)|≤t0(t0is the number in (2.2)),then (3.1)–(3.3) hold by (3.5) andψ∈Zµ(B).Ifthen for any 0w∈Bwith |ϕ(w)|>t0we take

wheregis the function in Lemma 2.3.

By Lemmas 2.3–2.4,it is clear that (∇fw)[ϕ(w)]=(0,0,···,0) and

By Lemma 2.3 and the definitions ofµandg,we have

This shows that ||fw||Zµ≤cby Lemma 2.1.

By the boundedness ofTϕ,ψ,Lemma 2.1 and (3.6),we have

(3.7) andψ∈Zµ(B) show that (3.3) holds.

Similarly,if we take

then we may obtain

This shows that (3.1) holds.

We writewhere 〈ϕ(w),ξ〉=0 withξ∈∂B.Take

It is clear thatfw[ϕ(w)]=0 and

Since |〈z,ξ〉|2+|〈z,z0〉|2≤|z|2<1 and |ϕ(w)|>t0>then

Therefore,by Lemma 2.3 and (3.9),we have

This means that ||fw||Zµ≤cby Lemma 2.1.

By the boundedness ofTϕ,ψand (3.8),Lemmas 2.3–2.4,we have

By (2.1),(3.1),(3.5) and (3.10),this means that (3.2) holds.

The proof is completed.

Corollary 3.1Letµbe a normal function on [0,1).Forn >1,supposeψ∈H(B) withψ(0)=0.Then the extendedoperatorTψis a bounded operator on Zµ(B) if and only if

ProofBy (2.1),it is clear that

Therefore,ifϕ(z)=z,then (3.2) is redundant in Theorem 3.1.

Note 3.1In general,the above two conditions in Corollary 3.1 are not independent.Letabe the parameter in the definition ofµ.If |z|→1−,then we have

This means thatTψis bounded on Zµ(B) if and only ifψ∈B(B) whena>2.Otherwise,it is clear thatTψis bounded on Zµ(B) if and only ifψ∈Zµ(B) when

4 Compactness of Tϕ,ψ

Theorem 4.1Letµbe a normal function on [0,1).Forn >1,suppose thatϕis a holomorphic self-map ofBandψ∈H(B) withψ(0)=0.

(1) If ||ϕ||∞<1thenTϕ,ψis a compact operator on Zµ(B) if and only ifψ∈Zµ(B) and

(2) If ||ϕ||∞= 1 andthenTϕ,ψis a compact operator on Zµ(B) if and only ifψ∈Zµ(B),(4.1) holds and

(3) If ||ϕ||∞= 1 andthenTϕ,ψis a compact operator on Zµ(B) if and only ifψ∈Zµ(B),(4.1)–(4.2) hold and

(4) If ||ϕ||∞= 1 andthenTϕ,ψis a compact operator on Zµ(B) if and only ifψ∈Zµ(B),(4.1)–(4.3) hold and

ProofFirst,we prove sufficiency.

Let {fj(z)} be a sequence which converges to 0 uniformly on any compact subset ofBand||fj||Zµ≤1.Then {|∇fj(z)|} has the same uniformly convergence.

(1) (i) Case ||ϕ||∞<1.

Ifψ∈Zµ(B) and (4.1) holds,then by Lemma 2.1 we have

Ifψ∈Zµ(B) and (4.1) holds,then by Lemmas 2.1 and 2.5 we have

(2) If (4.2) holds,then for anyε>0,there exists<δ <1 such that

whereξ∈∂Bwith 〈ϕ(z),ξ〉=0.

Ifψ∈Zµ(B) and (4.1)–(4.2) hold,then by Lemmas 2.1–2.2,(4.5)–(4.6) and

we have

(3) Ifψ∈Zµ(B),(4.1)–(4.3) hold,then by the proof in (2),Lemma 2.5,(3.4) and

Conversely,ifTϕ,ψis a compact operator on Zµ(B),thenTϕ,ψis bounded on Zµ(B).By Theorem 3.1,it is clear thatψ∈Zµ(B) and (4.1) holds.

This means that (1) is true.

Let {zj}⊂Bis a sequence withand |ϕ(zj)|>t0(j=1,2,···).

(4) Ifψ∈Zµ(B),(4.1)–(4.4) hold,then by the method of proof in (2),(3.4) and

(2) We just need to prove that(4.2)holds.Letgbe the function in Lemma 2.3.We choose function sequence as follows:

It is clear thatTherefore,it is easy to prove that ||fj||≤cand{fj(z)} converges to 0 uniformly on any compact subset ofBby Lemmas 2.1 and 2.3.At the same time,we have

By Lemma 2.1 andψ∈Zµ(B),(4.7) and Lemmas 2.3–2.5,the compactness ofTϕ,ψ,we have

This shows that (4.2) holds.

(3) We just need to prove (4.3).LetRϕ(zj) =with 〈ϕ(zj),ξj〉 = 0 andξj∈∂B(j=1,2,···).We take function sequence

It is easy to prove that ||fj||Zµ≤cand {fj(z)} converges to 0 uniformly on any compact subset ofBby (3.9),Lemmas 2.1 and 2.3–2.4.Otherwise,we have

By Lemma 2.1 and (4.8),ψ∈(B) and Lemmas 2.3–2.5,the compactness ofTϕ,ψ,we have

with 〈ϕ(z),ξ〉=0 andξ∈∂B.

By (2.1),(4.2),(4.9) and |Rϕ(z)| ≍|〈Rϕ(z),ϕ(z)〉|+|〈Rϕ(z),ξ〉| (|ϕ(z)|> t0),it is clear that (4.3) holds.

(4) All that remains is to prove (4.4).We take function sequence

Then ||fj||≤cand {fj(z)} converges to 0 uniformly on any compact subset ofBby simple calculation.At the same time,we have

By Lemmas 2.1,2.3 –2.4,ψ∈(4.2),(4.10) and the compactness ofTϕ,ψ,it is clear that

This shows that (4.4) holds.

The proof is completed.

Corollary 4.1Letµbe normal on [0,1).Forn>1,supposeψ∈H(B) withψ(0)=0.

ProofBy takingϕ(z)=zin Theorem 4.1,it is easy to obtain these results.Otherwise,if(4.12) holds,thenψ∈(B).

Note 4.1Ifa>2,thenTψis a compact operator on(B) if and only ifψ∈B0(B) (the little Bloch space onB).

Acknowledgement The authors thank the referees for their useful suggestions!