Báo cáo hóa học: "THE ESSENTIAL NORMS OF COMPOSITION OPERATORS BETWEEN GENERALIZED BLOCH SPACES IN THE POLYDISC AND THEIR APPLICATIONS" docx

ᏮpU n,Ꮾp 0U n, andᏮp 0∗U n denote thep-Bloch space, little p-Bloch space, and little star p-Bloch space in the unit polydisc U n, respectively, wherep, q > 0.. We recall that the essenti

BETWEEN GENERALIZED BLOCH SPACES IN THE POLYDISC AND THEIR APPLICATIONS

ZEHUA ZHOU AND YAN LIU

Received 27 December 2005; Revised 26 June 2006; Accepted 22 July 2006

LetU n be the unit polydisc ofCnandφ =(φ1, , φ n) a holomorphic self-map ofU n

Ꮾp(U n),Ꮾp

0(U n), andᏮp

0∗(U n) denote thep-Bloch space, little p-Bloch space, and little

star p-Bloch space in the unit polydisc U n, respectively, wherep, q > 0 This paper gives

the estimates of the essential norms of bounded composition operatorsC φinduced by

1 Introduction

The class of all holomorphic functions with domainΩ will be denoted by H(Ω), where

Ω is a bounded homogeneous domain inCn Letφ be a holomorphic self-map ofΩ, thecomposition operatorC φinduced byφ is defined by



C φ f(z) = f

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 90742, Pages 1 22

DOI 10.1155/JIA/2006/90742

Following Timoney [5], we say that f ∈ H(Ω) is in the Bloch space Ꮾ(Ω) if

Let∂Ω denote the boundary of Ω Following Timoney [6], forΩ= B nthe unit ball of

Cn,Ꮾ0(B n)= { f ∈ Ꮾ(B n) :Q f(z) →0, asz → ∂B n}; forΩ=Ᏸ the bounded symmetricdomain other than the ballB n,{ f ∈ Ꮾ(Ᏸ) : Q f(z) →0, asz → ∂Ᏸ}is the set of constantfunctions onᏰ So if Ᏸ is a bounded symmetric domain other than the ball, we denotethe Ꮾ0∗(Ᏸ)= { f ∈ Ꮾ(Ᏸ) : Q f(z) →0, asz → ∂ ∗Ᏸ} and call it little star Bloch space;here ∂ ∗Ᏸ means the distinguished boundary of Ᏸ The unit ball is the only boundedsymmetric domainᏰ with the property that ∂ ∗Ᏸ= ∂Ᏸ

LetU nbe the unit polydisc ofCn Timoney [5] shows that f ∈ Ꮾ(U n) if and only if

This definition was the starting point for introducing thep-Bloch spaces.

Letp > 0, a function f ∈ H(U n) is said to belong to thep-Bloch spaceᏮp(U n) if

It is an easy exercise to show thatᏮp(U n) is a Banach space with the norm  · p

forp ≥1; and for 0< p < 1,Ꮾp(U n) is a nonlocally convex topological vector space and

d( f , g) =  f − g  p pis a complete metric for it Its proof idea is basic, we refer the reader

to see the proof ofProposition 3.1or the statement corresponding the Bloch-type spacefor the unit ball in [13]

Just like Timoney [6], if

it is easy to show that f must be a constant Indeed, for fixed z1∈ U, (∂ f /∂z1)(z)(1 −

| z1|2)pis a holomorphic function inz =( 2, , z n)∈ U n −1 Ifz → ∂U n, thenz → ∂U n −1,which implies that

Hence, (∂ f /∂z1)(z)(1 − | z1|2)p ≡0 for everyz ∈ ∂U n −1, and for eachz1∈ U, and

con-sequently (∂ f /∂z1)(z) =0 for everyz ∈ U n Similarly, we can obtain that (∂ f /∂z j)(z) =0for everyz j ∈ U nand eachj ∈ {2, , n }; thereforef ≡const

So, there is no sense to introduce the corresponding little p-Bloch space in this way.

We will say that the littlep-Bloch spaceᏮp

0(U n) is the closure of the polynomials in the

p-Bloch space If f ∈ H(U n) and

we say f belongs to little star p-Bloch spaceᏮp

0∗(U n) Using the same methods as that

of [6, Theorem 4.15], we can show thatᏮp

0(U n) is a proper subspace ofᏮp

0∗(U n) and

Ꮾp

0∗(U n) is a nonseparable closed subspace ofᏮp(U n)

For the unit discU ⊂ C, Madigan and Matheson [1] proved thatC φis always bounded

onᏮ(U) and bounded on Ꮾ0( U) if and only if φ ∈Ꮾ0(U) They also gave the sufficientand necessary conditions thatC φis compact onᏮ(U) or Ꮾ0( U).

The analogues of these facts for the unit polydisc and classical symmetric domainswere obtained by Zhou and Shi in [8–10] They had already shown that C φ is alwaysbounded on the Bloch space of these domains, and also gave some sufficient and necessaryconditions forC φto be compact on those spaces For the results on the unit ball, we referthe reader to see [4,12]

We recall that the essential norm of a continuous linear operatorT is the distance from

T to the compact operators, that is,

In [7], Zhou stated and proved the corresponding compactness characterization for

Ꮾp(U n) for 0< p < 1, however, C φis not always bounded, and the test functions used

in [7] are only suitable for handling the case 0< p < 1 It is therefore natural to

won-der what results can be proven about boundedness and compactness ofC φ onp-Bloch

spaces for an arbitrary positive numberp or, more generally, between possibly different

p- and q-Bloch spaces of multivariable domains In this paper, we answer these questions

completely forU nwith essential norm approach, we give some estimates of the tial norms of bounded composition operatorsC φbetweenᏮp(U n)(Ꮾp

essen-0(U n) orᏮp

0∗(U n))andᏮq(U n)(Ꮾq

ideas of the proof are those used by Shapiro [3] to obtain the essential norm of a position operator on Hilbert spaces of analytic functions (Hardy and weighted Bergmanspaces) in terms of natural counting functions associated withφ This paper generalizes

com-the results on com-the Bloch space for com-the unit disc in [2] and the unit polydisc in [11].Throughout the remainder of this paperC will denote a positive constant, the exact

value of which will vary from one appearance to the next

Our main results are the following

Theorem 1.1 Let φ =(φ1,φ2, , φ n ) be a holomorphic self-map of U n and  C φe the essential norm of a bounded composition operator C φ:Ꮾp(U n)(Ꮾp

Remark 1.3 When n =1, p = q =1, onᏮ(U) we obtain [1, Theorem 2] Since∂U =

∂ ∗ U,Ꮾ0(U) =Ꮾ0∗(U), we can also obtain [1, Theorem 1]

Remark 1.4 When n > 1, p = q =1,C φis always bounded onᏮ(U n), so we can obtainthe corresponding results in [8,11]

The remainder of the present paper is assembled as follows: inSection 2, we state somelemmas for the proof ofTheorem 1.1 In terms of mapping properties of symbolφ, Lem-

mas2.3,2.4, and2.6will give some conditions forC φ to be bounded between possiblydifferent p- and q-Bloch spaces, “little” or “little star” p- and q-Bloch spaces, the methodsused are different from that of [7], since the test functions used in [7] are only suitablefor handling the p-Bloch space for the case 0 < p < 1, not others InSection 3, we givethe proof ofTheorem 1.1 InSection 4, as applications of Theorems1.1and1.2, we givesome corollaries forC φto be compact on those spaces

2 Some lemmas

In order to proveTheorem 1.1, we need some lemmas

Lemma 2.1 Let f ∈Ꮾp(U n ), then

(1) if 0 ≤ p < 1, then | f (z) | ≤ | f (0) |+ (n/(1 − p))  f p ;

(2) if p = 1, then | f (z) | ≤(1 + 1/n ln 2)(n

k =1ln(2/(1 − | z k|2))) f p ; (3) if p > 1, then | f (z) | ≤(1/n + 2 p −1/(p −1))n

k =1(1/(1 − | z k|2)p −1) f p Proof This Lemma can be easily obtained by some integral estimates, so we omit the

Lemma 2.2 For p > 0, set

f w(z) =

z l0

Now we prove that f w ∈Ꮾp

0(U n) Using the asymptotic formula

0∗(U n) Now the proof ofLemma 2.3is completed 

Lemma 2.4 Let φ =(φ1,φ2, , φ n ) be a holomorphic self-map of U n Then C φ:Ꮾp

In order to prove the converse, we first prove that ifφ l ∈Ꮾq

0∗(U n), for every l =

1, 2, , n, then f ◦ φ ∈Ꮾq

0∗(U n) for any f ∈Ꮾp

0∗(U n)

Without loss of generality, we prove this result whenn =2

For any sequence{ z j =( 1j,z2j)} ⊂ U nwithz j → ∂ ∗ U nasj → ∞, then

z j

1 −→1, z j

2 −→1. (2.14)Since| φ1( j)| < 1 and | φ2( j)| < 1, there exists a subsequence { z j s }in{ z j }such that

φ1

z j s  −→ ρ1, φ2

z j s  −→ ρ2, (2.15)

ass → ∞

It is clear that 0≤ ρ1,ρ2≤1 Then fork =1, 2, we have

Now we prove the left-hand side of (2.16)→0 ass → ∞according to four cases

Case 1 If ρ1< 1 and ρ2< 1, there exist r1andr2such thatρ1< r1< 1 and ρ2< r2< 1, so

asj is large enough, | φ1( j s)| ≤ r1and| φ2( j s)| ≤ r2.

ass → ∞

Case 2 If ρ1=1 andρ2=1, thenφ(z j s)→ ∂ ∗ U n, by (2.8) and, since f ∈Ꮾp

0∗(U n), (2.16)yields that

ass → ∞

Case 3 If ρ1< 1 and ρ2=1, similarly toCase 1, we can prove that

1 is a point of modulusR j swhere maximum ofF(w1) is attained This meansthat |(∂ f /∂w2)(φ1( j s),φ2( j s))| ≤ |(∂ f /∂w2)(w j s

1,w j s

2)| Since | w j s

1| →1, | w j s

2| → ρ2=1and f ∈Ꮾp

0∗(U n),



∂w ∂ f2

Combining Cases1,2,3, and4, we know there exists a subsequence{ z j s }in{ z j }suchthat

Remark 2.5 For the case C φ:Ꮾp(U n)→Ꮾq

0∗(U n), the necessity also holds, but we cannotguarantee that the sufficiency holds because we cannot be sure that Cφ f ∈Ꮾq

0∗(U n) forall f ∈Ꮾp(U n)

Lemma 2.6 Let φ =(φ1,φ2, , φ n ) be a holomorphic self-map of U n Then

is bounded if and only if φ γ ∈Ꮾq

0(U n ) for every multiindex γ, and ( 2.8 ) holds.

Proof (sufficiency) From (2.8) and byLemma 2.3we know thatC φ:Ꮾp(U n)→Ꮾq(U n)

The boundedness ofC φ:Ꮾp

0(U n)→Ꮾq

0(U n) directly follows, if we proveC φ f ∈Ꮾq

0(U n)whenever f ∈Ꮾp

0(U n) So, let f ∈Ꮾp

0(U n) By the definition ofᏮp

0(U n) it follows thatfor everyε > 0 there is a polynomial p εsuch that f − p εp < ε Hence

C φ f − C φ p ε q ≤ C φ Ꮾp(U n)→Ꮾq(U n) f − p ε p < ε C φ Ꮾp(U n)→Ꮾq(U n). (2.34)Since φ γ ∈Ꮾq

0(U n) for every multiindexγ, we obtain C φ p ε ∈Ꮾq

0(U n) From this and(2.34) the result follows

Remark 2.7 For the case C φ:Ꮾp(U n)(Ꮾp

0∗(U n))→Ꮾq

0(U n), in analogy toRemark 2.5,the necessity also holds, but we cannot guarantee that the sufficiency holds

Lemma 2.8 If { f k } is a bounded sequence inᏮp(U n ), then there exists a subsequence { f k l }

of { f k } which converges uniformly on compact subsets of U n to a holomorphic function f ∈

Ꮾp(U n ).

Proof Let { f k}be a bounded sequence inᏮp(U n) with f kp ≤ C ByLemma 2.1,{ f j}

is uniformly bounded on compact subsets ofU nand hence normal by Montel’s theorem

So we may extract a subsequence{ f j k }which converges uniformly on compact subsets of

U nto a holomorphic function f It follows that ∂ f j k /∂z l → ∂ f /∂z lfor eachl ∈ {1, 2, , n },so

Lemma 2.9 Let Ω be a domain inCn , f ∈ H( Ω) If a compact set K and its neighborhood

G satisfy K ⊂ G ⊂ G ⊂ Ω and ρ =dist(K, ∂G) > 0, then

3 The proof of Theorem 1.1

Now we turn to the proof ofTheorem 1.1 In the following, we are dealing with the caseforC φ:Ꮾp(U n)→Ꮾq(U n), but if we note that the test functions f mintroduced below be-long toᏮp

Ꮾ0∗(U n)⊂ Ꮾ(U n) for m =1, 2, , and this sequence converges to zero uniformly on

compact subsets of the unit polydiscU n Furthermore

opera-{ f m}to zero and the compactness ofK imply that  K f mq →0 It is easy to show that if

a bounded sequence that is contained inᏮp

0∗(U n) converges uniformly on compact sets ofU n, then it also converges weakly to zero inᏮp

whenever dist(φ(z), ∂U n)< δ0andl =1, 2, , n.

Sincer m →1 asm → ∞, we may choosem large enough so that r m > 1 − δ0 Ifφ(z) ∈

A m,r m ≤ | φ1(z) | ≤ r m+1, so 1− r m+1 < 1 − | φ1(z) | < 1 − r m < δ0; hence dist(φ1(z), ∂U) <

δ0 There existsw1with| w1| =1 such that dist(φ1(z), w1)=dist(φ1(z), ∂U) < δ0.

Letw =(w1,φ2(z), , φ n(z)) ∈ ∂U n Then

Letε →0, the low estimate follows.

To obtain the upper estimate we first prove the following proposition

Proposition 3.1 Let φ =(φ1, , φ n ) be a holomorphic self-map of U n Then for m ≥ 2, the operator K m on H(U n ) defined by K m f (z) = f (((m −1)/m)z) has the following properties For each f ∈ H(U n ),

(i)K m f ∈Ꮾp

0(U n)⊂Ꮾp

0∗(U n)⊂Ꮾp(U n );

(ii) if C φ:Ꮾp(U n)→Ꮾq(U n ) is bounded, then C φ K m f ∈Ꮾq(U n );

(iii) for fixed m, the operator K m is compact onᏮp(U n );

(iv) if C φ:Ꮾp(U n)→Ꮾq(U n ) is bounded, then C φ K m f ∈Ꮾq(U n ) is compact;

(v) I − K m  ≤ 2;

(vi) (I − K m)f converges to zero uniformly on compacta in U n

Proof (i) Let f ∈ H(U n),r m =(m −1)/m, and f m(z) = K m f (z) = f (r m z) First note that

On the other hand, f m ∈ H((1/r m)U n), and observe that (2/(1 + r m))U n ⊂(1/r m)U n

which implies that for fixedm, corresponding to each j =1, 2, , there is a polynomial

Let K = U n, G =(2/(1 + r m))U n, Ω=(1/r m)U n, then K ⊂ G ⊂ G ⊂ Ω and ρ =

dist(K, ∂G) =(1− r m)/(1 + r m)> 0, so for all w ∈ U n, k ∈ {1, , n }, it follows from

(ii) follows immediately from (i).

(iii) For any sequence{ f j} ⊂Ꮾp(U n) with f j p ≤ M, by (i), { K m f j} ∈Ꮾp

0(U n) ByLemma 2.8, there is a subsequence{ f j s }of{ f j}which converges uniformly on compactsubsets of U n to a holomorphic function f ∈Ꮾp(U n) and f p ≤ M The sequence

{ ∂ f j s /∂z i}, =1, 2, , n, also converges uniformly on compact subsets of U nto the morphic function∂ f /∂z i So ass is large enough, for any w ∈ E = {((m −1)/m)z : z ∈

(iv) follows immediately from (i) and (iii).

(v) follows from the fact that for any f ∈Ꮾp(U n), (I − K m)f (0) =0, so

Fort ∈[ m, 1] andz ∈ E, we have | tz k| = t | z k| ≤ | z k| < r, tz ∈ rU n, so there existsM > 0

such that|(∂ f /∂w k)(tz) | ≤ M for all t ∈[ m, 1] andz ∈ E Thus

Let us now return to the proof of the upper estimate For convenience, we remove thesubscriptp from  f p,

G = { w ∈ U n: dist(w, ∂U n)≥ δ }, and observe thatG is a compact subset ofCn

Then by Lemmas2.3,2.4, and2.6, and byProposition 3.1, we deduce

Since dist(G, ∂G3)= δ/2, then byLemma 2.9, (3.30) gives

I1≤2n

√

nC mδ

soI2≤ nC K /m →0 Thus letting firstm → ∞and thenδ →0 in (3.27), we get the upperestimate of C φe:

4 Some corollaries

The following three corollaries follow fromTheorem 1.2

Corollary 4.1 Let φ =(φ1, , φ n ) be a holomorphic self-map of U n Then C φ:

for all z ∈ U n and ( 1.12 ) holds.

Proof ByLemma 2.3, we knowC φ:Ꮾp(U n)(Ꮾp

The proof follows fromLemma 2.4

Corollary 4.3 Let φ =(φ1, , φ n ) be a holomorphic self-map of U n Then C φ:Ꮾp

0(U n)→

Ꮾq

0(U n ) is compact if and only if φ l ∈Ꮾq

0(U n ) for every l =1, 2, , n and ( 1.12 ) holds.

The proof follows fromLemma 2.6

Acknowledgments

The authors would like to thank the editor and referee(s) for helpful comments on themanuscript The first author is supported in part by the National Natural Science Foun-dation of China (Grants no 10671141 and no 10371091) and LiuHui Center for AppliedMathematics, Nankai University & Tianjin University

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Zehua Zhou: Department of Mathematics, Tianjin University, Tianjin 300072, China

E-mail address:zehuazhou2003@yahoo.com.cn

Yan Liu: Department of Mathematics, Tianjin University, Tianjin 300072, China

E-mail address:maple.ly@163.com