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Norms and essential norms of composition operators from Hto general weighted Bloch spaces in the polydisk

Abstract

Let Un be the unit polydisk of n and φ a holomorphic self-map of Un. H(Un) and B log α ( U n ) denote the space of bounded holomorphic functions and the space of general weighted Bloch functions defined on Un, respectively, where α > 0. This paper gives some estimates of the norm and essential norm of the composition operator C φ induced by φ from H(Un) to B log α ( U n ) . As applications, some characterizations of the boundedness and compactness of C φ from H(Un ) to B log α ( U n ) are obtained. Moreover, we also characterize the weak compactness of the composition operator C φ .

MSC(2000):

Primary 47B33; Secondary 32A37

1 Introduction

Let D be a bounded homogeneous domain in n , H (D) the class of all holomorphic functions on D. For φ, a holomorphic self-map of D, the linear operator defined by

C φ ( f ) = f φ , f H ( D ) ,

is called the composition operator with symbol φ. The study of composition operators is fundamental in the study of Banach and Hilbert spaces of holomorphic functions. We refer to the books [1] and [2] for an overview of some classical results on the theory of composition operators.

Let K (z, z) be the Bergman kernel function of D, and the Bergman metric H z (u, u) in D is defined by

H z ( u , u ) = 1 2 j , k = 1 n 2 log K ( z , z ) z j z k ¯ u j u k ¯ ,

where z = (z1, ..., z n ) D and u= ( u 1 , , u n ) n . A function f H (D) is said to be a Bloch function if β f = sup z D Q f ( z ) is finite, where

Q f ( z ) = sup u n \ { 0 } | ( f ) ( z ) u | H z 1 2 ( u , u ) ,

( f ) ( z ) u=<f ( z ) ,ū>= k = 1 n f z k ( z ) u k . By fixing a base point z0 D, the Bloch space B ( D ) of all Bloch functions on D is a Banach space under the norm f B =|f ( z 0 ) |+ β f [3]. For convenience, we assume the bounded homogeneous domain D to contain the origin and take z0 = 0. In [3], Timoney proved that the space H(D) of bounded holomorphic functions on a bounded homogeneous domain D is a subspace of B ( D ) and for each f H ( D ) ,f B C D f , where C D is a constant depending only on the domain D and f = sup z D |f ( z ) |.

Let Un be the unit polydisk of n . Timony [3] showed that fB ( D ) if and only if

sup z U n k = 1 n f z k ( z ) ( 1 - | z k | 2 ) < ,

and |f ( 0 ) |+ sup z U n k = 1 n | f z k ( z ) | ( 1 - | z k | 2 ) is equivalent to the Bloch norm f B . This characterization was the starting point for introducing α-Bloch spaces. For α > 0, the α -Bloch space B α ( U n ) is defined as follows.

B α ( U n ) = f H ( U n ) : sup z U n k = 1 n f z k ( z ) ( 1 - | z k | 2 ) α < .

Recently, Li and Liu [4] introduced the notation of general weighted Bloch spaces (Stević called these the logarithmic Bloch-type spaces in [5]) in polydisk. For α > 0, a function f H (Un) is said to belong to the general weighted Bloch space B log α ( U n ) if

sup z U n k = 1 n f z k ( z ) ( 1 - | z k | 2 ) α log 2 1 - | z k | 2 < .

It is easy to show that B log α ( U n ) is a Banach space with the norm

f B log α = | f ( 0 ) | + sup z U n k = 1 n f z k ( z ) ( 1 - | z k | 2 ) α log 2 1 - | z k | 2 .

Composition operators on various Bloch-type spaces have been studied extensively by many authors. For the unit disk U, Madigan and Matheson [6] proved that C φ is always bounded on B ( U ) . They also gave some sufficient and necessary conditions that C φ is compact on B ( U ) . Since then, there were many authors generalizing the results in [6] to the unit ball, polydisk and other classical symmetric domains, see, for example, [717]. At the same time, there were also many papers dealing with the composition operators between Bloch-type spaces and bounded holomorphic function spaces, refer to [18, 19] and the references therein for the details. Specially, Li and Liu [4] stated and proved the corresponding boundedness and compactness characterizations for C φ from H(Un) to B log α ( U n ) . But there is a little gap in the proof [[4], line 17, p. 1637]. In this paper, we apply methods developed by Montes-Rodriguez [9] to give some estimates of the norm and essential norm of C φ from H(Un) to B log α ( U n ) . Recall that the essential norm ||T|| e of a bounded operator T between Banach spaces X and Y is defined as the distance from T to the space of compact operators from X to Y. Notice that ||T|| e = 0 if and only if T is compact, so that estimates on || T || e lead to conditions for T to be compact. For convenience, we define || T|| e = || T|| = for any unbounded linear operator T. As an application of our estimates, we obtain the main results in [4] with new proofs. In addition, we also show the equivalence of the compactness and weak compactness of C φ : H ( U n ) B log α ( U n ) .

Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one occurrence to the others.

2 The norm of C φ

In this section, we give the following estimate of the norm of C φ : H ( U n ) B log α ( U n ) .

Theorem 1. Let α > 0 and φ = (φ1, ..., φ n ) be a holomorphic self-map of the unit polydisk Un, then

sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | ( φ l ( z ) | 2 log 2 1 | z k | 2 C φ : H ( U n ) log α ( U n ) 1 + sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | ( φ l ( z ) | 2 log 2 1 | z k | 2 .

Here and in the sequel, the symbol A B(or B A) means that A CB for some positive constant C independent of A and B. A ~ B means that A B and B A.

Proof. For the lower estimate: C φ : H ( U n ) log α ( U n ) sup z U n k , l = 1 n | φ ι l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 . If C φ : H ( U n ) B log α ( U n ) =, then the result is trivially true. Now suppose C φ : H ( U n ) B log α ( U n ) is bounded. For any fixed w Un and k {1, ..., n}, take the following test function

f ( z ) = ( 1 | ( φ k ( w ) | 2 ) α ( 1 ( φ k ( w ) ¯ z k ) α , z U n .

Then, f H (Un) and || f || 2α. Fix any δ (0, 1). If | φ k ( w )| ≥ δ, then

> C φ : H ( U n ) log α ( U n ) C φ f log α sup z U n j = 1 n | ( f ° φ ) z j ( z ) | ( 1 | z j | 2 ) α log 2 1 | z j | 2 = sup z U n j = 1 n | f w k ( φ ( z ) ) φ k z j ( z ) | ( 1 | z j | 2 ) α log 2 1 | z j | 2 = sup z U n j = 1 n | α φ k ( w ) ¯ ( 1 | ( φ k ( w ) 2 | ) α ( 1 φ k ( w ) ¯ φ k ( z ) ) α + 1 φ k z j ( z ) | ( 1 | z j | 2 ) α log 2 1 | z j | 2 j = 1 n α | φ k ( w ) | ( 1 | w j | 2 ) α 1 | ( φ k ( w ) | 2 | φ k z j ( w ) | log 2 1 | w j | 2 > ˜ j = 1 n | φ k z j ( w ) | ( 1 | w j | 2 ) α 1 | ( φ k ( w ) | 2 log 2 1 | w j | 2 .

If | φ k (w)| < δ, then

j = 1 n | φ k z j ( w ) | ( 1 | w j | 2 ) α 1 | ( φ k ( w ) | 2 log 2 1 | w j | 2 j = 1 n | φ k z j ( w ) | ( 1 | w j | 2 ) α log 2 1 | w j | 2 C φ z k log α C φ : H ( U n ) log α ( U n ) z k = C φ : H ( U n ) log α ( U n ) < .

That is, for any w Un,

j = 1 n | φ k z j ( w ) | ( 1 | w j | 2 ) α 1 | ( φ k ( w ) | 2 log 2 1 | w j | 2 C φ : H ( U n ) log α ( U n ) .

Since k {1, ..., n} is arbitrary, so

sup w U n k , j = 1 n | φ k z j ( w ) | ( 1 | w j | 2 ) α 1 | ( φ k ( w ) | 2 log 2 1 | w j | 2 C φ : H ( U n ) log α ( U n ) .

For the upper estimate: C φ : H ( U n ) log α ( U n ) 1 + sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ ι l ( z ) | 2 log 2 1 | z k | 2 . We also assume sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 < , since for the other case nothing needs to be proven. For any f H(Un),

k = 1 n | ( f φ ) z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 k = 1 n l = 1 n | f w l ( φ ( z ) ) φ l z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 l = 1 n | f w l ( φ ( z ) ) | ( 1 | φ l ( z ) | 2 ) k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ( k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ) f ( k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ) f ,

where f B f is used in the last line above.

Since

C φ f log α = | f ( φ ( 0 ) ) | + sup z U n k = 1 n | ( f φ ) z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 ( 1 + sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ) f .

Taking the supremum over all f H(Un) with || f ||∞ ≤ 1, we have

C φ : H ( U n ) log α ( U n ) 1 + sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ,

which completes the proof.

The following corollary is obtained immediately from Theorem 1.

Corollary 2. Let φ = (φ 1 , ..., φ n ) be a holomorphic self-map of Un and α > 0. Then, C φ : H ( U n ) B log α ( U n ) is bounded if and only if

sup z U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 < .

3 The essential norm of C φ

This section mainly gives the following estimate of the essential norm of C φ from H(Un) to B log α ( U n ) .

Theorem 3. Let φ = (φ1, ..., φ n ) be a holomorphic self-map of Un and α > 0, then

C φ : H ( U n ) log α ( U n ) e ~ lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 .

Proof. For the lower estimate:

C φ : H ( U n ) log α ( U n ) e lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 .

It is trivial when C φ is unbounded. So we assume that C φ is bounded. By Corollary 2,

k = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 1 , l = 1, , n .
(1)

Take f m ( z ) = z 1 m ( m 2 ) , then ||f m || = 1 and f m (z) converge to zero uniformly on any compact subset of Un. So K f m B log α 0 for any compact operator K: H ( U n ) B log α ( U n ) . Then

C φ K lim sup m ( C φ K ) f m log α = lim sup m C φ f m log α lim sup m sup z U n k = 1 n | ( f m φ ) z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 lim sup m sup z A m k = 1 n | ( f m φ ) z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 = lim sup m sup z A m k = 1 n | m φ 1 m 1 ( z ) φ 1 z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 = lim sup m sup z A m k = 1 n | φ 1 z k ( z ) | ( 1 | z k | 2 ) α 1 | φ 1 ( z ) | 2 log 2 1 | z k | 2 m | ( φ 1 ( z ) | m 1 ( 1 | ( φ 1 ( z ) | 2 ) lim sup m sup z A m k = 1 n | φ 1 z k ( z ) | ( 1 | z k | 2 ) α 1 | φ 1 ( z ) | 2 log 2 1 | z k | 2 lim inf m min z A m m | φ 1 ( z ) | m 1 ( 1 | ( φ 1 ( z ) | 2 ) ,
(2)

where A m = {z Un : r m ≤ |φ1(z)| ≤ rm + 1}, r m = ( m 1 m + 1 ) 1 / 2 . Since y = mxm-1(1 - x2), x [0, 1), is increasing on [0, r m ] and decreasing on [r m , 1), min z A m m | φ 1 ( z ) | m 1 ( 1 | φ 1 ( z ) | 2 ) = ( m m + 2 ) m 1 2 2 m m + 2 2 e ( as m ) . From (2), we have

C φ K lim sup m sup z A m k = 1 n | φ 1 z k ( z ) | ( 1 | z k | 2 ) α 1 | φ 1 ( z ) | 2 log 2 1 | z k | 2 .

It is from (1) that

lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 = a l < , l = 1, , n .

Then, for any ε > 0, there is δ0 (0, 1) such that

k = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 > a l ε ,

whenever dist (φ(z), ∂Un) < δ0. Again r m ↑ 1, so for m large enough,

C φ : H ( U n ) log α ( U n ) e l = 1 n a l ε , = lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ε .

So ||C φ K|| a1ε. Since K is arbitrary,

C φ : H ( U n ) B log α ( U n ) e a 1 - ε .

Similarly, considering the functions f ( z ) = z l m , l = 2, ..., n, we also have

C φ : H ( U n ) B log α ( U n ) e a l - ε .

Thus,

C φ : H ( U n ) log α ( U n ) e lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | ( φ l ( z ) | 2 log 2 1 | z k | 2 .

Again ε is arbitrary, and we obtain the desired lower estimate.

For the upper estimate:

C φ : H ( U n ) log α ( U n ) e lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | ( φ l ( z ) | 2 log 2 1 | z k | 2 .

If lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 = , then the estimate is trivial too. Now we suppose lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 = , then C φ : H ( U n ) B log α ( U n ) is bounded by Corollary 2. Define the operators K m (m ≥ 2) as follows

K m f ( z ) = f m - 1 m z .

It is easy to see that K m : H(Un) → H(Un) is compact since K m maps every bounded sequence in H( Un) converging to zero on compact subsets of Un to the sequence converging to zero in norm of H(Un). In addition, ||IK m : H(Un) → H(Un)|| ≤ 2. Therefore, C φ K m : H ( U n ) B log α ( U n ) is compact. Then

C φ : H ( U n ) B log α ( U n ) e C φ - C φ K m = C φ ( I - K m ) = sup f 1 C φ ( I - K m ) f B log α = sup f 1 | ( I - K m ) f ( φ ( 0 ) ) | + sup f 1 sup z U n k , l = 1 n ( I - K m ) f w l ( φ ( z ) ) φ l z k ( z ) ( 1 - | z k | 2 ) α log 2 1 - | z k | 2 = I 1 + I 2 .

Where I 1 = sup f 1 | ( I - K m ) f ( φ ( 0 ) ) | and I 2 = sup f 1 sup z U n k , l = 1 n | ( I K m ) f w l ( φ ( z ) ) φ l z k ( z ) | ( 1 | z k | 2 ) α log 2 1 | z k | 2 .

Fix δ (0, 1) and let G1 = {z Un : dist(φ(z), ∂Un) < δ}, G2 = Un\G1 = {z Un : dist(φ(z), ∂Un) ≥ δ}, which is a compact subset of Un.

I 2 = sup f 1 sup z G 1 k , l = 1 n ( I - K m ) f w l ( φ ( z ) ) φ l z k ( z ) ( 1 - | z k | 2 ) α log 2 1 - | z k | 2 + sup f 1 sup z G 2 k , l = 1 n ( I - K m ) f w l ( φ ( z ) ) φ l z k ( z ) ( 1 - | z k | 2 ) α log 2 1 - | z k | 2 = J 1 + J 2 .

Where

J 1 = sup f 1 sup z G 1 k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 | ( I K m ) f w l ( φ ( z ) ) | ( 1 | φ l ( z ) | 2 ) sup f 1 sup z G 1 k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ( I K m ) f sup f 1 sup z G 1 k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 ( I K m ) f sup z G 1 k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 I K m sup z G 1 k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | ( φ l ( z ) | 2 log 2 1 | z k | 2 .

And

J 2 = sup | | f | | 1 sup z G 2 k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | ( φ l ( z ) | 2 log 2 1 | z k | 2 | ( I K m ) f w l ( φ ( z ) ) | ( 1 | ( φ l ( z ) | 2 ) l = 1 n sup f 1 sup z G 2 | ( I K m ) f w l ( φ ( z ) ) | .

It is clear that the sequence of operators {IK m } m satisfies lim m ( I - K m ) f=0 for each f H (Un), and the space H(Un) endowed with the compact open topology τ is a Fréchet space. Further, D j : (H (Un ), τ ) → (H (Un), τ ) defined by D j f= f z j is a contionuous linear operator. Therefore, by the Banach-Steinhaus theorem, the sequence {D j ° (IK m )} m converges to zero uniformly on compact subsets of (H (Un), τ ). Since, by Montel's normal theorem, the closed unit ball of H(Un) is a compact subset of (H (Un), τ ), we conclude that

lim m sup f 1 sup z G 2 ( I - K m ) f w l ( φ ( z ) ) =0,l=1,,n.

Thence, J2 → 0 (as m → ∞).

Similarly, we know that I 1 = sup f 1 | ( I - K m ) f ( φ ( 0 ) ) |0, ( as m ) .

Consequently,

C φ : H ( U n ) log α ( U n ) e lim sup m C φ ( I K m ) lim sup m I 1 + lim sup m J 1 + lim sup m J 2 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 .

Thus

C φ : H ( U n ) log α ( U n ) e lim δ 0 sup d i s t ( φ ( z ) , U n ) < δ k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 .

The proof is complete.

As an application, we have the following corollary.

Corollary 4. Let φ = (φ1, ..., φ n ) be a holomorphic self-map of Un and α > 0. Then the following are equivalent.

  1. (1)

    C φ : H ( U n ) B log α ( U n ) is compact.

  2. (2)

    C φ : H ( U n ) B log α ( U n ) is weakly compact.

  3. (3)

    lim φ ( z ) U n k , l = 1 n | φ l z k ( z ) | ( 1 | z k | 2 ) α 1 | φ l ( z ) | 2 log 2 1 | z k | 2 = 0 .

Proof. (1) (2) is obvious, and (3) (1) follows immediately from Theorem 3. So it suffices to prove (2) (3). Now assume that C φ : H ( U n ) B log α ( U n ) is weakly compact. If (3) is not true, then there is a sequence {zj} Un and ε0 > 0 such that wj = φ(zj) → ∂Un (as j → ∞) together with

k , l = 1 n | φ l z k ( z j ) | ( 1 | z k j | 2 ) α 1 | ( φ l ( z j ) | 2 log 2 1 | z k j | 2 ε 0 ,
(3)

for each j ≥ 1. Since C φ is weakly compact, C φ is bounded. Then, by Corollary 2,

k , l = 1 n | φ l z k ( z j ) | ( 1 | z k j | 2 ) α 1 | φ l ( z j ) | 2 log 2 1 | z k j | 2 1.

Extracting a subsequence of {zj}, if needed, we may assume that lim j | φ l ( z j ) | exists for every l and

| φ l z k ( z j ) | ( 1 | z k j | 2 ) α 1 | ( φ l ( z j ) | 2 log 2 1 | z k j | 2 a l k [ 0, ) , ( as j ) .

From (3), there are k0 and l0 such that a l 0 k 0 >0, i.e.,

| φ l 0 z k 0 ( z j ) | ( 1 | z k 0 j | 2 ) α 1 | ( φ l 0 ( z j ) | 2 log 2 1 | z k 0 j | 2 a l 0 k 0 > 0.
(4)

If | w l 0 j |1, define f j ( z ) = 1 - | w l 0 j | 2 1 - z l 0 w l 0 j ¯ . Then, the sequence {f j } j H(Un) is bounded and converges to zero uniformly on any compact subset of Un. That is, {f j } weakly converges to zero in H( Un). Because H(Un) has Dunford-Pettis property (See Theorem 5.3 in [20] for H(U), and note the proof there works also for H(Un)), the weak compactness of C φ : H ( U n ) B log α ( U n ) implies that C φ f j log α 0 ( as j ) . But this is impossible since using (4) we may estimate that for each j ≥ 1,

C φ f j log α k = 1 n | ( f j φ ) z k ( z j ) | ( 1 | z k j | 2 ) α log 2 1 | z k j | 2 = k = 1 n | f j w l 0 ( φ ( z j ) ) φ l 0 z k ( z j ) | ( 1 | z k j | 2 ) α log 2 1 | z k j | 2 = | w l 0 j | k = 1 n ( 1 | z k j | 2 ) α 1 | φ l 0 ( z j ) | 2 | φ l 0 z k ( z j ) | log 2 1 | z k j | 2 | w l 0 j | | φ l 0 z k 0 ( z j ) | ( 1 | z k 0 j | 2 ) α 1 | φ l 0 ( z j ) | 2 log 2 1 | z k 0 j | 2 a l 0 k 0 > 0.

If | w l 0 j |ρ<1. Since wj∂Un, there is l1 {1, ..., n}\{l0} such that | w l 1 j |1. If there exists k1 such that

| φ l 1 z k 1 ( z j ) | ( 1 | z k 1 j | 2 ) α 1 | φ l 1 ( z j ) | 2 log 2 1 | z k 1 j | 2 a l 1 k 1 > 0,

then as in the last paragraph above we obtain the desired contradiction using the following test functions:

g j ( z ) = 1 - | w l 1 j | 2 1 - z l 1 w l 1 j ¯ .

Thus, we may assume that

| φ l 1 z k ( z j ) | ( 1 | z k j | 2 ) α 1 | ( φ l 1 ( z j ) | 2 log 2 1 | z k j | 2 0, ( as j ) ,
(5)

for each k. We now define the test functions h j as follows

h j ( z ) = ( z l 0 + 2 ) 1 - | w l 1 j | 2 1 - z l 1 w l 1 j ¯ .

Then, ||h j || 1 and h j converge to zero uniformly on any compact subset of Un. But for any j large enough

C φ h j log α k = 1 n | ( h j φ ) z k ( z j ) | ( 1 | z k j | 2 ) α log 2 1 | z k j | 2 = k = 1 n | φ l 0 z k ( z j ) + ( w l 0 j + 2 ) w l 1 j ¯ 1 1 | w l 1 j | 2 φ l 1 z k ( z j ) | | | ( 1 | z k j | 2 ) α log 2 1 | z k j | 2 k = 1 n ( 1 | z k j | 2 ) α | φ l 0 z k ( z j ) | log 2 1 | z k j | 2 k = 1 n | w l 1 j | | w l 0 j + 2 | | φ l 1 z k ( z j ) | ( 1 | z k j | 2 ) α 1 | w l 1 j | 2 log 2 1 | z k j | 2 k = 1 n ( 1 | z k j | 2 ) α | φ l 0 z k ( z j ) | log 2 1 | z k j | 2 ( 1 | z k 0 j | 2 ) α | φ l 0 z k 0 ( z j ) | log 2 1 | z k 0 j | 2 ( 1 | z k 0 j | 2 ) α 1 | ( φ l 0 ( z j ) | 2 | φ l 0 z k 0 ( z j ) | log 2 1 | z k 0 j | 2 a l 0 k 0 > 0 ,

the inequalities in the third and fourth lines above follow from (5), and the last line is due to| φ l 0 ( z j ) |ρ<1. This contradicts again C φ h j B log α 0, which completes the proof.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No.10901158,11071190).

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Correspondence to Maofa Wang.

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The authors declare that they have no competing interests.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Wang, M., Wu, H. Norms and essential norms of composition operators from Hto general weighted Bloch spaces in the polydisk. J Inequal Appl 2011, 46 (2011). https://doi.org/10.1186/1029-242X-2011-46

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Keywords

  • Composition operator
  • General weighted Bloch space
  • Essential norm
  • Boundedness
  • Weak compactness