Open Access

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

Journal of Inequalities and Applications20112011:46

https://doi.org/10.1186/1029-242X-2011-46

Received: 11 December 2010

Accepted: 2 September 2011

Published: 2 September 2011

Abstract

Let U n be the unit polydisk of n and φ a holomorphic self-map of U n . H(U n ) and B log α ( U n ) denote the space of bounded holomorphic functions and the space of general weighted Bloch functions defined on U n , 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 (U n ) to B log α ( U n ) . As applications, some characterizations of the boundedness and compactness of C φ from H (U n ) 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

Keywords

Composition operator General weighted Bloch space Essential norm Boundedness Weak compactness

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 U n be the unit polydisk of n . Timony [3] showed that f B ( 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 (U n ) 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 (U n ) 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(U n ) 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 U n , 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 U n 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 (U n ) and || f || 2 α . 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 U n ,
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(U n ),
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(U n ) 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 U n 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(U n ) to B log α ( U n ) .

Theorem 3. Let φ = (φ1, ..., φ n ) be a holomorphic self-map of U n 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 U n . 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 U n : 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), ∂U n ) < 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(U n ) → H(U n ) is compact since K m maps every bounded sequence in H ( U n ) converging to zero on compact subsets of U n to the sequence converging to zero in norm of H(U n ). In addition, ||IK m : H(U n ) → H(U n )|| ≤ 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 U n : dist(φ(z), ∂U n ) < δ}, G2 = U n \G1 = {z U n : dist(φ(z), ∂U n ) ≥ δ}, which is a compact subset of U n .
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 (U n ), and the space H(U n ) endowed with the compact open topology τ is a Fréchet space. Further, D j : (H (U n ), τ ) → (H (U n ), 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 (U n ), τ ). Since, by Montel's normal theorem, the closed unit ball of H(U n ) is a compact subset of (H (U n ), 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 U n 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 {z j } U n and ε0 > 0 such that w j = φ(z j ) → ∂U n (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 {z j }, 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(U n ) is bounded and converges to zero uniformly on any compact subset of U n . That is, {f j } weakly converges to zero in H ( U n ). Because H (U n ) has Dunford-Pettis property (See Theorem 5.3 in [20] for H (U), and note the proof there works also for H (U n )), 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 w j ∂U n , 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 U n . 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Wuhan University
(2)
Huaxia College, Wuhan University of Technology

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© Wang and Wu; licensee Springer. 2011

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