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General Toeplitz operators on weighted Bloch-type spaces in the unit ball of C n

Journal of Inequalities and Applications20132013:237

https://doi.org/10.1186/1029-242X-2013-237

  • Received: 17 October 2012
  • Accepted: 17 April 2013
  • Published:

Abstract

In this paper, we consider the weighted Bloch-type spaces B ω α , β ( B n ) with α > 0 and β 0 in the unit ball of C n . We present some basic properties of the spaces B ω α , β ( B n ) , then we consider the Toeplitz operator T μ α , β ; ω acting between B ω α , β ( B n ) spaces, where μ is a positive Borel measure in the unit ball B n . Moreover, we characterize complex measures μ for which the Toeplitz operator T μ α , β ; ω is bounded or compact on B ω α , β ( B n ) .

MSC: 47B35, 32A18.

Keywords

  • Toeplitz operators
  • weighted Bloch-type spaces
  • weighted Bergman spaces

1 Introduction

We start here with some terminology, notations and definitions of various classes of analytic functions defined on the unit ball of C n .

Let B n be the unit ball of the n-dimensional complex Euclidean space C n . The boundary of B n is denoted by S n and is called the unit sphere in C n . Occasionally, we will also need the closed unit ball B ¯ n . We denote the class of all holomorphic functions on the unit ball B n by H ( B n ) . The ball centered at z C n with radius r is denoted by B ( z , r ) . For α > 1 , let d ν α ( z ) = c α ( 1 | z | 2 ) α d ν , where is the normalized Lebesgue volume measure on B n and c α = Γ ( n + α + 1 ) n ! Γ ( α + 1 ) (where Γ denotes the gamma function) so that ν α ( B n ) 1 . The surface measure on S n is denoted by . Once again, we normalize σ so that σ α ( S n ) 1 . For any z = ( z 1 , z 2 , , z n ) , w = ( w 1 , w 2 , , w n ) C n , the inner product is defined by
z , w = k = 1 n z k w ¯ k .
For every point a B n , the Möbius transformation φ a : B n B n is defined by
φ a ( z ) = a P a ( z ) S a Q a ( z ) 1 z , a , z B n ,
where S a = 1 | a | 2 , P a ( z ) = a z , a | a | 2 , P 0 = 0 and Q a = I P a . The map φ a has the following properties that φ a ( 0 ) = a , φ a ( a ) = 0 , φ a = φ a 1 and
1 φ a ( z ) , φ a ( w ) = ( 1 | a | 2 ) ( 1 z , w ) ( 1 z , a ) ( 1 a , w ) ,
where z and w are arbitrary points in B n . In particular,
1 | φ a ( z ) | 2 = ( 1 | a | 2 ) ( 1 | z | 2 ) | 1 z , a | 2 .
For f H ( B n ) , the holomorphic gradient of f at z is defined by
f ( z ) = ( f z 1 ( z ) , f z 2 ( z ) , , f z n ( z ) )
and the radial derivative of f at z is defined by
R f ( z ) = f , z ¯ = j = 1 n z j f ( z ) z j .
Similarly, the Möbius invariant complex gradient of f at z is defined by
˜ f ( z ) = ( f φ z ) ( 0 ) .
For α > 0 , a function f H ( B n ) is said to belong to the α-Bloch spaces B α ( B n ) if (see [1])
b α ( f ) ( B n ) = sup z B n | f ( z ) | ( 1 | z | 2 ) α < .
The little Bloch space B 0 α ( B n ) consists of all f B α ( B n ) such that
lim | z | 1 | f ( z ) | ( 1 | z | 2 ) α = 0 .
With the norm f B α = | f ( 0 ) | + b α ( f ) ( B n ) , we know that B α ( B n ) becomes a Banach space and B 0 α ( B n ) is its closed subspace (see [1]). For α = 1 , the spaces B 1 ( B n ) and B 0 1 ( B n ) become the Bloch and the little Bloch space, respectively (see, for example, [25]). Zhu in [5] says that the norm f B ( B n ) is equivalent to
| f ( 0 ) | + sup z B n | R f ( z ) | ( 1 | z | 2 ) .
For α > 1 and 0 < p < , the weighted Bergman space A α p ( B n ) consists of holomorphic functions f L p ( B n , d ν α ) such that
f A α p ( B n ) p : = B n | f ( z ) | p d ν α ( z ) < ,
that is, A α p ( B n ) = L p ( B n , d ν α ) H ( B n ) . When the weight α = 0 , we simply write A p ( B n ) for A 0 p ( B n ) . In the special case when p = 2 , A α 2 ( B n ) is a Hilbert space. It is well known that for α > 1 , the Bergman kernel of A α 2 ( B n ) is given by
K α ( z , w ) = 1 ( 1 z , w ) n + α + 1 , z , w B n .
For α > 1 , a complex measure μ is such that
| B n ( 1 | w | 2 ) α d μ ( w ) | = | B n d μ α ( w ) | < .
The general Bergman projection P α is the orthogonal projection of the measure μ from L 2 ( B n , d ν α ) into A α 2 ( B n ) defined by
P α ( μ ) ( z ) = c α B n ( 1 | w | 2 ) α ( 1 z , w ) n + α + 1 d μ ( w ) = c α B n d μ α ( w ) ( 1 z , w ) n + α + 1 .
The general Bergman projection of the function f is
P α f ( z ) = c α B n f ( w ) ( 1 | w | 2 ) α ( 1 z , w ) n + α + 1 d ν ( w ) = c α B n f ( w ) d ν α ( w ) ( 1 z , w ) n + α + 1 .
Let ω : ( 0 , 1 ] ( 0 , ) be a right-continuous and nondecreasing function. For a complex measure μ , α > 1 , β 0 , and f L 1 ( B n , d ν α + β ) , define weighted general Toeplitz operator as follows:
T μ α , β ; ω f ( z ) = c α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n ( 1 | w | 2 ) α + β f ( w ) ( 1 z , w ) n + α + β + 1 d μ ( w ) = c α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n f ( w ) d μ α + β ( w ) ( 1 z , w ) n + α + β + 1 .

Thus P α + β ; ω ( μ ) ( z ) = T μ α , β ; ω ( 1 ) ( z ) , where 1 stands for a constant function.

Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see [6] and [7]. Boundedness and compactness of the general Toeplitz operators T μ α on the α-Bloch B α ( D ) spaces have been investigated in [8] on the unit disk D for 0 < α < . Also, in [9], the authors extended the general Toeplitz operator T μ α to B α ( B n ) with 1 α < 2 . Recently, in [10], the general Toeplitz operators T μ α on the analytic Besov B p ( D ) spaces with 1 p < have been investigated. Under a prerequisite condition, the authors characterized a complex measure μ on the unit disk for which T μ α is bounded or compact on the Besov space B p ( D ) . For more studies on the Toeplitz operator, we refer to [1117].

In this paper, we consider the weighted Bloch-type spaces B ω α , β ( B n ) with α > 0 and β 0 in the unit ball of C n . We prove a certain integral representation theorem that is used to determine the degree of growth of the functions in the space B ω α , β ( B n ) . It is also proved that the space B ω α , β ( B n ) is a Banach space for each weight α > 0 , β 0 , and the Banach dual of the Bergman space A 1 ( B n ) is B ω α , β ( B n ) for each α 1 , β 0 . Further, we extend the Toeplitz operator T μ α , β ; ω to B ω α , β ( B n ) in the unit ball of C n and completely characterize the positive Borel measure μ such that T μ α , β ; ω is bounded or compact in B ω α , β ( B n ) spaces with α + β 1 .

Throughout the paper, we say that the expressions A and B are equivalent, and write A B whenever there exist positive constants C 1 and C 2 such that C 1 A B C 2 A . As usual, the letter C denotes a positive constant, possibly different on each occurrence. Hereafter, ω stands for a right-continuous and nondecreasing function.

Theorem 1.1 (see [[5], Theorem 1.12])

Suppose b is real and s > 1 . Then the integrals
I b ( z ) = S n d σ ( ζ ) | 1 z , ζ | n + b , z B n
and
J b , s ( z ) = B n ( 1 | w | 2 ) s d ν ( w ) | 1 z , w | n + 1 + s + b , z B n ,

have the following asymptotic properties.

(1) If b < 0 , then I b ( z ) and J b , s ( z ) are both bounded in B n .

(2) If b = 0 , then
I b ( z ) I b , s ( z ) log 1 1 | z | 2 as | z | 1 1 .
(3) If b > 0 , then
I b ( z ) J b , s ( z ) ( 1 | z | 2 ) b as | z | 1 1 .

Lemma 1.1 (see [[5], Lemma 3.3])

Suppose γ is a real constant and g L 1 ( B n , d ν ) . If
u ( z ) = ( 1 | φ a ( z ) | 2 ) β B n g ( w ) d ν ( w ) ( 1 z , w ) γ , z B n ,
then
| ˜ u ( z ) | 2 | γ | ( 1 | z | 2 ) 1 2 B n g ( w ) d ν ( w ) | 1 z , w | γ + 1 2 , z B n .

Let β ( , ) be the Bergman metric on B n . Denote the Bergman metric ball at w ( j ) by B ( w ( j ) , r ) = { z B n : β ( w ( j ) , z ) < r } , where w ( j ) B n and r > 0 .

Lemma 1.2 (see [[5], Theorem 2.23])

For fixed r > 0 , there is a sequence { w ( j ) } B n such that:

j = 1 B ( w ( j ) , r ) = B n ;

there is a positive integer N such that each z B n is contained in at most N of the sets B ( w ( j ) , 2 r ) .

The following characterization of Carleson measures can be found in [6], or in [5].

A positive Borel measure μ on the unit ball B n is said to be a Carleson measure for the Bergman space A α p ( B n ) if
B n | f ( z ) | p d ν α ( z ) C f A α p ( B n ) p , f A α p ( B n ) .
It is well known that a positive Borel measure μ is a Carleson measure if and only if there is a positive constant C such that
sup w ( j ) B n μ ( B ( w ( j ) , r ) ) ν ( B ( w ( j ) , r ) ) < ,
where { w ( j ) } is the sequence in Lemma 1.2. If μ satisfies that
lim j μ ( B ( w ( j ) , r ) ) ν ( B ( w ( j ) , r ) ) = 0 ,

then μ is called a vanishing Carleson measure.

For a given reasonable function ω : ( 0 , 1 ] ( 0 , ) , the weighted Bloch space B ω of several complex variables is defined as the set of all analytic functions f on B n satisfying
( 1 | z | ) α | f ( z ) | C ω ( 1 | z | ) , z B n ,  where  α ( 0 , ) ,
for some fixed C = C f > 0 . In the special case where ω 1 , B ω reduces to the classical Bloch space in C n . This class of functions extends and generalizes the well known Bloch space. Now, we define the space B α , β ; ω ( B n ) in the unit ball B n . For α > 0 and β 0 , a function f H ( B n ) is said to belong to the ( α , β ; ω ) -Bloch space B α , β ; ω ( B n ) if
b α , β ; ω ( f ) ( B n ) = sup a , z B n ( 1 | z | 2 ) α + β ( 1 | φ a ( z ) | 2 ) β ω ( 1 | z | ) | f ( z ) | < .
The little ( α , β ; ω ) -Bloch space B α , β ; ω , 0 ( B n ) is a subspace of B α , β ; ω ( B n ) consisting of all f B α , β ; ω ( B n ) such that
lim | a | 1 lim | z | 1 ( 1 | z | 2 ) α + β ( 1 | φ a ( z ) | 2 ) β ω ( 1 | z | ) | f ( z ) | = 0 .

If β = 0 , ω ( 1 | z | ) = 1 , then we get the α-Bloch space B α ( B n ) and the little α-Bloch space B 0 α ( B n ) . If ω ( 1 | z | ) = 1 , α = 1 and β = 0 , then we get the classical Bloch space B ( B n ) and B 0 ( B n ) . These classes extend the weighted Bloch spaces defined in [18] to the setting of several complex variables.

The logarithmic ( α , β ; ω ) -Bloch space LB ω α , β ( B n ) is the space of holomorphic functions f such that
sup a , z B n ( 1 | z | 2 ) α + β ( 1 | φ a ( z ) | 2 ) β ω ( 1 | z | ) ( ln 2 1 | z | 2 ) | f ( z ) | < .
Correspondingly, the little logarithmic ( α , β ; ω ) -Bloch space LB ω ; 0 α , β ( B n ) is a subspace of LB ω α , β ( B n ) consisting of all functions f such that
lim | a | 1 lim | z | 1 ( 1 | z | 2 ) α + β ( 1 | φ a ( z ) | 2 ) β ω ( 1 | z | ) ( ln 2 1 | z | 2 ) | f ( z ) | = 0 .

If ω ( 1 | z | ) = 1 and β = 0 , then we get the logarithmic α-Bloch space LB α ( B n ) and the little logarithmic α-Bloch space LB 0 α ( B n ) . If ω ( 1 | z | ) = 1 , α = 1 and β = 0 , then we get the logarithmic Bloch space LB ( B n ) and LB 0 ( B n ) (see [19]).

2 Holomorphic ( α , β ; ω ) -Bloch space in the unit ball

In this section, we study the general ( α , β ; ω ) -Bloch space B ω α , β ( B n ) in the unit ball of C n by giving some characterizations of ( α , β ; ω ) -Bloch space, then we present several auxiliary results, which play important roles in the proofs of our main results.

Lemma 2.1 Let α , β ( 0 , ) and f B ω α , β ( B n ) . Suppose that
0 1 ω ( 1 t | z | ) | z | d t ( 1 t 2 | z | 2 ) α + β < .
(1)
Then
| f ( z ) | | f ( 0 ) | + f B ω α , β ( B n ) .
Proof Let z B n , 0 t < 1 and f B ω α , β ( B n ) . By the definition of B ω α , β ( B n ) and | z | > 1 2 , we have that
| f ( z ) f ( z 2 ) | = | 1 2 1 f ( t z ) , z d t | | 1 2 1 R f ( t z ) d t t | b α , β ; ω ( f ) 0 1 ( 1 | φ a ( t z ) | 2 ) β ω ( 1 t | z | ) ( 1 t 2 | z | 2 ) α + β | z | d t b α , β ; ω ( f ) 0 1 ( 1 | a | 2 ) β ω ( 1 t | z | ) | 1 t z , a | 2 β ( 1 t 2 | z | 2 ) α | z | d t .
Since ( 1 | a | ) | 1 t z , a | and ( 1 t | z | ) | 1 t z , a | , a , z B n , we get
| f ( z ) f ( z 2 ) | b α , β ; ω ( f ) 0 1 ( 1 | a | 2 ) β ω ( 1 | z | ) ( 1 | a | ) β ( 1 t | z | ) β ( 1 t 2 | z | 2 ) α | z | d t 4 β b α , β ( f ) 0 1 ω ( 1 t | z | ) | z | d t ( 1 t 2 | z | 2 ) α + β ,

from which the result follows. □

Theorem 2.1 For each 0 < α , β < , γ > 1 and f H ( B n ) . Then the following conditions are equivalent:

(i) f B ω α , β ( B n ) ;

(ii) The function ( 1 | z | 2 ) α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) | R f ( z ) | is bounded in B n ;

(iii) There exists a function g L ( B n ) such that
f ( z ) = ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n g ( w ) d ν γ ( w ) ( 1 z , w ) n + α + β + γ , z B n .
Proof By the Cauchy-Schwarz inequality in C n , we have
| R f ( z ) | | z | | f ( z ) | | f ( z ) | .

This proves that (i) (ii).

If (ii) holds, then the function
g ( z ) = c α + β + γ c γ ( 1 | z | 2 ) α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) ( f ( z ) + R f ( z ) n + α + β + γ )
is bounded in B n . For z B n consider the holomorphic function
F ( z ) = B n g ( w ) ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) d ν γ ( w ) ( 1 z , w ) n + α + β + γ = B n ω ( 1 | w | ) ( 1 z , w ) n + α + β + γ ( f ( w ) + R f ( w ) n + α + β + γ ) d ν α + β + γ ( w ) .

As in the proof of Theorem 7.1 in [5], we have F = f .

This shows that (ii) implies (iii). That (iii) implies (i) follows from differentiating under the integral sign and then applying Theorem 1.12 in [5]. □

Theorem 2.2 For each α > 0 , β 0 , α + β > 0 and s = α + β 1 . If s > 1 , then the Banach dual of A 1 ( B n ) can be identified with B ω α , β ( B n ) (with equivalent norms) under the following integral pairing:
f , g s = B n f ( z ) g ( z ) ¯ d ν s ( z ) , f A 1 ( B n ) , g B ω α , β ( B n ) .
(2)
Proof It is easy to see that 1 ( α + β ) + s > 1 . If g B ω α , β ( B n ) , then by Theorem 2.1, there exists a function h L ( B n ) such that
g ( z ) = 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n h ( w ) d ν 1 ( α + β ) + s ( w ) ( 1 z , w ) n + 1 + s , z , w B n ,
and h C g B ω α , β ( B n ) , where C is a positive constant independent of g. By Fubini’s theorem,
f , g s = B n f ( z ) h ( z ) ¯ ( 1 | z | 2 ) d ν 1 ( α + β ) + s ( z ) = c 1 ( α + β ) + s B n f ( z ) h ( z ) ¯ d ν ( z ) .
Applying Lemma 2.15 in [5] for all f A 1 ( B n ) , we have
B n | f ( z ) | d ν ( z ) f A 1 ( B n ) .
Combining this, we see that
| f , g s | h f A 1 ( B n ) C g B ω α , β ( B n ) f A 1 ( B n ) .
Conversely, if F is a bounded linear functional on A 1 ( B n ) and f A 1 ( B n ) , then
f r ( z ) = 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n f r ( w ) d ν s ( w ) ( 1 z , w ) n + 1 + s for  0 < r < 1 .
It is easy to verify (using the homogeneous expansion of the kernel function) that
F ( f r ) = B n f r ( w ) F z [ 1 ( 1 z , w ) n + 1 + s ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) ] d ν s ( w ) .
Define a function g on B n by
g ( w ) ¯ = F z [ 1 ( 1 z , w ) n + 1 + s ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) ] .
Then
F ( f r ) = B n f r ( w ) g ( w ) ¯ d ν s ( w ) = f , g s .

It remains for us to show that g B ω α , β .

We interchange differentiation and the application of F, which can be justified by using the homogeneous expansion of the kernel. The result is
R g ( w ) ¯ = ( n + 1 + s ) F z [ 1 ( 1 z , w ) n + 1 + s ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) ] .
Since F is bounded on A 1 ( B n ) , we have
| R g ( w ) | C F ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n d ν ( w ) | 1 z , w | n + 2 + s .
An application of Theorem 1.1 for s + 1 = α + β then shows that
| R g ( w ) | C F ( 1 | z | 2 ) α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) .

This shows that g B ω α , β ( B n ) and completes the proof of the theorem. □

Lemma 2.2 If n > 1 , α + β > 1 2 , then f B ω α , β ( B n ) if and only if the function
( 1 | z | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | ˜ f ( z ) |

is bounded in B n .

Proof Recall from Lemma 2.14 in [5] that
( 1 | z | 2 ) | f ( z ) | | ˜ f ( z ) | , z B n .
So, the boundedness of
( 1 | z | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | ˜ f ( z ) |
implies that of
( 1 | z | 2 ) α + β ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | f ( z ) | .
On the other hand, if f B ω α , β ( B n ) , then by Theorem 2.1,
f ( z ) = ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) B n g ( w ) d ν ( w ) ( 1 z , w ) n + α + β , z B n ,
where g is a function in L ( B n ) . Now we let f ( z ) = h ( z ) u ( z ) , where h ( z ) = ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) and
u ( z ) = B n g ( w ) d ν ( w ) ( 1 z , w ) n + α + β .
An application of Lemma 1.1 gives
| ˜ u ( z ) | | n + α + β | 2 ( 1 | z | 2 ) 1 2 B n g ( w ) d ν ( w ) | 1 z , w | n + α + β + 1 2 , z B n .
Since g ( z ) is bounded, by Theorem 1.1 we have
B n g ( w ) d ν ( w ) | 1 z , w | n + α + β + 1 2 ( 1 | z | 2 ) 1 2 ( α + β ) .
So,
| ˜ u ( z ) | C ( 1 | z | 2 ) 1 ( α + β ) .

It is easy to check that ˜ h ( z ) = ( h φ z ) ( 0 ) = 0 .

Using the product rule, we have
| ˜ f ( z ) | | ˜ h ( z ) | | u ( z ) | + | h ( z ) | | ˜ u ( z ) | | ˜ h ( z ) | | u ( z ) |
and we have
| ˜ f ( z ) | | n + α + β | 2 ( 1 | z | 2 ) 1 2 ( 1 | φ a ( z ) | 2 ) β B n g ( w ) d ν ( w ) | 1 z , w | n + α + β + 1 2 for all  z B n .
Hence,
( 1 | z | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | ˜ f ( z ) |

is bounded in B n . This completes the proof. □

Lemma 2.3 Let 0 < α + β 2 . Let λ be any real number satisfying the following properties:

0 λ α + β if 0 < α + β < 1 ;

0 < λ < 1 if α + β = 1 ;

α + β 1 λ 1 if 1 < α + β 2 .

Then, for all z , w B n a holomorphic function f B ω α , β ( B n ) if and only if
sup z w ( 1 | z | 2 ) λ ( 1 | w | ) α + β λ ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | f ( z ) f ( w ) | | z w | < .
(3)
Proof Let f B ω α , β ( B n ) . By a similar proof to the one for Theorem 3.1 in [20], we have
| f ( z ) f ( w ) | = n | z w | 0 1 | f ( t z ( 1 t ) w ) | d t
for any z , w B n with z w . We know that
f B ω α , β ( B n ) sup w , z B n ( 1 | z | 2 ) α + β ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | f ( z ) | .
Thus, there is a constant C > 0 such that
| f ( z ) f ( w ) | | z w | C f B ω α , β ( B n ) 0 1 ( 1 | φ a ( t z ( 1 t ) a ) | 2 ) β ω ( 1 | t z ( 1 t ) w | ) ( 1 | t z ( 1 t ) w | 2 ) α + β d t .
Since ( 1 | z | ) | 1 w , z | and
1 | t z + ( 1 t ) w | 2 1 | t z + ( 1 t ) w | 1 | w | + ( | w | | z | ) t ,
we get
( 1 | φ w ( t z ( 1 t ) w ) | 2 ) β = ( 1 | w | 2 ) β ( 1 | t z ( 1 t ) w | 2 ) β | 1 t z ( 1 t ) w , w | 2 β ( 1 | w | 2 ) β ( 1 | t z ( 1 t ) w | 2 ) β ( 1 | w | ) 2 β ( 1 + | w | ) β ( 1 | t z ( 1 t ) w | 2 ) β ( 1 | w | ) β ( 1 | t z ( 1 t ) w | 2 ) β ( 1 | w | ) β .
Thus
| f ( z ) f ( w ) | | z w | C f B ω α , β ( B n ) 0 1 1 ( 1 | w | + ( | w | | z | ) t ) α ( 1 | w | ) β d t .
(4)
If | z | = | a | , then
| f ( z ) f ( w ) | | z w | C f B ω α , β ( B n ) 0 1 1 ( 1 | w | ) α + β d t C f B ω α , β ( B n ) ( 1 | z | 2 ) λ ( 1 | w | 2 ) α + β λ .
(5)
Now suppose | z | | w | as in [21], there is a constant C > 0 such that this integral in (4) is dominated by
C ( 1 | z | 2 ) λ ( 1 | w | 2 ) α + β λ .
Combining with (5), we get that whenever z w ,
| f ( z ) f ( w ) | | z w | C f B ω α , β ( B n ) ( 1 | z | 2 ) λ ( 1 | w | 2 ) α + β λ .

This proves the necessity. The proof of the sufficiency condition is much akin to the corresponding one in [21], so the proof is omitted. □

Proposition 2.1 Suppose f B ω α , β ( B n ) and 1 α + β 2 . Let λ be any real number satisfying:

0 < λ < 1 if α + β = 1 ;

α + β 1 λ 1 if 1 < α + β 2 .

Then
sup z , w B n ( 1 | z | 2 ) 2 α + 2 β λ 1 ( 1 | w | 2 ) α + β λ | f ( z ) f ( w ) | ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | 1 w , z | 2 ( α + β ) ( 2 λ + 1 ) | z P z ( w ) S z Q z ( w ) | C f B ω α , β ( B n ) .
(6)
Proof Let z = 0 in (3), then we have
( 1 | w | 2 ) α + β λ | f ( 0 ) f ( w ) | | w | C b α , β ; ω ( f ) ( B n ) , w B n { 0 } .
Now, replacing f by f φ w , we get
( 1 | u | 2 ) α + β λ | f φ w ( 0 ) f φ w ( u ) | | u | C b α , β ; ω ( f φ w ) ( B n ) , u B n { 0 } .
(7)
Since
| ˜ ( f φ w ) ( z ) | = | ˜ f ( φ w ( z ) ) | ,
by Lemma 2.2, we obtain that
b α , β ; ω ( f φ w ) ( B n ) sup z , w B n ( 1 | w | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | ˜ ( f φ w ) ( z ) | = sup z , w B n ( 1 | w | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | ˜ f ( φ w ( z ) ) | = sup z , w B n ( 1 | w | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) α + β 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) | ˜ f ( φ w ( z ) ) | .
Then
b α , β ; ω ( f φ z ) ( B n ) C f B ω α , β ( B n ) ( 1 | w | 2 ) α + β 1 ( 1 | φ z ( w ) | 2 ) α + β 1 ω ( 1 | w | ) C f B ω α , β ( B n ) ( 1 | z | 2 ) α + β 1 .
Letting u = φ w ( z ) and w z in (7), we obtain
| f ( z ) f ( w ) | | φ w ( z ) | ω ( 1 | z | ) ( 1 | φ w ( z ) | 2 ) α + β λ C f B ω α , β ( B n ) ( 1 | z | 2 ) α + β 1 .
Since
1 | φ z ( w ) | 2 = ( 1 | z | 2 ) ( 1 | w | 2 ) | 1 w , z | 2
and
φ z ( w ) = z P z ( w ) S z Q z ( w ) 1 w , z .
Consequently,
( 1 | z | 2 ) 2 α + 2 β λ 1 ( 1 | w | 2 ) α + β λ | f ( z ) f ( w ) | ω ( 1 | z | ) ( 1 | φ z ( w ) | 2 ) α + β λ | 1 w , z | 2 ( α + β ) ( 2 λ + 1 ) | z P z ( w ) S z Q z ( w ) | C f B ω α , β ( B n ) .

 □

3 Boundedness of general Toeplitz operators

In this section, we study the boundedness of general Toeplitz operators acting on the weighted Bloch-type spaces B ω α , β ( B n ) in the unit ball of C n .

Theorem 3.1 Let μ be a positive Borel measure on B n . Then we have

(1) if α + β = 1 , then T μ α , β ; ω is bounded on B ω α , β ( B n ) if and only if P α + β 1 ; ω ( μ ) LB ω ( B n ) and μ is a Carleson measure;

(2) if α = β = 1 , then T μ α , β ; ω is bounded on B ω α , β ( B n ) if and only if P α + β 1 ; ω ( μ ) B ω ( B n ) LB ω 2 ( B n ) and μ is a Carleson measure;

(3) if α > 1 , β > 1 , then T μ α , β ; ω is bounded on B ω α , β ( B n ) if and only if P α + β 1 ; ω ( μ ) B ω α , β ( B n ) and μ is a Carleson measure.

Proof Since the Banach dual of A 1 ( B n ) is B ω α , β ( B n ) under the pairing (2), to prove the boundedness of T μ α , β ; ω , it suffices to show that
| f , T μ α , β ; ω ( g ) α | C f A 1 ( B n ) g B ω α , β ( B n )

for all f A 1 ( B n ) and g B ω α , β ( B n ) , where C is a positive constant that does not depend on f or g.

For s = α + β 1 , by Fubini’s theorem, we get
f , T μ α , β ; ω g s = B n f ( z ) T μ α , β ; ω g ( z ) ¯ d ν s ( z ) = c α + β 1 B n f ( z ) g ( z ) ¯ ( 1 | z | 2 ) α + β 1 d μ ( z ) .
Using the operator P α + β ; ω , we have
f , T μ α , β ; ω g s = c α + β 1 B n ( I z , w ; ω P α + β ; ω ) ( f g ¯ ) ( z ) ( 1 | z | 2 ) α + β 1 d μ ( z ) + c α + β 1 B n P α + β ; ω ( f g ¯ ) ( z ) ( 1 | z | 2 ) α + β 1 d μ ( z ) = I 1 + I 2 ,
where I z , w ; ω = I ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) , and I is the identity operator. Also,
( I z , w ; ω P α + β ; ω ) ( f g ¯ ) ( z ) = f ( z ) g ( z ) ¯ ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) c α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n f ( w ) g ( w ) ¯ ( 1 | w | 2 ) α + β ( 1 z , w ) n + α + β + 1 d ν ( w ) = c α + β ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) × B n ( g ( z ) ¯ g ( w ) ¯ ) f ( w ) ( 1 | w | 2 ) α + β ( 1 z , w ) n + α + β + 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) d ν ( z ) .
By Proposition 2.1, we have
| I 1 | = c α + β 1 | B n ( I z , w ; ω P α + β ; ω ) ( f g ¯ ) ( z ) ( 1 | z | 2 ) α + β 1 d μ ( z ) | = c α + β 1 c α + β | B n B n ( g ( z ) ¯ g ( w ) ¯ ) f ( w ) ( 1 | w | 2 ) α + β ( 1 | z | 2 ) α + β 1 ( 1 z , w ) n + α + β + 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) d ν ( w ) d μ ( z ) | = c α + β 1 c α + β | B n f ( w ) ( 1 | w | 2 ) α + β × B n ( g ( z ) ¯ g ( w ) ¯ ) ( 1 | z | 2 ) α + β 1 ( 1 z , w ) n + α + β + 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) d μ ( z ) d ν ( w ) | c α + β 1 c α + β B n | f ( w ) | ( 1 | w | 2 ) λ × B n ( 1 | z | 2 ) 2 ( α + β ) λ 1 ( 1 | w | 2 ) α + β λ | f ( z ) f ( w ) | | 1 w , z | 2 ( α + β ) ( 2 λ + 1 ) | z P z ( w ) S z Q z ( w ) | × | z P z ( w ) S z Q z ( w ) | ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n ( α + β ) + 2 λ + 2 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) d μ ( z ) d ν ( w ) C B n g B ω α , β ( B n ) | f ( w ) | ( 1 | w | 2 ) λ B n ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n ( α + β ) + 2 λ + 1 d μ ( z ) d ν ( w ) .
Since μ is a Carleson measure, taking λ ( α + β ) > 1 , then as in [9] or in [[22], Proposition 1.4.10], for fixed r > 0 , we get
B n ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n ( α + β ) + 2 λ + 1 d μ ( z ) j = 1 μ ( B ( z ( j ) , r ) ) ν ( B ( z ( j ) , r ) ) B ( z ( j ) , r ) ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n ( α + β ) + 2 λ + 1 d ν ( z ) C .
Therefore,
| I 1 | C f A 1 ( B n ) g B ω α , β ( B n ) .
Next considering I 2 , we have
| I 2 | = c α + β 1 | B n P α + β ; ω ( f g ¯ ) ( z ) ( 1 | z | 2 ) α + β 1 d μ ( z ) | = c α + β 1 c α + β | B n B n f ( w ) g ( w ) ¯ ( 1 | w | 2 ) α + β ( 1 | z | 2 ) α + β 1 ( 1 z , w ) n + α + β + 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) d ν ( w ) d μ ( z ) | c α + β B n | f ( w ) | ( 1 | w | 2 ) α + β | g ( w ) | × ( c α + β 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n ( 1 | z | 2 ) α + β 1 d μ ( z ) | 1 z , w | n + α + β + 1 ) d ν ( w ) C B n f A 1 ( B n ) ( 1 | w | 2 ) α + β | g ( w ) | Q μ α , β ; ω ( w ) d ν ( w ) ,
where
Q μ α , β ; ω ( w ) = c α + β 1 ( 1 | φ z ( w ) | 2 ) β ω ( 1 | w | ) B n ( 1 | z | 2 ) α + β 1 d μ ( z ) | 1 z , w | n + α + β + 1 .
As in [9], by simple calculation, we have
Q μ α , β ; ω ( w ) = P α + β 1 ; ω ( μ ) ( w ) + 1 n + α + β R P α + β 1 ; ω ( μ ) ( w ) .
(8)

It is easy to see that

(1) if α + β = 1 and P α + α 1 ; ω ( μ ) LB ω α , β ( B n ) , then
( 1 | w | 2 ) Q μ α , β ; ω ( w ) ( ln 2 1 | w | 2 ) L ( B n ) ;
(2) if α = β = 1 , P α + α 1 ; ω ( μ ) B ω ( B n ) LB ω 2 ( B n ) , then
( 1 | w | 2 ) Q μ α , β ; ω ( w ) L ( B n )
and
( 1 | w | 2 ) 2 Q μ α , β ; ω ( w ) ( ln 2 1 | w | 2 ) L ( B n ) ;
(3) if α > 1 , β > 1 , and P α + α 1 ; ω ( μ ) B ω α , β ( B n ) , then
( 1 | w | 2 ) α + β Q μ α , β ; ω ( w ) L ( B n ) .

This implies that | I 2 | C f A 1 ( B n ) g B ω α , β ( B n ) . Hence, T μ α , β ; ω is a bounded operator on B ω α , β ( B n ) with α > 0 , β 0 .

Conversely, suppose that T μ α , β ; ω is a bounded operator on B ω α , β ( B n ) . Take
f w ( z ) = ( 1 | w | 2 ) t ( 1 z , w ) n + t + 1 for  t > 0 .
It is clear that f w A 1 ( B n ) C . On the other hand, take
g w ( z ) = ( 1 | w | 2 ) n + 2 + t ( α + β ) ( 1 z , w ) n + t + 1 ; φ w ( z ) 1 and ω ( 1 | z | ) 1 for  t > 0 .
Then, we have g w B ω α , β ( B n ) C . Therefore
| f , T μ α g s | = c α + β 1 ( 1 | w | 2 ) n + 2 + 2 t ( α + β ) B n ( 1 | z | 2 ) α + β 1 d μ ( z ) | 1 z , w | 2 ( n + t + 1 ) C T μ α , β ; ω f w A 1 ( B n ) g w B ω α , β ( B n ) C .
Thus,
( 1 | w | 2 ) n + 2 + 2 t ( α + β ) B ( w , r ) ( 1 | z | 2 ) α + β 1 d μ ( z ) | 1 z , w | 2 ( n + t + 1 ) C
for every w B n . This implies that
sup w B n μ ( B ( w , r ) ) ν ( B ( w , r ) ) < .

Hence μ is a Carleson measure on B n .

From the proof of the sufficient condition, we find that there exists a constant C such that
| I 2 | = c α + β | B n f ( w ) ( 1 | w | 2 ) α + β g ( w ) Q μ α , β ; ω ( w ) ¯ d ν ( w ) | C f A 1 ( B n ) g B ω α , β ( B n ) .
This implies that
| g ( w ) Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) α + β C g B ω α , β ( B n ) .
If α + β = 1 , we have
| g ( w ) Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) C g B ω α , β ( B n ) .
Take g w ( z ) = ln 2 1 z , w ; φ w ( z ) 1 and ω ( 1 | z | ) 1 . It is clear that g w LB ω ( B n ) C . Taking z = w , then
| Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) ( ln 2 1 | w | 2 ) C .
From (8) we have P α + β 1 ( μ ) LB ω ( B n ) . Let α = β = 1 , we have
| g ( w ) Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) 2 C g B ω α , β ( B n ) .
Take g w ( z ) = 1 1 z , w + ln 2 1 z , w ; φ w ( z ) 1 and ω ( 1 | z | ) 1 . It is clear that
g w B ω ( B n ) LB ω 2 ( B n ) C .
Taking z = w , then
| g ( w ) Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) 2 = ( 1 1 | w | 2 + ln 2 1 | w | 2 ) | Q μ α , β ( w ) | ( 1 | w | 2 ) 2 | Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) + | Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) 2 ( ln 2 1 | w | 2 ) C .

By (8), then P α + β 1 ; ω ( μ ) LB ω 2 ( B n ) and P α + β 1 ; ω ( μ ) B ω ( B n ) .

When α , β > 1 , taking g w ( z ) = ( 1 z , w ) 1 ( α + β ) , we have g w B ω α , β ( B n ) C .

From Lemma 2.1, we get
| Q μ α , β ; ω ( w ) | ( 1 | w | 2 ) α + β C for  w B n .

By (8) it is obvious that P α + β 1 ; ω ( μ ) B ω α , β ( B n ) .

This completes the proof of Theorem 3.1. □

4 Compactness of general Toeplitz operators

In this section, we study the compactness of Toeplitz operators on the weighted Bloch-type spaces B ω α , β ( B n ) in the unit ball of C n . We need the following lemma.

Lemma 4.1 Let 0 < α < , 0 β < and T μ α , β ; ω be a bounded linear operator from B ω α , β ( B n ) into B ω α , β ( B n ) . When 0 < α < 1 , 0 β < 1 and α + β < 1 , then T μ α , β ; ω is compact if and only if
lim j T μ α , β ; ω f j B ω α , β ( B n ) = 0 ,

whenever ( f j ) is a bounded sequence in B ω α , β ( B n ) that converges to 0 uniformly on B ¯ n .

Proof This lemma can be proved by Montel’s theorem and Lemma 2.1. □

Theorem 4.1 Let μ be a positive Borel measure on B n . We have the following:

(1) if α + β = 1 , then T μ α , β ; ω is compact on B ω α , β ( B n ) if and only if P α + β 1 ; ω ( μ ) LB ω ; 0 ( B n ) and μ is a vanishing Carleson measure;

(2) if α = β = 1 , then T μ α , β ; ω is compact on B ω α , β ( B n ) if and only if P α + β 1 ; ω ( μ ) B ω ; 0 ( B n ) LB ω ; 0 2 ( B n ) and μ is a vanishing Carleson measure;

(3) if α > 1 , β > 1 , then T μ α , β ; ω is compact on B ω α , β ( B n ) if and only if P α + β 1 ( μ ) B ω ; 0 α , β ( B n ) and μ is a vanishing Carleson measure.

Proof For α + β 1 , let ( g j ) be a sequence in B ω α , β ( B n ) satisfying g j B ω α , β ( B n ) 1 and g j converges to 0 uniformly as j on B ¯ n . Suppose f A 1 ( B n ) . By duality, we have that T μ α , β ; ω is compact on B ω α , β ( B n ) if and only if
lim j sup f A 1 ( B n ) 1 | f , T μ α , β ; ω ( g j ) | = 0 .
Similarly, as in the proof of Theorem 3.1 for s = α + β 1 , we have
f , T μ α , β ; ω g j s = c α + β 1 B n f ( z ) g j ( z ) ¯ ( 1 | z | 2 ) α + β 1 d μ ( z ) = c α + β 1 B n [ ( I z , w ; ω P α + β ; ω ) ( f g ¯ j ) ] ( z ) ( 1 | z | 2 ) α + β 1 d μ ( z ) + c α + β 1 B n P α + β ; ω ( f g ¯ j ) ( z ) ( 1 | z | 2 ) α + β 1 d μ ( z ) = J 1 + J 2 .
For fixed 0 < ε < 1 , since μ is a vanishing Carleson measure, there exists 0 < η < 1 such that
( 1 | z | 2 ) λ B n η B n ( 1 | w | 2 ) λ ( α + β ) | 1 z , w | n + 2 λ + 1 ( α + β ) d μ ( w ) < ε ,
where η B n = { z C n , | z | < η } and λ ( α + β ) > 1 . For a positive constant 0 < δ < 1 , as in the proof of Theorem 3.1, by Proposition 2.1, we obtain
| J 1 | = c α + β 1 c α + β | B n B n ( g j ( z ) ¯ g j ( w ) ¯ ) f ( w ) ( 1 | w | 2 ) α + β ( 1 | z | 2 ) α + β 1 ( 1 z , w ) n + α + 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) d ν ( w ) d μ ( z ) | = c α + β 1 c α + β | B n f ( w ) ( 1 | w | 2 ) α + β × B n ( g j ( z ) ¯ g j ( w ) ¯ ) ( 1 | z | 2 ) α + β 1 ( 1 z , w ) n + α + 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) d μ ( z ) d ν ( w ) | c α + β 1 c α + β B n δ B n | f ( w ) | ( 1 | w | 2 ) α + β × B n | g j ( z ) g j ( w ) | ( 1 | z | 2 ) α + β 1 | 1 z , w | n + α + 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) d μ ( z ) d ν ( w ) + c α + β 1 c α + β δ B n | f ( w ) | ( 1 | w | 2 ) α + β × B n | g j ( z ) g j ( w ) | ( 1 | z | 2 ) α + β 1 | 1 z , w | n + α + 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) d μ ( z ) d ν ( w ) = L 1 + L 2 .
Since g j 0 as j on compact subsets of B n , we can choose j big enough so that
| f ( w ) | ( 1 | w | 2 ) α + β < ε .
Therefore,
L 2 ε C δ B n B n | g j ( z ) g j ( w ) | ( 1 | z | 2 ) α + β 1 | 1 z , w | n + α + 1 ( 1 | φ w ( z ) | 2 ) β ω ( 1 | z | ) d μ ( z ) d ν ( w ) ε C g j B ω α , β ( B n ) .
Now, taking δ such that 1 [ ε ( 1 η ) n + 1 + λ ] 1 λ δ < 1 , then
L 1 C g j B ω α , β ( B n ) B n δ B n | f ( w ) | ( 1 | w | 2 ) λ B n ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n + 2 λ + 1 ( α + β ) d μ ( z ) d ν ( w ) C B n δ B n | f ( w ) | ( 1 | w | 2 ) λ B n η B n ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n + 2 λ + 1 ( α + β ) d μ ( z ) d ν ( w ) + C B n δ B n | f ( w ) | ( 1 | w | 2 ) λ η B n ( 1 | z | 2 ) λ ( α + β ) | 1 z , w | n + 2 λ + 1 ( α + β ) d μ ( z ) d ν ( w ) C ε B n δ B n | f ( w ) | d ν ( w ) + C B n δ B n | f ( w ) | ( 1 δ ) λ ( 1 η ) n + 1 + λ d ν ( w ) C ε f A 1 ( B n ) .
Hence | J 1 | < C ε , where C does not depend on f ( z ) , and so
lim j sup f A 1 ( B n ) 1 | J 1 | = 0 .
Thus, T μ α , β ; ω is compact on B ω α , β ( B n ) if and only if
lim j sup f A 1 ( B n ) 1 | J 2 | = 0 .
Again, as in the proof of Theorem 3.1, we have
| J 2 | C B n f A 1 ( B n ) ( 1 | w | 2 ) α + β | g ( w ) | Q μ α , β ; ω ( w ) d ν ( w ) .

From (8) it is easy to see that

(1) if α + β = 1 and P α + α 1 ; ω ( μ ) LB ω , 0 α , β ( B n ) , then
lim | w | 1 ( 1 | w | 2 ) Q μ α , β ; ω ( w ) ( ln 2 1 | w | 2 ) = 0 ;
(2) if α = β = 1 , P α + α 1 ; ω ( μ ) B ω ; 0 ( B n ) LB ω ; 0 2 ( B n ) , then
lim | w | 1 ( 1 | w | 2 ) Q μ α , β ; ω ( w ) = 0
and
lim | w | 1 ( 1 | w | 2 ) 2 Q μ α , β ; ω ( w ) ( ln 2 1 | w | 2 ) = 0 ;
(3) if α > 1 , β > 1 , and P α + α 1 ; ω ( μ ) B ω ; 0 α , β ( B n ) , then
lim | w | 1 ( 1 | w | 2 ) α + β Q μ α , β ; ω ( w ) = 0 .
Combined with g j 0 as j on compact subsets of B n , we have
lim j sup f A 1 ( B n ) 1 | J 2 | = 0 .
Therefore,
lim j T μ α , β ; ω g j B ω α , β ( B n ) = 0 ,

which implies that T μ α , β ; ω is a compact operator.

Next assume that T μ α , β ; ω is a compact operator on B ω α , β ( B n ) . Again, as in the proof of Theorem 3.1, we take
f w ( z ) = ( 1 | w | 2 ) t ( 1 z , w ) n + t + 1 for  t > 0 .
We know that f w A 1 ( B n ) C . On the other hand, take
g w ( z ) = ( 1 | w | 2 ) n + 2 + t ( α + β ) ( 1 z , w ) n + t + 1 ; φ w ( z ) 1 and ω ( 1 | z | ) 1 for  t > 0 .
Then g w B ω α , β ( B n ) C and g w 0 uniformly on compact subsets of B n , as | w | 1 ,
| f , T μ α , β ; ω g s | = c α + β 1 ( 1 | w | 2 ) n + 2 + 2 t ( α + β ) B n ( 1 | z | 2 ) α + β 1 d μ ( z ) | 1 z , w | 2 ( n + t + 1 ) C f w A 1 ( B n ) T μ α , β g w B α , β ( B n ) .
From Lemma 4.1, we have
lim | w | 1 f w A 1 ( B n ) T μ α , β ; ω g w B ω α , β ( B n ) = 0 , w B n .

This implies that μ is a vanishing Carleson measure on B n .

Next let
f w ( z ) = ( 1 | w | 2 ) α + β ( 1 z , w ) n + α + β + 1 .
Then, we have f w A 1 ( B n ) C . Let { g j } be a bounded sequence in B ω α , β ( B n ) that converges to zero uniformly as j on B ¯ n . By the compactness of T μ α , β ; ω , we have
0 = lim j J 2 = lim j c α + β B n f w ( z ) ( 1 | z | 2 ) α + β g j ( z ) Q μ α , β ; ω ( z ) ¯ d ν ( z ) = lim j c α + β ( 1 | w | 2 ) α + β B n ( 1 | z | 2 ) α + β g j ( z ) Q μ α , β ; ω ( z ) ¯ ( 1 z , w ) n + α + β + 1 d ν ( z ) = lim j ( 1 | w | 2 ) α + β g j ( w ) Q μ α , β ; ω ( w ) ¯ .
When α + β = 1 , taking
g w ( z ) = ( ln 2 1 z , w ) 2 ( ln 1 1 | w | 2 ) 1 ; φ w ( z ) 1 and ω ( 1 | z | ) 1 ,

with | w | 1 2 , we have P α +