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Weighted composition followed and proceeded by differentiation operators from Q k ( p , q ) spaces to Bloch-type spaces

Journal of Inequalities and Applications20122012:160

https://doi.org/10.1186/1029-242X-2012-160

Received: 10 December 2011

Accepted: 10 July 2012

Published: 19 July 2012

Abstract

In this paper, we investigate boundedness and compactness of the weighted composition followed and proceeded by differentiation operators from Q k ( p , q ) spaces to Bloch-type spaces and little Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are obtained.

MSC:47B38, 30D45.

Keywords

Q k ( p , q ) spaces Bloch-type spaces weighted composition followed and proceeded by differentiation operators boundedness compactness

1 Introduction

Let Δ be an open unit disc in the complex plane, and let H ( Δ ) be the class of all analytic functions on Δ. The α-Bloch space B α ( α > 0 ) is, by definition, the set of all function f in H ( Δ ) such that
f B α = | f ( 0 ) | + sup z Δ ( 1 | z | 2 ) α | f ( z ) | < .
Under the above norm, B α is a Banach space. When α = 1 , B 1 = B is the well-known Bloch space. Let B 0 α denote the subspace of B α , for f
B 0 α = { f : ( 1 | z | 2 ) α | f ( z ) | 0  as  | z | 1 , f B α } .

This space is called a little α-Bloch space.

Assume that μ is a positive continuous function on [ 0 , 1 ) , having the property that there exist positive numbers s and t, 0 < s < t , and δ [ 0 , 1 ) , such that
μ ( r ) ( 1 r ) s is decreasing on  [ δ , 1 ) , lim r 1 μ ( r ) ( 1 r ) s = 0 , μ ( r ) ( 1 r ) t is increasing on  [ δ , 1 ) , lim r 1 μ ( r ) ( 1 r ) t = .

Then μ is called a normal function (see [9]).

Denote (see, e.g., [2, 4, 10])
B μ = { f : f B μ = | f ( 0 ) | + sup z Δ μ ( | z | ) | f ( z ) | < , f H ( Δ ) } .

It is known that B μ is a Banach space with the norm B μ (see [4]).

Let B μ , 0 denote the subspace of B μ , i.e.,
B μ , 0 = { f : μ ( | z | ) | f ( z ) | 0  as  | z | 1 , f B μ } .

This space is called a little Bloch-type space. When μ ( r ) = ( 1 r 2 ) α , the induced space B μ becomes the α-Bloch space B α .

Throughout this paper, we assume that K is a right continuous and nonnegative nondecreasing function. For 0 < p < , 2 < q < , we say that a function f H ( Δ ) belongs to the space Q k ( p , q ) (see, [11]), if
f = { sup z Δ Δ | f ( z ) | p ( 1 | z | 2 ) q K ( g ( z , a ) ) d A ( z ) } 1 p < ,
where dA denotes the normalized Lebesgue area measure on Δ, g ( z , a ) is the Green function with logarithmic singularity at a, that is, g ( z , a ) = log 1 | φ a ( z ) | , where φ a ( z ) = a z 1 a ¯ z for a Δ . When K ( x ) = x s , s 0 , the space Q k ( p , q ) equals to F ( p , q , s ) , which is introduced by Zhao in [13]. Moreover (see [13]), we have that F ( p , q , s ) = B q + 2 p and F 0 ( p , q , s ) = B 0 q + 2 p for s > 1 , F ( p , q , s ) B q + 2 p and F 0 ( p , q , s ) B 0 q + 2 p for 0 s < 1 . When p 1 , Q k ( p , q ) is a Banach space with the norm
f Q k ( p , q ) = | f ( 0 ) | + f .
From [11], we know that Q k ( p , q ) B q + 2 p , Q k ( p , q ) = B q + 2 p if and only if
0 1 K ( log 1 r ) ( 1 r 2 ) 2 r d r < .

Moreover, f B q + 2 p C f Q k ( p , q ) (see [11], Theorem 2.1]).

Throughout the paper, we assume that
0 1 K ( log 1 r ) ( 1 r 2 ) q r d r < ,

otherwise Q k ( p , q ) consists only of constant functions (see [11]).

Let φ be a nonconstant analytic self-map of Δ, and let ϕ be an analytic function in Δ. We define the linear operators
ϕ C φ D f = ϕ ( f o φ ) = ϕ f ( φ ) and ϕ D C φ f = ϕ ( f o φ ) = ϕ f ( φ ) φ , for  f H ( Δ ) .

They are called weighted composition followed and proceeded by differentiation operators respectively, where C φ and D are composition and differentiation operators respectively. The boundedness and compactness of D C φ on the Hardy spaces were investigated by Hibschweiler and Portnoy in [3] and by Ohno in [8]. In [6], Li and Stević studied the boundedness and compactness of the operator D C φ on the α-Bloch spaces. In [7], Li and Stević studied the boundedness and compactness of the composition and differentiation operators between H and α-Bloch spaces. In [12], Yang studied the boundedness and compactness of the operator D C φ (or C φ D ) from Q k ( p , q ) to the Bloch-type spaces.

In this paper, we investigate the operators ϕ D C φ and ϕ C φ D from Q k ( p , q ) spaces to Bloch-type spaces and little Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given. Our results also generalize some known results in [12].

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation A B means that there is a positive constant C such that B C A C B .

2 Statement of the main results

In this paper, we shall prove the following results.

Theorem 2.1 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, p > 0 , q > 2 , and K is a nonnegative nondecreasing function on [ 0 , ) such that
0 1 K ( log 1 r ) ( 1 r ) min { 1 , q } ( log 1 1 r ) χ 1 ( q ) r d r < ,
(2.1)
where χ A ( x ) denote the characteristic function of the set A. Then ϕ D C φ : Q k ( p , q ) B μ is bounded if and only if
sup z Δ μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p < , sup z Δ μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p < .
(2.2)
Theorem 2.2 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, p > 0 , q > 2 , and K is a nonnegative nondecreasing function on [ 0 , ) such that (2.1) hold. Then ϕ D C φ : Q k ( p , q ) B μ is compact if and only if ϕ D C φ : Q k ( p , q ) B μ is bounded, and
(2.3)
Theorem 2.3 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, p > 0 , q > 2 , and K is a nonnegative nondecreasing function on [ 0 , ) such that (2.1) hold. Then ϕ D C φ : Q k ( p , q ) B μ , 0 is compact if and only if
lim | z | 1 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p = 0 , lim | z | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p = 0 .
(2.4)

From the above three theorems, we get the following

Corollary 2.4 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Then the following statements hold.
  1. (i)
    ϕ D C φ : B B is bounded if and only if
    sup z Δ ( 1 | z | 2 ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) 2 < , sup z Δ ( 1 | z | 2 ) | ϕ ( z ) ( φ ( z ) ) + ϕ ( z ) φ ( z ) | 1 | φ ( z ) | 2 < .
     
  2. (ii)
    ϕ D C φ : B B is compact if and only if ϕ D C φ : B B is bounded, and
    lim | φ ( z ) | 1 ( 1 | z | 2 ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) 2 = 0 , lim | φ ( z ) | 1 ( 1 | z | 2 ) | ϕ ( z ) ( φ ( z ) ) + ϕ ( z ) φ ( z ) | 1 | φ ( z ) | 2 = 0 .
     
  3. (iii)
    ϕ D C φ : B B 0 is compact if and only if
    lim | z | 1 ( 1 | z | 2 ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) 2 = 0 , lim | z | 1 ( 1 | z | 2 ) | ϕ ( z ) ( φ ( z ) ) + ϕ ( z ) φ ( z ) | 1 | φ ( z ) | 2 = 0 .
     
Theorem 2.5 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal, p > 0 , q > 2 , and K is a nonnegative nondecreasing function on [ 0 , ) such that (2.1) hold. Then the following statements hold.
  1. (i)
    ϕ C φ D : Q k ( p , q ) B μ is bounded if and only if
    sup z Δ μ ( | z | ) | ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) p + q + 2 p < , sup z Δ μ ( | z | ) | ϕ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p < .
     
  2. (ii)
    ϕ C φ D : Q k ( p , q ) B μ is compact if and only if ϕ C φ D : Q k ( p , q ) B μ is bounded, and
    lim | φ ( z ) | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) p + q + 2 p = 0 , lim | φ ( z ) | 1 μ ( | z | ) | ϕ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p = 0 .
     
  3. (iii)
    ϕ C φ D : Q k ( p , q ) B μ , 0 is compact if and only if
    lim | z | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) p + q + 2 p = 0 , lim | z | 1 μ ( | z | ) | ϕ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p = 0 .
     

From Theorem 2.5, we get the following

Corollary 2.6 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Then the following statements hold.
  1. (i)
    ϕ C φ D : B B is bounded if and only if
    sup z Δ ( 1 | z | 2 ) | ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) 2 < , sup z Δ ( 1 | z | 2 ) | ϕ ( z ) | 1 | φ ( z ) | 2 < .
     
  2. (ii)
    ϕ C φ D : B B is compact if and only if ϕ C φ D : B B is bounded, and
    lim | φ ( z ) | 1 ( 1 | z | 2 ) | ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) 2 = 0 , lim | φ ( z ) | 1 ( 1 | z | 2 ) | ϕ ( z ) | 1 | φ ( z ) | 2 = 0 .
     
  3. (iii)
    ϕ C φ D : B B 0 is compact if and only if
    lim | z | 1 ( 1 | z | 2 ) | ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) 2 = 0 , lim | z | 1 ( 1 | z | 2 ) | ϕ ( z ) | 1 | φ ( z ) | 2 = 0 .
     

3 Proofs of the main results

In this section, we will prove our main results. For this purpose, we need some auxiliary results.

Lemma 3.1 Let φ be an analytic self-map of Δ, ϕ be an analytic function in Δ. Suppose p > 0 , q > 2 . Then ϕ D C φ (or  ϕ C φ D ) : Q k ( p , q ) B μ is compact if and only if ϕ D C φ (or  ϕ C φ D ) : Q k ( p , q ) B μ is bounded and for any bounded sequence { f n } n N in Q k ( p , q ) which converges to zero uniformly on compact subsets of Δ as n , and ϕ D C φ f n B μ 0 (or ϕ C φ D f n B μ 0 ) as n .

Lemma 3.1 can be proved by standard way (see [1], Proposition 3.11]).

Lemma 3.2 A closed set K of B μ , 0 is compact if and only if it is bounded and satisfies
lim | z | 1 sup f K μ ( | z | ) | f ( z ) | = 0 .
(3.1)
Proof First of all, we suppose that K is compact and let ε > 0 . By the definition of B μ , 0 , we can choose an ε 2 -net which center at f 1 , f 2 , , f n in K respectively, and a positive number r ( 0 < r < 1 ) such that μ ( | z | ) | f i ( z ) | < ε 2 , for 1 i n and | z | > r . If f K , f f i B μ < ε 2 for some f i , so we have
μ ( | z | ) | f ( z ) | f f i B μ + μ ( | z | ) | f i ( z ) | < ε ,

for | z | > r . This establishes (3.1). □

On the other hand, if K is a closed bounded set which satisfies (3.1) and { f n } is a sequence in K , then by the Montel’s theorem, there is a subsequence { f n k } which converges uniformly on compact subsets of Δ to some analytic function f, and also { f n k } converges uniformly to f on compact subsets of Δ. According to (3.1), for every ε > 0 , there is an r, 0 < r < 1 , such that for all g K , μ ( | z | ) | g ( z ) | < ε 2 , if | z | > r . It follows that μ ( | z | ) | f ( z ) | < ε 2 , if | z | > r . Since { f n k } converges uniformly to f and { f n k } converges uniformly to f on | z | r , it follows that lim k sup f n k f B μ ε , i.e., lim k f n k f B μ = 0 , so that K is compact.

Lemma 3.3 ([14])

Let α > 0 and f H ( Δ ) . Then we have
sup z Δ ( 1 | z | 2 ) α | f ( z ) | | f ( 0 ) | + sup z Δ ( 1 | z | 2 ) α + 1 | f ( z ) | .
Proof of Theorem 2.1 First, suppose that the conditions in (2.2) hold. Then for any z Δ and f Q k ( p , q ) , by use of the fact f B q + 2 p C f Q k ( p , q ) and Lemma 3.3, we have
(3.2)
Taking the supremum in (3.2) for z Δ , and employing (2.2), we deduce that
ϕ D C φ : Q k ( p , q ) B μ

is bounded.

Conversely, suppose that ϕ D C φ : Q k ( p , q ) B μ is bounded. Then there exists a constant C such that ϕ D C φ f B μ C f Q k ( p , q ) for all f Q k ( p , q ) . Taking the functions f ( z ) z , and f ( z ) z 2 2 , which belong to Q k ( p , q ) , we get
sup z Δ μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | <
(3.3)
and
sup z Δ μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 + φ ( z ) [ ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) ] | < .
(3.4)
From (3.3), (3.4), and the boundedness of the function φ ( z ) , it follows that
sup z Δ μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | < .
(3.5)
For w Δ , let
f w ( z ) = 1 | w | 2 ( 1 w ¯ z ) q + 2 p ,
by direct calculation, we get
f w ( w ) = q + 2 p w ¯ ( 1 | w | 2 ) q + 2 p , f w ( w ) = q + 2 p p + q + 2 p w ¯ 2 ( 1 | w | 2 ) p + q + 2 p .
From [5], we know that f w Q k ( p , q ) , for each w Δ . Moreover, there is a positive constant C such that sup w Δ f w Q k ( p , q ) C . Hence, we have
C ϕ D C φ ϕ D C φ f φ ( z ) B μ q + 2 p p + q + 2 p μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p + q + 2 p μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | | φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p ,
(3.6)
for z Δ . Therefore, we obtain
(3.7)
Next, for w Δ , let
g w ( z ) = ( 1 | w | 2 ) 2 ( 1 w ¯ z ) p + q + 2 p p + q + 2 q + 2 1 | w | 2 ( 1 w ¯ z ) q + 2 p .
Then from [5], we see that g w ( z ) Q k ( p , q ) and sup w Δ g w Q k ( p , q ) < . Since
g φ ( z ) ( φ ( z ) ) = 0 , | g φ ( z ) ( φ ( z ) ) | = p + q + 2 p | φ ( z ) | 2 ( 1 | φ ( z ) | 2 ) p + q + 2 p ,
we have
> C ϕ D C φ ϕ D C φ g φ ( z ) B μ p + q + 2 p μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p .
(3.8)
Thus
(3.9)
Inequality (3.5) gives
sup | φ ( z ) | 1 2 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p ( 4 3 ) p + q + 2 p sup | φ ( z ) | 1 2 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | < .
(3.10)
Therefore, the first inequality in (2.2) follows from (3.9) and (3.10). From (3.7) and (3.8), we obtain
sup z Δ μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | | φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p < .
(3.11)
Inequalities (3.3) and (3.11) imply
(3.12)
and
(3.13)

Inequality (3.12) together with (3.13) implies the second inequality of (2.2). The proof of Theorem 2.1 is completed. □

Proof of Theorem 2.2 First, suppose that ϕ D C φ : Q k ( p , q ) B μ is bounded and (2.3) hold. Let { f n } n N be a sequence in Q k ( p , q ) such that sup n N f n Q k ( p , q ) < , and f n converges to 0 uniformly on compact subsets of Δ as n . By the assumption, for any ε > 0 , there exists a δ ( 0 , 1 ) such that
μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p < ε
and
μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p < ε
hold for δ < | φ ( z ) | < 1 . Since ϕ D C φ : Q k ( p , q ) B μ is bounded, it follows from the proof of Theorem 2.1 that
M 1 : = sup z Δ μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | < , M 2 : = sup z Δ μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | < .
Let K = { z Δ : | φ ( z ) | δ } . Then we have
ϕ D C φ f n B μ = sup z Δ μ ( | z | ) | ( ϕ D C φ f n ) ( z ) | + | ϕ ( 0 ) f n ( φ ( 0 ) ) φ ( 0 ) | sup z Δ μ ( | z | ) | ϕ ( z ) f n ( φ ( z ) ) ( φ ( z ) ) 2 | + sup z Δ μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | | f n ( φ ( z ) ) | + | ϕ ( 0 ) f n ( φ ( 0 ) ) φ ( 0 ) | sup z K μ ( | z | ) | ϕ ( z ) f n ( φ ( z ) ) ( φ ( z ) ) 2 | + sup z K μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | | f n ( φ ( z ) ) | + sup z ( Δ K ) μ ( | z | ) | ϕ ( z ) f n ( φ ( z ) ) ( φ ( z ) ) 2 | + sup z ( Δ K ) μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | | f n ( φ ( z ) ) | + | ϕ ( 0 ) f n ( φ ( 0 ) ) φ ( 0 ) | sup z K μ ( | z | ) | ϕ ( z ) f n ( φ ( z ) ) ( φ ( z ) ) 2 | + sup z K μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | | f n ( φ ( z ) ) | + | ϕ ( 0 ) f n ( φ ( 0 ) ) φ ( 0 ) | + C sup z ( Δ K ) μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p f n Q k ( p , q ) + sup z ( Δ K ) μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p f n Q k ( p , q ) M 2 sup z K | f n ( φ ( z ) ) | + M 1 sup z K | f n ( φ ( z ) ) | + 2 C ε f n Q k ( p , q ) + | ϕ ( 0 ) f n ( φ ( 0 ) ) φ ( 0 ) | .
(3.14)

From the fact that f n 0 as n on compact subsets of Δ, and Cauchy’s estimate, we conclude that f n 0 and f n 0 as n on compact subsets of Δ. Letting n in (3.14) and using the fact that ε is an arbitrary positive number, we obtain lim n ϕ D C φ f n B μ = 0 . Applying Lemma 3.1, the result follows.

Conversely, suppose that ϕ D C φ : Q k ( p , q ) B μ is compact. Then it is clear that ϕ D C φ : Q k ( p , q ) B μ is bounded. Let { z n } be a sequence in Δ such that | φ ( z n ) | 1 as n . For n N , let
f n ( z ) = 1 | φ ( z n ) | 2 ( 1 φ ( z n ) ¯ z ) q + 2 p .
Then sup n N f n Q k ( p , q ) < and f n converges to 0 uniformly on compact subsets of Δ as n . Since ϕ D C φ : Q k ( p , q ) B μ is compact, by Lemma 3.1, we have lim n ϕ D C φ f n B μ = 0 . On the other hand, from (3.6) we have
C ϕ D C φ f n B μ q + 2 p p + q + 2 p μ ( | z n | ) | ϕ ( z n ) ( φ ( z n ) ) 2 ( φ ( z n ) ) 2 | ( 1 | φ ( z n ) | 2 ) p + q + 2 p + q + 2 p μ ( | z n | ) | ϕ ( z n ) φ ( z n ) + ϕ ( z n ) φ ( z n ) | | φ ( z n ) | ( 1 | φ ( z n ) | 2 ) q + 2 p ,
which implies that
(3.15)

if one of these two limits exists.

Next, for n N , set
g n ( z ) = ( 1 | φ ( z n ) | 2 ) 2 ( 1 φ ( z n ) ¯ z ) p + q + 2 p p + q + 2 q + 2 1 | φ ( z n ) | 2 ( 1 φ ( z n ) ¯ z ) q + 2 p .
Then { g n } n N is a sequence in Q k ( p , q ) . Notice that g n ( φ ( z n ) ) = 0 ,
| g n ( φ ( z n ) ) | = p + q + 2 p | φ ( z n ) | 2 ( 1 | φ ( z n ) | 2 ) p + q + 2 p .
And g n converges to 0 uniformly on compact subsets of Δ as n . Since ϕ D C φ : Q k ( p , q ) B μ is compact, we have lim n ϕ D C φ g n B μ = 0 . On the other hand, since
ϕ D C φ f n B μ p + q + 2 p μ ( | z n | ) | ϕ ( z n ) ( φ ( z n ) ) 2 ( φ ( z n ) ) 2 | ( 1 | φ ( z n ) | 2 ) p + q + 2 p ,
we have
(3.16)
From (3.15) and (3.16), we get
lim | φ ( z n ) | 1 μ ( | z n | ) | ϕ ( z n ) φ ( z n ) + ϕ ( z n ) φ ( z n ) | ( 1 | φ ( z n ) | 2 ) q + 2 p = 0 .
(3.17)

The proof of Theorem 2.2 is completed. □

Proof of Theorem 2.3 First, let f Q k ( p , q ) . By the proof of Theorem 2.1, we have
μ ( | z | ) | ( ϕ D C φ f ) ( z ) | C { μ ( | z | ) | | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p + μ ( | z | ) | | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p } f Q k ( p , q ) .
(3.18)
Taking the supremum in (3.18) over all f Q k ( p , q ) such that f Q k ( p , q ) 1 , we can get
lim | z | 1 sup f Q k ( p , q ) 1 μ ( | z | ) | ( ϕ D C φ f ) ( z ) | = 0 .

By Lemma 3.2, we see that the operator ϕ D C φ : Q k ( p , q ) B μ , 0 is compact.

Conversely, suppose that ϕ D C φ : Q k ( p , q ) B μ , 0 is compact. By taking f ( z ) z and using the boundedness of ϕ D C φ : Q k ( p , q ) B μ , 0 , we get
lim | z | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | = 0 .
(3.19)
From this, by taking the test function f ( z ) z 2 2 and using the boundedness of ϕ D C φ : Q k ( p , q ) B μ , 0 , it follows that
lim | z | 1 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | = 0 .
(3.20)

In the following, we distinguish two cases:

First, we assume that φ < 1 . From (3.19) and (3.20), we obtain
lim | z | 1 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p 1 ( 1 φ ) p + q + 2 p lim | z | 1 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | = 0
and
lim | z | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p 1 ( 1 φ ) q + 2 p lim | z | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | = 0 .

So the result follows in this case.

Secondly, we assume that φ = 1 . Let { φ ( z n ) } n N be a sequence such that lim n | φ ( z n ) | = 1 . From the compactness of ϕ D C φ : Q k ( p , q ) B μ , 0 , we see that ϕ D C φ : Q k ( p , q ) B μ is compact. According to Theorem 2.2, we get
lim | φ ( z ) | 1 μ ( | z | ) | ϕ ( z ) ( φ ( z ) ) 2 | ( 1 | φ ( z ) | 2 ) p + q + 2 p = 0
(3.21)
and
lim | φ ( z ) | 1 μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p = 0 .
(3.22)
For any ε > 0 , from (3.19) and (3.22), there exists r ( 0 , 1 ) such that
μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p < ε ,
for r < | φ ( z ) | < 1 , and there exists σ ( 0 , 1 ) such that
μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ε ( 1 r 2 ) q + 2 p ,
for σ < | z | < 1 . Therefore, when σ < | z | < 1 , and r < | φ ( z ) | < 1 , we obtain
μ ( | z | ) | ϕ ( z ) φ ( z ) + ϕ ( z ) φ ( z ) | ( 1 | φ ( z ) | 2 ) q + 2 p < ε .
(3.23)
On the other hand, if σ < | z | < 1 , and | φ ( z ) | r , we have
(3.24)

From (3.23) and (3.24), we get the second equality of (2.4). Similarly to the above arguments, by (3.20) and (3.21), we can get the first equality of (2.4). The proof of Theorem 2.3 is completed. □

Similarly to the proofs of Theorems 2.1-2.3, we can get the proofs of Corollary 2.4, Theorem 2.5 and Corollary 2.6. We omit the proofs.

Declarations

Acknowledgements

The work is supported by the National Natural Science Foundation of China (Grant No. 11171080), and Foundation of Science and Technology Department of Guizhou Province (Grant No. [2010] 07).

Authors’ Affiliations

(1)
School of Mathematics and Computer Science, Guizhou Normal University

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© Long and Wu; licensee Springer 2012

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