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 APM  Vol.9 No.10 , October 2019
QK Type Spaces and Bloch Type Spaces on the Unit Ball
Abstract: Different function spaces have certain inclusion or equivalence relations. In this paper, the author introduces a class of Möbius-invariant Banach spaces QK,0 (p,q) of analytic function on the unit ball of Cn, where K:(0,∞)→[0,∞) are non-decreasing functions and 0

1. Introduction

Q K spaces were first given by Hasi Wulan and Matts Essen around 2000. In recent years, Q K type spaces have caused extensive research (cf. [1] - [11] ). To study a new kind of function space, we usually need to establish the relationship between that and those known to all. The notion of the spaces Q K on the unit ball was defined by Xu Wen in his paper [4]. According to Hasi Wulan, Q K type spaces Q K ( p , q ) on unit disk were introduced and investigated, and the conditions on K such that Q K ( p , q ) become some known spaces were given (cf. [5] ). About multiple variables, the definition of Q K , 0 ( p , q ) on unit ball were given by Xu Wen (cf. [6] ), and the author has studied the inclusion relations between Q K ( p , q ) spaces and B q + n + 1 p spaces on the unit ball (cf. [7] ). In this paper, the author introduces the Q K , 0 ( p , q ) spaces and B 0 q + n + 1 p spaces on the unit ball of n , studies the inclusion relationship between them. Firstly, establish the relationship between the norm of the function which belongs to Q K , 0 ( p , q ) and the norm f B 0 α , proof that the Q K , 0 ( p , q ) is a subspace of B 0 q + n + 1 p ; and then obtain the necessary and sufficient condition of kernel functions K ( r ) when Q K , 0 ( p , q ) = B 0 q + n + 1 p .

2. Preliminaries

Let a B n and φ a be the involution of B n satisfied φ a ( 0 ) = a . d v ( z ) is the volume measure on B n , normalized so that v ( B n ) = 1 , and d λ = d v ( z ) ( 1 | z | 2 ) n + 1 is the Möbius invariant volume measure on B n (cf. [4] ), d σ is the normalized surface measure on S n , the measure v and σ are related by (cf. [12] )

B n f ( z ) d v ( z ) = 2 n 0 1 r 2 n 1 d r S n f ( r ζ ) d σ ( ζ ) . (1)

Let f ( z ) = ( f z 1 , f z 2 , , f z n ) denote the complex gradient of f, and ˜ f ( z ) = ( f φ z ) ( 0 ) is the invariant gradient of f (cf. [12] ). ˜ f ( z ) and f ( z ) are related by ( [12] )

( 1 | z | 2 ) | f ( z ) | | ˜ f ( z ) | ( 1 | z | 2 ) 1 2 | f ( z ) | . (2)

The Möbius invariant Green function is defined by G ( z , a ) = g ( φ a ( z ) ) , where

g ( z ) = n + 1 2 n | z | 1 ( 1 t 2 ) n 1 t 2 n + 1 d t . (3)

Definition 1 Let K : ( 0 , ) [ 0 , ) is a right-continuous, non-decreasing function, for 0 < p < , p 2 n 1 < q < , we say that a holomorphic function f belongs to the space Q K , 0 ( p , q ) if

lim | a | 1 B n | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) = 0 . (4)

Definition 2 B 0 α space is defined by

B 0 α = { f H ( B n ) : lim | a | 1 ( 1 | z | 2 ) α 1 | ˜ f ( z ) | = 0 } . (5)

The constant C can represent different values in different places in this paper.

3. Main Results

In this paper, the author demonstrates that Q K , 0 ( p , q ) is a subspace of B 0 q + n + 1 p as the first main result and it is of great help for the second one.

Theorem 1. Let 0 < p < , p 2 n 1 < q < , then Q K , 0 ( p , q ) B 0 q + n + 1 p .

Proof Let E ( a , r 0 ) = { z B n , | φ a ( z ) | < r 0 } , then

B n | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) E ( a , r 0 ) | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( g ( φ a ( z ) ) ) d λ ( z ) = | z | < r 0 | ˜ ( f φ a ) ( z ) | p ( 1 | φ a ( z ) | 2 ) q + n + 1 p K ( g ( z ) ) d λ ( z ) K ( g ( r 0 ) ) | z | < r 0 ( 1 | z | 2 ) p | ( f φ a ) ( z ) | p ( 1 | φ a ( z ) | 2 ) q + n + 1 p d v ( z ) ( 1 | z | 2 ) n + 1 C | z | < r 0 ( 1 | φ a ( z ) | 2 ) q + n + 1 p | ( f φ a ) ( z ) | p d v (z)

We have ( 1 | φ a ( z ) | 2 ) = ( 1 | z | 2 ) ( 1 | a | 2 ) | 1 z , a | 2 , when | z | r 0 , 1 r 0 2 ( 1 + r 0 ) 2 ( 1 | z | 2 ) | 1 z , a | 2 1 ( 1 r 0 ) 2 , and since | f ( z ) | p is subharmonic, that

B n | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) C ( 1 | a | 2 ) q + n + 1 p | z | < r 0 | ( f φ a ) ( z ) | p d v ( z ) = C ( 1 | a | 2 ) q + n + 1 p 0 r 0 r 2 n 1 d r S n | ( f φ a ) ( r ς ) | p d σ ( ς ) C ( 1 | a | 2 ) q + n + 1 p | ( f φ a ) ( z ) | p = C ( 1 | a | 2 ) q + n + 1 p | ˜ f ( a ) | p

Thus, we have lim | a | 1 ( 1 | a | 2 ) q + n + 1 p | ˜ f ( a ) | p = 0 when f Q K , 0 ( p , q ) , then f B 0 q + n + 1 p .

The following result is the further study on the equivalence between Q K , 0 ( p , q ) and B 0 q + n + 1 p .

Theorem 2. Let 0 < p < , p 2 n 1 < q < , Q K , 0 ( p , q ) = B 0 q + n + 1 p if and only if

0 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r < . (6)

Proof Sufficiency: By theorem 1, we only need to show that B 0 q + n + 1 p Q K , 0 ( p , q ) .

Since 0 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r < , for given ε > 0 , then there exists r 0 : 0 < r 0 < 1 , such that

r 0 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r < ε .

Let E ( a , r 0 ) = { z B n , | φ a ( z ) | < r 0 } , for any f B q + n + 1 p , z B n \ E ( a , r 0 ) , we have

B n \ E ( a , r 0 ) | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) f B q + n + 1 p p B n \ E ( a , r 0 ) K ( G ( z , a ) ) d λ ( z ) f B q + n + 1 p p r 0 < | z | < 1 ( 1 | z | 2 ) n 1 K ( g ( z ) ) d v ( z ) f B q + n + 1 p p r 0 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r S n d σ ( ς ) < ε f B q + n + 1 p p (7)

And when z E ( a , r 0 ) , we have

lim | a | 1 E ( a , r 0 ) | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) = lim | a | 1 | z | < r 0 | ˜ ( f φ a ) ( z ) | p ( 1 | φ a ( z ) | 2 ) q + n + 1 p K ( g ( z ) ) d λ ( z ) lim | a | 1 sup | z | < r 0 ( 1 | φ a ( z ) | 2 ) q + n + 1 p | ˜ ( f φ a ) ( z ) | p | z | < r 0 K ( g ( z ) ) ( 1 | z | 2 ) n 1 d V ( z ) = lim | a | 1 sup | z | < r 0 ( 1 | φ a ( z ) | 2 ) q + n + 1 p | ˜ ( f φ a ) ( z ) | p 2 n × 0 r 0 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r S n d σ ( ς ) C lim | a | 1 sup | z | < r 0 ( 1 | φ a ( z ) | 2 ) q + n + 1 p | ˜ ( f φ a ) ( z ) | p

( 1 | φ a ( z ) | 2 ) = ( 1 | z | 2 ) ( 1 | a | 2 ) | 1 z , a | 2 , and 1 r 0 2 ( 1 + r 0 ) 2 ( 1 | z | 2 ) | 1 z , a | 2 1 ( 1 r 0 ) 2 when | z | r 0 , so

lim | a | 1 ( 1 | φ a ( z ) | 2 ) q + n + 1 p p | ˜ ( f φ a ) ( z ) | = 0 ,

thus

lim | a | 1 E ( a , r 0 ) | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) = 0 ,

By formula(7), then we have lim | a | 1 B n | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) = 0 , i.e. f Q K , 0 ( p , q ) . It means B 0 q + n + 1 p Q K , 0 ( p , q ) .

Necessary: We only need to show that if 0 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r = , there exists a function f B 0 q + n + 1 p , but f Q K , 0 ( p , q ) .

Let α = ( α 1 , α 2 , , α n ) be an n-tuple of non-negative integers, and | α | = α 1 + α 2 + + α n satisfied | α | = 2 N where N is a integer. Let f = | α | q + n + 1 p p z α , it is easy to show that f B q + n + 1 p , and by the proof of theorem 3 in [7], we know that S n J ( r ς ) p 2 d σ ( ς ) C ( 1 r ) ( q + n + 1 ) + p 2 when r [ 3 4 , 1 ) , which

J ( r ς ) = r 2 | α | 2 | α | 2 ( q + n + 1 p ) p ( α 1 2 | ς 1 α 1 1 ς 2 α 2 ς n α n | 2 + + α n 2 | ς 1 α 1 ς 2 α 2 ς n α n 1 | 2 r 2 | α | 2 | ς α | 2 )

thus

B n | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) B n ( 1 | z | 2 ) p 2 ( J ( z ) ) p 2 ( 1 | z | 2 ) q + n + 1 p K ( g ( z ) ) d λ ( z ) = 2 n 0 1 ( 1 r 2 ) q p 2 r 2 n 1 K ( g ( r ) ) d r S n J ( r ς ) p 2 d σ ( ς ) C 3 4 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r

Since the conclusion of theorem 1 in [7], we have

0 3 4 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r C 0 1 ( 1 r 2 ) 2 n 1 r 2 n 1 K ( g ( r ) ) d r < ,

Then if 0 1 ( 1 r 2 ) n 1 r 2 n 1 K ( g ( r ) ) d r = , we can get

B n | ˜ f ( z ) | p ( 1 | z | 2 ) q + n + 1 p K ( G ( z , a ) ) d λ ( z ) = ,

which shows that f Q K , 0 ( p , q ) , the theorem is proved.

With the above conclusion, further study in this field of operator theory on Q K , 0 ( p , q ) can be conducted in the future.

Founding

Scientific Research Fund of Sichuan Provincial Education Department of China (18ZA0416).

Cite this paper: Hu, R. (2019) QK Type Spaces and Bloch Type Spaces on the Unit Ball. Advances in Pure Mathematics, 9, 857-862. doi: 10.4236/apm.2019.910042.
References

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