Advances in Pure Mathematics
Vol.09 No.10(2019), Article ID:95725,6 pages
10.4236/apm.2019.910042
QK Type Spaces and Bloch Type Spaces on the Unit Ball
Rong Hu
School of Mathematics, Sichuan University of Arts and Sciences, Dazhou, China
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: September 10, 2019; Accepted: October 12, 2019; Published: October 15, 2019
ABSTRACT
Different function spaces have certain inclusion or equivalence relations. In this paper, the author introduces a class of Möbius-invariant Banach spaces of analytic function on the unit ball of , where are non-decreasing functions and ,, studies the inclusion relations between and a class of spaces which was known before, and concludes that is a subspace of , and the sufficient and necessary condition on kernel function such that .
Keywords:
Unit Ball, Space, Space, Equivalence Relation
1. Introduction
spaces were first given by Hasi Wulan and Matts Essen around 2000. In recent years, 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 on the unit ball was defined by Xu Wen in his paper [4]. According to Hasi Wulan, type spaces on unit disk were introduced and investigated, and the conditions on K such that become some known spaces were given (cf. [5] ). About multiple variables, the definition of on unit ball were given by Xu Wen (cf. [6] ), and the author has studied the inclusion relations between spaces and spaces on the unit ball (cf. [7] ). In this paper, the author introduces the spaces and spaces on the unit ball of , studies the inclusion relationship between them. Firstly, establish the relationship between the norm of the function which belongs to and the norm , proof that the is a subspace of ; and then obtain the necessary and sufficient condition of kernel functions when .
2. Preliminaries
Let and be the involution of satisfied . is the volume measure on , normalized so that , and is the Möbius invariant volume measure on (cf. [4] ), is the normalized surface measure on , the measure v and are related by (cf. [12] )
. (1)
Let denote the complex gradient of f, and is the invariant gradient of f (cf. [12] ). and are related by ( [12] )
. (2)
The Möbius invariant Green function is defined by , where
. (3)
Definition 1 Let is a right-continuous, non-decreasing function, for ,, we say that a holomorphic function f belongs to the space if
. (4)
Definition 2 space is defined by
. (5)
The constant C can represent different values in different places in this paper.
3. Main Results
In this paper, the author demonstrates that is a subspace of as the first main result and it is of great help for the second one.
Theorem 1. Let ,, then .
Proof Let , then
We have , when ,, and since is subharmonic, that
Thus, we have when , then .
The following result is the further study on the equivalence between and .
Theorem 2. Let ,, if and only if
. (6)
Proof Sufficiency: By theorem 1, we only need to show that .
Since , for given , then there exists , such that
.
Let , for any ,, we have
(7)
And when , we have
, and when , so
,
thus
,
By formula(7), then we have , i.e. . It means .
Necessary: We only need to show that if , there exists a function , but .
Let be an n-tuple of non-negative integers, and satisfied where N is a integer. Let , it is easy to show that , and by the proof of theorem 3 in [7], we know that when , which
thus
Since the conclusion of theorem 1 in [7], we have
,
Then if , we can get
,
which shows that , the theorem is proved.
With the above conclusion, further study in this field of operator theory on can be conducted in the future.
Founding
Scientific Research Fund of Sichuan Provincial Education Department of China (18ZA0416).
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Hu, R. (2019) QK Type Spaces and Bloch Type Spaces on the Unit Ball. Advances in Pure Mathematics, 9, 857-862. https://doi.org/10.4236/apm.2019.910042
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