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).