Through this paper, is the unit ball of the n-dimensional complex Euclidean space , is the boundary of . We denote the class of all holomorphic functions, with the compact-open topology on the unit ball by .
For any , , the inner product is defined by , and write .
Let be the Lebesgue volume measure on , normalized so that and be the surface measure on . Once again, we normalize so that . For and let .
For the measures and are related by the following formula:
is called integration by slices, for all (see  ).
For every point the Möbius transformation is defined by
where and (see  or  ).
The map has the following properties that , , and
where z and w are arbitrary points in . In particular,
For the Möbius invariant Green function in the unit ball denoted by where is defined by:
For , we have
where is a constant depending on n only.
Let denote the Banach space of bounded functions in with the norm .
For , the Beurling-type space (sometimes also called the Bers-type space) in the unit ball consists of those functions for which
Let is a right-continuous, non-decreasing function and is not equal to zero identically. The space consists of all functions such that
Clearly, if , then . For it gives the Bergman space . If consists of just the constant functions, we say that it is trivial.
We assume from now that all to appear in this paper are right-continuous and nondecreasing function, which is not equal to 0 identically.
In  , several basic properties of are proved, in connection with the Beurling-type space . In particular, an embedding theorem for and is obtained, together with other useful properties. Hadamard gaps series and Hadamard product on spaces of holomorphic function in the case of the unit disk has been studied quite well in  and  .
Through this, paper, given two quantities and both depending on a function , we are going to write if there exists a constant , independent of , such that for all . When , we write . If the quantities and are equivalent, then in particular we have if and only if . As usual, the letter C will denote a positive constant, possibly different on each occurrence.
In this paper, we introduce spaces, in terms of the right continuous and non-decreasing function on the unit ball . We discuss the nesting property of . We prove a sufficient condition for
, (the Beurling-type space). Also we generalize
the necessary condetion to for a kind of lacunary series. As aplplication, we show that the sufficient condition is also a necessary to .
2. 𝓝K Spaces in the Unit Ball
In this section we prove some basic Banach space properties of space. A sufficient and necessary condition for to be non-trivial is given. We discuss the nesting property of spaces and prove a sufficient condition for .
Let be a non-constant function, where is an n-tuple of non-negative integers and .
Then, if .
Let k be such that Let k be such that and let . Suppose that
where . Then, we have
By Jensen’s inequality on convexity,
Because we have . Therefore,
and . The lemma is proved.
Theorem 2.1 The Holomorphic function spaces , contains all polynomials if
Otherwise, contains only constant functions.
First assume that (12) holds. Let be a polynomial i.e. (there exists a such that ). Then,
Since a is arbitrary, it follows that
Thus, and the first half of the theorem is proved.
Now, we assume that the integral in (12) is divergent. Let is an n-tuple of non-negative integers , .
Then, we have and
There exists such that , by the subharmonicity of ,
Combining (17) and (18), we see that (12) implies that .
It is proved that and, since is arbitrary, any non-constant polynomial is not contained in . Using Lemma 2.1, we conclude that contains only constant functions. The theorem is proved.
Let and satisfy (12). If there exist a constant such that for , then . As a consequence, . if for .
Proof: Let . We note that from the property of , there exists a constant , such that if . Then, we have
This show that and, consequently, .
Let be nondecreasing function, then .
Proof: The theorem proved in  .
Proof: Let . Then,
Thus, and . This shows that . By Theorem 2.3, we have . The proof of theorem is complete.
3. Hadamard Gaps in 𝓝K Spaces in the Unit Ball
In this section we prove a necessary condition for a lacunary series defined by a normal sequence to belong to space. As an implication of Theorem
2.4, we prove that (19) is also necessary for .
Recall that an written in the form where
is a homogeneous polynomial of degree , is said to have Hadamard gaps (also known as lacunary series) if there exists a constant such that (see e.g.  )
Let for The sequence of homogeneous polynomials
is called a normal sequence if it possesses the following property (see  ):
・ for ;
In what following, we will consider all lacunary series defined by normal sequences of homogeneous polynomials. To formulate our main result, we denote
Let be a normal sequence and let . Then a
lacunary series , belongs to if
Proof: Let . Then, we have
By (6) for , we have
Let be sufficiently large such that . Then, for ,
This shows (24) and the theorem is proved.
if and only if (18) holds.
Proof: The sufficient condition was proved by Theorem 2.4. Now we prove the necessary condition, assume that . Among lacunary series defined by normal sequences, we consider
where and for and .
This shows that and, consequently, . By Theorem
3.1, we have
By (6), we have
On the other hand,
since K is non-decreasing. Thus,
Combining this, we obtain (18). The theorem is proved.
Our aim of the present paper is to characterize the holomorphic functions with Hadamard gaps in -type spaces on the unit ball, where K is the right continuous and non-decreasing function. Our main results will be of important uses in the study of operator theory of holomorphic function spaces.
The authors are thankful to the referee for his/her valuable comments and very useful suggestions.