Hadamard Gaps and <i>N</i>_k-type Spaces in the Unit Ball

In this paper, we introduce a class of holomorphic Banach spaces N _K of functions on the unit ball B of C ⁿ. We develop the necessary and sufficient condition for N _K( B) spaces to be non-trivial and we discuss the nesting property of N _K ( B) spaces. Also, we obtain some characterizations of functions with Hadamard gaps in N _K ( B ) spaces. As a consequence, we prove a necessary and sufficient condition for that N _K ( B ) spaces coincides with the Beurling-type space.

• <i>N</i>_k-type Spaces
• Beurling-Type Space
• Hadamard Gaps

1. Introduction

Through this paper, B is the unit ball of the n-dimensional complex Euclidean space ℂ n , S is the boundary of B . We denote the class of all holomorphic functions, with the compact-open topology on the unit ball B by H ( B ) .

For any z = ( z 1 , z 2 , ⋯ , z n ) , w = ( w 1 , w 2 , ⋯ , w n ) ∈ ℂ n , the inner product is defined by 〈 z , w 〉 = ( z 1 w 1 ¯ , z 2 w 2 ¯ , ⋯ , z n w n ¯ ) , and write | z | = 〈 z , w 〉 .

Let d v be the Lebesgue volume measure on ℂ n , normalized so that v ( B ) ≡ 1 and d σ be the surface measure on S . Once again, we normalize σ so that σ ( B ) ≡ 1 . For z ∈ B and r > 0 let B r = { z ∈ B : | z | ≤ r } .

For ζ ∈ B the measures v and σ are related by the following formula:

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

The identity

∫ S f d σ = ∫ S d σ ( ζ ) 1 2π ∫ 0 2π f ( e i θ ζ ) d θ , (2)

is called integration by slices, for all 0 ≤ θ ≤ 2 π (see [1] ).

For every point a ∈ B the Möbius transformation φ a : B → B is defined by

φ a ( z ) = a − P a ( z ) − S a Q a ( z ) 1 − 〈 z , a 〉 , (3)

where S a = 1 − | z | 2 , P a ( z ) = a 〈 z , a 〉 | a | 2 , P 0 = 0 and Q a = I − P a ( z ) (see [1] or [2] ).

The map φ a has the following properties that φ a ( 0 ) = a , φ a ( a ) = 0 , φ a = φ a − 1 and

1 − 〈 φ a ( z ) , φ a ( w ) 〉 = ( 1 − | a | 2 ) ( 1 − 〈 z , w 〉 ) ( 1 − 〈 z , a 〉 ) ( 1 − 〈 a , w 〉 ) ,

where z and w are arbitrary points in B . In particular,

1 − | φ a ( z ) | 2 = ( 1 − | a | 2 ) ( 1 − | z | 2 ) | 1 − 〈 z , a 〉 | 2 , (4)

For a ∈ B the Möbius invariant Green function in the unit ball B denoted by G ( z , a ) = g ( φ a ( z ) ) where g ( z ) is defined by:

g ( z ) = n + 1 2 n ∫ | z | 1 ( 1 − t 2 ) n − 1 t 1 − 2 n d t . (5)

For n > 1 , we have

1 C n ( 1 − r 2 ) n t − 2 ( n − 1 ) ≤ C n ( 1 − r 2 ) n t − 2 ( n − 1 ) , (6)

where C n is a constant depending on n only.

Let H ∞ ( B ) denote the Banach space of bounded functions in H ( B ) with the norm ‖ f ‖ ∞ = sup z ∈ B | f ( z ) | .

For α > 0 , the Beurling-type space (sometimes also called the Bers-type space) H α ∞ ( B ) in the unit ball B consists of those functions f ∈ H ( B ) for which

‖ f ‖ H α ∞ ( B ) = sup z ∈ B | f ( z ) | ( 1 − | z | 2 ) α < ∞ . (7)

Let K : ( 0 , ∞ ) → [ 0 , ∞ ) is a right-continuous, non-decreasing function and is not equal to zero identically. The N K ( B ) space consists of all functions f ∈ H ( B ) such that

‖ f ‖ K 2 = sup z ∈ B ∫ B | f ( z ) | 2 K ( G ( z , a ) ) d v ( z ) < ∞ . (8)

Clearly, if K ( t ) = t p , then N K ( B ) = N p ( B ) . For K ( t ) = 1 it gives the Bergman space A 2 ( B ) . If N K ( B ) consists of just the constant functions, we say that it is trivial.

We assume from now that all K : ( 0 , ∞ ) → [ 0 , ∞ ) to appear in this paper are right-continuous and nondecreasing function, which is not equal to 0 identically.

In [3] , several basic properties of N K ( B ) are proved, in connection with the Beurling-type space H α ∞ ( B ) . In particular, an embedding theorem for N K ( B ) and H α ∞ ( B ) is obtained, together with other useful properties. Hadamard gaps series and Hadamard product on N K spaces of holomorphic function in the case of the unit disk has been studied quite well in [4] and [5] .

Through this, paper, given two quantities A f and B f both depending on a function f ∈ H ( B ) , we are going to write A f ≲ B f if there exists a constant C > 0 , independent of f , such that A f ≤ C B f for all f . When A f ≲ B f ≲ A f , we write A f ≈ B f . If the quantities A f and B f are equivalent, then in particular we have A f < ∞ if and only if B f < ∞ . As usual, the letter C will denote a positive constant, possibly different on each occurrence.

In this paper, we introduce N K ( B ) spaces, in terms of the right continuous and non-decreasing function K : ( 0 , ∞ ) → [ 0 , ∞ ) on the unit ball B . We discuss the nesting property of N K ( B ) . We prove a sufficient condition for

N K ( B ) = H α ∞ ( B ) , α = n + 1 2 (the Beurling-type space). Also we generalize

the necessary condetion to N K ( B ) for a kind of lacunary series. As aplplication, we show that the sufficient condition is also a necessary to N K ( B ) = H n + 1 2 ∞ ( B ) .

2. 𝓝 K Spaces in the Unit Ball

In this section we prove some basic Banach space properties of N K ( B ) space. A sufficient and necessary condition for N K ( B ) to be non-trivial is given. We discuss the nesting property of N K ( B ) spaces and prove a sufficient condition for N K ( B ) = H n + 1 2 ∞ ( B ) .

Lemma 2.1

Let f ( z ) = ∑ k = 1 ∞ a k z k be a non-constant function, where k = ( k 1 , k 2 , ⋯ , k n ) is an n-tuple of non-negative integers and z k = ( z 1 k 1 , z 2 k 2 , ⋯ , z n k n ) .

Then, z k ∈ N K ( B ) if a k ≠ 0 .

Proof:

Let k be such that Let k be such that a k ≠ 0 and let F k ( z ) = a k z k . Suppose that

U θ f ( z ) = f ( z 1 e i θ 1 , z 2 e i θ 2 , ⋯ , z n e i θ n ) = f ∘ U θ ( z ) ,

where U θ ( z ) = ( z 1 e i θ 1 , z 2 e i θ 2 , ⋯ , z n e i θ n ) . Then, we have

F k ( z ) = 1 ( 2 π ) n ∫ 0 2π ⋯ ∫ 0 2π f ( z 1 e i θ 1 , ⋯ , z n e i θ n ) e − i k 1 θ 1 ⋯ e − i k n θ n d θ n = 1 ( 2 π ) n ∫ 0 2π ⋯ ∫ 0 2π ( U θ f ) ( z ) e − i k 1 θ 1 ⋯ e − i k n θ n d θ n . (9)

By Jensen’s inequality on convexity,

| F k ( z ) | 2 ≤ 1 ( 2 π ) 2 n ∫ 0 2π ⋯ ∫ 0 2π | U θ f ( z ) | 2 d θ 1 ⋯ d θ n . (10)

Consequently,

∫ B | F k ( z ) | 2 K ( G ( z , a ) ) d λ ( z ) ≤ ‖ U θ f ‖ K 2 1 ( 2 π ) 2 n ∫ 0 2π ⋯ ∫ 0 2π d θ 1 ⋯ d θ n ≤ ‖ U θ f ‖ K 2 . (11)

Because U θ ( z ) ∈ A u t ( B ) we have ‖ U θ f ‖ K = ‖ f ‖ K . Therefore,

‖ F k f ‖ K = ‖ a k z k ‖ K ≤ ‖ f ‖ K

and z k ∈ N K ( B ) . The lemma is proved.

Theorem 2.1 The Holomorphic function spaces N K ( B ) , contains all polynomials if

∫ 0 1 r 2 n − 1 K ( g ( r ) ) d r < ∞ . (12)

Otherwise, N K ( B ) contains only constant functions.

Proof:

First assume that (12) holds. Let f ( z ) be a polynomial i.e. (there exists a M > 0 such that | f ( z ) | 2 ≤ M , ∀ z ∈ B ¯ = B ∪ S ). Then,

∫ B | f ( z ) | 2 K ( G ( z , a ) ) d v ( z ) = 2 n ∫ 0 1 r 2 n − 1 K ( g ( r ) ) d r ∫ S | f ( ϕ a ( r ζ ) ) | 2 d σ ( ζ ) ≤ 2 n M ∫ 0 1 r 2 n − 1 K ( g ( r ) ) d r . (13)

Since a is arbitrary, it follows that

‖ f ‖ K 2 ≤ 2 n M ∫ 0 1 r 2 n − 1 K ( g ( r ) ) d r < ∞ . (14)

Thus, f ∈ N K ( B ) and the first half of the theorem is proved.

Now, we assume that the integral in (12) is divergent. Let α = ( α 1 , α 2 , ⋯ , α n ) is an n-tuple of non-negative integers | α | = α 1 + α 2 + ⋯ + α n ≥ 1 , f ( z ) = z α .

Then, we have | f ( r ξ ) | 2 = r 2 | α | | ξ α | 2 and

∫ S | ( r ζ ) α | 2 d σ ( r ζ ) ≥ r 2 | α | ( n − 1 ) ! α ! ( n − 1 + | α | ) ! ≥ C r 2 | α | . (15)

Thus,

‖ f ‖ K ≥ n C 2 2 | α | − 1 ∫ 1 / 2 1 r 2 n − 1 K ( g ( r ) ) d r . (16)

There exists a ∈ B such that f ( a ) ≠ 0 , by the subharmonicity of | f ∘ φ a ( r ξ ) | ,

‖ f ‖ K ≥ 3 n 2 | f ( a ) | 2 ∫ 0 1 / 2 r 2 n − 1 ( 1 − r 2 ) n + 1 K ( g ( r ) ) d r . (17)

Combining (17) and (18), we see that (12) implies that ‖ f ‖ K = ∞ .

It is proved that f ∉ N K ( B ) and, since α is arbitrary, any non-constant polynomial is not contained in N K ( B ) . Using Lemma 2.1, we conclude that N K ( B ) contains only constant functions. The theorem is proved.

Theorem 2.2

Let K 1 and K 2 satisfy (12). If there exist a constant t 0 > 0 such that K 2 ( t ) ≲ K 1 ( t ) for t ∈ ( 0 , t 0 ) , then N K 1 ( B ) ⊆ N K 2 ( B ) . As a consequence, N K 1 ( B ) = N K 2 ( B ) . if K 2 ( t ) ≈ K 1 ( t ) for t ∈ ( 0 , t 0 ) .

Proof: Let f ∈ N K 1 ( B ) . We note that from the property of g ( z ) , there exists a constant δ > 0 , such that g ( z ) < t 0 if | z | > δ . Then, we have

∫ B | f ( z ) | 2 K 2 ( G ( z , a ) ) d v ( z ) = ∫ B δ + ∫ | z | ≥ δ | f ( ϕ a ( z ) ) | 2 K 2 ( g ( z ) ) d v ( z ) (18)

where

∫ B δ | f ( ϕ a ( z ) ) | 2 K 2 ( g ( z ) ) d v ( z ) ≤ ‖ f ‖ ∞ 2 ∫ B δ ( 1 − | z | 2 ) − n K 2 ( g ( z ) ) d v ( z ) ≤ 2 n ‖ f ‖ ∞ 2 ∫ 0 δ r 2 n − 1 K 2 ( g ( r ) ) d r < ∞ ,

and

∫ | z | ≥ δ | f ( ϕ a ( z ) ) | 2 K 2 ( g ( z ) ) d v ( z ) ≤ ∫ | z | ≥ δ | f ( ϕ a ( z ) ) | 2 K 1 ( g ( z ) ) d v ( z ) ≤ ‖ f ‖ K 1 2 < ∞ .

This show that ‖ f ‖ K 2 < ∞ and, consequently, f ∈ N K 2 ( B ) .

Theorem 2.3

Let K : ( 0 , ∞ ) → [ 0 , ∞ ) be nondecreasing function, then N K ( B ) ⊂ H n + 1 2 ∞ ( B ) .

Proof: The theorem proved in [3] .

Theorem 2.4

N K ( B ) = H n + 1 2 ∞ ( B ) if

∫ 0 1 r 2 n − 1 ( 1 − r 2 ) n + 1 K ( g ( r ) ) d r < ∞ . (19)

Proof: Let f ∈ H n + 1 2 ∞ ( B ) . Then,

∫ B | f ( z ) | 2 K ( G ( z , a ) ) d v ( z ) ≤ ‖ f ‖ H n + 1 2 ∞ ( B ) 2 ∫ B ( 1 − | z | 2 ) − n K ( g ( z ) ) d v ( z ) ( 1 − | z | 2 ) n + 1 ≤ 2 n ‖ f ‖ H n + 1 2 ∞ ( B ) 2 ∫ 0 1 r 2 n − 1 ( 1 − r 2 ) n + 1 K ( g ( r ) ) d r . (20)

Thus, ‖ f ‖ K < ∞ and f ∈ N K ( B ) . This shows that H n + 1 2 ∞ ( B ) ⊂ N K ( B ) . By Theorem 2.3, we have N K ( B ) ⊂ H n + 1 2 ∞ ( B ) . The proof of theorem is complete.

3. Hadamard Gaps in 𝓝 K 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 N K ( B ) space. As an implication of Theorem

2.4, we prove that (19) is also necessary for N K ( B ) = H n + 1 2 ∞ ( B ) .

Recall that an f ∈ H ( B ) written in the form f ( z ) = ∑ k = 0 ∞ P n k ( z ) where

P n k is a homogeneous polynomial of degree n k , is said to have Hadamard gaps (also known as lacunary series) if there exists a constant c > 1 such that (see e.g. [6] )

n k + 1 n k ≥ c ,   ∀ k ≥ 0. (21)

Let Λ n ⊂ S for n = n 0 , n 0 + 1 , ⋯ . The sequence of homogeneous polynomials

P n ( z ) = ∑ ζ ∈ Λ n 〈 z , ζ 〉 n , (22)

is called a normal sequence if it possesses the following property (see [7] ):

・ | P n ( z ) | ≤ C | z | n for z ∈ B ;

・ ∑ ξ , ζ ∈ Λ n ξ , ζ n ≥ n k + 1 C .

In what following, we will consider all lacunary series defined by normal sequences of homogeneous polynomials. To formulate our main result, we denote

L j = ∫ S | P n j ( ζ ) | 2 d σ ( ζ ) . (23)

Theorem 3.1

Let P n ( z ) be a normal sequence and let I K = { n ∈ ℕ : 2 k ≤ n ≤ 2 k + 1 } . Then a

lacunary series f ( z ) = ∑ k = 0 ∞ P n k ( z ) , belongs to N K ( B ) if

∑ k = 0 ∞ n k m 2 k K ( n k − m ) ∑ n j ∈ I k L j < ∞ . (24)

Proof: Let f ∈ N K ( B ) . Then, we have

∫ B | f ( z ) | 2 K ( G ( z , a ) ) d v ( z ) ≥ ∫ B | ∑ k = 0 ∞ P n k ( z ) | 2 K ( g ( | z | ) ) d v ( z ) ≥ ∑ k = 0 ∞ n k 2 k ∑ n j ∈ I k L j ∫ 0 1 r 2 m − 1 K ( g ( r ) ) d r , (25)

where

| ∑ k = 0 ∞ P n k ( z ) | 2 = ∑ k = 0 ∞ 1 2 k ∑ n j ∈ I k | P n k ( ζ ) | 2 . (26)

By (6) for 1 2 ≤ r ≤ 1 , we have

K ( g ( r ) ) ≥ K ( c − 1 ( 1 − r ) m ) . (27)

Consequently,

∫ 0 1 r 2 m − 1 K ( g ( r ) ) d r ≥ ∫ 1 2 1 r 2 m − 1 K ( c − 1 ( 1 − r ) m ) d r ≥ ∫ 0 log 2 e − 2 m t K ( c 1 − 1 t m ) d t ≥ K ( n k − m ) ∫ c 1 n k − 1 log 2 e − 2 m t d t ≥ n k m − 1 K ( n k − m ) ∫ c 1 n k log 2 e − 2 t d t . (28)

Let k ′ be sufficiently large such that n k ′ log 2 ≥ c 1 + 1 . Then, for k ≥ k ′ ,

∫ 0 1 r 2 m − 1 K ( g ( r ) ) d r ≥ n k m − 1 K ( n k − m ) . (29)

And

∫ B | f ( z ) | 2 K ( G ( z , a ) ) d v ( z ) ≥ C ∑ k = k ′ ∞ n k m 2 k K ( n k − m ) ∑ n j ∈ I k L j . (30)

This shows (24) and the theorem is proved.

Theorem 3.2

N K ( B ) = H n + 1 2 ∞ ( B ) if and only if (18) holds.

Proof: The sufficient condition was proved by Theorem 2.4. Now we prove the necessary condition, assume that N K ( B ) = H n + 1 2 ∞ ( B ) . Among lacunary series defined by normal sequences, we consider

f ( z ) = ∑ k = k 0 ∞ P 2 k ( z ) , (31)

where P 2 k = ∑ ζ ∈ Λ n 〈 z , ζ 〉 2 k and | P 2 k | = C | z | 2 k for k ≥ k 0 , 2 k 0 ≥ n 0 and z ∈ B .

Thus

| f ( z ) | ( 1 − | z | 2 ) n + 1 ≤ ( 1 − | z | 2 ) n + 1 ∑ k = k 0 ∞ | P 2 k ( z ) | ≤ C ∑ n = 1 ∞ | z | n ≤ C . (32)

This shows that f ∈ H n + 1 2 ∞ ( B ) and, consequently, f ∈ N K ( B ) . By Theorem

3.1, we have

∑ k = 1 ∞ 2 k ( m − 1 ) K ( 2 − m k ) < ∞ . (33)

By (6), we have

∫ 1 / 2 1 r 2 m − 1 ( 1 − r 2 ) m + 1 K ( g ( r ) ) d r ≤ ∫ 0 c 1 / m log 2 t − m − 1 K ( t m ) d t . (34)

On the other hand,

∫ 0 1 / 2 t − m − 1 K ( t m ) d t = ∑ k = 1 ∞ ∫ 2 − k − 1 2 − k t − m − 1 K ( t m ) d t = ∑ k = 1 ∞ 2 − ( k + 1 ) 2 − m − 1 K ( 2 − m k ) , (35)

since K is non-decreasing. Thus,

∫ 1 / 2 1 r 2 m − 1 ( 1 − r 2 ) m + 1 K ( g ( r ) ) d r < ∞ . (36)

Combining this, we obtain (18). The theorem is proved.

4. Conclusion

Our aim of the present paper is to characterize the holomorphic functions with Hadamard gaps in N K -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.

Acknowledgements

The authors are thankful to the referee for his/her valuable comments and very useful suggestions.

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