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Hadamard Gaps and Nk-type Spaces in the Unit Ball

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DOI: 10.4236/apm.2017.74017    658 Downloads   894 Views  

ABSTRACT

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

1. Introduction

Through this paper, B is the unit ball of the n-dimensional complex Euclidean space 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 ) 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 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 0 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. 𝓝K 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 0 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 0 ( 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 0 | 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 0 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 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 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 : 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Bakhit, M. and Shammaky, A. (2017) Hadamard Gaps and Nk-type Spaces in the Unit Ball. Advances in Pure Mathematics, 7, 306-313. doi: 10.4236/apm.2017.74017.

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