Skip to main content

Some new characterizations of the Bloch space

Abstract

We obtain some new characterizations for the Bloch space on the open unit disk in the complex plane and the open unit ball of C n .

MSC:32A18.

1 Introduction

Let D be the open unit disk in the complex plane , C, Bthe open unit ball of the complex vector space C n , H(D) the class of all holomorphic functions on D and H(B) the class of all holomorphic functions on B.

For f C 1 (B), the invariant gradient ˜ f is defined by

( ˜ f)(z)=(f φ z )(0),

where

f(z)= ( f z 1 ( z ) , , f z n ( z ) )

is the complex gradient of f.

A holomorphic function f in D is said to belong to the Bloch space B(D) if

f B = | f ( 0 ) | + sup z D ( 1 | z | 2 ) | f ( z ) | <.

Under the above norm, B(D) is a Banach space (see, e.g. [1]). On the unit ball, the Bloch space B(B), which was introduced by Hahn in [2], is the space of all fH(B) such that

f B = sup z B sup w C n { 0 } | f ( z ) , w ¯ | n + 1 2 ( 1 | z | 2 ) | w | 2 + | w , z | 2 ( 1 | z | 2 ) 2 <.

For some classical results on Bloch spaces see [3] and [4].

It is well known that fB(B) if and only if (see, e.g. [5])

sup z B ( 1 | z | 2 ) |f(z)|<.

For α>0, an fH(B) is said to belong to the α-Bloch space, denoted by B α (B), if

sup z B ( 1 | z | 2 ) α |f(z)|<.

When α=1, the α-Bloch space is the classical Bloch space.

It is of some importance to give new characterizations for a function space, since, for example, it can be useful in the study of operators acting on the space. For example, by using the last expression, it is difficult to study composition operators on α-Bloch space. However, in [6] Zhang and Xu introduced the metric

F z α (w)= n + 1 2 λ α ( | z | ) | w | 2 + ( 1 λ α ( | z | ) ) | w , z | 2 / | z | 2 ( 1 | z | 2 ) α

and proved that f B α (B) if and only if

sup z , w C n { 0 } | f ( z ) w | F z α ( w ) <,
(1)

and by using (1), they completely characterized the boundedness and compactness of composition operators on α-Bloch spaces. For some other results on operators on Bloch-type spaces see, for example, [721] and the references therein.

For fH(B), in [22] was proved that fB(B) if and only if

M 1 = sup z , w B z w ( 1 | z | 2 ) 1 / 2 ( 1 | w | 2 ) 1 / 2 | f ( z ) f ( w ) | | w P w z s w Q w z | <.
(2)

Somewhat later, in [23] it was proved that fB(B) if and only if

M 2 = sup z , w B z w ( 1 | z | 2 ) 1 / 2 ( 1 | w | 2 ) 1 / 2 | f ( z ) f ( w ) | | z w | <,
(3)

while in [24] it was proved that fB(B) if and only if

M 3 = sup z , w B ( 1 | z | 2 ) 1 / 2 ( 1 | w | 2 ) 1 / 2 | f ( z ) f ( w ) | | 1 z , w | <.

These characterizations can be seen as derivative-free characterizations of the Bloch space on the unit ball. For the case of the unit polydisk, see [25] and [26]. For more characterizations of Bloch-type spaces in the unit disk, unit polydisk and unit ball, see, for example, [5, 2224, 2749].

In this paper, we give some new characterizations for the Bloch space. In Section 2, we give some preliminary results which are used in the proofs of our main results. In Section 3, we give two new characterizations for the Bloch space on D. In Section 4, we give four new characterizations for the Bloch space in the unit ball B, which, among others, generalize the corollaries in Section 3.

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the next. We say that two quantities K 1 (x) and K 2 (x) are comparable, if there are positive constants C 1 and C 2 independent of variable x such that

C 1 K 1 (x) K 2 (x) C 2 K 1 (x).

2 Preliminaries and auxiliary results

Let z=( z 1 ,, z n ) and w=( w 1 ,, w n ) be points in the complex vector space C n and z,w= z 1 w ¯ 1 ++ z n w ¯ n . Let Aut(B) be the group of all biholomorphic selfmaps of B. It is well known that Aut(B) is generated by the unitary operators on C n and the involutions φ a of the form

φ a (z)= a P a z s a Q a z 1 z , a ,

where s a = ( 1 | a | 2 ) 1 / 2 , P a is the orthogonal projection into the space spanned by aB, i.e.,

P a z= z , a a | a | 2 , | a | 2 =a,a, P 0 z=0

and Q a =I P a . See [5, 50] for more properties of φ a (z).

Recall that the weighted Bergman space A α p (B), where 0<p< and α>1, consists of those functions fH(B) for which

f A α p p = B | f ( z ) | p d v α (z)= c α B | f ( z ) | p ( 1 | z | 2 ) α dv(z)<,

where c α = Γ ( n + α + 1 ) n ! Γ ( α + 1 ) , dv is the normalized Lebesgue measure of B (i.e. v(B)=1). When n=1, we denote d v α by d A α . When α=0, we get the classical Bergman space, which will be denoted by A p = A p (B).

Let

dλ(z)= d v ( z ) ( 1 | z | 2 ) n + 1 .

For any ψAut(B) and f L 1 (B),

B f(z)dλ(z)= B fψ(z)dλ(z).
(4)

Thus dλ(z) is a Möbius invariant measure (see, e.g. [36]).

Next, we quote some well-known results that will be used in the proofs of our main results. We begin with the following characterization of the Bloch space in the unit ball (see [5, 32, 34]).

Lemma 2.1 Let 0<p<. A holomorphic function f is in the Bloch space B(B) if and only if

sup a B f φ a f ( a ) A p <.

In [51], are proved the following two characterizations for the weighted Bergman space in the unit ball.

Lemma 2.2 Assume that 0<p<, α>1 and fH(B). If β and γ are real parameters such that

β+γ=α+p(n+1)
(5)

and

1<β<p(n+1),1<γ<p(n+1),
(6)

then the following statements are equivalent:

  1. (a)

    f A α p (B);

  2. (b)
    Q 1 (f):= B B | f ( z ) f ( w ) | p | z w | p d v β (z)d v γ (w)<;
    (7)
  3. (c)
    Q 2 (f):= B B | f ( z ) f ( w ) | p | 1 z , w | p d v β (z)d v γ (w)<.
    (8)

Moreover, the quantities Q 1 (f), Q 2 (f), and f A α p p are comparable.

Lemma 2.3 [32]

Assume that fH(B), 0<p<, 1<q<, 0s<, 0t<p+2n, and p+s>n. Then for aB,

B | f ( z ) f ( 0 ) | p | z | t ( 1 | z | 2 ) q ( 1 | φ a ( z ) | 2 ) s d v ( z ) C B | ˜ f | p ( 1 | z | 2 ) q ( 1 | φ a ( z ) | 2 ) s d v ( z ) .
(9)

Lemma 2.4 [32]

Assume that fH(B) and 0<p<. Then fB(B) if and only if

sup a B B | ˜ f ( z ) | p ( 1 | φ a ( z ) | 2 ) n + 1 dλ(z)<.
(10)

The following well-known result can be found in [50] or [5].

Lemma 2.5 Let 1<t< and cR. Then there is a positive constant C such that

B ( 1 | z | 2 ) t | 1 z , w | n + 1 + t + c dv(z) { C ( 1 | w | 2 ) c , if  c > 0 , C log e 1 | w | 2 , if  c = 0 , is bounded , if  c < 0 ,

for all wB.

3 Characterizations of the Bloch space in the unit disk

In this section, we give two characterizations for the Bloch space in the unit disk as follows.

Theorem 3.1 Assume that fH(D) and 0<p<. If β and γ are real parameters satisfying the following conditions:

β+γ=p2,1<β<p2,1<γ<p2,
(11)

then the following statements are equivalent:

  1. (a)

    fB(D);

  2. (b)
    sup a D D D | f ( z ) f ( w ) | p | z w | p ( 1 | a | 2 ) 2 ( 1 | z | 2 ) β ( 1 | w | 2 ) γ | 1 a ¯ z | β γ + 2 | 1 a ¯ w | γ β + 2 dA(z)dA(w)<;
  3. (c)
    sup a D D D | f ( z ) f ( w ) | p | 1 z ¯ w | p ( 1 | a | 2 ) 2 ( 1 | z | 2 ) β ( 1 | w | 2 ) γ | 1 a ¯ z | β γ + 2 | 1 a ¯ w | γ β + 2 dA(z)dA(w)<.

Proof By taking n=1, α=0 in Lemma 2.2, we see that f A p (D) if and only if

D D | f ( z ) f ( w ) | p | z w | p ( 1 | z | 2 ) β ( 1 | w | 2 ) γ dA(z)dA(w)<
(12)

and if and only if

D D | f ( z ) f ( w ) | p | 1 z ¯ w | p ( 1 | z | 2 ) β ( 1 | w | 2 ) γ dA(z)dA(w)<,
(13)

when the conditions in (11) hold.

Replacing f by f φ a f(a) in (12) and (13), and using Lemma 2.1, we conclude that fB(D) if and only if

sup a D D D | f φ a ( z ) f φ a ( w ) | p | z w | p ( 1 | z | 2 ) β ( 1 | w | 2 ) γ dA(z)dA(w)<,
(14)

which is equivalent to

sup a D D D | f φ a ( z ) f φ a ( w ) | p | 1 z ¯ w | p ( 1 | z | 2 ) β ( 1 | w | 2 ) γ dA(z)dA(w)<.
(15)

Using the change of variables z φ a (z), w φ a (w) and the following equalities (see, e.g. [1]):

| φ a ( z ) φ a ( w ) | = | z w | ( 1 | a | 2 ) | 1 a ¯ w | | 1 a ¯ z |

and

| 1 φ a ( z ) ¯ φ a ( w ) | = | 1 z ¯ w | ( 1 | a | 2 ) | 1 a ¯ w | | 1 a ¯ z | ,

we see that the double integrals on the left side of (14) and (15) are equivalent to

sup a D D D | f ( z ) f ( w ) | p | z w | p ( 1 | a | 2 ) 2 ( 1 | z | 2 ) β ( 1 | w | 2 ) γ | 1 a ¯ z | β γ + 2 | 1 a ¯ w | γ β + 2 dA(z)dA(w)

and

sup a D D D | f ( z ) f ( w ) | p | 1 z ¯ w | p ( 1 | a | 2 ) 2 ( 1 | z | 2 ) β ( 1 | w | 2 ) γ | 1 a ¯ z | β γ + 2 | 1 a ¯ w | γ β + 2 dA(z)dA(w),

respectively. Therefore, fB(D) if and only if (b) holds, and if and only if (c) holds, as desired. □

Taking β=γ=p/21 in Theorem 3.1, we easily get the following corollary.

Corollary 3.1 Assume that fH(D) and 2<p<. Then the following statements are equivalent:

  1. (a)

    fB(D);

  2. (b)
    sup a D D D ( | f ( z ) f ( w ) | | z w | ) p ( 1 | φ a ( z ) | 2 ) ( 1 | φ a ( w ) | 2 ) d A t (z)d A t (w)<;
  3. (c)
    sup a D D D ( | f ( z ) f ( w ) | | 1 z ¯ w | ) p ( 1 | φ a ( z ) | 2 ) ( 1 | φ a ( w ) | 2 ) d A t (z)d A t (w)<,

where t=(p4)/2.

Taking β=p/22, γ=p/2 in Theorem 3.1, we can easily get the following result.

Corollary 3.2 Assume that fH(D) and 4<p<. Then the following statements are equivalent:

  1. (a)

    fB(D);

  2. (b)
    sup a D D D ( | f ( z ) f ( w ) | | z w | ) p ( 1 | φ a ( w ) | 2 ) 2 d A t (z)d A t (w)<;
  3. (c)
    sup a D D D ( | f ( z ) f ( w ) | | 1 z ¯ w | ) p ( 1 | φ a ( w ) | 2 ) 2 d A t (z)d A t (w)<,

where t=(p4)/2.

4 Characterizations of the Bloch space in the unit ball

In this section, we generalize Corollaries 3.1 and 3.2 in the setting of the unit ball.

Theorem 4.1 Assume that fH(B) and n+1<p<. Then fB(B) if and only if

sup a B B B ( | f ( z ) f ( w ) | | 1 z , w | ) p ( 1 | φ a ( z ) | 2 ) n + 1 2 ( 1 | φ a ( w ) | 2 ) n + 1 2 d v t (z)d v t (w)<,

where t=(p2(n+1))/2.

Proof Let α=0 and β=γ=(p(n+1))/2 in Lemma 2.2. We see that f A α p (B) if and only if

B B ( | f ( z ) f ( w ) | | 1 z , w | ( 1 | z | 2 ) 1 / 2 ( 1 | w | 2 ) 1 / 2 ) p d v k (z)d v k (w)<,

where k=(n+1)/2.

From this and by Lemma 2.1 we see that fB(B) if and only if

sup a B B B ( | f φ a ( z ) f φ a ( w ) | | 1 z , w | ( 1 | z | 2 ) 1 2 ( 1 | w | 2 ) 1 2 ) p d v k (z)d v k (w)<.

Using the change of variables z φ a (z), w φ a (w) and (4), we see that the left side of the last inequality is equivalent to

sup a B B B | f ( z ) f ( w ) | p ( 1 | φ a ( z ) | 2 ) p + n + 1 2 ( 1 | φ a ( w ) | 2 ) p + n + 1 2 | 1 φ a ( z ) , φ a ( w ) | p dλ(z)dλ(w).
(16)

The result follows by using the equalities (see, e.g. [5])

1 φ a ( z ) , φ a ( w ) = ( 1 a , a ) ( 1 z , w ) ( 1 z , a ) ( 1 a , w )

and

1 | φ a ( z ) | 2 = ( 1 | a | 2 ) ( 1 | z | 2 ) | 1 z , a | 2 ,

in (16). □

Theorem 4.2 Assume that fH(B) and 2n<p<. Then fB(B) if and only if

sup a B B B ( | f ( z ) f ( w ) | | w P w z s w Q w z | ) p ( 1 | φ a ( z ) | 2 ) n + 1 2 ( 1 | φ a ( w ) | 2 ) n + 1 2 d v t (z)d v t (w)<,
(17)

where t=(p2(n+1))/2.

Proof Suppose that (17) holds. Since

1 | 1 z , w | 1 | w P w z s w Q w z | ,z,wB.
(18)

Then by Theorem 4.1 and (18) we see that fB(B).

Conversely, suppose that fB(B). By using the change of variables z φ w (u), Lemma 2.3 and the following equality (see [5]):

1 1 φ w ( u ) , w = 1 u , w 1 | w | 2 ,u,wB,

we have

B B | f ( z ) f ( w ) | p ( 1 | φ a ( z ) | 2 ) n + 1 2 ( 1 | φ a ( w ) | 2 ) n + 1 2 | w P w z s w Q w z | p d v t ( z ) d v t ( w ) = B B | f ( z ) f ( w ) | p ( 1 | φ a ( z ) | 2 ) n + 1 2 ( 1 | φ a ( w ) | 2 ) n + 1 2 | φ w ( z ) | p | 1 w , z | p d v t ( z ) d v t ( w ) = B B | f φ w ( u ) f φ w ( 0 ) | p | u | p ( 1 | φ a ( φ w ( u ) ) | 2 ) n + 1 2 d v t ( u ) ( 1 | φ a ( w ) | 2 ) n + 1 2 d λ ( w ) C B B | ˜ f φ w ( u ) | p ( 1 | φ a ( φ w ( u ) ) | 2 ) n + 1 2 d v t ( u ) ( 1 | φ a ( w ) | 2 ) n + 1 2 d λ ( w ) C B B | ˜ f ( z ) | p ( 1 | φ w ( z ) | 2 ) n + 1 + t ( 1 | φ a ( z ) | 2 ) n + 1 2 d λ ( z ) ( 1 | φ a ( w ) | 2 ) n + 1 2 d λ ( w ) C K B | ˜ f ( z ) | p ( 1 | φ a ( z ) | 2 ) n + 1 d λ ( z ) ,
(19)

where

K= sup a , z B B 1 ( 1 | φ a ( z ) | 2 ) n + 1 2 ( 1 | φ w ( z ) | 2 ) n + 1 + t ( 1 | φ a ( w ) | 2 ) n + 1 2 dλ(w).

Employing the change of variables w φ z (u) and using the fact that | φ z (w)|=| φ w (z)| we have

K= sup a , z B B 1 ( 1 | φ z ( a ) | 2 ) n + 1 2 ( 1 | u | 2 ) n + 1 + t ( 1 | φ a ( φ z ( u ) ) | 2 ) n + 1 2 dλ(u).

It follows from Lemma 2.5 and the fact that |( φ a φ z )(u)|=| φ φ z ( a ) (u)| (see [5]) that

K = sup a , z B B 1 ( 1 | φ z ( a ) | 2 ) n + 1 2 ( 1 | φ φ z ( a ) ( u ) | 2 ) n + 1 2 d v t ( u ) = sup a , z B B ( 1 | u | 2 ) n + 1 2 | 1 u , φ z ( a ) | n + 1 d v t ( u ) = sup w B B ( 1 | u | 2 ) n + 1 2 | 1 u , w | n + 1 d v t ( u ) < .
(20)

Combining (19) with (20), the result follows from Lemma 2.4. □

By choosing α=0, γ=p/2, and β=p/2(n+1) in Lemma 2.2, similarly to the proof of Theorem 4.1 is obtained the following result.

Theorem 4.3 Assume that fH(B) and 2(n+1)<p<. Then fB(B) if and only if

sup a B B B ( | f ( z ) f ( w ) | | 1 z , w | ) p ( 1 | φ a ( w ) | 2 ) n + 1 d v t (z)d v t (w)<,

where t=(p2(n+1))/2.

Using Theorem 4.3, similarly to the proof of Theorem 4.2 is proved the following result.

Theorem 4.4 Assume that fH(B) and 2(n+1)<p<. Then fB(B) if and only if

sup a B B B ( | f ( z ) f ( w ) | | w P w z s w Q w z | ) p ( 1 | φ a ( w ) | 2 ) n + 1 d v t (z)d v t (w)<,

where t=(p2(n+1))/2.

References

  1. Zhu K Pure and Applied Mathematics 139. In Operator Theory on Function Spaces. Dekker, New York; 1990.

    Google Scholar 

  2. Hahn K: Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem. Can. J. Math. 1975, 27: 446–458. 10.4153/CJM-1975-053-0

    Article  MathSciNet  MATH  Google Scholar 

  3. Timoney R: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 1980, 12: 241–267. 10.1112/blms/12.4.241

    Article  MathSciNet  MATH  Google Scholar 

  4. Timoney R: Bloch functions in several complex variables II. J. Reine Angew. Math. 1980, 319: 1–22.

    MathSciNet  MATH  Google Scholar 

  5. Zhu K: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York; 2005.

    MATH  Google Scholar 

  6. Zhang M, Xu W: Composition operators on α -Bloch spaces of the unit ball. Acta Math. Sin. Engl. Ser. 2007, 23: 1991–2002. 10.1007/s10114-007-0993-x

    Article  MathSciNet  MATH  Google Scholar 

  7. Colonna F: New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space. Cent. Eur. J. Math. 2013,11(1):55–73. 10.2478/s11533-012-0097-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Krantz S, Stević S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. TMA 2009, 71: 1772–1795. 10.1016/j.na.2009.01.013

    Article  MathSciNet  MATH  Google Scholar 

  9. Li S, Stević S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 2007,9(2):195–206.

    MathSciNet  MATH  Google Scholar 

  10. Li S, Stević S: Integral type operators from mixed-norm spaces to α -Bloch spaces. Integral Transforms Spec. Funct. 2007,18(7):485–493. 10.1080/10652460701320703

    Article  MathSciNet  MATH  Google Scholar 

  11. Li S, Stević S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci. 2007,117(3):371–385. 10.1007/s12044-007-0032-y

    Article  MathSciNet  MATH  Google Scholar 

  12. Li S, Stević S:Weighted composition operators from H to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007., 2007: Article ID 48478

    Google Scholar 

  13. Li S, Stević S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 2008, 338: 1282–1295. 10.1016/j.jmaa.2007.06.013

    Article  MathSciNet  MATH  Google Scholar 

  14. Li S, Stević S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 2008,206(2):825–831. 10.1016/j.amc.2008.10.006

    Article  MathSciNet  MATH  Google Scholar 

  15. Madigan K, Matheson A: A compact composition operators on the Bloch space. Trans. Am. Math. Soc. 1995,347(7):2679–2687. 10.1090/S0002-9947-1995-1273508-X

    Article  MathSciNet  MATH  Google Scholar 

  16. Stević S: Essential norms of weighted composition operators from the α -Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal. 2008., 2008: Article ID 279691

    Google Scholar 

  17. Stević S: Norms of some operators from Bergman spaces to weighted and Bloch-type space. Util. Math. 2008, 76: 59–64.

    MathSciNet  MATH  Google Scholar 

  18. Stević S: On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball. Appl. Math. Comput. 2008, 206: 313–320. 10.1016/j.amc.2008.09.002

    Article  MathSciNet  MATH  Google Scholar 

  19. Stević S: Norm and essential norm of composition followed by differentiation from α -Bloch spaces to H μ . Appl. Math. Comput. 2009, 207: 225–229. 10.1016/j.amc.2008.10.032

    Article  MathSciNet  MATH  Google Scholar 

  20. Stević S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 2009, 354: 426–434. 10.1016/j.jmaa.2008.12.059

    Article  MathSciNet  MATH  Google Scholar 

  21. Stević S: On an integral operator between Bloch-type spaces on the unit ball. Bull. Sci. Math. 2010, 134: 329–339. 10.1016/j.bulsci.2008.10.005

    Article  MathSciNet  MATH  Google Scholar 

  22. Nowak M:Bloch and Möbius invariant Besov spaces on the unit ball of C n . Complex Var. Theory Appl. 2001, 44: 1–12.

    Article  MATH  Google Scholar 

  23. Ren G, Tu C:Bloch space in the unit ball of C n . Proc. Am. Math. Soc. 2005, 133: 719–726. 10.1090/S0002-9939-04-07617-8

    Article  MathSciNet  MATH  Google Scholar 

  24. Li S, Wulan H: Characterizations of α -Bloch spaces on the unit ball. J. Math. Anal. Appl. 2008, 343: 58–63. 10.1016/j.jmaa.2008.01.023

    Article  MathSciNet  MATH  Google Scholar 

  25. Stević S: Boundedness and compactness of an integral operator on a weighted space on the polydisc. Indian J. Pure Appl. Math. 2006, 37: 343–355.

    MathSciNet  MATH  Google Scholar 

  26. Stević S: On Lipschitz and α -Bloch spaces on the unit polydisc. Studia Sci. Math. Hung. 2008, 45: 361–378.

    MathSciNet  MATH  Google Scholar 

  27. Avetisyan K: Hardy-Bloch type spaces and lacunary series on the polydisk. Glasg. Math. J. 2007, 49: 345–356. 10.1017/S001708950700359X

    Article  MathSciNet  MATH  Google Scholar 

  28. Avetisyan K: Weighted integrals and Bloch spaces of n -harmonic functions on the polydisc. Potential Anal. 2008, 29: 49–63. 10.1007/s11118-008-9087-3

    Article  MathSciNet  MATH  Google Scholar 

  29. Clahane D, Stević S: Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball. J. Inequal. Appl. 2006., 2006: Article ID 61018

    Google Scholar 

  30. Li B, Ouyang C: Higher radial derivative of Bloch type functions. Acta Math. Sci. 2002, 22: 433–445.

    MathSciNet  MATH  Google Scholar 

  31. Li S, Stević S: Some characterizations of the Besov space and the α -Bloch space. J. Math. Anal. Appl. 2008, 346: 262–273. 10.1016/j.jmaa.2008.05.044

    Article  MathSciNet  MATH  Google Scholar 

  32. Li S, Wulan H: Some new characterizations of Bloch spaces. Taiwan. J. Math. 2010,14(6):2245–2259.

    MathSciNet  MATH  Google Scholar 

  33. Meng X:Some results on Q K , 0 (p,q) space. Abstr. Appl. Anal. 2008., 2008: Article ID 404636

    Google Scholar 

  34. Nowak M:Bloch space on the unit ball of C n . Ann. Acad. Sci. Fenn., Math. 1998, 23: 461–473.

    MathSciNet  Google Scholar 

  35. Ouyang C, Yang W, Zhao R:Characterizations of Bergman spaces and the Bloch space in the unit ball of C n . Trans. Am. Math. Soc. 1995, 374: 4301–4312.

    MathSciNet  MATH  Google Scholar 

  36. Ouyang C, Yang W, Zhao R:Móbius invariant Q p spaces associated with the Green’s function on the unit ball of C n . Pac. J. Math. 1998, 182: 69–99. 10.2140/pjm.1998.182.69

    Article  MathSciNet  MATH  Google Scholar 

  37. Stević S: On Bloch-type functions with Hadamard gaps. Abstr. Appl. Anal. 2007., 2007: Article ID 39176

    Google Scholar 

  38. Stević S: Bloch-type functions with Hadamard gaps. Appl. Math. Comput. 2009, 208: 416–422. 10.1016/j.amc.2008.12.010

    Article  MathSciNet  MATH  Google Scholar 

  39. Stević S: Harmonic Bloch and Besov spaces on the unit ball. Ars Comb. 2009, 91: 3–9.

    MathSciNet  MATH  Google Scholar 

  40. Wulan H, Zhu K:Derivative-free characterizations of Q K spaces. J. Aust. Math. Soc. 2007, 82: 283–295. 10.1017/S1446788700016086

    Article  MathSciNet  MATH  Google Scholar 

  41. Wulan H, Zhu K:Lacunary series in Q K spaces. Stud. Math. 2007, 178: 217–230. 10.4064/sm178-3-2

    Article  MathSciNet  MATH  Google Scholar 

  42. Wulan H, Zhu K: Bloch and BMO functions in the unit ball. Complex Var. Theory Appl. 2008, 53: 1009–1019. 10.1080/17476930802429123

    Article  MathSciNet  MATH  Google Scholar 

  43. Wulan H, Zhu K: Q K spaces via higher order derivatives. Rocky Mt. J. Math. 2008, 38: 329–350. 10.1216/RMJ-2008-38-1-329

    Article  MathSciNet  MATH  Google Scholar 

  44. Yang W: Some characterizations of α -Bloch spaces on the unit ball of C n . Acta Math. Sci. 1997, 17: 471–477.

    MathSciNet  Google Scholar 

  45. Zhao R: On α -Bloch functions and VMOA. Acta Math. Sci. 1996, 16: 349–360.

    MathSciNet  MATH  Google Scholar 

  46. Zhao, R: On a general family of function spaces. Ann. Acad. Sci. Fenn., Math. Diss. 105, 56 pp. (1996)

  47. Zhao R:A characterization of Bloch-type spaces on the unit ball of C n . J. Math. Anal. Appl. 2007, 330: 291–297. 10.1016/j.jmaa.2006.06.100

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhou J:Lacunary series in Q K type spaces. J. Funct. Spaces Appl. 2008, 6: 293–301. 10.1155/2008/152321

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhu K: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 1993, 23: 1143–1177. 10.1216/rmjm/1181072549

    Article  MathSciNet  MATH  Google Scholar 

  50. Rudin W: Function Theory in the Unit Ball of Cn$\mathbb{C}^{n}$. Springer, New York; 1980.

    Book  Google Scholar 

  51. Li S, Wulan H, Zhao R, Zhu K:A characterization of Bergman spaces on the unit ball of C n . Glasg. Math. J. 2009, 51: 315–330. 10.1017/S0017089509004996

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author of this paper is supported by the project of Department of Education of Guangdong Province (No. 2013KJCX0170) and NSF of Guangdong, China (S2013010011978).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stevo Stević.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Stević, S. Some new characterizations of the Bloch space. J Inequal Appl 2014, 459 (2014). https://doi.org/10.1186/1029-242X-2014-459

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-459

Keywords