Characterizations of α-Bloch spaces on the unit ball

J. Math. Anal. Appl. 343 (2008) 58–63 www.elsevier.com/locate/jmaa

Characterizations of α-Bloch spaces on the unit ball Songxiao Li a,b,∗ , Hasi Wulan a a Department of Mathematics, Shantou University, Shantou 515063, GuangDong, China b Department of Mathematics, JiaYing University, 514015 Meizhou, GuangDong, China

Received 24 September 2007 Available online 16 January 2008 Submitted by R. Timoney

Abstract We obtain a characterization of the α-Bloch space for any α > 0 in the unit ball of Cn in terms of |f (z) − f (w)|/|z − w|. Moreover, a new characterization for the Bloch space is given. © 2008 Elsevier Inc. All rights reserved. Keywords: α-Bloch space; Little α-Bloch space; Derivative-free

1. Introduction Let B be the unit ball of Cn . Let z = (z1 , . . . , zn ) and w = (w1 , . . . , wn ) be points in Cn , we write  z, w = z1 w¯ 1 + · · · + zn w¯ n , |z| = |z1 |2 + · · · + |zn |2 . Thus B = {z ∈ Cn : |z| < 1}. Let Aut(B) be the group of all biholomorphic self maps of B. It is well known that Aut(B) is generated by the unitary operators on Cn and the involutions ϕa of the form ϕa (z) =

a − Pa z − sa Qa z , 1 − z, a

where sa = (1 − |a|2 )1/2 , Pa is the orthogonal projection into the space spanned by a ∈ B, i.e. Pa z =

z,aa , |a|2

|a|2 = a, a, P0 z = 0 and Qa = I − Pa (see [7,11]). It is well known that (see [11]) ρ(z, w) = |ϕz (w)| is a distance function on B, and we call it the pseudo-hyperbolic metric. We denote by H (B) the class of all holomorphic functions on the unit ball. For f ∈ H (B), ∇f denotes the complex gradient of f , i.e.   ∂f ∂f ∇f (z) = (z), . . . , (z) . ∂z1 ∂zn * Corresponding author at: Department of Mathematics, Shantou University, Shantou 515063, GuangDong, China.

E-mail addresses: [email protected], [email protected] (S. Li), [email protected] (H. Wulan). 0022-247X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2008.01.023

S. Li, H. Wulan / J. Math. Anal. Appl. 343 (2008) 58–63

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 denote the invariant gradient on B, i.e. (∇f  )(z) = ∇(f ◦ ϕz )(0). For f ∈ H (B) and z ∈ B, For f ∈ C 1 (B), let ∇f set |∇f (z), w| ¯ , Qf (z) = sup 1/2 n (H (w, w)) z w∈C \{0} where Hz (w, w) is the Bergman metric on B, i.e. Hz (w, w) =

n + 1 (1 − |z|2 )|w|2 + |w, z|2 . 2 (1 − |z|2 )2

The Bloch space B(B), introduced by Timoney (see [8,9]), is the space of all f ∈ H (B) such that f B = sup Qf (z) < ∞. z∈B

The little Bloch space B0 is the space of all f ∈ H (B) such that lim|z|→1 Qf (z) = 0. Let α > 0. The α-Bloch space B α is the space of all holomorphic functions f on B such that   α  bα (f ) = sup 1 − |z|2 ∇f (z) < ∞.

(1)

z∈B

It is clear that B α is a Banach space under the norm f Bα = |f (0)| + bα (f ). Let B0α denote the subspace of B α consisting of those f ∈ B α for which (1 − |z|2 )α |∇f (z)| → 0, as |z| → 1. B0α is called the little α-Bloch space. ∂f For f ∈ H (B), let Rf denote the radial derivative of f ; that is, Rf (z) = nj=1 zj ∂z (z). It is well known that j α f ∈ B if and only if (see for example [11])   α  sup 1 − |z|2 Rf (z) < ∞. z∈B

f ∈ B0α if and only if lim|z|→1 (1 − |z|2 )α |Rf (z)| = 0. See [1,3–6,8,9,11] for more characterizations of the α-Bloch space. Recently, Zhao [10] obtained a characterization for the α-Bloch space as follows. Theorem A. Let 0 < α  2. Let λ be any real number satisfying the following properties: (i) 0  λ  α if 0 < α < 1; (ii) 0 < λ < 1 if α = 1; (iii) α − 1  λ  1 if 1 < α  2. Then an analytic function f on B is in B α if and only if  λ  α−λ |f (z) − f (w)| < ∞. · Sλ (f ) := sup 1 − |z|2 1 − |w|2 |z − w| z,w∈B

(2)

z=w

Moreover, for any α and λ satisfying the above conditions two seminorms supz∈B (1 − |z|2 )|∇f (z)| and Sλ (f ) are equivalent. Zhao also showed that the above theorem fails for α > 2. The purpose of this paper is to give a characterization of the α-Bloch space for any α > 0 in terms of |f (z) − f (w)|/|z − w|, in which the assumption α  2 is not needed. A new derivative-free characterization for the Bloch space is also provided. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation A B means that there is a positive constant C such that B/C  A  CB. 2. Characterization of the α-Bloch space Our first result is the following Theorem 1, which gives a derivative-free characterization of α-Bloch functions for any α > 0.

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Theorem 1. Let α  s  0, r ∈ (0, 1), f ∈ H (B). Then f ∈ B α if and only if Ls =

 s  α−s |f (z) − f (w)| < ∞. · 1 − |z|2 1 − |w|2 |z − w| ρ(z,w)
(3)

z=w

Proof. Fixing a point w in (3), and letting z = w + ξ ∇f (w) → w

(4)

|w|2 )α |∇f (w)|

 Ls . This shows that the sufficiency of Theorem 1 is evident. So the with ξ ∈ C, we obtain (1 − remainder of the proof is to show that an α-Bloch function f possesses the characterization (3). Assume that f ∈ B α . If 0 < α  2, then from Theorem A, it is clear that (3) holds. Now we need only to show that (3) holds for α > 2. By  2 (1 − |z|2 )(1 − |w|2 ) 1 − ϕz (w) = , |1 − z, w|2 we get

ρ(z, w) |z − w|2 + |z, w|2 − |z|2 |w|2 = . |z − w| |z − w|2 |1 − z, w|2

(5)

(6)

Together with |z, w|2  |z|2 |w|2 and (6), we have ρ(z, w) 

|z − w| |1 − z, w|

(7)

for all z and w in B. Let E(z, r) = {w ∈ B: |ϕz (w)| < r}. Then the volume of E(z, r) is given by   r 2n (1 − |z|2 )n+1 v E(z, r) = . (1 − r 2 |z|2 )n+1

(8)

Set |E(z, r)| = v(E(z, r)). For r > 0 and w ∈ E(z, r), we have (see [4,11]) |1 − z, w| 2 1   , 2 1+r 1 − |w| 1 − r2

1 |1 − z, w| 1  .  2 2 1−r 1 − |z|

It follows that n+1    n+1  n+1  1 − |z|2 1 − |w|2 1 − z, w E(z, r),

(9)

(10)

whenever w ∈ E(z, r). For f ∈ H (B) and z ∈ B, we have f (z) − f (0) =

1 n 0

k=1

 ∂f zk (tz) dt. ∂zk

For r ∈ (0, 1) and ρ(0, z) < r, we have      f (z) − f (0)  |z| sup ∇f (w): w ∈ E(0, r) . It is easy to see that, in the relatively compact set E(0, r), the Euclidean metric is comparable to the pseudo-hyperbolic  (w)|. Hence there is a positive constant C depending only on r such metric and that |∇f (w)| is comparable to |∇f that      f (z) − f (0)  Cρ(z, 0) sup  ∇f (w): w ∈ E(0, r)

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for all z ∈ E(0, r). Replacing f by f ◦ ϕw and z by ϕw (z), using the Möbius invariance of the pseudo-hyperbolic metric and the invariant gradient, we obtain      f (z) − f (w)  Cρ(z, w) sup  (11) ∇f (v): v ∈ E(z, r) for all z and w in B with ρ(z, w) < r. For α > 2, (7), (9) and Theorem 7.2(a) of [11] give      f (z) − f (w)  Cρ(z, w) sup  ∇f (v): v ∈ E(z, r)    |z − w| C sup  ∇f (v): v ∈ E(z, r) |1 − z, w|     |z − w|   1 − |u|2 α−1 : u ∈ E(z, r) C sup ∇f (u) (1 − |z|2 )s (1 − |w|2 )α−s |z − w| f Bα C 2 (1 − |z| )s (1 − |w|2 )α−s

(12)

for all z and w in B with ρ(z, w) < r. Therefore,  s  α−s |f (z) − f (w)| 1 − |z|2 1 − |w|2  Cf Bα · |z − w| for all z and w in B with ρ(z, w) < r. The result follows.

(13) 2

Theorem 2. Let α  s  0, r ∈ (0, 1), f ∈ H (B). Then f ∈ B0α if and only if lim

|z|→1−

 s  α−s |f (z) − f (w)| = 0. 1 − |z|2 1 − |w|2 · |z − w| ρ(z,w)
(14)

sup

z=w

Proof. Sufficiency. Assume that (14) holds. For ε > 0, there exists a δ ∈ (0, 1) such that  s  α−s |f (z) − f (w)| < ε, 1 − |z|2 1 − |w|2 · |z − w| ρ(z,w)
z=w

whenever |z| > δ. Let w approach z in the radial direction, we have   α  1 − |z|2 Rf (z)  ε, whenever |z| > δ, which implies f ∈ B0α . Necessity. Now we assume that f ∈ B0α . Let fλ (z) = f (λz), λ ∈ (0, 1). By (13) we have  s  α−s |(f − fλ )(z) − (f − fλ )(w)|  Cf − fλ Bα 1 − |z|2 1 − |w|2 |z − w| and s  α−s |fλ (z) − fλ (w)|  1 − |z|2 1 − |w|2 |z − w| 2 s 2 α−s  s  α−s |f (λz) − f (λw)| λ(1 − |z| ) (1 − |w| ) = 1 − |λz|2 1 − |λw|2 |λz − λw| (1 − |λz|2 )s (1 − |λw|2 )α−s   Cλ s  1 − |z|2 f Bα (1 − |λ|2 )s for all z and w in B with ρ(z, w) < r. By the triangle inequality, we get  s  α−s |f (z) − f (w)|  s λ 1 − |z|2 1 − |w|2 1 − |z|2 f Bα .  Cf − fλ Bα + C 2 s |z − w| (1 − |λ| ) ρ(z,w)
z=w

In the above inequality, first letting |z| → 1, then letting λ → 1− , we get the desired result.

2

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3. Characterization of the Bloch space For f ∈ H (B), Nowak [4] proved that f ∈ B if and only if  1/2  1/2 M1 = sup 1 − |z|2 1 − |w|2 · z,w∈B z=w

|f (z) − f (w)| < ∞. |w − Pw z − sw Qw z|

(15)

In [6], Ren and Tu replaced the above condition by  1/2  1/2 |f (z) − f (w)| M2 = sup 1 − |z|2 < ∞. 1 − |w|2 · |z − w| z,w∈B

(16)

z=w

If n = 1, the conditions (15) and (16) become the same thing which was established by Holland and Walsh [2]. Moreover, Ren and Tu proved that f ∈ B0 if and only if  1/2  1/2 |f (z) − f (w)| = 0. lim sup 1 − |z|2 1 − |w|2 · |z − w|

|z|→1 w∈B z=w

(17)

In fact, we can claim that f ∈ B0 if and only if  1/2  1/2 lim sup 1 − |z|2 1 − |w|2 ·

|z|→1 w∈B z=w

|f (z) − f (w)| = 0. |w − Pw z − sw Qw z|

(18)

For all z, w ∈ B

  |w − Pw z − sw Qw z| = Pw (w − z) + sw Qw (w − z)  |w − z|,

therefore, from (17) we see that (18) implies f ∈ B0 . Carefully checking the proof of Theorem 3.2 of [6], we see that f ∈ B0 implies (18). In this section we give a new characterization of Bloch functions. Now we state our main result as follows. Theorem 3. A holomorphic function f is in the Bloch space B if and only if  1/2  1/2 |f (z) − f (w)| < ∞. M3 = sup 1 − |z|2 1 − |w|2 · |1 − z, w| z,w∈B

(19)

z=w

Proof. Assume that f ∈ B. Since 1 1  |1 − z, w| |w − Pw z − sw Qw z|

(20)

for all z, w ∈ B, it follows from (15) and (20) that (19) holds. Conversely, assume that (19) holds. Let B(z, r) be the ball centered at z with radius r. By the Cauchy estimate and the subharmonicity we have that for every ε ∈ (0, 1/4)      f (z) − f (a) sup 1 − |a|2 ∇f (a)  C z∈B(a,ε(1−|a|))



C (1 − |a|2 )n+1



  f (z) − f (a) dv(z).

(21)

B(a,2ε(1−|a|))

Let E(a, r) = {z ∈ B: |ϕa (z)| < r}. For a fixed r > 0, choose a ball B(a, 2ε(1 − |a|)) ⊂ E(a, r). In fact, for z ∈ B(a, 2ε(1 − |a|)), we have   |z − Pz a − sz Qz a|  |z − a|  2ε 1 − |a|       2ε 1 − |a||z|  2ε 1 − a, z    2ε 1 − a, z,

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i.e. |ϕa (z)| = |ϕz (a)|  2ε. Hence z ∈ E(a, 2ε), i.e. B(a, 2ε(1 − |a|)) ⊂ E(a, 2ε). It follows from (10) and (21) that there exists a constant C such that       C 2   f (z) − f (a) dv(z) 1 − |a| ∇f (a)  2 n+1 (1 − |a| ) E(a,r)

C  (1 − |a|2 )n+1



E(a,r)

1/2  1/2 |f (z) − f (a)|  1 − |z|2 1 − |a|2 dv(z). |1 − z, a|

(22)

It follows from (10) that     1/2  1/2 |f (z) − f (a)| sup 1 − |a|2 ∇f (a)  sup 1 − |z|2 1 − |a|2 · < ∞. |1 − z, a| a∈B z,a∈B z=a

Therefore f ∈ B.

2

Theorem 4. A holomorphic function f ∈ B0 if and only if  1/2  1/2 |f (z) − f (w)| lim sup 1 − |z|2 = 0. 1 − |w|2 · |1 − z, w|

(23)

|z|→1 w∈B z=w

Proof. Assume that f ∈ B0 . Then the result follows from (18) and (20). Conversely, we assume that (23) holds. For z ∈ B and r > 0, by (10) and (22) we have    1/2  1/2 |f (z) − f (w)|  1 − |w|2 · 1 − |z|2 ∇f (z)  C sup 1 − |z|2 . |1 − z, w| w∈B z=w

From the above inequality, we see that (23) implies f ∈ B0 , as desired.

2

Acknowledgments The authors of this paper are supported by the NNSF of China (No. 10671115), the Research Fund for the doctoral program of Higher Education (No. 20060560002) and NSF of Guangdong Province (No. 06105648). The authors thank the referee for his numerous helpful suggestions.

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