Let D = {z : |z| < 1} be the open unit disk in the complex plane C, and H(D) denote the set of all analytic functions on D. For f ∈ H(D), Let

MULTIPLIERS AND CYCLIC VECTORS ON THE WEIGHTED BLOCH SPACE

Shanli YE

Abstract. In this paper we study the pointwise multipliers and cyclic vectors on the weighted Bloch space βL = {f ∈ H(D) : supD(1−

|z|²) ln(_1−|z|² )|f⁰(z)| < +∞}. We obtain a characterization of multipliers on βLand little β_L⁰. Also, a sufficient condition and a necessary condition are given for which f is a cyclic vector in β_L⁰.

1. Introduction

Let D = {z : |z| < 1} be the open unit disk in the complex plane C, and H(D) denote the set of all analytic functions on D. For f ∈ H(D), Let

kfk

= sup {(1 − |z|

)

|f

(z) | : z ∈ D}, 0 < α < +∞, kfk

= sup {(1 − |z|

) ln( 2

1 − |z| ) |f

(z) | : z ∈ D}.

As in [7], [9], the α-Bloch space β

consists of all f ∈ H(D) satisfying kfk

< + ∞ and the little α-Bloch space β

consists of all f ∈ H(D) satisfying lim

|z|→1

(1 −|z|

)

|f

(z) | = 0; the logarithmic weighted Bloch space β

consists of all f ∈ H(D) satisfying kfk

< + ∞ and the little logarithmic weighted Bloch space β

consists of all f ∈ H(D) satisfying lim

|z|→1⁻

(1 −

|z|

) ln( 2

1 − |z| ) |f

(z) | = 0. It can easily proved that β

is a Banach space under the norm

kfk

= |f(0)| + kfk

and that β

is a closed subspace of β

. It is well known that with the norm kfk

= |f(0)| + kfk

β

is a Banach space and β

is a closed subspace of β

. It is easily proved that for 0 < α < 1, β

$ β

$ β

. For more information about β

, see, for example, [9].

Mathematics Subject Classification. Primary 30D05; Secondary 30H05, 46E15.

Key words and phrases. pointwise multipliers, cyclic vectors, weighted Bloch space.

The research was supported by the Foundation of Fujian Educational Committee (Grant No. JA02167, JA04171) and NSF of Fujian Province (Grant No. 2006J0201).

135

The space of analytic functions on D of bounded mean oscillation, denoted by BM OA, consists of f in H

for which

kfk

= sup

I

1

|I|

∫

S(I)

|f

(z)|

(1 − |z|

)dA(z) < +∞,

where dA(z) denotes the Lebesgue measure on D, I denotes a subarc of ∂D,

|I| denotes the arclength measure of I and S(I) = {re

: 1 −r ≤ |I|, e

∈ I}.

The subset of BM OA, denoted by V M OA, consists of f for which

|I|→0

lim 1

|I|

∫

S(I)

|f

(z) |

(1 − |z|

)dA(z) = 0.

For more details, see [5].

Let X be an analytic function space. We say a function φ is a pointwise multiplier on X, if φf ∈ X for all f ∈ X. Let M(X) denote the space of all pointwise multipliers on X. By M

we denote the operator of multiplication by φ: M

f = φf, f ∈ X. An application of the closed graph theorem shows that if φ ∈ M(X), then M

is a bounded linear transformation. Hence it has a finite norm kM

k.

In [2], K. R. M. Attle showed that for f ∈ L

(D), the Hankel operator H

: L

−→ L

is bounded if and only if f ∈ β

, and in [3], L. Brown and A. L. Shields proved that M

is bounded on the classical Bloch space β

(β

) if and only if φ ∈ β

∩ H

. R. Yoneda [7] studied the composition operator in β

space. In Section 2 we will characterize multiplier spaces M (β

) and M (β

).

Let Y be an analytic Banach function space and the polynomials are dense in it. For f ∈ Y and let [f] be the closure in Y of the polynomial multiples of f . Thus f is called a cyclic vector in Y if and only if [f ]=Y . In [1], [3], L.

Brown and A. L. Shields studied cyclic vectors in the classical Bloch space β

(β

). In the BM OA(V M OA) space, the author in [6] characterized the cyclic vectors. There are just the following theorem.

Theorem A.

(1) For f ∈ BMOA(V MOA), then f is a cyclic vector on BMOA(V MOA) if and only if f is an outer function.

(2) If f is an outer function in β

(β

), then f is cyclic in β

(β

).

(3) There exists a singular inner function that is cyclic in β

. In Section 3 we study cyclic vectors in β

.

2. Multipliers in the weighted Bloch space

In this section we shall characterize the pointwise multipliers space M (β

)

and M (β

). For this purpose, we need the following lemmas.

Lemma 2.1. If f ∈ β

, then (i) |f(z)| ≤ (2 + ln(ln 2

1 − |z| )) kfk

; (ii) |f(z) − f(tz)| ≤ ln( ln

ln

) kfk

, for every t with 0 ≤ t < 1.

Proof. Suppose f ∈ β

and z ∈ D, then

|f(z) − f(tz)| = |z

∫

t

f

(zt)dt | ≤ kfk

∫

t

|z|

(1 − |zt|

) ln

dt

≤ kfk

∫

t|z|

dx (1 − x) ln

= kfk

(ln ln

− ln ln

)

≤ ln( ln

ln

) kfk

.

Especially, |f(z) − f(0)| ≤ kfk

(ln ln

− ln ln 2), hence

|f(z)| ≤ (2 + ln ln 2

1 − |z| ) kfk

.

¤ Lemma 2.2. If f ∈ β

, then lim

|f(z)|

ln(ln

) = 0.

The proof is similar to Lemma 2.1. The details are omitted.

Lemma 2.3. Let f (z) = (1 − |z|) ln

|1 − z| ln

, z ∈ D. Then |f(z)| < 2.

Proof. Since r(x) = x ln

is increasing on (0,

], decreasing on [

, 1] and r(

) =

< 1, then |f(z)| < 1 where z ∈ D

= {z ∈ D : |1 − z| <

}.

On the other hand, for z ∈ D \ D

,

|f(z)| ≤ (1 − |z|) ln

2

e

ln 2 ≤

e

ln 2 < 2,

hence |f(z)| < 2. ¤

Theorem 2.4. The following are equivalent:

(a) φ ∈ M(β

);

(b) φ ∈ M(β

);

(c) φ ∈ H

and

(2.1) sup

D

(1 − |z|

) ln 2

1 − |z| ln(ln 2

1 − |z| )|φ

(z)| < +∞.

Proof. (c)⇒(a). Assume φ ∈ H

and (2.1) holds. For every f ∈ β

, by Lemma 2.1, we have

(1 − |z|

) ln 2

1 − |z| |(M

f )

(z) |

≤ (1 − |z|

) ln 2

1 − |z| |φ(z)||f

(z) | + (1 − |z|

) ln 2

1 − |z| |φ

(z) ||f(z)|

≤ kφk

kfk

+ (1 − |z|

) ln 2

1 − |z| (2 + ln(ln 2

1 − |z| )) |φ

(z) |kfk

< + ∞.

Thus f φ ∈ β

.

(a)⇒(c). Suppose that φ is a multiplier of β

. Then by [4, Proposition 3] φ ∈ H

and |φ(z)| ≤ kM

k. Let z

= re

. We take the test function

f (z) = ln(ln 4 1 − e

z ).

By Lemma 2.3 we know that f ∈ β

and kfk

≤ 5. We have kfφk

≤ kM

kkfk

≤ 5kM

k.

It follows that

(1 − |z|

) ln 2

1 − |z| | ln(ln 4

1 − e

z ) ||φ

(z) |

≤ (1 − |z|

) ln 2

1 − |z| |φ(z)||f

(z) | + 5kM

k

≤ 5(kφk

+ kM

k) < +∞.

Let z = z

. Hence (1 − |z

|

) ln 2

1 − |z

| | ln(ln 4

1 − |z

| ) |φ

(z

) | ≤ 5(kφk

+ kM

k) < +∞.

Thus

sup

D

(1 − |z|

) ln 2

1 − |z| ln(ln 2

1 − |z| ) |φ

(z) | < +∞.

(b) ⇒(c). Given z

= re

and α ∈ (0, 1). Let f

(z) = (ln(ln 4

1 − e

z ))

.

A calculation shows that f

∈ β

and sup

kfk

= k < + ∞. In a manner similar to the proof (a) ⇒(c), one obtains that if φ is a multiplier of β

, then for each α,

(1 − |z

|

) ln 2

1 − |z

| |(ln(ln 4

1 − |z

| ))

||φ

(z)| ≤ k(kφk

+ kM

k) < +∞.

Hence

(1 − |z

|

) ln 2

1 − |z

| | ln(ln 4

1 − |z

| ) ||φ

(z

) | < +∞, which shows that (1) holds.

(c) ⇒(b). Assume φ ∈ H

and sup

(1 − |z|

) ln

ln(ln

) |φ

(z) | = M < + ∞. For every f ∈ β

, by Lemma 2.2, we have

(1 − |z|

) ln 2

1 − |z| |(M

f )

(z) |

≤ (1 − |z|

) ln 2

1 − |z| |φ(z)||f

(z) | + (1 − |z|

) ln 2

1 − |z| |φ

(z) ||f(z)|

≤ kφk

(1 − |z|

) ln 2

1 − |z| |f

(z) | + M

ln(ln

) |f(z)| → 0 (|z| → 1).

Thus f φ ∈ β

. ¤

3. Cyclic vectors in the little weighted Bloch space Lemma 3.1. Let g(x) = (1 − x) ln 2

1 − x , x ∈ [0, 1). Then g(x)

g(tx) ≤ 2 for each t ∈ [0, 1].

Proof. Let x

= 1 − 2

e . A calculation shows that 4

3 x

< 1. We know that g(x) is increasing on [0, x

], and decreasing on [x

, 1).

First, suppose t > 3

4 and x > 4

3 x

. Then x ≥ tx > x

, hence g(x) ≤ g(tx).

Next, suppose t > 3

4 and x ≤ 4

3 x

. Then g(x)

g(tx) ≤ g(x

)

min(g(0), g(

x

)) = 2/e ln 2 < 2.

Finally, suppose t ≤ 3

4 . A calculation shows that 3

4 ln 2 ≤ g(tx) ≤ 2

e .

Then

g(x)

g(tx) ≤ 2/e

3

4

ln 2 < 2.

¤ Lemma 3.2. Let h(x) = (1 − x) ln

, x ∈ [0, 1). Then there exists a constant M > 0 such that h(x)

h(tx) ≤ M for each t ∈ [0, 1].

The proof is similar to Lemma 3.1. We omit the details.

Lemma 3.3. Suppose f ∈ β

, then f ∈ β

if and only if kf

− fk

−→

0(t → 1

), where f

(z) = f (tz).

Proof. Suppose f ∈ β

, then given any ² > 0, there exists δ ∈ (0, 1) such that (1 − |z|) ln 2

1 − |z| |f

(z) | ≤ (1 − |z|

) ln 2

1 − |z| |f

(z) | < ² for all δ

< |z| < 1.

Consider

kf

− fk

= sup

D

(1 − |z|

) ln 2

1 − |z| |tf

(tz) − f

(z) |

≤ sup

|z|>δ

(1 − |z|

) ln 2

1 − |z| |tf

(tz) − f

(z) | + sup

|z|≤δ

(1 − |z|

) ln 2

1 − |z| |tf

(tz) − f

(z)|

, I

+ I

.

If |z| > δ and t > δ, then |tz| > δ

. By Lemma 3.1 we have I

≤ sup

|z|>δ

(1 − |z|

) ln 2

1 − |z| |f

(z) | + sup

|z|>δ

(1 − |z|

) ln 2

1 − |z| |tf

(tz) |

≤ 2 sup

|z|>δ

(1 − |z|) ln 2

1 − |z| |f

(z) | + 2 sup

|z|>δ

(1 − |z|) ln 2

1 − |z| |f

(tz) |

≤ 2² + +4 sup

|z|>δ

(1 − |zt|) ln 2

1 − |zt| |f

(tz) |

≤ 2² + 4² = 6².

On the other hand, I

−→ 0 as t −→ 1

since tf

(tz) −→ f

(z) uniformly

for |z| ≤ δ. Thus lim

kf

− fk

= 0.

Conversely, suppose f ∈ β

and lim

kf

− fk

= 0. Then for ² > 0 there exists t ∈ (0, 1) such that kf

− fk

< ². It follows that

(1 − |z|

) ln

|f

(z)| ≤ kf

− fk

+ (1 − |z|

) ln

|(f

)

(z)|

< ² + (1 − |z|

) ln

|(f

)

(z) |.

Now let |z| −→ 1 then (1 − |z|

) ln

|(f

)

(z) | −→ 0 because f

∈ β

.

Hence f ∈ β

. ¤

Proposition 3.4. The polynomials are dense in β

. Proof. Let f ∈ β

and t

= 1 − 1

n , then f (t

z) is analytic in |z| ≤ 1. Hence there exists a polynomial p

(z) such that

|f(t

z) − p

(z) | < 1

n , |f

(t

z) − p

(z) | < 1 n for all |z| ≤ 1. Then by Lemma 3.3 we get

kf(z) − p

(z) k

≤ kf(z) − f(t

z) k

+ kf(t

z) − p

(z) k

< kf(z) − f(t

z) k

+ (1 +

)

n −→ 0 (n → ∞).

Thus the polynomials are dense in β

. ¤

Proposition 3.5. β

⊂ V MOA.

Proof. Let I is an arc in ∂D and S(I) is the Carleson box based on I, i.e, S(I) = {re

: 1 − r ≤ |I|, e

∈ I}. For f ∈ β

, it follows that

∫

S(I)

|f

(z) |

(1 − |z|

)dA(z)

≤

∫

S(I)

kfk

(1 − |z|

) ln

dA(z)

≤ kfk

|I|

∫

1−|I|

1

(1 − r) ln

dr = kfk

|I|

ln

. Then

1

|I|

∫

S(I)

|f

(z)|

(1 − |z|

)dA(z) ≤ kfk

ln

−→ 0 (|I| → 0).

Hence f ∈ V MOA.

¤

Since β

⊂ β

for 0 < α < 1, we have the following corollary.

Corollary 3.6. For 0 < α < 1, β

⊂ V MOA.

This fact was proved in [8, Theorem 3]. However this proof is much easier than the one in [8].

Theorem 3.7.

(1) Let f ∈ β

, if |f(z)| ≥ σ > 0 (|z| < 1), then f is a cyclic vector in β

. (2) If f is a cyclic vector in β

, then f is an outer function.

Proof. (1) For 0 < t < 1, f

(z) = f (tz). Since

t

is analytic in |z| ≤ 1 , we can easily prove that there exists polynomials p

such that kp

f − f

f

k

−→ 0 as n → ∞. Thus we have f

f

∈ [f]. If k f

f

− 1k

−→ 0 as t → 1

, then 1 ∈ [f], hence by Proposition 3.4, f is cyclic in β

. Now we are going to show that k f

f

− 1k

−→ 0(t → 1

).

We have k f

f

− 1k

≤ 1

σ kf − f

k

+ 1 σ

sup

D

(1 − |z|

) ln 2

1 − |z| |f(z) − f(tz)||tf

(tz) | , 1

σ I

+ 1 σ

I

.

By Lemma 3.3 we know I

−→ 0 (t → 1

), then we only prove I

−→

0 (t → 1

).

Since f ∈ β

, for a given any ² > 0, there exists δ ∈ (0, 1) such that (1 − |z|

) ln 2

1 − |z| |f

(z) | < ²

for all δ

< |z| < 1. If |z| > δ and t > δ, then |tz| > δ

. By Lemmas 2.1 and 3.2 it follows that

|z|>δ

sup (1 − |z|

) ln 2

1 − |z| |f(z) − f(tz)||tf

(tz)|

≤ ² (1 − |z|

) ln

(1 − |tz|

) ln

|f(z) − f(tz)|

≤ ²kfk

(1 − |z|

) ln

(1 − |tz|

) ln

ln( ln

ln

)

≤ ²kfk

2(1 − |z|) ln

(1 − |tz|) ln

≤ 2Mkfk

².

On the other hand, sup

|z|≤δ

(1 − |z|

) ln 2

1 − |z| |f(z) − f(tz)||tf

(tz) |

≤ kfk

sup

|z|≤δ

(1 − |z|

) ln

(1 − |tz|

) ln

ln( ln

ln

) −→ 0 (t −→ 1

).

Hence I

−→ 0 (t → 1

). Thus f is a cyclic vector in β

.

(2) If f is a cyclic vector in β

, then, according to Proposition 3.5 and [4, Proposition 6], f is a cyclic vector in V M OA. Hence f is an outer function by Theorem A. This completes the proof of Theorem 3.1. ¤

Acknowledgment

I thank the referee for numerous stylistic corrections.

References

[1] J. M. Anderson, J. L. Fernandez and A. L. Shields, Inner functions and cyclic vertors in the Bloch space, Trans. Amer. Math. Soc. 323(1991), 429-448.

[2] K. R. M. Attele, Toeplitz and Hankel on Bergman one space, Hokkaido Math J.

21(1992), 279-293.

[3] L. Brown and A. L. Shields, Multipliers and cyclic vectors in the Bloch space, Michigan Math. J. 38(1991), 141-146.

[4] L. Brown and A. L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math.

Soc. 285(1984), 269-304.

[5] J. Garnett, Bounded analytic functions, New York 1981.

[6] S. L. Ye, Cyclic vectors in the BM OA space, Acta Math. Sci. 17(1997), 356-360(in Chinese).

[7] R. Yoneda, The composition operators on weighted Bloch space, Arch. Math. 78(2002), 310-317.

[8] R. H. Zhao, On α− Bloch functions and V MOA, Acta Math. Sci. 16(1996), 349-360.

[9] K. H. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23(1993), 1143-1177.

Shanli Ye

Department of Mathematics Fujian Normal University Fuzhou 350007, P. R. China e-mail address: [email protected]

(Received June 21, 2005 )