IJMMS 2003:31, 1973–1979
PII. S0161171203207122
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
COMPOSITION OPERATORS FROM THE BLOCH SPACE
INTO THE SPACES QT
PENGCHENG WU and HASI WULAN
Received 10 July 2002
Suppose that ϕ(z) is an analytic self-map of the unit disk ∆. We consider the
boundedness of the composition operator Cϕ from Bloch space Ꮾ into the spaces
QT (QT ,0 ) defined by a nonnegative, nondecreasing function T (r ) on 0 ≤ r < ∞.
2000 Mathematics Subject Classification: 30D50, 47B38, 46E15.
1. Introduction. Let ∆ = {z : |z| < 1} be the unit disk of complex plane C
and let H(∆) be the space of all analytic functions in ∆. For a ∈ ∆, Green’s function with logarithmic singularity at a ∈ ∆ is denoted by g(z, a) = log |(1 − āz)/
(a − z)|. For 0 < p < ∞, the space Qp consists of all functions f analytic in ∆
for which
f (z)2 g(z, a) p dA(z) < ∞,
(1.1)
sup
a∈∆
∆
where dA(z) is the Euclidean area element on ∆.
Qp -spaces have been investigated by many authors (cf. [1, 2, 3, 9]). We know
that Q1 = BMOA, the space of all analytic functions of bounded mean oscillation (cf. [4]). Further, the spaces Qp are the same for each p ∈ (1, ∞), and each
space equals to the Bloch space Ꮾ, which is a Banach space with the norm
f Ꮾ := f (0) + f b := f (0) + sup 1 − |z|2 f (z).
(1.2)
z∈∆
Recently, we introduced a new space QT (cf. [5, 10]) by a nondecreasing
function T (r ) on 0 ≤ r < ∞ as follows.
Definition 1.1. Let T (r ) ≡ 0 be a nonnegative, nondecreasing function on
0 ≤ r < ∞. A function f ∈ H(∆) is said to belong to QT if
f (z)2 T g(z, a) dA(z) < ∞.
(1.3)
f 2QT := sup
a∈∆
∆
If
lim
|a|→1
∆
f (z)2 T g(z, a) dA(z) = 0,
then f is said to belong to QT ,0 .
(1.4)
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P. WU AND H. WULAN
For 0 < p < ∞, if we take T (r ) = r p , the space QT coincides with the space
Qp . We note that QT ⊂ Ꮾ for all nondecreasing functions T . We have previously
shown that QT = Qp under certain growth conditions on T (r ) (cf. [10]).
In the present paper, first we give some basic properties of QT spaces, some
of which are also new for the special case QT = Qp . For example, QT is a Banach
space with the norm f T defined by
f T := f (0) + f QT .
(1.5)
Then we investigate the boundedness of the composition operators from the
Bloch space Ꮾ into QT or QT ,0 . These results extend some previously known
results (cf. [6, 8]).
2. Basic properties of QT spaces. We give the following propositions.
Proposition 2.1. The space QT is a subspace of the Bloch space Ꮾ.
The proof of Proposition 2.1 can be found in [10].
Proposition 2.2. The space QT is a Banach space with the norm defined
in (1.5).
Proof. For f ∈ QT and a ∈ ∆, define
I 2 (f , a) :=
∆
f (z)2 T g(z, a) dA(z).
(2.1)
Let f1 , f2 ∈ QT . It follows from Schwarz’s inequality that
∆
f (z)f (z)T g(z, a) dA(z) ≤ I f1 , a I f2 , a ,
1
2
(2.2)
and then
I 2 f1 + f2 , a ≤ I 2 f1 , a + 2I f1 , a I f2 , a + I 2 f2 , a
2
= I f1 , a + I f2 , a .
(2.3)
Thus, I(f1 + f2 , a) ≤ I(f1 , a) + I(f2 , a) for all a ∈ ∆. Hence
f1 + f2 QT
≤ f1 QT + f2 QT .
(2.4)
Therefore,
2
f1 + f2 2 = f1 (0) + f2 (0) + f1 + f2 T
QT
2
≤ f1 (0) + f2 (0) + f1 QT + f2 QT
2
= f1 T + f2 T ,
(2.5)
COMPOSITION OPERATORS FROM THE BLOCH SPACE . . .
1975
that is, f1 +f2 T ≤ f1 T +f2 T . On the other hand, it is obvious that f T ≥
0 for each f ∈ QT and that f T = 0 if and only if f ≡ 0. It is obvious that
cf T = |c|f for any constant c. Thus, QT is a normed space.
Let f ∈ QT and let φa (w) = (a − w)/(1 − āw), a ∈ ∆. Then by changing a
variable w = φa (z), we obtain
f 2QT ≥
∆
f (z)2 T g(z, a) dA(z)
f ◦ φa (w)2 T log 1 dA(w)
|w|
∆
1
f ◦ φa (w)2 dA(w)
≥ T log
r
|w|<r
2
2 1 2
≥ π r T log
1 − |a|2 f (a) .
r
=
(2.6)
For r0 , 0 < r0 < 1, such that T (log(1/r0 )) = 0, we have
f b ≤
f QT
1/2 .
r0 π T log 1/r0
(2.7)
Since f ∈ QT ⊂ Ꮾ, we have for z ∈ ∆,
f (z) ≤ f (0) + f b log 1 + |z|
2
1 − |z|
f
Q T
1 + |z|
≤ f (0) +
1/2 log
1 − |z|
2r0 π T log 1/r0


1
1 + |z|
≤ f T 1 +
.
1/2  log
1 − |z|
2r0 π T log 1/r0
(2.8)
Suppose {fn } is a Cauchy sequence in QT . Then there is a constant M > 0 such
that
fn ≤ M,
T
n = 1, 2, . . . .
(2.9)
By the estimate (2.8) for a fixed r0 ∈ (0, 1), we obtain that

fn (z) ≤ M 1 +

1 + |z|
1/2  log
1
− |z|
2r0 π T log 1/r0
1
(2.10)
holds for all integral numbers n = 1, 2, . . . . Hence, there exist a subsequence
{fnj (z)} of {fn (z)} and an analytic function f defined on the unit disk ∆ such
that both {fnj (z)} and {fn j (z)} converge uniformly to f and f , respectively.
The conditions here are such that both the sequence of functions and the
sequence of derivatives converge since we know that {fn (z)} is bounded on
1976
P. WU AND H. WULAN
compact subsets of ∆ by inequality (2.10). By Fatou’s lemma, we get that
f (z)2 T g(z, a) dA(z)
∆
2 lim f (z) T g(z, a) dA(z)
=
∆ j→∞
≤ lim inf
j→∞
nj
∆
f (z)2 T g(z, a) dA(z)
nj
2
≤ lim inf fn j→∞
j
QT
(2.11)
≤ M2
holds for all a ∈ ∆, so that f ∈ QT . By a similar reasoning, we can prove that
fn − f T → 0 as n → ∞. The proof of Proposition 2.2 is complete.
3. Boundedness of composition operators. Let ϕ(z) be an analytic selfmap of the unit disk ∆. Let the composition operator Cϕ induced by ϕ from
H(∆) to itself be defined by Cϕ (f ) = f ◦ϕ for f ∈ H(∆). The boundednesses of
composition operators from Ꮾ to itself and from Ꮾ to Qp have been studied
in [6, 8], respectively. In this paper, we consider the same problems for the
general spaces QT .
Theorem 3.1. Let T (r ) ≡ 0 be a nonnegative, nondecreasing function on
0 ≤ r < ∞ and let ϕ be an analytic self-map of ∆. Then Cϕ : Ꮾ → QT is bounded
if and only if
sup
a∈∆
∆
ϕ (z)2
2 2 T g(z, a) dA(z) < ∞.
1 − ϕ(z)
(3.1)
Proof. Let (3.1) hold and let K12 (K1 > 0) be the supremum in (3.1). If f ∈ Ꮾ,
then for all a ∈ ∆, we have
Cϕ f (z)2 T g(z, a) dA(z)
∆
f ϕ(z) 2 ϕ (z)2 T g(z, a) dA(z)
=
∆
≤
f 2b
∆
ϕ (z)2
2 2 T g(z, a) dA(z)
1 − ϕ(z)
(3.2)
≤ K12 f 2b .
Consequently, Cϕ f QT ≤ K1 f b . Since f (z) ∈ Ꮾ, we obtain
2
Cϕ f 2 = f ◦ ϕ(0) + Cϕ f T
QT
2
f b
1 + ϕ(0)
+ K1 f b
≤ f (0) +
log
2
1 − ϕ(0)
2
2 ≤K
f (0) + f b = K 2 f 2Ꮾ ,
(3.3)
COMPOSITION OPERATORS FROM THE BLOCH SPACE . . .
1977
where K = max{1, K1 + (1/2) log(1 + |ϕ(0)|)/(1 − |ϕ(0)|)}. Thus, Cϕ f T ≤
Kf Ꮾ , which shows that Cϕ : Ꮾ → QT is bounded.
Conversely, assume that Cϕ : Ꮾ → QT is bounded, there exists a constant
K > 0 such that for each f ∈ Ꮾ, we have
Cϕ f ≤ Kf Ꮾ .
T
(3.4)
On the other hand, by a result in [7], there exist f1 , f2 ∈ Ꮾ such that
1
≤ f1 (z) + f2 (z)
2
1 − |z|
holds for all z ∈ ∆, so that
ϕ (z)2
2
2
2 2 ≤ 2 f1 ◦ ϕ (z) + 2 f2 ◦ ϕ (z) .
1 − ϕ(z)
Thus, the inequalities
ϕ (z)2
2 2 T g(z, a) dA(z)
∆
1 − ϕ(z)
f1 ◦ ϕ (z)2 + f2 ◦ ϕ (z)2 T g(z, a) dA(z)
≤2
∆
2 2
≤ 2K 2 f1 Ꮾ + f2 Ꮾ
(3.5)
(3.6)
(3.7)
hold for all z, a ∈ ∆, which establishes (3.1). The proof of Theorem 3.1 is completed.
Remark 3.2. Note that if Cϕ : Ꮾ → Ꮾ, then (3.1) holds for any increasing
function T satisfying QT = Ꮾ. Indeed, we know that QT = Ꮾ (see [5]) if and
only if
1 −2
1 1−r2
T log
r dr < ∞.
(3.8)
r
0
The Schwarz-Pick lemma guarantees that ((1 − |z|2 )/(1 − |ϕ(z)|2 ))|ϕ (z)| ≤ 1,
so that (3.8) leads easily to (3.1). It means that Cϕ : Ꮾ → Ꮾ is always bounded
(cf. [6]).
Remark 3.3. If one considers the composition operator Cϕ from the Bloch
space to the Dirichlet space
f (z)2 dA(z) < ∞ ,
Ᏸ = f ∈ H(∆) :
(3.9)
∆
then Cϕ : Ꮾ → Ᏸ is bounded if and only if
∆
ϕ (z)2
2 2 dA(z) < ∞.
1 − ϕ(z)
(3.10)
1978
P. WU AND H. WULAN
For the spaces QT ,0 , we have the following results.
Theorem 3.4. Let T (r ) be a nonnegative, nondecreasing function on 0 ≤
r < ∞ and let ϕ be an analytic self-map of ∆. Then Cϕ : Ꮾ → QT ,0 is bounded if
and only if
lim
|a|→1
∆
ϕ (z)2
2 2 T g(z, a) dA(z) = 0.
1 − ϕ(z)
(3.11)
Proof. Suppose Cϕ : Ꮾ → QT ,0 is bounded. Using a way similar to the proof
of Theorem 3.1, we choose functions f1 , f2 ∈ Ꮾ such that
1
≤ f1 (z) + f2 (z)
2
1 − |z|
(3.12)
for all z ∈ ∆. Then Cϕ f1 and Cϕ f2 belong to QT ,0 . Therefore,
ϕ (z)2
2 T g(z, a) dA(z)
2
|a|→1 ∆
1 − ϕ(z)
(3.13)
2 2 T g(z, a) dA(z) = 0,
≤ 2 lim
f1 ◦ ϕ (z) + f2 ◦ ϕ (z)
lim
|a|→1
∆
which shows that (3.11) holds.
Conversely, by Theorem 3.1, we know that Cϕ : Ꮾ → QT is bounded since
condition (3.11) implies that
sup
a∈∆
∆
ϕ (z)2
2 2 T g(z, a) dA(z) < ∞.
1 − ϕ(z)
(3.14)
We need only to prove that Cϕ f ∈ QT ,0 for each f ∈ Ꮾ, and this follows from
the inequality
Cϕ f (z)2 T g(z, a) dA(z)
∆
f ϕ(z) 2 ϕ (z)2 T g(z, a) dA(z)
=
(3.15)
∆
ϕ (z)2
≤ f 2b
2 2 T g(z, a) dA(z).
∆
1 − ϕ(z)
The proof of Theorem 3.4 is completed.
Acknowledgments. The essential part of this paper was written when the
authors were at the University of Joensuu, Finland, in 1998. This research is
now also supported in part by grants from the National Natural Science Foundation of China no. 10171058, Guangdong Province no. 010446, and Guizhou
Province of China. We would like to thank the referee for helpful comments.
COMPOSITION OPERATORS FROM THE BLOCH SPACE . . .
1979
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Pengcheng Wu: Department of Mathematics, Guizhou Normal University, Guiyang
550001, China
E-mail address: wupc@gznu.edu.cn
Hasi Wulan: Department of Mathematics, Shantou University, Shantou, Guangdong
515063, China
E-mail address: wulan@stu.edu.cn