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COMPOSITION OPERATORS FROM THE BLOCH SPACE
INTO THE SPACES
Q
TPENGCHENG 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 operatorCϕfrom Bloch spaceᏮinto the spaces QT(QT ,0) defined by a nonnegative, nondecreasing functionT (r )on 0≤r <∞.
2000 Mathematics Subject Classification: 30D50, 47B38, 46E15.
1. Introduction. Let∆= {z:|z|<1}be the unit disk of complex planeC and letH(∆)be the space of all analytic functions in∆. Fora∈∆, Green’s func-tion with logarithmic singularity ata∈∆is denoted byg(z, a)=log|(1−az)/¯
(a−z)|. For 0< p <∞, the spaceQpconsists of all functionsfanalytic in∆
for which
sup
a∈∆
∆
f(z)2g(z, a)pdA(z) <∞, (1.1)
wheredA(z)is the Euclidean area element on∆.
Qp-spaces have been investigated by many authors (cf. [1,2,3,9]). We know thatQ1=BMOA, the space of all analytic functions of bounded mean
oscilla-tion (cf. [4]). Further, the spacesQpare the same for eachp∈(1,∞), and each space equals to the Bloch spaceᏮ, which is a Banach space with the norm
fᏮ:=f (0)+fb:=f (0)+sup z∈∆
1−|z|2f(z). (1.2)
Recently, we introduced a new space QT (cf. [5, 10]) by a nondecreasing functionT (r )on 0≤r <∞as follows.
Definition1.1. LetT (r )≡0 be a nonnegative, nondecreasing function on
0≤r <∞. A functionf∈H(∆)is said to belong toQT if
f2
QT:=sup
a∈∆
∆
f(z)2Tg(z, a)dA(z) <∞. (1.3)
If
lim
|a|→1
∆
f(z)2Tg(z, a)dA(z)=0, (1.4)
For 0< p <∞, if we takeT (r )=rp, the spaceQT coincides with the space Qp. We note thatQT⊂Ꮾfor all nondecreasing functionsT. We have previously shown thatQT=Qpunder certain growth conditions onT (r )(cf. [10]).
In the present paper, first we give some basic properties ofQTspaces, some of which are also new for the special caseQT=Qp. For example,QTis a Banach space with the normfT defined by
fT:=f (0)+fQT. (1.5)
Then we investigate the boundedness of the composition operators from the Bloch spaceᏮintoQT orQT ,0. These results extend some previously known
results (cf. [6,8]).
2. Basic properties ofQT spaces. We give the following propositions.
Proposition2.1. The spaceQT is a subspace of the Bloch spaceᏮ.
The proof ofProposition 2.1can be found in [10].
Proposition2.2. The spaceQT is a Banach space with the norm defined
in (1.5).
Proof. Forf∈QT anda∈∆, define
I2(f , a):=
∆
f(z)2Tg(z, a)dA(z). (2.1)
Letf1, f2∈QT. It follows from Schwarz’s inequality that
∆
f
1(z)f2(z)T
g(z, a)dA(z)≤If1, aIf2, a, (2.2)
and then
I2f1+f2, a
≤I2f1, a
+2If1, a
If2, a
+I2f2, a
=If1, a+If2, a2.
(2.3)
Thus,I(f1+f2, a)≤I(f1, a)+I(f2, a)for alla∈∆. Hence
f1+f2QT ≤f1QT+f2QT. (2.4)
Therefore,
f1+f22T=
f1(0)+f2(0)+f1+f2QT 2
≤f1(0)+f2(0)+f1QT+f2QT 2
=f1T+f2T
2
,
that is,f1+f2T≤ f1T+f2T. On the other hand, it is obvious thatfT≥
0 for eachf ∈QT and thatfT =0 if and only iff≡0. It is obvious that cfT= |c|ffor any constantc. Thus,QT is a normed space.
Letf∈QT and letφa(w)=(a−w)/(1−aw)¯ ,a∈∆. Then by changing a variablew=φa(z), we obtain
f2
QT ≥
∆
f(z)2Tg(z, a)dA(z)
=
∆
f◦φa(w)2T
log 1
|w| dA(w) ≥T
log1
r |w|<r
f◦φa(w)2dA(w)
≥π r2T
log1
r 1−|a|
22f(a)2
.
(2.6)
Forr0, 0< r0<1, such thatT (log(1/r0))=0, we have
fb≤
fQT
r0π Tlog 1/r01/2
. (2.7)
Sincef∈QT⊂Ꮾ, we have forz∈∆,
f (z)≤f (0)+fb
2 log 1+|z|
1−|z|
≤f (0)+ fQT
2r0π Tlog1/r01/2
log1+|z| 1−|z|
≤ fT
1+ 1 2r0
π Tlog 1/r0 1/2
log1+|z| 1−|z|.
(2.8)
Suppose{fn}is a Cauchy sequence inQT. Then there is a constantM >0 such that
fnT≤M, n=1,2, . . . . (2.9)
By the estimate (2.8) for a fixedr0∈(0,1), we obtain that
fn(z)≤M
1+ 1
2r0π Tlog 1/r01/2
log1+|z|
1−|z| (2.10)
holds for all integral numbersn=1,2, . . . .Hence, there exist a subsequence
{fnj(z)}of{fn(z)}and an analytic functionfdefined on the unit disk∆such that both{fnj(z)}and{fnj(z)}converge uniformly tof andf
, respectively.
compact subsets of∆by inequality (2.10). By Fatou’s lemma, we get that
∆
f(z)2Tg(z, a)dA(z)
=
∆jlim→∞f
nj(z) 2
Tg(z, a)dA(z)
≤lim inf
j→∞
∆
f
nj(z) 2
Tg(z, a)dA(z)
≤lim inf
j→∞ fnj 2
QT ≤M 2
(2.11)
holds for alla∈∆, so thatf∈QT. By a similar reasoning, we can prove that
fn−fT→0 asn→ ∞. The proof ofProposition 2.2is complete.
3. Boundedness of composition operators. Letϕ(z) be an analytic self-map of the unit disk∆. Let the composition operatorCϕ induced byϕfrom
H(∆)to itself be defined byCϕ(f )=f◦ϕforf∈H(∆). The boundednesses of composition operators fromᏮto itself and fromᏮtoQphave been studied in [6, 8], respectively. In this paper, we consider the same problems for the general spacesQT.
Theorem 3.1. LetT (r )≡0be a nonnegative, nondecreasing function on
0≤r <∞and letϕbe an analytic self-map of∆. ThenCϕ:Ꮾ→QTis bounded if and only if
sup
a∈∆
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z) <∞. (3.1)
Proof. Let (3.1) hold and letK12(K1>0)be the supremum in (3.1). Iff∈Ꮾ,
then for alla∈∆, we have
∆
Cϕf(z)2Tg(z, a)dA(z)
=
∆
fϕ(z)2ϕ(z)2Tg(z, a)dA(z)
≤ f2
b
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z)
≤K2 1f2b.
(3.2)
Consequently,CϕfQT ≤K1fb. Sincef (z)∈Ꮾ, we obtain
Cϕf2T=f◦ϕ(0)+CϕfQT2
≤f (0)+fb
2 log
1+ϕ(0)
1−ϕ(0)+K1fb
2
≤K2f (0)+fb
2
=K2f2 Ꮾ,
whereK=max{1, K1+(1/2)log(1+ |ϕ(0)|)/(1− |ϕ(0)|)}. Thus,CϕfT ≤ KfᏮ, which shows thatCϕ:Ꮾ→QT is bounded.
Conversely, assume thatCϕ :Ꮾ→QT is bounded, there exists a constant
K >0 such that for eachf∈Ꮾ, we have
CϕfT≤KfᏮ. (3.4)
On the other hand, by a result in [7], there existf1, f2∈Ꮾsuch that
1
1−|z|2≤f
1(z)+f2(z) (3.5)
holds for allz∈∆, so that
ϕ(z)2
1−ϕ(z)22
≤2f1◦ϕ(z)2+2f2◦ϕ(z)2. (3.6)
Thus, the inequalities
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z)
≤2
∆
f1◦ϕ (z)2
+f2◦ϕ
(z)2
Tg(z, a)dA(z)
≤2K2f1 2 Ꮾ+f2
2 Ꮾ
(3.7)
hold for allz, a∈∆, which establishes (3.1). The proof ofTheorem 3.1is com-pleted.
Remark3.2. Note that ifCϕ:Ꮾ→Ꮾ, then (3.1) holds for any increasing
functionT satisfyingQT =Ꮾ. Indeed, we know that QT =Ꮾ (see [5]) if and only if
1
0T
log
1
r
1−r2−2r dr <∞. (3.8)
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 thatCϕ:Ꮾ→Ꮾis always bounded (cf. [6]).
Remark3.3. If one considers the composition operatorCϕfrom the Bloch
space to the Dirichlet space
Ᏸ=f∈H(∆):
∆
f(z)2dA(z) <∞, (3.9)
thenCϕ:Ꮾ→Ᏸis bounded if and only if
∆
ϕ(z)2
1−ϕ(z)22
For the spacesQT ,0, we have the following results.
Theorem3.4. LetT (r )be a nonnegative, nondecreasing function on 0≤ r <∞and letϕbe an analytic self-map of∆. ThenCϕ:Ꮾ→QT ,0is bounded if and only if
lim
|a|→1
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z)=0. (3.11)
Proof. SupposeCϕ:Ꮾ→QT ,0is bounded. Using a way similar to the proof
ofTheorem 3.1, we choose functionsf1, f2∈Ꮾsuch that
1
1−|z|2≤f1(z)+f2(z) (3.12)
for allz∈∆. ThenCϕf1andCϕf2belong toQT ,0. Therefore,
lim
|a|→1
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z)
≤2 lim
|a|→1
∆
f1◦ϕ (z)2
+f2◦ϕ
(z)2
Tg(z, a)dA(z)=0,
(3.13)
which shows that (3.11) holds.
Conversely, byTheorem 3.1, we know thatCϕ:Ꮾ→QT is bounded since condition (3.11) implies that
sup
a∈∆
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z) <∞. (3.14)
We need only to prove thatCϕf∈QT ,0for eachf∈Ꮾ, and this follows from
the inequality
∆
Cϕf(z)2
Tg(z, a)dA(z)
=
∆
fϕ(z)2
ϕ(z)2Tg(z, a)dA(z)
≤ f2
b
∆
ϕ(z)2
1−ϕ(z)22
Tg(z, a)dA(z).
(3.15)
The proof ofTheorem 3.4is completed.
Acknowledgments. The essential part of this paper was written when the
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Pengcheng Wu: Department of Mathematics, Guizhou Normal University, Guiyang 550001, China
E-mail address:[email protected]
Hasi Wulan: Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China