Ann. Funct. Anal. 2 (2011), no. 2, 138–152
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
COMPOSITION OPERATORS ACTING BETWEEN SOME
WEIGHTED MÖBIUS INVARIANT SPACES
A. EL-SAYED AHMED∗1 AND M. A. BAKHIT2
Communicated by K. Guerlebeck
Abstract. In this paper we investigate conditions under which a holomorphic self-map of the unit disk induces a composition operator Cφ with closed
range on the weighted Bloch space Blog . Also, we introduce a new class of
functions the so called Flog (p, q, s) spaces. Necessary and sufficient conditions
are given for a composition operator Cφ to be bounded and compact from Blog
to Flog (p, q, s). Moreover, necessary and sufficient conditions for Cφ from the
Dirichlet space D to the spaces Flog (p, q, s) to be compact are given in terms
of the map φ..
1. Introduction and preliminaries
Let ∆ = {z ∈ C : |z| < 1} be the unit disk in the complex plane C, ∂∆ it’s boundary, H(∆) be the class of all analytic functions on ∆ and dA(z) the normalized
area measure. For each w ∈ ∆, let ϕw (z) denote the Möbius transformations of
∆
w−z
ϕw (z) =
, f or z ∈ ∆.
1 − w̄z
Let Aut(∆) be the group of all conformal automorphisms of ∆. The pseudohyperbolic distance between z and w is given by σ(z, w) = |ϕz (w)|. The pseudohyperbolic distance is Möbius invariant, that is,
σ(gz, gw) = σ(z, w),
Date: Received: 8 March 2011; Revised: 15 August 2011; Accepted: 11 November 2011.
∗
Corresponding author .
2010 Mathematics Subject Classification. Primary 47B33; Secondary 46E15.
Key words and phrases. Composition operators, weighted logarithmic Bloch functions,
Flog (p, q, s) spaces.
138
MÖBIUS INVARIANT SPACES
139
for all g ∈ Aut(∆), the Möbius group of ∆, and all z, w ∈ ∆. It has the following
useful property:
1 − (σ(z, w))2 =
(1 − |z|2 )(1 − |w|2 )
= (1 − |z|2 )|ϕ0z (w)|.
|1 − z̄w|2
For 0 < α < ∞, the spaces of all analytic functions f on ∆ such that
kf kBα = sup(1 − |z|2 )α |f 0 (z)| < ∞,
z∈∆
are called α-Bloch spaces (see [27]). The space B 1 is called the Bloch space B
(see [4]).
The classical Dirichlet space D is the space of all functions f ∈ D such that
Z
2
kf kD =
|f 0 (z)|2 dA(z) < ∞.
∆
For p, s ∈ (0, ∞), −2 < q < ∞ and q + s > −1. An analytic function f : ∆ → C
defined in the unit disk ∆ belongs to the spaces F (p, q, s) (see [26]) if
Z
p
kf kF (p,q,s) = sup
|f 0 (z)|p (1 − |z|2 )q g s (z, a)dA(z) < ∞
a∈∆
∆
where g(z, a) = log |ϕa1(z)| is the Green’s function with logarithmic singularity at
a−z
a ∈ ∆, where ϕa (z) = 1−āz
, f or z ∈ ∆. For more information about F (p, q, s)
spaces, we refer to [26].
For 0 < α < ∞, the space of analytic functions f ∈ ∆ such that
2
2
α
α = sup (1 − |z| )
kf kBlog
log
|f 0 (z)| < ∞,
1 − |z|2
z∈∆
α
α
is called weighted α-Bloch space Blog
(see [16]). If α = 1 the space Blog
is just the
α
weighted Bloch space Blog . The little weighted Bloch space Blog,0 is a subspace of
α
α
Blog
consisting of all f ∈ Blog
such that
2
2 α
lim (1 − |z| ) log
|f 0 (z)| = 0.
|z|→1
1 − |z|2
Now, let 0 < h < 1, 0 ≤ θ < 2π, and
Ω(h, θ) = {reit : 1 − h < r < 1} and |t − θ| < h},
S(h, θ) = {reit : |reit − reiθ | < h}.
A positive measure µ on ∆ is a Carleson measure if there is a constant A with
µ(S(h, θ)) ≤ A h, where 0 < h < 1 and 0 ≤ θ < 2π.
For 0 < s < ∞, we say that a positive measure µ defined on ∆ is a bounded
s-Carleson measure (see [5, 26]) provided µ(S(I)) = O(|I|s ) for all subarcs I of
∂∆, where |I| denotes the arc length of I ⊂ ∂∆ and S(I) denotes the Carleson
box based on I, that is,
z
|I|
S(I) = z ∈ ∆ :
∈ I, 1 − |z| ≤
.
|z|
2π
140
A. EL-SAYED AHMED, M.A. BAKHIT
If µ(S(I)) = o(|I|s ) as |I| → 0, then we say that µ is a compact s-Carleson
measure.
A positive Borel measure µ on ∆ is called an s-logarithmic, p-Carleson measure
(p, s > 0) if
2 p
µ(S(I))(log |I|
)
sup
< ∞.
|I|p
I⊆∂∆
In [28] it is proved that µ is an s-logarithmic, p-Carleson measure on ∆ if and
only if
s Z
2
|ϕ0a (z)|p dµ(z) < ∞.
sup log
2
1 − |a|
a∈∆
∆
Definition 1.1. For p, s ∈ (0, ∞), −2 < q < ∞ and q + s > −1, a function
f ∈ H(∆) is said to belong to Flog (p, q, s) if
2 p Z
(log |I|
)
1 s
p
0
p
2 q
|f
(z)|
(1
−
|z|
)
log
dA(z) < ∞.
kf kFlog (p,q,s) = sup
|I|s
|z|
I⊂∂∆
S(I)
By the same proof as done in [26] and for 1 < p < ∞, −2 < q < ∞, 1 < s < ∞,
it is easy to see that Flog (p, q, s) are Banach spaces under the norm
Z
p1
2
kf kFlog (p,q,s) = |f (0)| + sup log
|f 0 (z)|p (1 − |z|2 )q g s (z, a)dA(z) .
1 − |a|2
a∈∆
∆
Remark 1.2. The interest in the Flog (p, q, s) spaces arises from the fact they cover
some well known function spaces, it is immediate that Flog (2, 0, 1) = BM OAlog
(see [3, 8]). Also, Flog (2, 0, p) = Qplog , where 0 < p < ∞ (see [11]).
The composition operator Cφ : H(∆) → H(∆) is defined by Cφ = f ◦ φ.
There have been several attempts to study compactness and boundedness of composition operators in many function spaces (see e.g. [1, 2, 6, 7, 9, 10, 13, 14, 15,
17, 18, 30] and others). There are also some studies in several complex variables
(see e.g. [21, 24, 29] and others). Most of the previous work in the theory of composition operators dealt with their compactness, relating it to classical function
theory. On the other hand there are some studies of closed range composition
operators (see [12, 19, 31, 32] and others).
In this paper, we determine when the composition operator Cφ has a closed range
on the weighted Bloch space Blog and we give a set of necessary conditions and
a partial converse for Cφ on the weighted Bloch space Blog . Also, we characα
terize boundedness and compactness of the composition operators Cφ : Blog
→
Flog (p, q, s). Finally, we consider the composition operators from the Dirichlet
space D into Flog (p, q, s) spaces.
Recall that a linear operator T : X → Y is said to be bounded if there exists
a constant M > 0 such that ||T (f )||Y ≤ M ||f ||X for all maps f ∈ X. Moreover,
T : X → Y is said to be compact if it takes bounded sets in X to sets in Y
which have compact closure. For Banach spaces X and Y of H(∆), T is compact
from X to Y if and only if for each bounded sequence {xn } ∈ X, the sequence
{T xn } ∈ Y contains a subsequence converging to some limit in Y.
MÖBIUS INVARIANT SPACES
141
Two quantities Af and Bf , both depending on an analytic function f on ∆,
are said to be equivalent, written as Af ≈ Bf , if there exists a finite positive
constant C not depending on f such that for every analytic function f on ∆ we
have:
1
Bf ≤ Af ≤ CBf .
C
If the quantities Af and Bf , are equivalent, then in particular we have Af < ∞
if and only if Bf < ∞.
2. Composition operator with closed range on Blog space
Let φ be a holomorphic self-map of the unit disk ∆. We write G = φ(∆), and
τφ (z) is defined by
0
2
(1 − |z|2 ) log 1−|z|
2 |φ (z)|
.
τφ (z) =
2
(1 − |φ(z)|2 ) log 1−|φ(z)|
2
Yoneda in [25] proved the following results:
Theorem 2.1. Let φ be a holomorphic function taking ∆ into ∆. Then Cφ is
bounded on Blog if and only if
2
(1 − |z|2 ) log 1−|z|
2
0
sup
|φ (z)| < +∞.
2
2
z∈∆ (1 − |φ(z)| ) log 1−|φ(z)|2
Lemma 2.2. If Cφ is bounded on Blog , then for all f ∈ Blog ,
2
0
2
|f (w)|, w ∈ G
kf kBlog ≤ k sup(1 − |w| ) log
1 − |w|2
for some constant k.
Now, we give the following result:
Theorem 2.3. If Cφ is bounded below on Blog , then there exist positive constants
ε, r with r < 1 such that, for all z ∈ ∆, σ(φ(Ωε ), z) ≤ r where
Ωε = {z ∈ ∆, |τφ (z)| > ε}.
Proof. Since Cφ : Blog → Blog is bounded below, then there is a constant k,
log 2 < k ≤ 1 such that
kCφ f kBlog ≥ kkf kBlog
for f ∈ B0,log , for each w ∈ ∆, let
fw (z) =
w−z
w − φ(0)
−
.
1 − w̄z 1 − w̄φ(0)
Clearly, fw (z) is a bounded and continuous analytic function on the closed unit
disk and so is in B0,log . Moreover an easy computation gives kfw klog ≥ 1. Thus
kCφ fw kBlog ≥ kkfw kBlog ≥ k.
142
A. EL-SAYED AHMED, M.A. BAKHIT
On the other hand, we also have
2
(1 − |z| ) log
2
|(Cφ fw )0 (z)|
1 − |z|2
2
2
= (1 − |ϕw (φ(z))| ) log
|τφ (z)|,
1 − |ϕw (φ(z))|2
and Cφ fw (0) = 0. Then there is a point zw ∈ ∆ such that
2
2
kCφ fw kBlog ≥ (1 − |zw | ) log
|(Cφ fw )0 (zw )|
1 − |zw |2
1
k
≥
kCφ fw kBlog ≥
.
2
2
So, we obtain that
(1 − |ϕw (φ(z))| ) log
2
k
|τφ (z)| ≥
.
2
1 − |ϕw (φ(z))|
2
2
Thus,
(1 − |ϕw (φ(z))| ) log
2
2
k
≥
,
1 − |ϕw (φ(z))|2
2
then,
2
|ϕw (φ(z))| ≤
r
Let r =
√
( 2−1)
k
e 2 −1
< 1 and ε =
√
( 2−1)
k
e 2 −1
√
( 2 − 1)
k
e2 − 1
.
. Noting σ(w, φ(zw )) = |ϕw (φ(zw ))|, we
conclude that
σ(w, φ(zw )) < r and |τφ (zw )| ≥ ε.
This completes the proof.
Theorem 2.4. If for some constants 0 < r < 13 , and ε > 0, for each w ∈ ∆,
there is a point zw ∈ ∆ such that
σ(w, φ(zw )) < r
and |τφ (zw )| > ε,
then Cφ : Blog → Blog is bounded below.
Proof. Let u = φ(0). Then φ = ϕu ◦ ϕu ◦ φ. Let ψ = ϕu ◦ φ. Thus ψ(0) = 0, and
Cφ = Cψ Cϕu . Since ϕu is a Möbus transform, Cϕu is isometry on Blog . So we need
only to prove that Cψ is bounded on Blog . Moreover ψ still satisfies the conditions
of the theorem. In order to prove that Cψ is bounded on Blog space it suffices to
prove
kCψ f klog ≥ k,
MÖBIUS INVARIANT SPACES
143
for some constant k > 0 and all f ∈ Blog with kf klog = 1. To do this, let f ∈ Blog
with norm kf klog = 1. For each zw ∈ ∆, we have
2 0
(C
f
)
(z)
ψ
1 − |z|2
0 2 0
ψ (z)
f
(ψ(z))
= (1 − |z|2 ) log
1 − |z|2
2
(1 − |ψ(z)|2 ) log 1−|ψ(z)|
0 2
2 0
2
ψ (z)
=
(1
−
|z|
)
log
f
(ψ(z))
2
1 − |z|2
(1 − |ψ(z)|2 ) log 1−|ψ(z)|
2
2
2
(1
−
|z|
)
log
0 2
2
1−|z|
ψ (z)
f 0 (ψ(z))
= (1 − |ψ(z)|2 ) log
2
1 − |ψ(z)|2
(1 − |ψ(z)|2 ) log 1−|ψ(z)|
2
2
f 0 (ψ(z))τψ(z) .
= (1 − |ψ(z)|2 ) log
2
1 − |ψ(z)|
(1 − |z|2 ) log
Since kf klog = 1, noting that kf klog = |f (0)|+kf kBlog , then kf kBlog = (1−|f (0)|),
there is a point w ∈ ∆ such that
0
2
(1 − |w| ) log
|f (w)| ≥ (1 −
1 − |w|2
2
1
1
3
−r
)(1 − |f (0)|),
2
−r
where 0 < r < 13 and 1 − 3 2 < 1. By Theorem 2.2, we have
0
0
2
2
2
2
(1 − |z| ) log
|f (z)| − (1 − |w| ) log
|f (w)| 2
2
1 − |z|
1 − |w|
≤ 3σ(z, w)kf ◦ ϕw kBlog .
Thus whenever σ(ψ(zw ), w) < r < 13 , we have that
0
2
|f (ψ(zw ))|
2
1 − |ψ(zw )|
0
2
≥ (1 − |w|2 ) log
|f (w)| − 3σ(ψ(zw ), w)(1 − |f (0)|)
2
1 − |w|
0
2
≥ (1 − |w|2 ) log
|f (w)| − 3r(1 − |f (0)|)
1 − |w|2
1
−r
≥ (1 − 3
− 3r)(1 − |f (0)|)
2
5
≥ (1 − 3r)(1 − |f (0)|).
6
(1 − |ψ(zw )|2 ) log
So,
2 (Cψ f )0 (z)
2
1 − |z|
0
2
f (ψ(z))τψ(z) , for all z ∈ ∆.
≥ |f (0)| + (1 − |ψ(z)|2 ) log
2
1 − |ψ(z)|
kCψ f klog ≥ |f (ψ(0))| + (1 − |z|2 ) log
144
A. EL-SAYED AHMED, M.A. BAKHIT
In particular,
2 0
(C
f
)
(z)
ψ
1 − |z|2
5
5
≥ |f (0)| + ε (1 − 3r)(1 − |f (0)|) ≥ ε (1 − 3r).
6
6
kCψ f klog ≥ |f (ψ(0))| + (1 − |z|2 ) log
Let k = 56 ε (1 − 3r). We have proved that
kCψ f klog ≥ k , whenever kf klog = 1.
This completes the proof.
3. Composition operators on Flog (p, q, s) spaces
α
Now we characterize the weighted logarithmic α-Bloch spaces Blog
by the
weighted Flog (p, q, s) spaces. The obtained result improve some previous results
due to Stroethoff [22] and Zhao [26].
Theorem 3.1. If 0 < p < ∞, −2 < q < ∞ , 1 < s < ∞ and α =
q + s > −1. Then the following statements are equivalent:
α
(A) f ∈ Blog
.
(B) f ∈ Flog (p, q, s).
p
R
2
(C) sup log 1−|a|2
|f 0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ∞.
∆
a∈∆ p
R
2
(D) sup log 1−|a|2
|f 0 (z)|p (1 − |z|2 )αp−2 g s (z, a)dA(z) < ∞.
∆
q+2
p
with
a∈∆
Proof. The proof is similar to the main results in [22, 27], so it will be omitted.
Theorem 3.1, will be needed to study composition operators between Flog (p, q, s)
α
and weighted Blog
spaces.
α
Lemma 3.2. Let 0 < α < ∞, there are two functions f1 , f2 ∈ Blog
such that
|f10 (z)| + |f20 (z)| ≥
C
(1 −
|z|2 )α (log
2
)
1−|z|2
where C is a positive constant.
Proof. The proof of this lemma is similar to that of Lemma 3.1 in [11] or Lemma 2.2
in [16] with some simple modifications, so it will be omitted.
We need the following notation.
p Z
2 αp−2
2
(1 − |ϕa (z)|2 )s
0
p (1 − |z| )
p dA(z),
Φφ (α, p, s; a) = log
|φ
(z)|
2
1 − |a|2
(1 − |φ(z)|2 )αp log 1−|φ(z)|
∆
2
for 0 < p, α < ∞ and 1 < s < ∞. Now, we will give the following theorem:
MÖBIUS INVARIANT SPACES
145
Theorem 3.3. Let 0 < p, α < ∞ let 1 < s < ∞. If φ is an analytic selfα
map of the unit disk, then the induced composition operator Cφ maps Blog
into
Flog (p, αp − 2, s) boundedly if and only if
sup Φφ (α, p, s; a) < ∞.
(3.1)
a∈∆
α
α ≤ 1, then in view of Theorem 3.1 , we obtain
Proof. Let f ∈ Blog
with kf kBlog
kCφ f kpFlog (p, αp−2,s)
p Z
2
= sup log
|(f ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
2
1 − |a|
a∈∆
p Z∆
2
= sup log
|f 0 (φ(z))|p |φ0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
1 − |a|2
a∈∆
∆
p Z
2 αp−2
2
(1 − |ϕa (z)|2 )s
p
0
p (1 − |z| )
p dA(z)
≤ kf kBα sup log
|φ
(z)|
2
log
1 − |a|2
(1 − |φ(z)|2 )αp log 1−|φ(z)|
a∈∆
∆
2
= kf kpBα sup Φφ (α, p, s; a) < ∞.
log
a∈∆
α
For the other direction we use the fact that for each function f ∈ Blog
, the analytic
function Cφ (f ) ∈ Flog (p, αp − 2, s). Then using the functions of Lemma 3.2 we
get the following:
p
p
p
2 kCφ f1 kFlog (p,αp−2,s) + kCφ f2 kFlog (p,αp−2,s)
p
2
p
= 2 sup log
1 − |a|2
a∈∆
Z 0
p
0
p
2 αp−2
2 s
×
|(f1 ◦ φ) (z)| + |(f2 ◦ φ) (z)| (1 − |z| )
(1 − |ϕa (z)| ) dA(z)
∆
p
2
≥ sup
log
1 − |a|2
a∈∆
p
Z 0
0
2 αp−2
2 s
×
|(f1 ◦ φ) (z)| + |(f2 ◦ φ) (z)| (1 − |z| )
(1 − |ϕa (z)| ) dA(z)
∆
p
2
≥ sup
log
1 − |a|2
a∈∆
p
Z 0
0
p
2 αp−2
2 s
×
|(f1 (φ(z))| + |(f2 (φ(z))| |φ (z)| (1 − |z| )
(1 − |ϕa (z)| ) dA(z)
∆
p Z
2 αp−2
2
(1 − |ϕa (z)|2 )s
0
p (1 − |z| )
p dA(z)
|φ
(z)|
≥ C sup log
2
1 − |a|2
(1 − |φ(z)|2 )αp log 1−|a|
a∈∆
∆
2
≥ C sup Φφ (α, p, s; a).
a∈∆
Hence Cφ is bounded, then (3.1) holds. The proof is completed.
Now, we describe compactness in the following result.
146
A. EL-SAYED AHMED, M.A. BAKHIT
Theorem 3.4. Let 0 < p, α < ∞ and let 1 < s < ∞. If φ is an analytic selfα
map of ∆, then the induced composition operator Cφ : Blog
→ Flog (p, αp − 2, s) is
compact if and only if φ ∈ Flog (p, αp − 2, s) and
lim sup Φφ (α, p, s; a) = 0.
r→1 a∈∆
(3.2)
α
Proof. Let Cφ : Blog
→ Flog (p, αp − 2, s) be compact. This means that
φ ∈ Flog (p, αp − 2, s).
zn
.
n
α
α
Let fn (z) =
Since kfn kBlog
≤ M (M = 2eα ) and fn (z) → 0 as n → ∞, locally
uniformly on ∆, then by the compactness of Cφ , kCφ (fn )kFlog (p,αp−2,s) → 0 as
n → ∞. This means that for each r ∈ (0, 1) and for all ε > 0, there exist N ∈ N
such that if n ≥ N , then
p Z
2
αp p(N −1)
N r
sup log
|φ0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε,
2
1
−
|a|
a∈∆
Ωr
where Ωr = {z ∈ ∆, |φ(z)| > r}, if we choose r so that (N αp rp(N −1) ) = 1, then
p Z
2
sup log
|φ0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε.
(3.3)
1 − |a|2
a∈∆
Ωr
α
Let now f with kf kBlog
≤ 1. We consider the functions ft (z) = f (tz), t ∈ (0, 1).
Then ft → f uniformly on compact subset of the unit disk as t → 1 and the
α
family (ft ) is bounded on Blog
, thus
k(ft ◦ φ) − (f ◦ φ)k → 0.
Due to compactness of Cφ we get that, for ε > 0 there is a t ∈ (0, 1) such that
p Z
2
sup log
|Ft (φ(z))|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε,
1 − |a|2
a∈∆
∆
where Ft (φ(z)) = (f ◦ φ)0 (z) − (ft ◦ φ)0 (z). Thus, if we fix t, then
p Z
2
sup log
|(f ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
2
1
−
|a|
a∈∆
Ωr
p Z
2
≤ 2p sup log
|Ft (φ(z))|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
2
1 − |a|
a∈∆
Ω
p Zr
2
+2p sup log
|(ft ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
2
1
−
|a|
a∈∆
Ωr
p
p
0 p
≤ ε2 + 2 kft kH ∞
p Z
2
× sup log
|φ0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
1 − |a|2
a∈∆
Ωr
≤ ε2p + ε2p |ft0 kpH ∞ ,
i.e.,
2
sup log
1 − |a|2
a∈∆
≤ ε2p (1 + |ft0 kpH ∞ ),
p Z
Ωr
|(f ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
MÖBIUS INVARIANT SPACES
147
α
where we have used (3.3). On the other hand, for each kf kBlog
≤ 1 and ε > 0,
there exists a δ depending on f, ε, such that for r ∈ [δ, 1),
p Z
2
sup log
|(f ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε. (3.4)
2
1 − |a|
a∈∆
Ωr
α
Since Cφ is compact, then it maps the unit ball of Blog
to a relatively compact
subset of Flog (p, αp − 2, s). Thus for each ε > 0 there exists a finite collection
α
α
of functions f1 , f2 , . . . , fn in the unit ball of Blog
such that for each kf kBlog
≤ 1,
there is k ∈ {1, 2, 3, . . . , n} such that
p Z
2
sup log
|Fk (φ(z))|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε,
2
1
−
|a|
a∈∆
∆
where Fk (φ(z)) = (f ◦ φ)0 (z) − (fk ◦ φ)0 (z).
Using also (3.4), we get for δ = max δ(fk , ε)andr ∈ [δ, 1), that
1≤k≤n
2
sup log
1 − |a|2
a∈∆
p Z
|(fk ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε.
Ωr
α ≤ 1, combining the two relations as above we get that
Hence for any f, kf kBlog
p Z
2
sup log
|(f ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z) < ε2p .
2
1 − |a|
a∈∆
Ωr
Therefore, we get that (3.2) holds.
For the sufficiency we use that φ ∈ Flog (p, αp − 2, s) and (3.2) holds. Let {fn }n∈N
α
be a sequence of functions in the unit ball of Blog
, such that fn → 0 as n → ∞,
uniformly on the compact subsets of the unit disk. Let also r ∈ (0, 1) and
Φr = {z ∈ ∆, |φ(z)| ≤ r}. Then
kfn ◦ φkpFlog (p,αp−2,s)
≤ 2p |fn (φ(0))|
p Z
2
p
|(fn ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
+2 sup log
2
1 − |a|
a∈∆
Φ
p Z r
2
+ 2p sup log
|(fn ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
2
1
−
|a|
a∈∆
Ωr
= 2p (I1 + I2 + I3 ).
Since fn → 0 as n → ∞, locally uniformly on the unit disk, then I1 = |fn (φ(0))|
goes to zero as n → ∞ and for each ε > 0 there is N ∈ N such that for each
n > N,
p Z
2
I2 = sup log
|(fn ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
1 − |a|2
a∈∆
Φr
≤ ε kφkpFlog (p, αp−2,s) .
148
A. EL-SAYED AHMED, M.A. BAKHIT
We also observe that
I3
2
= sup log
1 − |a|2
a∈∆
p
≤ kfn kBα sup log
log
a∈∆
p Z
|(fn ◦ φ)0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s dA(z)
p Z
|φ0 (z)|p (1 − |z|2 )αp−2 (1 − |ϕa (z)|2 )s
2
p
dA(z).
2
1 − |a|2
(1 − |φ(z)|2 )αp log 1−|φ(z)|
Ωr
2
Ωr
Under the assumption that (3.2) holds, then for every n > N and for every ε > 0
there exists r1 such that for every r > r1 , I3 < ε. Thus if φ ∈ Flog (p, αp − 2, s),
we obtain
kfn ◦ φkpFlog (p,αp−2,s) ≤ 2p {0 + ε kφkpFlog (p, αp−2,s) + ε} ≤ εC.
Combining the above, we get that kCφ (fn )kpFlog (p, αp−2,s) → 0 as n → ∞, which
proves compactness. The proof of our theorem is therefore established.
Now we consider the composition operators from the Dirichlet space D into
Flog (p, q, s) spaces. Our result is stated as follows.
Theorem 3.5. Let 2 ≤ p < ∞, 1 < s < ∞, −2 < q < ∞ and q + s > −1. If φ is
an analytic self-map of ∆, then the composition operator Cφ : D → Flog (p, q, s) is
compact if and only if
lim kCφ ϕa kFlog (p,q,s) = 0.
|a|→1
(3.5)
Proof. Assume that Cφ : D → Flog (p, q, s) is compact. Since {ϕa : a ∈ ∆} is
a bounded set in D and ϕa − a → 0 uniformly on compact sets as |a| → 1, the
compactness of Cφ yields that
kCφ ϕa kFlog (p,q,s) −→ 0 as |a| → 1.
Conversely, let {fn } ∈ D be a bounded sequence. Since fn ∈ D ⊂ B, for z ∈ ∆
|fn (z)| ≤ sup kfn kD
n
1
1 + |z|
1 + log
.
2
1 − |z|
Hence, {fn } is a normal family. Thus, there is a subsequence {fnk }, which converges to f analytic on ∆ and both fnk → f and fn0 k → f 0 uniformly on compact
subsets of ∆. It is easy to show that f ∈ D. We replace f by Cφ f, we remark
that Cφ is compact by showing
kCφ fnk − Cφ f kFlog (p,q,s) −→ 0
as |k| → ∞.
MÖBIUS INVARIANT SPACES
149
We write
kCφ ϕa kpFlog (p,q,s)
p Z
2
|(ϕa ◦ φ)0 (z)|p (1 − |z|2 )q g s (z, a)dA(z)
= sup log
2
1
−
|a|
a∈∆
p Z∆
2
(1 − |a|2 )p
= sup log
|φ0 (z)|p (1 − |z|2 )q g s (z, a)dA(z)
2
2p
1 − |a|
a∈∆
∆ |1 − aφ(z)|
Z
2 p
(1 − |a| )
a,p,q,s
= sup
Nlog
(φ, w) dA(w).
2p
a∈∆ ∆ |1 − aw|
Here,
a,p,q,s
Nlog
(φ, w)
=
2
log
1 − |a|2
p X
|φ0 (z)|p−2 (1 − |z|2 )q g s (z, a)
z∈φ−1 (w)
is the counting function. Thus (3.5) is equivalent to
Z
lim sup
|a|→1 a∈∆
∆
(1 − |a|2 )p a,p,q,s
N
(φ,
w)
dA(w) = 0.
log
|1 − aw|2p
Hence by [5] or [23], for any ε > 0 there exists δ, where 0 < δ < 1, such that for
0 < h < δ and all a ∈ ∆,
Z
sup
a∈∆
a,p,q,s
Nlog
(φ, w)dA(w) < ε hp ,
S(h,θ)
where S(h, θ) is a Carleson box. For Fnk (z) = fn0 k (z) − f 0 (z), the mean value
property for analytic functions fn0 k and f 0 yields that,
4
Fnk (w) =
π(1 − |w|)2
Z
|w−z|<
1−|w|
2
Fnk (z)dA(z).
Then by Jensen’s inequality (see [20] theorem 3.3), we have
4
|Fnk (w)| =
π(1 − |w|)2
p
Z
1−|w|
|w−z|< 2
|Fnk (z)|p dA(z).
Note that if |w − z| < 1−|w|
, then we have that w ∈ S(2(1 − |z|), θ) and also
2
1
C
≤
(see
[23]).
Then, by Fubini’s theorem (see [20] theorem 8.8), for
(1−|w|)2
(1−|z|)2
150
A. EL-SAYED AHMED, M.A. BAKHIT
Fnk (z) = fn0 k (z) − f 0 (z), we deduce that
Z
a,p,q,s
|Fnk (w)|p Nlog
(φ, w) dA(w)
a∈∆ ∆
Z Z
4
a,p,q,s
p
|Fnk (z)| dA(z) Nlog
(φ, w)dA(w)
≤ sup
2
1−|w|
a∈∆ ∆ π(1 − |w|)
|w−z|< 2
Z
Z
|Fnk (z)|p
a,p,q,s
Nlog
(φ, w)dA(w)dA(z)
≤ C sup
2
(1
−
|z|)
a∈∆ ∆
S(2(1−|z|),θ)
p
2
= C sup log
1 − |a|2
a∈∆
Z
Z
|Fnk (z)|p
a,p,q,s
×
Nlog (φ, w)dA(w)dA(z)
2
|z|>1− 2δ (1 − |z|)
S(2(1−|z|),θ)
p
2
+C sup log
1 − |a|2
a∈∆
Z
Z
|Fnk (z)|p
a,p,q,s
×
Nlog (φ, w) dA(w)dA(z) .
2
|z|≤1− 2δ (1 − |z|)
S(2(1−|z|),θ)
sup
For one hand, since fnk , f ∈ D ⊂ B, 2 ≤ p < ∞ and Fnk (z) = fn0 k (z) − f 0 (z), we
have
Z
Z
|Fnk (z)|p
a,p,q,s
sup
Nlog
(φ, w)dA(w)dA(z)
2
a∈∆ |z|>1− δ (1 − |z|)
S(2(1−|z|),θ)
2
Z
p
≤ ε 2 sup
|Fnk (z)|p (1 − |z|)p−2 dA(z)
a∈∆
|z|>1− 2δ
≤ εCkfnk −
≤
≤
f kp−2
B
Z
sup
a∈∆
εCkfnk − f kp−2
B kfnk
εC1 kfnk − f k2D ,
|z|>1− 2δ
|Fnk (z)|2 dA(z)
− f k2D
where C and C1 are positive constants. On the other hand,
Z
sup
a∈∆
|z|≤1− 2δ
Z
≤ C sup
a∈∆
∆
|Fnk (z)|p
(1 − |z|)2
Z
a,p,q,s
Nlog
(φ, w)dA(w)dA(z)
S(2(1−|z|),θ)
a,p,q,s
Nlog
(φ, w)dA(w)
Z
|z|≤1− 2δ
|Fnk (z)|p dA(z)
≤ εC,
for n large enough and since Fnk (z) = (fn0 k (z) − f 0 (z)) −→ 0 uniformly on
{z ∈ ∆ : |z| ≤ 1 − 2δ }. Therefore, for sufficiently large k, the above discussion
MÖBIUS INVARIANT SPACES
151
gives
kCφ fnk − Cφ f kpFlog (p,q,s)
p Z
2
|(fnk ◦ φ)0 (z) − (f ◦ φ)0 (z)|p (1 − |z|2 )q g s (z, a)dA(z)
= sup log
1 − |a|2
a∈∆
∆
Z
a,p,q,s
= sup
|fn0 k (z) − f 0 (z)|p (1 − |z|2 )q Nlog
(φ, w) dA(w) < εC.
a∈∆
∆
It follows that Cφ is a compact operator. Therefore, the proof is completed.
References
1. A. El-Sayed Ahmed and H. Al-Amri, A class of weighted holomorphic Bergman spaces, J.
Comput. Anal. Appl. 13 (2011), no. 2, 321–334.
2. A. El-Sayed Ahmed and M.A. Bakhit, Composition operators on some holomorphic Banach
function spaces, Math. Scand. 104 (2009), no. 2, 275–295.
3. K.R.M. Attele, Toeplitz and Hankel operators on Bergman one space, Hokkaido Math. J.
21 (1992), no. 2, 279–293.
4. J. Arazy, D. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math.
363 (1974), 110–145.
5. R. Aulaskari, D. Stegenga and J. Xiao, Some subclasses of BMOA and their charactrization
in terms of Carleson measures, Rocky Mountain J. Math. 26 (1996), 485–506.
6. R. Aulaskari and R. Zhao, Composition operators and closures of some Möbius invariant
spaces in the Bloch space, Math. Scand. 107 (2010), no. 1, 139–149.
7. B.S. Bourdon, J.A. Cima and A.L. Matheson, Compact composition operators on BMOA,
Trans. Amer. Math. Soc. 351 (1999), 2183–2169.
8. J. Cima and D.A. Stegenga, Hankel operators on H p , Analysis at Urbana. Vol. 1: Analysis
in function spaces, Proc. Spec. Year Mod. Anal./Ill. 1986-87, Lond. Math. Soc. Lect. Note
Ser. 137 (1989), 133–150.
9. Z. Cuckovic and R. Zhao,Weighted composition operators on the Bergman spaces, J. London
Math. Soc. (2)70 (2004), no. 2, 499–511.
10. Z. Cuckovic and R. Zhao,Weighted composition operators between different weighted
Bergman spaces and different Hardy spaces. Ill. J. Math. 51 (2007), no. 2, 479–498.
11. P. Galanopoulos,On Blog to Qplog pullbacks, J. Math. Anal. Appl. 337 (2008), 712–725.
12. P. Ghatage, D. Zheng and N. Zorboska, Sampling sets and closed range composition operators on the Bloch spac, Proc. Am. Math. Soc. 133 (2005), no. 5, 1371–1377.
13. P. Ghatage, J. Yan and D. Zheng, Composition operators with closed range on the Bloch
space, Proc. Amer. Math. Soc. 129 (2000), no. 7, 2039–2044.
14. L. Jiang and Y. He, Composition operators from B α to F (p, q, s), Acta Math. Sci. Ser B.
32 (2003), 252–260.
15. H. Li and P. Liu, Composition operators between generally weighted Bloch spaces and Qqlog
space, Banach J. Math. Anal. 3 (2009), no. 1, 99–110.
16. H. Li, P. Liu and M. Wang, Composition operators between generally weighted Bloch spaces
of polydisk, J. Inequal. Pure. Appl. Math. 3 (2007), Article 85, 1–8.
17. K. Madigan and A.Matheson, Compact composition operators on the Bloch space, Trans.
Amer. Math. Soc. 347 (1995), no. 7, 2679–2687.
18. S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral.
Math. Soc. 63 (2001), 177–185.
19. N. Palmberg, Weighted composition operators with closed range, Bull. Aust. Math. Soc.
75 (2007), no. 3, 331–354.
20. W. Rudin, Real and complex analysis, MeGraw-Hill, New York, 1987.
21. J. Shi and L. Luo, Composition operators on the Bloch space of several complex variables,
Acta Math. Sin. Engl. Ser 16 (2000), no. 1, 85–98.
152
A. EL-SAYED AHMED, M.A. BAKHIT
22. K. Stroethoff, Besov-type characterizations for the Bloch space, Bull. Austral. Math. Soc.
39 (1989), 405–420.
23. M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Thesis.
Michigan State University, 1996.
24. E. Wolf, Weighted composition operators on weighted Banach spaces of holomorphic functions on the unit polydisk with closed range, Numer. Funct. Anal. Optim. 31 (2010), no.
11, 1294–1300.
25. R. Yoneda, The composition operators on weighted Bloch space, Arch. Math. 78 (2002),
310–317.
26. R. Zhao, On a general family of function spaces, Annales Academiae Scientiarum Fennicae.
Series A I. Mathematica. Dissertationes. 105. Helsinki: Suomalainen Tiedeakatemia, (1996),
1–56.
27. R. Zhao, On α-Bloch functions and V M OA, Acta. Math. Sci. 3 (1996), 349–360.
28. R. Zhao, On logarithmic Carleson measures, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4,
605–618.
29. Z. Zhou and R. Che, On the composition operators on the Bloch space of several complex
variables, Sci. China, Ser A. 48 (2005), 392–399.
30. X. Zhu, Volterra composition operators on logarithmic Bloch spaces, Banach J. Math. Anal.
3 (2009), no. 1, 122–130.
31. N. Zorboska, Composition operators with closed range, Trans. Am. Math. Soc. 344 (1994),
no. 2, 791–801.
32. N. Zorboska, Closed range essentially normal composition operators are normal, Acta Sci.
Math. 65 (1999), no. 1-2, 287–292.
1
Sohag University, Faculty of Science, Department of Mathematics 82524 Sohag, Egypt;
Taif University, Faculty of Science, Mathematics Department, Box 888 El-Hawiyah,
El-Taif 5700, Saudi Arabia.
E-mail address: ahsayed80@hotmail.com
2
Jazan University, Faculty of Science, Department of Mathematics, Jazan
Saudi Arabia.
E-mail address: mabakhit2008@yahoo.com