CUBO A Mathematical Journal
Vol.14, No¯ 01, (09–19). March 2012

Integral composition operators between weighted Bergman
spaces and weighted Bloch type spaces

Elke Wolf

University of Paderborn,

Mathematical Institute,

D-33095 Paderborn, Germany,

email: lichte@math.uni-paderborn.de

ABSTRACT

We characterize boundedness and compactness of integral composition operators acting

between weighted Bergman spaces Av,p and weighted Bloch type spaces Bw.

RESUMEN

Caracterizamos la acotación y compacidad de operadores integrales compuestos ac-

tuando entre espacios de Bergman con peso Av,p y espacios Bw de tipo Bloch con

peso.

Keywords and Phrases: Weighted Bergman spaces, integral composition operator, weighted

Bloch type spaces

2010 AMS Mathematics Subject Classification: 47B33, 47B38.



10 Elke Wolf CUBO
14, 1 (2012)

1 Introduction

Let H(D) denote the set of all analytic functions on the open unit disk D of the complex plane. A
map g ∈ H(D) induces the Volterra type or Riemann-Stieltjes operator

Jg : H(D) → H(D), f 7→ ∫z
0

f(ξ)g′(ξ) dξ, z ∈ D.

This operator appears naturally in the study of pointwise multiplication operators since with

the companion integral operator

Ig : H(D) → H(D), f 7→ ∫z
0

f′(ξ)g(ξ) dξ, z ∈ D,

we have that

Jgf + Igf = Mgf − f(0)g(0),

where Mg denotes the pointwise multiplication operator given by

Mg : H(D) → H(D), (Mgf)(z) = g(z)f(z), z ∈ D.
See e.g. [1], [2], [3], [17] or [21].

Moreover, let v and w be strictly positive bounded and continuous functions (weights) on D. Then
the weighted Bergman space Av,p is defined as follows

Av,p = {f ∈ H(D); ‖f‖v,p :=
(∫

D
|f(z)|pv(z) dA(z)

) 1
p

< ∞}, 1 ≤ p < ∞,
where dA(z) is the area measure on D normalized so that area of D is 1. Furthermore, we consider
the weighted Bloch type spaces Bw of functions f ∈ H(D) satisfying ‖f‖Bw := supz∈D w(z)|f′(z)| <∞. Provided we identify functions that differ by a constant, ‖.‖Bw becomes a norm and Bw a
Banach space.

Let φ be an analytic self-map of D. In [13] Li characterized boundedness and compactness of
Volterra composition operators

(Jg,φf)(z) =

∫z
0

(f ◦ φ)(ξ)(g ◦ φ)′(ξ) dξ, z ∈ D,

and the integral composition operators

(Ig,φf)(z) =

∫z
0

(f ◦ φ)′(ξ)(g ◦ φ)(ξ) dξ, z ∈ D,

acting between weighted Bergman spaces and weighted Bloch type spaces, both generated by stan-

dard weights. In [19] we generalized his results related to the Volterra composition operators Jg,φ



CUBO
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Integral composition operators between . . . 11

to a more general setting. In this article our aim is to characterize boundedness and compactness

of the integral composition operators Ig,φ acting between weighted Bergman spaces and weighted

Bloch type spaces generated by a quite general class of weights.

2 The setting

This section is devoted to the description of the setting in which we are interested. Let ν be a

holomorphic function on D, non-vanishing, strictly positive on [0, 1[ and satisfying limr→1 ν(r) = 0.
Then we define the weight v as follows

v(z) := ν(|z|2) for every z ∈ D. (2.1)

Next, we give some illustrating examples of weights of this type:

(i) Consider ν(z) = (1 − z)α, α > 0. Then the corresponding weight is the so-called standard

weight v(z) = (1 − |z|2)α.

(ii) Select ν(z) = e
− 1

(1−z)α , α > 0. Then we obtain the weight v(z) = e
− 1

(1−|z|2)α .

(iii) Choose ν(z) = sin(1 − z) and the corresponding weight is given by v(z) = sin(1 − |z|2).

(iv) Let ν(z) = (1 − log(1 − z))β for some β < 0. Then we get v(z) = (1 − log(1 − |z|2))β.

For a fixed point a ∈ D we introduce a function va(z) := ν(az) for every z ∈ D. Since ν is
holomorphic on D, so is the function va.
We say that a weight v is radial if v(z) = v(|z|) for every z ∈ D. Moreover, radial weights are
typical if additionally lim|z|→1 v(z) = 0 holds. Thus, we introduced a class of typical weights. In
[15] Lusky studied weights satisfying the following condition (L1) which was renamed after the

author:

(L1) inf
n∈N

v(1 − 2−n−1)

v(1 − 2−n)
> 0.

Among others examples of weights satisfying condition (L1) are the standard weights (see Example

(i)) and the logarithmic weights (Example (iv)). Throughout this work condition (L1) will play a

great role, and we will need the following condition (A) which is equivalent to (L1):

(A) there are 0 < r < 1 and 1 < C < ∞ with v(z)
v(p)

≤ C for all p, z ∈ D with ρ(p, z) ≤ r.

The equivalence of the conditions (L1) and (A) was shown in [10]. See also [14].



12 Elke Wolf CUBO
14, 1 (2012)

3 Basic facts

We need some geometric data of the open unit disk. Fix a ∈ D and consider the authomorphism
ϕa(z) :=

z−a
1−az

, z ∈ D, which interchanges 0 and a. Moreover, we use the fact that

ϕ′a(z) =
|a|2 − 1

(1 − az)2
, z ∈ D.

Now, the pseudohyperbolic metric is given by

ρ(z, a) = |ϕa(z)|, z, a ∈ D.

One of the most important properties of the pseudohyperbolic metric is that it is Möbius invariant,

that is,

ρ(σ(z), σ(a)) = ρ(z, a) for every automorphism σ of D, z, a ∈ D.

The pseudohyperbolic metric is a true metric. In fact, it even satisfies a stronger version of the

triangle inequality, more precisely, for every z, a, b ∈ D we have that

ρ(z, a) ≤
ρ(z, b) + ρ(b, a)

1 + ρ(z, b)ρ(b, a)
.

4 Results

Before we are able to treat boundedness and compactness of operators Ig,φ we need a number of

auxiliary lemmas. The first lemma is taken from [18].

Lemma 1. Let v be a weight as defined in (2.1) such that supa∈D supz∈D
v(z)|va(ϕa(z))|

v(ϕa(z))
≤ C < ∞.

Then

|f(z)| ≤
C

1
p

v(0)
1
p (1 − |z|2)

2
p v(z)

1
p

‖f‖v,p

for all z ∈ D, f ∈ Av,p.

Calculations show that the examples (i) -(iv) which were listed up above satisfy the assump-

tions of the previous lemma. The next lemma was shown in [20].

Lemma 2. Let v be a radial weight as defined in (2.1) such that v additionally satisfies condition

(L1). Then for every f ∈ Av,p there is Cv > 0 such that

|f(z) − f(w)| ≤ Cv‖f‖v,p max

{
1

(1 − |z|2)
2
p v(z)

1
p

,
1

(1 − |w|2)
2
p v(w)

1
p

}
ρ(z, w)

for every z, w ∈ D.



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Integral composition operators between . . . 13

Lemma 3. Let v be a radial weight as defined in (2.1) such that v additionally satisfies condition

(L1) and supa∈D supz∈D
v(z)|va(ϕa(z))|

v(ϕa(z))
≤ C < ∞. Then

|f′(z)| ≤
C

1
p

v(0)
1
p (1 − |z|2)

2
p
+1

v(z)
1
p

‖f‖v,p

for every z ∈ D and every f ∈ Av,p.

Proof. Lemma 2 yields that for every f ∈ Av,p and every h, z ∈ D with z + h ∈ D, we have

|f(z + h) − f(z)| ≤ Cv‖f‖v,p max

{
1

(1 − |z + h|2)
2
p v(z + h)

1
p

,
1

(1 − |z|2)
2
p v(z)

1
p

}
|h|

|1 − z(z + h)|
.

Hence∣∣∣∣f(z + h) − f(z)h
∣∣∣∣ ≤ Cv‖f‖v,p max

{
1

(1 − |z + h|2)
2
p v(z + h)

1
p

,
1

(1 − |z|2)
2
p v(z)

1
p

}
1

|1 − z(z + h)|

and finally

|f′(z)| =

∣∣∣∣ lim
h→0

f(z + h) − f(z)

h

∣∣∣∣
≤ lim

h→0 Cv‖f‖v,p max
{

1

(1 − |z + h|2)
2
p v(z + h)

1
p

,
1

(1 − |z|2)
2
p v(z)

1
p

}
1

|1 − z(z + h)|

= Cv‖f‖v,p
1

(1 − |z|2)
2
p
+1

v(z)
1
p

for every z ∈ D, as desired.

Lemma 4. Let v be a radial weight as in Lemma 3. Then there exist 0 < r < 1 and a constant

M > 0 such that for f ∈ Av,p

|f′(z) − f′(w)| ≤
4MC

1
p

v(0)
1
p

‖f‖v,p
r(1 − |z|2)

2
p
+1

v(z)
1
p

ρ(z, w)

for every z, w ∈ D with ρ(z, w) ≤ r
2
.

Proof. By hypotesis, v has condition (L1), and, moreover, we know that (L1) is equivalent to

condition (A). Since the weight u(z) = 1 − |z|2 also satisfies condition (L1), we can find 0 < r < 1

and constants M1 < ∞ and M2 < ∞ such that
v(z)

v(w)
≤ M1 and

1 − |z|2

1 − |w|2
≤ M2 for every z, w ∈ D with ρ(z, w) ≤ r.

Let w ∈ D be fixed. Since
ϕw(ϕw(z)) = z and ϕw(0) = w,



14 Elke Wolf CUBO
14, 1 (2012)

we get that

|f′(z) − f′(w)| = |f′(ϕw(ϕw(z)) − f
′(ϕw(ϕw(w))|.

For |z| = ρ(ϕw(z), w) ≤ r we obtain by using Lemma 3

|f′(ϕw(z))| ≤
C

1
p ‖f‖v,p

v(0)
1
p (1 − |ϕw(z)|2)

2
p
+1

v(ϕw(z))
1
p

=
C

1
p ‖f‖v,p

v(0)
1
p (1 − |w|2)

2
p
+1

v(w)
1
p

(1 − |w|2)
2
p
+1

v(w)
1
p

(1 − |ϕw(z)|2)
2
p
+1

v(ϕw(z))
1
p

≤
C

1
p M

1
p

1
M

2
p
+1

2

v(0)
1
p

‖f‖v,p
(1 − |w|2)

2
p
+1

v(w)
1
p

.

Let us now consider gw(z) := f
′(ϕw(z)) for every z ∈ D. Thus, for ρ(z, w) = |ϕw(z)| ≤ r2 we can

find Θ ∈ D with |Θ| ≤ |ϕw(z)| ≤ r2 such that

|f′(z) − f′(w)| = |gw(ϕw(z)) − gw(0)|

≤ |ϕw(z)|

∣∣∣∣∣
∫1
0

[
∂

∂t
gw

]
(tϕw(z)) dt

∣∣∣∣∣
≤ |ϕw(z)|

∣∣∣∣ ∂∂zgw(Θ)
∣∣∣∣

= |ϕw(z)|
1

2π

∣∣∣∣∣
∫
|ξ|=r

gw(ξ)

(ξ − Θ)2
dΘ

∣∣∣∣∣
Finally,

|f′(z) − f′(w)| ≤
C

1
p M

1
p

1
M

2
p
+1

2

v(0)
1
p

|ϕw(z)|r‖f‖v,p
(r − |ϕw(z)|)2(1 − |w|2)

2
p
+1

v(w)
1
p

≤
4C

1
p M

1
p

1
M

2
p
+1

2

v(0)
1
p

ρ(z, w)‖f‖v,p
r(1 − |w|2)

2
p
+1

v(w)
1
p

.

We select M := M
1
p

1
M

2
p
+1

2
and obtain the claim.

Lemma 5. Let v be a weight as in Lemma 3. Then, there is Cv > 0 such that for every f ∈ Av,p

|f′(z) − f′(w)| ≤ Cv‖f‖v,p max

{
1

(1 − |z|2)
2
p
+1

v(z)
1
p

,
1

(1 − |w|2)
2
p
+1

v(w)
1
p

}
ρ(z, w)

for every z, w ∈ D.

Proof. By Lemma 4 we can find 0 < s < 1 and a constant M < ∞ such that
|f′(z) − f′(w)| ≤

4MC
1
p

v(0)
1
p

‖f‖v,p
s(1 − |z|2)

2
p
+1

v(z)
1
p

ρ(z, w)



CUBO
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Integral composition operators between . . . 15

for every z, w ∈ D with ρ(z, w) ≤ s
2
. Next, if ρ(z, w) > s

2
, then

|f′(z) − f′(w)| ≤ 2
C

1
p

v(0)
1
p

‖f‖v,p max

{
1

(1 − |z|2)
2
p
+1

v(z)
1
p

,
1

(1 − |w|2)
2
p
+1

v(w)
1
p

}

≤
4

s

C
1
p

v(0)
1
p

‖f‖v,p max

{
1

(1 − |z|2)
2
p
+1

v(z)
1
p

,
1

(1 − |w|2)
2
p
+1

v(w)
1
p

}
ρ(z, w).

Hence, with Cv := max

{
4MC

1
p

v(0)
1
p s

, 4C
1
p

sv(0)
1
p

}
, we conclude

|f′(z) − f′(w)| ≤ Cv max

{
1

(1 − |z|2)
2
p
+1

v(z)
1
p

,
1

(1 − |w|2)
2
p
+1

v(w)
1
p

}
ρ(z, w)

for every z, w ∈ D and the claim follows.

Inductively, we can show the following lemmas:

Lemma 6. Let v be a weight as in Lemma 3. Then there is Cv > 0 such that for every f ∈ Av,p

|f(n)(z)| ≤
Cv

(1 − |z|2)
2
p
+n

v(z)
1
p

‖f‖v,p

for every z ∈ D and every n ∈ N0.

Lemma 7. Let v be a weight as in Lemma 3. Then there exists Cv > 0 such that for every f ∈ Av,p

|f(n)(z) − f(n)(w)| ≤ Cv‖f‖v,p max

{
1

(1 − |z|2)
2
p
+n

v(z)
1
p

,
1

(1 − |w|2)
2
p
+n

v(w)
1
p

}
ρ(z, w)

for every z, w ∈ D and every n ∈ N0.

Now, we turn our attention to the operators Ig,φ and start with characterizing when they are

bounded.

Theorem 8. Let w be a weight and v be a weight as in Lemma 3 with M := supa∈D supz∈D
v(z)

|ν(az)|
<∞. If

sup
z∈D

w(z)|φ′(z)g(φ(z))|

(1 − |φ(z)|2)
2
p
+1

v(φ(z))
1
p

< ∞, (4.1)
then the operator Ig,φ : Av,p → Bw is bounded. If we assume additionally that

sup
z∈D

|ν′(|φ(z)|2)|w(z)|φ′(z)g(φ(z))|

v(φ(z))
1
p
+1(1 − |φ(z)|2)

2
p

< ∞, (4.2)
then the converse is also true.



16 Elke Wolf CUBO
14, 1 (2012)

Proof. We start with assuming that the operator Ig,φ is bounded and that the condition (4.2) is

satisfied. Fix a point a ∈ D and set

fa(z) :=
ϕ′a(z)

2
p

ν(az)
1
p

for every z ∈ D.

Then

‖f‖pv,p =
∫
D

|ϕ′a(z)|
2

|ν(az)|
v(z) dA(z) ≤ sup

z∈D

v(z)

|ν(az)|

∫
D
|ϕ′a(z)|

2 dA(z) ≤ sup
z∈D

v(z)

|ν(az)|
≤ M,

and the constant M is independent of the choice of the point a. For the derivative we have

f′a(z) =
2

p

ϕ′a(z)
2
p
−1

ϕ′′a(z)

ν(az)
1
p

−
1

p

aν′(az)ϕ′a(z)
2
p

ν(az)
1
p
+1

for every z ∈ D. Hence we can find a constant C∗ > 0 such that∣∣∣∣∣ w(a)|φ
′(a)||g(φ(a))|

(1 − |φ(a)|2)
2
p
+1

v(φ(a))
1
p

−
|ν′(|φ(a)|2)|w(a)|φ′(a)g(φ(a))|

v(φ(a))
1
p
+1(1 − |φ(a)|2)

2
p

∣∣∣∣∣ ≤
∣∣∣f′φ(a)(φ(a))|w(a)|g(φ(a))||φ′(a)|∣∣∣

≤ |(Ig,φfφ(a))′(a)|w(a)
≤ C∗‖Jg,φ‖‖fφ(a)‖v,p.

Finally, since (4.2) is fulfilled and the operator Ig,φ is bounded, the claim follows. Conversely, an

application of Lemma 3 yields for f ∈ Av,p

sup
z∈D

|(Ig,φf)
′(z)|w(z) = sup

z∈D
|f′(φ(z))||g(φ(z))||φ′(z)|w(z)

≤ sup
z∈D

C
1
p ‖f‖v,pw(z)|g(φ(z))||φ′(z)|

v(0)
1
p (1 − φ(z)|2)

2
p
+1

v(φ(z))
1
p

.

Hence the claim follows.

Next, we study, when such operators are compact. To do this we need a lemma which can

easily be derived from [9] Proposition 3.11.

Lemma 9. Let v and w be weights. Then the operator Ig,φ : Av,p → Bw is compact if and only if
it is bounded and for every bounded sequence (fn)n in Av,p which converges to zero uniformly on

the compact subsets of D, Ig,φfn tends to zero in Bw if n → ∞.
Theorem 10. Let w be a weight and v be a weight as in Theorem 8. Moreover, we assume that

Ig,φ : Av,p → Bw is bounded. If
lim
r→1 sup|φ(z)|>r

w(z)|φ′(z)g(φ(z))|

(1 − |φ(z)|2)
2
p
+1

v(φ(z))
1
p

= 0, (4.3)



CUBO
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Integral composition operators between . . . 17

then the operator Ig,φ : Av,p → Bw is compact. If we assume additionally
lim
r→1 sup|φ(z)|>r

|ν′(|φ(z)|2)|w(z)|φ′(z)g(φ(z))|

v(φ(z))
1
p
+1(1 − |φ(z)|2)

2
p

= 0, (4.4)

then the converse is also true.

Proof. Assume that the operator Ig,φ : Av,p → Bw is compact and that (4.4) is satisfied. To show
(4.3) let (zn)n be a sequence with |φ(zn)| → 1 and put

fk(z) :=
ϕ′

φ(zk)
(z)

2
p

ν(φ(zk)z)
1
p

for every z ∈ D and every k ∈ N.

Analogously to the proof of Theorem 8 we can show that (fn)n is a bounded sequence which tends

to zero uniformly on the compact subsets of D. Since Ig,φ is compact, by Lemma 9

‖Ig,φfn‖Bw → 0 if n → ∞.
Thus,

‖Ig,φfn‖Bw ≥

∣∣∣∣∣ w(zn)|φ
′(zn)||g(φ(zn)|

(1 − |φ(zn)|2)
2
p
+1

v(φ(zn))
1
p

−
|ν′(|φ(zn)|

2)|w(zn)|φ
′(zn)g(φ(zn))|

v(φ(zn))
1
p
+1(1 − |φ(zn)|2)

2
p

∣∣∣∣∣ ,
and, since (4.4) holds, condition (4.3) follows.

Conversely, suppose that (4.3) is satisfied. Let (fn)n be a bounded sequence in Av,p such that

‖fn‖v,p ≤ M1 < ∞ for every n ∈ N and such that (fn)n converges uniformly to zero on the
compact subsets of D if n → ∞. For a fixed ε > 0 we can find 0 < r0 < 1 such that if |φ(z)| > r0,
then

w(z)|g(φ(z))||φ′(z)|

(1 − |φ(z)|2)
2
p
+1

v(φ(z))
1
p

<
εv(0)

1
p

2C
1
p M1

.

Moreover, we can find M2 > 0 such that

sup
|φ(z)|≤r0

w(z)|g(φ(z))||φ′(z)| ≤ M2.

There is n0 ∈ N such that

sup
|φ(z)|≤r0

|f′n(φ(z))| ≤
ε

2M2
for every n ≥ n0.



18 Elke Wolf CUBO
14, 1 (2012)

We obtain applying Lemma 3

sup
z∈D

|(Ig,φfn)
′(z)|w(z) = sup

z∈D
w(z)|f′n(φ(z))||g(φ(z))||φ

′(z)|

≤ sup
|φ(z)|≤r0

w(z)|f′n(φ(z))||g(φ(z))||φ
′(z)|

+ sup
|φ(z)|>r0

w(z)|f′n(φ(z))||g(φ(z))||φ
′(z)|

≤ sup
|φ(z)|≤r0

|f′n(φ(z))| sup
|φ(z)|≤r0

w(z)|g(φ(z))||φ′(z)|

+ sup
|φ(z)|>r0

C
1
p ‖fn‖v,pw(z)|g(φ(z))||φ′(z)|

v(0)
1
p (1 − |φ(z)|2)

2
p
+1

v(φ(z))
1
p

≤ ε,

and the claim follows.

Received: March 2011. Revised: April 2011.

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