# Analytic Morrey Spaces and Bloch-Type Spaces

Authors: Ofori Samuel; Jianfei Wang; Yile Zhao
Journal: Journal of Function Spaces
 (2018)

---

## Abstract

This paper is devoted to characterizing the boundedness of the Riemann-Stieltjes operators from analytic Morrey spaces to Bloch-type spaces. Moreover, the boundedness of the superposition operator and weighted composition operator on analytic Morrey spaces is discussed, respectively.


---

## Body

## 1. Introduction

The open unit disk, the unit circle, and the area measure in the complexC are denoted by D, T, and d
A
(
z
), respectively. The space of all analytic functions in D will be written A
(
D
), and H
p denotes the Hardy space on the unit disk D.For(
p
-
1
,
η
)
∈
[
0
,
∞
)
×
[
0
,
∞
), the space of analytic Campanato space, denoted by A
L
p
,
η, is the set of all f
∈
H
p satisfying (1)
f
A
L
p
,
η
,
∗
=
s
u
p
I
⊆
T
I
-
η
∫
I
f
ξ
-
f
I
p
d
ξ
2
π
1
/
p
<
∞
,where s
u
p
I
⊆
T is taken over all subarcs I of T and (2)
d
ξ
=
d
e
i
θ
=
d
θ
I
=
2
π
-
1
∫
I
d
ξ
f
I
=
2
π
I
-
1
∫
I
f
ξ
d
ξ
.Table 1 will help us to understand the structure of A
L
p
,
η (see, e.g., [1–3] and [4, p. 209–217] for the real counterparts).Table 1
The structure of analytic Campanato space.
Index(
p
,
η
)
Analytic Campanato spaceA
L
p
,
η
η
=
0
Analytic Hardy spaceH
p
η
∈
(
0,1
)
Analytic Morrey spaceA
L
p
,
1
-
η
η
=
1
Analytic John-Nirenberg spaceB
M
O
A
η
∈
(
1,1
+
p
]
Analytic Lipschitz spaceA
(
η
-
1
)
/
p
η
∈
(
1
+
p
,
∞
)
Space of constantsCOf course, this value defines a seminorm onA
L
p
,
η. A complete norm on A
L
p
,
η can be equipped by (3)
f
A
L
p
,
η
=
f
p
+
f
A
L
p
,
η
,
∗
.Forα
>
0, the α-Bloch space, denoted by B
α
=
B
α
(
D
), is the set of all f
∈
A
(
D
) for which (4)
B
α
f
=
s
u
p
z
∈
D
1
-
z
2
α
f
′
z
<
∞
.The expressionB
α
(
f
) defines a seminorm while the natural norm is given by (5)
f
B
α
=
f
0
+
B
α
f
.The Riemann-Stieltjes operator J
g with analytic function symbol g is defined by (6)
J
g
f
z
=
∫
0
z
f
w
g
′
w
d
w
,
f
∈
A
D
.The corresponding integral operator is defined by (7)
I
g
f
z
=
∫
0
z
f
′
w
g
w
d
w
,
f
∈
A
D
.Obviously, the multiplication operator M
g is given as follows: (8)
M
g
f
z
=
f
z
g
z
=
f
0
g
0
+
I
g
f
z
+
J
g
f
z
.Continuing from [5–14], in this paper, we consider three operators associated with the analytic Morrey spaces. More precisely, in Section 2, we give two lemmas which are used to prove our main results in the paper. In Section 3, we obtain the boundedness of the Riemann-Stieltjes operators J
g and I
g from analytic Morrey spaces to Bloch-type spaces. In Section 4, the boundedness of the defined superposition operator on analytic Morrey spaces is proved. In Section 5, we consider the boundedness of the weighted composition operator from analytic Morrey spaces to Bloch-type spaces.Notation. Let A
≲
B and A
≳
B denote that there exists an absolute constant c
>
0 such that A
≤
c
B and A
≥
c
B, respectively. A
≈
B means that A
≲
B and A
≳
B hold.


## 2. Some Lemmas

We now recall some auxiliary results which will be used throughout this paper. Lemmas1 and 2 are proved, the first by Xiao and Yuan in [15] and the second by Wang in [16].Lemma 1.
Let(
p
-
1
,
η
)
∈
[
0
,
∞
)
×
(
0,1
). If f
∈
A
L
p
,
η, then (9)
f
z
≲
f
A
L
p
,
η
1
-
z
2
1
-
η
/
p
,
f
′
z
≲
f
A
L
p
,
η
1
-
z
2
1
+
1
-
η
/
phold for all z
∈
D.Lemma 2.
For any fixedw
∈
D. Let (
p
-
1
,
η
,
λ
)
∈
[
0
,
∞
)
×
(
0,1
)
×
(
1
/
p
,
∞
). Then functions (10)
f
w
z
=
1
-
w
2
λ
-
1
-
η
/
p
1
-
w
¯
z
λbelong to A
L
p
,
η. Moreover, f
w is uniformly bounded in A
L
p
,
η; that is,(11)
s
u
p
w
∈
D
f
w
A
L
p
,
η
<
∞
.


## 3. Boundedness of the OperatorsJ
g
,
I
g
:
A
L
p
,
η
→
B
α

In this section, we give the boundedness ofJ
g and I
g from A
L
p
,
η
→
B
α, respectively.Theorem 3.
Let(
p
-
1
,
η
)
∈
[
0
,
∞
)
×
[
0,1
). Suppose that g
∈
A
(
D
). Then, J
g
:
A
L
p
,
η
→
B
α is bounded if and only if (12)
s
u
p
z
∈
D
1
-
z
2
a
+
η
-
1
/
p
g
′
z
<
∞
.Moreover, (13)
J
g
A
L
p
,
η
→
B
α
=
s
u
p
z
∈
D
1
-
z
2
a
+
η
-
1
/
p
g
′
z
.Proof.
SupposeJ
g
:
A
L
p
,
η
→
B
α is bounded. Taking f
w
(
z
)
=
(
1
-
|
w
|
2
)
2
+
η
-
1
/
p
/
(
1
-
w
¯
z
)
2, then (14)
f
w
′
w
=
2
w
¯
1
-
w
2
2
+
η
-
1
/
p
1
-
w
¯
z
3
.Note that (15)
J
g
f
w
B
α
=
s
u
p
z
∈
D
1
-
z
2
α
J
g
f
w
′
z
=
s
u
p
z
∈
D
1
-
z
2
α
f
w
z
g
′
z
≳
1
-
w
2
α
f
w
w
g
′
w
=
1
-
w
2
α
1
-
w
2
η
-
1
/
p
g
′
w
=
1
-
w
2
α
+
η
-
1
/
p
g
′
w
.It implies that (16)
J
g
f
w
B
α
⩽
f
w
A
L
p
,
η
J
g
A
L
p
,
η
→
B
α
≲
J
g
A
L
p
,
η
→
B
α
,
s
u
p
z
∈
D
1
-
w
2
α
+
η
-
1
/
p
g
′
w
≲
J
g
A
L
p
,
η
→
B
α
.Hence, (17)
s
u
p
z
∈
D
1
-
z
2
a
+
η
-
1
/
p
g
′
z
<
∞
.Conversely, suppose (18)
s
u
p
z
∈
D
1
-
z
2
a
+
η
-
1
/
p
g
′
z
<
∞
.By Lemma 1, we have that (19)
f
z
≲
f
A
L
p
,
η
1
-
z
2
1
-
η
/
p
∀
z
∈
D
.Hence, (20)
J
g
f
B
α
=
s
u
p
z
∈
D
1
-
z
2
α
J
g
f
′
z
=
s
u
p
z
∈
D
1
-
z
2
α
f
z
g
′
z
≲
f
A
L
p
,
η
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
g
′
z
≲
f
A
L
p
,
ηimplies J
g
:
A
L
p
,
η
→
B
α is bounded and (21)
J
g
A
L
p
,
η
→
B
α
=
s
u
p
z
∈
D
1
-
z
2
a
+
η
-
1
/
p
g
′
z
.Theorem 4.
Let(
p
-
1
,
η
)
∈
[
0
,
∞
)
×
[
0,1
) and g
∈
A
(
D
). Then I
g
:
A
L
p
,
η
→
B
α is bounded if and only if(22)
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
<
∞
.Moreover, (23)
I
g
A
L
p
,
η
→
B
α
=
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
.Proof.
SupposeI
g
:
A
L
p
,
η
→
B
α is bounded. Let(24)
f
w
z
=
1
-
w
2
2
-
1
-
η
/
p
1
-
w
-
z
2
.Then(25)
f
w
′
w
=
2
w
¯
1
-
w
2
2
+
η
-
1
/
p
1
-
w
¯
z
3
.Lemma 2 shows that (26)
I
g
f
w
B
α
=
s
u
p
z
∈
D
1
-
z
2
α
I
g
f
w
′
z
=
s
u
p
z
∈
D
1
-
z
2
α
f
w
′
z
g
z
≥
1
-
w
2
α
f
w
′
w
g
w
≳
1
-
w
2
α
1
-
w
2
η
-
1
/
p
-
1
w
g
w
.Taking supremum in the last inequality over the set 1
/
2
≤
w
<
1 and applying to the maximum modulus principle we have (27)
s
u
p
w
≤
1
/
2
1
-
w
2
α
1
-
w
2
η
-
1
/
p
-
1
g
w
=
s
u
p
w
=
1
/
2
1
-
w
2
α
1
-
w
2
η
-
1
/
p
-
1
g
w
≲
s
u
p
1
/
2
≤
w
<
1
1
-
w
2
α
1
-
w
2
η
-
1
/
p
-
1
w
g
w
.Hence (28)
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
≤
I
g
A
L
p
,
η
→
B
α
.That is, (29)
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
<
∞
.Conversely, suppose (30)
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
<
∞
.By Lemma 1, we have that (31)
f
′
z
≲
f
A
L
p
,
η
1
-
z
2
1
+
1
-
η
/
p
∀
z
∈
D
.Then (32)
I
g
f
B
α
=
s
u
p
z
∈
D
1
-
z
2
α
I
g
f
′
z
=
s
u
p
z
∈
D
1
-
z
2
α
f
′
z
g
z
≲
f
A
L
p
,
η
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
≲
f
A
L
p
,
η
,which implies I
g
:
A
L
p
,
η
→
B
α is bounded and (33)
I
g
A
L
p
,
η
→
B
α
=
s
u
p
z
∈
D
1
-
z
2
α
+
η
-
1
/
p
-
1
g
z
.Remark 5.
Whenα
<
1
+
1
-
η
/
p in Theorem 4, the conclusion is equivalent to g
≡
0.As an application of Theorems3 and 4, we can obtain the following corollary.Corollary 6.
Let(
p
-
1
,
η
)
∈
[
0
,
∞
)
×
[
0,1
) and g
∈
A
(
D
). Then M
g
:
A
L
p
,
η
→
B
α is bounded if and only if g
∈
H
∞.


## 4. Superposition onA
L
p
,
η

LetX and Y represent two subspaces of A
(
D
). If ϕ is a complex-valued function C such that ϕ
∘
f
∈
Y whenever f
∈
X, then we call that ϕ acts by superposition from X into Y. If X and Y contain the linear functions, then ϕ must be an entire function. The superposition S
ϕ
:
X
→
Y with symbol ϕ is then defined by S
ϕ
(
f
)
=
ϕ
∘
f. A basic question is when S
ϕ map X into Y continuously. This question has been studied for many distinct pairs (
X
,
Y
)—see, for example, [17–20]. In this section, we are interested in the analytic Morrey space and have the following result which extends the case of p
=
2 in [3].Theorem 7.
Let(
p
-
1
,
η
)
∈
[
0
,
∞
)
×
(
0,1
). Then S
ϕ is bounded on A
L
p
,
η if and only if ϕ
(
z
)
=
a
z
+
b for some a
,
b
∈
C.Proof.
Lemma1 shows that(34)
f
w
z
=
1
-
w
2
2
+
η
-
1
/
p
1
-
w
¯
z
2
⟹
s
u
p
b
∈
D
f
w
A
L
p
,
η
<
∞
;
f
∈
A
L
p
,
η
⟹
f
′
z
≲
f
A
L
p
,
η
1
-
z
2
1
+
1
-
η
/
p
∀
z
∈
D
.
IfS
ϕ is bounded on A
L
p
,
η, then for f
∈
A
L
p
,
η we obtain(35)
S
ϕ
f
′
z
=
ϕ
′
f
z
f
′
z
≲
f
A
L
p
,
η
1
-
z
2
1
+
1
-
η
/
p
∀
z
∈
D
.Taking the following A
L
p
,
η-function (36)
f
θ
,
w
z
=
e
i
θ
1
-
w
2
2
+
η
-
1
/
p
1
-
w
¯
z
2in the last inequality, we have (37)
ϕ
′
f
θ
,
w
z
f
θ
,
w
′
z
=
ϕ
′
f
θ
,
w
z
w
1
-
w
2
3
+
η
-
1
/
p
1
-
w
-
z
3
≲
f
θ
,
w
A
L
p
,
η
1
-
z
2
1
-
η
/
p
.Since (38)
s
u
p
θ
,
w
f
θ
,
w
A
L
p
,
η
<
∞
,so there is a positive M independent of (
θ
,
w
) such that (39)
s
u
p
θ
,
w
f
θ
,
w
A
L
p
,
η
≤
M
.In particular, setting w
=
z yields (40)
ϕ
′
f
θ
,
w
w
f
θ
,
w
′
w
=
s
u
p
w
∈
D
ϕ
′
f
w
w
w
≤
M
⟹
ϕ
′
e
i
θ
1
-
w
2
1
-
η
/
p
≤
M
∀
w
∈
D
.Setting |
w
|
→
1 in the last estimate and noticing η
∈
(
0,1
), we obtain that the entire function ϕ is bounded on C. By using the maximum principle, we get that ϕ must be a linear function.
Conversely, ifϕ
(
z
)
=
a
z
+
b for some a
,
b
∈
C, then ϕ
′
(
z
)
=
a, and hence S
ϕ
(
f
)
=
a
f
+
b. Hence S
ϕ is bounded on A
L
p
,
η.


## 5. Weighted Composition Operator fromA
L
p
,
η
to
B
α

Suppose thatψ
∈
A
(
D
) and φ is an analytic self-map of D, φ
(
D
)
⊂
D. These maps induce a linear weighted composition operator W
ψ
,
φ which is defined by (41)
W
ψ
,
φ
f
z
=
M
ψ
C
φ
f
z
=
ψ
z
f
φ
z
,
z
∈
D
,where M
ψ is the operator of pointwise multiplication by ψ and C
φ is the composition operator f
↦
f
∘
φ. Our next result will be as follows.Theorem 8.
Let0
<
η
<
1, 0
<
α
,
p
-
1
<
∞, and φ be an analytic self-map of D. Then (42)
W
ψ
,
φ
f
B
α
≲
f
A
L
p
,
η
,
∀
f
∈
A
L
p
,
η
,if and only if(43)
s
u
p
z
∈
D
1
-
z
2
α
1
-
φ
z
2
ψ
′
z
+
ψ
z
φ
′
z
1
-
φ
z
2
p
-
η
+
1
/
p
<
∞
.Proof.
The growth of functions inA
L
p
,
η shows that if f
∈
A
L
p
,
η then (44)
1
-
z
2
α
W
ψ
,
φ
f
′
z
≤
1
-
z
2
α
ψ
′
z
f
φ
z
+
1
-
z
2
α
ψ
z
f
′
φ
z
φ
′
z
=
B
1
+
B
2
.It is easy to see (45)
B
1
=
1
-
z
2
α
ψ
′
z
f
φ
z
≤
1
-
z
2
α
ψ
′
z
f
A
L
p
,
η
1
-
φ
z
2
1
-
η
/
p
=
1
-
z
2
α
ψ
′
z
1
-
φ
z
2
1
-
η
/
p
f
A
L
p
,
η
,
B
2
=
1
-
z
2
α
ψ
z
f
′
φ
z
φ
′
z
≤
1
-
z
2
α
ψ
z
φ
′
z
1
-
φ
z
2
p
-
η
+
1
/
p
f
A
L
p
,
η
.If (46)
s
u
p
z
∈
D
1
-
z
2
α
1
-
φ
z
2
ψ
′
z
+
ψ
z
φ
′
z
1
-
φ
z
2
p
-
η
+
1
/
p
<
∞
,then B
1
<
∞ and B
2
<
∞. It yields that W
ψ
,
φ
(
f
)
∈
B
α which proves that the sufficiency holds. On the other hand, the necessary follows by taking the test function in A
L
p
,
η
(47)
f
w
z
=
1
-
φ
w
2
2
+
1
-
η
/
p
1
-
φ
w
¯
z
2
,
∀
z
∈
D
.Remark 9.
Whenψ
(
z
)
≡
1, the result reduces to Xiao and Yuan in [15]. When φ
(
z
)
≡
z, W
ψ
,
φ
f denote the multiplication operator. (48)
ψ
·
f
B
α
≲
f
A
L
p
,
η
,
∀
f
∈
A
L
p
,
η
,if and only if (49)
s
u
p
z
∈
D
1
-
z
2
ψ
′
z
+
ψ
z
1
-
z
2
p
-
η
+
1
/
p
-
α
<
∞
.



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*Source: 4983827-2018-03-11.xml*