# The Harmonic Bloch and Besov Spaces on the Real Unit Ball by an Oscillation

Authors: Xi Fu; Zhiyao Xu; Xiaoyou Liu
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

LetB be the real unit ball in R
n and f
∈
C
N
(
B
). Given a multi-index m
=
(
m
1
,
…
,
m
n
) of nonnegative integers with |
m
|
=
N, we set the quantity s
u
p
x
∈
B
,
y
∈
E
(
x
,
r
)
,
x
≠
y
(
1
-
|
x
|
2
)
α
(
1
-
|
y
|
2
)
β
|
∂
m
f
(
x
)
-
∂
m
f
(
y
)
|
/
|
x
-
y
|
γ
[
x
,
y
]
1
-
γ
,
x
≠
y
, where 0
≤
γ
≤
1 and α
+
β
=
N
+
1. In terms of it, we characterize harmonic Bloch and Besov spaces on the real unit ball. This generalizes the main results of Yoneda, 2002, into real harmonic setting.


---

## Body

## 1. Introduction

LetB be the real unit ball in R
n with n
≥
2, where d
v is the normalized volume measure on B and d
σ is the normalized surface measure on the unit sphere S
=
∂
B. We denote the class of all harmonic functions on the unit ball by H
(
B
). For f
∈
H
(
B
), ∇
f
(
x
) denotes the gradient of f. Given a multi-index m
=
(
m
1
,
…
,
m
n
) of nonnegative integers, we use the notations |
m
|
=
m
1
+
⋯
+
m
n and (1)
∂
m
f
=
∂
m
f
∂
x
m
=
∂
m
f
∂
x
1
m
1
⋯
∂
x
n
m
n
.For eachα
>
0, theharmonic
α
-Bloch space
B
α consists of all functions f
∈
H
(
B
) such that (2)
f
α
=
sup
x
∈
B
⁡
1
-
x
2
α
∇
f
x
<
∞
,and thelittle
α
-Bloch space
B
0
α consists of the functions f
∈
B
α such that (3)
lim
x
→
1
-
⁡
sup
x
∈
B
⁡
1
-
x
2
α
∇
f
x
=
0
.Theharmonic Besov space
B
p is the space of all functions in H
(
B
) for which (4)
∫
B
1
-
x
2
p
∇
f
x
p
d
τ
x
<
∞
,where p
>
n
-
1 and d
τ
(
x
)
=
(
1
-
|
x
|
2
)
-
n
d
v
(
x
) is the invariant measure on B.Letf be a continuous function in B. If there exists a constant C such that (5)
L
f
x
,
y
=
1
-
x
2
1
/
2
1
-
y
2
1
/
2
f
x
-
f
y
x
-
y
≤
C
,for any x
,
y
∈
B, then we say that f satisfiesweighted Lipschitz condition. By means of it, Ren and Kähler [1] obtained the following.Theorem A.
Letf
∈
H
(
B
). Then f
∈
B
1 if and only if it satisfies the weighted Lipschitz condition.Theorem B.
Letf
∈
H
(
B
) and p
∈
(
2
(
n
-
1
)
,
∞
). Then f
∈
B
p if and only if(6)
∬
B
L
f
p
x
,
y
d
τ
x
d
τ
y
<
∞
.LetD be the open unit disk in the complex plane C and let f be a continuous complex-value function in D. Denote by(7)
M
=
sup
z
∈
D
,
w
∈
D
z
,
r
,
z
≠
w
⁡
1
-
z
2
α
1
-
w
2
β
D
^
n
-
1
f
z
-
D
^
n
-
1
f
w
z
-
w
,where D
(
z
,
r
) is the hyperbolic disc with center z
∈
D and radius r, n
≥
1 an integer, α
+
β
=
n, and D
^
(
n
)
=
∂
n
/
∂
z
n
+
∂
n
/
∂
z
¯
n.In [2], Yoneda characterized harmonic Bloch and Besov spaces in D in terms of M as follows.Theorem C.
Letf be a complex-value harmonic function in D. Fix an integer n
≥
1 and a pair of real numbers α, β such that α
+
β
=
n. Then f
∈
B
1
(
D
) if and only if M is bounded.Theorem D.
Letf be a complex-value harmonic function in D. Fix an integer n
≥
2 and a pair of real numbers α, β such that α
+
β
=
n. Then for each p
≥
1, f
∈
B
p
(
D
) if and only if(8)
∫
D
M
p
d
λ
z
<
∞
,where d
λ
(
z
)
=
(
1
-
|
z
|
2
)
-
2
d
z.See [3–5] for various characterizations of the Bloch, little Bloch, and Besov spaces in the unit ball of C
n.The main purpose of this paper is to give some characterizations for the spacesB
α, B
0
α, and B
p in the real unit ball along Yoneda’s direction. In Section 2, we collect some known results that will be needed in the proof of our results. Our main results and their proofs are presented in Sections 3 and 4.Throughout this paper, constants are denoted byC; they are positive and may differ from one occurrence to the other. The notation A
≍
B means there is a positive constant C such that B
/
C
≤
A
≤
C
B.


## 2. Preliminaries

We will be using the same notation in [1, 6]: we write x
,
y
∈
R
n in polar coordinates by x
=
x
x
′ and y
=
y
y
′. For any a
,
b
∈
R
n, let (9)
a
,
b
=
a
b
-
a
′
.Then the symmetry lemma in [7] shows that (10)
a
,
b
=
b
,
a
.For any a
∈
B, denote by ϕ
a the Möbius transformation in B. It is an involution of B such that ϕ
a
(
0
)
=
a and ϕ
a
(
a
)
=
0, which is of the form (11)
ϕ
a
x
=
x
-
a
2
a
-
1
-
a
2
x
-
a
x
,
a
2
,
x
∈
B
.By simple computations, we have (12)
ϕ
a
x
=
x
-
a
x
,
a
,
1
-
ϕ
a
x
2
=
1
-
x
2
1
-
a
2
x
,
a
2
.For anya
∈
B and r
∈
(
0,1
), we define thepseudo-hyperbolic ball with center a and radius r as (13)
E
a
,
r
=
w
∈
B
:
ϕ
a
w
<
r
.Clearly, E
(
a
,
r
)
=
ϕ
a
(
B
(
0
,
r
)
).Lemma 1 (see [1, Lemma 2.1]).
Letr
∈
(
0,1
) and y
∈
E
(
x
,
r
). Then (14)
1
-
x
2
≍
1
-
y
2
≍
x
,
y
≍
E
x
,
r
,where |
E
(
x
,
r
)
| denotes the volume of E
(
x
,
r
).The following is a characterization of the spaceB
1 (resp., B
0
1) which is proved in [8].Lemma 2.
Leth
∈
H
(
B
) and N be a positive integer. Then h
∈
B
1 (resp., B
0
1) if and only if (15)
sup
x
∈
B
⁡
1
-
x
2
N
∂
m
h
x
<
∞
resp.
lim
x
→
1
-
⁡
sup
x
∈
B
⁡
1
-
x
2
N
∂
m
h
x
=
0for all multi-index m with |
m
|
=
N.As an application of Lemma2, we can obtain the following.Lemma 3.
Leth
∈
H
(
B
). Then h
∈
B
1 if and only if for each j
∈
{
1
,
…
,
n
}, (16)
sup
x
,
y
∈
B
,
x
≠
y
⁡
1
-
x
2
1
-
y
2
x
-
y
∂
h
∂
x
j
x
-
∂
h
∂
x
j
y
<
∞
.Proof.
Fixing a pointx and letting (17)
y
=
x
+
t
∇
∂
h
∂
x
j
x
⟶
xwith t
∈
R, we have (18)
1
-
x
2
2
∇
∂
h
∂
x
j
x
≤
L
,for each j
∈
{
1
,
…
,
n
}. By Lemma 2, we see that h
∈
B
1.
For the converse, we assume thath
∈
B
1. Let h
j
(
x
)
=
∂
h
(
x
)
/
∂
x
j; then for each j
∈
{
1
,
…
,
n
}
(19)
h
j
x
-
h
j
y
=
∫
0
1
d
h
j
d
s
s
x
+
1
-
s
y
d
s
≤
∑
k
=
1
n
x
k
-
y
k
∫
0
1
∂
h
j
∂
x
k
s
x
+
1
-
s
y
d
s
≤
n
x
-
y
∫
0
1
∇
h
j
s
x
+
1
-
s
y
d
s
≤
C
x
-
y
∫
0
1
d
s
1
-
s
x
+
1
-
s
y
2
2
.It follows from [9] that there exists C
>
0 such that (20)
∫
0
1
d
s
1
-
s
x
+
1
-
s
y
2
2
≤
C
1
-
x
2
1
-
y
2
.This implies that (21)
sup
x
,
y
∈
B
,
x
≠
y
⁡
1
-
x
2
1
-
y
2
x
-
y
h
j
x
-
h
j
y
<
∞
.So the result follows.Combining Theorem A and Lemma3, we extend [2, Corollary 2.4] into the real harmonic setting as follows.Corollary 4.
Leth
∈
H
(
B
). Then for i
=
1,2, h
∈
B
1 if and only if (22)
sup
x
,
y
∈
B
,
x
≠
y
⁡
1
-
x
2
i
/
2
1
-
y
2
i
/
2
x
-
y
∂
i
-
1
f
x
-
∂
i
-
1
f
y
<
∞
,where ∂
1
f
=
∂
f
/
∂
x
j for all j
∈
{
1
,
…
,
n
}.In the following, we give an example which shows that Corollary4 does not hold for i
≥
3.Example 5.
Letf
(
x
)
=
l
n
(
(
1
-
x
1
)
2
+
x
2
2
); then f
∈
B
1. By a simple computation we have (23)
∂
2
f
∂
x
1
2
=
2
1
-
x
1
2
+
x
2
2
-
4
1
-
x
1
2
1
-
x
1
2
+
x
2
2
2
,
sup
x
∈
B
⁡
1
-
x
2
3
/
2
x
∂
2
∂
x
1
2
f
x
-
∂
2
∂
x
1
2
f
0
=
∞
.


## 3. Results and Discussions

### 3.1. Harmonic Bloch Spaces

In this section, we give some characterizations of the spacesB
α which can be viewed as the generalizations of Yoneda’s results into the real unit ball B of R
n.For a continuous functionh in B and 0
≤
γ
≤
1, we write (24)
L
Λ
h
=
h
x
-
h
y
x
-
y
γ
x
,
y
1
-
γ
,
x
,
y
∈
B
,
x
≠
y
.By using the notation, we characterize B
1
,
B
0
1 as follows.Theorem 6.
Letf
∈
H
(
B
),  N
≥
0 be an integer and 0
<
r
<
1. Then f
∈
B
1 if and only if (25)
L
f
=
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
∞
,for all multi-index m with |
m
|
=
N, where α
+
β
=
N
+
1.Proof.
First we prove the sufficiency. Letf
∈
H
(
B
). Then for each multi-index m,  ∂
m
f
∈
H
(
B
). For r
∈
(
0,1
), it follows from [1] that (26)
∇
∂
m
f
x
≤
C
1
-
x
2
∫
E
x
,
r
∂
m
f
y
d
τ
y
.Fixing x
∈
B and replacing ∂
m
f by ∂
m
f
-
∂
m
f
(
x
), we have (27)
1
-
x
2
m
+
1
∇
∂
m
f
x
≤
C
1
-
x
2
m
∫
E
x
,
r
∂
m
f
y
-
∂
m
f
x
d
τ
y
.By Lemma 1, we can deduce that (28)
1
-
x
2
m
+
1
∇
∂
m
f
x
≤
C
1
-
x
2
m
+
1
∫
E
x
,
r
∂
m
f
y
-
∂
m
f
x
x
,
y
d
τ
y
≤
C
∫
E
x
,
r
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
d
τ
y
≤
C
L
f
τ
B
0
,
r
.Since τ
(
B
(
0
,
r
)
)
=
n
∫
0
r
t
n
-
1
(
1
-
t
2
)
-
n
d
t is a constant, we see that (29)
1
-
x
2
N
+
1
∇
∂
m
f
x
<
∞
,for any multi-index m with |
m
|
=
N. Hence Lemma 2 yields that f
∈
B
1.
Now we prove the necessity. Lety
∈
E
(
x
,
r
)
,
x
≠
y. Then for each multi-index m with |
m
|
=
N, we have(30)
∂
m
f
x
-
∂
m
f
y
=
∫
0
1
d
∂
m
f
d
s
s
x
+
1
-
s
y
d
s
≤
∑
k
=
1
n
x
k
-
y
k
∫
0
1
∂
∂
m
f
∂
x
k
s
x
+
1
-
s
y
d
s
≤
n
x
-
y
∫
0
1
∇
∂
m
f
s
x
+
1
-
s
y
d
s
≤
C
x
-
y
∫
0
1
d
s
1
-
x
z
+
1
-
s
y
N
+
1
.By Lemma 1 we infer that there exists λ
>
0 such that 1
-
|
y
|
=
λ
(
1
-
|
x
|
) and (31)
L
Λ
∂
m
f
≤
C
∫
0
1
d
s
s
1
-
x
+
1
-
s
1
-
y
N
+
1
≤
C
1
-
x
2
N
+
1
∫
0
1
d
s
s
+
λ
1
-
s
N
+
1
≤
C
1
-
x
2
α
1
-
y
2
β
.Thus, (32)
L
f
=
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
∞
.So the proof is complete.Theorem 7.
Letf
∈
H
(
B
), N
≥
0 be an integer and 0
<
r
<
1. Then f
∈
B
0
1 if and only if(33)
lim
x
→
1
-
⁡
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
=
0
,for all multi-index m with |
m
|
=
N, where α
+
β
=
N
+
1.Proof.
Sufficiency: assume that (33) holds. Then for any ϵ
>
0, there exists δ
∈
(
0,1
) such that (34)
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
ϵwhenever δ
<
|
x
|
<
1. It follows from an argument similar to that in proof of Theorem 6 that we have (35)
1
-
x
2
N
+
1
∇
∂
m
f
x
≤
C
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
C
ϵ
,whenever δ
<
|
x
|
<
1. Hence (36)
lim
x
→
1
-
⁡
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
=
0
,from which we see that f
∈
B
0
1.
Necessity: fort
∈
(
0,1
), let f
t
(
x
)
=
f
(
t
x
). By Lemma 1 and proof of Theorem 6, we see that, for each multi-index m with |
m
|
=
N, (37)
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
-
f
t
≤
C
1
-
ξ
2
N
+
1
∇
∂
m
f
-
f
t
ξ
,
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
t
≤
C
1
-
x
2
α
+
β
1
-
t
2
N
+
1
1
-
t
η
2
N
+
1
∇
∂
m
f
t
ηfor all x
∈
B and ξ
,
η
∈
E
(
x
,
r
). So (38)
L
f
≤
C
1
-
ξ
2
N
+
1
∇
∂
m
f
-
f
t
ξ
+
C
1
-
x
2
α
1
-
y
2
β
1
-
t
2
N
+
1
1
-
t
η
2
N
+
1
∇
∂
m
f
t
η
.First letting |
x
|
→
1
- and then letting t
→
1
-, we obtain the desired result.In the following, by removing the restrictiony
∈
E
(
x
,
r
) in Theorem 6, we obtain the following.Theorem 8.
Letf
∈
H
(
B
), 0
≤
β
<
1, and β
≤
α
<
1
+
β. Then f
∈
B
α if and only if(39)
Q
=
sup
x
,
y
∈
B
,
x
≠
y
⁡
1
-
x
2
β
1
-
y
2
α
-
β
L
Λ
f
<
∞
.Proof.
We only need to prove the necessity since the proof of sufficiency is similar to that in proof of Theorem6. Assume that f
∈
B
α. For x
,
y
∈
B, s
∈
[
0,1
],(40)
1
-
s
x
+
1
-
s
y
2
α
≥
s
1
-
x
2
2
+
1
-
s
1
-
y
2
2
α
≥
s
2
β
1
-
s
2
α
-
β
1
-
x
2
β
1
-
y
2
α
-
β
,which gives (41)
L
Λ
f
≤
C
∫
0
1
d
s
1
-
s
x
+
1
-
s
y
2
α
≤
C
∫
0
1
d
s
s
/
2
β
1
-
s
/
2
α
-
β
1
-
x
2
β
1
-
y
2
α
-
β
≤
C
1
-
x
2
β
1
-
y
2
α
-
β
∫
0
1
d
s
s
β
1
-
s
α
-
β
≤
C
1
-
x
2
β
1
-
y
2
α
-
β
,where the last integral converges since α
<
1
+
β. Thus (42)
1
-
x
2
β
1
-
y
2
α
-
β
L
Λ
f
<
∞
.Similarly, we can prove the following.Theorem 9.
Letf
∈
B
α, 0
≤
β
<
1, and β
≤
α
<
1
+
β. Then f
∈
B
0
α if and only if(43)
lim
x
→
1
-
⁡
sup
x
,
y
∈
B
,
x
≠
y
⁡
ω
1
-
x
2
β
1
-
y
2
α
-
β
L
Λ
f
=
0
.


### 3.2. Harmonic Besov Space

In order to prove our next result, we need the following lemma.Lemma 10.
Letf
∈
H
(
B
). Then f
∈
B
p if and only if (44)
sup
x
∈
B
⁡
1
-
x
2
N
+
1
∇
∂
m
f
x
∈
L
p
B
,
d
τfor all multi-index m with |
m
|
=
N.Proof.
This follows from [10, Theorem 3.7] by letting α
=
p
-
n.Lemma 11.
Leth
∈
H
(
B
) and 0
<
r
<
1. Then there exist constants C
>
0,  r
<
r
′
<
1 such that (45)
sup
y
∈
E
x
,
r
,
x
≠
y
⁡
h
x
-
h
y
x
-
y
≤
C
∫
E
x
,
r
′
∇
h
u
d
τ
u
.Proof.
By Cauchy’s estimates and Lemma1, for each x
∈
B, we have (46)
sup
y
∈
E
x
,
r
,
x
≠
y
⁡
h
x
-
h
y
x
-
y
≤
C
sup
ζ
∈
E
x
,
r
⁡
∇
h
ζ
≤
C
E
x
,
r
′
∫
E
x
,
r
′
∇
h
ζ
d
v
ζ
≤
C
∫
E
x
,
r
′
∇
h
ζ
d
τ
ζfor some r
′
>
r.Now, we come to state and prove the result for harmonic Besov spaces.Theorem 12.
Letf
∈
H
(
B
), N
≥
0 be an integer and 0
<
r
<
1. Then f
∈
B
p if and only if (47)
K
p
=
∫
B
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
p
d
τ
x
<
∞
,for all multi-index m with |
m
|
=
N, where α
+
β
=
N
+
1.Proof.
Letf
∈
H
(
B
). Suppose that (48)
K
p
=
∫
B
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
d
τ
<
∞
.Set(49)
L
f
x
=
lim
y
→
x
⁡
sup
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
.It follows from proof of Theorem 6 that we have (50)
1
-
x
2
N
+
1
∇
∂
m
f
x
≤
C
L
f
x
.Since L
f
(
x
)
≤
L
f, we see that (51)
∫
B
1
-
x
2
p
N
+
1
∇
∂
m
f
x
p
d
τ
x
≤
C
∫
B
L
f
p
x
d
τ
x
≤
C
∫
B
L
f
p
d
τ
x
=
C
K
p
,for all multi-index m with |
m
|
=
N. By Lemma 10, f
∈
B
p.
To prove the necessity, we suppose thatf
∈
B
p. By Lemmas 1 and 11, for each multi-index m and r
<
r
′
<
1, (52)
L
f
≤
C
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
+
β
L
Λ
∂
m
f
≤
C
∫
E
x
,
r
′
1
-
u
2
N
+
1
∇
∂
m
f
u
d
τ
u
.Since (53)
∫
E
x
,
r
′
d
τ
<
∞
,by Hölder’s inequality and Fubini’s theorem, we can obtain (54)
K
p
≤
C
∫
B
∫
E
x
,
r
′
1
-
u
2
N
+
1
∇
∂
m
f
u
d
τ
u
p
d
τ
x
≤
C
∫
B
1
-
u
2
N
+
1
∇
∂
m
f
u
p
d
τ
u
.It follows from Lemma 10 that we see that K
p is bounded. This completes the proof.


## 3.1. Harmonic Bloch Spaces

In this section, we give some characterizations of the spacesB
α which can be viewed as the generalizations of Yoneda’s results into the real unit ball B of R
n.For a continuous functionh in B and 0
≤
γ
≤
1, we write (24)
L
Λ
h
=
h
x
-
h
y
x
-
y
γ
x
,
y
1
-
γ
,
x
,
y
∈
B
,
x
≠
y
.By using the notation, we characterize B
1
,
B
0
1 as follows.Theorem 6.
Letf
∈
H
(
B
),  N
≥
0 be an integer and 0
<
r
<
1. Then f
∈
B
1 if and only if (25)
L
f
=
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
∞
,for all multi-index m with |
m
|
=
N, where α
+
β
=
N
+
1.Proof.
First we prove the sufficiency. Letf
∈
H
(
B
). Then for each multi-index m,  ∂
m
f
∈
H
(
B
). For r
∈
(
0,1
), it follows from [1] that (26)
∇
∂
m
f
x
≤
C
1
-
x
2
∫
E
x
,
r
∂
m
f
y
d
τ
y
.Fixing x
∈
B and replacing ∂
m
f by ∂
m
f
-
∂
m
f
(
x
), we have (27)
1
-
x
2
m
+
1
∇
∂
m
f
x
≤
C
1
-
x
2
m
∫
E
x
,
r
∂
m
f
y
-
∂
m
f
x
d
τ
y
.By Lemma 1, we can deduce that (28)
1
-
x
2
m
+
1
∇
∂
m
f
x
≤
C
1
-
x
2
m
+
1
∫
E
x
,
r
∂
m
f
y
-
∂
m
f
x
x
,
y
d
τ
y
≤
C
∫
E
x
,
r
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
d
τ
y
≤
C
L
f
τ
B
0
,
r
.Since τ
(
B
(
0
,
r
)
)
=
n
∫
0
r
t
n
-
1
(
1
-
t
2
)
-
n
d
t is a constant, we see that (29)
1
-
x
2
N
+
1
∇
∂
m
f
x
<
∞
,for any multi-index m with |
m
|
=
N. Hence Lemma 2 yields that f
∈
B
1.
Now we prove the necessity. Lety
∈
E
(
x
,
r
)
,
x
≠
y. Then for each multi-index m with |
m
|
=
N, we have(30)
∂
m
f
x
-
∂
m
f
y
=
∫
0
1
d
∂
m
f
d
s
s
x
+
1
-
s
y
d
s
≤
∑
k
=
1
n
x
k
-
y
k
∫
0
1
∂
∂
m
f
∂
x
k
s
x
+
1
-
s
y
d
s
≤
n
x
-
y
∫
0
1
∇
∂
m
f
s
x
+
1
-
s
y
d
s
≤
C
x
-
y
∫
0
1
d
s
1
-
x
z
+
1
-
s
y
N
+
1
.By Lemma 1 we infer that there exists λ
>
0 such that 1
-
|
y
|
=
λ
(
1
-
|
x
|
) and (31)
L
Λ
∂
m
f
≤
C
∫
0
1
d
s
s
1
-
x
+
1
-
s
1
-
y
N
+
1
≤
C
1
-
x
2
N
+
1
∫
0
1
d
s
s
+
λ
1
-
s
N
+
1
≤
C
1
-
x
2
α
1
-
y
2
β
.Thus, (32)
L
f
=
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
∞
.So the proof is complete.Theorem 7.
Letf
∈
H
(
B
), N
≥
0 be an integer and 0
<
r
<
1. Then f
∈
B
0
1 if and only if(33)
lim
x
→
1
-
⁡
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
=
0
,for all multi-index m with |
m
|
=
N, where α
+
β
=
N
+
1.Proof.
Sufficiency: assume that (33) holds. Then for any ϵ
>
0, there exists δ
∈
(
0,1
) such that (34)
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
ϵwhenever δ
<
|
x
|
<
1. It follows from an argument similar to that in proof of Theorem 6 that we have (35)
1
-
x
2
N
+
1
∇
∂
m
f
x
≤
C
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
<
C
ϵ
,whenever δ
<
|
x
|
<
1. Hence (36)
lim
x
→
1
-
⁡
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
=
0
,from which we see that f
∈
B
0
1.
Necessity: fort
∈
(
0,1
), let f
t
(
x
)
=
f
(
t
x
). By Lemma 1 and proof of Theorem 6, we see that, for each multi-index m with |
m
|
=
N, (37)
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
-
f
t
≤
C
1
-
ξ
2
N
+
1
∇
∂
m
f
-
f
t
ξ
,
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
t
≤
C
1
-
x
2
α
+
β
1
-
t
2
N
+
1
1
-
t
η
2
N
+
1
∇
∂
m
f
t
ηfor all x
∈
B and ξ
,
η
∈
E
(
x
,
r
). So (38)
L
f
≤
C
1
-
ξ
2
N
+
1
∇
∂
m
f
-
f
t
ξ
+
C
1
-
x
2
α
1
-
y
2
β
1
-
t
2
N
+
1
1
-
t
η
2
N
+
1
∇
∂
m
f
t
η
.First letting |
x
|
→
1
- and then letting t
→
1
-, we obtain the desired result.In the following, by removing the restrictiony
∈
E
(
x
,
r
) in Theorem 6, we obtain the following.Theorem 8.
Letf
∈
H
(
B
), 0
≤
β
<
1, and β
≤
α
<
1
+
β. Then f
∈
B
α if and only if(39)
Q
=
sup
x
,
y
∈
B
,
x
≠
y
⁡
1
-
x
2
β
1
-
y
2
α
-
β
L
Λ
f
<
∞
.Proof.
We only need to prove the necessity since the proof of sufficiency is similar to that in proof of Theorem6. Assume that f
∈
B
α. For x
,
y
∈
B, s
∈
[
0,1
],(40)
1
-
s
x
+
1
-
s
y
2
α
≥
s
1
-
x
2
2
+
1
-
s
1
-
y
2
2
α
≥
s
2
β
1
-
s
2
α
-
β
1
-
x
2
β
1
-
y
2
α
-
β
,which gives (41)
L
Λ
f
≤
C
∫
0
1
d
s
1
-
s
x
+
1
-
s
y
2
α
≤
C
∫
0
1
d
s
s
/
2
β
1
-
s
/
2
α
-
β
1
-
x
2
β
1
-
y
2
α
-
β
≤
C
1
-
x
2
β
1
-
y
2
α
-
β
∫
0
1
d
s
s
β
1
-
s
α
-
β
≤
C
1
-
x
2
β
1
-
y
2
α
-
β
,where the last integral converges since α
<
1
+
β. Thus (42)
1
-
x
2
β
1
-
y
2
α
-
β
L
Λ
f
<
∞
.Similarly, we can prove the following.Theorem 9.
Letf
∈
B
α, 0
≤
β
<
1, and β
≤
α
<
1
+
β. Then f
∈
B
0
α if and only if(43)
lim
x
→
1
-
⁡
sup
x
,
y
∈
B
,
x
≠
y
⁡
ω
1
-
x
2
β
1
-
y
2
α
-
β
L
Λ
f
=
0
.


## 3.2. Harmonic Besov Space

In order to prove our next result, we need the following lemma.Lemma 10.
Letf
∈
H
(
B
). Then f
∈
B
p if and only if (44)
sup
x
∈
B
⁡
1
-
x
2
N
+
1
∇
∂
m
f
x
∈
L
p
B
,
d
τfor all multi-index m with |
m
|
=
N.Proof.
This follows from [10, Theorem 3.7] by letting α
=
p
-
n.Lemma 11.
Leth
∈
H
(
B
) and 0
<
r
<
1. Then there exist constants C
>
0,  r
<
r
′
<
1 such that (45)
sup
y
∈
E
x
,
r
,
x
≠
y
⁡
h
x
-
h
y
x
-
y
≤
C
∫
E
x
,
r
′
∇
h
u
d
τ
u
.Proof.
By Cauchy’s estimates and Lemma1, for each x
∈
B, we have (46)
sup
y
∈
E
x
,
r
,
x
≠
y
⁡
h
x
-
h
y
x
-
y
≤
C
sup
ζ
∈
E
x
,
r
⁡
∇
h
ζ
≤
C
E
x
,
r
′
∫
E
x
,
r
′
∇
h
ζ
d
v
ζ
≤
C
∫
E
x
,
r
′
∇
h
ζ
d
τ
ζfor some r
′
>
r.Now, we come to state and prove the result for harmonic Besov spaces.Theorem 12.
Letf
∈
H
(
B
), N
≥
0 be an integer and 0
<
r
<
1. Then f
∈
B
p if and only if (47)
K
p
=
∫
B
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
p
d
τ
x
<
∞
,for all multi-index m with |
m
|
=
N, where α
+
β
=
N
+
1.Proof.
Letf
∈
H
(
B
). Suppose that (48)
K
p
=
∫
B
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
d
τ
<
∞
.Set(49)
L
f
x
=
lim
y
→
x
⁡
sup
⁡
1
-
x
2
α
1
-
y
2
β
L
Λ
∂
m
f
.It follows from proof of Theorem 6 that we have (50)
1
-
x
2
N
+
1
∇
∂
m
f
x
≤
C
L
f
x
.Since L
f
(
x
)
≤
L
f, we see that (51)
∫
B
1
-
x
2
p
N
+
1
∇
∂
m
f
x
p
d
τ
x
≤
C
∫
B
L
f
p
x
d
τ
x
≤
C
∫
B
L
f
p
d
τ
x
=
C
K
p
,for all multi-index m with |
m
|
=
N. By Lemma 10, f
∈
B
p.
To prove the necessity, we suppose thatf
∈
B
p. By Lemmas 1 and 11, for each multi-index m and r
<
r
′
<
1, (52)
L
f
≤
C
sup
x
∈
B
,
y
∈
E
x
,
r
,
x
≠
y
⁡
1
-
x
2
α
+
β
L
Λ
∂
m
f
≤
C
∫
E
x
,
r
′
1
-
u
2
N
+
1
∇
∂
m
f
u
d
τ
u
.Since (53)
∫
E
x
,
r
′
d
τ
<
∞
,by Hölder’s inequality and Fubini’s theorem, we can obtain (54)
K
p
≤
C
∫
B
∫
E
x
,
r
′
1
-
u
2
N
+
1
∇
∂
m
f
u
d
τ
u
p
d
τ
x
≤
C
∫
B
1
-
u
2
N
+
1
∇
∂
m
f
u
p
d
τ
u
.It follows from Lemma 10 that we see that K
p is bounded. This completes the proof.


## 4. Conclusions

In this paper, we characterize harmonic Bloch and Besov spaces by using the quantitys
u
p
x
∈
B
,
y
∈
E
(
x
,
r
)
,
x
≠
y
(
1
-
|
x
|
2
)
α
(
1
-
|
y
|
2
)
β
|
∂
m
f
(
x
)
-
∂
m
f
(
y
)
|
/
|
x
-
y
|
γ
[
x
,
y
]
1
-
γ
,
x
≠
y. Since |
x
-
y
|
≤
[
x
,
y
], our results can be viewed as the generalizations of Yoneda’s results (see [2]) into real harmonic setting. Furthermore, we obtain a characterization of the space B
α in terms of Q
=
s
u
p
x
,
y
∈
B
,
x
≠
y
(
1
-
|
x
|
2
)
β
(
1
-
|
y
|
2
)
α
-
β
L
Λ
(
f
), where 0
≤
β
<
1
,
β
≤
α
<
1
+
β.



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*Source: 6814326-2016-03-16.xml*