# Lower Estimates for Certain Harmonic Functions in the Half Space

Authors: Gang Xu; Xiaoyu Zhou
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We will give the growth properties of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Levin in a half plane.


---

## Body

## 1. Introduction and Main Theorem

LetR and R
+ be the sets of all real numbers and of all positive real numbers, respectively. Let R
n
(
n
≥
3
) denote the n-dimensional Euclidean space with points x
=
(
x
′
,
x
n
), where x
′
=
(
x
1
,
x
2
,
…
,
x
n
-
1
)
∈
R
n
-
1 and x
n
∈
R. The boundary and closure of an open set D of R
n are denoted by ∂
D and D
¯, respectively. The upper half space is the set H
=
{
(
x
′
,
x
n
)
∈
R
n
:
x
n
>
0
}, whose boundary is ∂
H.For a setE, E
⊂
R
+
∪
{
0
}, we denote {
x
∈
H
:
|
x
|
∈
E
} and {
x
∈
∂
H
:
|
x
|
∈
E
} by H
E and ∂
H
E, respectively. We identify R
n with R
n
-
1
×
R and R
n
-
1 with R
n
-
1
×
{
0
}, writing typical points x
,
y
∈
R
n as x
=
(
x
′
,
x
n
), y
=
(
y
′
,
y
n
), where y
′
=
(
y
1
,
y
2
,
…
,
y
n
-
1
)
∈
R
n
-
1, and putting
(1)
x
·
y
=
∑
j
=
1
n
x
j
y
j
=
x
′
·
y
′
+
x
n
y
n
,
|
x
|
=
x
·
x
,
|
x
′
|
=
x
′
·
x
′
,
|
x
′
|
=
|
x
|
cos
⁡
⁡
θ
,
x
n
=
|
x
|
sin
⁡
θ
(
0
<
θ
≤
π
2
)
.LetB
r denote the open ball with center at the origin and radius r
(
>
0
) in R
n. We use the standard notations u
+
=
max
⁡
⁡
(
u
,
0
) and u
-
=
-
min
⁡
⁡
(
u
,
0
). In the sense of Lebesgue measure d
y
′
=
d
y
1
⋯
d
y
n
-
1 and d
y
=
d
y
′
d
y
n. Let σ denote (
n
-
1
)-dimensional surface area measure and let ∂
/
∂
n denote differentiation along the inward normal into H.The estimate we deal with has a long history which can be traced back to Levin’s estimate of harmonic functions from below (see, e.g., [1, page 209]).Theorem A.
LetA
1 be a constant and let, u
(
z
) be harmonic in the upper half space C
+ and continuous on ∂
C
+. Suppose that
(2)
u
(
z
)
≤
A
1
R
ρ
,
z
∈
C
+
,
R
=
|
z
|
>
1
,
ρ
>
1
,
|
u
(
z
)
|
≤
A
1
,
|
z
|
≤
1
,
Im
z
≥
0
.
Then
(3)
u
(
Re
i
φ
)
≥
-
A
2
A
1
(
1
+
R
ρ
)
sin
-
1
φ
,
Re
i
φ
∈
C
+
,
where A
2 is a constant independent of A
1, R, φ, and the function u
(
z
).Further versions and refinements of Theorem1 may be found in [2, Chapter 1], [3, 4] and in the paper of Krasichkov-Ternovskiǐ [5].In this paper, we will consider functionsu
(
x
) harmonic in H and continuous on H
¯. In what follows we shall denote by M various values which do not depend on K, R
(
=
|
x
|
), θ, and the function u
(
x
).We prove in this note analogous estimates foru
(
x
) in H.Theorem 1.
Suppose that(4)
u
(
x
)
≤
K
R
ρ
(
R
)
,
x
∈
H
,
R
=
|
x
|
>
1
,
ρ
(
R
)
>
1
,
(5)
u
(
x
)
≥
-
K
,
|
x
|
≤
1
,
x
n
≥
0
.
Then
(6)
u
(
x
)
≥
-
M
K
(
1
+
ρ
(
R
)
R
ρ
(
R
)
)
sin
1
-
n
θ
,
where x
∈
H and ρ
(
R
) is nondecreasing on [
1
,
+
∞
).Remark 2.
Ifn
=
2 and ρ
(
R
)
≡
ρ, Theorem 1 is just the result of Theorem A.Theorem 3.
If (4) and (5) hold, then
(7)
u
(
x
)
≥
-
M
K
(
1
+
ρ
(
N
+
1
N
R
)
R
ρ
(
(
(
N
+
1
)
/
N
)
R
)
)
sin
1
-
n
θ
,
where x
∈
H, N
(
≥
1
) is a sufficiently large number, and ρ
(
R
) is defined in Theorem 1.


## 2. Main Lemmas

Carleman’s formula [6] connects the modulus and the zeros of a function analytic in C
+ (see, e.g., [7, page 224]). Nevanlinna’s formula (see [1, page 193]) refers to a harmonic function in a half disk. Ren obtained a generalized Nevanlinna-type formula in a half space and Poisson integral forumla for half balls, resepctively, which play important roles in our discussions.Lemma 4 (see [8]).
IfR
>
1, then one has
(8)
∫
{
x
∈
H
:
|
x
|
=
R
}
u
(
x
)
n
x
n
R
n
+
1
d
σ
(
x
)
+
∫
∂
H
(
1
,
R
)
u
(
x
′
)
(
1
|
x
′
|
n
-
1
R
n
)
d
x
′
=
c
1
+
c
2
R
n
,
where
(9)
c
1
=
∫
{
x
∈
H
:
|
x
|
=
1
}
(
(
n
-
1
)
x
n
u
(
x
)
+
x
n
∂
u
(
x
)
∂
n
)
d
σ
(
x
)
,
c
2
=
∫
{
x
∈
H
:
|
x
|
=
1
}
(
x
n
u
(
x
)
-
x
n
∂
u
(
x
)
∂
n
)
d
σ
(
x
)
.Lemma 5 (see [8]).
LetR
>
1 and let u
(
x
) be a function in B
R
+
=
B
R
∩
H and continuous in B
¯
R
+. Then
(10)
u
(
x
)
=
∫
{
y
∈
H
:
|
y
|
=
R
}
‍
R
2
-
|
x
|
2
ω
n
R
×
(
1
|
y
-
x
|
n
-
1
|
y
-
x
*
|
n
)
u
(
y
)
d
σ
(
y
)
+
2
x
n
ω
n
∫
∂
H
[
0
,
R
)
(
1
|
y
′
-
x
|
n
-
R
n
|
x
|
n
1
|
y
′
-
x
~
|
n
)
×
u
(
y
′
)
d
y
′
,
where x
∈
B
R
+, x
~
=
R
2
x
/
|
x
|
2, x
*
=
(
x
′
,
-
x
n
), and ω
n
=
π
n
/
2
/
Γ
(
1
+
(
n
/
2
)
) is the volume of the unit n-ball in R
n.


## 3. Proof of Theorem1

By applying Lemma4 to u
(
x
), we have
(11)
∫
{
x
∈
H
:
|
x
|
=
R
}
u
+
(
x
)
n
x
n
R
n
+
1
d
σ
(
x
)
+
∫
∂
H
(
1
,
R
)
u
+
(
x
′
)
(
1
|
x
′
|
n
-
1
R
n
)
d
x
′
=
∫
{
x
∈
H
:
|
x
|
=
R
}
u
-
(
x
)
n
x
n
R
n
+
1
d
σ
(
x
)
+
∫
∂
H
(
1
,
R
)
‍
u
-
(
x
′
)
(
1
|
x
′
|
n
-
1
R
n
)
d
x
′
+
c
1
+
c
2
R
n
.It immediately follows from (4) that
(12)
∫
{
x
∈
H
:
|
x
|
=
R
}
u
+
(
x
)
n
x
n
R
n
+
1
d
σ
(
x
)
≤
M
K
R
ρ
(
R
)
-
1
,
∫
∂
H
(
1
,
R
)
∫
‍
u
+
(
x
′
)
(
1
|
x
′
|
n
-
1
R
n
)
d
x
′
≤
M
K
R
ρ
(
R
)
-
1
.Hence from (11) and (12) we have
(13)
∫
{
x
∈
H
:
|
x
|
=
R
}
u
-
(
x
)
n
x
n
R
n
+
1
d
σ
(
x
)
≤
M
K
R
ρ
(
R
)
-
1
,
(14)
∫
∂
H
(
1
,
R
)
u
-
(
x
′
)
(
1
|
x
′
|
n
-
1
R
n
)
d
x
′
≤
M
K
R
ρ
(
R
)
-
1
.And (14) gives
(15)
∫
∂
H
(
1
,
R
)
u
-
(
x
′
)
|
x
′
|
n
d
x
′
≤
2
n
2
n
-
1
∫
∂
H
(
1
,
R
)
u
-
(
x
′
)
(
1
|
x
′
|
n
-
1
(
2
R
)
n
)
d
x
′
≤
M
K
ρ
(
R
)
(
2
R
)
ρ
(
2
R
)
-
1
.Since-
u
(
x
)
≤
u
-
(
x
), by applying Lemma 5 to -
u
(
x
), we have
(16)
-
u
(
x
)
≤
I
1
(
x
)
+
I
2
(
x
)
,
where
(17)
I
1
(
x
)
=
∫
{
y
∈
H
:
|
𝒴
|
=
R
}
‍
R
2
-
|
x
|
2
ω
n
R
×
(
1
|
y
-
x
|
n
-
1
|
y
-
x
*
|
n
)
u
-
(
y
)
d
σ
(
y
)
,
I
2
(
x
)
=
2
x
n
ω
n
∫
∂
H
[
0
,
R
)
(
1
|
y
′
-
x
|
n
-
R
n
|
x
|
n
1
|
y
′
-
x
~
|
n
)
×
u
-
(
y
′
)
d
y
′
.We remark that(18)
1
|
y
-
x
|
n
-
1
|
y
-
x
*
|
n
≤
2
n
x
n
y
n
|
y
-
x
|
n
+
2
,
|
y
-
x
|
n
≥
x
n
n
=
|
x
|
n
sin
n
θ
,
x
∈
H
,
y
n
=
0
.If we put|
x
|
=
r
>
1
/
2 and R
=
2
r in (16), then we finally have from (13) and (18)
(19)
I
1
(
x
)
≤
∫
{
y
∈
H
:
|
𝒴
|
=
R
}
R
2
-
r
2
ω
n
R
2
n
x
n
y
n
ω
n
|
y
-
x
|
n
+
2
u
-
(
y
)
d
σ
(
y
)
≤
M
K
ρ
(
R
)
R
ρ
(
R
)
,
I
2
(
x
)
≤
I
21
(
x
)
+
I
22
(
x
)
,
where
(20)
I
21
(
x
)
=
2
ω
n
x
n
n
-
1
∫
∂
H
(
1
,
R
)
u
-
(
y
′
)
d
y
′
,
I
22
(
x
)
=
2
ω
n
x
n
n
-
1
∫
∂
H
[
0,1
]
u
-
(
y
′
)
d
y
′
.We obtain that(21)
I
21
(
x
)
≤
2
R
n
ω
n
x
n
n
-
1
∫
∂
H
(
1
,
R
)
‍
u
-
(
y
′
)
|
y
′
|
n
d
y
′
≤
M
K
ρ
(
R
)
R
ρ
(
R
)
sin
1
-
n
θ
,
I
22
(
x
)
≤
2
K
ω
n
x
n
n
-
1
∫
∂
H
[
0,1
]
d
y
′
≤
M
K
ρ
(
R
)
sin
1
-
n
θ
,
from (15) and (5), respectively.From (16), (19), and (21), we have for |
x
|
>
1
/
2
(22)
-
u
(
x
)
≤
M
K
ρ
(
R
)
(
1
+
ρ
(
R
)
R
ρ
(
R
)
)
sin
1
-
n
θ
.For|
x
|
≤
1
/
2, we have from (5)
(23)
-
u
(
x
)
≤
K
≤
K
(
1
+
ρ
(
R
)
R
ρ
(
R
)
)
sin
1
-
n
θ
.Thus the conclusion immediately follows from (22) and (23).


## 4. Proof of Theorem3

By modifying (15), we have
(24)
∫
∂
H
(
1
,
R
)
u
-
(
x
′
)
|
x
′
|
n
d
x
′
≤
(
N
+
1
)
n
(
N
+
1
)
n
-
N
n
∫
∂
H
(
1
,
R
)
‍
u
-
(
x
′
)
×
(
1
|
x
′
|
n
-
1
(
(
(
N
+
1
)
/
N
)
R
)
n
)
d
x
′
≤
M
K
ρ
(
N
+
1
N
R
)
(
N
+
1
N
R
)
ρ
(
(
(
N
+
1
)
/
N
)
R
)
-
1
.Then (21), (22), and (23) are replaced accordingly by the following estimates:
(25)
I
21
(
x
)
≤
M
K
ρ
(
N
+
1
N
R
)
(
N
+
1
N
R
)
ρ
(
(
(
N
+
1
)
/
N
)
R
)
-
1
sin
1
-
n
θ
,
-
u
(
x
)
≤
M
K
(
1
+
ρ
(
N
+
1
N
R
)
R
ρ
(
(
(
N
+
1
)
/
N
)
R
)
)
sin
1
-
n
θ
,
-
u
(
x
)
≤
K
≤
M
K
(
1
+
ρ
(
N
+
1
N
R
)
R
ρ
(
(
(
N
+
1
)
/
N
)
R
)
)
sin
1
-
n
θ
.All (16), (19), (25), and (21) give
(26)
u
(
x
)
≥
-
M
K
(
1
+
ρ
(
N
+
1
N
R
)
R
ρ
(
(
(
N
+
1
)
/
N
)
R
)
)
sin
1
-
n
θ
,
from which the conclusion immediately follows.



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*Source: 248576-2014-02-23.xml*