SOME PROPERTIES OF POSITIVE HARMONIC FUNCTIONS
YIFEI PAN AND MEI WANG
Abstract. From a monotonicity property, we derive several results characterizing positive harmonic functions in the unit ball in Rn and positive measures
on the unit sphere S n−1 .
Let B n = {x ∈ Rn : |x| < 1}, n ≥ 2 be the unit ball in Rn and S n−1 =
∂B n be the unit sphere. From a monotonicity property, we derive several results
characterizing positive harmonic functions in B n and positive measures on S n−1 .
This paper has four sections. In the first section, we describe the monotonicity
property for positive harmonic functions in the unit ball. The rest sections utilize the monotonicity result to characterize different aspects of positive harmonic
functions. In Section 2 we derive a decomposition theorem for positive harmonic
functions. Section 3 is on the properties of positive measures on the sphere. In
Section 4, we obtain a precise asymptotic for spherical expansions of harmonic
functions.
1. On monotonicity of positive harmonic functions
It is known [1] that a positive harmonic function u in B n can be uniquely
represented by the Poisson kernel P (x, y) and a positive measure µ on S n−1 as
Z
Z
1 − |x|2
(1.1)
u(x) = P [µ](x) =
P (x, η)dµ(η) =
dµ(η).
n
S n−1
S n−1 |x − η|
The following is the monotonicity theorem for positive harmonic functions. Its
generalization to positive invariant harmonic functions can be found in [6].
Theorem 1.1. Let u be a positive harmonic function in B n , ζ ∈ S n−1 . Then the
function
(1 − r)n−1
u(rζ)
1+r
is decreasing and the function
(1 + r)n−1
u(rζ)
1−r
is increasing for 0 ≤ r < 1.
Proof. Firstly, for x ∈ B n , |x| = r, we claim that
µ
¶
n − (n − 2)r
∂
1 − r2
n + (n − 2)r
(1.2)
−
≤
≤
.
|x − ζ|n
∂r |x − ζ|n
|x − ζ|n
1
2
P
To prove the above claim, write x = rη, η ∈ S n and η · ζ = nj=1 ηj ζj . Notice
∂
that ∂r
|x − ζ|2 = 2(r − η · ζ). Direct calculation yields
µ
¶
∂
1−r2
−2r|x − ζ|2 − n(1 − r2 )(r − η · ζ)
(1.3)
=
,
∂r |x − ζ|n
|x − ζ|n+2
So the right inequality in (1.2) is equivalent to
−2r|x − ζ|2 − n(1 − r2 )(r − r · ζ) ≤ (n + (n − 2)r)|x − ζ|2 ,
or
−(1 − r)(r − r · ζ) ≤ |x − ζ|2 .
Since |x − ζ|2 = r2 − 2rη · ζ + 1, the above is equivalent to
η · ζ ≤ 1,
which is true since η, ζ ∈ S n−1 . Therefore the right inequality in (1.2) is true.
The proof of the left inequality in (1.2) is parallel.
Secondly, we claim that for any positive harmonic function u(x) defined in B n
with |x| = r,
(1.4)
−
n + (n − 2)r
n − (n − 2)r
∂u(x)
≤
u(x) ≤
u(x).
1 − r2
∂r
1 − r2
Because, by (1.2) and the Poisson kernel representation (1.1),
µ
¶
Z
Z
∂u(x)
∂ 1 − |x|2
n + (n − 2)r
=
dµ(ζ) ≤
dµ(ζ)
n
∂r
|x − ζ|
|x − ζ|n
S n−1 ∂r
S n−1
Z
n + (n − 2)r
1 − |x|2
n + (n − 2)r
=
dµ(ζ) =
u(x).
2
n
1−r
1 − r2
S n−1 |x − ζ|
Thus the right hand side of (1.4) is true. The proof of the left hand side is similar.
Finally, to complete the proof, let
ϕ(r) =
Then
(1 − r)n−1
,
1+r
ψ(r) =
ϕ0 (r)
n + (n − 2)r
=−
,
ϕ(r)
1 − r2
(1 + r)n−1
1−r
f or
0 ≤ r < 1.
n − (n − 2)r
ψ 0 (r)
=
.
ψ(r)
1 − r2
Given ω ∈ S n−1 , consider
I(r, ω) = ϕ(r)u(rω),
J(r, ω) = ψ(r)u(rω).
To prove Theorem 1.1, it suffices to show that I(r, ω) is decreasing and J(r, ω) is
increasing in r for 0 ≤ r < 1. By (1.4),
∂u(x)
ϕ0 (r)
n + (n − 2)r n + (n − 2)r
d
(log J(r, ω)) =
+ ∂r ≤ −
+
= 0.
dr
ϕ(r)
u(x)
1 − r2
1 − r2
3
Therefore log I(r, ω) is decreasing in r, and so is I(r, ω). Similarly,
∂u(x)
d
ψ 0 (r)
n − (n − 2)r n − (n − 2)r
(log I(r, ω)) =
+ ∂r ≥
−
= 0.
dr
ψ(r)
u(x)
1 − r2
1 − r2
Hence, J(r, ω) is increasing in r. This completes the proof of Theorem 1.1.
¤
Corollary 1.2. Let u be a positive harmonic function in B n defined by a positive
measure µ as in (1.1). Then
lim (1 − r)n−1 u(rζ) = 2µ({ζ})
r→1
and
u(rζ)
=
r→1 1 − r
Z
lim
S n−1
2
dµ(η).
|ζ − η|n
Proof. Applying Theorem 1.1 to the Poisson kernel we obtain
(
1, ζ = η
(1 − r)n−1
(1 − r)n
P (rζ, η) =
& δ(ζ, η) =
n
1+r
|rζ − η|
0, ζ 6= η
as
r → 1.
By the representation (1.1) and Lebesgue’s dominated convergence theorem,
Z
n−1
n−1
lim (1 − r)
u(rζ) = lim (1 − r)
P (rζ, η)dµ(η)
r→1
r→1
S n−1
¾
½
Z
(1 − r)n−1
P (rζ, η) dµ(η) = 2µ({ζ}).
= lim (1 + r)
lim
r→1
1+r
S n−1 r→1
(1 + r)n−1
(1 + r)n
P (rζ, η) =
increases as r → 1. By Lebesgue’s
1−r
|rζ − η|n
monotone convergence theorem,
Z
u(rζ)
1
lim
= lim
P (rζ, η)dµ(η)
r→1 1 − r
r→1 1 − r S n−1
½
¾
Z
1
(1 + r)n−1
= lim
lim
P (rζ, η) dµ(η)
r→1 (1 + r)n−1 S n−1 r→1
1−r
Z
Z
1
2n
2
=
dµ(η) =
dµ(η).
n−1
n
n
2
S n |ζ − η|
S n |ζ − η|
Similarly,
¤
Corollary 1.3. Let U be the potential function defined by a positive Borel measure
µ on S n−1 as follows:
Z
1
dµ(η),
x ∈ Bn.
U (x) =
n
|x
−
η|
n−1
S
4
Then for any ζ ∈ S n−1 , the function
(1 − r)n U (rζ)
is decreasing for 0 ≤ r < 1.
Proof.
(1 − r)n U (rζ) =
(1 − r)n−1
1+r
Z
S n−1
(1 − r)n−1
1 − r2
dµ(η)
=
u(rζ),
|rζ − η|n
1+r
which is decreasing for 0 ≤ r < 1 according to Theorem 1.1.
¤
Corollary 1.4. Let u be a positive harmonic function, 0 ≤ r0 ≤ r < 1. Then
(1 − r)n−1
(1 − r0 )n−1
max u(x) ≤
max u(x),
(1 + r) |x|=r
(1 + r0 ) |x|=r0
(1 + r)n−1
(1 + r0 )n−1
min u(x) ≥
min u(x).
(1 − r) |x|=r
(1 − r0 ) |x|=r0
Proof. By the maximum principle, there is ζ ∈ S n−1 such that u(rζ) = max|x|=r u(x).
Theorem 1.1 implies
(1 − r)n−1
(1 − r0 )n−1
(1 − r0 )n−1
(1 − r)n−1
0
max u(x) =
u(rζ) ≤
u(r
ζ)
≤
max u(x).
(1 + r) |x|=r
(1 + r)
(1 + r0 )
(1 + r0 ) |x|=r0
Similarly, there is ξ ∈ S n−1 such that u(rξ) = min|x|=r u(x). Theorem 1.1 yields
(1 + r)n−1
(1 + r0 )n−1
(1 + r0 )n−1
(1 + r)n−1
0
min u(x) =
u(rξ) ≥
u(r
ξ)
≥
min u(x).
(1 − r) |x|=r
(1 − r)
(1 − r0 )
(1 − r0 ) |x|=r0
¤
Corollary 1.5. Let u be a positive harmonic function in B n . Let
(1 − r)n−1
max u(x),
r→1
1 + r |x|=r
a = lim
(1 − r)n−1
min u(x).
r→1
1 + r |x|=r
b = lim
Then there exist η, ξ ∈ S n−1 such that
(1 − r)n−1
u(rη),
r→1
1+r
a = lim
(1 − r)n−1
u(rξ).
r→1
1+r
b = lim
Proof. For k = 1, 2, · · · , choose a sequence rk % 1 and ηk ∈ S n−1 such that
(1 − rk )n
(1 − rk )n
max u(x) = lim
u(rk ηk ).
rk →1 1 + rk
rk →1 1 + rk |x|=rk
a = lim
For a fixed r ∈ (0, 1), for any sufficiently large k with rk ≥ r, Theorem 1.1 implies
(1 − r)n−1
(1 − rk )n−1
u(rηk ) ≥
u(rk ηk ).
1+r
1 + rk
5
By the compactness of S n−1 , there exists a subsequence {ηkj } ⊂ {ηk } and η ∈ S n−1
such that ηkj → η as j → ∞. Therefore
(1 − rkj )n−1
(1 − r)n−1
(1 − r)n−1
u(rη) = lim
u(rηkj ) ≥ lim
u(rkj ηkj ) = a,
j→∞
j→∞
1+r
1+r
1 + rkj
thus
(1 − r)n−1
u(rη) ≥ a.
r→1
1+r
lim
On the other hand,
(1 − r)n−1
(1 − r)n−1
u(rη) ≤
max u(x),
1+r
1 + r |x|=r
so
(1 − r)n−1
(1 − r)n−1
u(rη) ≤ lim
max u(x) = a.
r→1
r→1
1+r
1 + r |x|=r
lim
Therefore
(1 − r)n−1
u(rη),
r→1
1+r
The proof for the minimum part is parallel.
a = lim
η ∈ S n−1 .
¤
The rest of the paper is on applications of the above monotonicity property.
2. Decompositions of positive harmonic functions
In this section, we apply the monotonicity property (Theorem 1.1) to obtain a
decomposition of positive harmonic functions in the unit ball.
Theorem 2.1. Let u be a positive harmonic function in B n . If
(1 − r)n−1
u(rζ) → a(ζ) > 0
1+r
as
r → 1,
then
u(x) = a(ζ)
1 − |x|2
+ v(x)
|ζ − x|n
where
v(x) ≥ 0,
∀x ∈ B n ,
The proof of Theorem 2.1 is based on the properties of positive Borel measures on
Rn . In the following proposition, it is sufficient for our purpose to state a property
of positive measures on S n−1 .
6
Proposition 2.2. Let µ be a positive Borel measure on S n−1 . If µ({xo }) >
0, xo ∈ S n−1 , then the set function µ̃ defined as
(
µ(A),
xo 6∈ A;
µ̃(A) =
∀A ∈ B(S n−1 )
µ(A) − µ({xo }), xo ∈ A;
is a positive Borel measure on S n−1 .
Proof. Since µ ≥ 0 is a Borel measure, µ can be decomposed uniquely into two
mutually singular measures [2]:
µ = µc + µd ,
µc ⊥ µd ,
where
µc (A) ≥ 0, ∀A ∈ B(S n−1 );
µc ({x}) = 0, ∀x ∈ S n−1
is a continuous measure,
µd =
X
µ({xk })δ{xk }
µ({xk })>0
is a discrete
measure, the set D = {x ∈ S n−1 : µ({x}) > 0} is at most countable,
P
and xk ∈D µ({xk }) < ∞. Consequently, the set D \ {xo } is countable, the set
function
X
µ̃d =
µd ({xi })δ{xi }
xi ∈D\{xo }
is a positive discrete measure, and µ̃d ⊥ µc . By the uniqueness of decomposition
of Borel measures,
µ̃ = µc + µ̃d = µ − µ({xo })δ{xo }
defines a Borel measure on S n−1 . By the positiveness of µc and µ̃d , µ̃ is a positive
Borel measure.
¤
Proof of Theorem 2.1. By the representation (1.1),
Z
1 − |x|2
u(x) =
dµ(η) > 0, f or
n
S n−1 |η − x|
x ∈ Bn.
n−1 , so the set D = {x ∈ S n−1 : µ({x}) > 0}
µ is a positive Borel measure
P on S
is at most countable and xk ∈D µ({xk }) < ∞. It follows from Corollary 1.2 that
a(ζ) = µ({ζ}), and µ({ζ}) > 0 implies ζ ∈ D. By Proposition 2.2,
µ̃ = µ − µ({ζ})δ{ζ}
7
is a positive Borel measure on S n−1 . Therefore
1 − |x|2
1 − |x|2
v(x) = u(x) − a({ζ})
=
u(x)
−
µ({ζ})
|ζ − x|n
|ζ − x|n
Z
Z
2
2
¡
¢
1 − |x|
1 − |x|
=
dµ(η) −
d µ({ζ})δ{ζ} (η)
n
n
S n−1 |η − x|
S n−1 |η − x|
Z
2
1 − |x|
=
dµ̃(η) ≥ 0,
∀x ∈ B n .
n
|η
−
x|
n−1
S
This completes the proof of Theorem 2.1.
¤
The following corollaries are immediate.
Corollary 2.3. Let u be a positive harmonic function in B n . If
(1 − r)n−1
u(rζ) → a(ζ) > 0
1+r
then
u(x) ≥ a(ζ)
1 − |x|2
,
|ζ − x|n
as
r → 1,
∀x ∈ B n .
Corollary 2.4. Let u be a positive harmonic function in B n . Then there are at
most countably many points ζ ∈ S n−1 such that
lim (1 − r)n−1 u(rζ) > 0.
r→1
Remarks.
(1) There is a measure point of view on the above results. As shown in the
proof of Theorem 2.1, the limit a(ζ) is the point mass µ({ζ}) (by Corollary
1.2).
(2) Theorem 2.1 can be generalized to the countable case: if in the theorem,
the limits a(ζj ) > 0 exist for j = 1, 2, · · · , then
u(x) =
∞
X
j=1
a(ζj )
1 − |x|2
+ v(x),
|ζj − x|n
v(x) ≥ 0,
∀x ∈ B n .
3. Positive measures on the sphere
From (1.1), any positive measure µ on the sphere S n−1 defines a positive harmonic function u. Corollary 1.2 shows that
Z
1
u(rζ)
=2
dµ(η).
lim
r→1 1 − r
|ζ
−
η|n
n−1
S
8
In this section, we use the monotonicity result to characterize
Z
1
(3.1)
G(ζ) =
dµ(η),
ζ ∈ S n−1 .
n
S n−1 |ξ − η|
G(ζ) may be viewed as radial limits of the potential function
Z
1
dµ(η),
x ∈ Bn.
n
|x
−
η|
n−1
S
We investigate a convergence property of G in Theorem 3.1. Proposition 3.3 and
Proposition 3.6 construct measures that induce examples for extreme cases of
G. Corollaries 3.5 and 3.7 give the corresponding results for positive harmonic
functions.
Theorem 3.1. For a positive measure µ on the unit sphere S n−1 , let
G∞ = {ζ ∈ S n−1 : G(ζ) = ∞}.
Then G∞ is a dense Gδ -set in the support of µ. In particular, the closure of G∞
(3.2)
G ∞ = supp(µ).
Before the proof of Theorem 3.1, we need a known result on the decomposition of
Borel measures (Corollary 6.44 in [1]) stated below as a lemma.
Lemma 3.2. If µ is a positive Borel measure on S n−1 and
dµ = dµa.c. + dµs = f dσ + dµs
— the Lebesgue decomposition of µ with respect to the Lebesgue measure σ, then
Z
1 − |x|2
u(x) =
dµ(η)
n
S n−1 |x − η|
has non-tangential limit f (ζ) at almost every ζ ∈ S n−1 .
Proof of Theorem 3.1. First notice that, if U ⊂ S n−1 is an open set and G(ζ) <
∞, ∀ζ ∈ U , then
lim u(rζ) = 0,
∀ζ ∈ U.
r→1
Because, by the definition of G and Corollary 1.2,
(3.3)
G(ζ) = lim
r→1
u(rζ)
,
1 − r2
therefore
G(ζ) < ∞,
∀ζ ∈ U
=⇒
lim u(rζ) = 0,
r→1
∀ζ ∈ U.
Next we show that G∞ is a Gδ set (a countable intersection of open sets). For
m = 1, 2, · · · , let
¾
½
Z
1
dµ(η).
Gm (ζ) =
min m,
|ζ − η|n
S n−1
9
The functions Gm (ζ) ≤ m × µ{S n−1 } and Gm (ζ) is continuous in ζ. By Lemma
3.2 and (3.3),
ζ ∈ S n−1
G(ζ) = sup Gm (ζ) = lim Gm (ζ),
m→∞
m
is well defined. Therefore G(ζ) is lower semi-continuous in ζ. Consequently {ζ ∈
S n−1 : G(ζ) > m} is an open set. By the definition,
G∞ = {ξ ∈ S n−1 : G(ξ) = ∞} =
∞
\
©
ª
ζ ∈ S n−1 : G(ζ) > m ,
m=1
therefore G∞ is a Gδ set.
In the following we prove that G ∞ = supp(µ). If ζ 6∈ supp(µ), then
D(ζ) = distance {ζ, supp(µ)} > 0,
then
Z
G(ζ) =
S n−1 ∩ supp(µ)
1
µ{S n−1 }
dµ(η)
≤
<∞
|ζ − η|n
D(ζ)n
=⇒
ζ 6∈ G∞ .
Hence
G∞ ⊂ supp(µ)
and
G ∞ ⊂ supp(µ) = supp(µ).
To prove G ∞ = supp(µ), we show that G∞ is dense in supp(µ). For any open set
U ⊂ S n−1 , U ∩ G∞ = ∅ and any ζ ∈ U , we have f (ζ) = limr→1 u(rζ) = 0 by
applying (3.3) and using the notation in Lemma 3.2. Consequently
µa.c. (U ) = 0.
From Corollary 1.2,
lim (1 − r)n−1 u(rζ) = 2µ({ζ}) = 2µs ({ζ}).
r→1
Thus
{ζ ∈ S n−1 : µs ({ζ}) > 0} ⊂ {ζ ∈ S n−1 : lim u(rζ) = ∞}.
r→1
Since limr→1 u(rζ) = 0, ∀ζ ∈ U , we obtain µs (U ) = 0. Hence
µ(U ) = µa.c. (U ) + µs (U ) = 0.
We have shown that U ∩ G∞ = ∅ implies µ(U ) = 0, i.e., G∞ is dense in supp(µ),
then (3.2) follows. This concludes the proof of Theorem 3.1.
¤
Proposition 3.3. There exists a discrete measure µ > 0 such that
G(ξ) < ∞,
ξ ∈ S n−1
almost everywhere with respect to the Lebesgue measure on S n−1 .
First we state a geometric property of Lebesgue measures on the sphere as a lemma
(omitting the proof).
10
Lemma 3.4. Let σ be the Lebesgue measure on S n−1 with σ(S n−1 ) = 1. Given
n ≥ 2, there exists a constant C > 0 such that
σ{ξ ∈ S n−1 : |ξ − η| < r} ≤ Crn−1 ,
∀η ∈ S n−1 .
n−1 , and
Proof of Proposition 3.3. Let {ζj }∞
j=1 be a dense countable set ⊂ S

−1
∞
∞
X
X
1
1

δ
,
Cn = 
µ = Cn
2nj {ζj }
2nj
j=1
j=1
be a measure on S n−1 . Then
Z
∞
X
1
1
1
dµ(η)
=
C
,
G(ξ) =
n
n
nj
2 |ξ − ζj |n
S n−1 |ξ − η|
ξ ∈ S n−1 .
j=1
Let
Am
m−1
n
o
[
n−1
−j/3
{ζk }.
= ξ∈S
, |ξ − ζj | > 2
,j ≥ m \
k=1
Then
Am =
∞
\
Bj \
j=m
m−1
[
{ζk },
where
n
o
Bj = ξ ∈ S n−1 , |ξ − ζj | > 2−j/3 ,
k=1
and

S n−1 \ Am = S n−1 \
∞
\

Bj 
[
{ζk }m−1
k=1
j=m
∞
[
¢[
¡ n−1
S
\ Bj
{ζk }m−1
k=1 .
=
j=m
Let σ be the normalized Lebesgue measure on S n−1 with σ(S n−1 ) = 1. Then
σ({ζj }) = 0, ∀j and by Lemma 3.4,
∞
∞
³
´n−1
¢ X
¢ X
¡ n−1
¡ n−1
.
σ S
\ Am ≤
σ S
\ Bj ≤
C 2−j/3
j=m
Hence
Ã
σ S
n−1
\
∞
[
m=1
!
Am
Ã
=σ
j=m
∞
\
©
S
n−1
\ Am
!
ª
m=1
Therefore
Ã
σ
∞
[
m=1
!
Am
= 1.
¡
¢
= lim σ S n−1 \ Am = 0.
m→∞
11
Notice that, for any m > 0,
G(ξ) = Cn
∞
m−1
∞
X
X 1
X
1
1
1
1
< ∞,
≤
C
+C
n
n
nj
n
nj
n
2nj/3
2 |ξ − ζj |
2 |ξ − ζj |
2
j=1
j=1
j=m
∀ξ ∈ Am .
Consequently,
G(ξ) < ∞
a. e.
This completes the proof of Proposition 3.3.
in
σ.
¤
Corollary 3.5. There exists a positive harmonic function u defined by a discrete
measure on S n−1 such that
lim u(rζ) = 0,
a. e.
r→1
ζ ∈ S n−1
with respect to the Lebesgue measure on S n−1 .
Proposition 3.6. There exists a discrete measure µ > 0 such that
G(ξ) = ∞,
∀ξ ∈ S n−1 .
To construct an example for the proof of Proposition 3.6, we use a minimum
energy concept in physics [5].
Definition. Let σ denote the normalized Lebesgue measure on S d of total mass
one. For a given s > 0, the discrete s-energy associated with a finite subset
ωN = {x1 , · · · , xN } of points on S d is
X
1
Ed (s, ωN ) =
.
|xi − xj |s
1≤i<j≤N
The minimal s-energy for N points on the sphere is
Ed (s, N ) = inf Ed (s, ωN ).
ωN ∈S d
Any configuration ωN for which the infimum is attained is called an s-extremal
configuration.
Proof of Proposition 3.6. Based on the above definition, for any N > 1, there
N
n−1 ⊂ Rn such that
exists X N = (xN
1 , · · · , xN ) ∈ S
X
X
1
1
¯
¯n =
min
,
E(N ) =
¯ N
¯
n−1
|xj − xk |n
N
(x1 ,··· ,xN )∈S
1≤j6=k≤N
1≤j6=k≤N ¯xj − xk ¯
i.e. X N is the s-extremal configuration of N points with s = n. For each N > 2,
define a measure
N
1 X
δ{xN }
µN =
i
N
i=1
12
and functions
Z
N
X
1
1
¯
¯ ,
dµN (y) =
n
¯x − xN ¯n
|x − y|
k
N
U (x) = N
S n−1
k=1
1
¯ ,
UiN (x) = U N (x) − ¯
¯x − xN ¯n
i
i = 1, · · · , N.
By the minimization property of X N , for any x ∈ S n−1 ,
X
1
¯
¯n
2 UiN (x) +
¯ N
N¯
1≤j6=k≤N ¯xj − x ¯
k
j,k6=i
X
=
X
1
1≤j≤N
j6=i
¯
¯n +
¯ N
¯
¯xj − x¯
1≤k≤N
k6=i
≥ E(N ) = 2 UiN (xi ) +
1
¯
¯n +
¯
¯
¯x − xN
k ¯
X
1≤j6=k≤N
j,k6=i
X
1≤j6=k≤N
j,k6=i
1
¯
¯n
¯ N
¯
¯xj − xN
k ¯
1
¯n .
¯
¯ N
¯
¯xj − xN
k ¯
Consequently
UiN (x) ≥ UiN (xi ),
i = 1, · · · , N.
By the definition of UiN (x),
N
U (x) =
1
¯n , i = 1, · · · , N
¯
x − xN
i
UiN (x)+ ¯
¯
N
=⇒
N U (x) =
N
X
UiN (x)+U N (x).
i=1
Therefore for x ∈ S n−1 ,
N
N
i=1
i=1
X
X
1
1
1
1 N
U (x) =
UiN (x) ≥
UiN (xi ) =
E(N ).
N
N (N − 1)
N (N − 1)
N (N − 1)
For ε > 0, define a measure
µ = Cε
∞
X
N =2
Then
Ã
1
N 1+ε
Z
G(ξ) =
S n−1
µN ,
with
Cε =
∞
X
N =2
1
!−1
N 1+ε
∞
X
1 1 N
1
dµ(η)
=
C
U (ξ).
ε
n
|ξ − η|
N 1+ε N
N =2
By Theorem 2 in [5],
1
E(N ) ∼ N 2+ n−1
as
N →∞
and there exist a constant C (depending on n only) such that
1
E(N ) ≥ CN 2+ n−1 .
.
13
Consequently, for x ∈ S n−1 ,
G(ξ) = Cε
≥ Cε
≥ Cε
∞
X
N =2
∞
X
N =2
∞
X
1
N 1+ε
1 N
U (ξ)
N
1
1
E(N )
N 1+ε N (N − 1)
1
CN 2+ n−1
N 2+ε (N − 1)
N =2
∞
X
= Cε C
N =2
1
N n−1 −ε
=∞
N −1
f or
µ
ε ∈ 0,
This completes the proof of Proposition 3.6.
1
n−1
¶
.
¤
Corollary 3.7. There exists a positive harmonic function u defined by a discrete
measure on S n−1 such that
u(rζ)
= ∞,
r→1 1 − r
lim
∀ζ ∈ S n−1 .
Remark. The two propositions are actually two examples that demonstrate the
extreme cases of Theorem 3.1. The construction of the examples follow the ideas
in Simon and Wolff ( [7], [8]) in R2 ; the second example utilizes the concept of
minimum energy and extreme configurations in Rn , n ≥ 3.
4. On spherical harmonic expansions
In this section, we apply the monotonicity property in Theorem 1.1 to obtain
an estimate on the spherical expansion of harmonic functions. First we review
some concepts and notations in spherical harmonic analysis used in this section.
Let Hm (S n−1 ) denote the complex vector space of spherical harmonics of degree
m. Hm (S n−1 ) is the restriction to S n−1 of the complex vector space Hm (Rn ) of
homogeneous harmonic polynomials of degree m in Rn . It is known [1] that
µ
¶ µ
¶
n+m−1
n+m−3
n
dim Hm (R ) =
−
,
n−1
n−1
R
and that under the inner product hp, qi = S n−1 p(x)q(x)dσ(x), where dσ is the
normalized Lebesgue measure on S n−1 , there exists an orthogonal decomposition
of the Hilbert space of square-integrable functions on S n−1 ,
n−1
L2 (S n−1 ) = ⊕∞
).
0 Hm (S
14
By the property of finite dimensional Hilbert space, ∀ζ ∈ S n−1 , there exists a
unique Zm (ζ, ·) ∈ Hm (S n−1 ) (the zonal function of pole ζ and order m) such that
Z
pm (ζ) =
pm (η)Zm (ζ, η)dσ(η), ∀pm ∈ Hm (S n−1 ).
S n−1
The above leads to a zonal expansion of the Poisson kernel (Theorem 5.33 in [1])
(4.1)
P (x, ζ) =
∞
X
1 − |x|2
=
Zm (x, ζ),
|x − ζ|n
∀x ∈ B n , ζ ∈ S n−1 .
m=0
Consequently, any complex measure on S n−1 has a spherical harmonic expansion
Z
∞
X
pm (ζ),
pm (ζ) =
Zm (ζ, η)dµ(η) ∈ Hm (S n−1 ), ζ ∈ S n−1 .
S n−1
m=0
If f ∈ L2 (S n−1 ) and dµ = f dσ, then the spherical harmonic expansion for µ
converges to f in L2 (S n−1 ). It is known [3] that if 1 ≤ p < 2 then there is an φ ∈
Lp (S n−1 ) with spherical harmonic expansion divergent almost everywhere. In the
following we provide a precise asymptotics for the spherical harmonic expansion
of complex measures on S n−1 .
Theorem 4.1. Let µ be a complex Borel measure on the unit sphere S n−1 . Let
P
∞
m=0 pm (ζ) be the spherical harmonic expansion of µ. Then
(4.2)
N
X
pm (ζ) ∼
m=0
2
µ({ζ}) N n−1
(n − 1)!
as
N → ∞.
The proof of Theorem 4.1 is an application of the well-known Hardy-Littlewood
Tauberian Theorem [4] stated below.
P
m
Hardy-Littlewood Tauberian Theorem. Assume that ∞
m=0 am x converges
on |x| < 1. Suppose that for some number α ≥ 0,
∞
X
am xm ∼
m=0
A
(1 − x)α
as
x%1
while
mam ≥ −Cmα ,
m ≥ 1,
then
∞
X
m=0
am ∼
A
N α.
Γ(α + 1)
Another known result crucial in our proof is stated as Lemma 4.2, which is a
modified version of Corollary 5.34 in [1].
15
Lemma 4.2. Let µ be a complex measure on S n−1 and u(x) = P [µ](x) as in
(1.1). Then there exist pm ∈ Hm (Rn ), m = 0, 1, 2, · · · such that
u(x) =
∞
X
pm (x),
x ∈ Bn
m=0
and the series converges absolutely and uniformly on compact subsets of B n . Furthermore, there is a positive constant C such that
|pm (x)| ≤ C|µ(S n−1 )|mn−2 |x|m ,
m = 0, 1, 2, · · · .
If x = |x|ζ, then pm (ζ) is given by
Z
pm (ζ) =
Zm (ζ, η)dµ(η) ∈ Hm (S n−1 ).
S n−1
Proof. Our proof of Lemma 4.2 is a modified version of the proof of Corollary 5.34
in [1] in terms of measures. By Theorem 5.33 of [1], the Poisson kernel expansion
by zonal harmonics (4.1) converges absolutely and uniformly on K × S n−1 for
every compact set K ⊂ B n . So for any x ∈ B n ,
Z
∞ Z
∞
X
X
u(x) =
P (x, ζ)dµ(ζ) =
P (x, ζ)Zm (x, ζ)dµ(ζ) =
pm (x)
S n−1
n−1
m=0 S
where
m=0
Z
pm (x) =
S n−1
Zm (x, ζ)dµ(ζ),
x ∈ Bn.
Since pm (x) ∈ Hm (Rn )., so for x = |x|η, we have Zm (x, ζ) = |x|m Zm (η, ζ). Furthermore, it is known that [1]
µ
¶ µ
¶
n+m−1
n+m−3
n
|Zm (η, ζ)| ≤ dim Hm (R ) =
−
n−1
n−1
By Pascal’s triangle,
µ
¶ µ
¶
µ
¶
n + 2m − 2 (n + m − 3)!
n+m−2
n+m−3
1
n
.
dim Hm (R ) =
+
=
(n − 2)!
m
(m − 1)!
n−2
n−2
Applying Stirling’s formula,
dim Hm (Rn )
2
→
n−2
m
(n − 2)!
as m → ∞.
Therefore there exists C = C(n) > 0 such that |Zm (x, ζ)| ≤ Cmn−2 and
¯
¯Z
¯
¯
¯
|Zm (x, ζ)|dµ(ζ)¯¯ ≤ C|µ(S n−1 )|mn−2 |x|m .
|pm (x)| ≤ ¯
S n−1
This completes the proof of Lemma 4.2.
¤
16
Proof of Theorem 4.1. From the above results, for x = rζ, ζ ∈ S n−1 , we can
write
∞
∞
X
X
u(x) = P [µ](x) =
pm (x) =
pm (ζ)rm ,
m=0
m=0
and the last series converges for |r| < 1, by Lemma 4.2 and Corollary 1.2,
∞
X
pm (ζ)rm ∼
m=0
2µ({ζ})
(1 − r)n−1
as r % 1
Taking real and imaginary parts we have
∞
X
Re {pm (ζ)} rm ∼
2 Re {µ({ζ})}
(1 − r)n−1
as r % 1
Im {pm (ζ)} rm ∼
2 Im {µ({ζ})}
(1 − r)n−1
as
m=0
and
∞
X
m=0
r % 1.
By Lemma 4.2, there exists a positive constant C so that
|pm (ζ)| ≤ Cmn−2
It follows that
m Re {pm (ζ)} ≥ −Cmn−1 ,
m Im {pm (ζ)} ≥ −Cmn−1 .
Applying Hardy-Littlewood Tauberian Theorem with α = n − 1 we obtain (4.2).
This completes the proof of Theorem 4.1.
¤
Corollary 4.3. Let µ be a complex Borel measure on S n−1 . If µ({ζ}) > 0 for
some ζ ∈ S n−1 then the spherical expansion series of µ is divergent:
∞
X
pm (ζ) = +∞.
m=0
If µ({ζ}) = 0 then
N
X
pm (ζ) = o(N n−1 ).
m=0
When the dimension of the space n = 2, the spherical expansion corresponds
to Fourier series, and Theorem 4.1 has the following form.
Corollary 4.4. Let µ be a complex Borel measure on S 1 . Let
Z π
∞
X
am eimθ ,
am =
e−imθ dµ(eiθ )
m=−∞
−π
17
be the Fourier series of µ. Then
N
X
am eimθ ∼ 2µ({eiθ })N
as
N → ∞.
m=−N
Proof. In R2 the zonal functions are given by
Zm (eiθ , eiφ ) = eim(θ−φ) + e−im(θ−φ)
for m > 0, and Z0 (eiθ , eiφ ) = 1. So Corollary 4.4 follows from Theorem 4.1.
¤
Acknowledgments. We thank Peter Drganev for the construction of the example
in Proposition 3.3.
References
[1] S. Axler, P. Bourdon and W. Ramey (2001) Harmonic Function Theory, 2nd ed.
Springer, New York.
[2] G. B. Folland, Real Analysis - Modern Techniques and Their Applications, 2nd ed. Willy,
New York, 1994.
[3] C. Kenig, R. Stanton and P. Tomas (1982) Divergence of eigenfunction expansions, J.
Funct. Anal. 46, no. 1, 28-44.
[4] J. Korevaar (2004) Tauberian Theory: a century of developments, Springer.
[5] A. B. Kuijlaars and E. B Saff (1998) Asymptotics for minimal discrete energy. Trans.
Amer. Math. Sci. 350 (2) 523–538.
[6] Y. Pan and M. Wang, Monotonicity of Harnack inequality for positive invariant harmonic
functions submitted.
[7] B. Simon (2005) Orthogonal Polynomials On The Unit Circle: Volume II, Colloquium
Publications (Amer. Mathematical Soc.).
[8] B. Simon and T. Wolff, (1986) Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math. 39(1) 75–90.
Yifei Pan, Department of Mathematical Sciences, Indiana University - Purdue
University Fort Wayne, Fort Wayne, IN 46805-1499
School of Mathematics and Informatics, Jiangxi Normal University, Nanchang,
China
E-mail address: [email protected]
Mei Wang, Department of Statistics, University of Chicago, Chicago, IL 60637
E-mail address: [email protected]