## A symmetry problem of elliptic differential operators in potential theory

### Yu-Ping Wang

Sacred Hearts High School

**Abstract**

This paper is a study of the equation(−_{∆}_{E})^{α}^{2}u(x) = f(x), where(−_{∆}_{E})^{α}^{2} is an
(elliptic pseudo-differential) operator defined by

(−_{∆}_{E})^{−}^{α}^{2}f = ^{1}
Γ(^{α}_{2})

Z_{∞}

0 t^{α}^{2}^{−1}(Ht∗f)(x)_{dt,}
Ht(x) ≡H(_{x, t}) = _{p} ^{1}

(*4πt*)^{n}*η*_{1}*η*_{2}· · ·*η*n

exp −

### ∑

i

x^{2}_{i}
*4η*_{i}t

! ,

*where η*_{1}*, η*_{2},· · ·*, η*nare a set of non-negative numbers that specify the operator. Note
that it is an extension of the fractional Laplacian operator(−_{∆})^{α}^{2}.

In this paper, we construct a solution, noted as J*α*f , by
J*α*f(x) ≡ ^{1}

*β*(*α*)
Z

**R**^{n}

f(y)

|*η*^{−1}· (x−y)|^{n−a}^{dy,}
where|*η*^{−1}· (x−y)|_{is}^{q}_{∑}^{n}_{i} _{η}^{−1}

i (x_{i}−y_{i})* _{, and β}*(

*)*

_{α}^{−1}

_{equals}

*β*(

*α*)

^{−1}= √

^{1}

*η*_{1}*η*_{2}· · ·*η*n

· ^{Γ}(^{n−α}_{2} )
*π*

n
22* ^{α}*Γ(

^{α}_{2})

^{.}

Then if we set f =*χ*_{Ω}*where χ*_{Ω}is the indicator function andΩ is some bounded
domain in**R**^{n}, then for all bounded domainΩ that is invariant under reflection trans-
formation Pm, namely PmΩ=Ω for all m=1, . . . , n, J* _{α}*f ≡J

*(x)satisfies*

_{α}J* _{α}*(x) = J

*(Pmx). The reflection transformation is defined as*

_{α}P_{m}x=P_{m}(x_{1},· · ·, x_{m},· · ·, x_{n}) = (x_{1},· · ·,−x_{m},· · ·, x_{n}),
where m=1, 2, . . . , n.

摘

摘摘要要要: 在這篇報告中, 我們要探討一個方程式(−_{∆}_{E})^{α}^{2}u=_{f ,}_{其中}(−_{∆}_{E})^{α}^{2} 是一個分數
次的橢圓形微分算子, 其定義為

(−_{∆}_{E})^{−}^{α}^{2}f = ^{1}
Γ(^{α}_{2})

Z_{∞}

0 t^{α}^{2}^{−1}(Ht∗f)(x)_{dt,}

Ht(x) ≡H(_{x, t}) = _{p} ^{1}

(*4πt*)^{n}*η*_{1}*η*_{2}· · ·*η*n

exp −

### ∑

i

x^{2}_{i}
*4η*_{i}t

! ,

*其中 η*1*, η*2,· · ·*, η*n是一群決定其算子特性的參數. 而它是從一般的分數次拉普拉斯算
子延伸而得到的.

在報告中, 我們也將找出其一個解, 記為 J*α*f ,為
J* _{α}*f(x) ≡

^{1}

*β*(*α*)
Z

**R**^{n}

f(y)

|*η*· (x−y)|^{n−a}^{dy,}
其中|*η*^{−1}· (x−y)|_{代表}^{q}_{∑}^{n}_{i} _{η}^{−1}

i (x_{i}−y_{i})* _{, 而 β}*(

*)*

_{α}^{−1}

_{等於}

*β*(

*α*)

^{−1}= √

^{1}

*η*_{1}*η*_{2}· · ·*η*n

· ^{Γ}(^{n−α}_{2} )
*π*

n
22* ^{α}*Γ(

^{α}_{2})

^{.}

如果在 J*α*f中令 f =*χ*_{Ω},*其中 χ*_{Ω}是**指示函數, 而 Ω 是一個在 R**^{n}中的有界區域,
則對於所有滿足鏡射變換 Pm的 Ω, 更精確的說, 對於 m =1, . . . , n, 都有 PmΩ =_{Ω,}
J*α*f ≡J(x)^{滿足}

J*α*(x) = J*α*(Pmx).
鏡射變換定義為

Pmx=Pm(x_{1},· · ·, xm,· · ·, xn) = (x_{1},· · ·,−xm,· · ·, xn)
其中 m=1, 2, . . . , n.

**1** **Introdution**

The basic idea of this paper is derived from an important concept in potential theory, the
Riesz potential I*α*f . It is known that Riesz potential is closely related to the fractional
Laplacian operator. It is actually the inverse operator of (−_{∆})^{α}^{2}, namely, u(x) = I* _{α}*f if
(−

_{∆})

^{α}^{2}u = f [1]. Now we let f ≡

*χ*

_{Ω}

*, where χ*

_{Ω}is the indicator function. Then this function denoted as I

*α*(x)in some bounded domainΩ has an interesting property. I

*α*(x) is radially symmetric to a center of a ball. In other words, u(x)|

*=const. if and only ifΩ is a ball [4].*

_{∂Ω}In this paper, we will extend the fractional Laplacian to an elliptic operator

(−_{∆}_{E})^{α}^{2}u= −

### ∑

n j*η*_{j} *∂*^{2}

*∂x*^{2}_{j}

!^{α}_{2}
u,

*where η*_{1}*, η*_{2},· · ·*, η*n > 0 and they are independent of the variables. The fractional ex-
ponent will be defined in the article. We hope to achieve the following things in the
paper:

1. Find the solution of(−∆E)^{α}^{2}u = f , which is denoted by J*α*f(x). Then u(x) = J*α*f
if(−_{∆}_{E})^{α}^{2}u(x) = f(x)_{.}

2. Discuss the integrability of J* _{α}*f .

3. Discuss the symmetry property of the solution of (−_{∆}_{E})^{α}^{2}u = *χ*_{Ω} where Ω is
an n-dimensional ellipsoid centered at origin point and axis parallel to the axis
(x1, x2,· · ·, xn)of some cartesian coordinate system.

4. Consider symmetry property of the solution of another equation (−_{∆}_{E})^{α}^{2}u =
*χ*_{Ω}xi, where i=1, 2,· · ·, n. (The antisymmetric property)

But before doing all this, we will first define some concepts.

**1.1** **Fractional Laplacian**

Now we turn to an important concept of this paper: the fractional Laplacian operator
(−_{∆})^{−}^{α}^{2}. Only the fractional exponent of a positive definite operator can be defined, so
we need to take a minus sign in front of the ordinary Laplacian∆.

One way to define(−∆)^{−}^{α}^{2} is to use the Gamma functionΓ(*α*). We can start from the
fact that for any number A [1, 3]:

A^{−s}= ^{1}
Γ(s)

Z _{∞}

0 t^{s−1}e^{−tA}dt. (1)

If we exchange A to a Laplacian, A7→ −_{∆, s}→ ^{α}_{2} , then we get the definition.

**Definition 1.** The fractional Laplacian(−_{∆})^{−}^{α}^{2} is defined by
(−_{∆})^{−}^{α}^{2} f = ^{1}

Γ(^{α}_{2})
Z _{∞}

0 t^{α}^{2}^{−1}e^{t∆}f dt, (2)

where

e^{∆t}f(x) =Gt∗f(x) =
Z

**R**^{n}Gt(x−y)f(y)dy (3)
and

G(x, t) =Gt(x) = (*4πt*)^{−n}exp(−|x|^{2}

4t ) ≥0. (4)

Gt(x) is called the Gauss-Weierstrass kernel [1]. It is the fundamental solution of
heat equation, and it is not difficult to see why we use it to define e^{t∆}

*∂G*t(x)

*∂t* =_{∆G}_{t}(x) ⇐⇒Gt(x) =e^{∆t}, t>_{0.} _{(5)}
*However, there is a problem in this definition. When α* = −2n, where n is a positive
integer, then the _{Γ(}^{1}*α*

2) = _{Γ(−n)}^{1} part will be zero, and the integral part diverges. We fix
this problem by taking the limit

*α→2n*lim
1
Γ(^{α}_{2})

Z _{∞}

0 t^{α}^{2}^{−1}e^{tA}f dt (6)

where A could be any number, and we find this limit to be A^{n}by using the equation
Γ(s+_{1})

A^{−s+1} =
Z _{∞}

0 t^{s}e^{−At}dt.

So it is reasonable to redefine the fractional Laplacian by taking limits in the definition of it. Now we can define the fractional Laplacian with a positive integer exponent by

(−_{∆})^{n}= lim

*α→2n*(−_{∆})^{−}^{α}^{2}. (7)

**1.2** **Riesz potential**

Riesz potential is closely related to the fractional Laplacian, for it can be seen as an inverse of the fractional Laplacian [1].

**Definition 2.** For any n≥2, 0<*α*<n, and x∈**R**^{n}the Riesz potential is
I* _{α}*f(x) = (K

*∗f)(x) =*

_{α}^{1}

*γ*(*α*)
Z

**R**^{n}

f(y)

|x−y|^{n−a}^{dy,} ^{(8)}
*where γ*(*α*)_{is}

*γ*(*α*) = ^{π}

n22* ^{α}*Γ(

^{α}_{2}) Γ(

^{n−α}_{2}) and

K* _{α}*=

^{1}

*γ*(*α*)|x|* ^{α−n}* (9)

is called the Risz kernel.

We are going to focus on Riesz potential in a compact domainΩ or 1

*γ*(*α*)
Z

Ω

f(y)

|x−y|^{n−a}^{dy}= ^{1}
*γ*(*α*)

Z
**R**^{n}

f(y)

|x−y|^{n−a}^{χ}^{Ω}^{dy,} ^{(10)}
*where χ*_{Ω} is the indicator function. The Riesz potential is a singular integral operator,
so the concept of integrability is important. In other words, the question will be for
f ∈ L^{p}(Ω), and I*α*f ∈ L^{q}(Ω), that p, q satisfy some condition which makes I*α* : L^{p}(Ω) →
L^{q}(_{Ω})a bounded operator.

This property can be seen by the Hardy-Littlewood-Sobolev inequality [2]:

**Theorem 1.** For 0<*α*<n, 1≤p, q≤∞, I*α* : L^{p}(Ω) →L^{q}(Ω)is a bounded operator:

kI*α*fkq ≤Ckfkp, if n
p ≤ ^{n}

q +*α.* (11)

Proof. See [2]. This theorem says that if f ∈ L^{p}(_{Ω}), then for x ∈ _{Ω, I}* _{α}*f(x)converges
absolutely.

We are going to see the relationship between fractional Laplacian and the Riesz po- tential.

**Theorem 2.** For any 0<*α*<n, if u(x)satisfies the equation

(−_{∆})^{α}^{2}u(x) =*ρ*(x), x∈_{R}^{n}, (12)
then u(** _{x}**)can be written as the convolution of K

*α*and f :

u(x) =I_{α}*ρ*(x) = (K* _{α}*∗

*ρ*)(x). (13) Proof. The proof is standard [1]. For convenience, we will recall it in the appendix.

**2** **Derive J**

*α*

**2.1** **Extending the fractional Laplacian**

Before extending the fractional Laplacian, we will start by looking at the normal Lalp- cian first:

∆≡

### ∑

i

*∂*^{2}

*∂x*^{2}_{i}. (14)

We will extend this to

−_{∆}_{E} = −

### ∑

i

*η*_{i} *∂*^{2}

*∂ξ*^{2}_{i}, where(*η*_{1}*, η*_{2},· · ·*η*n>0), (15)
*because it is positive definite, η*1*, η*2*, η*3· · · > *0. For the specified case η*1 =*η*_{2} = · · · =
*η*n=1, it reduces to the ordinary Laplacian.

The question is how to define this operator with a fractional exponent(−∆E)^{α}^{2}. We
can do the same as the original fractional Laplacian:

**Definition 3.** The fractional exponent for the elliptical operator can be written as
(−_{∆}_{E}f)^{−}^{α}^{2} = ^{1}

Γ(^{α}_{2})
Z _{∞}

0 t^{α}^{2}^{−1}e^{∆}^{E}^{t}f dt, (16)
where e^{∆}^{E}^{t}f = (H∗f)(*ξ*)_{, e}^{∆}^{E}^{t}*δ*(*ξ*) ≡H(*t, ξ*)*is the fundamental solution for ∂*tu=_{∆}_{E}u,
and

H(*ξ, t*) = _{p} ^{1}

(*4πt*)^{n}*η*_{1}*η*_{2}· · ·*η*n

exp −

### ∑

i

*ξ*^{2}_{i}
*4η*_{i}t

!

≥0. (17)

(17) can be easily calculated,

*∂H*(*ξ, t*)

*∂t* −_{∆}_{E}H(*ξ, t*) =0(t>0, lim

t→0=*δ*(x)) (18)

and we apply (18) to the Fourier transformation

*∂ b*H(*ξ, t*)

*∂t* +

### ∑

i

*η*i*ξ*^{2}_{i}

!

Hb(*ξ, t*) =0, (19)

Hb(*ξ, t*) =exp −

### ∑

i

*η*_{i}*ξ*^{2}_{i}

!

t=0, (20)

H(*ξ, t*) =

### ∏

i

1 2√

*πtη*_{i}exp − ^{ξ}

2i

*4tη*_{i}

!

= _{p} ^{1}

(*4πt*)^{n}*η*_{1}*η*2· · ·*η*n

exp −

### ∑

i

*ξ*^{2}_{i}
*4η*_{i}t

!

. (21)

**2.2** **The solution for** ∆

E
With all these definitions, we can start to derive the solution for fractional elliptic oper- ator associated to∆E.

**Theorem 3.** The solution for fractional elliptic operator

(−∆E)^{α}^{2}u(x) =*ρ*(x) (22)
can be taken as u(x) = J_{α}*ρ*(x), where

J*α*u(x) ≡ ^{1}
*β*(*α*)

Z
**R**^{n}

f(y)

|*η*^{−1}· (x−y)|^{n−a}^{dy,} ^{(23)}
and|*η*^{−1}(x−y)|stands for

q∑^{n}_{i} *η*^{−1}_{i} (xi−yi)*, and β*(*α*)^{−1}equals

*β*(*α*)^{−1}= √ ^{1}

*η*_{1}*η*_{2}· · ·*η*n · ^{Γ}(^{n−α}_{2} )
*π*

n

22* ^{α}*Γ(

^{α}_{2})

^{.}

^{(24)}

Proof. This theorem can be proved by some simple transformation of the variables.

For the equation

(−_{∆}_{E})^{α}^{2}u(x) =*ρ*(x), (25)
consider a transformation:

xi 7→ √^{ξ}^{i}

*η*_{i}. (26)

Then (25) is transformed to

(−∆)^{α}^{2}u˜(*ξ*) = *˜ρ*(*ξ*). (27)
This is just the ordinary fractional Laplacian, so its solution is just the Riesz potential:

˜

u(*ξ*) = ^{1}
*γ*(*α*)

Z
**R**^{n}

*˜ρ*(*ξ*)

|*ξ*−*ζ*|^{n−a}^{dζ,}^{(28)}

and we can transform it back to xivariable, so the solution will be
u(x) = √ ^{1}

*η*_{1}*η*2· · ·*η*n

· ^{Γ}(^{n−α}_{2} )
*π*^{n}^{2}2* ^{α}*Γ(

^{α}_{2})

Z
**R**^{n}

f(y)

|*η*^{−1}· (x−y)|^{n−a}^{dy.} ^{(29)}

It is also easy to define the solution for some compact domainΩ; simply set
J* _{α}*f ≡

^{1}

*β*(*α*)
Z

Ω

f(y)

|*η*^{−1}(x−y)|^{n−α}^{dy}= ^{1}
*β*(*α*)

Z
**R**^{n}

f(y)*χ*_{Ω}

|*η*^{−1}(x−y)|^{n−α}^{dy} ^{(30)}
*where χ*_{Ω}is the indicator function.

**3** **Integrability**

We have to discuss the integrability of J*α*f . Because J* _{α}*f can be turned to I

*f by changing variables, they should satisfy the same inequality. This has been proven to be true, so we can apply everything in the same way.*

_{α}**Theorem 4.** Let 0 ≤ q ≤ ∞, 0 < *α* < n. Then J*α* : L^{p}(Ω) → L^{q}(Ω)is an continuous
operator

kJ*α*fk_{L}q(Ω) ≤Ckfk_{L}p(Ω), for any 1
p ≤ ^{1}

q+^{α}

n. (31)

Proof. Before proofing this theorem we need some lemmas.

**Lemma 1.** If a function f(x)depends only on|*η*^{−1}x| ≡ r (where the norm stands for
(_{∑}^{n}_{i} *η*^{−1}_{i} x_{i})^{1/2}), then we have the integral equality

Z

**R**^{n} f(x)dx=*ω*n
Z _{∞}

0 f(r)r^{n−1}dr, (32)

*where ω*nis

*ω*n =√

*η*1· · ·*η*n

*π*^{n}^{2}

Γ(^{n}_{2}+1)^{.} ^{(33)}

Proof. We can start from the fact that [3]

Z
**R**^{n}+

f(x_{1}^{b}^{1}+x^{b}_{2}^{2}+ · · · +x^{b}_{n}^{n})x_{1}^{a}^{1}^{−1}x_{1}^{a}^{2}^{−1}· · ·x^{a}_{n}^{n}^{−1}dx

= ^{Γ}(^{a}_{b}^{1}

1)Γ(^{a}_{b}^{2}

2) · · ·Γ(^{a}_{b}^{n}

n)
b_{1}· · ·bnΓ(^{a}_{b}^{1}

1 +_{b}^{a}^{2}

2· · · + ^{a}_{b}^{n}

n)
Z _{∞}

0 f(t)t^{a1}^{b1}^{+}^{a2}^{b2}^{···+}^{an}^{bn}^{−1}dt.

(34)

**R**^{n}_{+}is defined as

**R**^{n}_{+}= {x∈**R**^{n} |x_{1},· · ·, xn>_{0}}. (35)
By setting b1=b2= · · · =bn=2, and a1=a2= · · · =an=1, and a transformation,

xi 7→√

*η*_{i}xi, i=1, 2,· · ·n, (36)
we get

Z
**R**^{n}+

f(|*η*^{−1}x|^{2})dx=√

*η*1· · ·*η*n

*π*^{n/2}
2^{n}Γ(^{n}_{2})

Z _{∞}

0 f(t)t^{n}^{2}^{−1}dt. (37)

Last, consider a change of variable t=r^{2}:
Z

**R**^{n}+

f(|*η*^{−1}x|)dx=√

*η*_{1}· · ·*η*_{n} *π*^{n/2}
2^{n}Γ(^{n}_{2} +1)

Z _{∞}

0 f(r)r^{n−1}dr. (38)
By the symmetry of f(|*η*^{−1}x|), it is easy to check

2^{n}
Z

**R**^{n}+

f(|*η*^{−1}x|)dx=
Z

**R**^{n} f(|*η*^{−1}x|)dx, (39)
then the lemma is proven.

**Lemma 2.** For some 1≤p, q, r≤∞, if they satisfy
1

r +_{1}= ^{1}
p +^{1}

q, (40)

then

*β*(*α*)kJ* _{α}*fk

_{r}≤ kfk

_{p}khk

_{q}, (41) where

h(x, y) ≡h(*η*^{−1}(x−y)) = ^{1}

|*η*^{−1}(x−y)|^{n−α}^{.} ^{(42)}
Proof. First, we set

|J*α*f| = ^{1}
*β*(*α*)

Z

**R**^{n} f(y)h(x, y)dy

≤ ^{1}

*β*(*α*)
Z

**R**^{n}|h(x, y)f(y)|dy

= ^{1}

*β*(*α*)
Z

**R**^{n}|f(y)|^{p}^{r}|f(y)|^{1−}^{p}^{r}|h(x, y)|^{q}^{r}|h(x, y)|^{1−}^{q}^{r}dy. (43)
We can see that,

1
r +^{ 1}

p −^{1}
r

+^{ 1}

q−^{1}
r

= ^{1}

r + ^{1}

pr/(p−r)+ ^{1}

qr/(q−r) =1. (44) Then we can apply the H ¨older inequality to it:

*β*(*α*)|J* _{α}*f| ≤

_{Z}

**R**^{n}|f(y)|^{p}|h(x, y)|^{q}dy

^{1}_{r}

·

_{Z}

**R**^{n}|f(y)|^{p}dy

^{1}_{p}−^{1}_{r}

·

_{Z}

**R**^{n}|h(x, y)|^{q}dy

^{1}_{q}−^{1}_{r}

. (45)

Take both sides to an exponent r, then integrate it by x, and we get

*β*(*α*)^{r}kJ* _{α}*fk

^{r}

_{r}≤

Z

**R**^{n}|f|^{p}dy

Z

**R**^{2n}|h|^{q}dxdy

kfk^{r−p}_{p} khk^{r−q}_{q} = kfk^{r}_{p}khk^{r}_{q} (46)
and the lemma is proven.

**Lemma 3.** For n≥2, 0<*α*<n, one has
Z

Ω

1

|*η*^{−1}(x−y)|^{α}^{dy}≤ ^{n}|En|
n−*α*

|_{Ω}|

|En|

1−^{α}_{n}

, (47)

where Enis

En=√

*η*_{1}· · ·*η*_{n} *π*^{n/2}

Γ(^{n}_{2}+1)^{.} ^{(48)}

It is the volume of a n-dimensional ellipsoid with axes√
*η*_{1},√

*η*_{2},· · ·,√
*η*n.

Proof. First, we set S ∈ _{R}^{n} an n-dimensional ellipsoid centered at x with axes√
*η*_{1}R,

√*η*_{2}R,· · ·,√

*η*_{n}R and each parallel to the axis x1, x2,· · ·, xn of coordinate, and |S| =
EnR^{n}is the volume of S. Then we set|Ω| = |S|, so that R= (|_{Ω}|/|S|)^{1/n}.

Z S

dy

|*η*^{−1}(x−y)|* ^{α}* =
Z

S∩Ω

dy

|*η*^{−1}(x−y)|* ^{α}*+
Z

S−(S∩Ω)

dy

|*η*^{−1}(x−y)|^{α}^{(49)}

and Z

Ω

dy

|*η*^{−1}(x−y)|* ^{α}* =
Z

S∩Ω

dy

|*η*^{−1}(x−y)|* ^{α}*+
Z

Ω−(S∩Ω)

dy

|*η*^{−1}(x−y)|^{α}^{.} ^{(50)}
Because S− (S∩Ω)is inside S,* _{|η}*−1(x−y)|

^{1}

*≤R*

^{α}*; therefore*

^{−α}Z

Ω−(S∩Ω)

dy

|*η*^{−1}(x−y)|* ^{α}* ≤R

*(|*

^{−α}_{Ω}| − |S∩

_{Ω}|). (51) Similarly,Ω− (S∩Ω)is outside S, so that

Z

S−(S∩Ω)

dy

|*η*^{−1}(x−y)|* ^{α}* ≥R

*(|S| − |S∩Ω|) =R*

^{−α}*(|*

^{−α}_{Ω}| − |S∩Ω|), (52) thus we get

Z

S−(S∩Ω)

dy

|*η*^{−1}(x−y)|* ^{α}* ≥
Z

Ω−(S∩Ω)

dy

|*η*^{−1}(x−y)|^{α}^{,} ^{(53)}
or

Z Ω

dy

|*η*^{−1}(x−y)|* ^{α}* ≤
Z

S

dy

|*η*^{−1}(x−y)|* ^{α}* =
Z

_{R}

0 r* ^{−α}*r

^{n−1}|nEn|dr

= ^{R}

*n−α*

n−*α*nEn= ^{n}|En|
n−*α*

|_{Ω}|

|En|

1−^{α}_{n}

. (54)

*Replace χ*_{Ω}f(x)by f(x)in lemma one in (54), and the lemma is proven.

The rest of the proof is obvious. For some r we can let 1

r +^{1}

p =1+^{1}

q (55)

and if 1≤r≤1/(1−*n/α*)then n/p≤n/q+n is satisfied. By Lemma 2

kJ*α*fk ≤*β*(*α*)^{−1}khk_{r}kfk_{p}. (56)
Note that we have replaced f(x)*χ*_{Ω}by f(x), and h(x, y)*χ*_{Ω}by h(x, y)because (57) only
integrates over a bounded domain.

Then by Lemma 3

khkr ≤

nEn

n− (n−*α*)r

^{1}_{r} |_{Ω}|

|En|

^{1}_{r}+^{α}_{n}−1

. (57)

So the theorem is proven.

**4** **The symmetry problem**

We know that for I*α*the solution of(−∆)* ^{α/2}*u=

*χ*

_{Ω}has some very interesting property,

*such as its the volume on ∂Ω is a constant if any only if Ω is a ball [4].*

(−∆E)^{α}^{2}u = *χ*_{Ω} is invariant under some “elliptical rotation” that preserves|*η*^{−1}x|,
just like the(−_{∆})* ^{α/2}*u=

*χ*

_{Ω}is invariant under rotations that preserve|x|, but the same property cannot carry over; that is, J

*(x)|*

_{α}_{Ω}will not be a constant whereΩ is the ellipsoid with axis parallel to the coordinate. It is because not all the transformation that preserves

|*η*^{−1}x| preserves an infinitesimal volume dV in **R**^{n}, (if we see this transformation as
a coordinate transformation, then it means the Jacobian does not equal one) [3], and
therefore J*α*(x)|* _{∂Ω}*does not satisfy this property.

There is a transformation that preserves|*η*^{−1}x| and infinitesimal volume. It is the
reflection transformation Pm(See Definition 4). It is a discrete transformation, so instead
of J*α*(x)|* _{∂Ω}* =const, we will get J

*α*(x)|

*=J*

_{∂Ω}*(Pmx)|*

_{α}*. (See Theorem 5.)*

_{∂Ω}**Definition 4.** We are going to introduce the reflection transformation Pm:**R**^{n} →**R**^{n}.
Pmx=Pm(x_{1},· · ·, xm,· · ·, xn) = (x_{1},· · ·,−xm,· · ·xn), (58)
where m=1, 2, . . . , n.

For an n-dimensional ellipsoid with axis√
*η*1,√

*η*2,· · ·,√

*η*n, and each parallel to
the axis of the coordinate(x1, x2,· · ·, xn), which will be noted asΩ is symmetric under
reflection transformation. That is, for a x∈_{Ω, then P}_{m}x∈Ω, and for some x∈* _{∂Ω, then}*
Pmx∈

*∂Ω.*

**Theorem 5.** Let

u(x) ≡ ^{1}
*β*(*α*)

Z Ω

dy

|*η*^{−1}(x−y)|* ^{n−α}* = J

*α*, (59) then there is a property of u(x)|

*.*

_{∂Ω}u(x)|* _{∂Ω}*=u(Pmx)|

*, for all n=*

_{∂Ω}_{1, 2,}· · ·, n. (60)

For convenience, we need to use a different kind of coordinate instead of the ordinary Cartesian coordinate.

**Definition 5.** We are going to define an elliptical coordinate (*ρ, φ*_{1}· · ·*φ*_{n−1})with the
center at some point p.

x1−p1 = √

*η*_{1}*ρcos φ*1

x2−p2 = √

*η*_{2}*ρsin φ*1*cos φ*2

...

x_{n−1}−p_{n−1} = √

*η*_{n−1}*ρsin φ*_{1}· · ·*sin φ*n−2*cos φ*_{n−1}
xn−pn = √

*η*n*ρsin φ*_{1}· · ·*sin φ*_{n−2}*sin φ*_{n−1}
We can set that p∈*∂Ω.*

Then another coordinate(*r, θ*1,· · ·*θ*_{n−1})at a point p^{0}.
x_{1}−p^{0}_{1} = √

*η*_{1}*r cos θ*_{1}
x2−p^{0}_{2} = ^{p}*η*2*r sin θ*_{1}*cos θ*2

...

x_{n−1}−p^{0}_{n−1} = √

*η*_{n−1}*r sin θ*_{1}· · ·*sin θ*n−2*cos θ*_{n−1}
xn−p^{0}_{n} = √

*η*n*r sin θ*_{1}· · ·*sin θ*_{n−2}*sin θ*_{n−1}
Then we can set that p^{0} =Pmp∈_{∂Ω.}

With this coordinate, we shall define a subset in**R**^{n}by

*τ*_{l}^{(k)}= {x ∈_{Ω}|2^{l−1}_{k} ≤*ρ*<2_{k}^{l}}, 1≤l ≤k. (61)
It is easy to check out that

[ 1≤l≤k

*τ*_{l}^{(k)}=_{Ω.} _{(62)}

We will do the same to coordinate(*r, θ*_{1}· · ·*θ*_{n−1})_{.}

*τ*_{l}^{0(k)} = {x∈Ω|2^{l−1}_{k} ≤r<2^{l}_{k}}, 1≤l≤k (63)
and

[ 1≤l≤k

*τ*_{l}^{0(k)} =_{Ω.} (64)

**Lemma 4.** For any k and any 1≤l≤k, it satisfies

|*τ*_{l}^{(k)}| = |*τ*

0(k)

l |. (65)

Proof. For any x∈*τ*_{l}^{(k)}it satisfies the condition

2l−1

k

2

≤ (x_{1}−p_{1})^{2}

*η*_{1} +(x2−p2)^{2}

*η*_{2} + · · ·(xm−pm)^{2}
*η*m

+ · · · + (xn−1−pn−1)^{2}
*η*n−1

+(xn−pn)^{2}
*η*n

≤

2l

k

2 (66)

and

x∈_{Ω.} (67)

If we transform any x ∈ *τ*_{l}^{(k)} with the reflection transformation Pm , then Pmx ≡ x^{0}
satisfies

2l−1

k

2

≤ (Pmx1−p1)^{2}

*η*_{1} +(Pmx2−p2)^{2}
*η*2

+ · · ·(Pmxm−pm)^{2}
*η*m

+ · · · + (Pmxn−pn)^{2}
*η*n

≤

2l

k

2 (68)

or

2l−1

k

2

≤ (x1−p1)^{2}

*η*_{1} +(x2−p2)^{2}
*η*2

+ · · ·(xm+pm)^{2}
*η*m

+ · · · + (x_{n−1}−p_{n−1})^{2}

*η*_{n−1} +(xn−pn)^{2}
*η*n

≤

2l

k

2

,

(69)

and x^{0}∈Ω. This is exactly the condition that satisfies for any x^{0} ∈*τ*_{l}^{0(k)}. Thus,

Pm*τ*_{l}^{(k)} =*τ*_{l}^{0(k)}. (70)

Since the reflection transformation preserves the volume, so that

|*τ*_{l}^{(k)}| = |*τ*_{l}^{0(k)}|. (71)

*We know that τ*_{l}^{(k)} *and τ*_{l}^{0}^{(k)} approach to zero as k approaches to infinity. But how
exactly and how rapidly it approaches to zero, we can see it by Lemma 5.

**Lemma 5.** For any integer k, and some 1≤l≤*k the volume of τ*_{l}^{(k)}*and τ*^{0}^{(k)}satisfy

|*τ*_{l}^{(k)}| ≤C(l)k^{−n}, (72)

|*τ*

0(k)

l | ≤C(l)k^{−n}, (73)

where C is a constant independent of k but dependent to l.

Proof. For convenience, we define

*σ*_{l}^{(k)} = {x∈**R**^{n}|2^{l−1}_{k} ≤*ρ*<2^{l}_{k}} (74)
and

*σ*

0(k)

l = {x∈_{R}^{n} |2^{l−1}_{k} ≤r<2_{k}^{l}}. (75)
where 1 ≤ l ≤ *k. By the definition of (74) and (75), we can see that τ*_{l}^{(k)} ⊆ *σ*_{l}^{(k)}, and
*τ*

0(k)⊆*σ*

0(k)

l ; therefore,|*τ*_{l}^{(k)}| ≤ |*σ*_{l}^{(k)}|, and|*τ*

0(k)
l | ≤ |*σ*

0(k)

l |*. Since the volume of σ*_{l}^{(k)}and
*σ*

0(k)

l can be computed

|*σ*_{l}^{(k)}| =
Z

**R**^{n}*χ*

*σ*_{l}^{(}^{k}^{)}dx=*ω*_{n}
Z ^{l}

k l−1

k

r^{n−1}dr= ^{ω}^{n}
n−1

l k

n

−^{ l}−1
k

n

. (76)

*We have used Lemma 1 in Equation (76). The ω*n has been defined in (33). Therefore,
we can see that

|*τ*_{l}^{(k)}| ≤ |*σ*_{l}^{(k)}| = ^{ω}^{n}
n−1

l k

n

−^{ l}−1
k

n

≡C(l)k^{−n}, (77)
where C(l)equals

C(l) = ^{ω}^{n}

n−1[l^{n}− (l−1)^{n}]. (78)
*The case for τ*_{l}^{0}^{(k)}can be proven in the same way.

Now, we can divide the function J* _{α}*(x)into
J

*(x) =*

_{α}^{1}

*β*(*α*)
Z

Ω

dy

|*η*^{−1}(x−y)|^{n−α}^{(79)}

=

### ∑

1≤l≤k Z

*τ*_{l}^{(}^{k}^{)}

dy

|*η*^{−1}(x−y)|* ^{n−α}* ≡

### ∑

1≤l≤k

j^{(l)}_{α}

=

### ∑

1≤l≤k Z

*τ*_{l}^{0 (}^{k}^{)}

dy

|*η*^{−1}(x−y)|* ^{n−α}* ≡

### ∑

1≤l≤k

j_{α}^{0}^{(l)},
where

j_{α}^{(l)}(x) ≡
Z

*τ*_{l}^{(}^{k}^{)}

dy

|*η*^{−1}(x−y)|^{n−α}^{,} ^{(80)}
and

j^{0}_{α}^{(l)}(x) ≡
Z

*τ*_{l}^{0 (}^{k}^{)}

dy

|*η*^{−1}(x−y)|^{n−α}^{.} ^{(81)}
We are going to define an approximation of j^{(l)}

j^{(l)}*appx.α*(x) = |*τ*_{l}^{(k)}| ^{1}

|*η*^{−1}(x−y)|^{n−α}^{,} ^{(82)}
j^{0}_{appx.α}^{(l)} (x) = |*τ*

0(k)

l | ^{1}

|*η*^{−1}(x−y^{0})|^{n−α}^{,} ^{(83)}
for some y∈*τ*_{l}^{(k)}and y^{0}∈*τ*

0(k) l .

**Lemma 6.** For any integer k and 1≤l≤k it satisfies

|j^{(l)}* _{α}* (p) −j

^{(k)}

*appx.α*(p)| ≤C

^{0}(l)k

*(84) and*

^{−α}|j^{0}_{α}^{(l)}(p^{0}) −j^{0}_{appx.α}^{(k)} (p^{0})| ≤C^{0}(l)k* ^{−α}*, (85)
where C

^{0}(l)is independent of k.

Proof. First, it is obvious that

|*τ*_{l}^{(k)}| min

*x∈τ*_{l}^{(}^{k}^{)}

1

|*η*^{−1}(p−y)|* ^{n−α}* ≤ j

^{(l)}

*(p), j*

_{α}^{(l)}

*appx.α*(p) ≤ |

*τ*

_{l}

^{(k)}| max

*x∈τ*_{l}^{(}^{k}^{)}

1

|*η*^{−1}(p−y)|^{n−α}^{,} ^{(86)}
where

max

*y∈τ*_{l}^{(}^{k}^{)}

1

|*η*^{−1}(p−y)|* ^{n−α}* =

^{1}

(2^{l−1}_{k} )^{n−α}^{,} ^{(87)}

and

min

*y∈τ*_{l}^{(}^{k}^{)}

1

|*η*^{−1}(p−y)|* ^{n−α}* =

^{1}

(2_{k}^{l})^{n−α}^{.} ^{(88)}

So for a sufficiently large k, it can satisfy

max

*y∈τ*_{l}^{(}^{k}^{)}

|*τ*_{l}^{(k)}|

|*η*^{−1}(p−y)|* ^{n−α}* − min

*x∈τ*_{l}^{(}^{k}^{)}

|*τ*_{l}^{(k)}|

|*η*^{−1}(p−y)|* ^{n−α}* = |

*τ*

_{l}

^{(k)}|

(2^{l−1}_{k} )* ^{n−α}* − |

*τ*

_{l}

^{(k)}|

(2^{l}_{k})^{n−α}^{,} ^{(89)}
so that

|j^{(l)}* _{α}* (p) −j

^{(k)}

*appx.α*(p)| ≤ |

*τ*

_{l}

^{(k)}|

^{1}

(2^{l−1}_{k} )* ^{n−α}* −

^{1}(2

^{l}

_{k})

^{n−α}!

≡C^{0}(l)k* ^{−α}*. (90)

We have used Lemma 5 in this equation and C^{0}(l)equals

C^{0}(l) =C(l)2* ^{α−n}*·

^{}(l−1)

*−l*

^{α−n}*. (91) Then the theorem is proven.*

^{α−n}Of course, we can basically do the same with|*τ*_{l}^{0(k)}|. Then we can get

|j^{0}_{α}^{(l)}(p^{0}) −j^{0}*appx.α*^{(k)} (p^{0})| ≤C^{0}(l)k* ^{−α}*. (92)
Note that C

^{0}(l)increases as l increases, and by (91) and (78), we can see that C

^{0}(l)in- creases in the order of k

^{n−1}·k

*=k*

^{α−n−1}

^{α−2}Now, back to the main theorem, we can see that j*appx.α*^{(k)} (p) = j^{0}*appx.α*^{(k)} (p^{0}) because
by Lemma 4|*τ*_{l}^{(k)}| = |*τ*

0(k)

l |, and * _{|η}*−1(p−y)|

^{1}

^{n}

^{−}

*=*

^{α}^{1}

*|η*^{−}^{1}(p^{0}−y^{0})|^{n}^{−}* ^{α}* for some y ∈

*τ*

_{l}

^{(k)}and Pmy=y

^{0}∈

*τ*

0(k)

l , therefore

|j^{0}_{α}^{(l)}(p^{0}) −j_{α}^{(l)}(p)| ≤ |j_{α}^{0}^{(l)}(p^{0}) −j^{0}_{appx.α}^{(k)} (p^{0})| + |j_{α}^{(l)}(p) −j^{(k)}* _{appx.α}*(p)| ≤

_{2C}

^{0}(l)k

*, (93)*

^{−α}thus, for sufficiently large k

|J* _{α}*(p) −J

*(p*

_{α}^{0})| =

### ∑

k lj^{(l)}* _{α}* (p) −j

^{0}

_{α}^{(l)}(p

^{0})

≤

### ∑

k l|j_{α}^{0}^{(l)}(p^{0}) −j_{α}^{(l)}(p)|

≤

### ∑

k lC^{0}(l)k* ^{−α}* ≤kC

^{0}(k) ·k

*.*

^{−α}(94)

Since kC^{0}(k)increases in the order of k* ^{α−1}*, so (94) will decrease in the order of k

^{−1}as k

approaches to infinity. So the theorem is proven.

**4.1** **Generalization**

In Theorem 5, we have assumedΩ to be an n-dimensional ellipsoid centered at the ori- gin point and it has axis of√

*η*_{1},√

*η*_{2}· · ·√

*η*n each parallel to the coordinate(x1· · ·xn).
But this assumption is superfluous, for all we need is the restriction forΩ is PmΩ =_{Ω}
and Ω is bounded. From (66) to (70) we can see that Lemma 4 still holds under this
restriction, and therefore, so does in Theorem 5.

Another assumption that is superfluous is that we only consider p∈ _{∂Ω and P}_{m}p∈

*∂Ω. That is, we only consider J**α*(x)under the restriction J*α*(x)|* _{∂Ω}*. We will extend it to
any point p∈

_{R}^{n}and p

^{0}=Pmp∈

_{R}^{n}.

We will redefine the coordinate(*ρ, φ*1· · ·*φ*n−1) and (*r, θ*1,· · ·*θ*n−1)in Definition 5
basically in the same way but this time the coordinate will be centered at any point p
and Pm*p which is not necessary on ∂Ω.*|*τ*_{l}^{(k)}|and|*τ*

0(k)

l |are now written as

*τ*_{l}^{(k)}= {x∈Ω|2^{l−1}_{k} ≤*ρ*<2_{k}^{l}}, kmin≤l≤kmax (95)
and

*τ*_{l}^{0(k)} = {x∈_{Ω}|2^{l−1}_{k} ≤r<2_{k}^{l}}, kmin≤l≤kmax, (96)
where kmax is defined as∀l > kmax*, τ*_{l}^{(k)} = _{∅. Since P}_{m}*τ*_{l}^{(k)} = *τ*

0(k)

l , so that∀l > kmax,
*τ*

0(k)

l =_{∅. Such k}_{max}exists because of the boundedness ofΩ. Similarly, kminis defined
as∀l<k_{min}*, τ*_{l}^{(k)}=∅. If such kmindoes not exist, then set k_{min}=1.

By this definition, we can get

Ω= ^{[}

k_{min}≤l≤k_{max}

*τ*_{l}^{(k)}= ^{[}

k_{min}≤l≤k_{max}

*τ*

0(k)

l , (97)

and therefore,

J*α*(x) = ^{1}
*β*(*α*)

Z Ω

dy

|*η*^{−1}(x−y)|^{n−α}^{(98)}

=

### ∑

k_{min}≤l≤kmax

Z
*τ*_{l}^{(}^{k}^{)}

dy

|*η*^{−1}(x−y)|* ^{n−α}* ≡

### ∑

k_{min}≤l≤kmax

j^{(l)}_{α}

=

### ∑

k_{min}≤l≤k_{max}
Z

*τ*_{l}^{0 (}^{k}^{)}

dy

|*η*^{−1}(x−y)|* ^{n−α}* ≡

### ∑

k_{min}≤l≤k_{max}

j_{α}^{0}^{(l)}.

The definition of j^{(l)}*α* (x)_{, j}_{α}^{0}^{(l)}(x)_{, j}_{appx.α}^{(l)} (x)_{, and j}^{‘(l)}* _{appx.α}*(x) are still the same, so that
we can see Theorem 5 still holds.

The generalization of Theorem 5 is:

**Theorem 6(generlization). Let**
u(x) ≡ ^{1}

*β*(*α*)
Z

Ω

dy

|*η*^{−1}(x−y)|* ^{n−α}* = J

*α*. (99) For all bounded domainΩ that satisfies Ω=PmΩ

u(x) =u(Pmx)_{.} _{(100)}

**4.2** **The antisymmetric property**

In the equation of (−_{∆}_{E})^{α}^{2}u(x) = f(x), we have set f(x) = *χ*_{Ω} and found out that
the solution, noted as J* _{α}*(x), has the symmetric property. Now, if we replace the func-
tion f(x) by another function g(x) =

*χ*

_{Ω}xi, where 1 ≤ i ≤ n, then the solution for (−

_{∆}

_{E})

^{α}^{2}u(x) =g(x), noted as J

*g(x) =J*

_{α}*(x)will satisfy another property.*

_{α}**Theorem 7.** Let

u(x) ≡ ^{1}
*β*(*α*)

Z Ω

xidy

|*η*^{−1}(x−y)|* ^{n−α}* =J

*(x), (101) then there is a property of u(x)|*

_{α}_{Ω}.

u(x)|* _{∂Ω}*=u(Pmx)|

*,for m6=i, (102) u(x)|*

_{∂Ω}*= −u(Pmx)|*

_{∂Ω}*,for m=i. (103) WhereΩ is an n-dimensional ellipsoid centered at origin point and axis parallel to the coordinate(x1, x2· · ·xn).*

_{∂Ω}Proof. First, we will redefine, j^{(l)}* _{α}* , j

^{0}

_{α}^{(l)}as j

_{α}^{(l)}(x) ≡

Z
*τ*_{l}^{(}^{k}^{)}

x_{i}dy

|*η*^{−1}(x−y)|^{n−α}^{(104)}

and

j^{0}_{α(x)}^{(l)} ≡
Z

*τ*_{l}^{0 (}^{k}^{)}

xidy

|*η*^{−1}(x−y)|^{n−α}^{.} ^{(105)}

*Since τ*_{l}^{(k)}*and τ*_{l}^{0}^{(k)}are the same as (61) and (63). so that

J* _{α}*(x) =

^{1}

*β*(

*α*)

Z Ω

xidy

|*η*^{−1}(x−y)|^{n−α}^{(106)}

=

### ∑

1≤l≤k Z

*τ*_{l}^{(}^{k}^{)}

xidy

|*η*^{−1}(x−y)|* ^{n−α}* ≡

### ∑

1≤l≤k

j^{(l)}_{α}

=

### ∑

1≤l≤k Z

*τ*_{l}^{0 (}^{k}^{)}

x_{i}dy

|*η*^{−1}(x−y)|* ^{n−α}* ≡

### ∑

1≤l≤k

j^{0}_{α}^{(l)}.

*Now we are going to do something different. Define a subset in τ*_{l}^{(k)},

*π*^{(k)}_{ml} = {x ∈*τ*_{l}^{(k)}|√

*η*_{i}^{m−1}_{k} ≤x_{i} <√

*η*_{i}^{m}_{k}}. (107)

Where m is an integer ranges from−k+1 to k. Similarly, we define
*π*

0(k)

ml = {x ∈*τ*

0(k) l |√

*η*_{i}^{m−1}_{k} ≤x_{i}<√

*η*_{i}^{m}_{k}}. (108)

By (107) and (108), we can see that Pm*π*_{ml}^{(k)} = *π*

0(k)

ml for m 6= i and Pm*π*^{(k)}_{ml} = *π*

0(k)

−m+1l for
m=i, therefore,|*π*_{ml}^{(k)}| = |*π*

0(k)

ml |for m6=i, and|*π*^{(k)}_{ml}| = |*π*

0(k)

−m+1l|for m=i.

We can see that|*π*^{(k)}_{ml}|decay to zero as k approach to infinity, and Lemma 7 told that
who rapidly does it approaches to zero.

**Lemma 7.** For any−k+1≤m≤k,|*π*^{(k)}_{ml}|and|*π*

0(k)

ml |satisfy

|*π*_{ml}^{(k)}| ≤Ck^{−n−1} (109)

and

|*π*

0(k)

ml | ≤Ck^{−n−1}, (110)

where C is a constant independent of k.

Proof. Set

|*π*|max≡max{|*π*^{(k)}_{−k+1l}|,|*π*_{−k+2l}^{(k)} | · · · |*π*^{(k)}_{k−1l}|,|*π*_{kl}^{(k)}|} (111)
and

|*π*|_{min}≡min{|*π*^{(k)}_{−k+1l}|,|*π*^{(k)}_{−k+2l}| · · · |*π*_{k−1l}^{(k)} |,|*π*_{kl}^{(k)}|}. (112)
Notice that,

2k|*π*|_{min}≤

### ∑

k m=−k+1|*π*^{(k)}_{ml}| = |*τ*_{l}^{(k)}|. (113)