放物型微分作用素について

On

　

Differendal

　

Operators

　

of

　

Parabolic

　

Type

Kyui

七

i

　

SAKUMA

放

_物

型

微 分

作

用

素

に

つ

い

て

佐

　

久

　

間

　

求

　

一

＊

Contents

§

1．

　

Introduction．

§2

．

　Differential　operator ∂_！∂r

．

§

3．

　 Reductions 　of　

A

　and ▽ to ∂_！∂r

．

§

4．

　

Definition

　of　_pZl

．

§5

．

　D1fferential　operator 　Ap

．

§

6．

　

Differential

　operator ρ

．

§7

．

Decomposition 　of _∬

．

目 次 §

L

　 導入 §

2．

微分 作 用素 ∂ノ∂r §

3．

∠ お よ び ▽ の ∂

1

∂r へ の誘導 §4

．

pZl の定 義 §5

。

微 分 作 用 素

Ap

§

6．

微 分作 用 素

9

§7

． H

の分解 ‘‘ 放 物 型微分 作用素にっい て’， の 要旨

　

こ の論 文で は主 として二つ の微 分 作 用 素 砺，ρ に つ い て考察 する。 た だ し

p

は正 の奇 数と す る

。Ap

は変 数 係 数の 放 物 型 作用素で

，

準 constant 　strength で ある

。

つ

ま り

，

p＞

1

な ら ば

，

　

Ap

は n 次 元の ク

ー

ク リッ ド空間

En

内で は

，

どんな直交座

標 系に対 し ても constant 　strength で は ない が， 適当な変数変換に よ り constant

strength _な作用素に帰着する。 （219 _{は リ}

ー

_{マ ソ} 多様 体

Mn

上 の微分 形式に対する微 分 作 用 素であり， 互に随 伴な定 係 数の放 物 型 作用 素 ∠，

A

’ の積， す なわ ち， 準楕 円 型 作用素 AA ’ の

一

般 化 と考え られ る。 尚

n

自 身 を 直

er

　

Mn

上に 拡 張 する ことは困 難 の ように思 わ れる。 順を追っ て次に内容を説明する。 　§

1

は導入 で， 以 下 の §の紹 介である。

　

§

2

で は基本解の台 が 半 空 間内に 入る微 分 作 用素の

一

つ の族が

，

ある条 件 を 満 足す る ような 碑 ∂西

k ＝1

，

2

，

…

で示 さ れ る事をの べ る。 　§

3

で は基 本 解の台 が半空 間内に入 る作用素とし て知られてい る二つ の_{作用素， す} な わ ち

，

熱の放物型 作 用 素

A

お よび波 動の双曲 型 作 用 素 ▽ が ∂_！∂7 に帰 着 するこ と を 示す

。

　 §

4

で は双曲型 作用素▽ の基 本 解を表 現する た め の

，

よく知られた

M ．Riesz

〔s｝ _の 超 函 数 Z‘に倣っ て， p

＝

奇 数＞0 の と き， 超 双 曲 型 作 用 素□の 基 本 解 を 表 現 するため ＊_{教 授　 　数 学 科}

Kyuiti

　

SAKUMA

に役立つ _{同様な超函数　pZt を考 察}し， その 明瞭な表 現を求め るe こ の基本解の表 現は de　Rhamte や その _他の表 現と多 少異 る 種類の もの で あ る

。

　

§

5

で は

p

≡ 奇

数

＞

0

の とぎ ， pZt を 用い て

ZipX

＝

δの解 炉

X2

（

1

）， な らびに

，一

般 に （Ap）ltx

＝

δ の解 pX2h （

h

）を求め る。　pX2h （

h

），　

h ＝

1，　2，

…

は半 空 間 内に台を もつ 超 函 数の

一

つ の族 を 形 成 する。 更に （Ap ’ ）i

・

g＝

f

，

　

fEI

）

，

ω _の解

p

_を p逓ん（

h

）に よ り 表 現するこ と につ い て考 察 する

。

な お

A

，’ は形式 的に

A

に よ く類 似して い るが

，

厳 密に は_{異 質}の _もので _あること を注 意する。 　§6 で は

P

に つ い て_{考 察}する。 小 平氏 （6）_{に よ り は じ め て}

Mn

_{上に定 義された} ， よ く知られて い る作 用素 △

＝d

δ＋δ

d

_とよ く似てい る が， 本 質 的に異る作用 素 ▽＝

d

。δ。＋ 。δ。

d

を定 義 する

。

つ ま り

，

△ と ちがい

，

▽ は 特 定の基 本 微 分 形 式の_系に_強く依 存す る。

9

はこれ らの △ と ▽ の組 合せに よ り定 義さ れ る。 次に

9

のパ ラメ トリッ ク ス q91 を 定 義 し

，

更に

9p ＝

0 を 満 足 す るような微 分形式 g の_測地座標に関する微 分可 能性につ いての べ る。 　 §7 で は コ ンパ ク トな リ

ー

V ン_{多 様 体}

Mn

上の

Cc °

の_{形 式}のつ くる準ヒ ル ベ ル ト 空間

H

の △ に よ る分 解

，

c5｝

・

｛6）

・

c7） _{に倣っ て}

_，9

_による

H

の分 解に つ い て考 察す る。 更に

9g ＝

0 を満 足 する形式

g

のつ く る部分 空 間

F1

の構 造に言 及 する。 し か し

9

はどの よ うな リ

ー

V ン_多様 体に対 して も

，

特に コ ソ パ ク トの場合に， その全体の上 に 大域 的に定 義で き る と は

，

必ず しも保 証 さ れ ない。 しか しなが ら， n 次元の ト

ー

P イ ド の場合に は， 明らか に， その全体の上に定 義され る。 したが っ て，

9

の定 義 され るコ ン パ ク トな リ

ー

v ン _{多 様}体_は確_かに_{存 在}する。 よっ て， §

7

に おける展開は十 分 意 味をもつ こ と がい える。

§

1

．



Introduction

　　　

Let

　

1

）

be

　

the

　 vector 　space 　

of

　

test

　

functions

，　

that

　

is

　

C

°

°

complex

valued 　

functions

　_{with 　compac} 七supports 　

in

　_a　rea1 ％

−

dimensional

　euclidean

space

　

En

．

　

Then

　

we

　

define



a



quasi

−

topology



in



1

）

，　so　

that

｛

Soi

（

x

）

｝

converges

to

　

O

，　whell 　and 　only 　when

｛

g

‘

（

x

）

｝

is

　

a

　

sequence

　of　

functions

∈

1

），　whose

supports 　_{are 　contained} 　

in

　_a　

fixed

　_compact 　

domain

　

in

　

En

，　

and

　

also

｛

9

‘

（

x

）

｝

，

｛

D

。

9i

＠）

｝

，

｛

1

）

茎

ψ‘

（

x

）

｝

，

…

　

respec 廿vely 　converge 　uniformly 　

to

　

O

，　

if

　

i

→ 。 。

（

1

・

．

Schwartz

【

11

）

。

　　　

If

五

is

　_a　

differential

　_{operator 　with 　constant} 　coefHcients 　and

　

X

　

is

　a

fundamental

　_{solution 　of}　

A

，　

that

　

is

，　a　

distributioll

　which 　

satisfies

∠

4X

＝

δ

for

　

9

∈

1

）_，　we 　ean 　represent _，　as　well

−

known

，　

the

　solution 　

g

　of　ari　equation

for

_！

∈

．

0

　　　　　　

A

望　＝＝

／

，

　　　　　　　

（

1

）

by

　

g

＝

X

 

∫

＝

X

（

ξ

）［

／（

x

一

ξ

）

】

．

　

But

　

if

　

A

　

is

　

an

　

operator

　 with 　variable

coe 缶 cients ，　we 　can 　not 　

always

　represent

．

the

　solution 　

g

　of

（

1）

by

　

X

 

f

，

　　　

If

　

A

　

is

　_{an 　operator 　of　cQnstant 　 streng 七}

h

　_with σ゜° eoe 缶 cients 　

in

　

a

neighborhood 　

of

　xo∈

i

樫 ，　

there

　exists 　a　

linear

　mapping 　

L

　of　

1

）

（

E

”

）

into

1

）

_（

EN

_）

such 　

that

　

g

　・ ＝　

Lf

　satisfies

（

1

）

in

　

a

　

suMciently

　small 　open 　neighbOrhood

of　_xo

_（

】

il

δrmander

_［

21

_）

．



If

　

A

　

is

　_reduced 　

to

　

an

　

operator

　of　constant 　strength

by

　

an

　

adequate

　

transformation

　of　variables ，　

A

　

will

　

be

　called 　an 　operatOr

of　

pre

−

constant 　strength

．

一

　

2

　

一

_納

・

On

diffbrential

operators of _parabotic

type

In

this

paper

we shall study

differential

operators

A.

and

9.

A,

is

a

_parabolic

operator with variable coefieients of

pre-eonstant

strength

which

has

a

fundamental

solution _,&(1)

with

support

in

a

half

space

for

A,X=6,

if

_p

is

an odd

integer>O.

But

li.

is

not of constant strength

in

eonnection with any orthogonal system

(x)

in

E",

if

p>1.

9

is

an

operator on

differential

forms

of a

Riemannian

manifold

M"

and

is

regarded

as

the

_{generalisation}

of a semi-elliptie operator

Azi',

where

A

is

a

_parabolic

operator with

constant

coefueients and

l'

is

an

adjoint

operator

of

zi.

In

E2,

we shall verify

that

the

sequence of operators

Oit10ric

h=1,

2,

･･･

constructs

a

certain

family

of

the

operators

whose

fundamental

solutions

have

their

supports

in

a

half

spaee,

if

some eonditions are

satisfied.

In

g3,

we shall

deal

with

the

reductions of

A

and a

hyperbolic

operator7

to

O/ar.

A

and

7,

as well-known,

have

respeetively

the

fundamental

solutions with supports

in

half

spaces.

In

g

4,

we

shall

deal

with

the

explicit

representation of a

distribution

,Zi, where

p

is

an odd

integer>O.

Our

representation

is

necessary

for

us

to

represent

the

fundamental

solution of

li..

As

well-known, .4

is

the

fundamental

solution of ultrahyperbolic

differential

operator

[]

of

2nd

order. .4

is

of

the

analogous

type

to

the

one

that

M.

Riesz

[3]

denoted

for

7,

and

it

differs,

more or

less,

from

the

representations of

de

Rham

and etc.

([4]).

In

g5,

we shall

deal

with

a

parabolic

operator

1,,

which

is

_generated

of

[I],

if

we restrict

the

support of .Zi

to

the

domain

r,

where xi

lO

(i=1,

･･･,

_p).

_Also

_{we shall verify}

_that

_A,

_is

_{easily reduced}

_to

_OfOr,

_and

the

iterations

of

],

eonstruct a

family

of

the

operators with variable

coeMcients whose

fundamental

solutions

have

their

supports

in

a

half

space

for

A2X==B.

Moreover

we shall

deal

with an

implicit

representation

o£

the

solution

_g

of

ASg=f

forfED.

A;

seems

to

be

formally

analogous

to

A,

but

it

differs

essentially

from

ri.

In

g

6,

we shall consider an operator

9

on a

Riemannian

manifold

M".

For

the

definition

of

9,

we

shall

define

7==de6o+eBed

as well as

a=al6+Sd.

A

is

a well-known operator,

but

7

differs

essentially

from

A,

for

the

definition

of

7

depends

closely on a certain

fixed

system of

fundamental

differential

forms

_(tu)

on

M",

while

ti

is

defined

independently

of

(a)).

Then

we can

define

9

byAand7.

Also

we shall researeh

into

the

local

_properties

of

differential

forms

_g

such

that

9g=O

on

M".

Therefore

we shall

define

a

_parametrix

,9i of

9.

Then

we ean obtain a

formula

between

9

and _,9i, and also we can

_prove

the

local

_properties

Kyuiti

SAKuMA

on a coordinate

neighborhood

of

PEM".

In

g

7

we shall

briefly

refer

to

the

_global

_properties

of

9

on a compact

Riemannian

manifold

M",

that

is,

the

direct

decomposition

of

H

on

M"

owing

to

9,

where

Il

is

a semi-Hilbert space which consists of

C"o

forms

EM".

Also

we

shall

deal

with

the

structure of

the

subspaee

Fl

which

consists

of

C"e

forms

_g

such

that

9g:=:O.

Finally

we note

that

gg6

and

7

owe

the

formulation

to

Y.

Akizuki,

K.

Kodaira

and

de

Rham,

([5],

[6],

[7]).

g2.

Differential

operator

OIOr

In

this

section we shall

deal

with

OIOr==Zi(aixilr)alaxi

that

is

an

operator with variable eoeMcients with respect

to

an orthogonal coordinate

system

(x)

in

E".

Now

we

assume

that

the

transformation

xt==rof

gi(t],

...,

t."i)

(i=1,

･..,

_n),

₍₁₎

satisfies

the

following

conditions

;

1)

ai>O

(i=1,..･,n)

2)

O<r<oe,

and

tED(t),

where

D(t)

is

an open

domain

of

t.

3)

gi,

･･･,g.

are

CeO

functions

on

D(t).

4)

There

exists at

least

one

gi

sueh

that

gii!O

on

D(t).

5)

O(x>fO(r,

t)=

TE"i-i17(t)

is

not

identieally

equal

to

O,

when

O<r<

oo

and

tED(t).

6)

There

exists at

least

one xi such

that

lim

lx`

== oo

if

T- _oo, when

tED(t).

7)

There

exists

a

function

G(t)

sueh

that

G(t)F<t)

is

integrable

on

the

closure

D(t).

Then

when

tED(t)

and

O<r<oo,

the

range

I"

of

(x)

is

contained

in

a

half

space of

E".

Now

we

define

a

distribution

X]t

as

follows,

.,,=Ig

k-Zdi`G(t)IK(kmui)!

E.Xf2;

where

_K=I.,,,

G(t)17'<t)dt.

_The

support of

.Xle

_is

contained

_in

a

half

spaee

of

En.

Also

we

define

.Ikof(x)

as

follows,

xL*.fKx>

==

I,"eS.,,,,

.x}(T',

t')

f((T-r'>th

g(t))

o(xF)!o(r',

t')

dr'

dt'

.

Then

we

have

the

following

theorem.

-4-t

On

did7lerentiat

operators

of

_paraboltc

type

Theorem

1.

Theequation

for

fED,

whose support

is

eontained

in

r,

. aixi

Oq

£

`-i r

6x,

=f'

(2)

t

is

fulfi11ed

by

_{g(x)==Xl*f(x),}

where

(x)

and

(T,

t)

satisfy

(1).

Proof.

By

(1),

we

have

te.,

"'iX`

&9,

-

8,

(x*f(x>)

=-!.,,,,i:Xl(r',t')

,O.,

f((r-r')di

g(t))

,(O.(,X,

'

l,)

･

dr'

dt'

,

By

the

condition

6),

we

have

lim,

fl(r-T')W

g)=O

for

fED,

if

r'

--

oo.

Therefore

it

follows

that

(2)

is

valid.

Also

we

have

the

following

theorem.

Theorem

2.

The

equation

for

fED,

whose support

is

contained

in

r,

OicglOric=f,

(3)

is

fulfi11ed

by

_{g(x)=.XL*f<x),}

where

(x)

and

(r,

t)

satisfy

(1).

Proof.

By

the

condition

6),

we

have

lim

r'ic-iOic'i.f7oT'ic-i=O

for

fED,

if

r'- oo.

Therefore

by

the

integrations

by

parts,

we

have

the

reduetion

formula

oic

sric

(Xl*f(X))=

oricmi

(Xi-'"f(X))

'

If

we repeat

the

integrations

by

parts,

it

follows

that

OicglOric=f

is

valid.

Therefore

we

can

conelude

that

the

operators

Oic/Or",

h=

1,

2,

･･･

have

respectively

the

fundamental

solutions

XL,

ic=1,

2,

･･･ with supports

in

a

half

space of

E".

g3.

Reductions

of

A

and

7

to

6!Or

In

this

section, we shall verify

that

the

operators

A

and

7

are easily

reduced

to

OIOT

by

the

integrations

by

parts,

if

we adopt

the

similar

transformations

to

(1)

in

_g

2.

At

first,

we

define

a

_parabolic

operator

ri

and

its

adjoint operator

A'

as

follows,

A=O/Ox.-02/6x?-

･

-

-

-a210x:-i

_,

Kyuiti

SAKUMA

The

fundamental

solution

E(x)

of

A,

as

well-known,

is

represented

as

follows,

E(.)=Isi/2i/iliii:.5)"-i

exp

(-Zi"-ix;･/4xn)

l.i

kioO

:

N=

Then

the

equation

zig(x)==.1"(x),

for

fED,

is

fulfi11ed

by

g(x)

==

gX

flx)

=

s(t)

[f(x-t)]

.

Now

we

define

a similar

transformation

to

(1)

in

g2,

_that

is,

IZ.ii.:･t,

(i-i,･･･,n-i),

,,,

where

O<T<oo,

D(t)=

{(t)

_;

-oo

_<t`<oo}

and

O(x)IO(T,

t>=r`"-i't2.

0wing

to

(1),

we can reduce

zi

to

OfOr

by

the

integrations

by

_parts,

_as

follows,

A=.

[q(x)1=-#[A'q(x)]=E[-Og(x)10r]

,

=O/Or･E[g(x)] _.

Then

the

fundamental

solution

S'(x)

of

A'

is

represented as

follows,

-.,(.)={g12ma1

.L)"-i exp(Zr-ix;･14xn)

lff.x".<oO

"=

:

The

operator

A'

is

also redueed

to

OIOr.

The

operator

ziA'=A'A

is

self-adjoint, and we can extend

AA'

to

an

operator

9

on

differential

forms

of a

Riemannian

manifold

M",

if

_we

represent

as

follows,

AA'

={(d-7)2-2(d+7)}/4

,

where

d

and

7

will

be

defined

later

on.

Next

we

define

a

hyperbolic

operator

7

as

follows,

l7

=o21ox:-o2/omr- ･

･

･

-o2/oxZ-i

.

7

is

self-adjoint,

that

is,

7

itself

is

its

adjoint operator.

The

fundamental

Xsolutions

4k

of

7k,

le=1,

･ ･ ･,

are

as

well-known

represented

respectively

as

follows,

Z}

=Rfsi-nlnC"-2)i2･2H

I'(l12)

r((l-n+2)/2)

,

= ¢

(t)

Rf'si-n

_,

for

e=2k,

k=1,

2,

･･･, where

Ill"

means "Partie

finie"

of

Hadamard

and we

define

-6-a

On

difurentint

operators

of

paraboeic type

,(.)=io xkpx{-'''-x:i

lff

X.:lg:

.a,>.O>'o,

where a=xZ-xZ-･･･-xZ.i.

Zl[g],

gED

is

a

regular

function

of

l(M.

Riesz

[3]),

Therefore

we

can

define

Z[g]

for

all eomplex values of

t,

by

the

analytieal continuation and

the

_passage

to

limits

coneerning with

tfrom

the

estimations of

Zl[g]

for

suMeiently

large

values of

Rl.

Now

we

define

another similar

transformation

to

(1)

in

g2,

that

is,

(E

･

== r.flt:

(i=1,

2,

･･t,

n-

1),

(2)

where

O<r<

+

oo,

D(t)

=

{(t);

O<

ti

<1,

1-t,

--

･･･

-t.-i>O}

and

O(x)10(r,

t)

= _±

2i'"

T"-'

t,-i/'

･･･t

Y,2.

(2)

is

deeomposed

into

2"-'

transfor-rnations.

Owing

to

(2),

we ean reduce

7

to

OIOr

by

the

integrations

by

parts,

as

follows,

7a[9].

r

.

'Z

g8.

"9

(

]

ti3)

-#Hlz

' ,[g].

I(3)

Moreover

we

define

Laplacian

d

and an ultrahyperbolic operator

[]

as

follows,

a=02/Ox:+

･ ･ ･

+o2/oxz.

,

Fl]

=02fOxl+ ･ ･ ･

+0210x2,-0210x2,.,-

･ ･ ･

-02/OxZ

,

where

_p

is

an odd

integer>O.

However

they

are similarly reduced

to

O/Or

by

the

integrations

by

parts,

if

we modify

the

conditions of

(1)

in

g2.

As

wellknown

they

ean not

have

respeetively

any

fundamental

solutions with supports

in

half

spaces.

g4.

Definition

of _,Zt

In

this

section, we shall

deal

with

the

definition

of a

distribution

.a, where once

for

all

we

note

that

we

assume

hereafter

that

p

is

an

odd

integer>O.

At

first,

we

define

the

analogous

transformation

to

(1)

in

g2,

that

is,

z;:.::l･:Sl::::t",e,;

--:;..,,...,..,.,,,,

l

(i)

Kyuiti

SAKVMA

where

D(T,

e,

y)

is

the

domain

such

that

O<r<

co _,

D(e)

={(e)

;

O<e,-i<2z,

O<e`<T

_(i=1,

･･･,p-2)}, and

D(y)

={(y>;O<y,<1

(i=1,

･･･,n-p),

O<Yi+

. . .

+y.-p<1},

and

O(x)10(r,

e,

y)

== _± rn-i

sinp-2

0,

. . ,, sin

e.-,

yrif2

y.--r!.2.

(1)

is

decomposed

into

2"-p

transformations.

Then

we

define

three

linear

operators

e,

Ul

and

Pt

for

ip,

_gED

as

follows,

e[di]

=j.-ce sin'-2

e,

sinph3

e,

-･- sin

e,-,.

¢

cle

_,

Ui[di]

=I..,,,

(1-

tiPyi)(i-n)i2"j?"yid/2-i.ipely,

(2)

PiLip]

-I:riei･ipdr

,

where we

define

di(x)

for

_gED,

as

follows,

¢

(x)=2P-"Zc,)g

(r

sin

ei

...

sin

e.-i

,

.･.

,rcos

ei

,

(-

1)ei

ryll' , ...

,

(-

1)en-p

ryi.!e.),

where

ei=O or

1

and

Z

means

the

sum over

2"-"

sets

(ei,

･..

, e.-,)

such

that

_(1,1,

O,

1,

･･.) etc.

Here

we note

that

e[ip]

is

independent

of

l.

Now

we refer

to

the

well-known

formula

that

is

important

in

gg4

and

5

(Takagi

[8]).

Lemma

1.

We

have

the

following

formula

for

pi>O

and

_q>O,

s

n-p

(1HYi-

' ' ･

-y.".)4-i

17

y,"･i-'

dy

DTCv) _i=1

=r(p,) ･ ･ ･

r(p.u.)

r(q)lr(p,+

･ ･ ･

+p.-p+q)

_･

Also

we

have

the

following

lemma.

Lemma

2.

1)

R[ip]

is

a regular

function

of

l

for

Rl>O.

2)

Ul[

¢

]

is

a regular

function

of

l

for

Rl>n-2.

3)

e,

Ui

and

Pi

are

commutative

one

another

for

Rl>n-2.

Proof.

1)

is

trivial.

2)

is

reduced of

Lemma

1.

3)

is

reduced of

the

definitions.

Then

we

define

differential

operators

D(l,

k),

k=1,

2,

..･

as

follows,

D(l,

ic)=

lle･=,

(e-p+slj+

y)

,

where

Y=2Z?･;i'ydOfOYd･

Then

we

have

the

following

lemma.

Lemma

3.

We

assume

that

_g

and

_¢

are contained

in

D,

and

_op'

is

a

bounded

continuous

funetion

on

D(r,0,y)

and vanishes

for

rlR.,,

where

R.,>O

is

suMciently

large.

8-･l

.,-.S).

On

dijt7lerentinl

operdtors

of

_par(thotictype

1)

di

is

aC"e

funetion

on

D(r,e,y),

but

is

not

differentiable

at

y`

==

o.

2)

¢

is

a

bounded

continuous

funetion

on

D(T,e,

_y)

and

there

exists

a number

RJ>O

sueh

that

a=O

for

rlRi.

3)

Va5="=rg'.

where

di=

_£

#..,.i

xiOglOxi･

4)

D(l,le)di=fl,le･=i

(l-p+2o')ip+"

where

il1=Tg'.

5)

D(t,

ic)ip==rg',

if

g=Zxtgj,

where

_giED.

6)

D(l,k)ip<O)=flf-..,

(t-p+2o')g(O).

7)

D(l,

le)

and

D(l',

k')

are eommutative eaeh other.

8)

Odi10r:=g'

and rOip/Or=".

9)

Oip10e=g'.

Proof.

1)

and

2)

are evident

by

the

definitions.

3)

In

faet

we

have

]Y?75:=(

_£

:;,"xPg!axj)=Tg',

where

g'=2"-"

Zc,]

2i(-1)ei

yl!2

Og/Oxi

(.

･ .,

(-1)"i

ryi,!2_, .

.

.)

.

Sinee

Oif/OxjciD

has

a

compact

support,

g'

is

a

bounded

continuous

function

and vanishes

for

rlR.,.

4),

5),

6)

and

7)

are also evident

by

the

definitions.

8)

and

9)

are eoncluded

by

the

same

way as

3).

Also

we

have

the

following

lemma.

Lemma

4.

We

have

the

reduction

formula

of

Ui

for

gED,

as

follows,

U,[<7;]

=

{1"((t-n+

2)/2)12LI"<(l+

2k-n+

2)12)}

Ui.,ktD(l,

k)4i]

(

3

)

where

Rt>n-2.

Proof.

By

the

integrations

by

parts

with respect

to

yi,

we

have

U,[4i]=

-2Ui

[y,

O<iilOyi]+(t-n)

U,.,

[y`

<Z5]

.

If

we sum up

from

1

to

n-p with respect

to

i,

we

have

(n-p)

q[ip]

:=

-

q

[

Ya5]+

(l-n)

(

U,-,

[ip]-

Ut[ip])

.

Therefore

if

we replaee

t

by

t+2,

we

have

U,[di]=(l-n+2)-i

U,.,

[D(l,

1)

¢

]

.

After

all

by

the

mathematical

induction,

we can

_prove

(3).

Here

we note

that

the

formula

(3)

defines

the

analytical continuation

of

Ui[ipl

with respect

to

l.

Now

we

define

a £unction s.(x) as

follows,

Kyuiti

SAKUMA

where

a,=xi+･..+x;-x}+i-･･.-xZ.

Then

if

Rl>n-2,

we can

define

a

distribution

_.S,(x)

for

_gED

as

follows,

pSi(X)

[9(X)]:t'i-u

"

L(X.)e[9[SX

])l'

]

(4)

However

Lemma

4

admit us

to

the

definition

of

Ul

if

Rl>O.

Therefore

we can

define

_.S,

for

Rl>O

as

follows.

Definition

1.

When

Rl>O,

2k>n

and

Rl-n#even

integer<O,

we

define

a

distribution

_.Si

for

_gED

as

follows,

,S,

[g]={r((l-n+2)12)/2ic

r((l+2k-n+2)12)}

a

U}.,,

e

[D(l,

k)ip]

.

Then

.Si

[g]

is

uniquely

decided

as

far

as

2k)n,

and

eoincides

with

the

definition

of

(4)

for

Rl>n-2

by

Lemma

4.

Moreover,

we ean

define

a

distribution

_,Zl

for

_gGD.

Definition

2.

We

define

.a

for

gED

as

follows

pZl

[sp]=popk

(l)

Pi

U}+2A

e

[D(l,

k)

45]

,

where

D(l,O)[=1,

Rt>O,

Rl+2k>n

and

,¢ ,(l)=

r((l

-p

+

2)/2)

r((p-

l)/2)

× sin

((l+1)T12)17r"'a

2i'it

l'(l12)

l"((l+2k-n

-l-

2)/2)

.

Then

if

.a

[g]

is

decided

through

the

analytical eontinuation and

the

passage

to

limits

with

respeet

to

tfor

RISn-2,

.Zl

[g]

is

uniquely

deeided

as

far

as

Rl>O

and

Rl+2le>n.

Therefore

_.Zl

[g]

is

an analytic

function

oflas

far

as

Rl>O

and

Rl+2le>n.

However

if

Rl$O,

we

need

Rlr,

for

the

integral

P`

[<b]

is

not

_generally

integrable.

But

we

shall

,

not

deal

with

the

ease of

RISO

in

this

_paper.

Finally

we note

that

,4

is

a

fundamental

solution of

D

as we can verify

by

the

similar

formula

to

(3)

in

g3,

if

p

is

odd.

:

g

5.

Differential

operator

A,

In

this

seetion we shall

deal

with

the

_parabolic

operator

is

_generated

of ultrahyperbolie operator

D,

if

we restrict

the

,Zi

to

the

closure of an open

domain

rsuch

that

s',>O

(i

=

1,

･･･,

p).

At

first

we

define

a

function

g.

as

follows,

g.(.)-Ig',

lli

X.Ef-

:

A,

which support of

and

xi>O

On

dQrlerentiat

operators

of

parabolic

type

g.

is

the

restriction of s,

to

I'

and

it

is

not continuous on

OI'.

Also

we

define

the

similar

transformation

_(A)

to

₍₁₎

in

_g2,

that

is

given

by

(1)

in

_g4,

when

D(r,e,

y)

is

the

domain

G

such

that

O<T<oo,

D(e)={(e);

O<ei<n12

_(i=1,

･･･

,

p-1)}

and

D(y)={(y);

O<y,<1

<al=1,

･･･ , n-p),

O<Yi+･･･+y.-p<1}･

By

_(A),

r

is

mapped on

G,

and

we

haveO(x)!o(r,

e,

_y)

= _±r"Hi sinP-'

e,

. . . sin

e,-,

yril'

. . t

y.m-ir.2.

Then

we

define

a

distribution

,ZNi

for

gED

as

follows,

-

N

,Zi

[gfr]=2"

,¢o

(l)

,Si

[g]

,

=:

2p

,¢ ,

(l>

1[lf'gi,-n

[gp]

,

=2P .di,

(l)

P,

u,

eN

[4i]

, where

Rl>n-2,

and

eN

[ip]=Ib,,m

di

sinp-2

_e,

･･･ sin

e,-,

do

_,

,2i

is

the

restrietion of .Zi

to

fi.

Now

we

have

the

following

theorem.

Theorem

3.

We

have

the

reduetion

formula

of ,21

for

tollows,

Ap

(l)

p2i+2

[9]

= p2t[9] ,

where

Rl

is

suMeiently

large,

and

AS

(Z)

=

-Z;.,

0210x;･

-(t+2-n)

Z7

(xi/a,)

O/Ox,

_,

A,

(t)

=

->i]:.,

021ax;･+(t+2-n)

£

f

(OIOxi)

xVa, _,

where

a, =x;+･-.+x2,-x},,-.･.-x;', and

AS(l)

is

an adjoint

Ap(l)･

Above

all we shall

_prove

a

lemma.

Then

there

exists a

C"

e.(x)

such

that

e.(x)-It

l.i:-tLO:.,

where ff>O, and

OSe.(m)gl

if

-a$x$O.

Also

we

define

follows,

.Si(a)=e,(a) .Si ,

where

e,(a)=e.(xi)

. . .

8.(xp>.

Iarge.

Then

it

results

that

_,S,<a)EiC2.

[Ilhen

we

have

the

following

lemma.

Lemma

5.

We

have

the

following

forrnulas.

(1)

gED

as

(2)

operater of

funetion

.St(a) as

Kyuiti

SAKuMA

1)

,g,[g]=lim･.s,(a)[g]. "e

2)

,S,(cr)[Mg]=(D,S,(cr))[g]･

'

where

_gED.

Proof.

1)

We

have

,Si(ff)[9]

-,g,[9]

=L(a> ･

9

,

where

Rl

is

suMciently

large

and

4`ff"9=

{jla

i:eo

"

iloilei:e"

"

' '

'

i:.

'(;)'

jl.j:..}e'(ff'

s}-"'g

dx

'

Then

Li}(a)g.O,

if

a.O.

Therefore

1)

is

valid.

2)

We

suppose

that

(li)

are

the

direction

eosines of a normal on

the

sphere

K

with

the

origin as

its

center, whose radius

R

is

suMeiently

large,

and

d"

is

the

surface element of

OK;

Then

by

the

theorem

of

Gauss,

we

have

.Si(e>

[ll9]-(D,Si(a))

[9]

=:

I,.

{gp(o')

s}-".

LN

Og!Ox-g

L. o(E.(.)

s$-n),tox}

dp

,

where

LN

OIOx

=

Zi'

li

O/Oxd

-Z;+i

li

O/Oxi.

Since

g

and

OglO

v vanish on

OK

it

follows

that

2>

is

valid.

_{

Now

we ean

_prove

Theorem

3

_as

follows.

Proof

of

Theorem

3.

At

first

we

have

[[]pSi

(if)

-

e.

(a)

[I]

s$-s

=

£

,p

{2(os}-"laxi)

Oe,(a>lo`vi

-f-

s$-"

6t

e.(cr)lox?}

.

Therefore

in

order

to

estimate

D,S(if)

[g],

we must

estimate

the

following

integral.

I={2(os}'nloxi)

oe.

(ff)loxi+s$-n

o2

e.

(.)fox:･}

[opl

.

･

Then

by

the

integrations

by

_parts,

we

have

i=-

j:..{

olll,

(

OoS.}ii."

'

sp)}6p(a')

dx

+

!:.{

oO.,

(s$-"

aa.9,.

)}

e,(a-)

dx

.

Sinee

the

lst

integral

of

I

vanishes,

it

follows

that

if

a--O, _we

have

the

following

formula

by

Lemma

5.

--!

-v

pZ}[09]-.Z}-,[91

=

2p

,¢ ,

(l)

y(x,)

･･･

y(x,)

[z,p

oloxi

(sS-n

agloxd)],

where

Y<x)

means "Heaviside _operator".

If

_we

_replacel

_by

l+2and

reform

both

sides

by

the

integrations

by

_parts,

we

have

,

On

dit71erential

operators

of

_parabolic

_type

N N

N

,Zt+2

[-Z",+i

02g/Ox;･]-.Zi

[gl=(l+2-ve)

,Z}.,

[aF'

Z7

xj

Oglaxj]

.

<3)

Therefore

it

follows

that

(2)

is

_generated

from

_(3).

However

AS(l)g,

gED

is

not always contained

in

D,

for

the

eoeMcients

of

AS(l)

are not continuous on

Ol'.

Therefore

the

analytical continuation of _.21.,

for

AS(l)g,

is

not admitted,

for

we can

not

always

define

D(l,

k)

<ziS(l)di)

for

_gED.

Then

we

define

a

distribution

_.Xi(h)

for

an

integer

h)O

as

follows,

,Xl(h)

[ge]=o'ig(aF"

.ZN,)

[so]

,

=

(aFh

.l)

[a;'

g]

.

Evidently

we

have

.Xl(h)=.Z

for

suMciently

large

Rl.

Sinee

atsziS(l)g,

wherehis an adequate

integer>O,

for

_gED,

is

contained

in

D,

the

analytieal continuation of .X}+,(h)

is

admitted

for

AS(t)g

by

Lemma

4.

Therefore

we can

define

.Xi(h)

for

Rl>O,

as

follows.

Definition

3.

We

define

.X<h) as

follows,

,Xl(h)

[g]

=:

(aik

,ZNi)

[a}'g]

=

2P-n,di,-,(t)

{r(e-2h-n+212)lr(l-n+212)}

･

P,

eA'

U}n,,.,,

[D(l-2h,

k)

(1-zr-p

y,)h･di]

,

where

Rt>O,

Rl-2h+2k>n

and

we

deeide

.X}(h)[g]

for

Rl-n=even

integer<O,

by

the

_passage

to

limits

with respect

to

l.

Then

we

have

the

following

theorem.

Theorem

4.

We

have

for

gED,

Ap'pXli(1)

[9]

=6

[9]

_,

where

A,=A.(O).

Proof.

By

the

integrations

by

parts

from

(3)

we

have

A,(t)

.Xl.2

(1)

[g]=

.l+2

[AS(l)g]=T(ifs'

.l.,)

[-(l-n+2)

OgfOr]

.

(5)

By

Definition

3,

Lemmas

1,2,3

and

4,

we

have

from

(4)

and

(5),

Ap(l)

pXl+2

(1)

[9]

=T({r};i

.Z'V,.,)

[-(l-n+2)

OsplOr]

=2"-'

.dik-,

(t+2)

(l+2-n)'t

eN

UI.,k

Pt.,

[-(l+2-n)

O/Or

(D(l,

k)

di)]

=2pmi

.di,-,

(l+2)

eN

Ut.,,[-j,ee

ri

oOr

<D(l,

ic)

¢

)

･

dr]

,

where

the

last

expression

is

defined

for

Rl>O,

if

Rt+2le>n.

Kyuiti

SAKVMA

zi,(O)

_,Xi

(1)

[g]=Ce

q,[-I,eO

aO.

(D(O,

k)ip)

ar]

=c

e'-

q,

[",k･.,

(2j'-p>

g(O)1

,

where

C=(V-1)p-3

2p'2"it/(V-)"-2

r(2k-n+2/2).

After

all,

by

Lemma

1

and

(2)

in

g4

we

have

zip･.X(1)

[g]=g(O)

･

Here

we note

that

(5)

is

the

reduction

formula

of

zi,(t)

to

O/aT.

Moreover

by

the

iterations

of

(2>

we

have

Ap(l)h

_p21+2k

[g]=pl

[g]

,

(6)

where

Rl

is

suMciently

large.

Then

we

have

the

following

theorem.

Theorern

5.

We

have

for

gED,

(A.)h..Xlih(h)

[g]=a

[g]

,

(7)

where

h

is

an arbitrary

integer>O.

Proof.

By

(4)',

(5)

and

(6)

we

ean

prove

(7)

similarly

to

Theorem

4.

Therefore

it

follows

that

the

iterations

of

A.

construet

a

family

of

the

operators which

have

fundamental

solutions

X=

.Xhh(h) with gupports

in

a

half

spaee of

E"

for

A;X=6,

when

_p

is

odd.

Next

we

shall

consider

an

implicit

representation

of

a

solution

that

satisfies

ziig=.L

for

fED.

Now

we

define

a

transformation

of

(x)

to

(p,

e,

t)

as

follows,

!:l:;gl:llI1I:L2ee11I･･･,x.-peose,,

l(s)

kxd

-'

tj-,

_(j'

=

p

+

1,

...

, n) ,

J

where

D(p,

0,

t)

is

an open

domain

Msuch

that

D(p,

t)={(p,

t)

_;

O<p<co,

Zt2<pt,

-oo<tj<oo,

_j'=p+1,

･･･, n} and

D(e)

=={(di;

O<ei<rr12,

i=1,

･･･,

p-1}.

Then

by

(8),

r

is

mapped on

Mwith

one

to

one correspondence, and

AS

is

redueed

to

an

operator

P(t,

_p,

D)

of

eonstant

strength

with

regard

to

(p,t)

in

r<e)

for

constant values of

(0)

as

follows,

Plp,

t,

M=

-Zr-P

0210tl-+(n-2)

aiip

Olap

.

Therefore

A;

is

an operator of

_pre-eonstant

strength.

Let

P<x,D)

have

Coe

coeMcients

and

be

of eonstant strength

in

a

neighborhood of x,EE".

If

V

is

a

suMciently

small

open

neighborhood

f

On

difflerentint

operators of paraboltc

tvpe

of xo,

there

exists a

linear

mapping

L

of

D(E")

into

D(E")

with

the

following

_properties.

P(x,

D)

Lf=f

in

V

if

fED(E")

,

LP

(x,

D)

n=u

in

V

if

uED(V) .

(H6rmander

l2]).

Therefore

there

exists a

linear

mapping

Lk(p,t)

for

{P(p,

t,

D)}ic

as

follows,

{P

(p,

t,

D)}ic

L,

(p,

t)

f=f

in

r(e)

if

fED,

L,

(p,

t)

{P(p,

t,

D)}ic

u=u

in

r(e)

if

uED(V) ,

where

fED(E")

is

considered as a

Cep

function

ED(E"-"'i)

with regard

to

(p,t).

Then

we can also

define

a

linear

mapping

A,(p,t)

on

D

as

follows,

{P(p-p',

t-t',

D)}ic

L,

f=

A,(p-p',

t-t')

f(p,

t)

.

Ak(p,t)

has

_generally

a eomplicated representation and

it

is

diMeult

to

represent explieitly

Ak(p,

t).

However

by

Ak(p,t),

we can represent a solution

_g

of

ASg=f

for

fED,

as

follows,

g=L,

(p,

t)f,

=pXh

(1)

[Ai

(P',

t')f(P+P',

t+t')]p'.t'

_,

where

(t',p')

satisfy

(8).

Also

we can

_generally

represent a solution

_g

of

(riS)hg=f

as

follows,

9==pXhh

(h)

[AA

(p',

t')f(p+p',t+t')]

,

where

(t',p')

satisfy

(8).

g6.

Differential

operator

9

In

this

section we shall

define

an operator

9

and

its

_parametrix

_,9i

on a

Riemannian

manifold

M".

If

Fli

is

a set whieh eonsists of

the

positive

normal orthogonal systems of veetors on

the

tangent

space of

PEILT",

and

B

is

a

set

whieh eonsists

of

the

elements

of

iiilo

when

P

removes on

M",

the

set

B

is

a

_principal

fibre

bundle

_whose

fundamental

group

is

composed of

_positive

orthogonal

transformations.

If

cai, .･.,

to.

are

a

system

of

linearly

independent

fundamental

differential

forms

which are

locally

defined

on

B,

they

are also eonsidered as

differential

forms

eontinued on

B

as

_globally

as

_possible.

Kyuiti

SAKUMA

differential

forms

of rth order

into

those

of

<n-r)th

order

as

follows,

'

(Wii

''' cair)

= sgn(z

l.,

2

IlI

'

i.

' '

il,

'

lil

I/IH.)(o,]

''' (D,nmr'

If

we

_define

an

_{inner product}

(a,P)=

I..a

-

P

between

_{Cpt formsaand}

P

of rth order on

M",

the

spaeeHof all

CW

forms

on

M"

is

eonsidered

as a semi-Hilbert space.

Also

Iet

an operator

d

be

an exterior

differential

operator

with

an

adjoint

operator

O

as

follows,

(dar-i,

Pr)

..

(ar-,

6fir)

_,

where

6=(-1)r

sc-i

el

x=(-1)"Cq-i)+i sc

dx

.

Mereover

we

define

loeally

an unitary operator e _which

is

_closely

conneeted

with a eertain

fixed

system

(to)

as

follows,

oF(tuir'''7tun)=F(Htuii'''sHto"-1fte")t

where

F'(tu)

is

a

_polynomial

of

(to)

with coeMcients ECeO.

Therefore

we

have

_(ooct,

_P)

==

(oat,

oB) ==

(a,

eoP)=

(ct,

B).

Then

if

M"

is

an euelidean space,

toi,

i

:1, . ･ ･

, n

coineides

respectively

with

dxi,

i=1,

･ ･ ., n

in

connection with an appropriate orthogonal system

<x).

If

we operate

7=doDe+e6ocl

onaCe

funetion

f,

that

is,

aform of

Oth

order, we

have

the

following

lemma.

Lemma

6.

We

have

the

following

formulas.

1)

7:f=-(o216x:-0211in:-･･･-0210x:.i)L

onE".

2)

7<rax`

･ ･ ･

dx..)

=

-

(02/ax:

-0210xr

-

･ ･ ･

-O!10xZ-i)

fdxii

. . ･

dx`,,

on

E".

3)

(7a,

P)

==

(at,

o7oB), where a and

3

are

C2

forrns

on

M".

4)

(7a,

oP>r(eat,7P), where a and

P

are

C2

forms

on

M".

Preof.

1),

2),

3)

and

4)

are respeetively

_proved

by

the

definition

of

7.

Therefore

7

is

regarded as an extension of _usual

hyperbolic

_partial

differential

operator of

2nd

order.

Now

we can

define

an operator

9

whieh

is

regarded as an extension

of

AA'

on

a eoordinate neighborhood of

PEM".

Definition

4.

We

define

an operator

9

and

its

adjoint operator

9'

as

follows,

9={(d--P7)2+2(A+7)}/4

,

9'

={(d-o7o)t+2(d+o7o)}/4 _,

-16-t

,r.

On

difilerential

operators

of

_parabolic type

where

A=

cl6+Dd.

ri==d6+6d

was,

as well

known,

for

the

first

time,

defined

on

M"

as

an

extension

of

252!Ox7･

by

K.

Kodaira.

d

is

independent

of a

system

(w).

However

9

is

closely related

to

a eertain

fixed

system

(tu)

as _well

as

7.

Then

if

we adopt _a

_geodesic

_coordinate system on

M",

we

have

also

the

following

lemma.

Lemma

7.

We

have

the

following

formulas.

1)

d=-==-Z02!OxZ･

mod

I2.

2>

7=-o7o=--(o'/ox2.-Zr-io2!ax3-)

mod

I'.

3)

9i-AA',

O'iiAA'

mod

I2.

where we set .(p={f;

fEf(M"),

f(P)==O,

PEiiM"},

when

f(II(")

is

a ring

of

Ctu

funetions

on

M".

Proof.

The

preof

is

reduced o £

the

definition

of

_geodesic

eoordinate

systems.

Next

we shall

define

the

_parametrix

of

9

on

M".

Let

V

be

_a

neighborhood of

P

on

M",

n"i(V)

be

its

inverse

image

on a

_principal

fibre

bundle

B

and

(di)

be

a

system

of

fundamental

differential

forms.

If

we adopt a

local

coodinate system

(x`)

on a eross sectien of nHi<V),

we ean represent tui=2?..i aij

(x)dxj

and

as2=Z:-.i

(toi,

toi)=Zij

_gii

dx`

dxJ,

where

_gij=2k

aki akd.

Then

we can

define

_geodesic

curves

by

d2

Xj/dt2+Z,,

I';f'i

(dXildt)

(dXic/dt)

=O _,

(1)

where

r;fiEC"o.

Therefore

if

a

_point

Q(y)

lies

in

a suMciently small

neighborhood of

P(x),

the

_geodesic

distanee

between

(x)

and

(y)

is

represented as

follows,

r(x･

y)

-I:

;,,

g,,(

d,X,`

)(

d,X,'

)

clt

,

=[v'Z

g,i

(dX`/dt)

(dX'ldt)],=,

, i"'

for

we

obtain

the

left-hand

side

of

(1)

if

we

differentiate

the

integrand

in

₍₂₎

with regard

to

t.

Also

we

have

T2(x,

_y)=

Z

(v`)

n`:=Zjaij

(x)ej,

and

g`=[clX`ldt],=,,

whie

Then

we

define

a

_geodesic

coordinate system z`=v`+Z

where c{n are appropriate

functions

ECe'.

If

represent

the

coordinates of

P

and

Q,

respectively

by

(zi)

and

<z2),

we

have

rt(x,

_y)=

Z,

(zi-z:)2+O(s`)

_,

v`

=(z:-zl)+O(s2) _,

square

et

h

are

Cop

funetions

of

(x)

and

(y)

Ciik ×

Vj

Vic,

geodesie

<2)

ofwhere

.

Kyuiti

SAKUMA

where s2=Z

(zS-z:)2.

Then

there

exists a neighborhood

U(P)

of

PEM"

such

that

we can

join

P

to

every

_point

E

U

by

one and only one

_geodesie

curve.

Therefore

there

exists a real number

_v>O,

such

that

has

the

following

_properties.

1)

If

the

length

of are which

joins

(x)

to

<y),

is

smaller

than

_v,

there

exists

always

one and only one

_geodesie

curve which

joins

(x)

to

(y)

and

the

_geodesie

distance

r(x,

_y)<T.

2)

T2(x,y>

is

a

Cco

funetion

with respect

to

<x)

and

(y).

Hereafter

we shall restriet ourselves

to

consider

in

the

domain

T(x,y)<v.

Of

course we can suppose

that

the

domain

T(x,y)<v

lies

in

the

same coordinate neighborhood.

Aecordingly

we

can

define

a

C"o

funetion

_p(x,y)

of

(x>

and

(y),

such

that

_p(x,y)=1

(if

r(x,y)5v12), =O

(if

TZv), and

OSpSl

(if

r<v).

Then

we ean

define

two

currents ,cos(x,y) and ,(Dl<x,y)

that

are

respectively

differential

forms

of

bi-qth

order with respect

to

(x)

and

(y)

as

follows,

Q(Vi(X,

Y)

=

='

t(rp)P(X,

Y)

ZAir--`,

s',･･.j, (IX`'' ' '

dXig

elYji

' . .

dyJg

,

where

A=T2(x,

_y)12,

Aij=02AIOx`

OYj,

and

Ai.･-i,

jr･･d,=Zck) eS'1:::e'g

Aii

ki ' ' '

A`,k,.

Also

we

define

Ei(v)

and

='l(v)

as

follows,

Ei(v)

(El(v))

..

{

rrv)

_(2.)i-niv.i

ei""-i' _exp

(-

EL

v;

i4]

ny"

)

lli

_Z:

s>

oO

[Zl

l.:

gl

l

Finally

,toI(x,

y)

is

obtained

from

,wi,

if

we replaee

gi(v)

by

El(v)

in

,tot.

Now

we can

define

a

_parametrix

_,9t, whieh

is

a current and

is

regarded as a

fundamental

solution of

9.

Definition

5.

We

define

,9i as

follows,

.s?i(x,

y)=j,(it)i(x,

e)

sc ,tu1(e,

y)

･

If

we adopt an adeqtiate

_geodesic

eoordinate systern

(2`),

we

have

Aii"'i,

ii'''i,=Zck) EkJii:::lr'g

(6ii

_thi+

4ii

_ki) × ' ' ' _×

(6i,

k,+Ci, ke) '

where

aij=1

(i

=j>, =O

(i

\j)

and

C,j=O(s2).

Then

we

have

the

following

lemma.

Lemma

8.

We

have

the

following

formulas.

1)

A

･_-",

(x)

==

A'

･

Ei

(x)

=

(l

-

1)

I"'i

(l)

･

(2

vtilYi

-n

×

lx.I

C2i-"-3'i2

exp

(-Z

x;･/41x.D.

2)

If

we adopt a

_geodesic

coordinate system

(z`)

such

that

<z:)=(O>

at

P

and

(zl)=(zi)

at

Q,

we

have

-18-g