stein uniform estimates of solutions for –equations on ... · chapter 1. stein manifolds and...

10384 Æ 200223009 UDC:

Stein ∂– Uniform Estimates of Solutions for∂–Equations on Stein Manifolds

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2005 ' 4 厦门大学博硕士论文摘要库

()**+,-Æ./0

( ):

厦门大学博硕士论文摘要库

1 2

3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

5 Stein 6789: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

§1.1 Stein Æ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

§1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

§1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

;<=>? @ (p, q) AB7C∂– DE5 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

§2.1 (p, q) ! Koppelman–Leray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

§2.2 " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

G ;H C1 <IJ@ (p, q) AB7C∂– E5 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

§3.1 (p, q) Koppelman–Leray–Norguet . . . . . . . . . . . . . . .24

§3.2 " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

#KL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


Contents

Chinese Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

English Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 1. Stein manifolds and basic lemma . . . . . . . . . . . . . . . . . . . . . . . 8

§1.1 Stein manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

§1.2 Basic lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

§1.3 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Chapter 2. Uniform estimates of solution with weight factors of

∂–equation for (p, q) differential form on a strictly

pseudoconvex domain with non-smooth boundary . 14

§2.1 Weighted Koppelman–Leray formula without boundary integral

for (p, q) differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

§2.2 The uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Chapter 3. Uniform estimates of solution of ∂-equation for(p, q)

differential form on an open set with piecewise C1

smooth boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

§3.1 Koppelman–Leray–Norguet formula for (p, q) differential form .24

§3.2 The uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Acknowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35


Stein ∂– $ 1

N

%& Stein Æ ' Æ Stein Æ

Cn Stein Æ Stein Æ

( )* ) Cauchy

Cn Stein Æ ∂–

!!" (0, q) ∂–

Holder " +#$ Demailly

Laurent–Thiebaut[8] Stein Æ (p, q) ∂–

"+ !%, !% %" SteinÆ "

&, ∂– "!" M n # Stein Æ

D ⊂⊂M

% SteinÆ "'"#

Stein Æ $( $$#-' "## !%)$ Demailly Laurent–Thiebaut $% SteinÆ

%&!"%& D ! Koppelman–Leray [11].

&'* &''# +

($ %")"''. "'&& ( )**+!,! )$ (-'./( )$) ( )!,+ Hormander

0"$#/)$* (p, q) ∂– ! "

%&' Leray–Norguet%( [4],*)$ Demailly Laurent–

Thiebaut[8] $% Stein Æ 1& C1 %& D

Koppelman–Leray–Norguet [9], $ Range Siu[15]

(p, q) ∂– "


Stein ∂– $ 2

2O3 SteinÆ * Hermitian++*0',* ∂– *

!*"


Stein ∂– $ 3

ABSTRACT

It is well known that a Stein manifold is a very important manifold on

which there are a lot of nonconstant holomorphic functions. Cn is just a Stein

manifold, so it is very natural to research into complex analysis in several

variables on Stein manifolds. The integral representation method is one of

main methods of complex analysis in several variables. In the past, many

integral formulas for solving the ∂–equations on Cn and Stein manifolds were

obtained. Based on these integral formulas, the Holder estimates and uniform

estimates of solutions of the ∂–equations for (0, q) differential form also were

given. On the basis of the former methods, by means of the ideas of Demailly

and Laurent–Thiebaut[8], the author researches into the uniform estimates of

the solutions of ∂–equations for (p, q) differential form on Stein manifolds.

The whole dissertation includes three chapters. In the second chapter and

the third chapter, the author discusses respectively the uniform estimates of

solutions of ∂–equations on the different domains on Stein manifolds. Suppose

that M is a Stein manifold with complex dimension n and D ⊂⊂ M is an

open set.

In the first chapter, the author introduces some definitions, the basic

lemma and notations on Stein manifolds, including a Stein manifold, the com-

plex tangent bundle, the complex cotangent bundle, fibre. Furthermore there

is the most important basic lemma and so on.

In the second chapter, applying the ideas of Demailly and Laurent–

Thiebaut[8], using the Hermitian metric and Chern connection the author gives

Koppelman–Leray formula[11] with weight factors on a strictly pseudoconvex

domain with non-smooth boundaries in a Stein manifold. By means of the

Hormander diameter and localization technique, the author admits the uni-

form estimate of solution with weight factors of ∂–equation for (p, q) differen-


Stein ∂– $ 4

tial form.

In the third chapter, the author firstly introduces the definition of Leray–

Norguet section[4]. Secondly, by means of the ideas of Demailly and Laurent–

Thiebaut[8], and by means of the Hermitian metric and Chern connection,

the author obtains Koppelman–Leray–Norguet formula[9] for (p, q)-form on an

open set with piecewise C1 smooth boundaries on Stein manifolds. Finally, by

means of the method introduced by Range and Siu[15], the uniform estimate

of solutions of ∂–equations for (p, q)–form is given.

Keywords: Stein manifold; Hermitian metric; Chern connection; ∂–

equation; weight factor; the uniform estimate.


Stein ∂– $ 5

Stein ∂– ∗

P

,--.,-.1/.450(//.)6027-/.*31.+212345,-/.4./0326/-0578,--.,4./8612--05/.,7-/122,-97-/1-030:Æ14/34248

5/./50Æ2-.1/./5976:608;<126.6787227=/9>:.1/./7?3;8

@A 1831 82 Cauchy <.5=9-9:-/7-/ Cauchy 3.=

82/.*4;9:3.;:A-.,4/:6Æ8;<<>;5 70 81

Henkin[1,2] 8 Grauert & Lieb[3] .=?:5 Cn 6744=<B/ ∂– 3>/

8/3.=8@2,-9//3.;:3;C=<D>-2?0,--.,/

863;602>/86EA-3@9// Cauchy 3.=80FA<B98

?<2 Cn 674?BC/8DEC//3.;::9G>H;?I,4. [1−6], ;< Koppelman[7] < 1967 8?:5 Cn 674 (0, q) [email protected]

8/ Koppelman =82?F Cn 674 (0, q) [email protected]/3.;:<@K

H;?I,4. [4−6], C-05</3.;:/4.L-=/8D>A/8

BC2 Henkin 8 Leiterer[4] 4.5 Stein 05< (0, q) [email protected]/3.

;:<@2?:5 (0, q) J/ Koppelman =82 Koppelman–Leray =88

Koppelman–Leray–Norguet =82EGF5 ∂– 3>/88?= Demailly 8

Laurent–Thiebaut[8] 4.5 Stein< (p, q)[email protected]/3.;:<@2?:5

(p, q)J/ Koppelman8 Koppelman–Leray=8=: ∂–3>/82>8 (0, q)

[email protected]/A5?G>/B=2MDHC3 Cn670FI> EuclidHD2N

0A Stein05< EuclidHDH-?B9IE/H9D805@JM>AE2

* KB?FCODE@AFG (FGHBL 10271097).


Stein ∂– $ 6

Demailly 8 Laurent–Thiebaut F> Hermitian HD8 Chern GI2GF5H

93.52M-0>6.:6/GH8AJHK<2JLC [9] ?:5 Stein05

<MIDJ C1 DE/>Æ< (p, q)[email protected]/ Koppelman–Leray–Norguet

=82EGF5 ∂– 3>/88/@MK B.Berndtsson 8 M.Andersson[10] N

PL?NKNE3.5/3;2JLC8LMQ [11] R?:5 Stein 05<L

?ODEMI4=<B</NKNE/ Koppelman–Leray =82KGF5 ∂–

3>NKNE/88IF4 [12] GF5 Stein 05< (p, q) [email protected]

E/ Koppelman–Leray–Norguet =88 ∂– 3>NKNE/883.;:<@<D?J0N0??2?E-4-0;3.=8GF/ ∂–

3>/8OSB98BCGF5 Cn 674? C2 MI/4=<>Æ84=<

B</ ∂– 3>8L? 1/2–Holder B98PTR?:0KM/ Cn 4MIODE8MIDJ C1 DE/4=<B=:4=<,NL</ ∂– 3>/8L?

0GB9 [4−6]. 1970 82 Henkin[2] F>; Henkin 8 Ramirez NP/3.=

8Q?5 Cn < C2 4=<>Æ/ ∂– 3>8/0GB92EO Holder B9

K; Henkin 8 Romanov[13] 7HP8 1971 8 Henkin[4] BCQ?5 Cn 48,,NL< (0, 1)– 58/B98 Poljakov[14] ROM>R@M6: (0, q)– 5

88 1973 82 Range 8 Siu[15] ?:5 Cn 4MIDJDE4=<B<8/0GB98 1991 82PQU [16] R?:5 Stein 05L? C1 DJDEMI/4=<B< (p, q)– 58/ Koppelman–Leray =88 ∂– 3>/HRMI3

./82EOGF5S80GB98GN/Q/-G> Demailly 8 Laurent–Thiebaut[8] /GH2F> Her-

mitian HD83GI2.=0 Stein 05<MIODE/4=<B< ∂– 3>

NKNE/88MIDJ C1 DE/>Æ< ∂– 3>/8OS0GB98T0V2ORCS@UV0E Stein 05<-SP2-WSP2PQWH

GXX=:HGY<2Y?0RTZ82Q<98MRRA[I-0@N/3.=82 ∂– 3>80GB9IJQ/8

T\V2T D ⊂⊂ M - ∂D ODE/4=<B2ρ - ∂D /UB θ </

4,S]8C/2GF?BKTC/ Φ(z, ζ) 8 Φ(z, ζ), F> Hermitian HD


Stein ∂– $ 7

83GI2S^ Leray UN s∗(z, ζ) NPH93.52<2?:5 Stein 05

<MIODE/4=<B D <HRMI3./NKNE/ Koppelman–Leray

=8 [11]. A Stein 05<2:ZXX Hormander V2 diamH(suppf) =:

dist(z,HΦξ (δ)), F>VU_WW2GF5 ∂– 3>NKNE8/0GB98A

0GB942; z ∈ D∩Uj , D∩Uj 7 Cn 4/X>ÆTY2=:<U< Uj/0> C∞ /@V.82<2OA Stein 05X0>Æ</0GB9` Cn 4TY/>Æ<-I8N05 Ω 7

n−k∧l=1

∂v∗jl∧ n∧

l=1dujl

∧ k∧l=1

dζjlLV0>aXW2

7=ORYI> (0, q)– 58/3;-B9 ωz,ζ,λ(v∗j ) ∧

n∧l=1

dujl∧ k∧

l=1dζjl

.

TZV2T D ⊂⊂ M L?DJ C1 DEMI ∂D />Æ2ρj - ∂D />b

c Ukj=1 </ C1 C/2F> Hermitain HD83GI2S^ Leray–Norguet

UN s∗1(z, ζ), · · ·, s∗k(z, ζ) NPH93.52<2?:5 Stein 05<L?DJ C1 DEMI/>Æ< (p, q) J/ Koppelman–Leray–Norguet =8 [9], EG

F5 ∂– 3>8/0GB98A0GB942F> Range 8 Siu[15] /3;2T

Ωk(ϕv, s, s∗1, · · · , s∗k, s)F< z 0 (p, q)J/.D0 Ωkp,q(z, ζ, λ),/@A SI ×σ0I

<B9 f(ζ) ∧ Ωkp,q(z, ζ, λ). BC[ AI,k

p,q−1 0 Ωkp,q−1 47?A λ 0 |I| X/,

W8/82?=[ ΩI,kp,q−1 =

∫σ0I

AI,kp,q−1, NJL[B9

∫SIf(ζ) ∧ ΩI,k

p,q−1.


Stein ∂– $ 8

M Stein NOQRPS

AM0Z[2ORBC\\ Stein 05</?FXX8HGY<20@

N/X<8B9IJQ8

§1.1 Stein TU

QV 1.1[4,5] T M -0- n Q/-052 O = O(M) 0 M </?B

C/8 M W0 Stein 052]d>^REMZ>X]4(i) M -?B</2^0 M 4_0_Æ K,

KO =

z ∈M : |f(z)| ≤ sup

K|f |, ∀f ∈ O(M)

K- M 4/_ÆÆ(ii) M 4/?BC/Y.` M </A2^0<_\/ z, w ∈ M, z = w, X

A f ∈ O(M), Y? f(z) = f(w);

(iii) M </?BC/YGF M /VUSY2^0<_\/ z ∈ M , XA

f1, · · · , fn ∈ O(M), (f1, · · · , fn) - z /0>UB/VUSY84 H.Grauert[1958] ?0>72/X<2^]d M -0-052L?

4,S]8àC/2ab M - Stein /2C-L6>X] (i) 8 (ii) 4Y=

HP Stein 05?04,S]8àC/8NJ; Grauert X<Y<A Stein

05/XX4X] (ii) -,W/8

QV 1.2[4,5] -SP8-WSP:9P/

T M -0>- n Q/-052[(Uj, hj)

- M /?BSYb8T

Uj ⊂⊂ M . [ Gij(z) := Jhih−1j

(z), z ∈ Ui ∩ Uj , M[ Jhih−1j

(z) - hi h−1j A

z / Jacobi c]8abA Ui ∩Uj ∩Uk ? GijGjk = Gik. ;êC/ Gij XX

M </?B4DP> T (M) ;:2WI M </-SP8;êC/ (Gtij)

−1

XX M </?B4DP> T ∗(M) ;:2WI M </-WSP2M[ (Gij)t


Stein ∂– $ 9

- Gij /TU8 T (M) 8 T ∗(M) A z ∈ M /PQ> Tz(M) 8 T ∗z (M) ;

:8 T (M) 8 T ∗(M) F<Z^ M ×M −→ M, (z, ζ) −→ z /dU.=T0

T (M ×M) 8 T ∗(M ×M).

T (M) 8 T ∗(M) /?BUN.=T0 s(z, ζ) 8 s∗(z, ζ), ^?

s(z, ζ) : M ×M −→ T (M ×M),

8

s∗(z, ζ) : M ×M −→ T ∗(M ×M).

_` M /0VVU?[>bcUj

, Y?0c0 j, ?0V?BSY

hj : Uj −→ Cn =:0?Bdf_ ψj : T (M)|Uj−→ Uj×Cn. 9S2: χj0

0<U<Uj

/ C∞@V.88abc0 T (M)W5? s(z, ζ)AÆg D ⊆ M

<Y=804DVs(j)WT2944DV

s(j);A hj

(Uj ∩D

)⊆ Cn <

/ s(j)r ?BC//4D s(j) =

(s(j)1 , · · · , s(j)

n

)7N?/8XX s(z, ζ) /P/0

∥∥∥s(z, ζ)∥∥∥ :=∑j

χj(z)n∑

r=1

∥∥∥s(j)r (hj(z))

∥∥∥ , z ∈ D.

94∥∥∥s(j)

r (hj(z))∥∥∥ -K/0 s(j)

r (hj(z)) /4D/e\ZH8

§1.2 WXX

EN/HGY<Y[eN] [4] 4/Y< 4.2.4.

Y 1.1[4] T M -0>- n Q/ Stein 052 T (M) - M </-S

P8afT M -0>h\ Stein 05<J0_>Æ8abXA0>?B8f

s : M ×M −→ T (M) 8 M ×M </?BC/ ϕ ^R=EX]4(i) 0<_\/ z, ζ ∈ M , s(z, ζ) ∈ Tz(M)(^ s(z, ζ) -P T (M) F<8f

M ×M (z, ζ) −→ z ∈M /dU</UN).

(ii) 0_\iX/ z ∈M , s(z, z) = 0,<M : Tz(M)/8f s(z, ζ)A ζ = z

/UB<-^?B/8

(iii) 0_\/ z ∈M , ϕ(z, z) = 1.


Stein ∂– $ 10

(iv) ]d Fs -; s g?/ M ×MΘ1 /8,E]2ab ϕ ∈ Fs

((M ×M) \

(z, z) : z ∈M).

(v) XA0>Y/ ℵ ≥ 0 Y? T (M) 40<_\/P/ ‖ · ‖, C/ ϕℵ‖s‖−2

A (M ×M) \ (z, z) : z ∈M - C2 /8

4 Y<4fT45 M 00h\/ Stein 05/J0_>EÆ7Y=H62Y<bF?g8

§1.3 Z[\Z]

: v = (v1, · · · , vn) 8 u = (u1, · · · , un) 0 C1– 05 M </- C1– C

/8XX

< v, u >= v1u1 + · · ·+ vnun,

ω(u) = du1 ∧ · · · ∧ dun,

ω′(v) =n∑

j=1

(−1)j−1vj

∧s =j

dvs,

Ω(v, u) =(n− 1)!

(2πi)n

ω′(v) ∧ ω(u)

< v, u >n.

]d ζ ;: M </9D2C/ vj 8 uj Ych<9_9D2LT0 ωζ,

ω′ζ 8 Ωζ . ]d vj 8 uj 4/jg9DH[0>2LMRjg9Dk6YF8EN-MR58/S@Æ>4

1. 58 Ω(v, u) -^/2^ dΩ(v(ζ), u(ζ)

)= 0, _ 〈v(ζ), u(ζ)〉 = 0.

2. ]d D - M 4/>Æ2C/ uj -?B/2L0 D </c0iTY

@C/ f ? d(fΩ(v, u)

)= ∂f ∧ Ω(v, u).

3. ]d D - M 4/>Æ2L0 D </c0- C1– C/ ψ, k?FK8

ω′(ψv) = ψnω′(v), Ω(ψv, u) = Ω(v, u).

Y 1.2[5] T M --0520A z ∈ M , u -hW<S67 Tz(M) /

8f2 v -hW<WS67 T ∗z (M) /8f8 D - M 4/>Æ2 Γ = (rij)


Stein ∂– $ 11

-0Hch< ζ ∈ D / n× n 3]2Y? detΓ = 0, : Γt = (rji), ab

ω′(Γ−1t v) ∧ ω(Γu) = ω′(v) ∧ ω(u), Ω(Γ−1

t v,Γu) = Ω(v, u).

=EORfX(Uj , hj)

- M /?BSYbÆY?07?/ j, Uj ⊂⊂

M . ]d D - M 4/>Æ2LT T (M) 8 T ∗(M) A D </[\0 T (D)

8 T ∗(D). MKXX 1.2, .AiX?Bdf_ ψj : T (Uj) −→ Uj × Cn 8

ψ∗j : T ∗(Uj) −→ Uj × Cn Y?

(z, (Gij(z))ζ

)= ψi ψ−1

j (z, ζ),(z, ((Gt

ij)−1(z))ζ

)= ψ∗

i (ψ∗j )

−1(z, ζ),

z ∈ Ui ∩ Uj , ζ ∈ Cn.

]d z ∈ Uj 8 a ∈ Tz(M)(a ∈ T ∗z (M)),abL? ψj(a) = (z, aj)(ψ

∗j (a) =

(z, aj)) /4D aj ∈ Cn W0 a F<(Uj , hj)

/;:8]d z ∈ Ui ∩ Uj ,

a ∈ Tz(M), b ∈ T ∗z (M) O ai, aj, bi, bj .=0 a, b F< (Ui, hi) 8 (Uj , hj) /

;:2ab?

ai = Gij(z)aj , bi = (Gtij)

−1(z)bj .

NJ2E`XX-]i/8QV 1.3[5] ]d z ∈ M , a ∈ Tz(M) 8 b ∈ T ∗

z (M), LY_\_`0 j

Y z ∈ Uj EXX

< b, a >:=< bj , aj >,

94 bj 8 aj - b 8 a F< (Uj , hj) /;:8

.A: D ⊆M 00>Æ2 N 00a/ C1– 052E: a : D ×N −→T (M) 8 b : D × N −→ T ∗(M) 0 C1– 8f2Y?07?/ (z, y) ∈ D × N

? a(z, y) ∈ Tz(M) 8 b(z, y) ∈ T ∗z (M). : aj : (D ∩ Uj) × N −→ Cn 8

bj : (D ∩ Uj) ×N −→ Cn 0 a 8 b F< (Uj , hj) /;:8L

ω′y(bi(z, y))∧ωy(ai(z, y)) = ω′

y(bj(z, y))∧ωy(aj(z, y)), z ∈ D∩Ui∩Uj , y ∈ N.

NJ?E`XX


Stein ∂– $ 12

QV 1.4[5] ]dD ⊆M 00>Æ2N 00a/ C1–052a : D×N −→T (M) 8 b : D × N −→ T ∗(M) 0 C1– 8f2Y?07?/ (z, y) ∈ D × N

? a(z, y) ∈ Tz(M) 8 b(z, y) ∈ T ∗z (M). : aj : (D ∩ Uj) × N −→ Cn 8

bj : (D ∩ Uj) × N −→ Cn 0 a 8 b F< (Uj, hj) /;:8LY_\_`0 j

Y z ∈ D ∩ Uj EXXA D ×N </[email protected] ω′y(b(z, y)) ∧ ωy(a(z, y))

ω′y(b(z, y)) ∧ ωy(a(z, y)) = ω′

y(bj(z, y)) ∧ ωy(aj(z, y)), z ∈ D ∩ Uj , y ∈ N.

ENORYOE`?F s(z, ζ) /XX8

QV 1.5[5] YO0`aPQ/ C∞– 8f

σ : T (M) −→ T ∗(M)

>JG< Cn 4/8f z −→ z, Y?^REMX]407? a ∈ T (M), <

σa, a >≥ 0 EO8f

‖a‖σ := (< σa, a >)12 , a ∈ T (M)

A T (M) /c0PQ<XX0P/8MF/8f σ Y=bE`38XX4]

d z ∈ Uj , a ∈ Tz(M) O aj - a F< (Uj , hj) /;:2LT σja 0 T ∗z (M) 4

/4D2>F< (Uj , hj) /;:0 aj . _`0<U< Uj / C∞– @V.8 χj

EXX

σa :=∑j

χj(z)σja, a ∈ Tz(M), z ∈M.

ET s(z, ζ) := σs(z, ζ).(MF s(z, ζ) 4Y=b1 Cn 4/8f ζ − z.)

5@ORGF Leray UN/XX8

QV 1.6[6] F< (D, s, ϕ)/ LerayUNXX00/0 (s∗,ℵ∗),94 ℵ∗ ≥0 -0Y/2 s∗(z, ζ) -0 ∂D /X0>UB4/ ζ 8 z ∈ D XX/hW<T ∗(M) / C1 8f2Y?^REMX]4(i) 07? z ∈ D 8 ∂D /X0>UB4/ ζ , s∗(z, ζ) ∈ T ∗

z (M);

(ii) 0 z ∈ D 8 ζ ∈ ∂D, ? 〈s∗(z, ζ), s(z, ζ)〉 = 0 8 ϕ(z, ζ) = 0, RC/

ϕℵ∗(z, ζ)

〈s∗(z, ζ), s(z, ζ)〉


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