On the Ohsawa-Takegoshi Extension Theorem

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On the Ohsawa-Takegoshi Extension Theorem

Kenz¯o A

DACHI

長崎大学教育学部紀要自然科学第 78 号別刷平成 22 年３月

Bulletin of Faculty of Education, Nagasaki University Natural Science No. 78 ( March 2010 )

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安達謙三教授退職記念

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安達謙三

略歴

昭和 42 年３月九州大学理学部数学科卒業

昭和 44 年３月九州大学大学院理学研究科修士課程修了

昭和 46 年３月九州大学大学院理学研究科博士課程中途退学

昭和 46 年４月茨城大学理学部助手

昭和 51 年４月長崎大学教育学部講師

昭和 53 年４月長崎大学教育学部助教授

昭和 54 年５月理学博士（九州大学）

平成元年４月長崎大学教育学部教授

学会における活動

昭和 45 年10月日本数学会会員（現在に至る）

平成５年４月九州数学教育学会会員（現在に至る）

平成 16 年４月日本数学会評議員（平成 17 年３月まで）

平成 21 年３月日本数学会代議員（平成 22 年２月まで）

研究業績

著書

１．微積分学，昭晃堂（共著），1987 年２．複素解析学，昭晃堂（共著），1988 年３．応用解析学，昭晃堂（共著），1992 年

４．複素解析学，東京電気大学出版局(共著)，1999 年５．解析学概論，開成出版，2001 年

６．解析学概論改訂版，開成出版，2003 年７．多変数複素関数論，開成出版，2003 年

８．数学概論 ― 数学史と微分積分 ―，開成出版，2005 年９．Principles of Real and Complex Analysis, 開成出版，2006 年

10．Several Complex Variables and Integral Formulas, World Scientific, 2007 年 11．Principles of Real and Complex Analysis, Second edition, 開成出版，2007 年 12．An Introduction to Analysis, 開成出版，2008 年

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主要論文

１．Extension of holomorphic mappings, Mem. Fac. Sci. Kyushu Univ. Ser. A, 24 (2) (1970), 238-241. (共著者 : M. Suzuki, M. Yoshida).

２．Cousin-II domains and domains of holomorphy, Mem. Fac. Sci. Kyushu Univ. Ser. A, 24(2)(1970), 242-248. (共著者 : M. Suzuki, M. Yoshida).

３．Continuation of holomorphic mappings, with values in a complex Lie group, Pacific J.

Math. 47(1)(1973), 1-4. (共著者 : M. Suzuki, M. Yoshida).

４．Extending bounded holomorphic functions from certain subvarieties of a strongly pseudoconvex domains, Bull. Fac. Sci., Ibaraki Univ., Math. (7)(1976), 1-7.

５．On the multiplicative Cousin problems for N^p(D) , Pacific J. Math. 80 (2) (1979) , 297-303.

６．Continuation ofA^∞functions from submanifolds to strictly pseudoconvex domains, J.

Math. Soc. Japan 32(2)(1980), 331-341.

７．Extending bounded holomorphic functions from certain subvarieties of a weakly pseudoconvex domain, Pacific J. Math. 110(1)(1984), 9-19.

８．Le problème de Lévi pour les fibrés grassmanniens et les variétés drapeaux, Pacific J. Math. 116(1)(1985), 1-6.

９．Continuation of bounded holomorphic functions from certain subvarieties to weakly pseudoconvex domains, Pacific J. Math. 130(1)(1987), 1-8.

10．ExtendingH^pfunctions from subvarieties to real ellipsoids, Trans. Amer. Math. Soc.

317(1)(1990), 351-359.

11．On the extension of Lipschitz functions from boundaries of subvarieties to strongly pseudoconvex domains, Pacific J. Math. 158 (2) (1993) , 201-222. (共著者 : H.

Kajimoto)

12．Continuation of holomorphic functions from subvarieties to pseudoconvex domains, Kobe J. Math. 11(1)(1994), 33-47.

13．Lipschitz and BMO extensions of holomorphic functions from subvarieties to a convex domain, Complex Variables, 36 (1997), 465-473. (共著者 : H. R. Cho) 14．H^p andL^p extensions of holomorphic functions from subvarieties to certain convex

domains, Math. J. Toyama Univ. (1997), 1-13. (共著者 : H. R. Cho)

15．L^p(1CpC∞) estimates for ̅∂ on a certain pseudoconvex domain inCⁿ, Nagoya Math. J. (1997), 127-136. (共著者 : H. R. Cho)

16．H^p and L^p extensions of holomorphic functions from subvarieties of analytic polyhedra, Pacific J. Math. 189(2)(1999), 201-210. (共著者 : M. Andersson, H. R.

Cho)

17．L^p extensions of holomorphic functions from submanifolds to strictly pseudoconvex domains with non-smooth boundary, Nagoya Math. J. 172 (2003), 103-110.

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On the Ohsawa-Takegoshi Extension Theorem

Kenz¯o A

^DACHI

Department of Mathematics, Faculty of Education, Nagasaki University Nagasaki, 852-8521, Japan

(Received October 30, 2009) Abstract

In this paper we give an elementary proof of the Ohsawa-Takegoshi extension theorem [OHT] by applying the method of Jarnicki-Pflug [JP].

1 Preliminaries

Let Ω ⊂⊂Cⁿbe a pseudoconvex domain and letH＝ {z∈Cⁿ|zn＝ 0 }. Then Ohsawa and Takegoshi [OHT] proved that every L² holomorphic function in H ∩ Ω can be extended to anL²holomorphic function in Ω. LetH^j,j ＝ 0, 1, 2, be Hilbert spaces. Let Djbe dense subsets ofH^j,j＝ 0, 1, respectively.

Let

T :D0→H¹, S:D1→H²

be closed linear operators such that ST ＝ 0. Let L : H¹ → H¹ be a linear bijection satisfying

(Lx,x)1B0 (x∈H¹). (1)

In this setting we have the following theorem.

Theorem 1Suppose

*(Lv,v)1*C**T^＊v**²0+**Sv**²2,

for every v∈DT^＊∩DS.Then forg∈ KerS,there exists u∈DTsuch that Tu＝g, **u**²0C*(L^-1g,g)1*.

Proof. It follows from (1) that

(L(x+y),x+y)1＝ (x+y,L(x+y))1, (L(x+iy),x+iy)1＝ (x+iy,L(x+iy))1. Then

(Lx,y)1+ (Ly,x)1＝ (x,Ly)1+ (y,Lx)1,

−(Lx,y)1+ (Ly,x)1＝ −(x,Ly)1+ (y,Lx)1. Bull. Fac. Educ., Nagasaki Univ. : Natural Science No.78，1 〜 16（2010.3）

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Thus we obtain

(Lx,y)1＝ (x,Ly)1 (x,y∈H¹).

It follows from (1) that fort∈Cwe obtain

(L(x+ty)1,x+ty)1B0.

Hence for every real numbert,

(L(x+ (Lx,y)1ty)1,x+ (Lx,y)1ty)1B0, which implies that for every real numbert,

(Lx,x)1+ 2*(Lx,y)1*²t+*(Lx,y)1*²(Ly,y)1t²B0.

Hence we have

*(Lx,y)1*²C(Lx,x)1(Ly,y)1 (x,y∈H¹).

SinceLis bijective, there exists ˜g ∈H¹such that L ˜g＝g. Thus forv∈DT^＊∩ KerS, we have

*(v,g)1*²＝*(v,L˜g)1*²C(Lv,v)1(L˜g, ˜g)1, C(L˜g, ˜g)1(**T^＊v**²0+**Sv**²2) ＝ (L˜g, ˜g)1**T^＊v**². Since (v,g)1＝ 0 forv∈DT^＊∩ (KerS)^h, we have

*(v,g)1*²C(L˜g, ˜g)1**T^＊v**²0 (2)

forv ∈DT^＊. Define a bounded linear functional ϕ:RT^＊ → C by ϕ(T^＊v) ＝ (v,g)1. Then by the Hahn-Banach theorem,ϕis extended to a bounded linear functional onH⁰. By the Riesz representaion theorem, there existsu0∈H⁰such that

ϕ(w)＝(w,u0)0, **ϕ**＝**u0**0 (w∈H⁰).

It follows from (2) that

*ϕ(T^＊v)*＝*(g,v)1*C(L˜g, ˜g)1**T^＊v**0, which implies that**ϕ**²C(L˜g, ˜g)1. Consequently,

**u0**²0C(L˜g, ˜g)1. On the other hand we have

ϕ(T^＊v)＝(T^＊v,u0)0＝(v,g)1 (v∈DT^＊). (3) Hence by (3) we have*(T^＊v, u0)0*C**v**1**g**1 for v ∈DT^＊, which implies that u0 ∈ DT^＊＊＝DT. By (3), (v,g)1＝(v,Tu0) for v ∈ DT^＊, which implies that Tu0＝g. This completes the proof of Theorem 1.

Kenzo

¯

^A^DACHI

2

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Let Ω ⊂⊂Rⁿbe a domain with C¹boundary and let ρ be a defining funtion for Ω, that is, ρ is a real-valuedC¹function in a neighborhoodGof ̅Ω and satisfies

Ω＝ {x∈G*ρ(x) < 0 },dρ(x): ＝

6

_j＝1ⁿ ^∂ρ_∂_x_j^(x)dx^j⁴^{0 (x}^∈∂Ω).

Define the surface elementdSby

dS＝

6

_j＝1ⁿ ^(-1)^j-1^ν^j^dx¹∧ · · · ∧ [dxj] ∧ · · · ∧dxn,

where, [dxj] means thatdxjis omitted, and ν＝(ν1, · · · , νn) is the unit outward normal vector for the boundary ∂Ω. If we set*dρ*＝⎧

⎨⎩

⎛⎝∂ρ

∂x1

⎞⎠

2+ · · · +⎛

⎝∂ρ

∂xn

⎞⎠

2⎫

⎬⎭

1/2, then ν can be written

ν＝ 1

*dρ*⎛

⎝∂ρ

∂x1, · · · , ∂ρ

∂xn

⎞⎠. Then we have the following:

^k＝1ⁿ ^(-1)^k-1 ^∂^∂ρ^x^k^d[x]^k

^{1 (tCl)}^{0 (tBl)}^, ^*^χ^˜^'^*^C ^1-l² ^.

For 0 < ε < 1 2, define

χε(z)＝ ˜χ⎛

⎝*zn*² ε² ⎞

⎠. Further, forf∈ ( ˜Ω), define

gε(z)＝ ̅∂⎛

⎝χε(z)f(z) zⁿ ⎞

⎠. Thengεis a ̅∂ closedC^y(0, 1) form on ˜Ω. We have

@

^Ω^*^g^ε^(z)*²^e^-^ϕ^(z)^dV(z)＝^ε¹⁴

@

^Ωε^*^f(z)*²

^*

^χ^˜^⎛^⎝^' ^*z^εⁿ²^*²^⎞^⎠

^*

²^e^-^ϕ^(z)^dV(z),

where

Ωε＝ {z∈Ω*lε²C*zn*²Cε²}, anddVis the Lebesgue measure inCⁿ. We chooseA> 1 such that

Ω ⊂C^n-1× {zn**zn*<A/2 }.

Define

γε(z)＝ 1

ε²+*zn*², ηε(z)＝ log(A²γε(z)).

Thenz∈Ω , and for ε∈(0, 1/2), ηε(z)Blog 2. Define σ(z)＝ *z*²

log 2, ψ＝ϕ+ σ.

Then we have

ηε(z)_j,k＝1

6

ⁿ _∂^∂_z_j²^σ_∂_z_̅_k^(z)w^j^w^̅^k^＝η^ε^(z)_{log 2}^*w*²^B*w*²

forz∈Ω,w∈Cⁿ, ε∈(0, 1/2). Consequently,

ηε(z)_j,k＝1

6

ⁿ _∂^∂_z_j²^ψ_∂_z_̅_k^(z)w^j^w^̅^k^B*w*² ^(z^∈Ω,^w^∈^Cⁿ^). ⁽⁴⁾

Kenzo

¯

^A^DACHI

4

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For 0Cε < 1/2, define

a

｛

ε＝ ^η^ε^{+ γ}¹ ^ε^{(ε＝ 0)}^{(ε > 0)}^. We set

H⁰＝L_(0,0)² (Ω, ψ), H¹＝L_(0,1)² (Ω, ψ), H²＝L_(0,2)² (Ω, ψ), and

Tε(u)＝ ̅∂(

^a^ε^u), ^Sε＝

^a^ε^∂,^̅ ^T^＝^T0, S＝S0. Then we have

DT_ε＝DT, DS_ε＝DS, DT_ε^＊＝DT^＊. Now we define a linear operatorLε:H¹→H¹by

Lε

（ ⁶

^j＝1^n-1^v^j^d¯^z^j⁺^vⁿ^d¯^z^j

）

^＝

⁶

^j＝1^n-1^v^j^d¯^z^j⁺ ^(ε²^+*z^ε²ⁿ^*²⁾²^vⁿ^d¯^zⁿ^.

ThenLε:H¹→H¹is bijective and satisfies

(Lε(x),x)1B0, for everyx∈H¹.

Lemma 1Let v＝6ⁿj=1vjd¯zj∈C(0,1)² ( ˜Ω).Then v∈DTε^＊if andonly if

6

_j＝1ⁿ ^v^j^(z)^∂ρ_∂_z_j^{(z)＝ 0 (z}^∈∂Ω).

Proof. Supposev＝

6

_j＝1ⁿ ^v^j^d¯^z^j^∈^C^(0,1)² ^{( ˜}^Ω)∩^D^T^ε^＊^{. Then}

(u,T^＊v)0＝ (Tu,v)1 (u∈DT), which means that

T^＊v＝−

6

_j＝1ⁿ ^e^ψ_∂^∂_z_j ^(v^j^e^-ψ^).

We set

˜

v(z)＝

6

ⁿ

j＝1vj(z)∂ρ

∂zj(z).

Suppose there existsz⁰∈∂Ω such that ˜v(z⁰)40. We may assume that Re ˜v> 0 in some neighborhoodWofz⁰. We choose a function ˜u∈C^*c(Cⁿ) with the properties that ˜uB0, On the Ohsawa-Takegoshi Extension Theorem 5

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˜

u(z⁰)> 0, supp (˜u) ⊂ W. Since ˜u∈DT, it follows from Greenʼs theorem (Theorem 2) that

(˜u,T^＊v)1＝(Tu,˜ v)2＝

@

^Ω

⁶

^j＝1ⁿ ^∂^∂^z^u^̅^˜^j^v^j^e^-ψ^dV

＝−

@

^Ω^u^˜

⁶

^j＝1ⁿ ê^ψ^∂(v^∂^jê^̅^z^-ψ^j ⁾ê^-ψ^dV⁺

@

^∂Ω

⁶

^j＝1ⁿ ^∂^∂ρ^̅^z^j^uv^˜ ^j^e^-ψ^dS

＝(˜u,T^＊v)1+

@

^∂Ω^u˜^˜^ve^-ψ^dS,

which implies that

@

^∂Ω^u˜^˜^ve^-ψ^dS^{＝ 0.}

This contradicts the choice of ˜vand ˜u. Thus we have ˜v*∂Ω＝ 0. Similarly we can prove the sufficiency. This completes the proof of Lemma 1.

Foru∈DT^＊andv∈DT, we have

(v,Tε^＊u)0＝(Tεv,u)1＝( ̅∂(

^a^ε^v),^u)¹^＝(v,

^a^ε^T^＊^u)⁰^,

which implies thatTε^＊u＝

aεT^＊u. Hence, foru＝6k=1ⁿ ukd¯zk∈C(0,1)² ( ˜Ω)∩DTε^＊, Tε^＊u＝−

aεe^ψ

6

_j＝1ⁿ _∂^∂_z_j^(u^j^e^-ψ^).

Theorem 3For0 < ε < 1/2andu∈C(0,1)² ( ˜Ω)∩DTε^＊,we have (Lεu,u)C**Tε^＊u**²0+**Sεu**²2. Proof. Using Greenʼs theorem, we have

**Tε^＊u**²0+**Sεu**²2

＝(aεT^＊u,T^＊u)0+ (aεSu,Su)2

ⁿ ^∂^∂^z^j^⎧^⎨⎩ηε⎛

⎝∂uk

∂z̅j−∂uj

∂̅zk

⎞⎠e^-ψ

⎫⎬

⎭u¯kdV +

@

j,k＝1u¯k⎛

⎝∂uj

∂̅zk

∂ρ∂zj+uj ∂²ρ

∂̅zk∂zj

⎞⎠＝Θ

6

_k＝1^￣￣￣￣ⁿ _∂^∂ρ_z_k^u^k^{＝ 0.}

Consequently,

@

^∂Ω^η^ε^j,k＝1

⁶

ⁿ ^⎛^⎝^∂^∂û^̅^z^k^j⁻^∂^∂^zû^̅^k^j^⎞^⎠^∂ρ^∂^z^jû^¯^kê^-ψ^dS

＝

@

^∂Ω^η^ε^j,k＝1

⁶

^j＝1ⁿ ^∂η^∂^̅^z^ε^j^T^＊^(u)¯^u^j^e^-ψ^dV

−

@

^Ω^j,k＝1

⁶

ⁿ ^∂η^∂^z^ε^j^⎛^⎝^∂^∂û^z^̅^k^j⁻^∂^∂^̅^zû^k^j^⎞^⎠e^-ψû^¯^k^dV

−

@

^Ω^j,k＝1

⁶

ⁿ ^η^ε^∂^∂^z^j^⎧^⎨⎩

⎛⎝∂uk

∂z̅j−∂uj

∂̅zk

⎞⎠e^-ψ

⎫⎬

⎭u¯kdV +

@

^Ω^j,k＝1

⁶

ⁿ û^¯^k^∂^∂^z^j^⎝^⎛^η^ε^∂^∂û^z^̅^k^jê^-ψ^⎞^⎠dV^.

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Since

(ηεTT^＊u,u)1＝

@

^Ω

⁶

^k＝1ⁿ ^η^ε^∂^∂^z^̅^k^(T^＊û)¯û^kê^-ψ^dV

ⁿ ^⎛^⎝^∂^∂^z^j²^η^∂^ε^̅^z^kû^jû^¯^kê^-ψ⁺ ^∂η^∂^z^ε^jû^j^∂^∂^z^̅^k^(¯û^kê^-ψ⁾^⎞^⎠dV^.

Therefore,

(*)B

@

^Ω

we obtain

(*)B

@

^Ω^j,k＝1

⁶

⁶

ⁿ ^η^ε^∂^∂^z^j²^ψ^∂^z^̅^kû^jû^¯^kê^-ψ^dV

−

@

^Ω^j,k＝1

⁶

ⁿ ^∂^∂^z^j^η²^∂^ε^̅^z^kû^jû^¯^kê^-ψ^dV

−

@

^Ω^ε^*zⁿ²^*^+*z²^*uⁿⁿ^*^*²²^e^-ψ^dV

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＝

@

^Ω^j,k＝1

⁶

ⁿ ^η^ε^∂^∂^z^j²^∂^ψ^z^̅^kû^jû^¯^kê^-ψ^dV

+

@

^Ω^γ^ε^*uⁿ^*²^(ε²^γ^ε⁻^*zⁿ^*²^)e^-ψ^dV

B

（ @

^Ω

⁶

^j＝1^n-1^u²^j⁺ ^(ε^ε²^+*z²^*uⁿⁿ^*^*²²⁾²

)

^e^-ψ^dV

＝(Lεu,u)1,

which completes the proof of Theorem 3.

The following theorem was proved by Hörmander [HR1]. We omit the proof.

Theorem 4For f＝6ⁿj=1fjd¯zj∈DT^＊∩DS,there exists a sequence{fν}with the following properties:

(a)fν∈L(0, 1)² (Ω, ψ).

(b)If fν＝6ν=1ⁿ fν,jd¯zj,then fν,j∈C²( ̅Ω).

(c)6ⁿj=1fν,j∂ρ

∂zj*∂Ω＝ 0,that is,fν∈DT^＊.

(d)**f−fν**1+**Sfν−Sf**2+**T^＊fν−T^＊f**0→ 0 (ν→∞).

Corollary 1Forgε＝ ̅∂(χεf/zn),there exists uε∈H¹such that Tεuε＝gε,and

@

^Ω^*^u^ε^*²^e^-ψ^dV^C ^(1−l)⁴ ²^ε⁶

@

^Ωε^(ε²⁺^*zⁿ^*²⁾²^*^f^*²^e^-ψ^dV.

Proof. Using Theorem 3 and Theorem 4, for 0 < ε < 1/2 andu∈DSε∩DTε^＊, we have (Lεu,u)1C**Tε^＊u**²0+**Sεu**²2.

By Theorem 1, there existsuε∈DTsuch that

Tεuε＝gε, **uε**0C*(Lε^-1gε,gε)1*.

On the other hand we have

Lε^-1gε＝ (ε²+*zn*²)² ε²

∂χε

∂̅zn

f znd¯zn, which implies that

¯

^A^DACHI

10

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This completes the proof of Corollary 1.

We set

Fε＝χεf−

^a^ε^zⁿ^u^ε^.

Since ̅∂Fε＝ 0,Fεis holomorphic in Ω. Moreover we haveFε*H∩Ω＝f. We set ˆΩε＝ {z

∈Ω**zn*Cε}. Then it follows from Minkowskiʼs inequality that

**Fε**0: ＝⎛

⎝

@

^Ω^*F^ε^*²^e^-ψ^dV⎞^⎠^1/2

C⎛

⎝

@

^Ωˆε^*^χ^ε^*²^*^f^*²^e^-ψ^dV⎞^⎠^½⁺^⎛^⎝

@

^Ω^*zⁿ^*²^*a^ε^**u^ε^*²^e^-ψ^dV⎞^⎠^½

C⎛

⎝

@

^Ωˆε^*^χ^ε^*²^*^f^*²^e^-ψ^dV⎞^⎠^½^{+ sup}^z∈Ω^*zⁿ^*

^*a^ε^*^⎛^⎝

@

^Ω^*u^ε^*²^e^-ψ^dV⎞^⎠^½^.

There exists a constantB> 0, such that

*zn*

*aε* C

^*zⁿ^*²^log

^r

^ε²^+*z^A²ⁿ^*²

^{+ 1} ^C^B.

It follows from Corollary 1 that

⎛⎝

@

^Ω^*F^ε^*²^e^-ψ^dV⎞^⎠^1/2^C^⎛^⎝

@

^Ωˆε^*^χ^ε^*²^*^f^*²^e^-ψ^dV⎞^⎠^½ ⁽⁷⁾

+ 2B (1−l)ε³⎛

⎝

@

^Ωε^(ε²⁺^*zⁿ^*²⁾²^*^f^*²^e^-ψ^dV⎞^⎠^½^.

The first term in the right side of (7) converges to 0 as ε→ 0. In order to investigate the second term in the right side of (7), we need the following lemma.

Lemma 2For ϕ∈C^*( ̅Ω),we have

ε→0+lim

@

^Ωε^(*z^ϕⁿ^*^(z)²^+ε)²dV(z)＝(1−l)π

@

^{^zⁿ^＝0}∩Ω^ϕ^(z)dV^n-1^(z),

where dV and dVn-1are the Lebesgue measures inCⁿandC^n-1,respectively.

Proof. Let 0 <εC 1/2. If we choose ε sufficiently small, then there exist a constant a> 0 and compact setsE^(ε),F^(ε)⊂C^n-1with the following properties:

E^(ε)× {lεC*zn*Cε} ⊂Ωε⊂F^(ε)× {lεC*zn*Cε} (8) and

μ(F^(ε)−E^(ε))Caε, (9)

(16)

where μ is the Lebesgue measure in C^n-1. We set z' ＝ (z1, · · · , zn-1) , z＝ (z', zn) . We define τ by τ(z)＝ϕ(z)−ϕ(z', 0). Then there exists a constant C > 0 such that

*

τ(z)*CC*zn*. On the other hand we have

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^Ωε^(ε²⁺^*zⁿ^*²⁾²^*^f^*²^e^-ψ^dV

C16

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^Ωε^(ε+*z^*^f^*²^eⁿ^-ψ^*²⁾²^dV

→ 16(1−l)π

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^H∩Ω^*^{f(z', 0)*}²^e^{-ψ(z', 0)}^dV^n-1

C16(1−l)π sup

z∈H∩Ωe^-σ(z)

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^H∩Ω^*^{f(z', 0)*}²^e^{-ψ(z', 0)}^dV^n-1^.

Consequently,

lim sup

ε→0

@

^Ω^*F^ε^*²^e^-ψ^dV^C^C

@

^H^∩Ω^*^{f(z', 0)*}²^e^{-ψ(z', 0)}^dV^n-1^, ⁽¹⁰⁾

whereC＝(64B²π)/(1−l)supz∈H∩Ωe^-σ(z).

The following lemma is well known. So we omit the proof.

Lemma 3 (Montels theorem) Let {uk} be a sequence of holomorphic functions in Ω which are uniformly bounded on every compact subset of Ω. Then there exists a subsequence{ukj}of{uk}which converges uniformly on every compact subset ofΩ.

Lemma 4 Let Ω be a bounded pseudoconvex domain in Cⁿ with C² boundary whose defining functionρsatisfies*dρ*＝ 1on∂Ω.Then there exists a constant C> 0such that for every holomorphic function f in H∩Ω,there exists a holomorphic function F in Ω which satisfies F*H∩Ω＝f and

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^Ω^*F*²^e^-ψ^dV^C^C

@

^H∩Ω^*^{f(z', 0)*}²^e^{-ψ(z', 0)}^dV^n-1^.

Kenzo

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^A^DACHI

12

On the Ohsawa-Takegoshi Extension Theorem

On the Ohsawa-Takegoshi Extension Theorem

Kenz¯o A

安 達 謙 三 教 授 退 職 記 念

安 達 謙 三

略 歴

学会における活動

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On the Ohsawa-Takegoshi Extension Theorem

Kenz¯o A

1 Preliminaries

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安達謙三教授退職記念

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