数理解析研究所講究録
1155 巻 2000 年 128-142

128

Continuation of Holomorphic Functions from Subvarieties toPseudoconvex Domains
安達謙三

長崎大学教育学部(KENZ6 ADACHI)

1. Introduction
a subvariety of $D$ . In theandLet $D$ be a bounded pseudoconvex domain inpresent paper, we give some recent results concerning holomorphic extensions from $V$ to $D$
estimates for the problenl inin some function spaces. In 1965, H\"ormander obtained. In 1970, Henkin, Grauert-Lieb and Lieb obtailled
bounded pseudoconvex domains inwithproblem in strictly pseudoconvex domains in
the uniform estimates for the
two
by
were
studiedresults,
problems
extensionsmooth boundary. Corresponding to thesedifferent methods. The one is the extension using the integral formula in the case where $D$
is a bounded pseudoconvex domain with a support function (for example, bounded strictly
pseudoconvex domains or bounded convex domains with smooth boulldary). The other istheextension using the Hilbert space theory in the case where $D$ is a general bounded
pseudoconvex domain. The main purpose of the present paper is to introduce Berndtssoll
extension theorem of Ohsawa-Takegoshi in bounded pseudoconvex
another proof of thedomains.
$V$$\mathrm{C}^{\mathrm{n}}$$L^{2}$$\overline{\partial}$$\mathrm{C}^{\mathrm{n}}$$\mathrm{C}^{n}$$\overline{\partial}$$L^{2}$$\mathrm{s}$$L^{2}$

2. Some recent resultsa real function. We denoteandDefinition. Let $D$ be an open set inwith respect to the measure
the space of square-integrable functions inby
the space of. We denote by, whereis the Lebesgue measure ill$(p, q)$ -forms with coefficients in(D. ),$\varphi\in C^{\infty}(D)$$\mathrm{C}^{n}$$D$$L^{2}(D, \varphi)$$\mathrm{C}^{n}$$cl\mu$$e^{-\varphi}d\mu$$L_{(p,q)}^{2}(D, \varphi)$$L^{2}$$\varphi$$f= \sum_{\backslash }f_{I},JdZ^{I}\wedge d\overline{z}^{J}|I|=p|J’|=q$ ’

where
We set$\sum’$

means that the summation is performed only over strictly increasing multi-irldices.
$|f|^{2}= \sum_{JI\backslash }|f_{I},J|^{2}/$,$||f||=(.

\int D|f|^{2}e^{-\varphi}d\mu)^{\frac{1}{\underline{)}}}$

129

For $f,$$g\in L_{(p,q)}^{2}(D, \varphi)$

product in

’

with

$L_{(p,q)}^{2}(D, \varphi)$A$f= \sum_{I,J}f,,Jd_{\mathcal{Z}^{I}}$’$d\overline{z}^{J},$$.q= \sum_{I.J}g_{I,J}dz^{I}$$L_{(p,q)}^{\mathit{2}}(D, \varphi)$$d\overline{z}^{J}$, we define the innerby$(f.g)= \sum_{I,J}\prime JDf_{I,J\overline{gI}}.,Je^{-}d\varphi\mu$

Then

A

.

is a Hilbert space with this inner product.

Theorem
Let $D$ be a bounded pseudoconvex open set in
the diameter of D. and let be a plurisubharmonic function in D. For every, one can then find,such thatand$1.(\mathrm{H}\ddot{\mathrm{o}}\mathrm{r}\mathrm{I}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}r[14])$$\mathrm{t}’$

’

,

let

$\delta$be$f\in L_{p,q}^{2}(D, \varphi)$$w\iota th\overline{\partial}f=0$$\overline{(}\mathit{1}>0,$$\mathrm{C}^{n}$$u\in L_{(p,q-1)}^{2}(D, \varphi)$,$\overline{\partial}u=f$$C]\prime_{D}|\mathit{1}l|2-ed\varphi V\leq e\delta^{2}./D^{\cdot}|f|^{\mathit{2}}e^{-}d\varphi V$

Theorem
Let be a bounded strictly pseudoconvex domaininwith smooth boundary. Then th, $ere$ exist a pseudoconvex domain
and functions
andandsuch thatefined for$2.(\mathrm{H}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{i}\mathrm{n}[10].

$D$\mathrm{R}\mathrm{a}\mathrm{m}\mathrm{i}r\mathrm{e}\mathrm{z}[1^{\vee}/])$$\mathrm{C}^{n}$$\tilde{D}\supset\overline{D}$$\Phi(\zeta.

$I\iota’(\zeta, \approx)$(1)$I\iota^{-}(\mathrm{t}, z)$

and(2) For every$\approx=(.$z)d$$z\in\tilde{D}$$\zeta\in\partial D$

are$\Phi(\zeta, \sim)\sim$the fnnctz on$\zeta\in\partial D$.

and continuous in$hol_{\text{ノ}}omorph_{\text{ノ}}?,c\uparrow nZ\in\tilde{D}$$\Phi(\zeta.z)$

vanishes on the(3) For any holomorphic function $f$ in $D$ that$\iota s$$\zeta\in\partial D$$cloSure\overline{D}$continuous on$\overline{D}$only at the point

and any $z\in D.$ the$\uparrow\eta te_{Jf}\mathrm{c}\gamma al\subset)rmu\prime a$$f.(z)= \int_{cJD}.f(_{\zeta}\llcorner)\frac{I\iota’((,z)}{\Phi((_{\backslash \sim}\wedge)^{\gamma 1}}.\cdot d\sigma(\zeta)$

holds. where$d\sigma\tau,s$the$(^{\mathit{6}},dn- \mathit{1})d\tau mens?onal$

Definition. Let $f(x)$ be a function 011$D$Lebesgue measure on

$f$

be a

$(0,\mathrm{c}_{1})$-forlrl with the coefficiellts.

. Then we define$|f.|_{0}=\mathrm{s}x\in J\mathrm{t}1\mathrm{p}_{J}|\mathit{1}^{\cdot}(_{i\chi}\cdot)|$Let$\partial D$$f_{i_{1}\cdots.j_{q}}$.

. Then we define.

$|f.|_{(\rangle}=_{i_{1\backslash q}^{\mathrm{m}}},\cdot.\mathrm{a}_{?}.\wedge.\mathrm{X}|f_{i_{1}}.\cdots.\mathrm{t}.q|_{(})$

Theoremconvex domain in$3.(\mathrm{H}\mathrm{c}\mathrm{n}\mathrm{k}\mathrm{i}\mathrm{I}\mathrm{l}[11].

$f/..5$$a$

Let $D$ be a bounded strictly pseudo.
smooth $boun(Jar.y$ . Then there exists a constantsuch-form, on D. then th, $ere$ exkgts-form, on $D$ zmth\mathrm{G}_{\mathrm{l}\mathrm{a}}\iota 1\mathrm{e}\mathrm{r}\lceil-\mathrm{L}\mathrm{i}\mathrm{c}[)[8],$$\mathrm{C}^{7l}?\mathit{4}^{f}i\dagger h$$\overline{\partial}(,l,dc’\infty(\mathit{0}.q+l)$$\mathrm{L}\mathrm{i}\mathrm{e}\mathrm{l})[15])$$I_{1}’$$(r, C^{\mathrm{Y}}\infty(0.q)$$\overline{\partial}n=f$and$|n|_{0}\leq I\iota^{arrow}|f|_{0}$.

$u$$that\uparrow f$

130

be a
alld letwith slnooth )(pseudoconvex domaill ill]$-?I=-$
lll‘\supset
cfl‘,
.
set
We
wlliclltransversallv.
in a neighborhoodof allthe )nlarlif ld. We dellote by
be a dolIlain in sonie
Let
)
of
tlle
bounded hololllorphic fullctiolls in . XVe also denote
havp
4
or.
tlleorPlll
we.
. Ill thisfunctions in that are
Let $D$ be$-\mathrm{t}\tilde{I}$$\dagger$$\mathrm{C}^{\prime 1}$$1\ln(1\dot{\mathrm{f}}\mathrm{t}\mathrm{r}.\mathrm{Y}^{r}$$\mathrm{a};,\mathrm{t}_{1}\cdot \mathrm{i}_{\mathrm{C}\mathrm{f}\mathrm{l}\mathrm{y}}$$\tilde{D}\mathrm{t}\mathrm{f}\overline{D}$$\Omega$$\tilde{I}\cap D$$\dot{c}^{1D}$$\mathrm{s}\mathrm{u}\mathrm{I})\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}$$\mathrm{c}\cdot \mathrm{o}\mathrm{m}_{\mathrm{I}^{)}}1\mathrm{G}\mathrm{x}$$H^{\infty}(())$$\langle)$$\Omega$$\mathrm{a}\mathrm{c}\mathrm{c}$$\backslash ‘,1$$\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{o}1_{\mathrm{o}\mathrm{I}}\iota 1\mathrm{o}\mathrm{r}\mathrm{p}\iota_{1\mathrm{i}}\langle$$[_{)1^{\tau},\mathrm{L}}\lrcorner 4^{\infty}(\mathrm{f}l)$$\mathrm{a}\mathrm{C}^{\cdot}\mathrm{e}$$\mathrm{s}\mathrm{l}$$C^{\infty}(11\overline{\Omega}$$\Omega$

Theorem

Theorem$A^{\infty}(D)$$\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{r}\mathrm{l}\mathrm{g}$$4.(\mathrm{H}(^{\mathrm{Y}}\mathrm{n}\mathrm{k}\mathrm{i}11[12]\rangle Thc^{J}re\mathrm{C}^{\lrcorner}X?\backslash 9t,.S$. Moreover,$H^{\infty}(D)$$\mathrm{a}11(1$$Ef\iota.\mathrm{s}$$5.(\mathrm{A}(\mathrm{l}\mathrm{a}c\mathrm{h}\mathrm{i}[1].

continuou8 on\mathrm{E}\mathrm{l}\mathrm{b}^{\rangle}\mathrm{t}\iota \mathrm{C}\mathrm{t}\mathrm{a}[\overline{(}])$

a$\overline{D}\dot{l}ff\dot{l}6$’There: $H^{\infty}(M)arrow$$lme(\mathit{1}7^{\cdot}\zeta’xt(^{\lrcorner}nS?on(J\beta)c\lrcorner r(r\dagger’)rE$

continuous on$\overline{M}$$C’,X?_{!}.9ts(rl?(’\mathrm{o}r\cdot\rho.\prime ctens\rho on$.

operator$E:_{-}4^{\infty}(-\prime lI)arrow$.

proved $thc^{J}orem\mathit{5}u$ $h,enD?Spse?ldoCor’ vex$ .
Remark.
.
proved theorem 4 unthouf $assum?ng$ the $trans\tau$$Henkt\eta-Leiterc’r[\mathit{1}\mathit{3}l$)$Amar[\mathit{4}J$) $er.\mathrm{s}at?ty$

donlaill in
Let $D$ be a boullded psetldo(
be a smooth mapping stlch that$\mathrm{C}^{\gamma 1}$$\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}$xvith smooth. Let$\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{r}1(1\mathrm{a}\mathrm{r}_{\mathrm{V}}$$\gamma$:$\partial D\cross$$Darrow \mathrm{C}^{n}$$((-z, \wedge)’\cdot\sum_{1}^{\prime l}=((^{-}j-\approx_{j})\wedge((\prime jzj=\{-.)\neq \mathrm{r})$Let$h_{1},$$\cdots,$$h_{l?},(777<n)\})\mathrm{e}\mathrm{h}\mathrm{o}1_{0}\mathrm{m}\mathrm{o}\mathrm{r}1)\mathrm{h}\mathrm{i}\langle$

filnctions in a$\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}$$\tilde{V}=\{z\in\tilde{D}|h1(z)=\cdots=h\gamma’ 7(\mathcal{Z})=0\}$ ,We say$V$intersects$\partial D$$\partial D\cross D$$()\mathrm{I}1$).

$\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{l}\mathrm{O}\mathrm{o}\mathrm{d}\tilde{D}$$V=\hat{\mathrm{T}}^{f}\cap D$of$\overline{D}$. Define.

tranSVerSall.v if
on$d\rho\wedge\partial h_{\mathrm{j}}\wedge\cdots\wedge\partial h_{m}\neq 0$.

$\partial V$

In the above setting, we have the following:. on
There i..s a smooth form
Theorem$mV$ andholomorphic
(such
is
that
in
which is of type (0.0) in and (n-m-l, n-m)
ifcontznuous on , then for$6.(\mathrm{S}\mathrm{t}_{\mathrm{o}\mathrm{u}}\mathrm{t}[19].

$I\iota_{\mathrm{t}}’(($\mathrm{H}\mathrm{a}\mathrm{t}_{7}\lrcorner \mathrm{i}\mathrm{a}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}[9])$(1)Moreover,$\partial V\cross\overline{1^{r}}$

$f$

$z$

$\overline{\mathrm{T}^{\gamma}}$$z)$$z\in l^{\tau}$$f(_{Z})=J_{\hat{\zeta}} \in\partial Vf(\mathrm{t}\mathrm{I}\frac{I_{1_{1}}’(\hat{\mathrm{t}}\backslash \approx)}{((-z,\gamma((,Z))n-m}\cdot$$I1_{V}^{\wedge}((, Z)$is holomorphic in $z\in D$ promded that$\gamma(\zeta, z)$is holom,orphic in $z\in D$ .

Let $D$ be a bounded convex domain with a defining function . Then we can choose$p$$\gamma_{i}(\mathrm{t}^{\mathrm{b}}, Z)=\frac{\partial\rho}{\partial(_{i}}((^{-})$

.

131

Let

$\mathrm{E}(\mathrm{f})(7_{I})\dagger)\mathrm{e}$the right hand side of (1). Then we have

Theoremboundary and letwith real analytic
Let $D$ be a bounded convex domazn in
one dimensional subvariety of $D$ defined above. Then we have$\mathrm{C}^{n}$$7.(\mathrm{A}\mathrm{d}\mathrm{a}\mathrm{c}\mathrm{l}\mathrm{l}\mathrm{i}-\mathrm{C}\mathrm{h}\mathrm{o}[3])$

$V$

be

$a$

(1) Let $1\leq p<\infty$ . If $f\in H^{p}(V)$ , then $E(f)\in H^{p}(D)$ .

(2) Suppose that$Vha\mathit{8}$no singular point8 and $1\leq p<\infty$ . If $f\in O(V)\cap L^{p}(V)$ , then$E(f.)\in O(D)\cap L^{p}(D)$.

where $O(V)$ (resp. $O(D)$ ) denotes the
A bounded domain$\zeta$).

$\subset \mathrm{C}^{n}$$\mathit{8}pace$of all holomorphzc functions in$V$(resp. $D$ ).

is an analytic polyhedroll with defining functions$\phi_{j}$if$\Omega=\{z\in \mathrm{C}n||\phi_{j}(Z)|<1,j=1, \cdots, N\}$ ,are holomorphic in some neighborhood of . We seton. We say that S2 is non-degenerat if$I=\{i_{1},\cdots,i_{k}\}$$|I|=k\leq
n$
that
is
say
strongly
.
We
that
such
for everv nlultiindex
be a regularonfor all multiindices . LetA
non-degenerate if. of codimension given by
subvariety

where the defining$\tilde{\Omega}$$\overline{\Omega}$$\mathrm{f}\iota 1\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{011\mathrm{S}}\phi_{j}$$\sigma_{I}=\{z\in\overline{\mathrm{I}f}||\mathrm{c}\rho_{j}(Z)|=1.j\in I\}$$\partial\phi_{i_{1}}\wedge\cdots\wedge\partial\phi_{i_{k}}\neq 0$$e$$\Omega$$\sigma_{T}$$\partial\varphi_{i_{k}}\neq 0$$\partial\phi_{i_{1}}\mathrm{A}\cdots$$\tilde{V}$$I$$\sigma_{I}$$\mathrm{o}\mathrm{f}.\tilde{\cap}$$7n$$\tilde{V}=\{z\in\tilde{\Omega}|h1(_{Z})=\cdots=h_{m}(Z)=0\}$

, and
where
transversal assumption that$h_{j}\in O((^{\sim}l)$$\partial h_{1}\wedge\cdots$

A $h_{rn}\neq 0$ on$\partial h_{1}\wedge\cdots\wedge\partial h_{rn}\wedge\partial\phi_{i_{1}}\wedge\cdots$A$\tilde{V}$.

We set$\partial\phi_{i_{k}}\neq 0$,$V=\tilde{V}\cap\Omega$on$\overline{V}\cap\sigma_{I}$. We impose the,multiilldex such that $|I|=k\leq n-m$ . For a strongly non-degenerate polyhedronfor
we can define the Hardy spaces$I$$\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}_{v}\mathrm{v}$$\Omega$$H^{p}( \Omega)=\{f\in O(\Omega)|\sup||f||L\mathrm{p}(\sigma_{\xi})<\infty\in>0\}$.

In the above setting, we have by applying the integral formula obtained by

Theorem$\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{d}\mathrm{t}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}[5]$:$8.(\mathrm{A}\mathrm{d}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}- \mathrm{A}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}- \mathrm{C}\mathrm{h}\mathrm{o}[3])$be a, non-degenerate analytic polyhedron. For each $f\in O(V)\cap L^{p}(V),$ $1\leq$$p<\infty$ , there exnsts $F\in O(\Omega)\cap L^{p}(\Omega)$ such that $F(z)=f(z)$ for $z\in V$ and(1) Let$\mathrm{f}f$$||F||_{L^{p}(\Omega)}\leq C||f||_{L^{p(V}})$

.

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(2) Letbe a strongly non-degeneate analytic $pof,yhedron$ . Then for all $f\in H^{p}(V)$ .
theresuch that $F(\approx)=f(z)$ for$c||f||_{H}p(V)$ .
$\Omega$$1<p\leq\infty$ ,$ex?,\mathit{8}t\mathit{8}F\in H^{p}(\Omega)$$z\in\iota_{/}^{r}and||F||_{H^{P}()}\Omega\leq$

3. Outline of the proof of the theorem of Ohsawa-Takegoshi
due to Berndtsson
In this section, we shall prove the extension theorem of Ohsawa-Takegoshi by following
the Berndtsson’s. Using
space techniques. Ohsawa and Takegoshi obtained thefollowing:$L^{2}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}[6]$$)$TheoremLetbe a bounded pseudoconvex domazn in.
We $\mathit{8}etH=\{z\in \mathrm{C}^{n}|z_{1}=0\}$ . Then there exists a $con\mathit{8}tantC$ which depends only on
the diameter ofsuch that, for any plurisubharmonzc function
on , and for any
holomorphic function on $H\cap D$ , therea holomorphic function in
that
$D$

$9.(\mathrm{O}\mathrm{h}\mathrm{s}\mathrm{a}\mathrm{w}\mathrm{a}-\mathrm{T}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{g}_{0}\mathrm{s}\mathrm{h}\mathrm{i}[16])$$C^{r\iota}$$D$$F|_{H\cap D}=f$where$d\mu$

andLemma
and let$d\mu_{1}$$F$

are Lebesgue,$\int_{D}|F|^{2}e^{-\varphi}d\mu\leq C\int_{H\cap D}|f|^{2}e^{-\varphi}d\mu_{1}$$mea\mathit{8}ures$

in

and

$\mathrm{C}^{n}$$\mathrm{C}^{n-1}$$\rho$$D\mathrm{s}?/ch$,. $re.9pect?vely$ .

Let $D$ be a bounded open set in
be a smooth defining function for $D$ . For$1.(\mathrm{H}_{\ddot{\mathrm{O}}}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}[14])$$\partial D$$D$$\varphi$$exi\mathit{8}tS$$f$$\mathrm{C}^{n}$with smooth bounda
and$r\mathrm{y}$$f= \sum_{J}’f_{J}d\overline{Z}^{J}\in C_{(q)}^{1}(0.\overline{D})$$u= \sum_{\mathrm{A}},u_{K}d_{\overline{\mathcal{Z}}}\prime K\in C_{(0,q-1}^{1}())\overline{D}$, the following equalitv is valid.

$( \overline{\partial}u, f)=-\int D\sum_{1}\sum_{\mathrm{A}’j=}uK\overline{\delta jfj\mathrm{A}\prime}e-\varphi d\mu/n+\int\partial D\prime f\sum_{\mathrm{A}}\mathrm{t}\iota K\sum_{j=1}j\mathrm{A}^{-\frac{\partial\rho}{\partial z_{j}}}/\gamma\iota‘\supset-\varphi_{\frac{dS}{|\partial\rho|}}$Definition. For $u\in C^{1}(D)$ , define,$\delta_{j}u=e^{\varphi}\frac{\partial}{\partial\approx_{j}}(ue-\varphi)=\frac{\partial u}{\partial\approx_{\mathrm{j}}}-\frac{\partial\varphi}{\partial\approx_{j}}u$

For $C^{1}(0, q)$ -form$f= \sum_{|J|=q}f_{J}d\overline{\approx}\prime J$, define$\partial_{k}u=\frac{\partial u}{\partial\tilde{4}k}$$\overline{\partial}_{k}u=.\frac{?u}{\partial\overline{z}_{k}}($.

. We define$\overline{\partial}^{*}f=-\sum_{J\mathrm{i}’}\sum_{j}’=’?1\delta jf_{j}Kd\overline{\approx}^{K}$$f \in \mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial}^{*})\Leftrightarrow\sum_{j=1}^{n}f_{jK}\frac{\partial\rho}{\partial z_{j}}=0$.

on$\partial D$

for all$I\iota^{-}$

.

133

We say satisfies the boundary condition if
condition, we have from lemma 1$f$$f\in \mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial}^{*})$$(\overline{\partial}_{\mathrm{t}/,f}.)=(\mathrm{t}\iota,\overline{\partial}*f)$Lemma$\mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial}^{*})$$2.(\mathrm{H}_{\ddot{\mathrm{O}}\mathrm{r}\mathrm{D}1}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{r}[14])$. For$\varphi\in C^{\infty}(\overline{D})$$=$$||\overline{\partial}^{*}\alpha||+||\overline{\partial}\mathfrak{a}||^{2}$$+$$\backslash h^{\gamma}\mathrm{e}\mathrm{a}_{\rangle}^{1}‘,\mathrm{s}\iota 1111‘)$$L^{2}(D, \backslash p)$tllatlneansLemma 3. Let$\overline{D}$$\mathrm{e}$a smooth$(0_{C},1)$-form in$\overline{D}$

and a

$\in$

we have$\sum_{\mathrm{A}k}’\sum_{=j,1}^{n}./D^{\cdot}\alpha jh^{r}\overline{\mathrm{o}kJ\mathrm{i}\prime}\frac{\partial^{2}\hat{\Psi}}{\partial\approx_{j}\partial\overline{z}_{k}}e^{-}\varphi d\mu+\sum,\sum_{1j=}^{n}\int_{D}’|\frac{\partial\alpha_{J}}{\partial_{\sim j}^{-}\sim}|^{2}e^{-}d\varphi\mu$.

$\sum_{\mathit{1}\mathrm{i}’k}\sum_{=j,1}^{n}\int\prime j\alpha K\partial D\overline{(X_{\mathrm{A}}.K}\frac{\partial^{2}\rho}{\partial_{\wedge j}\sim\partial\overline{z}_{k}}e^{-}\varphi_{\frac{dS}{|\partial\rho|}}$is a snloothinfrom lemnla 3 to lemma 7. Thus $f\in$. We omit the proof of. since the detailed proof of.
$\mathrm{f}_{\mathrm{l}\mathrm{n}\mathrm{C}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\varphi$satisfies the boundary$\overline{D}$$\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{I}\mathrm{n}\mathrm{a}3$$\mathrm{i}_{11}[6]$

${ }$

in

)

$f$

.

$\alpha=|.’|=\sum_{q}\mathrm{o}_{J}d\overline{\approx}’;1$$f\in L^{\mathit{2}}(D)$lelIlma .3 is givell$(0.1)$ -formLet. When$uj\dagger$)$\mathrm{e}$a real valued smooth function insatisfvillg the boun($\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{y}$$\overline{D}$.

$\alpha=\sum_{j=1}^{n}\alpha_{j}d\overline{z}_{j}$is a smoothcondition. Then we have

$J_{D}^{\cdot}u’ \sum_{kj_{\backslash }=1}\hat{\Psi}jk\alpha j\overline{\alpha_{k}}e-\varphi d\mu-n.\int DdwjkC\iota_{j}\overline{\alpha_{k}.}e^{-}\mu\varphi$$+. \prime_{\int y}u\cdot|\overline{o}*()\mathrm{i}|^{2_{(}-\varphi}\rangle Cl\mu+\int I)u’ j,\sum_{k=\mathit{1}}^{n}|\frac{\partial 0_{\mathrm{A}}}{\partial\overline{\approx}_{j}}.|^{2}C^{J^{-\varphi}}d\mu+.\int_{CJ}Ij.-w\sum_{1j,k=}^{n}\rho?^{k}.\varphi\frac{dS}{|\partial\rho|}\alpha_{j}\overline{\alpha_{k}}e$$=\underline{\nu}\mathrm{R}\mathrm{e}./,\cdot)w\overline{o}\overline{\partial}^{*-}\alpha\cdot\overline{\alpha}ed\varphi\mu+./D^{\cdot}u’|\overline{\partial}_{C\}}|2-\varphi d\epsilon\mu$Definition. Letof$\psi\in C^{\infty}(\overline{D})$and.

$\alpha=\sum_{j=1}^{n}\alpha_{j}d\overline{\approx}j\in C_{(0_{\backslash }\mathrm{J})}^{\infty}(\overline{D})$. We define the inner product$.q=\iota’\overline{\partial}(_{\wedge}\sim_{1}-)1\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{d}(\}\mathrm{b}.\mathrm{Y}$$<g.r \supset>=\sum_{j=}’\gamma \mathrm{J}<?/\cdot\frac{\partial}{\partial_{\sim j}^{-}\sim}(\sim-^{1})\vee 1’\alpha_{j}>=1\mathrm{i}_{\mathrm{I}\mathrm{n},arrow 0}./_{/}arrow)c\psi(_{\sim}\wedge)\frac{\partial}{\partial_{\sim 1}^{-}\wedge}(\frac{\overline{z}_{1}}{|_{\sim 1}\sim|^{2}+\hat{\mathrm{c}}})\overline{\alpha_{1}(z)}e^{-}d\varphi \mathrm{t}\approx\rangle\mu(z)$

.

134

Moreover. if we defille$h_{\sim}.(_{\sim}-)= \psi(_{\sim}\sim)\overline{o}(\frac{\overline{\approx_{1}}}{|_{\sim 1}^{\sim}|^{\underline{y}}+\overline{\mathrm{C}}}.)$.

then we obtaill(2)$<.(J \cdot\alpha>=\mathrm{i}_{1\iota 1}(h\overline{\vee}\alpha\frac{1}{\mathrm{c}\tilde}arrow 0arrow’)$.

In view of lemma 6, tlle right hand side of (2) exists. For $u\in L^{1}(D)$ and a$D$ with compact support, we define$<\overline{\partial}?l.\alpha>=(u.\overline{\partial}^{*}(\mathrm{b})$$(()$. -fornl

(

$1)$

$\}$

ill

.

Then we have the following::,moothLemma 4. Let $D$ be a bounded strictlv pseudocollvex domain ill). Let {( $\in L^{1}(D)$ . If theandfunctioll inboundary. Let be a$\mathrm{C}^{tl}\tau\backslash ’\mathrm{i}\mathfrak{f}\mathrm{h}$$f$$\overline{D}$$\mathrm{h}\mathrm{i}\mathrm{c}$$g=f \overline{\dot{\mathrm{c}}1}(\frac{1}{z_{1}})$$\mathrm{h}_{0}1_{01\mathrm{n}\mathrm{o}\mathrm{r}}1$

equality$<.c/,$ $\alpha>=\int_{D}u\overline{\overline{\partial}^{*}\alpha}e-\varphi dl\iota$

holds for any closed
the sense of distribution.
$\overline{\partial}$$\alpha\in C_{(0.1)}^{\infty}(\overline{D})_{1\mathrm{v}\mathrm{h}\mathrm{i}}\mathrm{c}1_{1}$Proof. Let

$a$

be a

$C^{(\infty}(0,1)$

satisfies thethell.q$\mathrm{I}_{\mathrm{J}\mathrm{O}11\mathrm{I}}1(1\mathrm{a}\mathrm{r}\mathrm{y}((11\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n},$$=\overline{\partial}n$in-form in $D$ with compact support. We define$\mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial})=\{g\in L_{(}2(0,q)D,\hat{\Psi})|\overline{\partial}g\in L^{2}(\mathrm{t}0.q+1)D,\dot{\Psi})\}$

For Laplace-Beltrami operator$\square =\overline{\partial}\overline{\partial}^{*}+\overline{\partial}^{*}\overline{\partial}$:.

$L_{()}^{2}0,1(D,\hat{\Psi})arrow L_{(0,1)}^{\mathit{2}}(D,\wedge,))^{\llcorner}$’

defille$\mathrm{D}\mathrm{e}\mathrm{f}(\square )=\{\alpha\in L_{()}^{2}0,1(D, \varphi)|\alpha\in \mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial}),\overline{\partial}a\in \mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial}*), \mathfrak{a}\in \mathrm{D}\mathrm{e}\mathrm{f}(\overline{\partial}^{*}),\overline{\partial}^{*}\alpha\in \mathrm{D}\mathrm{e}\mathrm{f}(\overline{o})\}$,.

$\mathcal{H}=\{\alpha\in \mathrm{D}\mathrm{e}\mathrm{f}(^{\coprod)1} \coprod_{(\}=}0\}$

is a closed subspace of the Hilbert space
Then
be the orthogonal projection. From the theory of thesuch that:
Neumann operator. Let $H$ :Neumann problem, there exists$\Lambda/’$$\overline{\partial}$$L_{()}^{2}0_{\nu}1(D, \varphi)arrow \mathrm{D}\mathrm{e}\mathrm{f}(\square )$$\alpha=\overline{\partial}\overline{\partial}^{*}\Lambda’\alpha+\overline{\partial}*\overline{\partial}N\alpha+Ha$

For$\beta\in \mathcal{H}$$L_{(0,1)}^{2}(D,\dot{\Psi})arrow \mathcal{H}$$L_{(0,\mathrm{l})}^{2}‘(D, \varphi)$$\mathcal{H}$.

. we have$0=(\square \mathcal{B}./\mathit{3})’=(\overline{\partial}\overline{\partial}^{*}\beta./\mathit{3})+(\overline{\partial}*\overline{\partial}/\mathit{3}, \theta)=||\overline{\partial}^{*}\beta||2+||\overline{\partial}\theta||^{2}$

.

135

Hence we obtairl

Frorfi lemma 2, it holds that$\overline{\partial}\beta=\overline{\partial}^{*}i=0.$$0=||\overline{\partial}\beta||^{2}+||\overline{\partial}^{*}\beta||^{2}$Thus( $j_{j}\mathrm{i};$, holomorphic in $D$ and$\geq$$j,k= \sum_{1}^{\iota}\gamma||\frac{\partial/’\mathit{9}_{j}}{\partial\overline{z}_{k}}||2\int_{\partial D}+j,k=1\sum^{n}\frac{\partial^{2}\rho}{\partial z_{i}\partial_{\overline{Z}_{k}}}\beta_{j}\overline{\beta}_{k^{\frac{dS}{|\partial\rho|}}}$$\geq$$\sum_{j,k=1}^{n}||\frac{\partial\beta_{j}}{\partial\overline{z}_{k}}||2+c\int_{\partial}D|\beta|^{2}\frac{dS}{|\partial\rho|}$

$0$

in

$\partial D$.

so that $\beta=0$ . Therefore$\alpha_{1}=\overline{\partial}\overline{\partial}^{*\iota^{r}},/\alpha$,$\alpha_{2}=\overline{\partial}^{*}\overline{\partial}.\iota_{\alpha}’$$\mathcal{H}=0$. We set.

Sillce Neumann operator maps smooth $(0,1)$ -forms to smooth $(0,1)$ -forms in the strictly.
are both smooth $(0,1)$ -forms in . Obviously,andpseudocollvex domain $D,$
. On the other. Hence, then by lenlma 1
Ifhand, froln lemma 1, for an. smooth function on , we have$\overline{\partial}\alpha_{1}=0$$\overline{D}$$\alpha_{2}$$\alpha_{1}$$\alpha_{2}\perp \mathrm{K}\mathrm{e}\mathrm{r}(\overline{\partial})$$(\beta, \alpha_{2})=(\overline{\partial}\mathcal{B},\overline{\partial}\Lambda^{r_{\alpha}})=0$$\overline{\partial}\beta=0$$\overline{D}$$\beta$$\mathrm{v}$$0=( \overline{o}_{\beta,\alpha_{2})}=(\beta,\overline{\partial}^{*}\alpha 2)+\int_{\partial D}\beta\overline{\alpha_{2}\cdot\partial p}e^{-}\frac{dS}{|\partial\rho|}\varphi$. ThereforeThusboundary condition. If we set$\overline{\partial}^{*}\mathrm{c}\nu_{2}=0$$\alpha_{2}$.

satisfies the boundary condition. Hence.

$h_{\in}(z)=f(z) \overline{\partial}(\frac{\overline{z_{1}}}{|z_{1}|^{2}+\epsilon})$$\alpha_{1}$satisfies the,then we have$<g,$ $\alpha_{2}>=\lim_{\inarrow 0}(h\alpha)\Xi’ 2=^{\mathrm{o}}$.

Thus we have$<g,$ $\alpha>=<g,$which means$g=\overline{\partial}u$$\alpha_{1}>=\int_{D}u\overline{\overline{\partial}^{*}\alpha_{1}}e^{-\varphi}d\mu=\int_{D}u\overline{\overline{\partial}^{*}\alpha}e-\varphi d\mu=(u,\overline{\partial}*\alpha)=<\overline{\partial}u,$$\alpha>$,.

be a non-negative real valued
be the same as in lemma 4. Let
is integrable. If the inequality
function in $D$ with the property that
Lemma 5. Let$\lambda$$g$$\frac{1}{\lambda}$$|<g,$$\alpha>|^{2}\leq C\int_{D}|\overline{\partial}^{*}\alpha|^{2}\frac{e^{-\varphi}}{\lambda}d\mu$which satisfies the boundary condition, then there

holds for any closed
exists $u\in L^{1}(D, \varphi)$ such that$\overline{\partial}$$\alpha\in C_{(0,1)}^{\infty}(\overline{D})$$\overline{\partial}\mathrm{z}\iota=g$,$\int_{D}|u|^{2}\lambda e^{-\varphi}d\mu\leq C$

.

136

Proof. Let
be the space of all
boundary condition. We set$C_{b}^{\infty}(\overline{D})$$\overline{\partial}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}c\infty(\mathrm{O},1)$$F=\{\overline{\partial}*\alpha|(y\in C^{\infty}(b)\overline{D}\},$

Then

$F$

$w=\overline{\partial}^{*},\alpha$is a vector subspace of. We define$L^{2}(D, \varphi_{1})$-forms in$\varphi_{1}=\frac{e^{-\varphi}}{\lambda}$

which satisfies the.

. For $w\in F$ . there exists$\Phi(u’)=<g,$$\overline{D}$such that$\alpha\in C_{b}^{\infty}(\overline{D})$$\alpha>_{\mathrm{A}}$Then $\Phi(w)$ is independent of the choice of . Also. we have$\alpha$$|\Phi(u\mathit{1})|2\leq c||w||_{\varphi 1}^{2}$,$||\Phi||\leq\sqrt{C}$.

Thus
is a bounded anti-linear operator on . From the Hahn-Banach theorem.
isextended to a bounded anti-linear operator on. From the Riesz representation
theorem, there exists
such that$\Phi$$F$$\Phi$$L^{\mathit{2}}(D, \varphi_{1})$$v\in L^{2}(D, \varphi_{1})$$\Phi(w)=(v, w)_{\varphi_{1}}$,$||v||_{\varphi_{1}}=||\Phi||\leq\sqrt{C}$.

Therefore we have$<g,$ $\alpha>=\Phi(w)=(\mathrm{C}),$If we set$u= \frac{v}{\lambda}$$w)_{\varphi_{1}}= \oint_{D}\uparrow)\overline{\overline{\partial}^{*}\alpha}\frac{e^{-\varphi}}{\lambda}$,.

$\int_{D}|\iota||^{\mathit{2}}\frac{e^{-\varphi}}{\lambda}d\mu=||1^{\cdot}||_{\varphi_{1}}^{\mathit{2}}.\leq C’.$

, then$\int_{D}|u|^{2}\lambda e-\varphi d\mu\leq C’$

,

$<g,$

$\alpha>=\int_{D}u\overline{\overline{\partial}^{*}\alpha}\theta^{-}d\varphi\mu$.

On the other hand, we have$\int_{D}|u|e^{-}\varphi d_{l^{\mathit{1}}}\leq\int_{D}\frac{|v|^{2}}{\lambda}e^{-\varphi}d\mu\int_{D}\frac{e^{-\varphi}}{\lambda}d\mu\leq C\int_{D}\frac{e^{-\varphi}}{\lambda}d\mu<\infty$

Thus, $u\in L^{1}(D, \varphi)$ . From lemma 4, we obtain

Lemma 6. For$\overline{\partial}u=g$.

. it holds that$\varphi\in C\infty(\overline{D})\text{ノ}$$\lim_{\epsilonarrow 0}\int_{D}\frac{\epsilon}{(|z_{1}|2+\epsilon)^{2}}\varphi(z)d\mu(Z)=\tau_{\mathrm{I}}\int_{\mathrm{f}\}}z_{1}=0\mathrm{n}D\tilde{z}\varphi(Z)d\mu_{1}()$

where$d\mu$

and$d\mu_{1}$are Lebesgue measures in$\mathrm{C}^{n}$and$\mathrm{C}^{n-1}$.

. respectively.

.

137

Lemma 7. Let $D$ be a bounded strictly pseudoconvex donlain
with smoothboundary and $D\subset\{z||Z_{1}|\leq 1\}$ . Let be a smooth )function inand$(0.1)$
let be a-forln in which satisfies the boulldary collditioll. Then,smoothfor $0<\delta<1$ . we have$\mathrm{i}\mathrm{I}\mathrm{l}\mathrm{C}^{n}$$\overline{D}$$1_{\mathrm{t}\mathrm{l}\mathrm{r}}i\mathrm{S}\mathrm{U}\mathrm{b}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c}$$\varphi$$\overline{\partial}\mathrm{c}\cdot \mathrm{l}\mathrm{o}S\mathrm{e}\mathrm{d}$$\mathrm{I}$$\overline{D}$$\alpha$$./_{\{\}D} \cdot\approx_{1}=0\cap|01|^{\mathit{2}}e^{-}d\varphi\mu_{1}\leq\frac{2}{7,}(1+\frac{1}{\delta^{2}})./_{D}\cdot\frac{|\overline{\partial}^{*}\alpha|^{2}}{|z_{1}|^{2\delta}}e^{-\varphi}d\mu$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}.\cdot$.

For $0<\delta<1$ . we set$u^{\delta}.=1-|_{\sim}-|1^{\cdot}=1-2\delta(_{Z)}1^{\wedge}\sim-1\delta$.

Fronl lemma 3: we llave$\int_{\iota)}\iota l)\sum_{=}^{\prime l}\grave{\delta}.\varphi_{jj}k\alpha j,k\mathrm{l}\overline{\alpha_{k}}e-\varphi d\mu+\delta^{2}\int D\mu|z\mathrm{l}|^{2\delta 2}-2|_{C\mathrm{J}}1|e^{-}d+\varphi.\int_{D}w^{\delta}|\overline{\partial}^{*}\alpha|^{2}e^{-\varphi}d\mu$$+ \mathit{1}_{\mathit{1})}^{u^{\delta}\sum^{n}}.,j\backslash k\cdot=\downarrow|\frac{\partial\alpha_{j}}{\partial\overline{\approx}_{k}}|^{2}c^{-\varphi}d\mu+.\int\partial D\rho w\sum_{1}^{n}\delta\cap jk.j\overline{0k}e-\varphi\frac{dS}{|\partial\rho|}j,k=$.
$=2{\rm Re} \int_{D}u^{\delta}’\overline{\partial}\overline{\partial}^{*}\alpha\cdot\overline{\alpha}e^{-\succ^{\neg}}d\mu$$\mathrm{H}\mathrm{e}\mathrm{I}\mathrm{l}\mathrm{t}\cdot \mathrm{e}$.

we llave$\delta^{2}./D|\approx_{1}|^{\mathit{2}\overline{\delta}-}\mathit{2}|(\lambda 1|^{2}e^{-\varphi}dfr+./Du^{\delta}|\partial^{*}\alpha|2-c’\hat{\mathrm{v}}d\mu\leq 2{\rm Re}\int_{D}w^{\delta}\overline{\partial}\overline{\partial}^{*}\alpha\cdot\overline{\alpha}e-\varphi d/l$$=2{\rm Re}( \overline{\partial}^{*}l1.\overline{\partial}^{*}(1l’ lx)\delta)=2{\rm Re}(\overline{\partial}*,\delta-\mathfrak{n}.\tau l\overline{o}*(\}j\mathrm{t}\sum_{=}^{l}\gamma\frac{\partial u^{\delta}}{\partial z_{j}},\alpha_{j})$$\overline{\partial_{U^{1\delta}}}$$=^{\underline{\eta}} \int_{D}\mathrm{t};^{\overline{\delta}}’|\overline{\partial}^{*}\cap|^{2,\varphi}C-(t\ell\iota-2\mathrm{R}\{^{\mathrm{Y}}\oint D1^{\mathrm{t}^{\overline{\mathrm{J}}}}\overline{\partial}*\alpha_{\overline{\partial_{\sim 1}^{\sim}}}\cap-\varphi_{(l\mu}$$\leq^{\underline{\eta}}\int_{Ij}\mathrm{t}l/’|\overline{\partial}^{*}\alpha|^{2}.‘ \mathit{2}^{-\varphi}d\delta \mathit{1}\mu+^{\underline{\eta}}\cdot|\overline{\partial}*|(1\delta|\sim 1|^{\delta 3}\underline{.)}-|\prime 1_{1}|D\wedge \mathrm{e}-\varphi d\mu$$\leq 2\oint_{L)}u^{\grave{\delta}}’|\overline{\partial}^{*}\alpha|^{2\varphi}rJ-d\mu+2./I^{\cdot}J|\overline{\partial}^{*}\alpha|^{2}|_{\sim}\sim 1|^{2\delta}e^{-\varphi}d\mu+\underline{\frac{1}{9}}\int_{D}\grave{\delta}^{2}|\alpha_{1}|2|_{Z_{1}}|2\delta-2e-\varphi d\mu$.

Thus we have$\frac{1}{9,arrow}\tilde{\delta}^{\mathit{2}}J_{lJ}^{\cdot}|_{\sim}-|^{2}1|_{1_{1}}‘|^{2}C’\delta-\mathit{2}-\varphi d\mu\leq./D^{\cdot}(1-|\approx_{1}|^{\mathit{2}}\tilde{\delta}\rangle|\overline{\partial}*|0(\supset-\varphi d22\mu+./\mathit{1}^{\cdot})|\overline{\partial}^{*}O|^{2}|_{Z}1|^{2}\delta\rho^{-}d\varphi\mu$.

$=J_{D}|\overline{\partial}^{*}’\iota|2_{t^{J}}-_{\vec{r}}d\mu+./D^{\cdot}|\overline{\partial}^{*}a|^{\mathit{2}}‘|\mathcal{Z}1|^{2}\delta\varphi d_{l}e^{-}l\leq 2J_{D}|\overline{\partial}*(\lambda|^{2}e^{-\varphi}d\mu$138Therefore. for $\circ<\delta<1$ . we(3)$\mathrm{o}\mathrm{l}$)$\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}$.

$\delta^{2}./D^{\cdot}|_{\sim}^{-}1|^{2}\delta-2|a1|^{2-\varphi}ed_{l}l\leq 4.//\cdot J|\overline{\partial}^{\star}(\mathrm{t}|^{2-\succ}\epsilon Jd\neg\mu$

On the other hand, we set$w_{\epsilon \mathrm{i}}= \frac{1}{\pi}\log\frac{1}{|\approx_{1}|^{2}+\epsilon}$We$\mathrm{a}_{\mathrm{I}^{)}}\mathrm{P}^{1}\mathrm{y}$lemma 3 to’$u_{\epsilon}$

alld let$\epsilonarrow 0$.

$u’– \frac{1}{\pi}\log\frac{1}{|_{\sim 1}^{\sim}|^{2}}$.

, then by leInnla 6.

$\int\{^{-_{\mathrm{i}}},=0\}\cap D\mu|\alpha_{1}|2-C\lrcorner\varphi d1+.[D’ \mathrm{t}u\cdot|\overline{\partial}*\alpha|2(--\varphi d_{l}\leq\underline{9}{\rm Re}\int_{D}\{l^{1}\overline{\partial}\overline{\partial}^{*}\alpha\cdot\overline{rX}cJ^{-}\varphi d\mu$

By the same calculatioll as the first llart alld)$\mathrm{V}1\mathrm{l}(.3)$$\dot{r}\mathrm{t}1^{)}1^{1\mathrm{i}\mathrm{g}}$to $0<\delta<1$ . we llave$./_{\{=0\}\cap} \cdot\approx_{1}O|\alpha_{1}|2e-\varphi d\mu_{1}\leq\frac{1}{\pi}\int_{D}\log\frac{1}{|_{\sim 1}^{\sim}|^{2}}|\vec{\partial}*\alpha|^{\mathit{2}}e^{-}\varphi dl^{x+\frac{2}{\pi}}./r.)|\overline{\partial}^{*}\alpha|\frac{|(\lambda_{1}|}{|\approx_{1}|}\epsilon-\varphi d\mu$$\leq\frac{1}{\pi}\int_{D}\log\frac{1}{|_{\sim 1}\sim|^{2}}|\overline{\partial}^{*}\alpha|2d\mu e^{-_{\hat{\mathrm{v}}}}+\frac{2}{\Gamma_{1}}\int_{D}\frac{|\overline{\partial}^{*}\alpha|^{2}}{|\approx_{1}|^{2\delta}}.e^{-\varphi}d_{l^{l}}+\frac{1}{2\pi}\int_{y},|_{Z_{1}||_{0|(}}2\delta-212-\supseteq\hat{\prime}d_{l}l$$\leq\frac{1}{\Gamma \mathrm{t}}./_{D}\cdot\log\frac{1}{|z_{1}|^{\mathit{2}}}|\overline{\partial}^{*}\alpha|2e^{-}\varphi d\mu+\frac{2}{\pi}\int f).\frac{|(\overline{)}^{*}\alpha|^{2}}{|\approx\iota|^{2\delta}}e-\varphi d\mu+\pi.\overline{\delta^{2}}\int_{r}\underline{\supset}|\overline{\partial}*C)\iota|^{\mathit{2}}e^{-\vee^{\wedge}}d\mu$.

$\leq\frac{1}{\overline{/\downarrow}\delta^{2}}\int_{D}\log\frac{1}{|z_{1}|^{\mathit{2}}\delta}|\overline{\partial}^{*}\alpha|2(^{\overline{J}}-\varphi d\mu+\frac{2}{\pi}J^{\cdot}D\frac{|\overline{\partial}^{*}\alpha|^{2}}{1_{\sim\iota 1^{2\grave{\delta}}}^{\sim}}\epsilon^{J}-\varphi d\mu+\frac{\underline{9}}{\overline{/1}\delta^{2}}.[_{D}|\overline{\partial}*\Omega|2\mathrm{e}\lrcorner-\varphi d\mu$Using the fact that $x( \log\frac{1}{x}+2)\leq 2$ for $0<x\leq 1$ , we have$\int_{\{_{\mathcal{Z}_{1}}=0}\}\cap D\mu_{1}|(\mathrm{y}_{1}|^{2}e^{-}d\varphi\leq\frac{2}{\pi}./_{D}\cdot\frac{|\overline{\partial}^{*}(\}|^{2}}{|z_{1}|^{2}\delta}e^{-\varphi}d\mu+\frac{1}{\pi\delta^{2}}\int D\frac{2|\overline{\partial}^{*}\alpha|^{2}}{|_{\sim 1}\sim|^{2\delta}}e-\hat{\Psi}d\mu$,$= \frac{2}{\pi}(1+\frac{1}{\delta^{2}}).\int_{D}\frac{|\overline{\partial}^{*}\alpha|^{2}}{|\tilde{4}1|^{2\delta}}e^{-}’\varphi l\mu$which completes the proof.
Lemma 8. Let $D$ be a pseudoconvex domaill ina holomorphic function in $X$ . If $H$ is$1_{\mathrm{o}(\mathrm{a}}11_{\mathrm{L}}\mathrm{v}$then there exists a holomorphic function$\tilde{H}(z)=f(z)$ for.
$\sim-\in X$$\mathrm{C}^{n}$

and $X=\{z\in D|z_{1}=0\}$ . Let

integrable in$\tilde{H}$$D$

and satisfies

in $D$ such that

$f$

be

$\overline{\partial}H=f\overline{\partial}(_{\wedge}^{\underline{1}}\sim 1)$$\tilde{H}(z)=\approx_{1}H(z)\mathrm{a}.\mathrm{e}$

.

. and

139

Proof. There exists a neighborhood of $X$ in $D$ such that can be extended to be
in . Let X
be a function such that $\chi=1$ in a neighborhood of $X$.
andin $D$ . We set$f$$\omega$$1101_{0}\mathrm{r}\mathrm{r}\mathrm{l}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{c}$$\mathrm{i}11\ ^{1}$)$,\mathrm{C}$$\in C^{\infty}(D)$$\omega^{1}$$0\leq\backslash \leq 1$$\mathrm{u}\mathrm{p}\mathrm{p}(\chi)\subset\omega$$\mathrm{L}v=.\frac{f\overline{\partial}\chi}{z_{1}}$Tllell$\sim|$

satisfies that.

. Define$\mathrm{A}’\in C_{(0,1}^{\infty}())D.\overline{\partial}\omega=0$$G= \frac{\chi f}{z_{1}}-H$

then

$G$

,

is locally integrable. Since we have,$\overline{\partial}G=\overline{\partial}(\iota f.)+_{\mathrm{t}}f\overline{\partial}\approx_{1}\underline{1}(\frac{1}{z_{1}})-\overline{\partial}H=f\overline{\partial}_{\lambda^{\frac{1}{z_{1}}}}+\mathrm{x}f\overline{\partial}(\frac{1}{z_{1}})-\overline{\partial}H=\overline{\partial}x\frac{f}{z_{1}}=\omega$

there existb a smooth’$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\cdot \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\tilde{G}$in $D$ such that$\tilde{G}=G\mathrm{a}.\mathrm{e}.$. We set

X $(z)f(Z)-Z1\tilde{c}_{(\approx})=\tilde{H}(Z)$ ,thell we have$z_{1}H(z)=\tilde{H}(z)\mathrm{a}.\mathrm{e}$. and $\tilde{H}(z)=f(z)$ for $z\in X$ . Moreover we have$\overline{\partial}\tilde{H}(z)=(\overline{\partial}_{\lambda}(\mathcal{Z}))f(z)-Z_{1}\overline{\partial}\tilde{G}(Z)=(\overline{\partial}x(\approx))f(Z)-z_{1}\omega(z)=0$

Hellce$\tilde{H}(z)$is$\mathrm{h}\mathrm{o}1_{01}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}_{\mathrm{C}}$.

in $D$ .

Lemma 9. Let $D$ be an open set
and let $K\subset D$ be a compact set. Then there
exists a constantthat for any holomorphic function in $D$ and any neighborhoodof$\mathrm{i}_{11}\mathrm{C}^{n}$$C_{\mathrm{S}11}\mathrm{c}’ \mathrm{h}$$f$$I_{1^{-}}$$d^{\prime)}$$\sup_{h-}|f|\leq C||f||_{L^{1}(\omega})$.

Lemma 10. Let
be a sequence of holomorphic functions in $D$ which are uniformly
bounded on any compact subset of $D$ . Then there exists a subsequence
of
such
$D$
that
converges uniformly on any conlpact subset of to a holomorphic function in$\{u_{k}\}$$\{u_{k_{j}}\}$$\{u_{k}\}$$\{u_{k_{j}}\}$$D$.

Theorem
Let $D$ be a bounded pseudoconvex domain inand$D$
let be a plurisubharmonic function ill . We set $X=\{z\in D|z_{1}=0\}$ . Suppose that$10.(\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{d}\mathrm{t}_{\mathrm{S}}\mathrm{S}\mathrm{o}\mathrm{n}[6])$$\varphi$$C_{\text{ノ}^{}n}$140$D\subset\{z\in \mathrm{C}^{n}||z_{1}|\leq-4\}$

$F$

in

$D$

. If

$f$

is holomorphic in X. then there exists a holomorphic function

such that$F|_{X}=f$ ,

where

and

$d\mu$$\int_{D}|F|^{2-}ed\mu\leq 4A2\overline{\prime|}\varphi\int_{X}|f|^{2}e^{-\varphi}d\mu 1$

are Lebesgue measures in$d\mu_{1}$and$\mathrm{C}^{n}$$\mathrm{C}^{n-1}$,. respectively.

Proof. Without loss of generality, we may assume that $A=1$ . There exists an increasing
with smooth boundary such thatsequence of bounded strictly pseudoconvex domains inplurisubharmonic functions in
be a sequence of
. Let
and
$\mathrm{C}^{n}$$\overline{D_{n}}\subset\subset D$. We set
the boundary condition on$\overline{D_{n}}$such that$\varphi_{n}\downarrow\varphi$$g=f \overline{\partial}(\frac{1}{z_{1}})$$\partial D_{n}$$|<g,$$C^{\infty}$$\{\varphi_{n}\}$$\infty\bigcup_{n=1}D_{n}=D$. Let a be a$\overline{\partial}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{S}\mathrm{e}\mathrm{d}(0,1)$-form which satisfies. From lemma . we have$\overline,$$\alpha>_{\varphi_{n}}|^{2}=|\lim_{\epsilonarrow 0}\int D_{n}f\frac{\epsilon}{(|z_{1}|2+\epsilon)^{2}}\overline{\alpha_{1}}e^{-\varphi}dn\mu\int_{i}\approx_{1}=0\}\cap D_{n}\mu_{11}\pi f\overline{\alpha 1}e^{-}d\varphi_{n}2$$\leq\pi^{2}\int_{\{z_{1}=0\}}\cap D_{n}d|f|^{2}e^{-}\varphi_{n}\mu 1\int_{\{}z_{1}=0\}\cap Dn\mu_{1}|\alpha_{1}|\mathit{2}-e\varphi nd$$\leq 2\pi(1+\frac{1}{\delta^{2}})\int_{\{_{Z_{1}}=}0\}\cap D_{?b}1|f|2-e\varphi nd\mu\int_{D_{n}}\frac{|\overline{\partial}^{*}\alpha|^{2}}{|\approx_{\mathrm{J}}|^{2\delta}}e^{-}d\varphi_{n}\mu$

From lemma 5, there exist integrable functions$\overline{\partial}u_{\delta}^{n}=g$

We set$F_{\delta}^{n}=u_{\delta}^{n}z_{1}$$f|_{\{\}D_{n}}z_{1}=0\cap\cdot$,$u_{\delta}^{n}$in.

such that$D_{n}$$\int_{D_{n}}|u_{\delta}^{n}|2|z1|2\delta e^{-}d\mu\leq\varphi_{n}2\pi(1+\frac{1}{\delta^{2}})\int_{\{z_{1}=0}\}\cap D_{n}|f|2\epsilon\lrcorner-\varphi\eta d\mu 1$. Then, froln lemma 8,

Suppose that

$F_{\delta}^{n}$are holomorphic in$D_{n}$and satisfy.

$F_{\delta}^{n}|_{\mathrm{t}\}D_{2}}- 1-=0\cap\iota=$.
$\int_{X}$$|f|^{2}e^{-\varphi}d\mu 1=C<\infty$,

then it holds that$\int_{D_{n}}|F_{\delta}^{n}|^{2-\varphi}edn\mu$$=$$\int_{D_{n}}|u_{\delta}^{n}|^{2}|z1|2e-\varphi_{n}d\mu\leq\int_{D_{n}}|v_{\delta}^{n}|^{2}|\approx 1|2\delta e-\varphi_{n}dl^{\ell}$$\leq$$2 \pi(1+\frac{1}{\delta^{2}})\int_{\mathrm{i}\approx 1}=0\}\cap D_{n}|f|^{2}e^{-\varphi}dn\mu_{1}\underline{<}2\pi(1+\frac{1}{\delta^{2}})$

Therefore. for some fixed , there exists a constant$n$$C_{1}$$\int_{D_{n}}|F_{\delta}^{n}|^{2}d\mu\leq C1$.

such that

C.

141

From lemma 9,10. there exists a sequencewith
fornlly on any compact subset of. Thento. Moreover, we llave$\{\delta_{j}\}$$F^{n}$$D_{n}$

stlch tllat
converges ulliare llolomorphic inand$\delta_{j}arrow 1$$F^{n}$$F_{\delta_{j}^{7l}}$$D_{\eta}$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}_{\mathrm{L}}\mathrm{v}$$F^{71}|_{\{\approx_{1}0\}}=\cap D_{n}=f|_{\mathrm{t}^{-=0}\}}.1\cap Dn$$\int_{D_{n}}|F^{n}|^{2}e^{-}d\varphi\eta l^{\iota}\leq 4\pi C$.

be a compact subset of $D$ . There exists a nattlral rlumber$N)$ . If we set

Let$I\iota’$$\wedge lI,,$$=1_{\frac{11\mathrm{i}}{D_{n}}}\mathrm{n}e-\varphi_{\eta}$

then, for $n\geq N$ , there exist a constant$\wedge\backslash ^{\mathrm{Y}}$such that$K\subset D_{r\iota}$.

$(r\not\supset\geq$,

such that$C_{2}$$4 \pi C’\geq\int_{D_{n}}|F^{\Gamma}’|‘ 2-e\varphi_{r}id_{\ell l}\geq \mathrm{A}\}I_{}\backslash 7./I^{\cdot}J_{N}\mathrm{A}|F^{n}|^{2}d\mu\geq C^{!}2\mathrm{S}\mathrm{u}_{1,r})|F\gamma l|^{2}$.

are uIlifornlly
allv coIllpa( subset of D. Tllell we can find a
subsequenceofwhich converges uniforlnly on any compact subset of $D$ . We
set
Thell $F$ is holomorphic in $D$ and atisfies $F|_{X}=f$ . For any compact$D$
subsetof , we llave$\mathrm{T}\}_{1\mathrm{U}\iota}‘,,$$\{F^{n}\}$$\dagger_{)\mathrm{o}\mathrm{u}\mathrm{I}\mathrm{l}\mathrm{c}}1\mathrm{e}(1011$$\mathrm{t}$$\{F^{k_{n}}\}$$\{F^{n}\}$$1\mathrm{i}_{\mathrm{l}}\mathrm{n}_{narrow}F^{k}\infty’|=F.$$s$$I\iota^{-}$$\oint_{h’}|F|^{2_{J}}‘arrow-_{\overline{Y}}d\mu=1\mathrm{i}_{\mathrm{I},arrow}11r|\infty\int_{K}|F^{k,}.\iota|^{2\varphi_{k_{n}}}\mathrm{r}^{-}\lrcorner Cl\mu\leq 4\pi C$($\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{t}\cdot \mathrm{h}\mathrm{C}\mathrm{o}111\mathrm{I}^{1\mathrm{y}})$$\mathrm{t}\mathrm{P}\mathrm{S}$the )$\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}$$1$Remark.
also
the constant$\mathrm{w}\mathrm{h}\mathrm{i}$(.

$\mathrm{o}\mathrm{l}$$\mathrm{S}\mathrm{i}\iota\iota[18]$)$\mathrm{f}\mathrm{a}\mathrm{i}\mathrm{I}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{l}\mathrm{a}11\mathrm{o}\mathrm{t}1_{1}\mathrm{e}\mathrm{r}\mathrm{p}_{\mathrm{l}\mathrm{O}}\mathrm{o}\mathrm{f}$$C=‘ \frac{)4}{\backslash )}\pi 442(1+\frac{1}{1}‘)^{1/2}\mathrm{I}$$\mathrm{h}$,of the theorem of Ohsawa-Takegoshi in.

) $\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{i}(\mathrm{l}\mathrm{e}\mathrm{d}D\subset\{z||Z|\leq\wedge 4\}$

References[1] K.

$\mathrm{A}_{(}1\mathrm{a}c\cdot \mathrm{h}\mathrm{i}$($l_{om(J,?n},\mathrm{t}9$ ,.

J.

$conti\eta?/_{\text{ノ}}(\iota t_{\dot{t}\mathit{0}}n$$\mathrm{h}/\mathrm{I}\mathrm{a}\mathrm{t}1_{1}$. Soc.

of$\lrcorner 4^{\infty}hnct\prime on.gf\dot{r}om$$.\mathfrak{s}‘ \mathrm{i}1^{)\mathrm{a}\mathrm{I}}1$, 32(1980),[2] K. Adaclli. M. Alldersson ffild H.R. Cho,polyhedra.
of ($l\gamma lal’.?/^{t?}c$$f\gamma\cdot orn.\mathrm{S}ul)var\prime i(^{\lrcorner}Jf\uparrow(^{\text{ノ}}S$

submanZfold8 to strzctly$p_{\mathit{8}eudocon}veX$

331-341.

$L^{p}$

and$H^{p}$$\mathrm{P}\mathrm{e}\backslash \mathrm{t}\cdot \mathrm{i}\mathrm{f}\mathrm{i}\mathrm{t}\cdot.\mathrm{T}$ertensions of holomorphic functions. Matll., 189(1999). 201-210.

[3] K. Ad. Cllo.
anoffrorn subvarieties to certain convex domains. Math. J. Toyama Univ.. 20(1997). 1-13
$H^{l^{J}}$$i\iota \mathrm{t}\cdot 1_{1}\mathrm{i}\mathrm{a}11(1\mathrm{H}.\mathrm{R}$[4] E. Amar. Cohom

140.

$()lo.(\text{ノ^{}\eta(^{\mathrm{J}}}$,$‘ fL^{l^{J}}(’’..\Gamma t(^{\prime r}ls?\mathit{0}nS$$h\mathrm{o}l\mathit{0}morph?c\cdot f\dot{u}nCttons$

complexe et applications. J. London Math. Soc.. 29(1984). 127-

142

[5] B.

A formuf,0 for$\mathrm{B}\mathrm{e}\Gamma 11(1\mathrm{t}_{S\mathrm{s}}\mathrm{O}11.$399-418.

$\uparrow n_{\text{ノ}}t.\mathrm{c}\prime rpolat’/,on$and$(l\uparrow v’\ell.sl,(nm\mathrm{c}^{\gamma 1}$.

$\perp\backslash \mathrm{I}\mathrm{a}\mathrm{t}\mathrm{h}$and tl\prime c\lrcorner $th,eor(,Jm$ of[6] B. Berndtssoll. The extenszon theorern ofDonnelly-Fefferman, Alln. Inst. Fourier, 46(1996). 1083-1094.

$Oh.gC\iota?v(\prime \mathrm{A}^{-}\tau(xk‘-g_{\mathit{0}_{}9l}/?$[7] M. Elgueta.
a

to 8triCtly I) $seud()convex$ (lomains
to theartd
in general$ExtenS\dot{l}on\mathit{8}$$\mathit{8}ufman?^{f\zeta)}ld$$po6^{\cdot}?t7on$$c^{1}\infty?l\beta$ ). Allll..$263(1983)$ ,$()ffunCt\dot{l,}\mathrm{O}7bShol_{\mathit{0}},morph?C\iota n$$l,\mathit{0}?l,n\mathrm{c}lrJ,r.l/$,III. .T.

$\mathrm{M}\mathrm{a}\mathrm{t}1_{1}..24(1981))$

.

1-17.
der
[8] H. Grauert and I. Lieb. Das $Ram?r\cdot eZgChe$ Irit egra, l.
Formen, Rice Univ. Studies, 56(1970). 29-5().
im Bereich der$Gl,e\iota t,’ hur|g\overline{\partial}f=(\}$$?\mathit{1},rld(f?(^{J}L\dot{\mathrm{C}}\rangle\backslash un.q$$b_{C.g}chr\ddot{a}n,kten$.

Integral representatzon formulas on[9] T.
123(1986), 71-91.
$\mathrm{H}\mathrm{a}\mathrm{t}\mathrm{z}\mathrm{i}\mathrm{a}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}1$$anal_{\text{ノ}}y\dagger’/,c\cdot var\dot{l}(^{\lrcorner}’\dagger’\iota es,$

Pae.ifi([10] G. . Hellkill. $Inte_{Jp}(r.alrerC\mathit{8}entation$ of functions holornorphzc m strzctly
Math. USSR Sb., 7(1969), 597-616.
vex domains and some applt$\perp \mathrm{N}\mathrm{I}$J. Math.,’-$pp_{9(^{y}u}.,d_{\mathit{0}}Cor$$cation6_{\}^{\cdot}$

doconvex dorrt,ains[11] G.M. Henkin. Integral representati,on of fnnctions ?, strongly
to tfic -pro blcm. Math. USSR Sb.. 11 (1970). 273-281.
and ($p_{\mathit{8}}eu$$\overline{\partial}$$xppl,i,cation\iota\sigma$[12] G.M. Henkin.
functzons from .suf)of boundcd holomo
Izv., 6(1972), 536-563.
general position in a strlctly pseudoconvex domazn, Math.

$manlf(Jld\prime g\uparrow r\iota$$rph\uparrow c$$Con\dagger_{\text{ノ}}?nuat\iota on$$\mathrm{L}^{\mathrm{T}}\mathrm{s}\mathrm{S}\mathrm{R}$[13] G.M. Henkin and J. Leiterer, Theory of function.9 on Compl,ex manifol&, Birkh\"auser.

1984.

[14] L. H\"ornlallder,
113 (1965). 89-152.

$L^{2}$

estzmates and existence tfieorcms$fo7^{\cdot}th,e\overline{\partial}$[15] I. Lieb. $D\iota eC_{\text{ノ}}auCh,y-R?er\Gamma\iota \mathit{0}nnSch,enDlfferentlalgle?chungen$ auf$Geb?eten$ , Math. Ann., 190(1970), 6-44.
[16] T. Ohsawa aIld K. Takegoshi, On
Z.. 195(1987). 197-204.

$tf\iota e$

extenszon of$L^{2}$-operator, Acta Math..

stren..g lJ8e-. udokonvexen

holomorphic functions , Math.

[17] E. Ranlirez. Em Dzvt8tons problem und Randintegraldarstellungen m der $K_{omp}.leXe\gamma$’, Math. Ann., 184(1970), 265-275.
$Analy_{\mathit{8}}\dot{\iota}s$[18] Y.T. Siu, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi,, Proc.
.
3rd Illt. RIMSJ.. Geometric Complex Analysz8, World Sci.. 1996.
$5\overline{/}7- 592$

[19] E.L. Stout. An zntegral formula for holomorphic functz oris on strzc\prime tly pseudoconvex
hypersurfaces. Duke Math. J.. 42(1975), 347-356.