数理解析研究所講究録 1211 巻 2001 年 66-75

66
Hartogs’ phenomena for microfunctions withholomorphic parameters
Otto Liess (Universit\‘a di Bologna)
Yasunori Okada (Chiba University) (岡田靖則)
Nobuyuki Tose (Keio University) (戸瀬信之)

0Introduction
Hyperfunctions and microfunctions with holomorphic parameters of form $u(x’, z”)$ havebeen considered in classical microlocalization and they play an important role in second
microlocalization. It is standard to define them in acohomological way: cf. in particular(1.1) and (1.2) below. The reason why they are not just defined as “holomorphic functionsin the variables $z’$ with values in hyper- or microfunctions in the variables $x’”$ , is thatspaces of hyperfunctions, respectively microfunctions, admit no natural topology. On
with for every fixed $z’$ .
the other hand, it is quite easy to associate avalue
Since hyper- and microfunctions with holomorphic parameters share many properties incommon with standard, complex-valued holomorphic functions, the following problem
arises:$u(\cdot, z’)$

Problem 0.1. Do the “values”
detemine the original$u(\cdot,\dot{z}’)=u|_{z’=\dot{z}’}$$u$of u with respect to the$z’$-variable$u^{\mathit{9}}$It is aresult of K. Kataoka and T. Oshima that the answer to this problem is affirmativein the hyperfunction case and we shall show in this note that the same is true also in thecase of microfunctions: see Theorem 2.1 below. Actually, Kataoka and Oshima consideredaslightly more general situation in which the parameter is assumed to be “real-analytic”,
rather than “holomorphic”. For this reason we shall also consider aresult similar to
Theorem 2.1 for the case of real-analytic parameters: see Corollary 1.2 and Theorem 2.8.

Remark 0.2.. K. Kataoka and T. Oshima did not publish their result themselves, but
A. Kaneko included it (with proof) in his book [3] as Theorem 4.4.7’.

1Statement of the results
1. Let us fix the situation in which we work

67
Let $M’$ be areal analytic manifold with complexification $X’$ and let $X’$ be acomplex, $X’$ , and $X’$ are denoted by $x’$ , , and $z’$ respectively.
manifold. Local coordinates of
We consider the embedding $N:=M’\cross X’\mathrm{c}arrow X:=X’\cross X’$ and identify the conormal
to $N$ (resp.
along $N$ with $T_{M}^{*},X’\cross X’$ . The canonical projection frombundleof microfunctions). We introduce the sheaf(resp.
to $M’$ ) is denoted bybywith holomorphic parameter ” on$z’$$\mathrm{M}’$$T_{N}^{*}X$$T_{N}^{*}X$$\mathrm{C}\mathcal{O}_{N}$$T_{M}^{*},X’$$\pi_{M’}$$\pi_{N}$$z$$T_{N}^{*}X$(1.1)$\mathrm{C}\mathcal{O}_{N}:=\mu_{N}(\mathcal{O}_{X})\otimes or_{N/X}[\dim_{\mathbb{R}}M’]$and the sheaf$B\mathcal{O}_{N}$of hyperfunctions with holomorphic parameter $z’$ on $N$ by$B\mathcal{O}_{N}:=\mathrm{C}\mathcal{O}_{N}|_{N}$

.

(1.2)

Sato’s microlocalization
denotes the sheaf of holomorphic functions on $X$ ,Here,functor along $N$ , and $or_{N/X}$ the relative orientation sheaf. For any fixed point
we can define the restriction morphisms$\mu_{N}$$\mathcal{O}_{X}$$\dot{z}’\in X’$$B\mathcal{O}_{N}|_{\{z’=\dot{z}’\}}arrow B_{M’}$,$u\mapsto u|_{z’=\dot{z}’}$,$u\mapsto u|_{z’=\dot{z}’}$and$\mathrm{C}\mathcal{O}_{N}|_{\{z’=\dot{z}’\}}arrow \mathrm{C}_{M’}$under the identifications $M’\cross\{\dot{z}’\}\simeq M’$ andby $Bm1$ the sheaf of hyperfunctions on $M’$ and by
Now we state our main theorem:$T_{M}^{*},X’\cross\{\dot{z}’\}\simeq T_{M}^{*},X’$$\mathrm{C}_{M’}$. Here we denotethat of microfunctions on$T_{M’}^{*}X’$.

be a point, $U’\subset X^{n}$ an open subset andTheorem 1.1. Let.
\"U ) a microfuntion with holomorphic parameter $z’$ defined in a neighborhood of
is 0at . Then $u=0$ in, the restriction
Assume that for any fixed.
neighborhood of$u\in \mathrm{C}\mathcal{O}_{N}(\{\dot{q}’\}\cross$$\dot{q}’\in T_{M}^{*},X’$$\{\dot{q}’\}\cross U’$$\dot{z}’\in U’$$u|_{z’=\dot{z}’}$$a$$\dot{q}’$$\{\dot{q}’\}\cross U’$

We give explicitly two corollaries of Theorem 1.1.

be a point, $U’\subset X’$ an open subset andCorollary 1.2. Let\"U ) a hyperfunction with holomorphic parameter $z’$ defined in a neighborhood of, the restriction $u|_{z=\dot{z}’}=0$ at $i’$ . Then $u=0$ in”. Assume that for any$u\in B\mathcal{O}_{N}(\{\dot{x}’\}\cross$$\dot{x}’\in M’$$\{i’\}\cross$$\dot{z}^{n}\in U’$$U$neighborhood of$\{\dot{x}’\}\cross U’$$a$.

and $U’\subset X’$ be two open subsets andCorollary 1.3. Let, the restriction
hyperfunction with holomorphic parameter. Assume that for any,
is real analytic. Then itself is an analytic function onwithand a holomorphic function
there exist a neighborhoodof$u\in B\mathcal{O}_{N}(\Omega’\cross U’)a$$\Omega’\subset M’$$\dot{z}’\in U’$$u$$u|_{z=\dot{z}’}\in B_{M’}(\Omega’)$$\tilde{U}\subset X$$u=f|_{\Omega’\mathrm{x}U’}$.

$\Omega’\cross U’$$\Omega’\cross U_{f}’i.e.$$f\in \mathcal{O}_{X}(\tilde{U})$68
We remark that Corollary 1.2 can be obtained from aresult due to Kataoka andOshima given in Theorem 2.8, in which $u(x’, x’)$ is assumed to have $x’$ as real analytic
parameters.

Let us consider moreover the case where $X’$ is the complexification of some real
analytic manifold $M’$ . In this situation, we give astronger result:

Theorem 1.4. Let
be a point, $U’\subset X’$ a connected open subset and $u\in$a microfunction with holomorphic parameters defined in a neighborhood of. Assume that $M’\cap U’$ is non-empty and that $u|_{z’=i’}=0$ for any\"U.
Then $u=0$ in a neighborhood of.
$\dot{q}’\in T_{M}^{*},X’$$\mathrm{C}\mathcal{O}_{N}(\{\dot{q}’\}\cross U’)$$\{\dot{q}’\}\cross U^{u}$$\dot{x}’\in \mathrm{M}" \mathrm{n}$$\{\mathrm{q}’\}\cross U’$

Note that since we can argue locally, Theorem 1.1 will be aconsequence of Theorem
1.4.

2. Let us next consider the sequence of embeddings$M:=M’\cross M’rightarrow Narrow X$,

which defines the sheaf$A_{\Sigma}^{2}:=\mathrm{C}\mathcal{O}_{N}|_{\Sigma}$of second analytic functions defined on the real regular involutive submanifold$\Sigma:=T_{M}^{*}X\cross\tau*\mathrm{x}T_{N}^{*}X$.

It is again aconsequence of Theorem 1.4 that the sections ofwisely:$A_{\Sigma}^{2}$are determined point-

Corollary 1.5. Let
be a point,
an open subset, anda second analytic function. Assume that $u|_{x’=i’}=0$ for any fixed $i’\in\Omega’$ . Then$\dot{q}’\in T_{M}^{*},X’$$\Omega’\subset M’$$u\in A\mathrm{j}(\{\dot{q}’\}$$\cross$$\Omega’)$

$u=0$

.

The following particular case of Theorem 1.4 will be the main intermediate step in theargument. We denote in it by
the conormal bundle to $M’$ with thezero section removed and by
the canonical projection from
to $M’$ .
$\dot{T}_{M}^{*},X’=T_{M}^{*},X’\backslash M’$$\dot{\pi}_{M’}$

Theorem 1.6. Let$U’\cap M’\neq\phi$ , andthe intersection$\dot{T}_{M}^{*},X’$be an open subset, $U’\subset X’$ a connected open subset witha closed conic subset such that for each base pointconsists of only one direction. Also consider a fixed point$\Omega’\subset M’$$Z\subset\dot{\pi}_{M}^{-1},(\Omega’)$$Z\cap\dot{\pi}_{M}^{-1},(i’)$

in

$Z$

$\dot{x}’\in\Omega’$$\dot{q}’$.

Assume that a section$u\in \mathrm{C}\mathcal{O}_{N}(\dot{\pi}_{M}^{-1},(\Omega’)\cross\text{\"{U}})$satisfies the condition$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(u)\subset Z\cross U’$and that $u|_{z’=\dot{x}’}=0$ atof$\{\dot{q}’\}\mathrm{x}$

$U’$

.

$\dot{q}’$for any $i’\in U’\cap M’$ . Then we have $u=0$ in a neighborhood

69
2Local forms of the main results
1. The theorems above in Section 1are of alocal nature. We may argue therefore in local
coordinates and assume that
, $X’=\mathbb{C}^{n-d}$ for some and . We shall identify$X’\cross X’$ within anatural way: if $z=$
are the coordinates in , we write, ) and $z’$ for $(z_{d+1}, \ldots,z_{n})$ . Thus, $z=(z’, z’)$ and $X’=\{z\in \mathbb{C}^{n};z’=0\}$ ,for {zl$X’=\{z\in \mathbb{C}^{n};z’=0\}$ . We denote $M=\{z\in X;{\rm Im} z=0\}=\mathbb{R}\mathrm{n}$ , regarded as areal
analytic submanifold inand consider its partial complexification $N=\{z\in X;{\rm Im} z’=$. Variables in $M$ shall be written as,,$x’=$ $(x_{d+1}, \ldots, x_{n})$ . Coordinates ofare denoted byor $(x’, z’;\xi’)$ under the identification.
It is instructive to rewrite Theorem 1.1 in local variables and in terms of defining
functions:$X’=\mathbb{C}^{d}$$\ldots$$n$$(z_{1}, \ldots, z_{n})$$\mathbb{C}^{n}$

$z’$

$d$

$\mathbb{C}^{n}$$z_{d}$$\mathbb{C}^{n}$$0\}=\mathbb{R}^{d}\cross \mathbb{C}^{n-d}$$x=(\mathrm{x}\mathrm{f}, x’)$

$T_{N}^{*}X=T_{\mathrm{R}^{d}}^{*}\mathbb{C}^{d}\cross \mathbb{C}^{n-d}$$x’=(\mathrm{x}\mathrm{f}, \ldots, x_{d})$$(x’, z’;\xi’\cdot \mathrm{d}\mathrm{x}\mathrm{f})$$T_{\mathbb{R}^{d}}^{*}\mathbb{C}^{d}\simeq\sqrt{-1}T^{*}\mathbb{R}^{d}\simeq T^{*}\mathbb{R}^{d}$${\rm Im} z’\in G’,z’\in \mathbb{C}^{n-d}$ , $|z’|<$
Theorem 2.1. Assume thatand denote by the hyperfunction with holomorphic parameters on $\{(x’, z’);|x’|<$, $|z’|<\delta\}$ associated with . Also
assume that,for any $z’$ with $|z’|<\delta$ . Then there are $\epsilon’>0_{f}\delta’>0$ , open convex cones , ,inso thatholomorphic functions, $y’\in$defined on, $|z’|<\delta’\}$ so that is for $|z’|<\delta’$ equal to.
$h\in \mathrm{O}(\{\mathrm{z}\in \mathbb{C}^{n};|z’|<\epsilon,$$\delta\})$$u$$consider\dot{\xi}’\in G^{\prime[perp]}and$$h$$\epsilon$$(0, (’)\not\in \mathrm{W}\mathrm{F}_{A}u(\cdot,z’)$$G_{1}’$$\mathbb{R}^{d}$$\dot{\xi}’\not\in G_{j^{[perp]}}’and$$G_{s}’$$\{z\in \mathbb{C}^{n};|z’|<\epsilon’$$h_{j}$$\sum_{j=1}^{s}\dot{b}(h_{j})$$u$$G_{j}’$$\ldots$

Theorem 2.1 may be considered as amicrolocal variant of atheorem of MalgrangeZerner. For the classical situation, see H. Komatsu [6]. We also give aversion of Theorem
1.6 in local coordinates:
Theorem 2.2. Let
be an open convex cone and a holomorphic function defined
on $\{z\in \mathbb{C}^{n};|z’|<\epsilon, y’\in G’, |z’|<\epsilon\}$ . Assume that for every
with $|i’|<$, the holomorphic function
defined on $\{z’\in \mathbb{C}^{d};|z’|<\epsilon, y’\in G’\}$
extends holomorphically to a neighborhood of. Then extends holomorphically toa neighborhood of the set $\{z\in \mathbb{C}^{n};z’=0, |z’|<\epsilon\}$ .
$G’\subset \mathbb{R}^{d}$$h$$i’\in \mathbb{R}^{n-d}$$h(\cdot,\dot{x}’)=h|_{z’=\dot{x}’}$$\epsilon$$0\in \mathbb{C}^{d}$$h$Remark 2.3. As a consequence of Corollary 1.2 it is possible to recast the definitionof hyperfunctions (respectively microfunctions) with holomorphic parameters considered
above as follows. Let again
be some open subset. A function $h:U’arrow B_{0}(B_{0}$denotes the set of germs of hyperfunctions at the point) is then a hyperfunction
with holomorphic parameter $z’$ precisely iffor any
there is an open neighborhood, a finite collection of open convex cones, $j=1$ , , , and”of ,holomorphic functions, so that for any,
is equal to.
$U’\subset \mathbb{C}^{n-d}$$0\in \mathbb{R}^{d}$$\tilde{z}’\in U’$$\tilde{U}\prime\prime=\tilde{U}_{\overline{z}}$$\tilde{z}’$$\epsilon>0$$G_{j}’\subset \mathbb{R}^{d}$$\tilde{h}_{j}\in \mathcal{O}(z;|z’|<\epsilon, {\rm Im} z’\in G_{j}’, z’\in\tilde{U}’)$$h(z^{JJ})$$\ldots$$s$$z’\in\tilde{U}’$$\sum_{j=1}^{s}b(h_{j}(\cdot, z’))$,denotes here the set of germs of microfuncLikewise, afunction :tions at the point
will be amicrofunction with holomorphic parameter $z’$ precisely$h$$(0, \dot{\xi}’))$$U’arrow \mathrm{C}_{(0,\dot{\xi})},(\mathrm{C}_{(0,\dot{\xi})}$

70, an open cone”of ,there is an open neighborhood
if for any, $z’\in$which contains 4’ and aholomorphic function,is the microfunction defined by the microfunctional, so that for any. We shall,,boundary value of the holomorphic function
call alocal defining function for (near ). When we want to make the dependence.
of on explicit, we shall occasionally write$\tilde{U}\prime\prime=\tilde{U}_{\overline{z}}$$\tilde{z}’\in U’$$\epsilon>0$$\tilde{h}\in \mathcal{O}(z;|z’|<\epsilon,$ ${\rm Im} z’\in G’$$G’\subset \mathbb{R}^{d}$$\dot{z}’\in\tilde{U}’$$\tilde{U}’)$$h(\dot{z}’)$$z’arrow\tilde{h}(z’,\dot{z}’)$$h$$\tilde{h}$$\tilde{h}$$\tilde{z}’$$|z’|<\epsilon$${\rm Im} z’\in G’$$\tilde{z}’$$\tilde{h}_{\tilde{z}’},$$\tilde{z}’$Remark 2.4. It is a significant fact that the local defining functions ”associated withdo not always admit a common holomorphic extension for all $z’\in U’$ .
the various(I. , in general there will exist no $f\in \mathcal{O}(z;|z’|<\epsilon, {\rm Im} z’\in G’, z’\in U’)$ with,$\tilde{h}_{\overline{z}}$$\tilde{z}’$$(0, \dot{\xi}’)$$e.$$\mathrm{W}\mathrm{F}_{A}\theta[f(\cdot, z’)-h(z’)]$$($$\forall z’\in U’.)$

Remark 2.5. Let us consider Corollary 1.3 again. This corollar says that the real animplies the real analyticity of . It is important inalyticity of $u(x’, z’)|_{z’=\dot{z}’}$ for each. Indeed, there is no analogous
this result that $z’$ is allowed to vary in an open set inresult when we only have assumptions for $z’$ real. This is the content of the following$ry$$\dot{z}’$$u$$\mathbb{C}^{n}$

Example 2.6. Let$u$be the hyperfunction on$\mathbb{R}^{2}$

defined by$u(x_{1}, x_{2})= \frac{x_{2}}{x_{1}+ix_{2}^{2}+i0}$.

Then$\mathrm{W}\mathrm{F}_{A}(u)=\{(0,0;1,0)\}$In particular the restrictions of with respect to therestrictions are real analytic (in one variable), but$u$.

variable are well-defined. All theseitself is not real analytic.

$x_{2}$$u$

We remark that asimilar example was already obtained in A. Kaneko [2].
We also give the following
Example 2.7. Consider a holomorphic function$h(z_{1}, z_{2})= \sum_{j=1}^{\infty}\frac{i(-iz_{2})^{j}}{j^{j}(z_{1}+i(z_{2}^{2}+j^{-3j}))}$

on$\{(z_{1},z_{2})\in \mathbb{C}^{2};{\rm Im} z_{1}>({\rm Im} z_{2})^{2}-({\rm Re} z_{2})^{2}\}$(1) for any i2 and any $k=0,1,2$ ,$\ldots$,. The boundary value$\partial_{x_{2}}^{k}u|_{x_{2}=\dot{x}_{2}}$$u(x_{1}, x_{2})$of satisfies$h$is well-defined and real analytic,(2) $\mathrm{W}\mathrm{F}_{A}(u)=\{(0,0;1,0)\}$ .

2. We give aremark. In view of Theorem 4.4.7’ in A.KanekO[3] mentioned in the introduction, we have the following result due to Kataoka and Oshima concerning hyperfunctions
with holomorphic parameters

71and ” be open subsets of $M’$ andTheorem 2.8 (K.Kataoka and T.Oshima). Let$M’$ respectively. Let $u(x’, x^{u})\in BA(\Omega’\cross\Omega’)$ and assume that$\Omega’$$\Omega$(2.1)$u(x’, x’)|_{x’=x_{\acute{\acute{0}}}}=0$

for any$x_{0}’\in\Omega’$. Then it follows that $u=0$ .

Here we have used the notation$BA$ $:=H_{M}^{d}(\mathcal{O}_{X}|_{X’\mathrm{x}M’},)$

(2.2)

,

and recall that the sheaf $BA$ was considered by M.Sato[lO]. (We remark that M. Satoused the sheaf $BA$ to discuss restriction of hyperfunctions in [10], before the notion of
singular spectrum came into being.)
3. It follows from the above dicussion that all results mentioned so far will be reduced to
Theorem 2.2. We shall therefore give the proof of this theorem in the next section.
The reduction of 1.6 to 2.2 will be done by using the characterization of real-analyticity
of hyperfunctions with single defining functions and the unique continuation property
along holomorphic parameters. Likewise, the reduction of Theorem 1.4 to Theorem 1.6
will be done by using two morphisms due to Kashiwara and which were used in the proofof the flabbiness of the sheaf of microfunctions. We omit these two steps in this note.

3Proof of Theorem 2.2.
1. In this section, we give abrief sketch of the proof of Theorem 2.2. The argument will
be based on several tools: acharacterization of extendibility of holomorphic functions
by duality, Baire’s principle, atheorem of Hartogs’ type, and the unique continuation
property of singularities along holomorphic parameters.
First we shall give avery simple result on extendibility of holomorphic functions.
Before we can state the result we need to introduce some additional notations and conandinthe polydisc
ventions. We shall in fact denote byshall use, for subsets $U’$ ,$V’\subset \mathbb{C}^{d}$,, the conventions:${\rm Re}(-i\langle z’, \zeta’\rangle)$$H_{U’}( \zeta’):=,\sup_{z\in U}$$\mathbb{C}^{d}$$\{z’\in \mathbb{C}^{d};|z_{j}|<\delta,\forall j\}$$B’(\delta)$,$U’+V’:=\{z’+\tilde{z}’;z’\in U’,\tilde{z}’\in V’\}$

.

(3.1)

is an open
. Finally, if
It is immediate that
. For simplicity
the space of analytic functionals on
set, we denote by
we shall assume that $U’$ is convex. It is well-known that analytic functionals inare characterized by the fact their Fourier-Borel transform$U’\subset \mathbb{C}^{d}$$H_{U’+B’(\mathit{5})(\zeta’)=H_{U’}(\zeta’)}+ \delta\sum_{j=1}^{d}|(_{j}|$$\mathcal{O}_{\mathbb{C}^{d}}(U’)$$\mathcal{O}_{\mathbb{C}^{d}}’(U’)$$v$$\zeta’arrow\hat{v}(\zeta’)=F(v)(\zeta’)=v(\exp[-i\langle z’, \zeta’\rangle])$$\mathcal{O}_{\mathbb{C}^{d}}’(U’)$

72satisfies an estimate of form$|\hat{v}((’)|\leq c\exp$ $[H_{Q}(\zeta’)]$for some constant$C$and some compact set $Q\subset U’$ .

a bounded
Theorem 3.1 (Holomorphic extensions and Duality). Leta holomorphic function defined on $U’$ . Then
open convex domain andextends holomorphically to $U’+B’(\delta)$ if and only if for any with $0<\delta’<\delta$ there exists
satisfying
a constant$U’\subset \mathbb{C}^{d}$

$be$

$h$

$h\in \mathcal{O}_{\mathbb{C}^{d}}(U’)$$\delta’$$c_{\delta’}$$\forall v\in \mathcal{O}_{\mathbb{C}^{d}}’(U’)$,$|\hat{v}(\zeta’)|\leq\exp\{H_{U’+B’(\delta)},(\zeta’)\}\Rightarrow|v(h)|\leq c_{\delta’}$.

We next prove amodified version of Hartogs’ theorem. We only consider the case of
convex sets.

be bounded open
andTheorem 3.2 (Hartogs-type theorem). Leta holomorphic function,
andconvex domains with, there exists a positive number
defined on $U’\cross U’$ . Assume that for any$\delta(x’)>0$
extends holomorphically to $B’(\delta(x’))$ .
for which the function
Then we can find an open ball $B’\subset U’$ centered at some point $i”\in U’\cap \mathbb{R}^{n-d}$ and.
constant $\delta>0$ in such a way that extends holomorphically to$U’\subset \mathbb{C}^{n-d}$$U’\subset \mathbb{C}^{d}$$0\in\partial U’$$U^{n}\cap \mathbb{R}^{n-d}\neq\emptyset$$h\in \mathcal{O}_{\mathbb{C}^{n}}(U’\cross U’)$$x’\in U’\cap \mathbb{R}^{n-d}$$h(\cdot, x’)\in \mathcal{O}_{\mathbb{C}^{d}}(U’)$$a$$B’(\delta)\cross B’$$h$

Before we enter the proof of Theorem 3.2, we recall alocal variant of the Phragm\’enLindel\"of principle.
be a plur isubharmonic
and letLemma 3.3. Let $B’$ be the unit disc in”$B’$$)\leq 1$ on
and that $\rho(x’)\leq 0$ for $x’\in \mathbb{R}^{n-d}\cap B’$ .
function on $B’$ . Assume that
Then there is a constant independent of satisfying $\rho(z’)\leq C|{\rm Im} z’|$ for $|z’|\leq 1/2$ .
$\mathbb{C}^{n-d}$$\rho:B’arrow \mathbb{R}$$\rho(z$$C$$\rho$

Note that the lemma implies

For aproof of this result cf. e.g. Meise-Taylor-Vogt
in atrivial way the following remark:is plurisubhar monic but assume now that
Remark 3.4. Assume that :$\rho(z’)\leq c$ on
whereas $\rho(x’)\leq d$ forfor some constants , .
. In particular it follows that if we fix $c’>c’$Thenforthat there is (which depends on and but not on ) so that $\rho(z’)\leq c’if|z’|<\epsilon’$ .
$\rho$$B’(\epsilon)arrow \mathbb{R}$$x’\in \mathbb{R}^{n-d}\cap B’(\epsilon)$$B’(\epsilon)$$\mathrm{p}(\mathrm{z}")\leq c’+cC|{\rm Im} z’|$$\epsilon’$

$C$

$c$

\bullet\bullet$0\in\partial\hat{K}’$$\hat{K}’$$\rho$$K’$

CC $U’$ whose interior Int If’ isthe convex hull of the set {0},h is holomorphic in Int$c’$$|z’’|\leq\epsilon/2$

Proof of Theorem 3.2. Take any compact convex set
non-empty and denote by$c$$\hat{K}’\cross U’$,$\mathrm{U}K’$in$\mathbb{C}^{d}$. Then we have:

73$\bullet$for any$x’\in U’\cap \mathbb{R}^{n-d}$Thus by shrinking $U’$ to Int
the beginning, that for any
to $U’+B’(\delta(x’))$ .
Set,$\hat{K}’$$h(\cdot, x’)$extends holomorphically to aneighborhood of$\hat{K}’$.

suitably, we may assume, fromand by also shrinking, the function
extends holomorphically$\delta(x’)$$x’\in U’\cap \mathbb{R}^{n-d}$$h(\cdot, x’)$

$E_{j}’:= \bigcap_{v\in \mathcal{O}_{\acute{\mathfrak{c}}^{d}}(U’),|\hat{v}(\zeta’)|\leq\exp H_{U’+B’(1/j)}(\zeta’)}\{x’\in U’\cap \mathbb{R}^{n-d};|v(h(\cdot, x’))|\leq j\}$.

. We can also see that every
By Theorem 3.1, we have thatmust include an open ballis closed. Thus from Baire’s principle, some
$E^{JJ}:=\{x’\in \mathbb{R}^{n-d};|x’-\dot{x}^{n}|<\epsilon\}$ . By shrinking, we can assume thatCC $U’$ .
, and also take apositivewith
We define by $\delta=1/(4j_{0})$ , take apoint
constant with
CC $U’$ . From the considerations above, our function satisfies
the following two properties.
$U’ \cap \mathbb{R}^{n-d}=\bigcup_{j}E_{j}’$$E_{j}’\subset$$U’\cap \mathbb{R}^{n-d}$$E_{j_{0}}’$$\dot{x}’+B’(\epsilon)$$\epsilon$$\dot{z}’\in U’$$\delta$$\delta’$

(PI)

$h$

$|\dot{z}’|<\delta$$h$$\dot{z}’+B’(\delta’)$is holomorphic in aneighborhood of the closure of $(\dot{z}’+B’(\delta’))\cross(i’+B’(\epsilon))$ ,(P2) each$h(\cdot, x’)$satisfies $|v(h(\cdot, x’))|\leq j_{0}$ for any.

$v\in \mathcal{O}_{\mathbb{C}^{d}}’(\dot{z}’+B’(\delta))$with$|\hat{v}(\zeta’)|\leq$$\exp H_{\dot{z}’+B’(3\mathit{5}\rangle}(\zeta’)$

Take the Taylor expansion of$h$in the variables$z’$around$h(z)= \sum_{\alpha}a_{\alpha}(z^{n})(z’-\dot{z}’)^{\alpha}$

$z.’$

:

.

(3.2)

Now we will estimate the functions $z’arrow|a_{\alpha}(z’)|$ in two ways. Our aim is to show thatsatisfy estimates which are good enough to ensure that the function is analyticon alarger domain than its initial domain of definition.
$h$$a_{\alpha}(z’)$

First we apply Cauchy’s integral formula to the functions
for all $z’\in\dot{x}’+B’(\epsilon)$ . Using the property (PI), we obtain:$|a_{\alpha}(z’)|\leq C_{1}\delta^{\prime-|\alpha|}$for anywhere
On the other hand, the expression:$C_{1}:= \sup_{z\in(\dot{z}’+B’(\mathit{6}’))\mathrm{x}(\dot{x}’+B’(\epsilon))}|h(z)|$$z’\in\dot{x}^{u}+B’(\epsilon)$$h(\cdot, z’)$and anyin the variables$\alpha$

,

(3.3)

.

$a_{\alpha}(z’)=v_{\alpha}(h(\cdot, z’))$,is the analytic functional va:
whereproperty (P2) give us the following estimate:$f\mapsto v_{\alpha}(f):=(1/\alpha!)((\partial/\partial z’)^{\alpha}f)|_{z’=\dot{z}’}$$v_{\alpha}$$|a_{\alpha}(x’)|\leq C_{2}(3\delta)^{-|\alpha|}$$z’$

for any$x’\in B’(\epsilon)\cap \mathbb{R}^{n-d}$

and any ,$\alpha$, and the

(3.3)

74
where $C_{2}:=j_{0}e^{-d}$ .
In this situation we apply Remark 3.4 to the plurisubharmonic functions$z’ arrow\frac{1}{|\alpha|}(\log|a_{\alpha}(z^{u}+i’)|-\log\max(C_{1}, C_{2}))$,

From the estimates (3.3) and (3.4) we obtainfor any $z’\in i’+B’(\epsilon’)$ ,$|a_{\alpha}(z’)|\leq C(2\delta)^{-|\alpha|}$with some constant . This estimate shows that for any $z’\in i’’+B’(\epsilon’)$ , the Taylor series
and that our function extends holomorphically(3.2) converges at least on. Thusto the domain $(\dot{z}’+B’(2\delta))\cross(i’+B’(\epsilon’))$ , which includes
we have the desired result if we take $i’+B’(\epsilon’)$ for $B’$ .
$\epsilon’$$h$$\dot{z}’+B’(2\delta)$$B’(\delta)\cross(\dot{x}\prime\prime+B’(\epsilon’))$$\square$2. Now we give aproof of Theorem 2.2.

Proof of Theorem 2.2. Let us assume that is aholomorphic function satisfying the as$h$with $|i’|<\epsilon$ , the restriction
sumption of Theorem 2.2. For eachextends holomorphically to aset of type $\{z’;|z’|<\delta(i’)\}$ with some positive number$i’\in \mathbb{R}^{n-d}$$\delta(i’)$$h(\cdot,\dot{x}\prime\prime)$.

Then from Theorem 3.2, we can take apositive constant and an open ball $B’\subset\{z’\in$such that extends holomorphically tocentered at some point$B’(\delta)\cross B’$ . Let us consider the boundary value $u=\dot{b}(h)\in B\mathcal{O}(\{(x’, z’);|x’|<\epsilon,$ $|z’|<$. From the domain of holomorphy of , we can see that is real analytic on the domain$\{(\#’, \mathrm{z}");|x’|<\delta, z’\in B’\}$ . Then from the unique continuation property for the analytic
wave front sets along holomorphic parameters, we can conclude that is real analytic on
the domain $\{(x’, z’);|x’|<\delta, |z’|<\epsilon\}$ . Since there is only one defining function in theboundary value representation $u=\dot{b}(h)$ , the real analyticity of asserts that extends
holomorphically to the domain $\{(x’, z’);|x’|<\delta, |z’|<\epsilon\}$ . This completes the proof.
$\delta$$\mathbb{C}^{n-d};|z’|<\epsilon\}$$\epsilon\})$$i’\in \mathbb{R}^{n-d}$

$h$

$h$

$u$

$u$

$h$

$u$

$h$

$\square$75
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