数理解析研究所講究録 1211 巻 2001 年 122-128122

調和関数の Fourier-Ehrenpreis 積分表示
千葉工業大学自然系 山根英司

Hideshi YAMANE, Chiba Institute of Technology

Ehrenpreis の基本原理

1
$z$

ら

の多項式 $P_{1}(z)$ ,, $P_{m}(z)$ を考える. $P=(P_{1}, \ldots, P_{m})$ はへの写像を定める.
の有界凸開集合 において定数係数線形偏微分方程式系$\in \mathrm{C}^{n}$$\mathrm{C}^{n}$$\ldots$$\mathrm{C}^{m}$か$\Omega$$\mathrm{R}_{t}^{n}$$(*)$$P_{j}(D)u(t)=0$ ,$1\leq j\leq m$,$t\in\Omega\subset \mathrm{R}_{t}^{n}$とし, $D=D_{t}=(D_{1}, \ldots, D_{n})$ ,の座標をの符号 [ま慣用と違うが, Berndtsson$D_{j}=i\partial/\partial t_{j}(1\leq j\leq n)$とする. (
Passare に倣う. ); $P_{j}(z)=$このとき Ehrenpreis の基本原理によれば $P^{-1}(0)= \bigcap_{j=1}^{m}\{z$が存在してに台を持つ測度を考える. ここで,$\mathrm{R}_{t}^{n}$

$t$

$=$

$(t_{1}, \ldots, t_{n})$$D_{j}$$\in \mathrm{C}^{n}$$0\}$$\mu_{k}$$u(t)$$= \sum_{k}\int_{P^{-1}(0)}A_{k}(z, t)e^{-i\langle z,t\rangle}d\mu_{k}(z)$,$t$$\in\Omega$の関は多項式で, 各 $z\in P^{-1}(0)$ を固定したとき,が成り立つ. ここでは $(*)$ の解である.
数
証明は構成的でなく, Hahn-Banach の定理を用いる. (Hahn-Banach は
Zorn の補題を用いて示されることに注意しよう. )$t$$A_{k}$$A_{k}(z, t)e^{-:\langle z,t\rangle}$2

基本原理の具体的バージョン
もともと基本原理は抽象的な存在定理で, 測度を具体的な式で書くことは出来なかった. しかし, 多変数複素解析の積分公式の発達により, 測度の代わりにカレントを用いて, 解が具体的に書けるようになった. ここでは$\mu_{k}$Berndtsson-Passare の結果を紹介しよう.
は 0 を含み, 有界かつ狭義凸で, 境界は滑らかとする.
ます
とおく.
を [こ dual な変数とし,$\Omega\subset \mathrm{R}\mathrm{p}$$z\in \mathrm{C}^{n}$を$\Omega$$t$の台関数とする.

$y={\rm Im} z\in \mathrm{R}^{n}$$\psi(y)=\sup_{t\in\Omega}\langle t, y\rangle$

123

6

tjt Ii 1($\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{X}\mathrm{N}\mathrm{h}$. y$5\supset \mathrm{r}\mathrm{b}6$$\mathrm{r}\mathrm{i}^{\ovalbox{\tt\small REJECT}}\mathrm{J}(\ovalbox{\tt\small REJECT}\{\mathrm{g}\mathrm{i}\mathrm{L},$$\ovalbox{\tt\small REJECT}$0C,

Qrh 6.

$\mathrm{r}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}$$\mathrm{Q}(\mathrm{A}\mathrm{y})\ovalbox{\tt\small REJECT}$

A$(t7), A $>0$ )$\mathrm{E}\ovalbox{\tt\small REJECT} \mathrm{f}^{\ovalbox{\tt\small REJECT}}3+\mathrm{f}6^{\ovalbox{\tt\small REJECT}}\mathrm{F}\mathrm{E}^{\mathrm{J}}\mathrm{f}\mathrm{t}\mathrm{t}\mathrm{i}ovalbox{\tt\small REJECT} 66^{;}’ \mathrm{P}\mathrm{b}^{>}\mathrm{O}\mathrm{r}$R;$\ovalbox{\tt\small REJECT}’\ovalbox{\tt\small REJECT}$e$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\ovalbox{\tt\small REJECT}$$\#$ $<$$\ddagger\check{\vee J}\#\acute{-}\varphi k\dagger T\mathcal{X}\mathrm{b}6$$\ovalbox{\tt\small REJECT} \mathrm{I}\ovalbox{\tt\small REJECT}_{\backslash }\mathrm{X}\prime x\mathit{0})\ovalbox{\tt\small REJECT}$

$6$

.

.

Q

$\ovalbox{\tt\small REJECT}’\ovalbox{\tt\small REJECT}$

if
if$\varphi’(y)\in\Omega$$f_{\mathrm{c}}\zeta\xi$p,$ct>\{e$$\ovalbox{\tt\small REJECT} e\ovalbox{\tt\small REJECT}$y$\ovalbox{\tt\small REJECT}$

0$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{C}^{\ovalbox{\tt\small REJECT}}1\mathrm{W}$$ct\ovalbox{\tt\small REJECT}\rangle i$. |y|$\mathrm{e}*\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\mathrm{c}\ovalbox{\tt\small REJECT} 7\mathrm{t}66>7\ovalbox{\tt\small REJECT}|\ovalbox{\tt\small REJECT} \mathrm{S}\ovalbox{\tt\small REJECT}*\#\mathrm{c}!_{\mathrm{i}}^{2}\mathrm{f}\#\mathrm{E}\ovalbox{\tt\small REJECT}(y)>0\ovalbox{\tt\small REJECT}?\mathrm{f}^{\ovalbox{\tt\small REJECT}}\mathrm{f}^{\mathrm{E}}6$$\varphi’(y)\in\partial\Omega$&, y f-0$\mathrm{r}\mathrm{W}\mathrm{A}\mathrm{A}/\ovalbox{\tt\small REJECT}|\ovalbox{\tt\small REJECT} \mathrm{s}\mathrm{f}4\mathrm{E}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\mathrm{c}^{\ovalbox{\tt\small REJECT}}+’)$$\ovalbox{\tt\small REJECT}$

$<$

$0$

$\ovalbox{\tt\small REJECT}+!\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 5>"\rangle$$|y|>>0$ ,$0\leq|y|\ll 1$.

$\{gjk(z, \zeta)\}j=1,\ldots,m;k=1,\ldots,n\epsilon$$\{Pj\}t\acute{-}(1\backslash \ovalbox{\tt\small REJECT} \mathrm{T}6$

Hefer map &$\tau$$T/x\triangleright \mathrm{b}$$P_{j}(()-P_{j}(z) = \sum_{k=1}^{n}g_{jk}(z, \zeta)(\zeta_{k}-z_{k})$$kT6$ .

$1\leq j\leq m$$(\{g_{jk}\}\iota \mathrm{x}-,\overline{-\Leftarrow}rightarrow \text{で^{}\backslash }\backslash [] \mathrm{f}fx\backslash \hslash\grave{\grave{>}}, -*\mathrm{f}1k_{\grave{\mathrm{J}}}\ovalbox{\tt\small REJECT} k^{\backslash }\backslash .)$$g_{j}(z, \zeta)=\sum_{k}g_{jk}(z, \zeta)dz_{k}$.

$\ k\grave<$,$g\#\mathrm{f}$$z\emptyset$$(m, 0)-\#./\nearrow/\mathrm{f}\mathrm{i}^{-}C^{\backslash }\backslash$,,$g(z, \zeta)=g_{m}\Lambda\cdots\Lambda g_{1}$$\zeta k\nearrow\backslash ^{\mathrm{O}}\overline{7}\nearrow-Pl’arrow\Leftrightarrow\vee\supset$:complete intersection,.

$Pt’arrow 5\neq 51_{\vee}^{-}T\mathit{1}R\sigma|)2\vee\supset C)R\text{定}\#\mathrm{k}^{\backslash }<$(HI): $P$ :$\mathrm{C}^{n}arrow \mathrm{C}^{m}[] \mathrm{f}$.

$\ovalbox{\tt\small REJECT} t\vec{-}m\leq n$(H2):$P\mathrm{I}\mathrm{i}*7_{\mathrm{B}}^{\mathrm{g}}\mathrm{E}\pi 4^{\mathfrak{l}-}C^{\backslash }\backslash h6$$l_{arrow}^{=}\ovalbox{\tt\small REJECT} \mathrm{a}\mathrm{e}\ovalbox{\tt\small REJECT} \mathrm{b}\mathrm{T}$$\#\mathrm{f}^{\grave{\backslash }}$,$\tau rx\mathrm{b}\:\mathrm{i}\mathrm{E}\ovalbox{\tt\small REJECT}$$|z|^{\gamma}\leq A(1+|y|)p_{\grave{\grave{1}}}ffi$$l\mathrm{f}_{\Xi}^{\mathrm{B}},\backslash \sigma)t$$(\#)$$\in\Omega$

El

$A$

,

,

$\mathrm{c}\mathrm{o}\dim_{\mathrm{C}}P^{-1}(\mathrm{O})=m$$\gamma\hslash\grave{\grave{>}}\Gamma+\#\mathrm{b}^{-}C$.

.

$\not\in_{)}1_{\vee}u(t)$$\in \mathrm{C}^{\infty}(\overline{\Omega})\hslash\grave{\grave{>}}\Omega\vee C^{\backslash }\backslash (*)k\mathit{6}7_{-}’\mathrm{T}$.

$arrowarrow C^{\backslash }>- R\backslash (z)=\overline{\partial}[1/P_{1}(z)]\Lambda\cdots\Lambda\overline{\partial}[1/P_{m}(z)]rightarrow C^{\backslash }\backslash h6$$\mathrm{T}^{\backslash }\backslash hU$$\mathit{0}\supset k\sim 6^{-}C^{\backslash }\backslash \overline{\partial}[1/P_{j}]=0f_{arrow}’\backslash \backslash \hslash^{\}}\mathrm{b}$$\ovalbox{\tt\small REJECT}\neq 5\mathcal{F}6’\ovalbox{\tt\small REJECT}_{JJ}^{\prime\backslash }\mathrm{k}$@D$\simk$),$\hslash\grave{\grave{>}}hV$,$\mathrm{f}\{\mathrm{F}\llcorner\emptyset$$\ddagger \mathit{0}\vee rx$$\not\in_{)}\sigma\supsetrightarrow C^{\backslash }h6([2], [7])$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}R(z)\subset P^{-1}(0)[] \mathrm{f}\mathrm{B}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{b}$$\Phi^{-\tau-\mathrm{E}\{\mathrm{b}@\gamma \mathrm{b}\mathrm{T}\mathrm{V}}$$\hslash>$

.

’

6.

.

.

$P_{j}\neq$$R(z)\ovalbox{\tt\small REJECT} \mathrm{f}$$k\emptyset\pm\ovalbox{\tt\small REJECT}^{\mathrm{B}\bigwedge_{\coprod}}te^{-i\langle z,t\rangle}\emptyset_{\epsilon}\mathrm{k}\check{\mathit{0}}fj\mathrm{I}\ovalbox{\tt\small REJECT}\hslash\grave{\grave{>}}\mathrm{a}\mathrm{e}\mathrm{n}6$$l_{arrow}^{=}\uparrow\Xi^{\backslash }\mathrm{g}\mathrm{T}6\ovalbox{\tt\small REJECT}\mp \mathfrak{p}_{\grave{\grave{1}}}\ovalbox{\tt\small REJECT}\hslash:t1_{-}\mathrm{b}\mathrm{a}\mathrm{e}\gamma_{\mathrm{b}}r_{j\mathrm{V}}\backslash \sigma)\#\mathrm{f}^{}\sim \mathcal{X}\mathrm{L}\hslash\grave{\grave{>}}\Phi \mathrm{f}\mathrm{f}\mathrm{i}^{-}C^{\backslash }\backslash h6$$t_{arrow}^{arrow}\ddagger$$(\mathrm{f}_{\overline{\mathrm{u}^{1}}\backslash }^{\mathrm{B}},\in\sigma)z\in P^{-1}(0)$$\mathrm{b}^{\vee}C$$R(z)\#\mathrm{f}\mathrm{V}^{\backslash }\lambda \mathcal{D}\emptyset)6_{\mathrm{f}\mathrm{f}1}^{\mathrm{f}\mathrm{f}D}\ovalbox{\tt\small REJECT} \mathcal{X}\triangleright\sqrt[\backslash ]{}\triangleright$

Rigat

,

.

$u(t)= \frac{1}{(2\pi i)^{n}}R(z).g(z, D_{t})u(\varphi’(y))\Lambda e^{-i\langle z,t-\varphi’(y)\rangle}(2\overline{\partial}\partial\varphi(y))^{n-m}$$\not\supset\grave{>}\Re\backslash V)\overline{\backslash _{-}\backslash \backslash }[perp]^{-}\vee\supset \mathrm{I}$$0$$\mathfrak{y}arrow[perp]\backslash \vee\supset$(Berndtsson-Passare)$\hat{\mathrm{g}}^{r}\mathrm{I}21$$\# x\mathrm{b}$.

$T^{\gamma}x*_{\mathit{2}}\mathrm{b}$.

$z[]_{\mathrm{c}}^{=}$$A_{k}(z, t)$$fp\mathrm{k}_{\mathrm{c}}^{>}\emptyset^{J}\backslash \mathrm{f}-\ovalbox{\tt\small REJECT} l\mathrm{f}$

124

(5”

$\mathrm{S}^{\ovalbox{\tt\small REJECT}}\mathrm{C}_{\mathrm{t}}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$”\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} l\ovalbox{\tt\small REJECT} 7\cdot\ovalbox{\tt\small REJECT}(7)*\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\mathrm{C}\ovalbox{\tt\small REJECT} (’)\mathrm{K}\ovalbox{\tt\small REJECT} \mathrm{i})’$$z\ovalbox{\tt\small REJECT}$$+\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$+z\ovalbox{\tt\small REJECT}:\rangle g(Z_{\rangle}(’)\ovalbox{\tt\small REJECT}$”$\mathrm{I}:_{\ovalbox{\tt\small REJECT} 1}(z_{k}+(_{7\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}’)d_{\ovalbox{\tt\small REJECT} \mathrm{Z})_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}}.\mathrm{b}$’tl$\mathrm{b}\mathrm{B}^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT} \mathrm{C}\mathrm{b}\mathrm{J}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$.

$P(z)\ovalbox{\tt\small REJECT}$$\rangle$$g(z, D_{t})u( \varphi’(y))=\sum_{k=1}^{n}\{z_{k}u(\varphi’(y))+(D_{k}u)(\varphi’(y))\}dz_{k}$$\mathfrak{x}rx6$.

$u(\varphi’(y))[] \mathrm{f}$Dirichlet$\mathrm{t}_{\mathrm{R}}^{\mathrm{g}}ffl$$[] \mathrm{f}*\{*\mathrm{N}\mathrm{e}\mathrm{u}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}$ $\mathrm{g}ffll\ovalbox{\tt\small REJECT}[]_{-}\prime hf_{arrow}’6$$\hslash\backslash \mathrm{b}$,$(\#)\#\acute{-}[] \mathrm{f}_{l\backslash \backslash }^{\mathrm{f}\mathrm{f}1}\Re\hslash\grave{\grave{:}}h6$.

$\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}9\ovalbox{\tt\small REJECT}^{-}\overline{\prime\rfloor\backslash }k\acute{\mathrm{r}}\ovalbox{\tt\small REJECT}\gamma--\psi\backslash$$\leq\emptyset$\d ag er.

((&u(1)$\llcorner F$.

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} t\mathrm{f}$$\check{\mathit{0}}rx\mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT} k$.

, 3$\mathrm{P}\pi$$(D_{k}u)(\varphi’(y))$

ffl ($\llcorner Ff_{arrow}^{-}\backslash \backslash \# f^{-}C^{\backslash }\backslash \mathrm{a}\mathrm{e}\backslash$\yen 6

Fourier-Ehrenpreis$\pi \mathrm{x}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}_{\square }^{\mathrm{A}}l\veearrow\ovalbox{\tt\small REJECT}$b て,$JR\sigma\supset\ovalbox{\tt\small REJECT}\tau_{1’}^{\backslash }\backslash -\backslash \underline{|\frac{\backslash }{\backslash }}$$\mathrm{f}l|\mathrm{P}_{\mathrm{B}}\ovalbox{\tt\small REJECT}$$n$$=3\ i6$ . $\Omega=\{t\in \mathrm{R}^{3};|t|<1\}kT6\ \partial\Omega=S^{2}$$(z_{1}, z_{2}, z_{3})\in \mathrm{C}^{3};z^{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=0\}$$\triangleright\nearrow\backslash \mathrm{b}k$$\}_{-}’\grave{l}’\mathfrak{o}^{\backslash }\check{\mathcal{D}}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}_{J7}^{\prime\backslash }\emptyset X$$\mathrm{g}$,$-C^{\backslash }h9$$\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}1\mathrm{e}\mathrm{t}-\mapsto \mathrm{E}\hslash\dagger\ovalbox{\tt\small REJECT} f_{arrow}\underline{\backslash }\backslash lfk_{\mathrm{B}}^{r}\theta$$\wedge^{\backslash }6\backslash .2\pi\grave{\mathrm{x}}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}_{\square }^{\mathrm{A}}\#\mathrm{f}\not\in 5\ovalbox{\tt\small REJECT}- \mathrm{c}_{\grave{1}}\not\subset\backslash \wedge^{\backslash }6\backslash$3)
Dirichlet$\Omega^{\vee}C^{\backslash }U)\{\ovalbox{\tt\small REJECT}$$\hslash\grave{\grave{\backslash }}h6$$h\hslash[] \mathrm{f}$,.)$2\pi i[V]=\overline{\partial}[1/z^{2}]\Lambda d(z^{2})$$g\ovalbox{\tt\small REJECT}_{X}$$\triangleright\nearrow\backslash \mathrm{b}\overline{\partial}[1/z^{2}]\ |\acute{-}\ddagger$$\mathrm{K}\hslash\grave{\grave{>}}\mathrm{f}^{\sqrt}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{-}C^{\backslash }h6$$u(t)$$\not\in\Phi 2(\lfloor \mathrm{U}\mathrm{f}\mathrm{f}\mathrm{l})$$u(t)$$\mathrm{E}’\supset$.

$=[V]$.

$*\}’.$

,

.

&k‘ .

$VU$)

. $V=\{z$smooth locus$=$$V\backslash \{0\}$$(V\backslash \{0\}[] \mathrm{f}\mathfrak{B}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\Phi f_{arrow}\underline{\backslash }\backslash \hslash\backslash \mathrm{b}$ $\Xi,*_{\iota\backslash \backslash }\# x\overline{|\mathrm{p}\rfloor}$て$\backslash \backslash h$$6$$\hslash 1\mathrm{b}$, $[V]l’-$ \ddagger$6\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}_{J7}^{\prime\backslash }\ovalbox{\tt\small REJECT}_{\overline{/\rfloor\backslash }}^{-}\hslash\grave{\grave{>}}\acute{\tau}_{\mathrm{f}\mathrm{f}}^{\mathrm{B}}\mathrm{b}$$\mathcal{X}\iota f_{arrow\sim}^{-\vee}k$$6\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}’.\varpi^{\backslash }g\overline{J\mathrm{J}}-\backslash \mathrm{b}\overline{\overline{\mathrm{p}}\mathrm{H}}\mathbb{H}\}\acute{|-}\acute{\mathrm{f}}\ovalbox{\tt\small REJECT} \mathrm{b}$$\mathrm{t}_{arrow}’\# x6$.
$\in \mathrm{C}^{0}(\overline{\Omega})\hslash\grave{\grave{>}}\Omega \mathfrak{P}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 0k\mathrm{i}\text{定}\mathrm{j}6$

Dirichlet F8{ $kT6$ .

$\hslash\grave{\grave{>}}RV$$[V]k\ovalbox{\tt\small REJECT} T$$\text{く}$$\text{で^{}\backslash }h6$.

.

$v=u|_{\partial\Omega}\in \mathrm{C}^{0}(S^{2})\geq$$\sim-\emptyset\pm@$$\frac{1}{4\pi^{2}}(2-\frac{1}{|y|})v(y/|y|)e^{-:(z,t-y/|y|\rangle}(\overline{\partial}\partial|y|)^{2}$,$t$$\in\Omega$$u(t)$ $[] \mathrm{f}\{\exp(-i\langle z, t -y/|y|\rangle)\}_{z^{2}=0,y/|y|\in\sup \mathrm{p}v}\sigma)\ovalbox{\tt\small REJECT} d2_{\square }^{\mathrm{A}}\mathfrak{b}$#て ‘‘\hslash 6.

4$–\overline{\mathrm{Q}}\pi-$Hfi$\star\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}1\}’.\mathrm{t}\backslash ’\supset$て$V[]_{arrow}^{r}\mathrm{E}\ovalbox{\tt\small REJECT} k\lambda \mathrm{Y}\iota^{-}C_{\mathrm{p}}^{\ni}+\Leftrightarrow T6$.

$\ovalbox{\tt\small REJECT}rightarrow \mathrm{F}z^{2}=0[] \mathrm{f}|x|^{2}-|y|^{2}=\langle x, y\rangle=\mathrm{O}k\overline{\mathrm{p}}\Pi(_{\llcorner}^{\mathrm{g}-}C^{\backslash }h6.$x&y&!

$K\hat{x}T\mathit{6}$.

$\vee\supset\not\in V$$|x|=|y|-C^{\backslash }\backslash$

,

125
,

$\ovalbox{\tt\small REJECT} \mathrm{e}\mathrm{C}\mathrm{B}\ovalbox{\tt\small REJECT} \mathrm{N}\mathrm{i}\ovalbox{\tt\small REJECT}/\mathrm{n}1\mathrm{f}^{1\ovalbox{\tt\small REJECT}}$$y\ovalbox{\tt\small REJECT}^{=}0$

V Ii

x

$\mathrm{R}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$!\ovalbox{\tt\small REJECT}$$\mathrm{S}\ovalbox{\tt\small REJECT} \mathrm{j}$!if$$ft^{\ovalbox{\tt\small REJECT}}g\{\mathit{8}_{\ovalbox{\tt\small REJECT}}k*\#\mathit{4}\ovalbox{\tt\small REJECT} tn$$\mathrm{W}*\mathrm{f}$$\mathrm{L}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}_{t^{\mathrm{A}}}^{\ovalbox{\tt\small REJECT}}$.

6x CI)#*1’$\mathrm{E}\ovalbox{\tt\small REJECT}$$\mathrm{L}\ovalbox{\tt\small REJECT} 6\supset$$\mathrm{L}/\mathrm{E}\ovalbox{\tt\small REJECT}!$E$$S^{\mathit{1}}$&*’t\rangle$\mathrm{L}_{\ovalbox{\tt\small REJECT}}\mathrm{k}^{\mathrm{A}}\mathrm{t}$()$\ovalbox{\tt\small REJECT}-\mathit{7}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT} t\mathit{0}\mathit{1}’ tLimr^{\ovalbox{\tt\small REJECT}}(_{\ovalbox{\tt\small REJECT}}NWb^{S}\ovalbox{\tt\small REJECT} lh\mathit{6}\ovalbox{\tt\small REJECT}.$).

$\ ^{\ovalbox{\tt\small REJECT}}\mathrm{J}^{-}\mathrm{C}1\ovalbox{\tt\small REJECT}/\ovalbox{\tt\small REJECT}|\mathrm{C}^{\ovalbox{\tt\small REJECT}_{t}\mathrm{s}}$T V&R$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} f\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$. & zl$\mathrm{m}\mathrm{l}(\mathrm{a}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{c}\mathrm{h}6\ovalbox{\tt\small REJECT}$$(\{\mathrm{O}$$\ovalbox{\tt\small REJECT} \mathrm{J}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\mathrm{a}^{\ovalbox{\tt\small REJECT}}\mathrm{u}\mathrm{a}^{\ovalbox{\tt\small REJECT}}\mathrm{L}(/)$$\ovalbox{\tt\small REJECT} \mathrm{j}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{x}|\mathrm{f}^{\ovalbox{\tt\small REJECT}}\mathrm{J}*\ovalbox{\tt\small REJECT}*$$\hat{V}$

$=$

$E$

$k_{\mathrm{J}}^{\underline{\mathrm{g}}}\underline{\backslash ,,}\lambda \mathcal{F}6$$E$$=\mathrm{R}_{y}^{3}\backslash \{y_{1}=y_{2}=0\}$$(\mathrm{k}\mathrm{g}_{1\backslash \backslash },\Xi\hslash\grave{\grave{>}}f\mathit{1}\mathrm{V}\backslash )\hslash>\mathrm{b}$$\cross S_{\theta}^{1}l_{\mathrm{c}}^{=}l\mathrm{f}dy_{1}\Lambda dy_{2}\Lambda dy_{3}\Lambda d\theta\tau_{\mathrm{H}}^{\backslash }\backslash \cap \mathrm{g}$,,$.\hat{V}\#\mathrm{f}\not\in\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}^{-}C^{\backslash }h6$

$E\cross S^{1}\hslash^{\}}\mathrm{b}\hat{V}\wedge \mathit{0})$$\tau^{\backslash }$$V\backslash \{y_{1}=y_{2}=0\}\subset \mathrm{C}^{3}$,$\xi\vee\supset t\mathrm{J}6$$\Xi,\mathfrak{R}_{1\backslash \backslash }^{\backslash }fx\mathrm{p}\cap\doteqdot$.

$\cap \mathrm{p}\mathrm{g}\Leftrightarrow \mathrm{k}’\supset’\ovalbox{\tt\small REJECT}_{J7}^{/\backslash }\overline{|\overline{\mathrm{p}}\rfloor}\ddagger \mathrm{B}\Phi k7g\Re \mathrm{b}$\d ag er$\check{:)}$.

$\epsilon\Leftrightarrow\vee\supset$.

$\not\in\sigma\supset\gamma_{arrow}’\emptyset\#\proptoarrow\not\in$$y\in El_{\acute{\mathrm{c}}\nearrow}\mathrm{T}_{\iota^{\backslash }}^{1_{\vee}}$$v=v(y)= \frac{|y|}{\sqrt{y_{1}^{2}+y_{2}^{2}}}(-y_{2}, y_{1},0)$,$w=w(y)= \frac{1}{|y|}y\mathrm{x}v=\frac{(-y_{1}y_{3},-y_{2}y_{3},y_{1}^{2}+y_{2}^{2})}{\sqrt{y_{1}^{2}+y_{2}^{2}}}$.

$\ k\grave<$”$\mathit{1}\backslash Rl-$$\vee \mathrm{C}^{\backslash }\backslash h$$\mathrm{Y}j$$\langle y, v\rangle=\langle v, w\rangle=\langle w, y\rangle=0\hslash>\vee\supset|y|=|v|=|w|\vee C^{\backslash }\backslash h6$$x=x(y, \theta)=v(y)\cos\theta+w(y)\sin\theta k$:$E\cross S^{1}arrow\hat{V}$$l\mathrm{f}\cap \mathrm{p}\xi\Leftrightarrow\{\ovalbox{\tt\small REJECT}’\supset’\ovalbox{\tt\small REJECT}_{J\mathrm{J}}^{\prime\backslash }\overline{|\overline{\mathrm{p}}\rfloor}\uparrow \mathrm{B}krx6$$\hat{V}l\mathrm{f}$,$(y, \theta)\vdash\not\simeq z=x(y, \theta)+iy$.
,$V\sigma)\ovalbox{\tt\small REJECT}_{\acute{\acute{\tau}}\mathrm{f}1^{\backslash }\mathit{1}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}f_{arrow}’\hslash>\mathrm{b}}^{\mapsto f}\backslash \backslash$$\acute{L_{\ulcorner\backslash }^{\mathrm{t}}}T\prime \mathcal{X}\mathrm{b}l\mathrm{f}^{\grave{\backslash }}$\d ag er$\backslash .\acute{(}\ovalbox{\tt\small REJECT}\prime \mathrm{g}\mu$$[V]k_{\mathrm{p}}^{arrow}\equiv+\mathrm{i}\mathrm{f}$$(\mathrm{R}_{y}^{3}\backslash \{0\})\cross S^{1}$$h6$ .
$7\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{i}^{\mathrm{B}}\mathrm{L}}\mathrm{F}$

A )
$U$

$6$ $l_{\acute{\mathrm{c}}}\#\mathrm{f}E\cross S^{1}$$\mathrm{F}_{7J}^{\prime\backslash }\prime k$$\sigma$)A$\sigma\supset\ovalbox{\tt\small REJECT}_{JJ}^{J\backslash }\Leftrightarrow\Leftrightarrow$$\mathrm{D}\cross$ $\mathrm{R}^{1}\mathrm{J}k\prime A^{\backslash }\backslash \ovalbox{\tt\small REJECT} t’arrow r\Gamma^{-}\llcorner_{4}\backslash \mathrm{I}_{\vee}^{\backslash }\backslash C_{r}^{E_{\underline{\backslash }\backslash }}\vee$$1$$\underline{\backslash }"[perp]^{\nearrow}\supset$$t\in\Omega$\d ag ermE 2 fi$\mathrm{T}6$, $|x|--|y|$,$\Phi$$p_{\grave{1}}\backslash \Re \mathfrak{y}$.

$\mathrm{k}^{\mathrm{Y}}l1\#\mathrm{f}^{\backslash \backslash }\mathrm{g},\mathfrak{R}_{\iota\backslash \backslash }^{\backslash }\langle x, y\rangle=0$0’ $(-i\langle z, t-y/|y|\rangle)$$=$$\langle y, t\rangle$$\Phi^{*}((\overline{\partial}\partial|y|)^{2})$$=$$\frac{1}{2|y|}dy_{1}\Lambda dy_{2}\Lambda dy_{3}\Lambda d\theta$$-|y|-i\langle x(y, \theta), t\rangle$,.
$U\ovalbox{\tt\small REJECT}$$1XU$)$k_{\grave{\mathrm{J}}}\underline{7\mathrm{J}}\emptyset \mathrm{F}_{\mathrm{p}}#\mathrm{f}y\neq 0$$\mathit{0}$)$\not\simeq \mathrm{g}\mathrm{g}$て$\backslash \backslash h6^{>}\sim k$$t_{\acute{\mathrm{c}}}\backslash \mathrm{f}\mathrm{f}^{\mathrm{B}},\Leftrightarrow-\backslash \mathrm{b}$\d ag er$\mathcal{D}\vee$$k_{\overline{\overline{\mathrm{p}}}}^{-}-i\mathrm{E}\mathrm{B}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{T}6\#\acute{-}\#\mathrm{f}\ovalbox{\tt\small REJECT} \mathrm{B}^{1}\mathrm{J}rx7/(^{\backslash ^{\backslash }}\overline{7}^{-}7\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{V} ’ 6fp_{\mathrm{V}} \backslash \hslash\grave{\grave{\}}}, 1\mathrm{H}^{\backslash }4\sigma)_{\mathrm{p}}\overline{\overline{\Rightarrow}}+\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} k\ovalbox{\tt\small REJECT}$. $f$l\mathrm{f}$Maple$\epsilon$

ffl$\mathrm{v}\backslash \gamma’-\cdot$

.

126

$\not\in$$2\mathrm{f}\mathrm{i}\sigma)^{\sqrt}\mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT} \mathfrak{p}_{\grave{\grave{1}}}-arrow q)$$\gamma_{-}\prime \mathrm{g}\mathrm{g}$.

$\not\equiv FJJ\backslash$$\{y, v, w\}1^{\backslash }A\% U)$&ffLIf ( @Ii@J$1_{\vee}^{\backslash }\backslash kT6$$\cap \mathrm{p}$\d ag er$\epsilon\Xi\lambda \mathrm{b}$.

$f_{X6}$$\backslash$$\mathrm{V}$て$\not\in \mathrm{f}$$\mathcal{D}\vee$5$\sim-k$

2

$k$

,

$\ovalbox{\tt\small REJECT}\Phi\#\mathrm{f}\overline{\triangleleft\backslash }\mathrm{B}\mathrm{f}\mathrm{f}\mathrm{l}^{-}C^{\backslash }\backslash ,-\frac{--}{\mathrm{p}}+\ovalbox{\tt\small REJECT} \mathrm{b}^{-}C^{ff}\star$A

A$\mathrm{d}\mathrm{y}2$$\mathrm{d}\mathrm{y}3$A-c$\theta\sigma)_{\grave{\mathrm{J}}}\ovalbox{\tt\small REJECT} \mathrm{O}^{\backslash ^{\backslash }}E\emptyset\grave{\grave{1}}\mapsto l’-\nearrow \mathrm{W}^{\backslash }$orthogonal frame kffl$\mathrm{V}^{\backslash }\vee CE\mathrm{E}$$\backslash 6$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\eta(y)\not\supset\grave{\grave{>}}\Gamma\mp\#\mathrm{T}6$$\mathrm{E}’\supset$$\mathrm{d}9\hslash\grave{\grave{:}}\Re 0$

‘て

$k$

$\pi|\int-\Rightarrow C^{\backslash }\overline{\mathrm{a}^{\backslash }},\ovalbox{\tt\small REJECT}$.

,$\mathrm{B}fl$.
.

$\}_{\acute{\iota}}-\overline{\overline{\mathrm{p}\Rightarrow}}+\ovalbox{\tt\small REJECT}^{-}C^{\backslash }\mathrm{g}6$$\mathfrak{B}6$ $\emptyset[] \mathrm{f}s\in S^{2}t\acute{|}$-$\hslash\grave{\grave{>}}_{\overline{\overline{\beta}}}^{\Rightarrow}\not\subset \mathrm{B}f\mathrm{f}\mathrm{l}^{-}C^{\backslash }\mathrm{g}6$$\hslash\grave{\grave{>}}$$t-” \ovalbox{\tt\small REJECT}\Phi\wedge\Gamma\pm \mathrm{A}\mathrm{j}\mathrm{F}^{\mathrm{i}},\frac{\mathrm{I}\mathrm{F}}{\tau}\backslash$$dy1dy2dy3d6=q^{2}dqd\theta ds$

Laplace\mbox{\boldmath $\tau$}6ffi/J]\て ‘‘’$\emptyset \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}_{J\mathrm{J}}^{\prime\backslash }1\mathrm{f}$i&J$\xi k^{\mathrm{g}}\backslash \Leftrightarrow\lambda \mathrm{b}\gammarightarrowarrow$$\backslash \}\mathrm{f}(\mathrm{R}_{y}^{3}\backslash \{0\})\cross S^{1}\downarrow\emptyset\ovalbox{\tt\small REJECT}_{JJ}^{\prime\backslash }k_{\mathrm{p}}^{\overline{\overline{\simeq}}}+\ovalbox{\tt\small REJECT} \mathrm{T}6f_{arrow\emptyset}’$$\sim\mathcal{X}\iota\hslash\grave{\grave{>}}$&$\mathit{1}^{\backslash }Rl’arrow\theta l_{\check{\mathrm{c}}}\mathrm{c}$Poisson$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{J\mathrm{J}}^{\prime\backslash }l_{arrow}^{-}-$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}^{\bigwedge_{-}}$$\Omega=\{t\in \mathrm{R}^{2};|t|<1\}kT6k$& # く.

$\mathrm{C}^{2};z^{2}=z_{1}^{2}+z_{2}^{2}=0\}$$\nearrow^{\backslash }\}\backslash \#$$\ovalbox{\tt\small REJECT} E$. $q=|y|$ , $s=y/|y|\in S^{2}$ &k‘ く$q[]_{arrow’}’\supset \mathrm{V}$$\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{J}7}^{\prime\backslash }\mathcal{F}6$$\mathrm{a}\tau\epsilon$, $h6$.

$\sigma$) $\xi=\theta+\eta(y)fx6$$\hslash>[]=dy_{1}\Lambda dy_{2}\Lambda dy_{3}\Lambda d\xi=\mathrm{d}\mathrm{y}\mathrm{i}$$E\cross S^{1}\downarrow\sigma)$)$\check{\mathcal{D}}\}_{\acute{\mathrm{c}}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathfrak{l}\acute{-}fx6*\mathrm{g}$$\hslash \mathrm{V}\backslash l\acute{-}rX’\supset f’.\mathfrak{x}\mathrm{b}\hslash>\overline{\overline{\equiv}}\grave{\chi}fx\mathrm{v}\backslash$$rightarrow \mathrm{C}[] \mathrm{f}fX\iota\backslash$$\mathrm{b}$\d ag er

& T.

$[V]$

(Lhffl)$\not\in\Phi 3$

Dirichlet$V\emptyset$. $V=\{z$$\partial\Omega=S^{1-}C^{\backslash }h6$smooth locus$V\backslash \{0\}[]_{-}’\grave{l}’\mathfrak{o}^{\backslash }\check{\mathcal{D}}\not\in_{J\mathrm{J}}^{\prime\backslash }U\supset fi$$u(t)\in \mathrm{C}^{0}(\overline{\Omega})\hslash\grave{\grave{:}}\Omega-\mathrm{C}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square kR\hat{E}T6$$\mathrm{F}\hslash l\ovalbox{\tt\small REJECT} kT6$.

$\sim-\emptyset\not\simeq$$=(z_{1}, z_{2})\in$.

$\triangleright$$v=u|_{\partial\Omega}\in \mathrm{C}^{0}(S^{1})\not\simeq$$\mathrm{g}$$Q[v](t)= \frac{-1}{16\pi^{2}}[V].v(y/|y|)e^{-i\langle z,t-y/|y|\rangle}\overline{\partial}\partial|y|$$k$,$\mathrm{k}^{\backslash }\#\ddagger\#\mathrm{f}\cdot$n6$l\mathrm{f},\ovalbox{\tt\small REJECT}\emptyset$$t\in\Omega[]_{\acute{\mathrm{c}}}k\backslash \mathrm{b}$て $u(t)=2Q[v](t)-Q[v](0)\hslash\grave{\grave{:}}ffiV$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\wedge\circ-\mathrm{E}1\mathrm{b}\not\in u$$\epsilon^{\backslash }\mathrm{c}$$\emptyset \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}9^{J}\Delta^{\backslash }\mathrm{f}\mathrm{f}\hslash\grave{\grave{:}}h$ $6k*_{\mathrm{J}\mu\backslash$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\Pi\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\sigma)$$\ovalbox{\tt\small REJECT}’\phi l\acute{|-},$z}^{\mathrm{a}\mathrm{e}\mathrm{g}}$$\cap\overline{-}\ovalbox{\tt\small REJECT}[]’-$$\hslash 6$

Fourier-Ehrenpreis

.

,

n-c

$=x+iyk\mathrm{k}^{\backslash }<\mathrm{g}\mathrm{g}$$\ovalbox{\tt\small REJECT} \mathrm{A}$$\text{く^{}\backslash ^{\backslash }}\}_{arrow}’*b$$\mathrm{f}\mathrm{f}1_{JJ}^{\prime\backslash g_{\overline{\prime \mathrm{J}}}}-\backslash \#\mathrm{f}\mathrm{T}$$\mathit{0}\vee$$\iota$$\backslash rx\mathrm{v}$$\backslash$.

$\ovalbox{\tt\small REJECT}_{\dot{R}}^{\mathrm{E}}[]_{arrow}’[] \mathrm{f}\ovalbox{\tt\small REJECT}_{\grave{\mathrm{A}}}^{\Psi}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\hslash\Pi+\mathrm{f}\mathrm{f}\mathrm{i}$$k\emptyset \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{f}1_{\mathrm{A}^{\backslash }}’\mathrm{R}\# n$\d ag er, z,$\Leftrightarrow\yen\emptyset X\mathrm{b}$$\ovalbox{\tt\small REJECT} \mathfrak{o}\mathrm{f}\mathrm{f}1-\ovalbox{\tt\small REJECT} \mathrm{X}\emptyset/\backslash \mathrm{A}X\#\mathrm{f}\mathrm{f}\mathrm{E}[]’.\ovalbox{\tt\small REJECT}_{\hat{J1}}|,$$\in \mathrm{C}^{d+1}[]’.X\backslash \mathrm{f}\mathrm{b}$.

$\tau$$n\grave{\wedge}\ovalbox{\tt\small REJECT}\varpi^{\wedge}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}[]_{-\overline{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\emptyset^{\prime\backslash }}\prime \mathrm{p}\Delta \mathrm{f}\mathrm{i}[] \mathrm{f}_{\urcorner}^{\mathrm{A}}\emptyset\mu-arrow 6\acute{\mathrm{r}}\ovalbox{\tt\small REJECT} \mathrm{b}$Berndtsson-Passare \emptyset ff&$\underline{\backslash }" L^{\vee}\supset$\check C‘\not\in R(b て ‘‘$\mathrm{b}\hslash$

6&

.

,

$N=\{z;z_{1}^{2}+\cdots+z_{d+1}^{2}=0, |x|=|y|=1\}$$\mathrm{E}o\vee$.

$\mathrm{g}$$\hslash$$l\mathrm{f}^{\grave{\backslash }}$

,

127
$k$

$oe\emptysetrightarrow 6k$$f_{\mathrm{c}}\mathrm{c}h\mathrm{b}$,$N\mathrm{l}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}\backslash \ovalbox{\tt\small REJECT}\backslash \mathrm{f}\mathrm{f}\mathrm{i}^{\gamma}x\mathrm{f}^{\backslash }\mathrm{f}\mathrm{i}^{1}\mathrm{J}P>$$dNk\Leftrightarrow\vee\supset$.

$\mathrm{b}$$\mathrm{I}_{\vee}f(z)\emptyset\grave{\grave{1}}(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT})\ovalbox{\tt\small REJECT}-\ovalbox{\tt\small REJECT}\square \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$(\partial^{2}/\partial z_{1}^{2}+\cdots+\partial^{2}/\partial z_{d+1}^{2})f(z)=0r_{j}\mathrm{b}$$\#\mathrm{f}^{\mathrm{Y}}$,$\tau$,$f(z)= \int_{N}f(\rho z’/2)\frac{1+\overline{z}’\cdot(z/\rho)}{\{1-\overline{z}’\cdot(z/\rho)\}^{d}}dN(z’)$$\not\supset\grave{>}\infty^{\backslash }\backslash \gamma j\backslash _{\underline{\overline{\backslash \backslash }}}L\mathrm{C}\mapsto$.

$\sim\sim>>$て$\backslash \backslash$$\rho\#\mathrm{f}\Phi \mathrm{g}r_{j}\mathrm{E}\text{定}\ovalbox{\tt\small REJECT}$.

$N\subset\{z\in \mathrm{C}^{d+1}; z_{1}^{2}+\cdots+z_{d+1}^{2}=0\}\gamma_{-}’\hslash^{\}}\backslash \backslash \mathrm{b}$ $\mathrm{E}\mathrm{T}\overline{J,\mathrm{F}_{\backslash }}\Phi k$$\not\in_{)}\infty\gamma_{\mathrm{b}fj^{1_{\sqrt}\backslash }\hslash\grave{\grave{1}}}$$\frac{\rho}{2}N=$,$\exists^{:_{\mathrm{B}}^{\mathrm{a}}}\ovalbox{\tt\small REJECT} 5\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} k$$\{\rho z/2;z\in N\}$$\Rightarrow\sigma\supset\nearrow\Delta^{\backslash }\mathrm{f}\mathrm{i}k$)$k\backslash ^{\backslash }\check{\mathit{0}}\backslash \beta_{\mathrm{D}}J\Phi^{\backslash }\backslash ’\supset\langle\sigma$-b て “\emptyset$f\emptyset \mathrm{t}_{\mathrm{L}C_{\Xi}^{\backslash \geqq}}^{\mathrm{g}-}\backslash$$\emptyset 5\mathrm{H}\Gamma\not\simeq_{\backslash }l\mathrm{f}’\supset\doteqdot\epsilon\check{\mathcal{D}}\}_{\acute{\mathrm{c}}}fX$$\mathrm{v}\backslash$.

$\hslash>\# 3$l\ddagger て$\mathrm{V}$: Bffl$\backslash 6$$\mathrm{b}$$\mathfrak{p}_{\grave{\grave{1}}}$,\hslash 1 て ‘‘$r_{X^{}}$’.

$5\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{f}\backslash \emptyset\grave{\grave{>}}h6$$\ovalbox{\tt\small REJECT}$$\gamma_{=}$,$\hslash>$$\mathrm{f}\mathrm{i}_{\grave{1}}\underline{7\mathrm{J}}l3$$e_{N\not\subset \mathrm{R}^{d+1}f_{arrow}’\hslash 1\mathrm{b}}2\backslash \backslash$,;$\ovalbox{\tt\small REJECT}$

References[1] C. A. Berenstein, R. Gay, A. Vidras and A. Yger, “Residue currents andBezout identities”, Birkh\"auser, Basel, 1993.

[2] B. Berndtsson and M. Passare, Integral formulas and an explicit versionof the fundamental principle, J. Fund. Anal, 84 (1989), 358-372.
[3] J. E. Bj\"ork, “ Rings of differential operators”, North-Holland Publ. Co.,
Amsterdam, New York, Oxford, 1979.
[4] N. R. Coleff and M. E. Herrera, ”Les courants r\’esiduels associes \‘aune forme m\’eromorphe’’, Lect. Notes in Math. 633, Springer, BerlinHeidelberg-New York, 1978.

[5] L. H\"ormander, “An introduction to complex analysis in several variables,
third edition (revised)”, North-Holland Publ. Co., Amsterdam, London,
New York, Tokyo, 1990.
[6] A. Meril and A. Yger, Probl\‘eme de Cauchy globaux, Bull. Soc. math.
France, 120 (1992), 87-111.
[7] M. Passare, Acalculus for meromorphic currents, J. reine angew. Math.,
392 (1988), 37-56.

[8] M. Passare, Residue solutions to holomorphic Cauchy problems, ”Seminar in Complex Analysis and Geometry 1987 (Rende, 1987)”, EditEl,
Rende, 1988, 101-105

128[9] S. Rigat, Version explicite du principe fondamental d’EhrenpreisMalgrange-Palamodov dans le cas non homogene, J. Math. Pures Appl.,
76 (1997), 777-799.
[10] H. Yamane, Residue currents and aFourier integral representation ofharmonic functions, preprint.
[11] H. Yamane, Fourier integral representation of harmonic functions interms of acurrent, preprint.
[12] A. Yger, Formules de division et prolongement meromorphe, Springer
Lecture Notes in Math., 1295 (1987), 226-283.

[13] R. Wada and M. Morimoto, Auniqueness set for the differential operator, Tokyo Journal of Math., 10(1987), 93-10$\triangle_{z}+\lambda^{2}$