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
1
Ehrenpreis
の基本原理
$z$
$\in \mathrm{C}^{n}$
の多項式
$P_{1}(z)$
,
$\ldots$
,
$P_{m}(z)$
を考える
.
$P=(P_{1}, \ldots, P_{m})$
は
$\mathrm{C}^{n}$か
ら
$\mathrm{C}^{m}$
への写像を定める
.
$\mathrm{R}_{t}^{n}$の有界凸開集合
$\Omega$
において定数係数線形偏微分方程式系
$(*)$
$P_{j}(D)u(t)=0$
,
$1\leq j\leq m$
,
$t\in\Omega\subset \mathrm{R}_{t}^{n}$
を考える
.
ここで
,
$\mathrm{R}_{t}^{n}$の座標を
$t$
$=$
$(t_{1}, \ldots, t_{n})$
とし
,
$D=D_{t}=(D_{1}, \ldots, D_{n})$
,
$D_{j}=i\partial/\partial t_{j}(1\leq j\leq n)$
とする
.
(
$D_{j}$
の符号
[
ま慣用と違うが
,
Berndtsson-Passare
に倣う
.
)
このとき
Ehrenpreis
の基本原理によれば
$P^{-1}(0)= \bigcap_{j=1}^{m}\{z$
$\in \mathrm{C}^{n}$
;
$P_{j}(z)=$
$0\}$
に台を持つ測度
$\mu_{k}$
が存在して
$u(t)$
$= \sum_{k}\int_{P^{-1}(0)}A_{k}(z, t)e^{-i\langle z,t\rangle}d\mu_{k}(z)$
,
$t$
$\in\Omega$
が成り立つ
.
ここで
$A_{k}$
は多項式で
, 各
$z\in P^{-1}(0)$
を固定したとき
,
$t$
の関
数
$A_{k}(z, t)e^{-:\langle z,t\rangle}$
は
$(*)$
の解である
.
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
. )
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
$\mu_{k}$
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
Berndtsson-Passare
の結果を紹介しよう.
ます
$\Omega\subset \mathrm{R}\mathrm{p}$
は
0
を含み,
有界かつ狭義凸で
, 境界は滑らかとする
.
$z\in \mathrm{C}^{n}$
を
$t$
[
こ
dual
な変数とし
,
$y={\rm Im} z\in \mathrm{R}^{n}$
とおく.
$\psi(y)=\sup_{t\in\Omega}\langle t, y\rangle$
を
$\Omega$
の台関数とする
.
数理解析研究所講究録 1211 巻 2001 年 122-128
tjt
Ii
1
$\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{X}\mathrm{N}\mathrm{h}$(
$\mathrm{Q}(\mathrm{A}\mathrm{y})\ovalbox{\tt\small REJECT}$
A$(t7),
A
$>0$
)
$\mathrm{r}\mathrm{W}\mathrm{A}\mathrm{A}/\ovalbox{\tt\small REJECT}|\ovalbox{\tt\small REJECT} \mathrm{s}\mathrm{f}4\mathrm{E}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\mathrm{c}^{\ovalbox{\tt\small REJECT}}+’)$,
y
f-0
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{C}^{\ovalbox{\tt\small REJECT}}1\mathrm{W}$6
$5\supset \mathrm{r}\mathrm{b}6$
.
y
$\ovalbox{\tt\small REJECT}$0
$\mathrm{E}\ovalbox{\tt\small REJECT} \mathrm{f}^{\ovalbox{\tt\small REJECT}}3+\mathrm{f}6^{\ovalbox{\tt\small REJECT}}\mathrm{F}\mathrm{E}^{\mathrm{J}}\mathrm{f}\mathrm{t}\mathrm{t}\mathrm{i}ovalbox{\tt\small REJECT} 66^{;}’ \mathrm{P}\mathrm{b}^{>}\mathrm{O}\mathrm{r}$,
p
$ct>\{e$
$\ovalbox{\tt\small REJECT} e\ovalbox{\tt\small REJECT}$y
$\ovalbox{\tt\small REJECT}$
0
$ct\ovalbox{\tt\small REJECT}\rangle i$$<$
$\mathrm{r}\mathrm{i}^{\ovalbox{\tt\small REJECT}}\mathrm{J}(\ovalbox{\tt\small REJECT}\{\mathrm{g}\mathrm{i}\mathrm{L},$
C,
R;
$\mathrm{e}*\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\mathrm{c}\ovalbox{\tt\small REJECT} 7\mathrm{t}66>7\ovalbox{\tt\small REJECT}|\ovalbox{\tt\small REJECT} \mathrm{S}\ovalbox{\tt\small REJECT}*\#\mathrm{c}!_{\mathrm{i}}^{2}\mathrm{f}\#\mathrm{E}\ovalbox{\tt\small REJECT}(y)>0\ovalbox{\tt\small REJECT}?\mathrm{f}^{\ovalbox{\tt\small REJECT}}\mathrm{f}^{\mathrm{E}}6$
.
|y|
$\ovalbox{\tt\small REJECT}$$0$
$\mathrm{r}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}$Qrh
6.
$\ovalbox{\tt\small REJECT}’\ovalbox{\tt\small REJECT}$ $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\ovalbox{\tt\small REJECT}$e
$\#$
$<$
.
$\ovalbox{\tt\small REJECT}’\ovalbox{\tt\small REJECT}$Q
$\ovalbox{\tt\small REJECT}+!\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 5>"\rangle$
$\varphi’(y)\in\partial\Omega$
if
$|y|>>0$
,
$\varphi’(y)\in\Omega$
if
$0\leq|y|\ll 1$
&
$f_{\mathrm{c}}\zeta\xi$$\ddagger\check{\vee J}\#\acute{-}\varphi k\dagger T\mathcal{X}\mathrm{b}6$
.
$\ovalbox{\tt\small REJECT} \mathrm{I}\ovalbox{\tt\small REJECT}_{\backslash }\mathrm{X}\prime x\mathit{0})\ovalbox{\tt\small REJECT}$
$\{gjk(z, \zeta)\}j=1,\ldots,m;k=1,\ldots,n\epsilon$
$\{Pj\}t\acute{-}(1\backslash \ovalbox{\tt\small REJECT} \mathrm{T}6$
Hefer map
&
$\tau$
$6$
.
$T/x\triangleright \mathrm{b}$
$P_{j}(()-P_{j}(z) = \sum_{k=1}^{n}g_{jk}(z, \zeta)(\zeta_{k}-z_{k})$
,
$1\leq j\leq m$
$kT6$
.
$(\{g_{jk}\}\iota \mathrm{x}-,\overline{-\Leftarrow}rightarrow \text{で^{}\backslash }\backslash [] \mathrm{f}fx\backslash \hslash\grave{\grave{>}}, -*\mathrm{f}1k_{\grave{\mathrm{J}}}\ovalbox{\tt\small REJECT} k^{\backslash }\backslash .)$$g_{j}(z, \zeta)=\sum_{k}g_{jk}(z, \zeta)dz_{k}$
,
$g(z, \zeta)=g_{m}\Lambda\cdots\Lambda g_{1}$
$\ k\grave<$
.
$g\#\mathrm{f}$
$z\emptyset$
$(m, 0)-\#./\nearrow/\mathrm{f}\mathrm{i}^{-}C^{\backslash }\backslash$
,
$\zeta k\nearrow\backslash ^{\mathrm{O}}\overline{7}\nearrow-Pl’arrow\Leftrightarrow\vee\supset$
.
$Pt’arrow 5\neq 51_{\vee}^{-}T\mathit{1}R\sigma|)2\vee\supset C)R\text{定}\#\mathrm{k}^{\backslash }<$
:
(HI):
$P$
:
$\mathrm{C}^{n}arrow \mathrm{C}^{m}[] \mathrm{f}$
complete
intersection,
$T^{\gamma}x*_{\mathit{2}}\mathrm{b}$
,
$\mathrm{c}\mathrm{o}\dim_{\mathrm{C}}P^{-1}(\mathrm{O})=m$
.
$\ovalbox{\tt\small REJECT} t\vec{-}m\leq n$
.
(H2):
$P\mathrm{I}\mathrm{i}*7_{\mathrm{B}}^{\mathrm{g}}\mathrm{E}\pi 4^{\mathfrak{l}-}C^{\backslash }\backslash h6$.
$\tau rx\mathrm{b}\:\mathrm{i}\mathrm{E}\ovalbox{\tt\small REJECT}$
$A$
,
$\gamma\hslash\grave{\grave{>}}\Gamma+\#\mathrm{b}^{-}C$
,
$(\mathrm{f}_{\overline{\mathrm{u}^{1}}\backslash }^{\mathrm{B}},\in\sigma)z\in P^{-1}(0)$
$l_{arrow}^{=}\ovalbox{\tt\small REJECT} \mathrm{a}\mathrm{e}\ovalbox{\tt\small REJECT} \mathrm{b}\mathrm{T}$
$|z|^{\gamma}\leq A(1+|y|)p_{\grave{\grave{1}}}ffi$
$\mathfrak{y}arrow[perp]\backslash \vee\supset$.
$\hat{\mathrm{g}}^{r}\mathrm{I}21$
(Berndtsson-Passare)
$\not\in_{)}1_{\vee}u(t)$
$\in \mathrm{C}^{\infty}(\overline{\Omega})\hslash\grave{\grave{>}}\Omega\vee C^{\backslash }\backslash (*)k\mathit{6}7_{-}’\mathrm{T}$
$\# x\mathrm{b}$
$\#\mathrm{f}^{\grave{\backslash }}$,
$l\mathrm{f}_{\Xi}^{\mathrm{B}},\backslash \sigma)t$$\in\Omega$
El
$\mathrm{b}^{\vee}C$
$(\#)$
$u(t)= \frac{1}{(2\pi i)^{n}}R(z).g(z, D_{t})u(\varphi’(y))\Lambda e^{-i\langle z,t-\varphi’(y)\rangle}(2\overline{\partial}\partial\varphi(y))^{n-m}$
$\not\supset\grave{>}\Re\backslash V)\overline{\backslash _{-}\backslash \backslash }[perp]^{-}\vee\supset \mathrm{I}$
.
$arrowarrow C^{\backslash }>- R\backslash (z)=\overline{\partial}[1/P_{1}(z)]\Lambda\cdots\Lambda\overline{\partial}[1/P_{m}(z)]rightarrow C^{\backslash }\backslash h6$
.
$R(z)\#\mathrm{f}\mathrm{V}^{\backslash }\lambda \mathcal{D}\emptyset)6_{\mathrm{f}\mathrm{f}1}^{\mathrm{f}\mathrm{f}D}\ovalbox{\tt\small REJECT} \mathcal{X}\triangleright\sqrt[\backslash ]{}\triangleright$$\mathrm{T}^{\backslash }\backslash hU$
,
$\mathrm{f}\{\mathrm{F}\llcorner\emptyset$$\ddagger \mathit{0}\vee rx$
$\not\in_{)}\sigma\supsetrightarrow C^{\backslash }h6([2], [7])$
.
$P_{j}\neq$
$0$
$\mathit{0}\supset k\sim 6^{-}C^{\backslash }\backslash \overline{\partial}[1/P_{j}]=0f_{arrow}’\backslash \backslash \hslash^{\}}\mathrm{b}$
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}R(z)\subset P^{-1}(0)[] \mathrm{f}\mathrm{B}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{b}$
$\hslash>$
.
$R(z)\ovalbox{\tt\small REJECT} \mathrm{f}$$z[]_{\mathrm{c}}^{=}$
$\ovalbox{\tt\small REJECT}\neq 5\mathcal{F}6’\ovalbox{\tt\small REJECT}_{JJ}^{\prime\backslash }\mathrm{k}$
@D
$\simk$
$\hslash\grave{\grave{>}}hV$
),
$k\emptyset\pm\ovalbox{\tt\small REJECT}^{\mathrm{B}\bigwedge_{\coprod}}te^{-i\langle z,t\rangle}\emptyset_{\epsilon}\mathrm{k}\check{\mathit{0}}fj\mathrm{I}\ovalbox{\tt\small REJECT}\hslash\grave{\grave{>}}\mathrm{a}\mathrm{e}\mathrm{n}6$.
$A_{k}(z, t)$
$l_{arrow}^{=}\uparrow\Xi^{\backslash }\mathrm{g}\mathrm{T}6\ovalbox{\tt\small REJECT}\mp \mathfrak{p}_{\grave{\grave{1}}}\ovalbox{\tt\small REJECT}\hslash:t1_{-}\mathrm{b}\mathrm{a}\mathrm{e}\gamma_{\mathrm{b}}r_{j\mathrm{V}}\backslash \sigma)\#\mathrm{f}^{}\sim \mathcal{X}\mathrm{L}\hslash\grave{\grave{>}}\Phi \mathrm{f}\mathrm{f}\mathrm{i}^{-}C^{\backslash }\backslash h6$.
$fp\mathrm{k}_{\mathrm{c}}^{>}\emptyset^{J}\backslash \mathrm{f}-\ovalbox{\tt\small REJECT} l\mathrm{f}$Rigat
$t_{arrow}^{arrow}\ddagger$$\Phi^{-\tau-\mathrm{E}\{\mathrm{b}@\gamma \mathrm{b}\mathrm{T}\mathrm{V}}$
’
6.
$\mathrm{S}^{\ovalbox{\tt\small REJECT}}\mathrm{C}_{\mathrm{t}}\ovalbox{\tt\small REJECT}$
$”\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} l\ovalbox{\tt\small REJECT} 7\cdot\ovalbox{\tt\small REJECT}(7)*\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\mathrm{C}\ovalbox{\tt\small REJECT} (’)\mathrm{K}\ovalbox{\tt\small REJECT} \mathrm{i})’$
(5”
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$
’tl
$\mathrm{b}\mathrm{B}^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT} \mathrm{C}\mathrm{b}\mathrm{J}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$
.
$P(z)\ovalbox{\tt\small REJECT}$
$z\ovalbox{\tt\small REJECT}$ $+\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$+z\ovalbox{\tt\small REJECT}:\rangle g(Z_{\rangle}(’)\ovalbox{\tt\small REJECT}$ $\mathrm{I}:_{\ovalbox{\tt\small REJECT} 1}(z_{k}+(_{7\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}’)d_{\ovalbox{\tt\small REJECT} \mathrm{Z})_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}}.\mathrm{b}$
”
$\rangle$$g(z, D_{t})u( \varphi’(y))=\sum_{k=1}^{n}\{z_{k}u(\varphi’(y))+(D_{k}u)(\varphi’(y))\}dz_{k}$
$\mathfrak{x}rx6$
.
$u(\varphi’(y))[] \mathrm{f}$
Dirichlet
$\mathrm{t}_{\mathrm{R}}^{\mathrm{g}}ffl$(
$\llcorner F$(&u(1)
$\Omega^{\vee}C^{\backslash }U)\{\ovalbox{\tt\small REJECT}$)
$-C^{\backslash }h9$
,
$(D_{k}u)(\varphi’(y))$
$[] \mathrm{f}*\{*\mathrm{N}\mathrm{e}\mathrm{u}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}$ $\mathrm{g}ffll\ovalbox{\tt\small REJECT}[]_{-}\prime hf_{arrow}’6$
.
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} t\mathrm{f}$Dirichlet
$\mathrm{P}\pi$ffl
(
$\llcorner Ff_{arrow}^{-}\backslash \backslash \# f^{-}C^{\backslash }\backslash \mathrm{a}\mathrm{e}\backslash$\yen
6
$\hslash\backslash \mathrm{b}$
,
$(\#)\#\acute{-}[] \mathrm{f}_{l\backslash \backslash }^{\mathrm{f}\mathrm{f}1}\Re\hslash\grave{\grave{:}}h6$
.
$\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}1\mathrm{e}\mathrm{t}-\mapsto \mathrm{E}\hslash\dagger\ovalbox{\tt\small REJECT} f_{arrow}\underline{\backslash }\backslash lfk_{\mathrm{B}}^{r}\theta$Fourier-Ehrenpreis
$\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}9\ovalbox{\tt\small REJECT}^{-}\overline{\prime\rfloor\backslash }k\acute{\mathrm{r}}\ovalbox{\tt\small REJECT}\gamma--\psi\backslash$
.
$\leq\emptyset$
\ddagger
$\check{\mathit{0}}rx\mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT} k$,
3
$\pi \mathrm{x}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}_{\square }^{\mathrm{A}}l\veearrow\ovalbox{\tt\small REJECT}$b て,
$JR\sigma\supset\ovalbox{\tt\small REJECT}\tau_{1’}^{\backslash }\backslash -\backslash \underline{|\frac{\backslash }{\backslash }}$$\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
$\mathrm{f}l|\mathrm{P}_{\mathrm{B}}\ovalbox{\tt\small REJECT}$$n$
$=3\ i6$
.
$\Omega=\{t\in \mathrm{R}^{3};|t|<1\}kT6\ \partial\Omega=S^{2}$
$\text{で^{}\backslash }h6$
.
$V=\{z$
$=$
$(z_{1}, z_{2}, z_{3})\in \mathrm{C}^{3};z^{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=0\}$
&k‘
$\text{く}$.
$VU$
)
smooth locus
$V\backslash \{0\}$
$\}_{-}’\grave{l}’\mathfrak{o}^{\backslash }\check{\mathcal{D}}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}_{J7}^{\prime\backslash }\emptyset X$
$\triangleright\nearrow\backslash \mathrm{b}k$
$[V]k\ovalbox{\tt\small REJECT} T$
.
$(V\backslash \{0\}[] \mathrm{f}\mathfrak{B}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\Phi f_{arrow}\underline{\backslash }\backslash \hslash\backslash \mathrm{b}$ $\Xi,*_{\iota\backslash \backslash }\# x\overline{|\mathrm{p}\rfloor}$
$\mathrm{g}$ $\hslash\grave{\grave{\backslash }}h6$
.)
$2\pi i[V]=\overline{\partial}[1/z^{2}]\Lambda d(z^{2})$
て
$\backslash \backslash h$$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}$$h\hslash[] \mathrm{f}$
,
$g\ovalbox{\tt\small REJECT}_{X}$
$\triangleright\nearrow\backslash \mathrm{b}\overline{\partial}[1/z^{2}]\ |\acute{-}\ddagger$
$6\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}’.\varpi^{\backslash }g\overline{J\mathrm{J}}-\backslash \mathrm{b}\overline{\overline{\mathrm{p}}\mathrm{H}}\mathbb{H}\}\acute{|-}\acute{\mathrm{f}}\ovalbox{\tt\small REJECT} \mathrm{b}$$\mathcal{X}\iota f_{arrow\sim}^{-\vee}k$
$\mathrm{t}_{arrow}’\# x6$
.
$\mathrm{K}\hslash\grave{\grave{>}}\mathrm{f}^{\sqrt}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{-}C^{\backslash }h6$.
$\not\in\Phi 2(\lfloor \mathrm{U}\mathrm{f}\mathrm{f}\mathrm{l})$
$u(t)$
$\in \mathrm{C}^{0}(\overline{\Omega})\hslash\grave{\grave{>}}\Omega \mathfrak{P}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 0k\mathrm{i}\text{定}\mathrm{j}6$
.
$v=u|_{\partial\Omega}\in \mathrm{C}^{0}(S^{2})\geq$
Dirichlet
F8{
$kT6$
.
$\sim-\emptyset\pm@$
$u(t)$
$=[V]$
.
$\frac{1}{4\pi^{2}}(2-\frac{1}{|y|})v(y/|y|)e^{-:(z,t-y/|y|\rangle}(\overline{\partial}\partial|y|)^{2}$
,
$t$
$\in\Omega$
$\hslash\grave{\grave{>}}RV$
$\mathrm{E}’\supset$
.
$*\}’.$
,
$u(t)$
$[] \mathrm{f}\{\exp(-i\langle z, t -y/|y|\rangle)\}_{z^{2}=0,y/|y|\in\sup \mathrm{p}v}\sigma)\ovalbox{\tt\small REJECT} d2_{\square }^{\mathrm{A}}\mathfrak{b}$
#て ‘‘\hslash
6.
4
$–\overline{\mathrm{Q}}\pi-$Hfi
$\star\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}1\}’.\mathrm{t}\backslash ’\supset$
て
$V[]_{arrow}^{r}\mathrm{E}\ovalbox{\tt\small REJECT} k\lambda \mathrm{Y}\iota^{-}C_{\mathrm{p}}^{\ni}+\Leftrightarrow T6$
.
$\ovalbox{\tt\small REJECT}rightarrow \mathrm{F}z^{2}=0[] \mathrm{f}|x|^{2}-|y|^{2}=\langle x, y\rangle=\mathrm{O}k\overline{\mathrm{p}}\Pi(_{\llcorner}^{\mathrm{g}-}C^{\backslash }h6.$
$\vee\supset\not\in V$
$|x|=|y|-C^{\backslash }\backslash$
,
x&y&!
$K\hat{x}T\mathit{6}$
.
$y\ovalbox{\tt\small REJECT}^{=}0$
$\ovalbox{\tt\small REJECT} \mathrm{e}\mathrm{C}\mathrm{B}\ovalbox{\tt\small REJECT} \mathrm{N}\mathrm{i}\ovalbox{\tt\small REJECT}/\mathrm{n}1\mathrm{f}^{1\ovalbox{\tt\small REJECT}}$
,
$\mathrm{W}*\mathrm{f}$
6x
CI)#*1’
$\mathrm{E}\ovalbox{\tt\small REJECT}$ $S^{\mathit{1}}$&*’t\rangle
$\mathrm{m}\mathrm{l}(\mathrm{a}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{c}\mathrm{h}6\ovalbox{\tt\small REJECT}$.
&
zl
V
Ii
$\mathrm{R}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$x
$\mathrm{S}\ovalbox{\tt\small REJECT} \mathrm{j}$ $!\ovalbox{\tt\small REJECT}$
!if$
$\mathrm{L}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}_{t^{\mathrm{A}}}^{\ovalbox{\tt\small REJECT}}$.
$\mathrm{L}\ovalbox{\tt\small REJECT} 6\supset$$\mathrm{L}/\mathrm{E}\ovalbox{\tt\small REJECT}!$E$
$\mathrm{L}_{\ovalbox{\tt\small REJECT}}\mathrm{k}^{\mathrm{A}}\mathrm{t}$)
$\ ^{\ovalbox{\tt\small REJECT}}\mathrm{J}^{-}\mathrm{C}1\ovalbox{\tt\small REJECT}/\ovalbox{\tt\small REJECT}|\mathrm{C}^{\ovalbox{\tt\small REJECT}_{t}\mathrm{s}}$.
$(\{\mathrm{O}$
$\ovalbox{\tt\small REJECT} \mathrm{J}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$ft^{\ovalbox{\tt\small REJECT}}g\{\mathit{8}_{\ovalbox{\tt\small REJECT}}k*\#\mathit{4}\ovalbox{\tt\small REJECT} tn$
(
$\ovalbox{\tt\small REJECT}-\mathit{7}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT} t\mathit{0}\mathit{1}’ tLimr^{\ovalbox{\tt\small REJECT}}(_{\ovalbox{\tt\small REJECT}}NWb^{S}\ovalbox{\tt\small REJECT} lh\mathit{6}\ovalbox{\tt\small REJECT}.$)
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} f\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$T
V&R
$\mathrm{a}^{\ovalbox{\tt\small REJECT}}\mathrm{u}\mathrm{a}^{\ovalbox{\tt\small REJECT}}\mathrm{L}(/)$ $\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}*$$\hat{V}$
$=$
$V\backslash \{y_{1}=y_{2}=0\}\subset \mathrm{C}^{3}$
,
$E$
$=\mathrm{R}_{y}^{3}\backslash \{y_{1}=y_{2}=0\}$
$k_{\mathrm{J}}^{\underline{\mathrm{g}}}\underline{\backslash ,,}\lambda \mathcal{F}6$ $.\hat{V}\#\mathrm{f}\not\in\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}^{-}C^{\backslash }h6$
$(\mathrm{k}\mathrm{g}_{1\backslash \backslash },\Xi\hslash\grave{\grave{>}}f\mathit{1}\mathrm{V}\backslash )\hslash>\mathrm{b}$ $\Xi,\mathfrak{R}_{1\backslash \backslash }^{\backslash }fx\mathrm{p}\cap\doteqdot$ $\epsilon\Leftrightarrow\vee\supset$
.
$E$
$\cross S_{\theta}^{1}l_{\mathrm{c}}^{=}l\mathrm{f}dy_{1}\Lambda dy_{2}\Lambda dy_{3}\Lambda d\theta\tau_{\mathrm{H}}^{\backslash }\backslash \cap \mathrm{g}$
$\xi\vee\supset t\mathrm{J}6$
.
$E\cross S^{1}\hslash^{\}}\mathrm{b}\hat{V}\wedge \mathit{0})$
,
$\cap \mathrm{p}\mathrm{g}\Leftrightarrow \mathrm{k}’\supset’\ovalbox{\tt\small REJECT}_{J7}^{/\backslash }\overline{|\overline{\mathrm{p}}\rfloor}\ddagger \mathrm{B}\Phi k7g\Re \mathrm{b}$\ddagger
$\check{:)}$.
$\not\in\sigma\supset\gamma_{arrow}’\emptyset\#\proptoarrow\not\in$
$\tau^{\backslash }$
,
$y\in El_{\acute{\mathrm{c}}\nearrow}\mathrm{T}_{\iota^{\backslash }}^{1_{\vee}}$$v=v(y)= \frac{|y|}{\sqrt{y_{1}^{2}+y_{2}^{2}}}(-y_{2}, y_{1},0)$
,
$w=w(y)= \frac{1}{|y|}y\mathrm{x}v=\frac{(-y_{1}y_{3},-y_{2}y_{3},y_{1}^{2}+y_{2}^{2})}{\sqrt{y_{1}^{2}+y_{2}^{2}}}$
$\ k\grave<$
.
$\langle y, v\rangle=\langle v, w\rangle=\langle w, y\rangle=0\hslash>\vee\supset|y|=|v|=|w|\vee C^{\backslash }\backslash h6$
.
$\mathit{1}\backslash Rl-$
”
$x=x(y, \theta)=v(y)\cos\theta+w(y)\sin\theta k$
$\mathrm{k}^{\mathrm{Y}}l1\#\mathrm{f}^{\backslash \backslash }\mathrm{g},\mathfrak{R}_{\iota\backslash \backslash }^{\backslash }\langle x, y\rangle=0$,
$|x|--|y|$
$\vee \mathrm{C}^{\backslash }\backslash h$ $\mathrm{Y}j$
,
$\Phi$
:
$E\cross S^{1}arrow\hat{V}$
,
$(y, \theta)\vdash\not\simeq z=x(y, \theta)+iy$
$l\mathrm{f}\cap \mathrm{p}\xi\Leftrightarrow\{\ovalbox{\tt\small REJECT}’\supset’\ovalbox{\tt\small REJECT}_{J\mathrm{J}}^{\prime\backslash }\overline{|\overline{\mathrm{p}}\rfloor}\uparrow \mathrm{B}krx6$.
$\hat{V}l\mathrm{f}$
$V\sigma)\ovalbox{\tt\small REJECT}_{\acute{\acute{\tau}}\mathrm{f}1^{\backslash }\mathit{1}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}f_{arrow}’\hslash>\mathrm{b}}^{\mapsto f}\backslash \backslash$,
$[V]k_{\mathrm{p}}^{arrow}\equiv+\mathrm{i}\mathrm{f}$
$6$
$l_{\acute{\mathrm{c}}}\#\mathrm{f}E\cross S^{1}$
A
$\sigma\supset\ovalbox{\tt\small REJECT}_{JJ}^{J\backslash }\Leftrightarrow\Leftrightarrow$$\acute{L_{\ulcorner\backslash }^{\mathrm{t}}}T\prime \mathcal{X}\mathrm{b}l\mathrm{f}^{\grave{\backslash }}$
\ddagger
$\backslash .\acute{(}\ovalbox{\tt\small REJECT}\prime \mathrm{g}\mu$$(\mathrm{R}_{y}^{3}\backslash \{0\})\cross S^{1}$
A
$U$
)
$\mathrm{F}_{7J}^{\prime\backslash }\prime k$ $\sigma$)
$\mathrm{D}\cross$ $\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$$h6$
.
$7\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{i}^{\mathrm{B}}\mathrm{L}}\mathrm{F}$
$1$
0’
$(-i\langle z, t-y/|y|\rangle)$
$=$
$\langle y, t\rangle$
$-|y|-i\langle x(y, \theta), t\rangle$
,
$\Phi^{*}((\overline{\partial}\partial|y|)^{2})$
$=$
$\frac{1}{2|y|}dy_{1}\Lambda dy_{2}\Lambda dy_{3}\Lambda d\theta$
$p_{\grave{1}}\backslash \Re \mathfrak{y}$ $\underline{\backslash }"[perp]^{\nearrow}\supset$
.
$t\in\Omega$
\ddagger
$U\ovalbox{\tt\small REJECT}$$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
$\backslash \backslash h6^{>}\sim k$
$t_{\acute{\mathrm{c}}}\backslash \mathrm{f}\mathrm{f}^{\mathrm{B}},\Leftrightarrow-\backslash \mathrm{b}$\ddagger
$\mathcal{D}\vee$.
mE
2
fi
$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}$$\mathrm{T}6$
. $f
$l\mathrm{f}$Maple
$\epsilon$ffl
$\mathrm{v}\backslash \gamma’-\cdot$$\not\in$
$2\mathrm{f}\mathrm{i}\sigma)^{\sqrt}\mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT} \mathfrak{p}_{\grave{\grave{1}}}-arrow q)$\ddagger
$\check{\mathcal{D}}\}_{\acute{\mathrm{c}}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathfrak{l}\acute{-}fx6*\mathrm{g}$$\sigma$)
$\ovalbox{\tt\small REJECT}\Phi\#\mathrm{f}\overline{\triangleleft\backslash }\mathrm{B}\mathrm{f}\mathrm{f}\mathrm{l}^{-}C^{\backslash }\backslash ,-\frac{--}{\mathrm{p}}+\ovalbox{\tt\small REJECT} \mathrm{b}^{-}C^{ff}\star$ $\gamma_{-}\prime \mathrm{g}\mathrm{g}$$\hslash \mathrm{V}\backslash l\acute{-}rX’\supset f’.\mathfrak{x}\mathrm{b}\hslash>\overline{\overline{\equiv}}\grave{\chi}fx\mathrm{v}\backslash$
.
$\ovalbox{\tt\small REJECT} E$ $\theta\sigma)_{\grave{\mathrm{J}}}\ovalbox{\tt\small REJECT} \mathrm{O}^{\backslash ^{\backslash }}E\emptyset\grave{\grave{1}}\mapsto l’-\nearrow \mathrm{W}^{\backslash }$-c
$\backslash 6$
i&J
$rightarrow \mathrm{C}[] \mathrm{f}fX\iota\backslash$.
$\not\equiv FJJ\backslash$$\{y, v, w\}1^{\backslash }A\% U)$
orthogonal
frame
kffl
$\mathrm{V}^{\backslash }\vee CE\mathrm{E}$
$\xi k^{\mathrm{g}}\backslash \Leftrightarrow\lambda \mathrm{b}\gammarightarrowarrow$&ffLIf
(
$\cap \mathrm{p}$@Ii@J
$1_{\vee}^{\backslash }\backslash kT6$
)
$\xi=\theta+\eta(y)fx6$
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\eta(y)\not\supset\grave{\grave{>}}\Gamma\mp\#\mathrm{T}6$
$\hslash\grave{\grave{>}}$,
$\mathrm{B}fl$
$\mathrm{b}$
$\hslash>[]=dy_{1}\Lambda dy_{2}\Lambda dy_{3}\Lambda d\xi=\mathrm{d}\mathrm{y}\mathrm{i}$
A
$\mathrm{d}\mathrm{y}2$A
$\mathrm{d}\mathrm{y}3$A
$\mathrm{d}9\hslash\grave{\grave{:}}\Re 0$
$\mathrm{E}’\supset$
.
$E\cross S^{1}\downarrow\sigma)$
,
$h6$
$\backslash \}\mathrm{f}(\mathrm{R}_{y}^{3}\backslash \{0\})\cross S^{1}\downarrow\emptyset\ovalbox{\tt\small REJECT}_{JJ}^{\prime\backslash }k_{\mathrm{p}}^{\overline{\overline{\simeq}}}+\ovalbox{\tt\small REJECT} \mathrm{T}6f_{arrow\emptyset}’$
$t-” \ovalbox{\tt\small REJECT}\Phi\wedge\Gamma\pm \mathrm{A}\mathrm{j}\mathrm{F}^{\mathrm{i}},\frac{\mathrm{I}\mathrm{F}}{\tau}\backslash$$\epsilon\Xi\lambda \mathrm{b}$
\ddagger
$\mathcal{D}\vee$.
$q=|y|$
,
$s=y/|y|\in S^{2}$
&k‘
く
$k$
$dy1dy2dy3d6=q^{2}dqd\theta ds$
&
$f_{X6}$
.
$\not\in \mathrm{f}$
$q[]_{arrow’}’\supset \mathrm{V}$
‘
て
$\emptyset \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}_{J\mathrm{J}}^{\prime\backslash }1\mathrm{f}$Laplace
$\pi|\int-\Rightarrow C^{\backslash }\overline{\mathrm{a}^{\backslash }},\ovalbox{\tt\small REJECT}$$\}_{\acute{\iota}}-\overline{\overline{\mathrm{p}\Rightarrow}}+\ovalbox{\tt\small REJECT}^{-}C^{\backslash }\mathrm{g}6$.
$\mathit{1}^{\backslash }Rl’arrow\theta l_{\check{\mathrm{c}}}\mathrm{c}$$\mathrm{V}$$\backslash$
て
$\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{J}7}^{\prime\backslash }\mathcal{F}6$$k$
,
$\mathfrak{B}6$
$\emptyset[] \mathrm{f}s\in S^{2}t\acute{|}$
-
\mbox{\boldmath $\tau$}6ffi/J]\
て
‘‘’
$\sim\mathcal{X}\iota\hslash\grave{\grave{>}}$
Poisson
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{J\mathrm{J}}^{\prime\backslash }l_{arrow}^{-}-$$\mathrm{a}\tau\epsilon$
$\sim-k$
$\hslash\grave{\grave{>}}_{\overline{\overline{\beta}}}^{\Rightarrow}\not\subset \mathrm{B}f\mathrm{f}\mathrm{l}^{-}C^{\backslash }\mathrm{g}6$.
5
2
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}^{\bigwedge_{-}}$$\Omega=\{t\in \mathrm{R}^{2};|t|<1\}kT6k$
$\partial\Omega=S^{1-}C^{\backslash }h6$
.
$V=\{z$
$=(z_{1}, z_{2})\in$
$\mathrm{C}^{2};z^{2}=z_{1}^{2}+z_{2}^{2}=0\}$
&
#
く
.
$V\emptyset$
smooth locus
$V\backslash \{0\}[]_{-}’\grave{l}’\mathfrak{o}^{\backslash }\check{\mathcal{D}}\not\in_{J\mathrm{J}}^{\prime\backslash }U\supset fi$ $\triangleright$$\nearrow^{\backslash }\}\backslash \#$
$[V]$
&
T.
$\not\in\Phi 3$
(Lhffl)
$u(t)\in \mathrm{C}^{0}(\overline{\Omega})\hslash\grave{\grave{:}}\Omega-\mathrm{C}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square kR\hat{E}T6$
.
$v=u|_{\partial\Omega}\in \mathrm{C}^{0}(S^{1})\not\simeq$
Dirichlet
$\mathrm{F}\hslash l\ovalbox{\tt\small REJECT} kT6$
.
$\sim-\emptyset\not\simeq$
$\mathrm{g}$$Q[v](t)= \frac{-1}{16\pi^{2}}[V].v(y/|y|)e^{-i\langle z,t-y/|y|\rangle}\overline{\partial}\partial|y|$
$k$
$\mathrm{k}^{\backslash }\#\ddagger\#\mathrm{f}\cdot$,
$l\mathrm{f},\ovalbox{\tt\small REJECT}\emptyset$$t\in\Omega[]_{\acute{\mathrm{c}}}k\backslash \mathrm{b}$
て
$u(t)=2Q[v](t)-Q[v](0)\hslash\grave{\grave{:}}ffiV$
$\underline{\backslash }" L^{\vee}\supset$.
6
n
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\wedge\circ-\mathrm{E}1\mathrm{b}\not\in u$
$\epsilon^{\backslash }\mathrm{c}$$\tau$
$n\grave{\wedge}\ovalbox{\tt\small REJECT}\varpi^{\wedge}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\emptyset\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}[]_{-\overline{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\emptyset^{\prime\backslash }}\prime \mathrm{p}\Delta \mathrm{f}\mathrm{i}[] \mathrm{f}_{\urcorner}^{\mathrm{A}}\emptyset\mu-arrow 6\acute{\mathrm{r}}\ovalbox{\tt\small REJECT} \mathrm{b}$
n-c
$\iota$$\backslash rx\mathrm{v}$$\backslash$.
Berndtsson-Passare
\emptyset ff&
$\cap\overline{-}\ovalbox{\tt\small REJECT}[]’-$,
$\Leftrightarrow\yen\emptyset X\mathrm{b}$
,
$\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}$$\emptyset \mathrm{f}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}9^{J}\Delta^{\backslash }\mathrm{f}\mathrm{f}\hslash\grave{\grave{:}}h$$6k*_{\mathrm{J}\mu\backslash }^{\mathrm{a}\mathrm{e}\mathrm{g}}$
$\hslash 6$
.
$k\emptyset \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{f}1_{\mathrm{A}^{\backslash }}’\mathrm{R}\# n$
$\ovalbox{\tt\small REJECT} \mathrm{A}$\check C‘\not\in R(b
て
‘‘
$\mathrm{g}$$\hslash$
$l\mathrm{f}^{\grave{\backslash }}$,
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\Pi\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\sigma)$
Fourier-Ehrenpreis
$\mathrm{f}\mathrm{f}1_{JJ}^{\prime\backslash g_{\overline{\prime \mathrm{J}}}}-\backslash \#\mathrm{f}\mathrm{T}$ $\text{く^{}\backslash ^{\backslash }}\}_{arrow}’*b$$\mathrm{b}\hslash$
6&
$\mathrm{E}o\vee$
.
$\ovalbox{\tt\small REJECT}’\phi l\acute{|-},$ $\ovalbox{\tt\small REJECT} \mathfrak{o}\mathrm{f}\mathrm{f}1-\ovalbox{\tt\small REJECT} \mathrm{X}\emptyset/\backslash \mathrm{A}X\#\mathrm{f}\mathrm{f}\mathrm{E}[]’.\ovalbox{\tt\small REJECT}_{\hat{J1}}|,$\ddagger
$\mathit{0}\vee$.
z
$\in \mathrm{C}^{d+1}[]’.X\backslash \mathrm{f}\mathrm{b}$
,
z
$=x+iyk\mathrm{k}^{\backslash }<\mathrm{g}\mathrm{g}$
,
$N=\{z;z_{1}^{2}+\cdots+z_{d+1}^{2}=0, |x|=|y|=1\}$
$k$
$oe\emptysetrightarrow 6k$
,
$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}$
,
$\tau$
$f_{\mathrm{c}}\mathrm{c}h\mathrm{b}$
$(\partial^{2}/\partial z_{1}^{2}+\cdots+\partial^{2}/\partial z_{d+1}^{2})f(z)=0r_{j}\mathrm{b}$
$\#\mathrm{f}^{\mathrm{Y}}$,
$f(z)= \int_{N}f(\rho z’/2)\frac{1+\overline{z}’\cdot(z/\rho)}{\{1-\overline{z}’\cdot(z/\rho)\}^{d}}dN(z’)$
$\not\supset\grave{>}\infty^{\backslash }\backslash \gamma j\backslash _{\underline{\overline{\backslash \backslash }}}L\mathrm{C}\mapsto$
.
$\sim\sim>>$
て
$\backslash \backslash$$\rho\#\mathrm{f}\Phi \mathrm{g}r_{j}\mathrm{E}\text{定}\ovalbox{\tt\small REJECT}$
.
$N\subset\{z\in \mathrm{C}^{d+1}; z_{1}^{2}+\cdots+z_{d+1}^{2}=0\}\gamma_{-}’\hslash^{\}}\backslash \backslash \mathrm{b}$
$\mathrm{E}\mathrm{T}\overline{J,\mathrm{F}_{\backslash }}\Phi k$ $5\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{f}\backslash \emptyset\grave{\grave{>}}h6$$\hslash>$
$\not\in_{)}\infty\gamma_{\mathrm{b}fj^{1_{\sqrt}\backslash }\hslash\grave{\grave{1}}}$,
$\exists^{:_{\mathrm{B}}^{\mathrm{a}}}\ovalbox{\tt\small REJECT} 5\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} k$ $k\backslash ^{\backslash }\check{\mathit{0}}\backslash \beta_{\mathrm{D}}J\Phi^{\backslash }\backslash ’\supset\langle\sigma$)
$\hslash>\# 3$
:
Bffl
$\mathrm{b}$\hslash 1
て
‘‘
$r_{X^{}}$
’.
$\ovalbox{\tt\small REJECT}$$\gamma_{=}$,
$\mathrm{f}\mathrm{i}_{\grave{1}}\underline{7\mathrm{J}}l3$;
$\frac{\rho}{2}N=$
$\{\rho z/2;z\in N\}$
-b
て
“\emptyset
$f\emptyset \mathrm{t}_{\mathrm{L}C_{\Xi}^{\backslash \geqq}}^{\mathrm{g}-}\backslash$l\ddagger
て
$\mathrm{V}$$\backslash 6$
$\mathfrak{p}_{\grave{\grave{1}}}$,
$e_{N\not\subset \mathrm{R}^{d+1}f_{arrow}’\hslash 1\mathrm{b}}2\backslash \backslash$
,
$\ovalbox{\tt\small REJECT}$$\Rightarrow\sigma\supset\nearrow\Delta^{\backslash }\mathrm{f}\mathrm{i}k$$\emptyset 5\mathrm{H}\Gamma\not\simeq_{\backslash }l\mathrm{f}’\supset\doteqdot\epsilon\check{\mathcal{D}}\}_{\acute{\mathrm{c}}}fX$$\mathrm{v}\backslash$
