数理解析研究所講究録1090 巻 1999 年 33-45

33

Lagrangian properties for the diffractionin the complex dornain
Pascal Laubin
University of Li\‘ege, Belgium

1Introduction

Iaet $M$ be a real manifold with boundary anda second order differential operator withsmooth coefficients and real principal symbol . We assume that is of real principal typeon the boundary. Let us consider the classical Dirichlet
and
$P$

$p$

$p$

$\mathrm{n}\mathrm{o}_{v}^{+}\ _{\mathrm{m}\mathrm{a}\mathrm{C}_{0}}\circ \mathrm{r}+\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{S}\mathrm{t}\mathrm{i}\mathbb{C}$$\mathrm{p}^{r_{\mathrm{G}}}.\mathrm{h}.1\mathrm{e}\Phi\wedge$$Pu=0$ in$M$,$u_{|\partial M}=0$

If the equation of the boundary is $f=0$ with $f>0$ inby$M_{\tau}$.
the diffractive region is defined$\mathcal{G}_{-\mathrm{t}^{-}}=\{\rho\in\dot{T}^{*}\partial M : p(pl=0, \{p_{i}.f\}=0, \frac{\{p,\{p,f\}\}_{p}}{\{\{p,f\},f\}_{p}}>0\}$ofand corresponds to rays tangent to the boundary. The propagatiort of[ [ However. very
Gevrey and analytic singularities is known in this setting, seealong diffractive rays. In [9] Lebeau proves that,
lagrangian properties are
far away frorn the data, the operator mapping the DirichIet data to the normal derivative
Gevrey 3 distributions with weight.
of the solution belongs to a class of
We review a result on the lagrangian properties of the solution at the transition fromframework. Using thhe canonical invarithe shadow to the illuminated region in theance, we prove that the solution belongs to a class of lagrang.ian distributions associateddata,
to a pair of lagrangian submanifolds. As a consequence. we see that, for a
the second wave front lies in a lagrangian submanifold.
Here we use thenext investigate the same problem in thethespaces of Sj\"ostrand. We
etry of complex canonical trarrsforms
the
FBI
the
transforrn
definition of bilagrangian distributions in this framework aiid describe.
of the solntion the boundary$C^{\infty_{2}}$$\sin_{\mathrm{b}}\mathrm{m}^{\mathrm{Y}}1C\mathrm{q}r\mathrm{i}_{\dot{l}}\mathrm{a}\mathrm{e}\mathrm{s}$.

$\mathfrak{s}_{12}1\mathrm{L}\lrcorner.,$$7_{4}\rceil,$$8\mathrm{j}$,$t\gamma \mathrm{t}\mathrm{e}\mathrm{v}$$\mathrm{p}_{\perp}^{\mathrm{r}}\mathrm{e}@^{\wedge}.\mathrm{r}\backslash _{\mathrm{v}}\oplus$$\}$$1\mathrm{a}\mathrm{f}\mathrm{f}^{\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{a}\mathrm{n}}$$C^{\infty}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\overline{\mathrm{l}}1d$$\mathrm{V}_{\mathrm{v}}^{T}\mathrm{e}$$\mathrm{a}\mathrm{n}\mathrm{a}\iota \mathrm{y}\mathrm{t}\mathrm{i}_{\mathbb{C}}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}_{\Leftrightarrow}\sigma \mathrm{o}\mathrm{r}_{f}1$$\circ\sigma \mathrm{e}\mathrm{c}_{-^{\mathrm{Y}}}..\sim 1^{-}$$\mathrm{a}\iota \mathrm{l}\mathrm{d}$$\mathrm{c}_{\mathrm{A}}^{\mathrm{f}}$2
2.1$\mathrm{P}_{\epsilon 1\mathrm{i}\mathrm{r}\mathrm{s}}^{\sigma}$$\mathrm{s}\mathrm{u}\iota:$)$\mathrm{n}\mathrm{z}\mathrm{a}\mathrm{n}i\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}$phase

Let $X$ be amanifold of real dimension alld with local coordinates
the cotangent bundle $T^{*}X$ , we consider the canonical 2-.form$C^{\infty}$$\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{e}\Gamma \mathrm{a}1_{\dot{\overline{\mathrm{L}}}\mathrm{Z}}\mathrm{e}$$\backslash \omega \mathrm{u}^{\iota 1}.\mathrm{p}\mathrm{r}\mathrm{c}\mathrm{b}\iota\Delta.\mathrm{m}$of lagrangian$\backslash \perp \mathrm{I}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\iota \mathrm{o}\mathrm{C}\mathrm{a}\mathrm{l}$$H_{\varphi}$$n$$\sigma=\sum_{J^{=\wedge}}d\epsilon d1dj^{\wedge}X_{\mathrm{i}}$$x_{1},$$\ldots,$$x_{n}$. On

34. Thisis conic for)bywhere the dual coordinates$T^{*}X=T^{*}X$(:
the cotangent
the multiplication. We denote by}with the zero section. It is said conic if itof d\’imension is lagrangian if
A submanifold A of
$t>0$
every
.
is invariant through for
The classical definition of a phase function for a conic Iagrangian submanifold is the. For sinplicity, we restrict ourself to the case of a real non-degenerate phasefunction.
$\Gamma_{1\grave{1}}\mathrm{a}\mathrm{n}\mathrm{i}_{\mathrm{A}}t1\mathrm{o}\mathrm{d}$$\mathrm{a}\iota’\hat{\mathrm{c}}\{\mathrm{i}\mathrm{e}\epsilon \mathrm{n}\mathrm{e}\mathrm{d}$$=.\delta_{ik}$$d_{\wedge\dot{s}^{r}}^{\mathrm{f}}\backslash D\prime x_{k}$$x,\xi\rangle\mapsto(x,t\xi)$$M_{t}$$\mathrm{h}\eta\sim^{1\mathrm{L}\mathrm{d}}1‘\}$$\{0\}$$\mathrm{r}\mathrm{f}_{\vee}^{1}\mathrm{m}\mathrm{o}\mathrm{V}\mathrm{e}\mathrm{d}-$,$\dot{T}^{*}X$$n$$\sigma_{|\Lambda}=0$$T_{t}$$\mathrm{f}\mathrm{o}\mathrm{g}\zeta m\mathrm{i}\mathrm{n}\mathrm{g},$$[1]$

Definition 1 Let $X$ be aconic subset ofa localwhich isin

$X\cross \mathbb{R}^{N}\backslash .\{0\}$$\Gamma$and$C^{\infty}\prime trl_{\text{ノ}}amf_{\mathit{0}}td$$\varphi$$\Gamma$and$C_{\varphi}=\{_{(’}X,\theta)\in \mathrm{r} :$\varphi^{\mathit{1}}$is a$\wedge|\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}$phase$\mathrm{f}\mathrm{u}\mathrm{r}_{\wedge}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{H}}$$j_{\varphi}$:real valued function in an open
is calledof $de,gree1$ . The
in the set$C^{\infty}$$h_{\mathit{0}\gamma}no_{-}.qeneous$$pb_{\nu}aS_{\vee} \rho \mathrm{f}unct’lon\mathrm{w}Oj.,\mathrm{f}_{--}Y?_{J}.\mathrm{f}d\varphi\neq\bigcap_{\vee}$

If

be a

$funct\dot{i}on\varphi$$\mathrm{r}\mathrm{g}(\{\swarrow_{\theta x}’,\prime l\varphi_{\theta\theta})=N$\varphi_{\theta}’(X,\theta)=0\}$.

then the differential of the map$C_{\varphi}arrow\dot{T}^{*}X$

is of rank . If it is an embedding then$n$:$\varphi$$(x, \theta)\vdash\Rightarrow(x, \varphi_{x}(\prime x,\theta)\backslash )$is called a phnse function. Since$j_{\varphi}^{*}o\cdot=j_{\varphi}^{*}d_{\langle}’\xi dx\grave{J}=d\langle\varphi_{x}d\prime x)=d\{d\varphi_{\{}c\rangle\varphi=0$

its image$*\mathrm{b}=j_{\varphi\nwarrow\varphi}\prime C$} is a lagrangian submanifold of$\dot{T}^{*}X$,.

2.2 2-microlocal phase}wave front set along a lagrangian submanifold A is defined as a subset of theThecotangent bundle of A. To define lagrangian distributions associated to this.
setting, we iutroduce new phase
If A is a conic iagrangian submanifold of
we have the identification$\Re(\mathrm{x}$$\mathrm{n}\mathrm{d}$$0^{Y}\{\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\Gamma \mathrm{i}\mathrm{c}$$\mathrm{f}\mathrm{o}\mathrm{e}_{\grave{1}}\mathrm{C}\mathrm{t}i\{\}\mathrm{n}\mathrm{s}$$\dot{T}^{*}X,$$\mathrm{t}^{\mathrm{Y}}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}$$\dot{T}^{*}\Lambda\sim\tau_{\mathrm{A}}\dot{\tau}^{*}x$where the right hand side is the normal bundle of A. Indeed, if is a normal to A at apoint thenis a well-defined l-forn.
Morecver this manifold has twa homogeneities: one inherited from A and another one
is said conic bitagranqian if it
as a cotangent bundle. A lagrangian submanifold ofais conic for both homogeneities. We introduce phase functions that parameterize.
andsuch thatLet
be an open subset of$S_{\}}t>0$ imply (. Suctt ; open set is called a profile. An open subset ofuJ$t’h ノ pwfile
ifis said$k$$T_{\rho}\Lambda\ni h\vdasharrow\sigma(h$$p$$k\rangle$}$\dot{T}^{*}\Lambda$$\mathrm{s}\mathrm{t}\iota\Lambda$$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{l}\mathrm{f}_{1}\wedge.1\mathrm{d}$$(x,\theta,l\})\in\Gamma_{0}$$X\cross \mathbb{R}^{N}\backslash \{^{\mathrm{f}\}}.\}\mathrm{X}\mathbb{R}^{M\backslash }0\backslash ^{\mathrm{I}_{1}}\}$$\Gamma_{0}$$x,t\theta,st^{r}q1\in\Gamma_{0}$$X\cross \mathbb{R}^{N}\backslash _{\backslash }\{0\}\cross\Re^{M}\backslash \mathrm{i}_{\mathrm{L}}\{\}\}$$\Gamma$$\mathrm{a}x$$b\check{w}_{J}anic$$\Gamma_{0}$$\sim$,$\bullet(x,\theta,\eta)\in \mathrm{r}_{\mathrm{a}}\mathrm{n}\mathrm{d}t>0\mathrm{i}\mathrm{m}_{\mathrm{P}^{\}}}\mathrm{y}(x,t\theta,t\eta)\in\Gamma$$\bullet$forand $0<s<\epsilon$ .

$\epsilon \mathrm{a}\mathrm{d}_{1\infty \mathrm{m}\mathrm{p}\mathrm{c}}\mathrm{a}\mathrm{t}$subset $K$ of$\Gamma_{0}$, there is$\epsilon>0$such that (ffj,$\theta,$$s\eta$)$\in\Gamma$

if $(x,\theta,\eta)\in K$

35

respect to a family of
If is biconic
union. The profile of is the largest profile
We also introduce$\Gamma$$\mathrm{w}\underline{i}\mathrm{t}\mathrm{h}$$\Gamma${$\mathrm{r}_{1}\mathrm{J}.=$$(x,\theta):\overline{\lrcorner\lrcorner}\eta$This is an open conic subset of
. A
and
Let
$(p,q;r)$ if$r\in\perp \mathrm{Y}_{0}$$p,$ $q\in \mathbb{R}$$\bullet$$\bullet$

it is also biconic with respect to theirsuch that the last condition is satisfied.

$\mathrm{p}\mathrm{r}\bigcap_{-\mathrm{f}}\underline{\mathrm{i}}1\mathrm{e}\mathrm{s}.$$f(x,t\theta,t\eta)=t^{p}fi_{\backslash }’X,\theta,\eta\grave{)}$}$\Gamma_{0}$such that$X\cross \mathbb{R}^{l\mathrm{V}}\backslash \{0\}$$C^{\infty}$iffunctionfor every $\gamma\hslash))/\in 1" 0$ , there is a
$F$ ining$\langle x_{0},\theta_{0},$

}.

.

$f$

$(x,\theta,\eta)\in\Gamma,$$(x,\theta,\eta)\in\Gamma$:$\Gammaarrow \mathbb{R}^{m}$is said $b^{J}i’ ho7rwgeneous$ of degree$t>0$ ,$\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{l}\mathrm{z}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{c}’ \mathrm{d}V\mathrm{o}1\mathrm{r}(x_{0},\theta 0,\eta_{0})\backslash$and a$C^{\infty}$function$V\mathrm{x}_{\mathit{1}^{-\epsilon\epsilon[\mathrm{S}\mathrm{a}\mathrm{t}}}\rceil\backslash r\mathrm{i}_{\mathrm{S}}\Re_{\mathrm{V}}$$f(x,\theta, s\eta)=s^{q}F(X,\theta,\eta, S^{1,/r})$if$(x,\theta,\eta, s)\in V\mathrm{x}\overline{|}0,\epsilon[$.

. In the, it
The integer . inserted here essentialJy forlagrangian
microlocal
the
on
has
some
geometrybut
effects
does not affect the 2-microlocal
submanifolds involved. We say that has the regularity .
$\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{a}\underline{\ddagger}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{S}\cap \mathrm{B}^{\underline{\mathrm{q}}_{\sim}}-$$r\underline{i}\mathrm{q}$$f$$\mathrm{a}_{-}\mathrm{p}\mathrm{p}^{\underline{|}}iC\lrcorner \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{i}\mathrm{o}\mathrm{n}.$$r$Deflnition 2 Let$\bullet$A be a conic lagrangian submanifotd of$\bullet$($\rho$$\epsilon\psi^{i}$$\dot{T}^{*}X$,real vatued function which is homogeneous of degree 1 in

be a$C^{\infty}$

be a$\rho.\infty rea_{\epsilon}$}valued function $uJhich$ is$bif!\sim oqery\wedge meo^{\phi l}.s$of

The pair$(_{\backslash }\varphi,\psi,\rangle$

is a local 2-phase $func\hslash on$ of A ($u\prime itf\iota$. a local phase function that$\epsilon\varphi’i^{Q}$$\bullet$\varphi^{f}\theta(x.\theta)’\psi^{;}=\mathrm{Q}1,\eta(x,\theta,\eta)=0\}$

at $e,ach$, point of$C_{\varphi,\eta_{\acute{J}}}$,;$d.\rho.gY\epsilon e(\iota,$

and$C_{\varphi,\psi}=\{\mathrm{t}x,\theta,\eta)\in \mathrm{r}_{0} :$\Gamma_{1}$$1;\wedge\backslash ;r$in$\Gamma\wedge$.

regular\’ity ) if$r$$paramet\rho.\dot{n}_{\sim}’\beta.\mathrm{e}.\Lambda_{f}$,. the vector$(\psi_{1,.r}’,,\psi 1,\theta)\mathit{1}J$

is different ffim

$r_{\vee}\mathrm{k}(\psi_{\varphi_{\theta}}^{JJ}1\eta xj,\psi_{\varphi_{\theta\theta}}’’+\eta\theta j’\psi_{\iota_{\eta\eta 0}}’’,\}=N+M$

$0$

and

.

If is a phase function, the last condition means that the nap
is a local phase function of A. This definition has the folowing consequences.
a) The map$(p,\eta)\mapsto\psi_{1}(j_{\varphi}^{-}1\backslash pr,\eta;\searrow 1$$\varphi$$j_{\Psi,\Psi^{l}}$:.

$c-arrow\dot{T}*\Lambda:(_{X,\theta,\eta)}\varphi,\varphi’rightarrow(\langle x, \varphi’x\rangle,j_{\varphi*}((\psi\prime 1,x’\psi_{1}’,\theta)_{1\varphi}\tau C))$is a lagrangian immersion.

36

Following the identification$C_{\varphi.\psi}arrow\dot{\tau}_{\mathrm{A}}\tau^{*}x$

where

$h,$

:

$\dot{T}^{*}\Lambda\sim\tau_{\Lambda}\dot{\tau}*x$, the map$j_{\varphi,\psi}$

can be identified with$(x, \theta,\eta)rightarrow((x, \varphi_{\alpha}’\rangle, \langle h,\tilde{\psi}_{1,x}’+\varphi_{xx}’’.h+\varphi_{x\theta}’’.k))$$kS’\grave{u}tis^{\mathrm{A}}/\dot{y}$$\varphi_{\theta}^{\prime_{f}}x.+h\varphi^{;}\theta \mathrm{f}’.k+\tilde{\psi}_{1};’,0\theta=$,

in a biconic set andregularity
be a local 2-phase functionin. By the definition, is a local phaseand there isa biconic open subset of whose profile contains
such thatis called
is a local phase function in . A local 2-phase function
a 2-phase function ifandare embeddings.
that if
is a local
One canfunction in andis asuch thatthen there is a biconic open set whose profile contains.
in2-ph&9e function
Hence, if
is a 2-pht,se function then$\mathrm{b}_{\text{ノ}}\backslash \mathrm{L}\underline{\mathrm{r}}*_{\backslash },$$(\Psi^{\tau^{\int_{r}}},’)$$(x_{00},\theta,\eta_{0},)\in C_{\varphi,\psi}$$\Gamma$$r\rangle$$l\mathrm{h}\backslash ^{\mathrm{W}\mathrm{i}\mathrm{t}_{}}-$$\mathrm{f}\mathrm{u}\mathrm{I}\mathrm{l}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$\Gamma_{1}$$\varphi$$\tilde{\Gamma}$$\Gamma$$(x, (\theta,\eta))\vdasharrow$$(x_{0},\theta_{0},\eta_{0})$$\tilde{\Gamma}$$\varphi(x, \theta)+\psi(X,\theta,\eta)$$(\varphi, \psi_{J}\backslash ]$$j_{\varphi},$$j_{\varphi+\psi}$$j_{\varphi,\psi}$$(\varphi,\psi)$$\mathrm{v}\mathrm{e}\mathrm{r}i\alpha$$\Gamma$$\mathit{2}- phX^{\mathrm{p}}e$$(x_{0}, \theta_{0},\eta_{(3})\in C_{\varphi,\psi}^{\mathrm{Y}}$$\tilde{\Gamma}$$(x_{0},\theta_{0}, \eta 0)$$(\varphi,\dot{\psi})$$\tilde{\Gamma}$$(\varphi,?\mathit{1}J)$$\{((x, \varphi_{x})’, (h,\psi’1,x\varphi xx\varphi_{x\theta}.k)+\prime\prime.h-\vdash r’) :is a conic bilagrangian submanifold ofIf$\mathrm{C}_{\text{ノ}})$$(\varphi., \emptyset)$(_{X},\theta)\in C\psi\}\psi_{1,\theta}\varphi,+\varphi_{\theta x}^{l\prime}.h’\varphi^{J\prime}+\theta\theta\cdot 0k=\}$. It is denoted$\dot{T}^{*}\Lambda_{\varphi}$$\Lambda_{\varphi,\psi}$.

is a 2-phase function, then$n-\mathrm{r}\mathrm{g}(\pi_{\mathrm{A},X}\varphi)=N-\mathrm{r}\mathrm{g}(\varphi_{\theta\theta}\prime\prime)$,$n-\mathrm{r}\mathrm{g}(\mathcal{T}\downarrow \mathrm{A}_{\varphi},\psi,\Lambda\varphi 1=M-\mathrm{r}\mathrm{g}(_{\hat{\dot{\Psi}}_{1},m}\prime\prime l)$

,

and,

.

$n-\mathrm{r}\mathrm{g}(\pi_{\Lambda_{\varphi,\psi}},X)=N+M-\mathrm{r}\mathrm{k}(_{\{\mathrm{I}}^{\psi_{1}^{\prime;}},\eta\eta\psi_{\varphi’’}’’1,\eta\theta\theta\theta)$

Pairs of lagrangian submanifolds

2.3

We now describe the geometric setting associated to a -phase. If, the blowup of $X$ along
is
a

$Y$

is a

$\mathrm{s}\mathrm{u}\mathrm{b}\iota \mathrm{n}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}_{\mathrm{o}\mathrm{l}\mathrm{d}}$

of

$Y$

$c_{\text{ノ}}\infty_{\mathrm{m}\mathrm{a}}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{C}’ 1\mathrm{d}X$.

$\hat{X}_{\mathrm{Y}}=(X\backslash \mathrm{Y})\mathrm{U}T_{Y}^{*}X$

The sets$\cap$($\{x\in\omega :f_{j}(x)>0\}\cup\{(x, h)\in\dot{T}_{Y}X : x\in\omega, df_{j_{\backslash }}[x).h>\theta\})$$1\underline{\prec}j\leq \mathrm{P}$where$\omega$

is an open subset of $X$ and. For this topology, the projectionof$f_{j}\in C^{\infty}(\omega),$$\hat{X}_{\mathrm{Y}}$$\mathrm{t}\mathrm{o}_{\mathrm{P}^{\mathrm{o}1}\mathfrak{B}^{\gamma}}\mathrm{o}$Definition 3 A pair ($\Lambda_{0},\Lambda_{1\text{ノ}^{}\}}$

is a$\mathit{2}-m_{v}^{i}\prime_{7v}.to\mathrm{t}nl$

is a conuic lagrangian submaiiifolds of$\bullet$$\Lambda_{0}$$\bullet$$\Lambda_{1}\cap(\dot{T}^{*}X\backslash \Lambda_{0})$is a conic lagrangianfor } , form a basis of
is continuous.
$\mathrm{a}1$$d\mathrm{f}_{j_{1}}\mathrm{Y}\cap\omega=0$$\pi:\hat{X}_{\mathrm{Y}}arrow X$

pair of lagrangian$\dot{T}^{*}X,$$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{m}\mathrm{a}\mathrm{I}\mathrm{L}\mathrm{i}\mathrm{f}01\mathrm{d}_{\mathrm{S}}$$\Lambda_{1}\subset(_{\backslash }’\dot{T}^{*}X)_{\grave{\Lambda}0}’$$\mathrm{b}^{\backslash }11\mathrm{b}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{n}i\$)$1\mathrm{d}$of$\dot{T}^{*}X_{f}$

$j$

,

of

$\dot{T}^{*}X$

if

37

$\bullet$for each
and a 2-phase function$(\rho, h)\in \mathrm{A}_{1}\cap\dot{T}_{\Lambda}\tau*x\mathrm{o}$$\langle$’there is an open neighborhood$\varphi,\psi)$$V$$(\rho,h)$in$(\dot{T}^{*}X)^{\wedge}\mathrm{A}_{0}$such that

and$\Lambda_{\mathrm{O}}\cap\pi(V)=\Lambda_{\varphi}$$\Lambda_{1}\cap V---\Lambda_{\varphi+\psi}\cup\Lambda_{\varphi_{)}\psi}$

In this situation. we say that the 2-phase function (deflnes (. This is a conic bilagrangian submanifold of.
$\varphi,\psi\rangle$$\Lambda_{1}\cap\dot{T}_{\Lambda 0}(\tau^{*x)}$.

$\Lambda_{0},\Lambda_{1\text{ノ}^{})}$. Let$T_{\mathrm{A}\mathrm{o}}\Lambda_{\mathrm{i}}=$$\dot{T}^{*}\Lambda_{0}$Example 4 In$\dot{T}^{*}\mathbb{R}^{n}$, consider$\varphi(x,\xi_{)}^{\backslash }=x.\xi,$

where

of

$\xi=(\xi’,\xi_{n})$$\psi_{\backslash ^{X,\xi,\eta}}^{/\prime}\rangle=\frac{\eta’.\xi^{p}}{\xi_{n}}-H(\eta\xi_{n}’,)$

and $H$ is bihomogeneous of degree$\Lambda_{\vee^{\wedge}}.=\{(\backslash \mathrm{o},\xi) :$(_{\backslash }1, \mathrm{L}r)$.

. We have\xi n\neq 0\}$and$\Lambda_{\varphi+\psi=}\{((-\frac{\eta^{f}}{\xi_{n}},\frac{\eta’.H_{\eta}^{f}}{\xi_{n}}, +H_{i,\backslash l\iota}’,), (\xi nH_{\eta}’,,\xi-n.,).)If

in$fI(\eta’,\xi_{n})=\eta_{1}^{3}f/\eta_{2}^{2}$$\mathbb{R}^{3}$, the projection of$\{(0_{7}\xi) :on$T_{\mathrm{A}_{\varphi}}\mathrm{A}_{\varphi\psi}+$(_{\frac{\xi_{1}}{3}\backslash ^{3}}\backslash \text{ノ}=\xi_{3} :$\Lambda_{\varphi}$\xi_{3}-\neq 0\}$:\epsilon n\neq 0\}$.

is the cusp.

It can be shown, see [4], that the property of being a microlocal pair of lagrangiansubmanifolds is preserved by an homogeneous canonical transformation.
Let us describe the equivalence offunctions.
Two 2-phaseand (defined in biconic open subsets and of:diffeomorphism
are said $equivaf,ent$ if there is asuch that$2- \mathrm{p}\mathrm{h}\mathrm{l}‘ \mathfrak{B}\mathrm{e}$$\tilde{\Gamma}$$\mathrm{f}\mathrm{i}\ln c,\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}(_{(\beta}.\theta)\backslash$$\Gamma$$\tilde{\varphi},\tilde{\emptyset}\rangle$$\Gammaarrow\overline{\Gamma}$$C^{\alpha)}$$X,\cross \mathbb{R}^{N}\backslash \{0\}\mathrm{x}\mathbb{R}^{M}\backslash \{0\}$$(x,\theta,\eta)\vdasharrow(x, f(x, \theta,\eta),g(X,\theta,\eta))$$\bullet\varphi(x, f(_{X,\theta},\eta))+\psi(x, f(X,\theta,\eta),g(X,\theta,\eta))=\tilde{\varphi}(X,\theta)+\tilde{\psi}(_{X,\theta},\eta)$$\bullet$$f$is strictly bihonogeneous off degree 1,0; $r)$ and
,
$($

$g$

,

is bihomogeneous of degree$\iota(1,1;r_{\text{ノ}^{}\backslash }$are invertible in
These two pairs define the same$\bullet$$D_{\theta}f_{0}$and$D_{\eta}pj_{1}$$\Gamma_{0}$

.

pair.
is a diagonal real invertible matrix, the pair of phases$2-\mathrm{I}\mathrm{r}_{\mathrm{l}1}^{\mathrm{i}_{\mathbb{C}}}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{o}^{\wedge}.\mathrm{a}1$

If$\Delta$$\varphi(x,\theta)=\tilde{\varphi}(x,\theta’’)+\frac{\langle\Delta\theta’,\theta’\rangle}{\underline{?}\}\theta^{;\prime}||},$

defines the same

$1\mathrm{a}\mathrm{g}_{\Gamma \mathrm{a}\mathrm{n}}\mathrm{g}iC\gamma \mathrm{n}$submanifolds as$\varphi(X,\theta)---\tilde{\varphi}\{.X,$ $\theta),$$\tilde{\varphi}$$\psi(x,\theta’’,\eta)=/l\tilde{p}(x,$and$\dot{\check{\psi}}$$\theta\prime\prime,\eta\rangle$. In the same way,$\psi(.x, \theta,\eta)=\tilde{\psi}(_{X},\theta,\eta\rangle r’+\frac{\langle\Delta\eta_{;}’\eta}{2|\eta’’|}.\underline{\prime\rangle}$defines the same lagrangian submanifolds as and .
defining the same
It can be shomm that the
between two 2-phase2-microlocal pair of iagrangian submanifolds can be obtained by a composition of the$\tilde{\psi}$$\tilde{\varphi}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}_{\vee}^{\mathrm{q}}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$previous$\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}$ $\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$).

$\mathrm{f}^{1}\mathrm{J}\mathrm{n}\mathrm{C}\mathrm{t}\underline{\mathrm{i}}\mathrm{o}\mathrm{n}-\mathrm{s}$383

Bilagrangian distributions

3.1

Sym.bols

We use only classical symbols. This is enough for the applications that we consider here.
Definition 5 If
the set of all
multiorders

denote by
and $X$ is an open subset ofthat for every compact subset $K$ of $X$ and all
there is a $C>0$ satisfying$m,p\in \mathbb{R}$$\mathbb{R}^{n},$$S^{m_{\mathrm{P}}},(X, \mathbb{R}N,\mathbb{R}M)$$u\prime e$$a\in c_{\text{ノ}}\infty(X\mathrm{x}\mathbb{R}^{N}\mathrm{x}\mathrm{R}^{M_{)}}-\text{ノ}.\veearrow \mathrm{Q}q/Ch$$\alpha,$$\beta,$$\gamma$$|D_{x}^{\alpha_{D_{\theta}D_{\eta}}}\beta\gamma a(x,\theta,\eta)|\leq C(1+|\theta|+|\eta|\rangle^{m-\mathrm{I}\beta|(1}+|\eta|)^{\mathrm{p}-\mathrm{t}\gamma}\mathrm{I}$for all$(x,\theta,\eta)\in K\cross \mathrm{R}^{N}\cross \mathbb{R}^{M}$.

$\mathrm{W}\mathrm{r}\mathrm{i}\mathrm{t}_{J}\mathrm{e}$,$S_{2}^{\infty}= \bigcup_{m,p\in \mathrm{R}}s^{m.,p}$$s^{m,-\infty}= \bigcap_{p\in \mathrm{R}}S^{m,p}$.

is a R\’echet space with semi-norms given by the smallest constants
It is clear thatwhich can be used in the definition.
-phawe functions.
integrals can be defined using symbols in$\mathrm{t}(\backslash ,m,\mathrm{p}$$S^{m,p_{r}}\mathrm{c}\iota \mathrm{n}\mathrm{d}2$$\mathrm{O}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{Y}$Tbeorem 6 Let
closed conic subset$(\varphi,\psi)$an open biconic set and let
be a 2-phase function, the linear $fo7m$.
that $F\ll\Gamma$ . For e’uery$\Gamma$$\dot{i}n$

$F$

be a

$u\in C_{0}^{\infty}(x)$$o_{j^{\wedge\vee}}^{f\mathrm{r}t}\mathrm{q}0.ch$$a rightarrow \mathrm{c}[\int\int e^{i}-(\varphi\langle x,\theta)\dashv\psi\langle x,\theta,\eta\rangle)a(_{X},\theta,\eta)u(X)dxd\theta d\eta$defined in the set of all$a\in S^{-\infty}(X;\mathbb{R}^{N},\vee\backslash \mathbb{R}^{M})s\alpha tmathit{8}f\# ing\mathrm{s}^{\mathrm{t}}\tau \mathrm{d}\mathrm{P}\mathrm{p}(a1\subset F_{f}$in a unique way such th.at it isonsatisfying supp(a)for every $m,p$ .
$S_{2}^{\infty}$$\alpha$) $nt\dot{i}nuc\prime us$on the set ofcan$b\epsilon$

extended$a\in \mathrm{c}\gamma^{n2}’\backslash ^{X,\mathbb{R}^{N},\mathbb{R}}\cap \mathrm{p}\prime M$)$\mathfrak{c}_{-}^{-}F$

3.2

Distribution classmanifold of dimension
Let $X$ be a.
lagrangian submanifolds of$C^{\infty}$$n$and let ($\Lambda_{0},$$\Lambda_{\iota)}$

be a 2-microlocal pair of$\dot{T}^{*}X$

Definition 7 The spaceof $I^{m}(X,$ , an element of$I^{m,\mathrm{p}}(x, s\mathrm{t}_{0},\Lambda_{1})$$\Lambda_{0}\grave{J}$locally finite sums of an element
is the set ofand distributions of the forrn$al_{u}^{l}$$I^{m+}\prime p(x, \Lambda_{1}\mathrm{n}T^{*}x)$$I_{\varphi,\psi,a}(u)=(2 \pi)^{-}(n+2\langle N+M\rangle\}f4\int\int\oint e^{i(\varphi(\rangle}a(X,\theta x,\theta+\psi(x,\theta,\eta\rangle),\eta)u(x)dXd,\theta d\eta$where (

$U_{t}.\chi i$

is a chart ofin, an $o_{1}nenbi,c\prime on,ic$ subset$X_{2}u\in C_{0}^{\infty}(x),$$\Gamma$of$(\varphi,\psi)$

is a 2-phase function ofand$\chi(U)\cross \mathrm{I}\mathrm{R}^{N}\backslash \{0\}\mathrm{x}\mathbb{R}^{\backslash }\wedge i\backslash \{0\}$$a\in S^{m+(2}n-N)/4,\mathrm{P}^{-}M/2(x(U),\mathbb{R}^{N},\mathbb{R}M)$

satisfies$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(a)\ll\Gamma$.

$(\Lambda_{l3}, \mathrm{A}_{1})defir\iota 6d$

39

It can bc shown that this space is invariant by composition with a Fourier integral
near a point. Moreover, any 2-phase function defining the pairnear .
can be used to define any element of
The singularities of an element ofare included in the lagrangialy subinvolved, [4].
$\rho_{0}\in\Lambda_{0}$$(\Lambda_{0}, \Lambda_{1})$$\mathrm{o}\mathrm{p}\mathrm{e},\mathrm{r}\mathrm{a}\mathrm{t}_{0}\mathrm{r}\mathrm{s}$$I^{m,p}(X,\Lambda_{0},\mathrm{A}_{1})$$p_{0}$$I^{m,p}(X, \Lambda 0,\Lambda 1)$$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\overline{\mathrm{l}}\mathrm{d}\mathrm{S}$Theorem 8$I_{i}fu\in I^{m,p}(X,\Lambda_{0}\backslash ’\Lambda_{1})\mathrm{A}$

the ,
$??$

$WF(u\rangle\subseteq\Lambda_{0}\cup\Lambda_{1} , WF_{\Lambda_{0}}^{\langle 2)}(u)\subseteq T_{\mathrm{A}_{0}}\mathrm{A}_{1}$.

Application to diffraction

4

Let us consider the boundary value problen

where we use the decomposition
strictly diffiactive problems in theLet. This is a model for the$(t.x^{l}" X_{n})\in \mathbb{R}\mathrm{X}\mathbb{R}^{n-1}\mathrm{x}\mathbb{R}_{+}$$C_{\text{ノ}^{}\infty}$

category, see$[11\grave{\mathrm{J}}$.

$p\backslash ^{x_{n},T,\xi}/\}=1{}^{\mathfrak{t}}\xi|2-(1-_{ix}1)nT^{2}$be the boundary hamilbe the principal symbol of the operator and}are involved here. On one hand, we consider thetonian. Two lagrangian
flowout$r(\tau,\xi^{f})=|\xi’|^{2}-\tau^{2}$$\mathrm{m}\mathrm{a}\mathrm{n}^{\frac{\backslash }{1}}\mathrm{f}\mathrm{o}\ddagger \mathrm{d}\mathrm{S}$$\mathrm{s}\mathrm{u}\dot{\mathrm{t}}$$\Lambda_{\mathrm{O}}=\Lambda_{0,+}\mathrm{U}\Lambda 0,-\mathrm{o}\mathrm{f}$$\{((!t,0_{\text{ノ}?\backslash }\backslash \prime j\mathcal{T},\xi)):\tau=\pm|\xi’|\neq\alpha, \xi*--0\}$

throughon the boundary and followed by
the other hand, the flowotit$H_{r}$$H_{p}$

intersected with $t>0$ and $x_{n}>0$ . On$\Lambda_{1}=\Lambda_{1,+^{1j},-^{\mathrm{o}\mathrm{f}}}\Lambda_{1}$$\{_{\backslash }^{(}(0_{\gamma \text{ノ}}-\mathrm{o}), (\tau,\xi\backslash \rangle) :f \mathcal{T}=\pm|\xi|7\xi_{n}\neq 0\}$are smooth but arewith $t>0$ and $x_{n}>0$ . These twotangent at their intersection.
pair of lagrangian submanifolds
can be checked that
is a
with
through$\mathrm{m}\mathrm{a}\mathrm{n}_{1}^{;_{\mathrm{f}\Phi^{1}}}\mathrm{z}\mathrm{d}\mathrm{S}$$H_{p}\mathrm{i}\mathcal{B}_{\mathrm{t}}^{\mathrm{A}}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}$$\underline{\tau}\iota$$\langle$$\mathrm{A}_{0,\pm\}}\Lambda_{\mathrm{A}}9,\pm’\backslash$$2-\mathrm{m}_{-\mathrm{i}}\mathrm{c}\mathrm{r}010\mathrm{c}\mathrm{a}.1$$\tau_{\mathrm{A}\mathrm{n},\vee},\pm^{\Lambda_{1,\pm}}=\{(((\frac{\underline{9}}{3}x_{n}+\mathrm{a}/22\Gamma X_{n},Xx_{n})’,,$( $(0,0,0),$ (A 2-phase function$\langle\varphi_{\pm_{r}}.\psi_{\mathrm{A}}/)$

of$\pm_{\underline{\frac{1}{9}}\sigma},0,$$(\pm|\epsilon;|,\epsilon^{;},$$\mp|\epsilon’|\sqrt\overline{x_{n}}\rangle\rangle$$\mp\frac{1}{2}\sigma(\sqrt{x_{n}}+\frac{1}{\sqrt{x_{n}}}\rangle)\rangle$$(\Lambda_{0,\pm,1,\pm}\Lambda\cup\tau_{\mathrm{A}_{0,\pm}}\Lambda_{1,\pm})$:$\sigma,$,$x_{n}>0,\xi’\neq 0$}.

is given by$\varphi\pm(t,X,\xi’)=x’.\xi’-"\llcorner|\xi’|(t-\frac{2}{3}x)3n/2$

and$(\varphi_{\pm}+\psi\pm)\{t_{X},,$$\sigma,\xi 1)’=x^{;}.\xi’\pm|\xi’|(:-\frac{\sigma}{|\xi’|})^{-}1/2(t-\frac{2}{3}((X_{n}+\frac{\sigma}{|\xi’|})s/2-(\frac{\sigma}{|\xi’|})3/2))$.

40This 2-phase function has the regularity 2.
the set of all lagrangian distributions on
We denote by. This means that the symbol satisfies the following inequalities$I_{p}^{m}(X, \Lambda_{0})$$\mathrm{A}_{0}$with symbol in$S_{\rho}^{m}$$|D_{x}^{\alpha}D_{\theta}^{\beta}a(X,\theta)|\leq c\alpha.\beta(1+|\theta|)rn-\mathfrak{l}\beta 1+(1-\rho)\langle \mathrm{t}\alpha \mathrm{I}+|\beta_{1}^{1}\rangle$.

An analysis of the solution of the initial boundary value problem given in [2] leads tothe following result.
Theorem 9 The solutionof the-poenious boundary value problem belongs to$u$$I^{\frac{n}{4}-1,\frac{3}{4}}(\mathbb{R}\mathrm{x}\mathbb{R}^{n}-1\mathrm{x}\mathbb{R}_{+}, \mathrm{A}_{0}, \Lambda 1\cup\tau_{\mathrm{A}_{0}}\Lambda 1\rangle)$$+$.

$I^{\frac{n}{2’4}-\frac{\iota}{2}},(\mathrm{a}\mathbb{R}\cross \mathbb{R}^{n}-1\cross \mathbb{R}_{+’ 0}\Lambda)$

lI’be geometry in the complex domain

5

used to characterize the bilagrangian disOur purpose is to define the phasetransform. In the microlocal
of
the
in
the
formalism
tributions
case, we closely follow [6] and collect some material lrom [9], see also [13].
As usual, we identify$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\langle)\dot{\mathrm{n}}\mathrm{s}$$\mathrm{F}_{0}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\Gamma^{-}\mathrm{B}\mathrm{r}0\mathrm{S}- \mathrm{I}\mathrm{a}\mathrm{g}\mathrm{Q}\mathrm{l}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{z}\mathrm{e}\mathrm{r}$$\bullet$$\mathbb{C}^{n}$with$\bullet$$‘’,$$\bullet$$T_{z}^{*}\mathbb{C}^{n}$$\mathrm{R}^{n}\cross \mathbb{R}^{n}$$\in T_{z}^{*}\mathbb{C}^{n}$with (

withand write $z=x+\dot{i}?j$ ,$\sim c_{1,\}}\ldots\zeta_{n}\rangle\in \mathbb{C}^{n}$$\tau_{(x,y)}*\mathbb{R}2n$$harrow\Sigma-\tau_{\mathrm{c}\vee}\sim_{\mathrm{c}}\zeta(h)$using$\zeta\{h$)$= \sum_{j}\zeta_{j}h_{j}$by mapping the -linear form$\mathbb{C}$,$\zeta\in T_{z}^{*}\mathbb{C}^{n}$to the-linear form$\mathbb{R}$.

is endowed with the usual canonical 2-form andThis map is symplectic if
defined below.
with theis identified with $d(-s\propto f)\in$
It folows
if is a holomorphic function,
since $d(-\Im f)=-\Leftrightarrow s(df)=-\circ s(\partial f)$ .
is identified withIn the same way, if is a real function then$T^{*}\mathbb{C}^{n}$$T^{*}\mathbb{R}^{2n}$$2- \mathrm{f}_{\mathrm{o}\mathrm{f}}\mathrm{n}\iota-^{\alpha}s\sigma$$\mathrm{t}\mathrm{h}_{\dot{\mathrm{t}}}\tau \mathrm{t}$$f$$\partial f\in T_{z}^{*}\mathbb{C}^{n}$$T_{(x,y}^{*}\mathbb{R}^{2})n$$\mathbb{C}^{n}$.

$\frac{2}{i}D_{z}\varphi\in$$d\varphi\in T_{(x,\mathrm{y})}^{*}\mathbb{R}^{2}n$$\varphi$

All the constructions described in this section are local even this is not stated

5.1

FBI transform

Writing $z=x+_{\dot{i}}y$ and $\zeta=\xi+i\eta.$, t,he canonical 2-form on$\sigma=\sum_{j}d\zeta_{j}\wedge d_{Z_{j}}$Its real and$\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\eta^{r}$$T^{*}\mathbb{C}^{n}$

is

.

parts

(
$\Re_{\sigma}=\sum_{j}id$$d\xi_{j}\wedge dX_{j}-\eta J’$A$dy_{j}$),$\circ\triangleleft\sigma=\sum$($dj\eta_{\vee}\gamma$

A $dx_{j}+d\xi_{j}$ A$dy_{-}?.$)$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}_{\mathrm{C}}\mathrm{i}\mathrm{t}_{\lrcorner}1\mathrm{y}$

.

41

are

.
forms onbe a realfunction defined in a neighborhood of$\mathbb{R}^{2n}$$\mathrm{s}\mathrm{y}\overline{\iota 1}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{C}\mathrm{t}\dot{i}\mathrm{C}$Let$\varphi$$C_{1}$$A\mathrm{t}.--\{(_{Z,\frac{2}{\prime i-}}D_{z}\varphi(Z)) : Z\in \mathbb{C}n\}$This manifold$j_{\varphi}$and.

-lagrangian since it is identified with$\mathrm{i}\mathrm{s}srightarrow$$\{(z,d\varphi(z\rangle) :If$z_{0}\in \mathbb{C}^{n}$denotes thez\in \mathbb{C}^{n}\}\subseteq T^{*}\mathbb{R}^{2}n$$i\mathrm{f}\mathrm{f}_{1}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{S}\mathrm{i}\mathrm{o}_{\overline{\mathrm{u}}z}\mapsto\backslash z/_{z,\frac{2}{i}D\varphi}(Z)_{\mathit{1}}\backslash$

.

then

$j_{\varphi}^{*}( \Re_{\sigma})=_{\acute{J}_{\varphi}^{*}}(\sigma)=j_{\varphi}^{*}\backslash d/\mathfrak{i}_{\backslash }/\zeta d_{\sim}^{\gamma})\rangle=d(\frac{2}{l}’\partial\varphi\rangle=-\frac{2}{i}\overline{\partial}\partial\varphi$.

is a symplectic map from (It follows that,is non degenerate,. Its inverse is the projection.
is proven in [ , see also .
The$\mathrm{i}_{1Uo\varphi}^{c\overline{\mathfrak{Q}}}\mathrm{q}$$j_{\varphi}$$\frac{2}{i}f\backslash n.\overline{\partial}\partial\varphi_{f}^{\backslash }$$\backslash \cdot,$

onto$(\Lambda_{\varphi}, \Re_{\sigma},)$$7\mathrm{J}$$\mathrm{f}\mathrm{o}_{\wedge}^{\iota 1}\wedge\Re \mathrm{v}_{\wedge}^{\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{t}}\mathrm{r}\mathrm{e}\mathrm{S}1_{-}^{1}1$$[3_{\mathrm{J}}^{\rceil}$and X :
be a strtctly
Theorem 10 Letfunction near
.
such
that
near
a
transform defined). There is a unique $holomo7phic$ function
is endowed with theHere, such thatnear$\dot{T}^{*}\mathbb{R}^{n}arrow\Lambda_{\varphi}$$z_{0}\in \mathbb{C}^{n}$$plu\tau\dot{\eta}SubharY\gamma\iota onic$$\varphi$$ca,non\dot{i}cal$$\langle$$\chi(y_{0},m\rangle$$y_{0},\eta_{0\}}$$=(z_{0}, \frac{2}{i},D_{z}\varphi(Z_{\mathrm{t}\}}))\backslash$$2arrow f_{D}\gamma m\Re\sigma$$z^{\prime*}$$g_{\backslash }^{(}\chi,y$$(z_{0},y_{0})$$\bullet$the complexifioetion of:$\chi^{\mathbb{C}}$$\bullet\dot{i}g(_{\circ},\sim,y_{0}0)=\varphi(z_{0)},$$\bullet$$\chi$

is$\mathit{2}^{\urcorner*}\mathbb{C}narrow T^{*}\mathbb{C}^{n}$$-D_{v}g(_{Z_{0}},y_{0}\mathit{1}=\eta 0\backslash$

the functicm$(0, n)$, and $crd$ ical value$yrightarrow-\sim\triangleleft g(z,y)$$\varphi(z),$$(y(z),$For example, if
The FBI transform: $(y,$$-Dgy(z,y)_{\grave{l}}\mapsto(z,D_{z}g(z,y\mathrm{i})$,,has a non degenerate criticnl point $y(z,)$ uith signature
Moreover: we have$\cdot$$-Dv_{arrow y,)(}q( \vee\backslash z,l_{\backslash }z\rangle, =\chi^{-\hat{1}}\backslash Z, \frac{\underline{9}}{i}\prime Dz\varphi(Z))$$\chi:(x,\xi_{J}\backslash \overline{\prime},\langle x-i_{\searrow}^{\zeta},\xi)$$\mathrm{a}\mathrm{s}_{\llcorner}\mathrm{s}\mathrm{o}\mathrm{C}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$to$\varphi,$$\chi$

and.

.

$\varphi(z)=\frac{1}{2}||\alpha_{Z}s|^{2*_{v\mathrm{h}}},\mathrm{e}\mathrm{n}g_{\mathrm{t}}^{(_{Z}}\backslash ’ y)=\frac{i}{2}(.z-y)^{2}$

near the points ($y_{0},\eta_{0}1,$$z_{0}$

is$\tau_{-\chi}u(_{Z}, \lambda)--\int e^{i}a(z,y, \lambda\rangle u(y)d8/\lambda g\mathrm{t}z,y)$

where

5.2

$a$

is a classical s.ymbol.

Lagrangian submanifolds

In thi$ setting, lagrangian submanifold$\mathrm{C}|\delta \mathrm{K}1$

be parameterized by a holomorphic function.

42

A be aProposition 11
A near and be a local$Le_{\iota}^{l}$$p_{0}$

$FBI$

map

$canon\dot{i}\alpha\iota l$$\chi$

of$lag_{in}^{\alpha}ng\dot{i}ans\iota bman\dot{i}f_{\mathit{0}}\dot{w}ld$$\dot{T}^{*}\mathbb{R}^{n},$to$fwmT^{*}\mathbb{R}^{n}$$h$be a phase function of
to . If the$\Lambda_{\varphi}mapp\dot{i}ngp\mathrm{C}|$phase defined in theorem 10 and$g$$z_{0}$$\phi_{\mathrm{A}}(Z\rangle=\mathrm{c}\mathrm{v}_{(}\alpha,\theta)(g(_{Z,x)}+h(x,\theta))$then$\varphi_{\Lambda}=-^{\mathrm{t}_{\tilde{\mathrm{L}}}}S\emptyset\Lambda$. The critical points are given by$(_{\backslash }x, \theta)=j^{-}\mathrm{c}^{1-}\mathrm{O}\chi \mathbb{C}(1z, D_{z}\emptyset \mathrm{A}(z))$

Hereis the immersion$j$$(x, \theta\rangle\mapsto(_{X}b’\nu)\backslash ’ x$

and

. its

.

$comple\dot{\mathfrak{M}}fiCat_{\dot{i}}on$$j_{\mathbb{C}}\dot{i}^{\mathrm{Q}}$.

We have$x^{\mathrm{c}_{(\Lambda}\mathbb{C}})=\{(z\backslash ’ D\mathcal{Z}\phi_{\Lambda}(_{Z})) :Z\in \mathbb{C}n\}$and$\iota/r’\Lambda(z\iota,\leq\varphi(_{Z_{\text{ノ}}})$The
In this. holds if and$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}_{\mathrm{V}}$$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\overline{\mathrm{t}}\mathrm{i}_{\mathrm{S}\mathrm{m}}$if$\mathrm{o}\mathrm{n}\mathrm{l}_{y}\mathrm{V}$.

$\langle_{Z_{\mathrm{Y}}}.\frac{2}{i}D_{z}\varphi\langle_{Z}))\in\chi\langle\Lambda)$.

, the lagrangian distributions are defined in the following way.

a $lagmng\dot{i}an$ subbe a distribution in an open subset ofif, in a. W\’ith the notations of propositionis said lagmngian atofneighborhood of , we haveDeflnition 12 Let$man\dot{i}fold$$\Omega$$u$$\mathbb{R}^{n},$$\Lambda$$\ulcorner\dot{I}^{*}\Omega$$\mathit{1}\mathit{1}_{\mathrm{Z}}u$$p_{0}$$z_{0}$$(\tau_{\chi}u)(_{Z,\lambda)=eb}i\lambda\phi_{\mathrm{A}}(z\rangle(z, \lambda)$

whereis a $classi\infty l$ analytic symbol.

$b$This is equivalent to the fact that can be written $u=u_{1}+u_{2}$ with
not in the singular spectrum of and$u$$p_{0}=j_{h}(x0,\theta 0\grave{)}$$u_{2}$$\prime u_{1}(X)=\int_{\Gamma}ea(x,\theta)\theta ih(x.\theta)$

where$\Gamma$

is a conic neighborheood of$\theta_{(\}}$

and . is a-cal ana.lytic symbol near$\mathrm{C}\iota_{\mathrm{a}_{\lrcorner}\mathrm{s}\mathrm{s}}\mathrm{i}$$\mathit{0}$$(x_{0},$$\theta_{0}\rangle$.

Pairs of lagrangian submanifolds

5.3

Let us consider the FBI transforn of a 2-phaseparameter.
to the case of one$\mathrm{f}\mathrm{f}\mathrm{i}_{1\mathrm{C}}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$. For simplicity, we restrict ourself$2-\mathrm{m}\mathrm{i}_{\mathrm{C}\mathrm{r}}\mathrm{o}\overline{1}\mathrm{o}\mathrm{c}\mathrm{a}1$

Proposition 13 Let (pair of lagrangia submanifol&and $(h,\psi)$be a. We assume thatbe a 2-phase $fu\dot{n}ction$ for the pairnear a point,
analytic and that is anfunction of$2- mi_{C}rol\delta Cal$$\Lambda_{0,\Lambda_{1})}$$n$$\psi$$analyt\dot{i}C$$h\dot{i}S$$p_{0}\in\Lambda_{0}$$(\Lambda_{0},\Lambda_{1},)$$(x,\theta, \sigma)\underline{1}/2$.

$\psi(x,\theta,\sigma)=\psi 1(_{X},\theta)\sigma+\psi_{3}/2(_{X},\theta)\sigma^{3}+/2\psi_{(X,\theta},p))\sigma+\mathcal{O}2(\sigma^{5/2})$

If$g\dot{i}S$$Z\mathrm{n}\vee\in \mathrm{C}^{\backslash n},$an $FBI$ phase function associn, we have$\phi(z,\sigma)$$t\mathrm{e}d$to a local canonical map$\chi$

such that$=\mathrm{C}\mathrm{V}_{(x,\theta\rangle}(g(z,x)+h(_{X,\theta)}-\vdash\psi(x, \theta,\sigma))$$=\Phi_{\Lambda_{0}}(z)+\Phi 1(Z)\sigma+\Phi_{3}/2(z)\sigma+\Phi_{2}\mathrm{s}/2(z)\sigma+O2(\sigma^{5})/2$

Here$\Phi_{1}$and$\Phi_{3/2}$$\chi(\rho_{0)}^{\backslash }/=$

are real on$\pi \mathrm{c}’\chi\langle\Lambda_{0}),$$\Phi_{1}(Z_{0})=0,$.

$D_{z}\Phi_{1}(Z_{0})\neq 0and_{S\Phi}^{\alpha}2(z_{0})>0$.

43With the notations of the proposition 13, a
at with respect to
if, in a neighborhood of.

is said $analyti’$ bilagmngian, we have$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{f}\mathrm{i}\mathrm{b}\mathrm{q}\star \mathrm{i}\mathrm{b}\mathrm{o}\mathrm{n}u$$(\Lambda_{0},\Lambda_{1})$$p_{0}$$z_{0}$$(\tau_{\chi}u_{J\backslash })(Z,\lambda)\text{ノ}=I_{0}^{e^{i}a(\gamma}\delta\phi \mathrm{t}z,\sigma\rangle \text{ノ}.J^{\cdot}\sigma,\lambda)d\sigma$

where a‘$\mathrm{i}_{\grave{\mathrm{o}}}$holomorphic in an open set of the form$\{(_{Z_{7}\sigma})\in \mathbb{C}n\mathrm{x}\mathbb{C}:|z-z_{0}|<\epsilon_{\backslash ,\prime}|^{\Im}\sigma|<c\Re\sigma\}$and is bounded by
Since

for
and

$C\lambda^{m}$$S^{\triangleright_{\Phi_{2}(z_{0)}}}>0$$\lambda>1$.

$\Phi_{1}(_{\triangleleft\}}\sim),$$\Phi_{3}/2\{Z\mathrm{o}\rangle$

are real, we can choose$-\Im\phi(z_{0,\delta})<-\alpha(S\varphi\Lambda_{0}z\mathrm{o})$For$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}_{\mathrm{s}}$$5\backslash /\backslash \mathrm{o}_{\mathrm{s}\mathrm{m}}\underline{\mathrm{D}\mathrm{t}\iota}$such that.

if$\Lambda_{0}=\{((\mathrm{o},X_{n}), (\xi’,0))\}$,$\Lambda_{1}---\{((0,0\rangle, (\xi’,\xi_{n}))\}$

and $g(z,y)=i(z-y\rangle^{2}/2$ , we have.

$\Phi_{\Lambda 0}(_{Z})=\frac{i}{2}z^{p_{\sim}}’$$\Phi_{\Lambda_{1}}(z)=\frac{\acute{l}}{\dot{2}}Z^{2}$and$\phi_{\backslash ^{\sim}}^{(’}\cdot,’,$6$\sigma)=\frac{iz^{J2}}{2}\dashv-\sigma\sim’+\sim_{n}\frac{\dot{i}\sigma^{2}}{2}$.

Bilagrangian structure of the parametrix

Let us show how, at the transition of the shadow arid the ilIurninated region, the parametrix
a bilagrangian
if thedata is conormal.
Using [11], we
assume that the operator can be written$\mathrm{d}\mathrm{e}\mathrm{h}^{\mathrm{a}}\mathrm{n}\mathrm{e}\mathrm{s}$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\overline{\mathrm{b}}\mathrm{u}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$$\dot{\mathrm{b}}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}$$\mathrm{n}_{\wedge}1\mathrm{a}\mathrm{y}$$P(x,$ $D\rangle=D_{x_{n}}^{2}+R(x,D_{x^{\prime)}}$in the half space $\{x_{n}>0\}$ . Its principal symbol is$p’\backslash x,\xi)=\xi_{n}^{2}+r(x,\zeta’\rangle$.

. We assume that the point (Let
is diffractive. This means
that
.
and
Following [7], we first perform a complex canonical transform. We choose the weightfunctionand a caiionical map$r_{0}(X_{7\xi^{l})}’=r(x_{J}^{t}.,0,\xi’)$$r_{0}(x_{0}’,\xi\prime \mathfrak{g})=0$$dr_{()}\neq 0,$$X_{0’\xi_{0}’)}’$$\partial_{x_{n}}r<0$$\varphi_{0}(Z’)=|\mathrm{s}z^{f}\propto|^{2}/2$$\chi_{0:\dot{\tau}*}\mathbb{R}n-1arrow(\Lambda_{\varphi 0},\Re\sigma)/$mappingto (and the giancing region $\{r_{0}=0\}$ to
nap is associated a FBI transform.
$(x_{0_{7}}’\xi_{0}’)$$0,0\rangle$$\{\propto sz_{1}=0\}$. To this canonical44

After this transform, we obtain a$\mathrm{p}_{\mathrm{S}\mathrm{e}\mathrm{u}\mathrm{d}_{0}\mathrm{d}\mathrm{i}}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}^{\mathrm{f}}\mathrm{b}-\dot{i}\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}0^{\mathrm{r}}\mathrm{A}$$P(x,\tilde{D}, \lambda)=\tilde{D}_{x_{n}}^{2}+R(x,\tilde{D}_{x^{f}}, \lambda)$

near$\mathrm{t}^{\{\}}/,$

Its principai symbol $p(x,\xi)=\mathrm{t}nc2+r(x,\xi’)$ is real on$\mathrm{o})_{0}\Gamma 1\Lambda_{\varphi}0^{\cdot}$

is equivalent to$x_{n}+q|x\xi_{\grave{2}}\text{ノ}\prime,=0$$H_{\varphi}$

and $p\langle x,\xi)=0$

with$q_{\backslash ?}^{(_{X’}}\xi)=\xi_{1^{-}}e_{\backslash }(x’,\xi’\text{ノ}\xi)2\neq \mathcal{O}^{(}n\backslash \xi_{n}^{4})$In the$\Lambda_{\varphi 0}$,$e_{\backslash }^{(}0,0)>0$.

space, the problem is reduced to find an outgoing solution to$P_{(X}^{(},\tilde{D},$Define, as above,$\Lambda_{\{i}$$\lambda)u\langle X,$$-0,$$\lambda)_{--},$$u_{1^{x_{h}\mathrm{c}}}|==g$.

(1)

as the flowout of the set of diffractive points through thebyandas the fiowout of all the characteristic points at$\mathrm{b}_{0}\mathrm{t}\mathrm{m}\mathrm{d}\dot{C}\iota_{\Im^{\gamma}}$hamiltonian$x=0$ through.
In the boundary value problem , we consider the boundary data
corresponding to a Dirac mass. Using the Lebeau construction of the parametrix, we obtain the following estimation.
$H_{r}\mathrm{f}\mathrm{t}4\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{d}$$H_{p}$$\Lambda_{1}$$H_{\mathrm{p}}$$g(x’, \lambda)=\exp(i\lambda z^{J}\sim)0$$(1)’$$\mathrm{T}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{l}4$

The $fu,ncti_{on}$$\varphi(Z_{\rangle}\sigma 1,$$= \mathrm{C}\mathrm{V}_{\langle_{\vee}^{\varphi}},\cdot,\prime l\rangle(_{\backslash }\frac{i}{2}(_{\backslash }\prime \mathrm{a}n-X_{r\mathrm{g}},)2_{-\vdash}H(_{\backslash }Z\sigma,\eta’\sqrt{x_{n}+}’,’,)\theta$$-x_{1} \sigma-x\prime\prime.J’\eta+F(X^{J}, \sqrt{\sigma},\eta\prime\prime)+\frac{\dot{i}X^{;2}}{\cap,\angle})$the conditions of proposition 13. Moreover, the solut\’ion of the boundary value
problem (1) can be wnitten $u_{1}+u_{2}$ whereis anallytic $bd$agrangian and$sat_{\dot{i}}SfieS$$u$$u_{1}$$|/u_{2}(z, \lambda\grave{l}|\leq^{r^{\gamma}}\cup\epsilon e^{\lambda(}\varphi\Lambda 0\langle z)+cd(z,\pi \mathrm{o}x-\langle \mathrm{A}_{0}.).)^{3})+\in\lambda$

near for every$0$$\epsilon>0$.

References[1]$\mathrm{H}_{\ddot{\mathrm{O}}\mathrm{r}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{n}}\mathrm{d}\mathrm{e}\mathrm{r}_{\mathrm{y}}$L.,$\tau-heanalysi_{\vee}\mathrm{q}\sim\cap f\vee$linear$part\dot{i}al\prime xiffe7\mathrm{e}n\cdot t?a\vee\iota$

operators I-IV, Springer-

Verlag, 1983-85.
[2] Friedlander, F.G. and Melrose, R.B., The wave front set of the solution of a simple
value problem with glancing rays JI, Math. Proc. Camb. Phil. Soc.,
87,, 97-120.
$init’\dot{i}alrightarrow boundar\mathrm{t}/$$19_{l}^{\kappa}7$[3] Laubin, P., Etude

295-416.

$\mathrm{L}^{4]}\lceil$$\mathit{2}- m\dot{i}Crolocale$de la diffiaction, Bull. Soc. lioy. Sc. Li\‘ege, 4., 1987,

to a 2-microlocal pair of lagrangian
Laubin P., Willems B.,
manzfolds, Comm. in Part. Diff. Eq., 19, 1994, 1581-1610.
$D\dot{i}str^{i}l\mathrm{a}uti\theta nSasS\alpha \mathrm{i}iated$

45$\mathrm{r}1^{5]}$Laubin P.,
and lagmngian properties in diffiactive boundanj value problems, Microlocal analysis and spectral theory, 91-113, Kluwer Academic Press, 1997.
$c_{ono\mathit{7}}mality$

同 Lebeau, G., Demi\‘eme micmlocalisation sur tes
Fourier, Grenoble, 35, 1985, 145-216.
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