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Analytic

smoothing

effects

for

aclass

of

dispersive

equations

Hideki

TAKUWA

*

1Introduction

and the

main results

There

are

many reserches about smoothing effects for dispersive equations. We

can

find many

results about smoothing effects for the Schodinger equation. However the study for general

class of dispersive operators is not enough when

we

compare the results for

a

general class

of dispersive operators with that of Sch\"odinger operators. One of our aim is to know the

dependence of the order of operators. Our class of operators we define later include not only

Schf..fiinger operators but also linearized $\mathrm{K}\mathrm{d}\mathrm{V}$ operators.

First let us describe

our

problem.

Let $m$be anintegergreater thanorequalto2. Let $P(y, D_{y})$ be alineardifferentialoperator

of order $m$ in $\mathrm{R}^{rl}$,

(1. 1) $P(y, D_{y})= \sum_{|a|\leq|n}c_{a}(y)D_{y}^{a}$

.

We

assume

that $P(y, D_{y})$ has analytic coefficients in $\mathrm{R}^{n}$ and areal principal symbol. And

we

assume

that $P(y, D_{y})$ is the real $\mathrm{p}\mathrm{i}\cdot \mathrm{c}\mathrm{i}\mathrm{p}\mathrm{a}\mathrm{l}$ type (in strong senae). That is, for

$\mathrm{d}1$ $(y,\eta)\in$

$T^{t}\mathrm{R}^{n}\backslash 0$ there exists

an

integer $j$ with $l\leq j\leq n$ such that

we

have

$\partial_{\mathrm{V}j}p(y,\eta)\neq 0$, where

$p(y, \eta)=\sum_{|a|=m}c_{a}(y)\eta^{a}$ be the principal symbol of$P(y,D_{y})$

.

Let

us

consider the initial valueproblem

(1. 2) $\{$

$D_{t}u+P(y,D_{y})u=0$,

$u|_{t=0}=u_{0}(y)$

.

We

can

study

more

gereral situations, however we consider the simpler

case

that the space

dimension $n$ equals to 1in this note.

(1. 3) $P(y, D_{y})u= \sum_{0\leq l\leq m}c_{l}(y)D_{y}^{l}$,

Department of And$\mathrm{M}\theta \mathrm{h}$ uioe andPhysics, Graduate SchoolofInformatics,Kyoto University, Kyoto

$\mathrm{R}8\mathrm{S}01$,Japa

数理解析研究所講究録 1261 巻 2002 年 133-139

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where $c_{m}(y)=1$ and $\mathrm{q}(y)$

are

analytic in $\mathrm{R}$, that is,

the coefficient of the principal part $\mathrm{i}$

constant.

Moreover we shall make the folowing assumption.

One

can

findpositive $C_{0}>0,R$ $>0,K_{0}>0$, and $\sigma_{0}\in(0,1)$ such that for $y\in \mathrm{R}$with $|y|>R$

and $k$ $\in \mathrm{N}\cup\{0\}$,

(1. 4) $\sum_{0\leq l\leq m-1}|D_{y}^{k}\mathrm{q}(y)|\leq C_{0}\frac{K_{0}^{k}k!}{|y|^{1+\sigma_{0}+k}}$

.

Let $\rho=(y,\eta)\in T.\mathrm{R}\backslash 0$, $\mathrm{m}\mathrm{d}$ let

$(\mathrm{Y}(s;y,\eta),\Theta(s;y,\eta))$ be the solution to the equation,

(1. 5) $\{\frac\Theta(s)\frac{d}{au\mathrm{r}}\mathrm{Y}(s)=-\alpha_{(\mathrm{Y}(\epsilon),\Theta(s))}=\infty m_{\Phi}^{(\mathrm{Y}(s),\Theta(s))}’,\mathrm{Y}(0)=y\Theta(0)=’\eta$

In our

case

$p(y,\eta)=p(\eta)=\eta^{m}$

.

Therefore

(1. 6) $\{\begin{array}{l}\mathrm{Y}(s)=y+m\epsilon l|^{m-1}\Theta(s)=\Theta(0)=\eta\end{array}$

We remark that for $\eta\neq 0$

.

$\lim_{larrow\infty}|\mathrm{Y}(s,y,\eta)|=+\infty$

.

The nontrappingcondition is satisfied.

Let $u(t, \cdot)\in C(\mathrm{R},L^{2}(\mathrm{R}))$ be the solution of the $\mathrm{i}\mathrm{n}\cdot \mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$

value problem (1.2).

Let us introduce aspace of the initial data,

(1. 7) $\Gamma_{h}^{+}=\{\mathrm{Y}(s;y0,\mathrm{b}) \in \mathrm{R};s \geq 0\}$

(1. 8) $X_{h}^{+}=\{v\in L^{2}(\mathrm{R});\exists\epsilon_{0}>0,\Re$ $>0,e^{h|y|^{n}\star_{-}}.v(y)\in L^{2}(\Gamma_{h}^{+})\rangle$

.

The next theorem is the main result of this paper. This is

one

of the expression for the

microlocal smoothing effect.

$\mathrm{T}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}1.1$ Let

$P(y,D_{y})$ be

defined

in(1.3) satisfying (1.4) and

$h$ $=(y\mathit{0},f\hslash)$ $\in P\mathrm{R}\backslash \mathrm{O}$

.

Let

$u_{0}\in L^{2}(\mathrm{R})$ be in $X_{h}^{+}$

.

$\mathfrak{M}en$

for

all

$t<0m$

does not belong to Me andytic

wave

$hM$ set

$WF_{A}[u$($t$,$\cdot$)$]$

of

the solution $u(t$,

$\cdot$$)$

for

(1.2). We give asimple application of thistheorem.

Let $u(t,y)$ be the solution to the next initial value problem,

(1. 9) $\{\begin{array}{l}D_{t}u+D_{l}^{m}u=0u|_{\llcorner-0}=\mathrm{u}(y)\end{array}$

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Corollary 1.1 Let the initial data $u_{0}\in L^{2}(\mathrm{R})$ satisfy that there nish a positive constant $\delta_{0}$

such that

(1. 10) $\int_{0}^{\infty}e^{2\delta_{0}|y|^{n*}\star_{-}}|u_{0}(y)|^{2}dy<\infty$,

in the

case

$m$ is odd, or

(1. 11) $\int_{-\infty}^{\infty}e^{\mathfrak{B}0|u|^{m}\star_{-}}|u_{0}(y)|^{2}dy<\infty$,

in $d\iota e$

case

$m$ is odd.

Then the solution to the initial value problem (1.9) becomes analytic with respect to the spase

variable$y$

for

$t<0$

.

Our approach is based

on

FBI transform which

was

used by Robbiano and Zuily in [15].

The reader

can

see the details about this problem and historical results in [15] and [17].

2Analytic

wave

front

set

and FBI

transform

In this section

we

define FBI transform and analytic

wave

front set. The reader had better

refer to [15] and [20].

Let $h$ $=(y0,\mathrm{I})$ $\in T^{*}\mathrm{R}\backslash \mathrm{O}$

.

Let $\varphi(x, y)$ beaholomorphicfunction in aneighborhood $U_{0}\mathrm{x}V_{0}$

of $(0, y_{0})$ in $\mathbb{C}\cross \mathbb{C}$which satisfies

(2. 1) $\frac{\partial\varphi}{\partial y}(0, y)=-\eta_{0}$,

(2. 2) ${\rm Im} \frac{\partial^{2}\varphi}{\partial y^{2}}(0,y)>0$,

(2. 3) $\frac{\partial^{2}\varphi}{\partial x\partial y}(0,y)\neq 0$

.

For above $\varphi(x,y)$

we can

define,

(2. 4) $\Phi(x)=\max_{y\in V_{0}},(-{\rm Im}\varphi(x,y))$,

for$x$ $\in U_{0}$

.

Let $a(x,y, \lambda)=\sum_{k\geq 0}a_{k}(x, y)\lambda^{-k}$ be aanalytic symbol of order zero, elliptic in

aneigh-borhood of $(0, y_{0})$

.

Let $\chi\in C_{0}^{\infty}$ be acutoff function with support in

a

$\mathrm{n}$

eighberhoQ}d

of $y0$,

$0\leq\chi\leq 1$, and $\chi\equiv 1$

near

$y_{0}$

.

The FBI transform of adistribution $u\in X(\mathrm{R})$ is defined by

(2. 5) Tu(x, A) $=\langle\chi(\cdot)u,e^{\dot{|}\lambda\varphi(x,\cdot)}a(x, \cdot,\lambda)\rangle$, $\lambda>1$

.

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According to [20]

we can

characterize the analytic

wave

front set of$u\in \mathcal{D}(\mathrm{R})$ by using FBI

transform. Next (2.6) and (2.7)

are

equivalent,

(2. 6) $n$ $\not\in WF_{A}[u]$

.

$\exists C>0,\exists\mu>0,\exists\lambda_{0}\geq 1$ such that

(2. 7)

$e^{-\lambda\bullet(l)}|Tu(x,\lambda)|\leq Ce^{-\mu\lambda}$, for$\forall x\in U_{0},\forall\lambda\geq\lambda_{0}$

.

Assume

that$u(t, ’)$is elementofa family

distribution

on

$\mathrm{R}$depending

ofa realparameter

$t$

.

Let $t_{0}\in \mathrm{R}$

.

We shaU say that a point

$n$ $\in T^{*}\mathrm{R}\backslash 0$ does not belong to the locally

uniform

analytic

wave

front set $\overline{WF_{A}}[u$(

$t_{0}$,$\cdot$)$]$ it there exist

an

FBI transform

$T$, positive constants

$C,\mu$,$\lambda_{0},\epsilon$, and aneighberhood

$U_{0}$ of 0such that

$e^{-\lambda\bullet(\sim)}|Tu(t,x,\lambda)|\leq Ce^{-\mu\lambda}$, (2. 8)

for $\forall x\in U_{0}$, $\forall\lambda\geq\lambda_{0}$, $\forall t\in(t_{0}-\epsilon,t_{0}+\epsilon)$

.

3Idea

of

proof

We make

use

ofFB1 transform in order to make achange

our

operator $P(y,D_{l})$ into

more

simpler

one.

This idea has already been known by Egorov’s theorem in the theory of

Fourier

integral operators. Thanks tothis theorem

we

can

transformanyfirst order real principal type operatorinto $D_{l}$ by Fourier integraloperators.

Since

our

operator is of order $m\geq 2$,

so we

make

use

ofFBI transform instead ofFourier

integral operators in order to transform the operator of order $m$ into the first order operator

$D_{x}$

.

The parameter Ain the FBI transform

means

$|\xi|$ in

some sense.

To

balance

the order

of two operators

we

intrMuoe the parmeter $\lambda$

.

In $\Re \mathrm{t}$ we

can

realize the next relationship by

introducing asuitable FBI transform $F$,

(3. 1) $F \frac{1}{\lambda^{m}}P(y,D_{y})=\frac{1}{\lambda}D_{l}F$, (mod

analytic).

This is the main idea of this approach.

Let us give asketch ofproofrelated to the assumption for the initial data.

We introduce aFBI type transformation.

Let $u(t, z)$ is the solution of the initial value problem,

(3. 2) $\{\begin{array}{l}[D_{t}+P(z,D_{z})]u(t,z)=0u|_{t\fallingdotseq 0}=u_{0}(z)\end{array}$

kt $\chi\in C_{0}^{\infty}(\mathrm{R})$ with $0\leq\chi\leq 1$ and

$\chi(r)=\{\begin{array}{l}10\end{array}$

$|r|$ $\leq\frac{1}{2}\epsilon_{0}$,

$|r|\geq\epsilon_{0}$

.

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We introduce

(3. 3) $Su(t,x, \lambda)=\int_{\mathrm{R}}e^{:\lambda\varphi(x,z)}f(x, z, \lambda)\chi(\frac{z-\mathrm{Y}({\rm Re} x)}{1+|x|})u(t, z)dz$.

where$\mathrm{Y}(s)$ is given in (1.5).

By operating the operator$S$to (3.2) we have

(3. 4) $( \frac{\partial}{\partial t}+\lambda^{m-1}\frac{\partial}{\partial x})Su(t,x, \lambda)=i\lambda^{m}(\frac{1}{\lambda}D_{t}Su-\frac{1}{\lambda^{m}}SPu)$

.

We define

$I(t,x, \lambda)=\frac{1}{\lambda}D_{x}Su-\frac{1}{\lambda^{m}}SPu$

(3. 5)

$= \int_{\mathrm{R}}(\frac{1}{\lambda}D_{x}-\frac{1}{\lambda^{m}}{}^{t}P(z,D_{z}))(e^{:\lambda\varphi}f\chi)u(t,z)dz$,

where${}^{t}P(z,D_{z})w= \sum_{l=0}^{m}(-D_{z})^{l}(a_{l}(z)w(z))=(-D_{z})^{m}w+\sum_{l=0}^{m-1}b_{l}(z)D_{z}^{l}$ . Thecoefficientsalso

satisfythe condition (1.4).

We define

(3. 6) $J(x, z, \lambda)=(\frac{1}{\lambda}D_{l}-\frac{1}{\lambda^{m}}{}^{t}P(z, D_{z}))(e^{:\lambda\varphi}f)$.

If$J(x, z, \lambda)$ is smallenoughin somesense, then wehaveonlyto consider the equation

(3. 7) $( \frac{\partial}{\partial \mathrm{t}}+\lambda^{m-1}\frac{\partial}{\partial x})Su(t,x, \lambda)=0$.

Thisequation iseasilysolved

$Su(t,x, \lambda)=\mathrm{S}\mathrm{u}(\mathrm{t}, -\lambda^{m-1}t, \lambda)$.

Since$x$ isnear0, wehave $|z| \geq\frac{1}{4}|\dagger n|^{m-1}|t_{0}|\lambda^{m-1}$ onthesupport of$\chi$.

Then wehave

$|Su(0, x -\lambda^{m-1}t, \lambda)|$

$=| \int_{\mathrm{R}}e^{\dot{l}\lambda\varphi(x-\lambda^{m-1}t,z)}f(x-\lambda^{m-1}t, z, \lambda)$

(3. 8) $\chi(\frac{z-\mathrm{Y}({\rm Re} x-\lambda^{m-1}t)}{1+|x-\lambda^{m-1}t|})e^{-^{s\star}}+|z|^{n}*-e^{s\star}+|z|^{m-}u_{0}(z)dz|$

$\leq Ce^{\lambda C(ae)-*1m||t_{0}|^{\underline{L}}}\overline{n*}\mathrm{r}_{\delta_{0}\lambda}\int_{\mathrm{B}}|\chi(\cdots)e^{\underline{s}_{\mathrm{f}|z|}arrow n*-}u_{0}(z)|dz$

$\leq Ce^{\lambda\Leftrightarrow(x)-*1m1\mathfrak{l}u|^{m}\star_{-}\mathrm{r}\lambda}||e^{l_{0}|z|\mathrm{A}_{-}}u_{0}||_{L^{2}(\Gamma_{0})}.$

.

Thisimplies

$e^{-\lambda\Phi(x)}|Su(t,x, \lambda)|\leq Ce^{-\mu\lambda}$,

(3. 9)

for $\forall x\in U_{0}$, $\forall\lambda\geq\lambda_{0}$, $\forall t\in(t_{0}-\epsilon,t_{0}+\epsilon)$

.

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Since we

can

get

same

properties for FBI transform with ausual cutoff function

as

the

operator S introduced in (3.3),

we can

prove Theorem 1.1.

The $8\mathrm{m}\mathrm{a}\mathrm{l}\ln \mathrm{a}\mathrm{e}\mathrm{s}$ of

$J(x,z,\lambda)$ is important for the above approach. In fact

we

can

globally

constract aparametrix along the

bicharacteristics.

Lemma 3.1 There eist $\epsilon_{1}>0$, and a holomorphic

function

$\varphi(x, z)$ in the set

(3. 10) $E=\{(x,z)\in \mathbb{C} \mathrm{x}\mathrm{C};{\rm Re} x\geq-\epsilon_{1}, |{\rm Im} x| <\epsilon_{1}, |z-\mathrm{Y}(x;y0,f\hslash)|<\epsilon_{1}(1+|x|)\}$,

such

mat

(3. 11) $\frac{\partial\varphi}{\partial x}(x,z)=p_{m}(z, -\frac{\partial\varphi}{\partial z}(x,z))$ in $E$,

(3. 12) $\frac{\partial\varphi}{\partial z}(0,y\mathrm{o})=-m$,

(3. 13) ${\rm Im} \frac{\partial^{2}\varphi}{\partial z^{2}}(0,m)$ $>0$,

(3. 14) $\frac{\partial^{2}\varphi}{\partial x\partial z}1^{\mathrm{o}},m)$

$\neq 0$

.

Lemma

3.2 There exists

an

analytic symbol $f$

of

order

zero

defined

in $E$ such that

(3. 15) $|J(x,z,\lambda)|\leq Ce^{\lambda\bullet(\sim)-\mu 0\lambda}(1+|x|)^{N_{0}}$,

where$r$ $>0$ and $N_{0}$ is $a$ integer.

In order to prove Lemma 3.1 and Lemma 3.2

we

have to solve the eikonalequation and the

transport equation. Since we make $\lambda$ large, we

introduce the aet $E$ which is global along the

bicharacteristic.

The way to constract phase and amplitude functionsis discussed in [17].

References

[1] H.

Chihara,

Gain of $\mathrm{r}\mathrm{e}\infty \mathrm{a}\mathrm{r}\mathrm{i}\forall$ for semiloear $\mathrm{M}\mathrm{r}\overline{\mathrm{d}}$

inger equations, Math. Ann. 315,

(1999), 529-567.

[2] H. Chihara, Gain of analyticity for semilnearSchrodinger equations, preprint.

[3] W. Craig, T. KappelerandW. Strauss,Miclolocal

disper8ive8moothing

for the$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{r}\tilde{\mathrm{o}}\mathrm{d}\dot{\mathrm{m}}$ger

equation, Comm. Pure. Appl. Math. 48, (1995), 76uffl.

[4] J-M. Delort, F.B.I,transformation, Lecture Notes in Math. StudiesN0.122. 1992,

Springer-Verlag.

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428-469.

[6] S. Doi, Smoothingeffects forSchodinger evolutionequation and globalbehavior of

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J. Punct. Anal. 128 (1995), no. 2, 253-277.

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[9] L. Hormander, The Analysis of Linear Partial

Differential

Operators, Springer Verlag,

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[10] L. T’Joen, Microlocal smoothing properties for plates equation, preprint.

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173-186.

[13] Y. Morimoto, L. Robbiano, and C. Zuily, Remark

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Schrdinger equation, Indiana Math. J., to appear.

[14] T. $\overline{\mathrm{O}}\mathrm{h}\mathrm{j}\mathrm{i}$, Propergation of

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[15] L. Robbiano, and C. Zuily, Miclolocal analytic smoothingeffect for the Schr”dingerequ&

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