数理解析研究所講究録
1156 巻 2000 年 11-26

11-boundedness of wave operators fortwo dimensional Schr\"odinger operators$L^{p}$

Kenji Yajima$(\nearrow 4\backslash \ovalbox{\tt\small REJECT}\backslash \ovalbox{\tt\small REJECT}_{-}^{-})$

Department of Mathematical Sciences, University of Tokyo

3-8-1 Komaba, Meguro-ku, Tokyo, 153 Japan

Introduction, Theorems

1

This lecture is concerned with the boundedness in$L^{p}$or Sobolev spaces of

the wave operators for the Schr\"odinger operators. Let $H_{0}=-\triangle$ be the free

Schr\"odinger operator on$\mathrm{R}^{d},$$d\geq 1$, and $H=H_{0}+V$ be its perturbation bya multiplication operator with a real valued function$V$. It is well known$\mathrm{A}_{\wedge\wedge}|$the spectral and scattering theory for Schr\"odinger operators $([1], [4], [5], [6])$that if$V$

is short range,$\mathrm{v}\mathrm{i}\mathrm{z}$. $V(x)$ decays at infinity$1\mathrm{i}\mathrm{k}\mathrm{e}\sim C|x|-1-\epsilon,$$\epsilon>0$

,

then:
1. The operator $H$ is selfadjoint in the Hilber spacedomain$H^{2}(\mathrm{R}^{d})$$L^{2}(\mathrm{R}^{d})$with the, the Sobolev space of order 2.

2. The spectrum of $H$ consists of non-positive eigenvalues and the absolutely continuous spectrum$[0, \infty)$. The singular continuous spectrumof $H$ is absent.

3. The wave operators defined by the limits $W_{\pm}u= \lim_{tarrow\pm\infty}eeitH-itH0u$ exist.
The wave operators$W_{\pm}$

are unitary fromcontinuous spectral subspace
absolutely continuous partis the projection onto

$L_{\mathrm{a}\mathrm{c}}^{2}(H)$$HP_{\mathrm{a}\mathrm{c}}$$L_{\mathrm{a}\mathrm{c}}^{2}(H)$.

of$L^{2}(\mathrm{R}^{d})$onto the absolutely

for $H$ and intertwine$H:W_{\pm}H_{0}W_{\pm}^{*}=HP_{\mathrm{a}\mathrm{c}}$$H_{0}$and the, where$F_{\mathrm{a}\mathrm{c}}1$

12

It follows from the peoperty (3) thatfunction$f$on

and the mapping properties of$\mathrm{R}^{1}$or Sobolev spaces

any Borel$f(H)P_{\mathrm{a}\mathrm{c}}=W_{\pm}f(H_{0})W^{*}\pm \mathrm{f}\mathrm{o}\mathrm{r}$between If spaces$f(H)P_{\mathrm{a}\mathrm{c}}$may be derived from those of $f(H_{0})$ if$W^{k,p}(\mathrm{R}^{d})$$W_{\pm}$andare bounded in theor Sobolev spaces. Note that $f(H_{0})$ is the convolution operator by the Fourier transform $K(x)$ of the function
and the$L^{p}$$W_{\pm}^{*}$$f(\xi^{2})$$If-L^{q}$ continuity of $f(H)$ may

be derived by studying the function $K(x)|$.

In particular, we can obtain the following $L^{p}-L^{q}$ estimates for the propagator $e^{-itH}P_{C}(H)$ of the time dependent Schr\"odinger equations $i\partial_{t}u=Hu$

on the continuos spectral subspace and for the$\sin t\sqrt{H}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{t}_{\mathrm{o}\mathrm{r}}\overline{\sqrt{H}}P_{C}(H)$ofthe wave equation with potential $\partial_{t}^{2}u+Hu=0$ by applying our theorems to
the well-known estimates for free equations:$||e^{-itH}P(cH)u||L^{p}\leq C|t|^{-}d(1/2-1/p)||u||_{L}p’$;$|| \frac{\sin t\sqrt{H}}{\sqrt{H}}P_{c}(H)u||_{p}\leq C|t|-(d-1)(1/2-1/p)||u||_{W^{()}/2}d-1-(d+1)/p,p\rangle$’

both for $2\leq p$ and $1/p+1/p’=1$ .
When the spatial dimensions $d\geq 3$ , we have shown in our previous papers ([15], [16]) that the wave operators$1\leq p\leq\infty$$W_{\pm}$are bounded in$L^{p}(\mathrm{R}^{d})$for allunder suitable conditions. For small potentials, thecovers rather general potentials and when $d=3$ , the wave$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{W}\mathrm{i}\mathrm{n}_{!}\mathrm{g}$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}([15])$

operator is bounded in If for any $1\leq p\leq\infty$ whensome$\sigma>1,$$\langle x\rangle=(1+x^{2})^{1/2}$

and

in

for all$L^{p}(\mathrm{R}^{d})$$||\mathcal{F}(\langle x\rangle\sigma)V||_{L^{d_{*}}}$$1\leq p\leq\infty$

is small for. We write $d_{*}=(d-1)/(d-2)$ .

Theorem 1.1 Let $d\geq 3$ . Suppose that$\sigma>2/d_{*}$$||\langle x\rangle^{\sigma}V||_{L^{2}}$$\mathcal{F}(\langle x\rangle^{\sigma})V\in L^{d_{*}}(\mathrm{R}^{d})$is sufficiently small. Then

$W_{\pm}$for someare bounded.

For larger potentials, we need an additional spectral condition for

be bounded in$L^{p}$

and we obtain the following theorem $([16])$ .

$W_{\pm}$to13

Theorem 1.2 Let $d\geq 3$ . Suppose that there exists a constant $C>0$ suchthat, for some $\rho>d/2,$

satisfies$V$$||V||_{L\rho(|}x-y|<1)\leq C(1+|X|)-(3d+2+\epsilon)/2$.

Suppose, in addition, that zero is neither eigenvalue nor resonance of theoperator H. Then, the wave operators$1\leq p\leq\infty$

are bounded in If$W_{\pm}$

has a solution
$0$

$u$

such that$\langle x\rangle^{-1\epsilon}-u\in L^{2}(\mathrm{R}^{d})$is not a resonace for $H$ . If

$0$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}-\triangle u+V(X)u(x)=0$

but$u\not\in L^{2}(\mathrm{R}^{d})$Remark 1.3 If$D^{\alpha}V,$$|\alpha|=0,1,$ $\ldots$$\ln$

for all$1\leq p\leq\infty$$i$$L^{p}$

for all $1<p<\infty$ .

, , satisfy the conditions of Theorem$\ell$

1.1 or Theorem 1.2, then the wave operators$W^{k,p}(\mathrm{R}^{d})$. It is well know

is a resonance or eigenvalue of $H$ , itisknown that the wave operators cannot be bounded in

space

for all.

Here, zero is said to be a resonance of $H$ if thethat$(\mathrm{R}^{d})$$W_{\pm}$

and $k=0,1,$

are bounded in the Sobolev, . See [15] for the details.
$\ell$$\ldots$

the lower dimensions $d=1$ and $d=2$ , however, the high singularitiesof the resolvent $(H_{0^{-}}Z)-1$ at $z=0$ prevents us to apply the analysis valid forhigher dimensions. The one dimensional case, however, can be treated as follows by writing the wave operators in terms of the generalized eigenfunctionsin the form
$W_{\pm}u(x)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi_{\pm}$

(

$X,$

k)\^u(k)dk.

We estimate the integral kernel$K(x, y)= \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i}\phi_{\pm(X}ky,$$k)dk$

by using the information on the eigenfunctions obtained in the ODE theory,and then appply standard$L^{p}$

estimates for the singular integral operators on

14

the line.
$f_{\pm}(x, k)$this way, we can prove the following theorem $([3])$ . We denote by$\ln$

the solution$\mathrm{o}\mathrm{f}-f^{\prime/}+Vf=k^{2}f$as $xarrow\pm\infty$ . We say that$V$

which satisfies $|f_{\pm}(x, k)-e^{\pm i}xk|arrow 0$$0\neq 0,$
is of generic type if $[f_{+}(x, 0),$ $f_{-()]}X,$$0=0$ , whereof exceptional type if [ $f_{+}(x, 0),$ $f_{-()]}X,$$\mathrm{a}11\subset\lambda 1$$[u, v]=u’v-uv/_{\mathrm{i}_{\mathrm{S}}}$

theWronskian.
Theorem 1.4$L^{1}(\mathrm{R}^{1})$inif$L^{p}(\mathrm{R}^{d})$$V$$Suppo\mathit{8}e\langle x\rangle^{3}V\in L^{1}(\mathrm{R}^{1})$

if

$V$

is of generic type andis of exceptional type. Then, the wave operators$W_{\pm}$$\langle x\rangle^{4}V\in$are bounded

for all $1<p<\infty$ .

Remark 1.5 The decay conditions on the potential has been relaxed by Wederin respective cases. Moreover,[14] toor to$\langle x\rangle^{2}V\in L^{1}(\mathrm{R}^{1})$$W_{\pm}$$\langle x\rangle^{3}V\in L^{1}(\mathrm{R}^{1})$are bounded in the Hardy space$H^{1}(\mathrm{R}^{d})$and $BMO$ space. See, [14] forthe details.

The purspose of this lecture is to extend these results to two dimensions.

We assume that$V$

is bounded and satisfies the following decay condition.

Assumption 1.6 The potential $V(x)$ satisfies

some$\delta>6$$x\in \mathrm{R}^{2}$$|V(x)|\leq C\langle_{X}\rangle^{-}\delta,$for.

For stating the main result, we need some notation which we introduce

now. For$s\in \mathrm{R}$and integral $k\geq 0,$

is the weighted Sobolev space, and
$X$

and

$\mathrm{Y},$$B(X, X)$ .

vents$B(X, \mathrm{Y})$$H^{k,s}( \mathrm{R}^{2})=\{f :\sum_{|\alpha|\leq k}||\langle x\rangle^{s_{D^{\alpha}f|}}|_{2}<\infty\}$$L^{2,s}(\mathrm{R}^{2})=H^{0,s}(\mathrm{R}^{2})$. For Banach spaces

is the space of bounded operators from $X$ to$\mathrm{Y},$$B(X)=$We denote the boundary values on the positive reals of the resol-$R_{0}(z)$and $R(z)=(H-z)^{-}1$ by$R_{0}^{\pm}( \lambda)\equiv\lim_{\epsilonarrow+0}R_{0}(\lambda\pm i\epsilon)$,$R^{\pm}( \lambda)\equiv\lim_{\epsilonarrow+0}R(\lambda\pm i\epsilon)$.

15

These limits exist in$B(L^{2,\sigma}(\mathrm{R}^{2}), H2,-\sigma(\mathrm{R}^{2})),$$\sigma>1/2$

and they are locally

H\"older continuous with respect to $\lambda\in(0, \infty)$ (cf. [1]). In two dimensions,has the logarithmic singularities at $k=0$ and has the$R_{0}^{\pm}(k^{2})$asymptotic expansion as a

$B(L^{2,S}(\mathrm{R}2), H^{2},-s(\mathrm{R}2))$-valued function, $s>3$ :$R_{0}^{\pm}(k^{2})=c^{\pm}(k)P_{0}+G_{0}+O(k^{2}\log k)$

where$c^{\pm}(k)=1 \pm i\frac{2}{\pi}\gamma\pm i\frac{2}{\pi}\log\frac{k^{2}}{2},$$\gamma$.ng$\mathrm{f}\mathrm{o}\mathrm{l}1_{\mathrm{o}\mathrm{W}}.\mathrm{i}$

(1.1)

,

is the Euler number,$P_{0}$

is the rank

one operator defined by$P_{0}u(x)= \int_{\mathrm{R}^{2}}u(x)dx$

and$G_{0}$

is the mini.m$\mathrm{a}1$

Green function$\mathrm{o}\mathrm{f}-\triangle$:$G_{0}u(x)= \frac{-1}{2\pi}\int_{\mathrm{R}^{2}}(\log|x-y|)u(y)dy$

We write $c_{0}= \int V(X)dx$ and set $V_{0}(x)=c_{0}^{-1}V(X),$ $P=P_{0}V_{0}$ and $Q=1-P$ .
We have $P^{2}=P$ and $Q^{2}=Q$ . We assume

Assumption 1.7$c_{0}\neq 0$

and$1+QG_{0^{VQi\mathit{8}}}$invertible in$L^{2,-S}(\mathrm{R}^{2})$for some$1<s<\delta-1$ .

The main theorem in this lecture may be stated as follows:
Theorem 1.8 Suppose that Assumption 1.6 and Assumption 1.7 are satis-

fied. Then, for any $1<p<\infty$ , there exists a constant $C>0$ such that$||W_{\pm}u||p\leq C_{p}||u||_{\mathrm{P}}$,$u\in L^{2}(\mathrm{R}^{2})\cap L^{p}(\mathrm{R}^{2})$where the constant $C>0$ is independent of .
$u$Some remarks are in order:

16

Remark 1.9 If Assumption 1.7 is satisfied, then $1+QG_{0}VQ$ is invertible
infor all $1<s<\delta-1$ (cf. [7]). Assumption 1.7 is satisfied if and$L^{2,-S}(\mathrm{R}^{2})$only if there are no non-trivial solutions$u\in H_{1_{\mathrm{o}\mathrm{C}}}^{2}(\mathrm{R}^{2})of-\triangle u+V(x)u=0$

which satisfy the asymptotic behaviour at infinity$\frac{\partial^{\alpha}}{\partial x^{\alpha}}(u-a-\frac{b_{1}X_{1}+b2x_{2}}{|x|^{2}})=O(|x|^{-1-}|\alpha|-\epsilon)$

for some$\epsilon>0$$con\mathit{8}tants$a,$b_{1}$, where a,and$b_{2}$$u$andare constants.

$b_{2}$

does not vanish, thena half bound state andthen$b_{1}$$0$$u$

since $-\triangle Q=-\triangle$ . Moreover,$0$

is an eigenvalue of $H$ .

$u\in L^{2,-S}(\mathrm{R}^{2})$$u=Qu\mathrm{a}\mathrm{n}\mathrm{d}-\triangle u+Vu=\mathrm{U}l$for any $s>1$ and lettingin the integral expression $G_{0}Vu(x)= \frac{-1}{2\pi}\int\log|x-y|V(y)u(y)dy$

and using $Pu= \int V_{0}(x)u(x)dx=0$ , we see thatthe other hand if$u\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{S}-\triangle u+V(X)u=0$$u$satisfies$\mathcal{F}(Vu)(\mathrm{O})=0$$(1.2)(\mathrm{c}\mathrm{f}.

[2])$

. On

and (1.2), then, by comparing

the singularities at $\xi=0$ of the Fourier transforms$Rarrow\infty$

If at least one of theis called a resonant solution or

Indeed, if $u\in L^{2,-S}$ satisfies $u+QG_{0}Vu=0$ , thenwe have(1.2)$|\alpha|\leq \mathrm{I}$is the resonance of H. If all these constants vanish,

is an eigenfunction of $H$ and$|x|arrow\infty$,$\mathcal{F}(Vu)(\xi)$and$\xi^{2}\mathcal{F}u(\xi)$,or $Qu=u$ . And, in virtue of (1.2), the limit asof the boundary integral in the right hand side of$\lim_{Rarrow\infty}\frac{-1}{2\pi}\int_{y|\leq R}|-(\triangle u)(y)\log|x-y|dy$$=u(x)+ \mathrm{l}\mathrm{i}\mathrm{m}Rarrow\infty\frac{1}{2\pi}\int_{|y|=R}(\frac{\partial u}{\partial n}(y)\log|_{X}-y|-u(y)\frac{\partial\log|x-y|}{\partial n})dy$

converges to the constant $-a$ . It follows that $G_{0}Vu=-u(x)+a$ and$QG_{0}VQu+u=0$ , since $Qa=0$ .

Remark 1.10 As in the higher dimensional case, we can prove by applying

the commutator method of [15] that

$W_{\pm}$are bounded in the Sobolev space

17

for any $1<p<\infty$ and $k=0,$
and Assumption 1.7.
for$W^{k,p}(\mathrm{R}^{2})$$\ldots$

,

if

$p$

$V$

satisfies$|D^{\alpha}V(X)|\leq$$|\alpha|\leq\ell$$C_{\alpha}\langle_{X}\rangle^{-}\delta$

Remark 1.11 Likewise, if $z=0$ is a resonance or an eigenvalue of $H,$

cannot be bounded inthat$e^{-itH}P_{\mathrm{a}\mathrm{c}}$$L^{p}(\mathrm{R}^{2})$in this case

$W_{\pm}$

for all $1<p<\infty$ . Indeed Murata [7] has shown$sati_{\mathit{8}}fies$$\lim_{tarrow\infty}||(\log t)e$

-itHPCa$f$$-c_{0f}||_{L^{2}},-s=0$,$s>3$ ,

(1.3)

where $C_{0}\neq 0$ is an explicitly computable finite rank operator. This clearly$contradiCt\mathit{8}$with the$L^{p}$$||(\log t)e$boundedness of-itHPaC

because the latter would imply$W_{\pm}$$f||L^{2,-s}\leq||(\log t)W_{+}e-itH0W_{+^{f||_{L}}}^{*}p$$\leq C_{p}||f||_{Lp}’(\log t)t^{-}2(1/2-1/p)arrow 0$$(tarrow\infty)$

for sufficiently large $p>2$ and $p’=p/(p-1)$ and becausein$L^{2,-S}$$L^{2,-S}\cap L^{p}$is dense.

In what follows we deal with

$W_{+}$

only.

$W_{-}$

use the following notation and convention.

may be treated similarly. We$D_{j}=-i\partial/\partial x_{j},$we use the vector notation $D=(D_{1}, D_{2}),$$1\leq p\leq\infty$ .
norm ofis the unit circle$j=1,2$ , and$\langle D\rangle=(1+D^{2})^{1/2}$$L^{p}$$\Sigma$$u,$

the standard line element of$\Sigma$.

$S^{1}\subset \mathrm{R}^{2}$.

and$||u||_{p}$$d\omega$is thedenotes$\mathcal{F}u(\xi)=\hat{u}(\xi)=\frac{1}{2\pi}\int_{\mathrm{R}^{2}}e^{-ix\cdot\xi}u(X)dx$

is theFourier transform of , Various constants are denoted by the same letter
$u$

$C$

if their specific values are not important, and these constants may differ

from one place to another. We take and fix throughout this paper the cut-off
functions$\chi(t)\in C_{0}^{\infty}(\mathrm{R}^{1})$$\chi(t)=\chi(-t),$$|t|\geq 2c$and$\tilde{\chi}(t)\in C^{\infty}(\mathrm{R}^{1}),$$0\leq\chi(t),\tilde{\chi}(t)\leq 1,$$\chi(t)=1$$\chi(t)+\tilde{\chi}(t)\equiv 1$, such$\mathrm{t}\mathrm{h}\cap Jl-$

for $|t|\leq c$ and $\chi(t)=0$ for, where $0<c<1$ is the sufficiently small constant to be specified

below. We note that$\chi(H_{0})$is the convolution operator with the Fourier

18transform ofin$I\nearrow(\mathrm{R}^{2})$$\chi(\xi^{2})\in C_{0}^{\infty}(\mathrm{R}^{2})$$\chi(H_{0})$for any $1\leq p\leq\infty$ . For$\langle f, g\rangle=\int f(x)\overline{g(_{X)}}d_{X}$

2

and

$f$

and

and

$\tilde{\chi}(H_{0})$$g$

are bounded operators

in suitable function spaces,.

Outline of the Proof

We outline the proof of Theorem 1.8. The basic strategy is similar to the one
employed in [15] and [16] for proving the corresponding property in higher

dimensions $d\geq 3$ : We start from the stationary representation formula ([6]):(2.4)$W_{+}u=u- \frac{1}{\pi i}\int_{0}^{\infty}R^{-}(k^{2})V\{R_{0}^{+}(k^{2})-R-(\mathrm{o}k^{2})\}kudk$

and expand$W_{+}$

into the sum of a few Born terms and the remainder$W_{+}= \sum_{j=0}^{\ell}W(j)W++\ell+1$

by successively replacing$R^{-}(k^{2})$by $R^{-}(k^{2})=R0-(k^{2})-R0-(k2)VR-(k^{2})$ in

the right of (2.4): $W_{+}^{(0)}=I$ is the identity operator and for $j=1,$$W^{(j)}u= \frac{(-1)^{j}}{\pi i}\int_{0}^{\infty}R_{0}^{-(}k^{2})V(R^{-}(k^{2})V)j-1\{R+(k2)-R-(\mathrm{o}00k^{2})\}kudk$$W_{\ell+1}u= \frac{(-1)^{\ell+}1}{\pi i}\int_{0}^{\infty}R_{0}^{-}(k^{2})VF_{\ell}(k^{2})\{R+(k^{2})-R-(00)k^{2}\}kudk$, ,$\ell$$\ldots$, (2.5),where $F_{\ell()}k^{2}=(R_{0}^{-}(k^{2})V)\ell_{-1}R-(k^{2})V$ . We prove that the Born terms

are bounded in$L^{p}(\mathrm{R}^{2})$for all$\mathrm{I}<p<\infty$(2.6)$W^{(j)}$by showing that they are superpo-

sitions of compositions of essentially one dimensional convolution operators;
the remainder term

$W_{\ell+1}$

has the integral kernel $K(x, y)$ which satisfies thecondition of Schur’s lemma$\sup_{x\in \mathrm{R}^{2}}\int_{\mathrm{R}^{2}}|K(x, y)|dy<\infty$,$\sup_{y\in \mathrm{R}^{2}}\int_{\mathrm{R}^{2}}|K(x, y)|dx<\infty$

19
and, therefore$W_{\ell+1}$

is bounded in If$(\mathrm{R}^{2})$for all $1\leq p\leq\infty$ . We explain

here the difficulties which we encounter in this approach, in two dimensions inparticular,$\mathrm{a}\dot{\mathrm{n}}\mathrm{d}$the ideas how to overcome these difficulties. As the difficulties

are of different kinds in the low energy part and the high energy part, we
split
into the highand the low energy partsby using$W_{+}$$W_{+}\chi(H_{0})$$W_{+}\tilde{\chi}(H_{0})$the cut-off functions introduced above.
First, we prove that the first two Born terms

arebounded infor any $1<p<\infty$ . We write $W^{(1)}=W^{(1)}(V)$ when wewant to make the dependence on explicit.
$W^{(1)}$and$W^{(2)}\tilde{x}(H_{0})$$L^{p}(\mathrm{R}^{2})$$V$

Lemma 2.12 The operators$W^{(1)}$and$W^{(2)}$

may be written in the form;

$W^{(1)}u(x)= \frac{i}{4\pi}\int_{\Sigma}d\omega\int_{0}^{\infty}K(t+2_{X}\omega, \omega)u(x+t\omega)dt$(2.7)$W^{(2)}u(x)=C \int_{\Sigma^{2}}d\Omega\int_{[0,\infty)}22\hat{K}(t1, t_{2}+2x\omega_{2}, \omega_{1}, \omega_{2})u(x+t_{1}\omega_{1}+t_{2}\omega_{2})dt1dt_{2}$

,

(2.8)

where $C=(i/4\pi)^{2},$$d\Omega=d\omega_{1}d\omega_{2}$and$K(t, \omega)=\int_{0}^{\infty}\hat{V}(r\omega)editr/2r$,$\hat{K}_{2}(t_{1}, t_{2}, \omega_{1}, \omega 2)=\int_{[0,\infty)}2\hat{V}e^{i}((t1^{S}1+t2s2)/2S1\omega_{1})\hat{V}(S_{22}\omega-S_{1}\omega 1)ds1ds2$(2.9). (2.10)

Proof. By writing $V(x)=(2 \pi)^{-1}\int e^{ix\xi}\hat{V}(\xi)d\xi$ we have$( \mathcal{F}W^{(1)}u)(\xi)=-\int 0\frac{1}{2\pi}\infty(\int\frac{\hat{V}(\eta)}{\xi^{2}-\lambda+i\mathrm{o}}\delta((\xi-\eta)2-\lambda)\hat{u}(\xi-\eta)d\eta \mathrm{I}^{d}\lambda$

Computing the Fourier inverse transform in$W^{(1)}u(X)=- \frac{1}{(2\pi)^{2}}\int\int\frac{e^{ix\xi}\hat{V}(\eta)\hat{u}(\xi-\eta)}{2\xi\cdot\eta-\eta^{2}+i0}d\eta d\xi$.

20

we obtain (2.7). For obtaining (2.8), we repeat similar computations. See theproof of Proposition 2.2, Lemma 2.3 and Lemma 2.4 of [15] for the details. 1

When $d\geq 3$ , the similar computation produces expressions (2.7) and(2.8) for$W^{(1)}$and$W^{(2)}$

with$K\in L^{1}(\mathrm{R}\cross\Sigma)$andthe classical Minkowski inequality implies thatin$L^{p}(\mathrm{R}^{2})$$\hat{K}_{2}\in L^{1}(\mathrm{R}^{2}\cross\Sigma^{2})$$W^{(1)}$and. Hence,are bounded$W^{(2)}$for any $1\leq p\leq\infty$ if $d\geq 3$ . If $d=2$ , this is obviously not thecase, however, we can show
$K_{1}(t, \omega)=K(t, \omega)-2\hat{V}(\mathrm{o})\tilde{x}(t)/it\in L^{1}S>1(\mathrm{R}\cross\Sigma)$$||K_{1}||_{L^{1}}\leq C||\langle x\rangle^{S}V||_{2}$,and that the integral operator which arises when $K$ is replaced by(2.7) is a superposition$\int_{\Sigma}F_{\omega}u(X)d\omega$over$\omega\in\Sigma$.

$\omega$

inof(2.11)$F_{\omega}u(X)= \int_{0}^{\infty}\frac{\tilde{\chi}(t+2_{X}\omega)}{t+2x\omega}u(x+t\omega)dt$

After rotating the coordinates by , we estimate
for $x_{1}>0$ and for $x_{1}<0$ :$\tilde{\chi}(t)/it$$F_{\mathrm{e}_{1}}u(x)$as follows separately$|F_{\mathrm{e}_{1}}u(x)| \leq\theta(x_{1})\int_{0}^{\infty}\frac{|u(t,x_{2})|}{t+x_{1}}dt$$+ \theta(-x_{1})\int-\infty 0\frac{|u(t,x_{2})|}{|t+x_{1}|}dt+\theta(-X_{1})|\int_{0}^{\infty}\frac{u(t,X_{2})}{t+x_{1}}dt|$

We then apply$L^{p}$.

boundedness theorem for the one-dimensional Hardy-

Littlewood operators on the half lines$(0, \pm\infty)$

to the first two integrals on

the right and for one dimensional singular integral operator of the CalderonZygmund type to the third, and conclude thatuniformly bounded operators in$L^{p}$$\{F_{\omega} :\omega\in\Sigma\}$

is a family of

for any 1 $<p<\infty$ . In this way, we

obtain the estimate$||W^{(1)}(V)u||_{p}\leq C_{p_{S}}||\langle x\rangle^{s}V||2||u||_{p}$,for any $s>1$ .

(2.12)

21

The proof of theWe write$\hat{K}_{2}$$L^{p}$

boundedness of

is a bit more involved.

$W^{(2)}\tilde{x}(H_{0})$as a sum of three functions $K_{21}+K_{22}+K_{23)}$$K_{21}\in L^{1}(\mathrm{R}^{2}\cross\Sigma^{2})$,$K_{22}=C(\tilde{\chi}(t_{1})/t_{1})\cross K(t_{2}, \omega_{2})$,with $K(t, \omega)$ being defined by (2.9), and$K_{23}=(\tilde{\chi}(t_{2})/t_{2})\cross K’(t_{1}, \omega_{1})$,$K’\in L^{1}(\mathrm{R}^{1}\cross\Sigma)$

We show that the operators which are produced by replacing$K_{2j}$are bounded in If for any

1. The operator arising from$1<p<\backslash \infty$$K_{21}$.

$\hat{K}_{2}$in (2.8) by

as follows.

can be estimated by using the Minkowski

inequality.

2. If we denote by $M$ the convolution operator with
operator arising from

$K_{22}$$\tilde{\chi}(|x|)/|x|^{2}$, then theis of the form $W^{(1)}M$ . The operator$M\tilde{\chi}(H_{0})$is bounded in $IP$ by Calderon-Zygmund theory;

3. The operator arising from

$K_{23}$

may be written in the form$\int_{\Sigma}\int_{0}^{\infty}K’(t_{1,1}\omega)(\int_{\Sigma}(F_{\omega}u)2(x+t_{1}\omega_{1})d\omega_{2})d\omega 1dt_{1}$and the estimate for (2.11) mentioned above and the Minkowski inequality imply that this also is bounded in$L^{p}$.

of the remainder ;We then prove that the high energy partfor any $1\leq p\leq\infty$ by showing that its integral kernel
is bounded in$W_{3}\tilde{\chi}(H_{0})$$W_{3}$$L^{p}$$T(x, y)$ is bounded by a constant times $\langle x\rangle^{-}1/2\langle y\rangle^{-1}/2\langle|x|-|y|\rangle^{-2}$$F(k)=R_{0}^{-}(k^{2})VR^{-}(k^{2})$. Because$R_{0}^{\pm}(k^{2})$. We write

is the convolution operator with

22

$G^{\pm}(x, k)=(\pm i/4)H_{0}\pm(k|x|)$function of the j-th kind,, where $H_{0}^{\pm}(z)=H_{0}^{(j)}(z)$ is the O-th order Hankel$\pm \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$to $(-1)^{j+1}$ (cf. [12]), $T(x, y)$ isgiven as $T(x, y)=T^{+}(X, y)-\tau^{-(x,y)}$ :$T^{\pm}(x, y)=- \frac{1}{\pi i}\int_{0}^{\infty}\langle F(k)VG^{\pm}(y-\cdot , k), VG^{+}(x-\cdot , k)\rangle\tilde{\chi}(k2)kdk$$\ln$virtue of the classical estimate for Hankel functions. (2.13)$H_{0^{\pm}}(k|x|) \sim\frac{Ce^{\pm il\mathfrak{i}}||}{\sqrt{k|x|}}$and the decay property of the resolvent at high energy$||\langle x\rangle^{-\sigma}-j(d/dk)^{j}F(k)\langle x\rangle^{-\sigma}-j||_{B}(L^{2})\leq Ck^{-2}$

for $j=0,1,2$ and $\sigma>1/2$ , the integral (2.13) is absolutely convergent.
However, a simple minded estimate by using these facts only would yield$|T^{\pm}(x, y)|\leq C\langle x\rangle^{-}1/2\langle y\rangle-1/2$

which is far from being sufficient to concludethat$L^{p}$$W_{3\tilde{x}}(H_{0)}$is bounded infor all $1<p<\infty$ . This difficulty can be

resolved by exploiting the old method in [15] and [16]: We write $G^{\pm}(x-$$y,$ $k)=e^{\pm ik||}G_{k}^{\pm}x,x(y)$so that$T^{\pm}(x, y)=- \frac{1}{\pi i}\int_{0}^{\infty}e^{-}|\mp i(|x|y|)k\langle F(k)VG\pm VGy,k’ xk+,\rangle k\tilde{x}(k^{2})dk$,(2.14)

and apply the integration by parts twice to the -integral in the right by$k$using the identity$\frac{1+i(|X|\mp|y|)(\partial/\partial k)}{1+(|x|\mp|y|)^{2}}e^{-i(|}x|\mp \mathrm{I}y|)ki(=e^{-}|x|\mp|y|)k$.

This yields the desired estimate$|T^{\pm}(_{X}, y)|\leq C\langle|_{X}|\mp|y|\rangle-2\langle x\rangle-1/2\langle y\rangle-1/2$.

The estimate of the low energy part of the wave operator

little more involved. Here we write$R^{-}(k^{2})V=R_{0}^{-}(k^{2})V(1+R_{0}^{-}(k^{2})V)^{-1}$$W_{+}\chi(H_{0})$

is a

23

in (2.4) and investigate the low energy behavior of $(1+R_{0}^{-}(k^{2})V)^{-1}$ following
the argument in [7] and [2]. We find that, for $0<k<2c,$ being a sufficiently$c$small constant, which is the constant to be used for defining the cut off ,$\chi$$(1+R_{0}^{-}(k^{2})V)^{-1}$can be written as the sum$(1+R_{0}^{-}(k^{2})V)^{-1}= \sum_{j=0}^{4}d_{j(}k)K_{j}+N(k)$1. For $0\leq j\leq 4,$$K_{j}(x, y)$$K_{j}$

is an integral operator with the integral kernel

which satisfies for some $s>1$$\int_{\mathrm{R}^{2}}||\langle x\rangle^{s}VKjy||2dy<\infty$2.

$d_{j}(k)$.

satisfies,$K_{jy}(x)=K_{j}(x, x-y)$ .

$|(\partial/\partial\xi)^{\alpha}d_{j}(|\xi|)|\leq C_{\alpha}|\xi|^{-}|\alpha|$(2.15).

3. The remainder $N(k)$ is an operator valued function which satisfies$\mathrm{t}_{\perp}^{1}‘$estimate for $j=0,1,2$ :$||(d/dk)^{j}N(k)||_{B(}L2,-s)\leq C_{j}k^{2-j}|\log k|$(Actually $d_{0}(k)=1$ and$K_{j}$$s>3$ ,,

for $1\leq j\leq 4$ are rank one operators.)

The operator which is produce by inserting $R_{0}^{-}(k^{2})VN(k)x(k^{2})$ in place of,$R^{-}(k^{2})V$in (2.4) is an integral operator with the kernel$\tilde{T}^{\pm}(x, y)$being given by the right hand side of (2.14) with $N(k)\chi(k^{2})$ in$\tilde{T}^{+}(x, y)-^{\tilde{\tau}}-(x, y)$place of $F(k)V\tilde{\chi}(k2)$ . The method employed for estimating $T^{\pm}(x, y)$ applies

because $N(k)$ vanishes at$k=\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}$derivative, and yields$|\tilde{T}^{\pm}(_{X}, y)|\leq C\langle|_{X}|\mp|y|\rangle-2\langle X\rangle-1/2\langle y\rangle^{-}1/2$

and the operator in question is bounded in$L^{p}$

for any $1\leq p\leq\infty$ . The

operator produced by inserting $R_{0}^{-}(k^{2})Vdj(k)K_{j}$ in place of $R^{-}(k^{2})V$ in$(2.l\pm 1)$

24

may be written as$\frac{-1}{\pi i}\int_{0}^{\infty}R_{0}^{-(}k^{2})VK_{j}d_{j}(k)\{R_{0}^{+}(k^{2})-R_{0}-(k2)\}\chi(k^{2})kudk$.

(2.16)

Observing that$d_{j}(k)\{R_{0}^{+}(k2)-R_{0}-(k^{2})\}=\{R_{0}^{+}(k^{2})-R_{0}-(k2)\}d_{j}(|D|)$and that the integral operator may be written as$\int A(x, y)u(y)dy=\int A(x, x-y)u(x-y)dy=\int A_{y}(x)\mathcal{T}u(yx)dy$ ,$\mathrm{v}\mathrm{i}\mathrm{z}$. the superposition of the composition of the multiplication by $A_{y}(x)=$$A(x, x-y)$ and the translation$\tau_{y}$by , we rewrite (2.16) in the form$y$$\int_{\mathrm{R}^{2}}(\frac{-1}{\pi i}\int_{0}^{\infty}R_{0}^{-(}k^{2})VKjy\{R^{+}(0)2-R_{0}-(kk^{2})\}kdk)dj(|D|)\chi(H\mathrm{o})\tau_{y}udy$.

(2.17)The operator in the parenthesis is nothing but(2.12), the$L^{p}$$W^{(1)}$(VKjy) and, in virtue of-norm of (2.17) may be estimated as follows:$|| \int_{\mathrm{R}^{2}}W^{(}1)(VKjy)d_{j}(|D|)x(H_{0})\mathcal{T}_{y}udy||p$$\leq C||u||_{p}||dj(|D|)\chi(H\mathrm{o})||B(LP)\int_{\mathrm{R}^{2}}||\langle x\rangle^{s}VKjy||2dy$.

Because Fourier multipliers $d_{j}(|D|)\chi(H_{0})$ are bounded inby the wellknown theorem in the Fourier analysis and because (2.15) implies that the$L^{p}$

i..ntegral in the rig.

$\mathrm{h}.\mathrm{t}$is finite, the operators arising from $d_{j}(k)K_{j},$ $j=0.’\ldots,$$4i$

are all bounded in..L for any $1<p<\infty$ . Combining these all, we completes

the proof of Theorem 1.8.

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[14] R. Weder, The$W_{k,p}$-continuity of the Schr\"odinger wave operators on$\mathrm{t}\mathrm{h}^{l}\mathrm{e}$line, preprint, UNAM (1999).
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