数理解析研究所講究録
1491 巻 2006 年 25-35

25Uniqueness of solutions for Schr\"odinger maps andrelated estimates for the product of functions
京都大学大学院理学研究科

加藤淳 (Jun Kato)’

Department of Mathematics, Kyoto University

1

Introduction

In this note, we consider the uniqueness of solutions of the Cauchy problem forthe system of the nonlinear Schr\"odinger equations arising form the Schr\"odinger map,
which is called the modified Schr\"odinger map.
We briefly explain the Schr\"odingre map and the derivation of the modified
Schr\"odinger map. The Schr\"odinger map fromtois described by the map$\mathbb{R}\cross \mathbb{R}^{n}$$s:\mathbb{R}\cross \mathbb{R}^{n}arrow S^{2}\subset \mathbb{R}^{3}$$S^{2}$

satisfying$\partial_{t}s=s\cross\Delta s$

,

(1.1)

the exterior product in. In general, the Schr\"odinger map isformulated as the Schr\"odingerlike evolution of the harmonic map. In the case where

where$\mathbb{R}^{3}$$\cross \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s}$the target manifold ishave a special importance, because they naturally arise
from Landau-Lifshitz equations governing the static as well as dynamical propertiesof magnetization.
. to each term of the equation (1.1), we have
Applying$S^{2}$$(s\cross\partial_{t}s)$$\partial_{t}s\cdot\Delta s=0$

,

(1.2)

since
$(s\cross\partial_{t}s)\cdot\partial_{t}s=\det(s\partial_{t}s\partial_{t}s)=0$,$(s\cross\partial_{t}s)\cdot(s\cross\Delta s)=|s|^{2}\partial_{t}s\cdot\Delta s-(s\cdot\Delta s)(s\cdot\partial_{t}s)=\partial_{t}s\cdot\Delta s$

現在の所属 : 名古屋大学大学院多元数理科学研究科

Department of Mathematics, Nagoya University.

26

Thus, integrating (1.2) over$\mathbb{R}^{n}$, we obtain$\partial_{t}\int_{\mathrm{R}^{n}}|\nabla s(t,x)|^{2}dx=0$

,

(1.3)

-norm. So, it is natural towhich implies that the solution of (1.1) conserves the
expect the well-posedness of the Cauchy problem for (1.1) in such a energy class. We
also notice that the equation (1.1) is invariant with respect to the scale transformation$\dot{H}^{1}$$s(t,x)\mapsto s(\lambda^{2}t, \lambda x)$. Observing the relation between the size of -norm of the initial data
for all
and life span of the time-local solution by using the scale transformation above, it is
only if $r\geq n/2$ . So,
considered that the Cauchy problem of (1.1) is well-posed inis critical in two space dimensions, and this critical case provides
the energy classinteresting problems similar to the wave maps [7].
In what follows, we consider the local well-posedness of the Cauchy problem of (1.1)
with the small as long as possible. To
in two space dimensions for the data in
begin with, we rewrite the equation as follows. By using the stereographic projection$\dot{H}^{r}$$\lambda>0$$H^{f}$$H^{1}$$H^{f}$$\mathbb{C}\ni zrightarrow(\frac{2{\rm Re} z}{1+|z|^{2}},$$r$$\frac{2{\rm Im} z}{1+|z|^{2}},$$\frac{1-|z|^{2}}{1+|z|^{2}})\in S^{2}$,the equation (1.1) is rewritten as the nonlinear Schr\"odinger equation

.

(1.4)

$\partial_{t}z=i\sum_{j=1}^{2}(\partial_{j}-\frac{2\overline{z}\partial_{j}z}{1+|z|^{2}})\partial_{j}z$Then, in [5], Nahmod, Stefanov and Uhlenbeck used the $U(1)$ gauge invariance of (1.4)
to transform it into a system of nonlinear Schr\"odinger equations, called the modified

Schr\"odinger map:$i\partial_{t}u_{1}+\Delta u_{1}=-2iA\cdot\nabla u_{1}+A_{\mathit{0}}u_{1}+|A|^{2}u_{1}+4i{\rm Im}(u_{2}\overline{u}_{1})u_{2}$

,
,

(1.5)

$i\partial_{t}u_{2}+\Delta u_{2}=-2iA\cdot\nabla u_{2}+A\mathit{0}u_{2}+|A|^{2}u_{2}+4i{\rm Im}(\mathrm{u}_{1}\overline{u}_{2})u_{1}$where$u_{j}=e^{:\psi} \frac{\partial_{j}z}{1+|z|^{2}}$,$j=1,2$ ,

(1.6)

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and the Coulomb gauge has been chosen. Then, $A=A[u]=(A_{1}[u],A_{2}[u])$ and$A_{0}=A_{0}[u]$are determined from$\mathrm{d}\mathrm{i}\mathrm{v}A=0$,$\partial_{j}A_{k}-\partial_{k}A_{j}=-4{\rm Im}(u_{j}\overline{u}_{k})$,$- \Delta A_{0}=4\sum_{j,k=1}^{n}\partial_{j}\partial_{k}{\rm Re}(u_{j}\overline{u}_{k})-2\Delta|u|^{2}$,

where we denoted $u=(u_{1}, u_{2})$ . For the explicit representation of $A$ and , see (2.1),(2.2). In the rest of this note we devote to the local well-posedness of the Cauchy$A_{0}$

problem for the modified Schr\"odinger map.
Remark 1.1. (1) Although in general it is quite intricate to work out the equivalence
between the Schr\"odinger map problem (1.1) and the modified Schr\"odinger map problem (1.5), for the sphere and in cases which cover the range of solutions consideredin this paper such equivalence has been discussed in Kenig and Nahmod [3].
(2) It is important to notice that due to the relation (1.6) the well-posedness of themodified Schr\"odinger map problem (1.5) incorresponds to the well-posedness of
the Schr\"odinger map problem (1.1) in $H^{s+1}$ . Thus, the energy class of the Schr\"odinger
map problem (1.1) corresponds to thefor the modified Schr\"odinger map problem(1.5). In fact, we can compute
directly from (1.5).
$H^{s}$$L^{2}$$\partial_{t}||u(t)||_{L^{2}}^{2}=0$

2 Main Result
In the following we consider the well-posedness of the Cauchy problem for themodified Schr\"odinger map in two space dimensions,$(\mathrm{M}\mathrm{S})\{$$i\partial_{t}u_{1}+\Delta u_{1}=-2iA[u]\cdot\nabla u_{1}+A_{0}[u]u_{1}+|A[u]|^{2}u_{1}+4i{\rm Im}(u_{2}\overline{u}_{1})u_{2}$,$i\partial_{t}u_{2}+\Delta u_{2}=-2iA[u]\cdot\nabla u_{2}+A_{0}[u]u_{2}+|A[u]|^{2}u_{2}+4i{\rm Im}(u_{1}\overline{u}_{2})u_{1}$,$u_{1}(0,x)=u_{0}^{1}(x)$,$u_{2}(0, x)=u_{0}^{2}(x)$

,

28

where

$u_{1},$$u_{2}$are complex valued functions, and $A[u]=(A_{1}[u], A_{2}[u]),$$A_{0}[u]$aredetermined by$A_{j}[u]=4G_{j}*{\rm Im}(u_{1}\overline{u}_{2})$$G_{1}(x)= \frac{1}{2\pi}\frac{x_{2}}{|x|^{2}}$,,$j=1,2$ ,$G_{2}(x)=- \frac{1}{2\pi}\frac{x_{1}}{|x|^{2}}$

(2.1)

,

$A_{0}[u]=-4 \sum_{j,k=1}^{2}R_{j}R_{k}{\rm Re}(u_{j}\overline{u}_{k})-2|u|^{2}$

.

(2.2)

denotes the Riesz transforms.
have the niceinstead of (1.1) is that
One of the advantage to consider
structure in the nonlinear term, which enable us to construct energy estimate for. In fact, Nahmod, Stefanov, and Uhlenbeck [6] showed the local
the solution towhen $s>1$ by using the energy method. Then,well-posedness for the data in
independently, the author [1], and Kenig and Nahmod [3], showed the existence ofwhen $s>1/2$ .
the solution for the data in

Here,$R_{j}=\partial_{j}(-\Delta)^{-1/2}$$(\mathrm{M}\mathrm{S})$$(\mathrm{M}\mathrm{S})$$(\mathrm{M}\mathrm{S})$$H^{s}$$H^{s}$

with $s>1/2$ . Then, there enist $T>0$ andTheorem 2.1 ([1], [3]). Let
at least one solution $u\in L^{\infty}(\mathrm{O},T;H^{s})\cap C_{w}([0,T];H^{s})$ to $(MS)$ satisfying$u_{0}\in H^{s}(\mathbb{R}^{2})$(2.3)$J^{s-1/2-e}u\in L^{2}(0,T;L^{\infty})$where $0<\epsilon<s-1/2,$ $J=(I-\Delta)^{1/2}$ . The solution is unique when $s\geq 1$ .
The improvement of regularity comes form the use of a variant of the Strichartz
estimates which was first introduced by Koch and Tzvetkov [4] in the context of the
Benjamin-Ono equation. However, the uniqueness of solutions could not be provedon the equationin the same class due to the lack of the good structure such as
satisfied by the difference of the two solutions.
The purpose of this note is to show the idea of the proof of our following recent. In the following we use the notation
result on the uniqueness of solutions to$(\mathrm{M}\mathrm{S})$$(\mathrm{M}\mathrm{S})$$L_{T}^{p}X$

to denote $L^{p}(0, T;X)$ for a Banach space $X$ .

Theorem 2.2 ([2]). Letdata satishing
$u,$

$u$

and

$v$

be smooth solutions to $(MS)$ unth the same smooth$v\in L^{\infty}(\mathrm{O}, T;H^{1/2})\cap L^{\mathrm{p}}(0, T;B_{q,2}^{1/2})$

(2.4)

29

for some $q>4$ with $1/p=1/2-1/q$ . Then,$u\equiv v$

holds. Moreover, the estimate(2.5)$||u(t)-v(t)||_{H^{-1/2}}\leq C||u(t’)-v(t’)||_{H^{-1/2}}$

holds when $t>t’$ , where the constant $C$ depends on
, and
, and
is the Besov space.
$||u||_{L_{T}^{p}B_{q,2}^{1/2}}$$||v||_{L_{T}^{p}B_{\mathrm{q},2}^{1/2}}$$||u||_{L_{\mathcal{T}}^{\infty}H^{1/2}},$$||v||_{L_{T}^{\infty}H^{1/2}}$,$B_{p,q}^{\mathit{8}}$Theorem 2.1 was proved by using the compactness argument based on a priori
estimates of the solution to:$(\mathrm{M}\mathrm{S})$$|1J^{s-1/2-\epsilon}u||_{L_{T}^{2}\iota\infty}\leq C||u_{0}||_{H}\cdot$,$||u||_{L_{T}^{\infty}H^{f}}\leq C(||u_{0}||_{H^{1/2+e’}})||u_{0}||_{H}\cdot$

(2.6)

,

(2.7)

for $s>1/2,$ $\epsilon’\in(0, s-1/2)$ . When $s>3/4$ , we observe that the solution tosatisfy the condition (2.4) by interpolating a priori estimates (2.6), (2.7). Indeed, if
we set $s=3/4+2\epsilon$ , we have$(\mathrm{M}\mathrm{S})$$\epsilon,$$||u||_{L^{2}\tau^{B_{\infty,2}^{1/4}\sim}}<||J^{1/4+\epsilon}u||_{L_{T}^{2}L}\infty\leq M$,,$||u||_{L_{T}^{\infty}B_{2.2}^{3/4+2\epsilon}}\sim<||u||_{L_{\mathrm{T}}^{\infty}H^{3/4+2e}}\leq M$

for some constant $M>0$ , thus we obtain,$||u||_{L_{T}^{p}B_{q,2}^{1/2}}\leq||||u||^{1-2/q}||u||_{B_{2,2}^{3/4+2e}}^{2/q}|B_{\infty,2}^{1/4}|_{L_{T}^{p}}\leq||u||^{2/p}||u||^{1-2/p}L_{T}^{2}B_{\infty,2}^{1/4}L_{T}^{\infty}B_{2,2}^{l/4+2e}\leq M$

where $1/q=1/(4+16\epsilon)$ and $1/p=1/2-1/q$ . Therefore, in the case $s>3/4$ weare able to apply Theorem 2.2 in the proof of Theorem 2.1, and obtain the following
corollary.
with $s>3/4$ . Then, there nts $tT>0$ and a uniquesolution $u\in C([0, T];H^{s})$ to $(MS)$ satisfying

Corollary 2.3. Let$u_{0}\in H^{s}(\mathbb{R}^{2})$$J^{s-1/2-\epsilon}u\in L^{2}(0, T;L^{\infty})$where $0<\epsilon<s-1/2$ .
Remark 2.4. In Corollary 2.3 we could improve the condition of the regularity on theinitial data which was already known, $s=1$ . There is still a gap from the condition$s>1/2$ which the existence of a solution is known. There is also a gap from thecritical space$L^{2}$.

30

3 ldea of Proof of Theorem 2.1
In this section we describe the idea of the proof of Theorem 2.1. For simplicity, we,
consider the more simple problem which contains the essential part of$(\mathrm{M}\mathrm{S})$$i\partial_{t}u+\Delta u=iA[u]\cdot\nabla u$

(P)

$\{$

$u(0, x)=u_{0}$ ,$x\in \mathbb{R}^{2}$,$(t, x)\in(\mathrm{O},T)\mathrm{x}\mathbb{R}^{2}$,,

where $A[u]=G*|u|^{2}$ . Here, we denoted $G=(G_{1}, G_{2})$ . We notice that
$A[u]\sim|x|^{-1}*|u|^{2}\sim D^{-1}|u|^{2}$,

where $D^{s}=(-\Delta)^{s/2}$ .
be the solutions to (P), then $w\equiv u-v$ satisfy
Let
$u,$

$v$

.

(3.1)

$i\partial_{t}w+\Delta w=iA[u]\cdot\nabla w+i(A[u]-A[v])\cdot\nabla v$

The usual way to show the uniqueness is to estimate the -norm of . In fact,
to both sides of the equation (3.1), taking the imaginary part, andmultiplying
then integrating over , we obtain$L^{2}$$w$$\mathrm{d}$$\mathbb{R}^{2}$.

$\frac{1}{2}\partial_{t}||w(t)||_{L^{2}}^{2}={\rm Re}\int_{\mathrm{R}^{2}}(A[u]-A[v])\cdot\nabla v\overline{w}dx$

If we consider the solutions in the class$u,$ $v\in C([0, T];H^{\epsilon})$with $s>1$ , then the uniqueness of solutions is easily obtained as follows. Let $1<$
$s_{0}< \min(s, 2)$ , and set $1/p=1-s_{0}/2,1/2=1/p+1/q$ , and $1/r=1/q+1/2$ . Then,
applying the H\"older inequality and the Sobolev embedding we obtain$\frac{1}{2}\partial_{t}||w(t)||_{L^{2}}^{2}\leq|\int_{\mathrm{R}^{2}}(A[u]-A[v])\cdot\nabla v\overline{w}dx|$$\leq||D^{-1}(|u|^{2}-|v|^{2})||_{L^{q}}||\nabla v||_{L^{\mathrm{p}}}||w||_{L^{2}}$$\sim<|||u|^{2}-|v|^{2}||_{L^{r}}||v||_{\dot{H}^{\epsilon_{0}}}||w||_{L^{2}}$$\sim<(||u||_{L^{q}}+||v||_{L^{q}})||v||_{\dot{H}^{f}0}||w||_{L^{2}}^{2}$$\sim<(||u||_{\dot{H}^{\mathrm{z}+||v||_{\dot{H}^{2-s}0})||v||_{\dot{H}^{\delta}0}||w||_{L^{2}}^{2}}}-\cdot 0^{\cdot}$

31

Since

$H^{s}‘arrow\dot{H}^{2-s_{0}},$$H^{s_{\mathrm{c}}}arrow\dot{H}^{s_{0}}$, by using the Gronwall inequality we obtain$||w(t)||_{L^{2}}\leq C||w(0)||_{L^{2}}$,which implies the uniqueness of solutions.
To show the uniqueness of less regular solutions, we consider the estimate of in$H^{-1/2}$
instead ofto overcome the loss of the derivative the nonlinearity. We use
the following energy estimate.
$w$$L^{2}$

Lemma 3.1. Let$w$

be a solution to
(3.2)

,

$i\partial_{t}w+\Delta w-ia\cdot\nabla w=F$

where

$a$

is

$\mathbb{R}^{2}$-valued function. Then, for $0<s<1,0<t<T$ , we have$||w(t)||_{H-\iota} \leq\exp\{C\int_{0}^{t}||\nabla a(t’)||_{L\infty}dt’\}(||w(0)||_{H}-s+\int_{0}^{t}||F(t’)||_{H}-\cdot dt’)$

.

(3.3)

Idea of Proof of Lemma 3.1. For $0\leq\tau<T$ , we denote by $S(t, \tau)f$ the solution to$\{$$i\partial_{t}v+\Delta v-ia\cdot\nabla v=0$,$v(lr, x)=f(x)$ ,.

$x\in \mathbb{R}^{2}$$(t, x)\in(\tau, T)\cross \mathbb{R}^{2}$,

Then, the solution to (3.2) is written as$w(t)=S(t, \mathrm{O})w(\mathrm{O})-i\int_{0}^{t}S(t, \tau)F(\tau)d\tau$.

Thus, to prove (3.3) it suffices to show$||S(t, \tau)f||_{H}-\iota\leq\exp\{C\int_{0}^{t}||\nabla a(t’)||_{L}\infty dt’\}||f||_{H}$ -..

To prove (3.4) we consider the dual problem for fixed$i\partial_{\tau}\tilde{v}+\Delta\tilde{v}-i\nabla$. (a ) $=0$ ,$\tilde{v}$$\{$$\overline{v}(t, x)=g(x)$,$x\in \mathbb{R}^{2}$$t\in(\mathrm{O}, T]$

(3.4)

,

$(\tau, x)\in(\mathrm{O},t)\mathrm{x}\mathrm{R}^{2}$,.

We denote by
the solution to the problem above. Then,
to $S(t, \tau)$ . In fact, the simple calculation shows that$\overline{S}(\tau, t)g$$\partial_{t’}\langle S(t’, \tau)f,\tilde{S}(t’,t)g\rangle=0$$\tilde{S}(\tau, t)$

is dual operator

32

by using the equation, and integrating this from$\tau$

to we derive$t$$\langle S(t, \tau)f,g\rangle=\langle f,\tilde{S}(\tau, t)g\rangle$.

Meanwhile, from the equation we have$\partial_{\tau}||\tilde{v}(\tau)||_{L^{2}}^{2}\leq\int a\cdot\nabla|\tilde{v}|^{2}dx\leq||\nabla a||_{L\infty}||\tilde{v}(\tau)||_{L^{2}}^{2}$

Similarly, we have$\partial_{\tau}||\nabla\tilde{v}(\tau)||_{L^{2}}\leq C||\nabla a||_{L}\infty||\nabla\tilde{v}(\tau)||_{L^{2}}$.

.

Thus, interpolating them we obtain$|| \tilde{S}(\tau,t)g||_{H}\cdot\leq\exp\{C\int_{0}^{t}||\nabla a(t’)||_{L\infty}dt’\}||g||_{H^{\iota}}$

,

(3.5)

for $0\leq s\leq 1$ . Therefore, by using the duality we obtain$||S(t, \tau)f||_{H^{-\epsilon}}=\sup_{||\varphi||_{H}\cdot=1}|\int S(t, \tau)f\varphi dx|$$= \sup_{||\varphi||_{H}*=1}|\int f\tilde{S}(\tau, t)\varphi dx|$$\leq|\sup$$||f||_{H^{-}}.||\tilde{S}(\tau, t)\varphi||_{H^{*}}$

llgll $H.=1$$\leq\exp\{C\int_{0}^{t}||\nabla a(\tau)||_{L\infty}d\tau\}||f||_{H}-\cdot$.

Thus we obtain (3.4).

$\square$

Applying Lemma 3.1 to (3.1) with $s=1/2$ we obtain$||w(t)||_{H^{-1/2}} \leq\exp\{C\int_{0}^{T}||\nabla A[u](t’)||_{L^{\infty}}dt’\}$(3.6)$\cross(||w(0)||_{H^{-1/2}}+\int_{0}^{t}||(A[u]-A[v])\cdot\nabla v||_{H}-1/2dt’)$Since$\nabla A[u]\sim R_{j}R_{k}|u|^{2}$, for sufficiently small$||\nabla A[u|||_{L\infty}\sim<||J^{\delta}R_{j}R_{k}|u|^{2}||_{L^{\overline{q}}}\sim<$I$\delta>0$

and$\delta>2/\tilde{q}$.

, we have$J^{\delta}|u|^{2}||_{L^{\overline{q}}}\leq||J^{\delta}u||_{L^{\overline{q}}}^{2}\leq||u||_{B_{\mathrm{q},2}^{1/2}}^{2}$.

So, the problem is to estimate the product of functions in the Sobolev spaces ofnegative order which appears in the last term in (3.6).

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Remark 3.2. One might think there would be another possibility to apply Lemma
3.1 instead of $H^{-1/2}$ . However, from the general version of the lemma below, andfrom the structure of the nonlinear term, the space $H^{-1/2}$ provides the best result in
our method.
Lemma 3.3. Suppose $n=2$ and $q>4$ . Then the following estimates hold.
$||fg||_{H^{-1/2}}\sim<||g||_{B_{q,2}^{1/2}}||f||_{H/2}-1$

,

(3.7)

.

(3.8)

$||(G*(fg))\nabla h||_{H}-1/2\sim<(||g||_{H^{1/2}}||h||_{H^{1/2}}+||g||_{B_{q,2}^{1/2}}||h||_{B_{q,2}^{1/2}})||f||_{H^{-1/2}}$

If we apply (3.8) to estimate the last term of (3.6), then we obtain$||w(t)||_{H^{-1/2}} \leq C(||w(0)||_{H^{-1/2}}+\int_{0}^{t}(||u(\tau)||_{X}^{2}+||v(\tau)||_{X}^{2})||w(\tau)||_{H^{-1/2}}d\tau)$,

where we denoted $X=H^{1/2}\cap B_{q,2}^{1/2}$ . Thus, by the Gronwall inequality we obtain$||w(t)||_{H^{-1/2}}\leq C||w(t)||_{H^{-1/2}}$.

Thus, Theorem 2.2, the uniqueness of the solution, follows.

Finally we describe the idea of the proof of Lemma 3.3.

Idea of Proof of Lemma 3.3. To prove (3.8) we first show that
(3.9)$|\mathrm{I}fg||_{H^{1}/2}\sim<||g||_{B_{q,2}^{1/2}}||f||_{H^{1/2}}$

holds. In fact, by using fractional Leibniz rule we have$||fg||_{H^{1/2}\sim}<||f||_{H^{1/2}}||g||_{B_{\infty 2}^{0}}+||f||_{B_{\tau,2}^{0}}||g||_{B_{q,2}^{1/2}}’$ ’

where $1/2=1/q+1/r$ . Then, the embeddings
give (3.9). Thus, by using the duality we obtain$B_{q,2}^{1/2}-B_{\infty,2}^{0}$

and $H^{1/2}arrow H^{2/q}-$$B_{\mathrm{r},\mathit{2}}^{0}$$||fg||_{H}-1/2= \sup_{||\varphi||_{H^{1/2}}=1}|\int fg\varphi dx|$$\leq\sup_{||\varphi||_{H^{1/2}}=1}||f||_{H/2}-1||g\varphi||_{H^{1/2}}$$\sim<||f||_{H/2}-\iota||g||_{B_{q,2}^{1/2}}$

Now we turn to the proof of (3.8). Since.
, we have$\mathrm{d}\mathrm{i}\mathrm{v}G*(fg)=0$$||(G*(fg))\nabla h||_{H}-1/2=||\mathrm{d}\mathrm{i}\mathrm{v}\{(G*(fg))h\}||_{H^{-1/2}}\sim<||(G*(fg))h||_{H^{1}/2}$.

(3.10)

34

To estimate the right hand side of (3.10) we divide $G*(fg)$ into the high frequency
part and the low frequency part,$G*(fg)=S_{0}(G*(fg))+(1-S_{0})(G*(fg))$ .

(3.11)

withis defined as the Fourier multiplier by , where
near the origin.
As for the high frequency part, the second term on the right hand side of (3.11),
we easily obtain

Here,$S_{0}$$\varphi\equiv 1$$\varphi\in C_{0}^{\infty}(\mathbb{R}^{n})$$\varphi$$||\{(1-S_{0})(G*(fg))\}h||_{H^{1/2}}\sim<||h||_{B_{q,2}^{1/2}}||(1-S_{0})(G*(fg))||_{H^{1/2}}$$\sim<||h||_{B_{q_{1}2}^{1/2}}$

I$fg||_{H^{-1/2}}$$\sim<||h||_{B_{q,2}^{1/2}}||g||_{B_{q,2}^{1/2}}||f||_{H^{-1/2}}$,

by using (3.9), (3.7).

As for the low frequency part, the first term on the right hand side of (3.11), we
estimate$||\{S_{0}(G*(fg))\}h||_{H^{1/2}\sim}<||S_{0}(G*(fg))||_{W^{1,\infty}}||h||_{H^{1/2}}$.

(3.12)

To complete the proof we have to estimate $S_{0}(G*(fg))$ and its gradient. By translation invariance it suffices to do this at the origin. The argument for $S_{0}(G*(fg))$
and for its gradient is the same. We observe that
$S_{0}(G*(fg))(0)=\{\Phi*(fg)\}(0)$$= \int_{\mathrm{R}^{2}}\Phi(y)f(y)g(y)dx$

where we set $\Phi=F^{-1}[\varphi]*G$ . Note that$\Phi\in L^{f}(\mathbb{R}^{2})$,for $2<r<\infty$ . Thus,$|S_{0}(G*(fg))(0)|=| \int_{1\sim|f||_{H^{-1/2}}}\leq<||\Phi g||_{H^{1/2}}||f||_{H}||\Phi||_{B_{q,2}^{1/2}}||g||_{H^{1’ 2^{-1/2}}}\mathrm{R}^{2}\Phi(y)f(y)g(y)dy|$.
Finally, we notice that$||\Phi||_{B_{q,2}^{1/2}}\leq||\Phi||_{B_{q,q}^{1/2+e}}\leq||\Phi||_{B_{q,q}^{0}}\sim||\Phi||_{L^{q}}<\infty$,

since is supported in the low frequency part in the Fourier space, andfor $2<r<\infty$ . This completes the proof of (3.8).
$\Phi$$\Phi\in L^{f}(\mathbb{R}^{2})$$\square$

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map, Math. Res. Lett. 12 (2005), 171-186.
, to[2] J. Kato, H. Koch, Uniqueness of modified Schr\"odinger maps inappear in Comm. Partial Differential Equations.
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system and Schr\"odinger maps, Nonlinearity 18 (2005), 1987-2009.
[4] H. Koch, N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation, Int. Math. Res. Not. 26 (2003), 1449-1464.
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