Local well-posedness for the Schrödinger-KdV system in $H^{s_{1}}\times H^{s_{2}}$ , II

Yingzhe Ban The Graduate School of China Academy of Engineering Physics, Beijing, 100088, P.R. China Jie Chen School of Sciences, Jimei University, Xiamen 361021, P.R. China Ying Zhang Academy of Mathematics and Systems Science, CAS, Beijing 100190, P.R. China The Graduate School of China Academy of Engineering Physics, Beijing, 100088, P.R. China

Abstract

In this paper, we continue the study of the local well-posedness theory for the Schrödinger-KdV system in the Sobolev space $H^{s_{1}}\times H^{s_{2}}$ . We show the local well-posedness in $H^{-3/16}\times H^{-3/4}$ for $\beta=0$ . Combining our work [2], we also have the local well-posedness for $\max\{-3/4,s_{1}-3\}\leq s_{2}\leq\min\{4s_{1},s_{1}+2\}$ . The result is sharp by using the contraction mapping argument.

1 Introduction

We study the Cauchy problem for Schrödinger-KdV system

\left\{\begin{aligned} &i\partial_{t}u+\partial_{xx}u=\alpha uv+\beta|u|^{2}u,\\ &\partial_{t}v+\partial_{xxx}v=\partial_{x}(\gamma|u|^{2}-v^{2}/2),\\ &(u,v)|_{t=0}=(u_{0},v_{0})\in H^{s_{1}}\times H^{s_{2}}.\end{aligned}\right.

(S-KdV)

In this paper we concern the local well-posedness theory for (S-KdV) with $\beta=0$ . The constants $\alpha,\gamma\neq 0$ in the model is not essential. We would assume $\alpha=\gamma=1$ for brevity.

This model appears in the study of resonant interaction between solitary wave of Langmuir and solitary wave of ion-acoustic. See Appert-Vaclavik [1]. It also appears in plasma physics and a diatomic lattice system. See [6, 4] and reference therein.

We recall some early results for this model. Tsutsumi showed the well-posedness in $H^{s+1/2}\times H^{s}$ , $s\in\mathbb{N}$ in [9]. In [5], Guo–Miao showed the well-posedness in $H^{s}\times H^{s}$ , $s\in\mathbb{N}$ . Guo–Wang [6] obtained the well-posedness in $H^{s_{1}}\times H^{-3/4}$ , $s_{1}>-1/16$ and Wang–Cui showed the well-posedness in $H^{s_{1}}\times H^{-3/4}$ , $s_{1}>-3/16$ in [11] . See also [3, 4, 2] for other results.

In this paper, we obtain the following result.

Theorem 1.1.

(S-KdV) is local well-posed in $H^{-3/16}\times H^{-3/4}$ .

In fact combining the results in our work [2] we have

Corollary 1.2.

(S-KdV) is local well-posed in $H^{s_{1}}\times H^{s_{2}}$ for

\max\{-3/4,s_{1}-3\}\leq s_{2}\leq\min\{4s_{1},s_{1}+2\}.

Recall the proof of Theorem 1.1 in [2] one has

Theorem 1.3.

The data-to-solution mapping of (S-KdV) can not be $C^{2}$ except $\max\{-3/4,s_{1}-3\}\leq s_{2}\leq\min\{4s_{1},s_{1}+2\}$ .

Thus it is reasonable to say than our local well-posedness result is sharp by using contraction mapping argument.

To show the local well-posedness in $H^{-3/16}\times H^{-3/4}$ , there are two difficulties which come from the terms $\partial_{x}(|u|^{2})$ and $\partial_{x}(v^{2})$ . In some sense, $H^{-3/16}\times H^{-3/4}$ is the “double critical” space for this model. We need some techniques that are different from before.

The paper is organized as follows. In Section 2, we rewrite (S-KdV) into a new form and construct the workspace. Then we show linear estimates, bilinear estimates related the Schrödinger equation, and tirlinear estimate related the KdV equation. In Section 3, we estimate $F[u_{0}]$ (see the definition (2.1)). In Section 4, we give some refinement for the bilinear estimates related to the KdV equation and finally conclude the proof.

Notations. . For $a,b\in\mathbb{R}^{+}$ , $a\lesssim b$ means that there exists $C>0$ such that $a\leq Cb$ and $a\sim b$ means $a\lesssim b\lesssim a$ where $C$ is a constant. We use $\langle\xi\rangle$ to denote $(1+\xi^{2})^{1/2}$ . Let $\varphi$ be an even, smooth function and $\varphi|_{[-1,1]}=0$ , $\varphi|_{[-2,2]^{c}}=0$ . $\varphi_{N}=\varphi(\cdot/N)$ , $\psi_{N}=\varphi_{N}-\varphi_{N/2}$ . We use inhomogeneous Littlewood-Paley projection: $P_{N}:=\mathscr{F}^{-1}\psi_{N}\mathscr{F}$ , $N\geq 2$ ; $P_{1}=\mathscr{F}^{-1}\varphi\mathscr{F}$ ; $P_{>N}=\sum_{M>N}P_{M}$ , $etc.$ . We always use $N,L$ to denote a dyadic number larger than $1$ .

Let $S_{\lambda}(t)=e^{i\lambda t\partial_{xx}},K(t)=e^{-t\partial_{xxx}}$ ,

\mathscr{A}_{\lambda}(f):=i\lambda\int_{0}^{t}S_{\lambda}(t-s)f(s)~{}ds,\quad\mathscr{B}(f):=\int_{0}^{t}K(t-s)\partial_{x}f(s)~{}ds.

We always assume $0<\lambda\ll 1$ in this paper. We define the modulation decomposition $Q^{S_{\lambda}}_{L}$ , $Q_{L}^{K}$ by $\mathscr{F}^{-1}_{\tau,\xi}\psi_{L}(\tau+\lambda\xi^{2})\mathscr{F}_{t,x}$ , $\mathscr{F}^{-1}_{\tau,\xi}\psi_{L}(\tau-\xi^{3})\mathscr{F}_{t,x}$ respectively. Similarly, we define $Q_{>L}^{S_{\lambda}}$ , $Q_{>L}^{K}$ , $Q_{1}^{S_{\lambda}}$ , and $Q_{1}^{K}$ .

2 Function spaces and multilinear estimates

To show the well-posedness of (S-KdV) with $s_{2}=-3/4$ , we use the argument in [6]. By rescaling, we consider the equation

\left\{\begin{aligned} &i\partial_{t}u+\lambda\partial_{xx}u=\lambda uv,\\ &\partial_{t}v+\partial_{xxx}v=\partial_{x}(|u|^{2}-v^{2}),\\ &(u,v)|_{t=0}=(u_{0,\lambda},v_{0,\lambda})\in H^{-3/16}\times H^{-3/4}\end{aligned}\right.

where $0<\lambda\ll 1$ , $\|u_{0,\lambda}\|_{H^{-3/16}}\lesssim\lambda^{21/16}$ , $\|v_{0,\lambda}\|_{H^{3/4}}\lesssim\lambda^{3/4}$ . Recall the definition of $U^{p}-V^{p}$ spaces in [7]. We define

\|u\|_{X_{\lambda}}=\|N^{-3/16}P_{N}u\|_{l^{2}_{N}U^{2}_{S_{\lambda}}},~{}~{}\|v\|_{Y}=\|P_{1}v\|_{L_{x}^{2}L_{t}^{\infty}}+\|N^{-3/4}P_{N}v\|_{l^{2}_{N\geq 2}U^{2}_{K}}.

For the brevity, we still denote $(u_{0,\lambda},v_{0,\lambda})$ by $(u_{0},v_{0})$ . The corresponding integral equation is

\left\{\begin{aligned} u(t)&=S_{\lambda}(t)u_{0}-\mathscr{A}_{\lambda}(uv)(t),\\ v(t)&=K(t)v_{0}+\mathscr{B}(|u|^{2}-v^{2})(t).\end{aligned}\right.

We replace the function $u$ in second equation with the first one and obtain

\left\{\begin{aligned} u(t)&=S_{\lambda}(t)u_{0}-i\mathscr{A}_{\lambda}(uv)(t),\\ v(t)&=K(t)v_{0}+\mathscr{B}(|S_{\lambda}(t)u_{0}-\mathscr{A}_{\lambda}(uv)|^{2}-v^{2})(t).\end{aligned}\right.

Let $\eta,\tilde{\eta}\in C_{0}^{\infty}$ with $\eta|_{[-1,1]}=1$ , $\tilde{\eta}|_{\mathrm{supp}(\eta)}=1$ . Define

F[u_{0}](t):=\eta(t)\mathscr{B}(|S_{\lambda}(t)u_{0}|^{2})(t).

(2.1)

In fact we can not show $F[u_{0}]\in Y$ . Let $w(t)=v(t)-F[u_{0}](t)$ . We consider the equation

\left\{\begin{aligned} u(t)&=\eta S_{\lambda}(t)u_{0}-\eta\mathscr{A}_{\lambda}(u(w+F[u_{0}]))(t),\\ w(t)&=\eta K(t)v_{0}-\eta\mathscr{B}((w+F[u_{0}])^{2})(t)+\eta\mathscr{B}(T_{\lambda}(u,w,u_{0}))(t)\end{aligned}\right.

(2.2)

where $T_{\lambda}(u,w,u_{0})=2\mathrm{Im}(\mathscr{A}_{\lambda}(u(w+F[u_{0}]))\overline{S_{\lambda}(t)u_{0}})-|\mathscr{A}_{\lambda}(u(w+F[u_{0}]))|^{2}$ . Define

\|v\|_{Z}:=\|P_{\lesssim 1}v\|_{L_{x}^{2}L_{t}^{\infty}}+\|N^{-3/4}P_{N}v\|_{l^{2}_{N\geq 2}V^{2}_{K}}+\|N^{1/4}P_{N}v\|_{l^{2}_{N}L_{x}^{\infty}L_{t}^{2}}.

(2.3)

Note that for $v$ supported on $[0,1]\times\mathbb{R}$ , $\|v\|_{Z}\lesssim\|v\|_{Y}$ by Strichartz, maximal function, and local smoothing estimates. We would show $F[u_{0}]\in Z$ in Section 3.

Directly, one has the following estimates due to the maximal function estimate and the definition of $U^{2}$ .

Lemma 2.1.

$\|\eta S_{\lambda}(t)u_{0}\|_{X_{\lambda}}\lesssim\|u_{0}\|_{H^{-3/16}}$ , $\|\eta K(t)v_{0}\|_{Y}\lesssim\|v_{0}\|_{H^{-3/4}}$ .

Then, we show the bilinear estimate related to the Schrödinger equation.

Lemma 2.2.

$\|\eta\mathscr{A}_{\lambda}(uv)\|_{X_{\lambda}}\lesssim\lambda^{1/2}\|u\|_{X_{\lambda}}\|v\|_{Z}$ .

Proof..

Firstly, for low frequency part of $v$ we have

	$\displaystyle\left\|\int_{\mathbb{R}^{2}}P_{N}(uP_{1}v)\tilde{\eta}^{2}\bar{w}~{}dxdt\right\|$	$\displaystyle\lesssim\sum_{N_{1}\sim N}\\|\tilde{\eta}P_{N_{1}}u\\|_{L_{x}^{\infty}L_{t}^{2}}\\|P_{1}v\\|_{L_{x}^{2}L_{t}^{\infty}}\\|\tilde{\eta}w\\|_{L_{t,x}^{2}}$		(2.4)
		$\displaystyle\lesssim(\lambda N)^{-1/2}\sum_{N_{1}\sim N}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|v\\|_{Z}\\|w\\|_{V^{2}_{S_{\lambda}}}.$		(2.4)

By duality one has

	$\displaystyle\\|\eta\mathscr{A}_{\lambda}(uP_{1}v)\\|_{X_{\lambda}}$	$\displaystyle\lesssim\lambda\sum_{N}N^{-3/16}(\lambda N)^{-1/2}\sum_{N_{1}\sim N}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|v\\|_{Z}$
		$\displaystyle\lesssim\lambda^{1/2}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}.$

Then by triangle inequality,

\displaystyle\|\eta\mathscr{A}_{\lambda}(uP_{>1}v)\|_{X_{\lambda}}\leq\sum_{N}\sum_{N_{1}}\sum_{N_{2}\geq 2}N^{-3/16}\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\|_{U^{2}_{S_{\lambda}}}.

If $N_{2}^{2}\nsim\lambda N_{1}$ , for $\tau_{1}+\tau_{2}=\tau$ , $\xi_{1}+\xi_{2}=\xi$ , $|\xi_{1}|\sim N_{1}$ , $|\xi_{2}|\sim N_{2}$ , $|\xi|\sim N$ we have

\displaystyle|\tau_{1}+\lambda\xi_{1}^{2}|+|\tau_{2}-\xi^{3}_{2}|+|\tau+\lambda\xi^{2}|\gtrsim N_{2}\max\{N_{2}^{2},\lambda N_{1}\}.

Let $L=cN_{2}\max\{N_{2}^{2},\lambda N_{1}\}$ . Typically we need to estimate

	$\displaystyle I$	$\displaystyle=\int_{\mathbb{R}^{2}}Q_{>L}^{S_{\lambda}}P_{N_{1}}uP_{N_{2}}vP_{N}\bar{w}~{}dxdt,$
	$\displaystyle II$	$\displaystyle=\int_{\mathbb{R}^{2}}P_{N_{1}}uQ_{>L}^{K}P_{N_{2}}vP_{N}\bar{w}~{}dxdt$

where $u,v,w$ are supported on $[-1,1]\times\mathbb{R}$ . By the Strichartz estimate one has

\|u\|_{L_{t,x}^{6}}\lesssim\lambda^{-1/6}\|u\|_{U^{2}_{S_{\lambda}}}\lesssim\lambda^{-1/6}\|u\|_{V^{2}_{S_{\lambda}}},~{}\|u\|_{L_{t,x}^{2}}\lesssim\|u\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim\|u\|_{V^{2}_{S_{\lambda}}}.

Thus $\|u\|_{L_{t,x}^{4}}\leq\|u\|_{L_{t,x}^{6}}^{3/4}\|u\|_{L_{t,x}^{2}}^{1/4}\lesssim\lambda^{-1/8}\|u\|_{V^{2}_{S_{\lambda}}}$ . Similarly,

\|P_{N_{2}}v\|_{L_{t,x}^{4}}\lesssim N_{2}^{-1/8}\|v\|_{V^{2}_{K}},\quad\|P_{N}w\|_{L_{t,x}^{4}}\lesssim\lambda^{-1/8}\|w\|_{V^{2}_{K}}.

By the Hölder inequality we have

	$\displaystyle\|I\|$	$\displaystyle\lesssim\\|Q_{>L}^{S_{\lambda}}P_{N_{1}}u\\|_{L_{t,x}^{2}}\\|P_{N_{2}}v\\|_{L_{t,x}^{4}}\\|P_{N}w\\|_{L_{t,x}^{4}}$
		$\displaystyle\lesssim L^{-1/2}\lambda^{-1/8}\\|u\\|_{V^{2}_{S_{\lambda}}}N_{2}^{-1/8}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim\lambda^{-1/8}N_{2}^{-5/8}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\\|u\\|_{U^{2}_{S_{\lambda}}}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}.$

Also,

\displaystyle|II|\lesssim\lambda^{-1/4}N_{2}^{-1/2}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\|u\|_{U^{2}_{S_{\lambda}}}\|v\|_{V^{2}_{K}}\|w\|_{V^{2}_{K}}.

Thus by the duality one has

		$\displaystyle\quad\\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\\|_{U^{2}_{S_{\lambda}}}$		(2.5)
		$\displaystyle\lesssim\lambda\lambda^{-1/4}N_{2}^{-1/2}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim\lambda^{3/4}N_{1}^{3/16}N_{2}^{1/4}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}.$

Let $N_{\max},N_{\mathrm{med}}$ be the maximum, medium of $N_{1},N_{2},N$ . Note that

\sum_{N_{\max}\sim N_{\mathrm{med}}}N^{-3/16}\lambda^{3/4}N_{1}^{3/16}N_{2}^{1/4}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\lesssim\lambda^{3/4}\log\lambda^{-1}.

Thus we obtain

\sum_{N_{2}\geq 2,N_{2}^{2}\nsim\lambda N_{1},N_{1},N}N^{-3/16}\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\|_{U^{2}_{S_{\lambda}}}\lesssim\frac{\log\lambda^{-1}}{\lambda^{-3/4}}\|u\|_{X_{\lambda}}\|v\|_{Z}.

If $N_{2}^{2}\sim\lambda N_{1}\sim\lambda N$ , by Lemma 4.12, (4.8) in [2], we have

\displaystyle\left|\int_{\mathbb{R}^{2}}P_{N_{1}}uP_{N_{2}}vP_{N}\bar{w}~{}dxdt\right|\lesssim N_{2}^{-1}\|u\|_{U^{2}_{S_{\lambda}}}\|v\|_{U^{2}_{K}}\|w\|_{U^{2}_{S_{\lambda}}}.

On the other hand, by Hölder’s inequality and Strichartz estimates, one has

	$\displaystyle\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}uP_{N_{2}}vP_{N}\bar{w}~{}dxdt\right\|$	$\displaystyle\lesssim\\|u\\|_{L_{t,x}^{3}}\\|P_{N_{2}}v\\|_{L_{t,x}^{3}}\\|w\\|_{L_{t,x}^{3}}$
		$\displaystyle\lesssim\\|u\\|_{L_{t}^{12}L_{x}^{3}}\\|P_{N_{2}}v\\|_{L_{t}^{12}L_{x}^{3}}\\|w\\|_{L_{t}^{12}L_{x}^{3}}$
		$\displaystyle\lesssim\lambda^{-1/6}N_{2}^{-1/12}\\|u\\|_{U^{12}_{S_{\lambda}}}\\|v\\|_{U^{12}_{K}}\\|w\\|_{U^{12}_{S_{\lambda}}}.$

Then by the interpolation (cf. Proposition 2.20 in [7]), we have

\displaystyle\left|\int_{\mathbb{R}^{2}}P_{N_{1}}uP_{N_{2}}vP_{N}\bar{w}~{}dxdt\right|\lesssim N_{2}^{-1}\lambda^{-1/6}(\log N_{2})^{3}\|u\|_{U^{2}_{S_{\lambda}}}\|v\|_{V^{2}_{K}}\|w\|_{V^{2}_{S_{\lambda}}}.

By the duality one has

	$\displaystyle\\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\\|_{U^{2}_{S_{\lambda}}}$	$\displaystyle\lesssim\lambda N_{2}^{-1}\lambda^{-1/6}(\log N_{2})^{3}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$		(2.6)
		$\displaystyle\lesssim\lambda^{5/6}N_{2}^{-1/4}N_{1}^{3/16}(\log N_{2})^{3}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}.$		(2.6)

Note that

\sum_{N_{2}^{2}\sim\lambda N_{1}\sim\lambda N}N^{-3/16}N_{2}^{-1/4}N_{1}^{3/16}\lambda^{5/6}(\log N_{2})^{3}\lesssim\lambda^{5/6}.

Overall one has

\displaystyle\quad\sum_{N}\sum_{N_{1}}\sum_{N_{2}\geq 2}N^{-3/16}\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\|_{U^{2}_{S_{\lambda}}}\lesssim\lambda^{1/2}\|u\|_{X_{\lambda}}\|v\|_{Z}.

We finish the proof of this lemma. ∎

Next we show the trilinear estimate related to the KdV equation.

Lemma 2.3.

$\|\eta\mathscr{B}(\mathscr{A}_{\lambda}(uv)\bar{w})\|_{Y}\lesssim\lambda^{-1/2}\|u\|_{X_{\lambda}}\|v\|_{Z}\|w\|_{X_{\lambda}}$ .

Proof.

Without loss of the generality we would assume that $u,v,w$ are supported on $[-1,1]\times\mathbb{R}$ . For the low frequency we have

	$\displaystyle\\|\eta P_{1}\mathscr{B}(\mathscr{A}_{\lambda}(uv)\bar{w})\\|_{L_{x}^{2}L_{t}^{\infty}}$	$\displaystyle\lesssim\\|P_{1}(\mathscr{A}(uv)\bar{w})\\|_{L_{x}^{2}L_{t}^{1}}$
		$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}}\\|P_{1}(P_{N_{1}}\mathscr{A}_{\lambda}(uv)P_{N_{2}}\bar{w})\\|_{L_{x}^{2}L_{t}^{1}}$
		$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}}\\|\tilde{\eta}P_{N_{1}}\mathscr{A}_{\lambda}(uv)\\|_{L_{t,x}^{2}}\\|P_{N_{2}}w\\|_{L_{x}^{\infty}L_{t}^{2}}$
		$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}}\\|\tilde{\eta}P_{N_{1}}\mathscr{A}_{\lambda}(uv)\\|_{U^{2}_{S_{\lambda}}}(\lambda N_{2})^{-1/2}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}$
		$\displaystyle\lesssim\lambda^{-1/2}\sum_{N_{1}\sim N_{2}}N_{1}^{3/16}N_{2}^{-5/16}\\|\tilde{\eta}\mathscr{A}_{\lambda}(uv)\\|_{X_{\lambda}}\\|w\\|_{X_{\lambda}}$
		$\displaystyle\lesssim\lambda^{-1/2}\\|\tilde{\eta}\mathscr{A}_{\lambda}(uv)\\|_{X_{\lambda}}\\|w\\|_{X_{\lambda}}.$

By Lemma 2.2 we obtain

\displaystyle\|\eta P_{1}\mathscr{B}(\mathscr{A}_{\lambda}(uv)\bar{w})\|_{L_{x}^{2}L_{t}^{\infty}}\lesssim\|u\|_{X_{\lambda}}\|v\|_{Z}\|w\|_{X_{\lambda}}.

(2.7)

Since $\|P_{N}Q_{L}^{K}v\|_{L_{x}^{\infty}L_{t}^{2}}\lesssim N^{-1}L^{1/2}\|P_{N}Q_{L}^{K}v\|_{L_{t,x}^{2}}$ , by the interpolation one has

\|P_{N}Q_{L}^{K}v\|_{L_{x}^{2/(1-\theta)}L_{t}^{2}}\lesssim(N^{-1}L^{1/2})^{\theta}\|P_{N}Q_{L}^{K}v\|_{L^{2}_{t,x}},\quad\forall~{}0<\theta<1.

Thus ( $v$ is supported on $[-1,1]\times\mathbb{R}$ )

	$\displaystyle\\|P_{N}v\\|_{L_{x}^{2/(1-\theta)}L_{t}^{2}}$	$\displaystyle\lesssim\sum_{L}(N^{-1}L^{1/2})^{\theta}\\|P_{N}Q_{L}^{K}v\\|_{L^{2}_{t,x}}$
		$\displaystyle\lesssim\sum_{L}(N^{-1}L^{1/2})^{\theta}L^{-1/2}\\|P_{N}v\\|_{V^{2}_{K}}\lesssim N^{-\theta}\\|P_{N}v\\|_{V^{2}_{K}}.$

If $N_{2}\lesssim N$ (which implies $N_{1}\lesssim N$ ), by the duality, Minkowski, and Bernstein inequalities one has

	$\displaystyle\quad\sum_{N_{1},N_{2}\lesssim N}\\|\eta P_{N}\mathscr{B}(\mathscr{A}_{\lambda}(P_{N_{1}}uP_{1}v)P_{N_{2}}\bar{w})\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N_{1},N_{2}\lesssim N}NN^{-\theta}\\|P_{N_{1}}\mathscr{A}_{\lambda}(uP_{1}v)P_{N_{2}}w\\|_{L_{x}^{2/(1+\theta)}L_{t}^{2}}$
	$\displaystyle\lesssim\sum_{N_{1},N_{2}\lesssim N}N^{1-\theta}\\|\tilde{\eta}P_{N_{1}}\mathscr{A}_{\lambda}(uP_{1}v)\\|_{L_{x}^{2/\theta}L_{t}^{2}}\\|\tilde{\eta}P_{N_{2}}w\\|_{L_{x}^{2}L_{t}^{\infty}}$
	$\displaystyle\lesssim N^{3/2-\theta}\sum_{N_{1},N_{2}\lesssim N}(\lambda N_{1})^{(\theta-1)/2}\\|\tilde{\eta}P_{N_{1}}\mathscr{A}_{\lambda}(uP_{1}v)\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}.$

By (2.4) one has $\|\tilde{\eta}P_{N_{1}}\mathscr{A}_{\lambda}(uP_{1}v)\|_{U^{2}_{S_{\lambda}}}\lesssim\lambda(\lambda N_{1})^{-1/2}N_{1}^{3/16}\|u\|_{X_{\lambda}}\|v\|_{Z}$ . Thus we can control the former term by

\displaystyle N^{3/2-\theta}\sum_{N_{1},N_{2}\lesssim N}(\lambda N_{1})^{(\theta-1)/2}\lambda(\lambda N_{1})^{-1/2}N_{1}^{3/16}\|u\|_{X_{\lambda}}\|v\|_{Z}N_{2}^{3/16}\|w\|_{X_{\lambda}},

By choosing $15/16<\theta<1$ , then

\lambda\sum_{N}N^{-3/4}N^{3/2-\theta}\sum_{N_{1},N_{2}\lesssim N}(\lambda N_{1})^{(\theta-1)/2}(\lambda N_{1})^{-1/2}N_{1}^{3/16}N_{2}^{3/16}\lesssim\lambda^{\theta/2}.

We obtain the desired estimate since we always assume $0<\lambda\ll 1$ .

If $N_{2}\gg N$ (which implies $N_{1}\sim N_{2}$ ) similar to the former argument, we have $\|P_{N}v\|_{L_{t,x}^{30/7}}\lesssim N^{-2/15}\|v\|_{V^{2}_{K}}$ . Then by the duality one has

	$\displaystyle\quad\sum_{N_{1}\sim N_{2}\gg N}\\|\eta P_{N}\mathscr{B}(\mathscr{A}_{\lambda}(P_{N_{1}}uP_{1}v)P_{N_{2}}\bar{w})\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}\gg N}NN^{-2/15}\\|\mathscr{A}_{\lambda}(P_{N_{1}}uP_{1}v)P_{N_{2}}w\\|_{L_{t,x}^{30/23}}$
	$\displaystyle\lesssim\sum_{N_{1}\sim N_{3}\gg N}N^{13/15}\\|\tilde{\eta}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{1}v)\\|_{L_{t,x}^{2}}\\|P_{N_{2}}w\\|_{L_{t,x}^{2}}^{3/10}\\|P_{N_{2}}w\\|_{L_{t,x}^{6}}^{7/10}$
	$\displaystyle\lesssim\lambda^{53/60}\sum_{N_{1}\sim N_{2}\gg N}N^{13/15}\\|P_{N_{1}}u\\|_{L^{\infty}_{x}L_{t}^{2}}\\|P_{1}v\\|_{L_{x}^{2}L_{t}^{\infty}}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}$
	$\displaystyle\lesssim\lambda^{53/60}N^{13/15}\sum_{N_{1}\sim N_{2}\gg N}N_{2}^{3/16}(\lambda N_{1})^{-1/2}N_{1}^{3/16}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}\\|w\\|_{X_{\lambda}}$
	$\displaystyle\lesssim\lambda^{23/60}N^{89/120}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}\\|w\\|_{X_{\lambda}}.$

Note that $89/120<3/4$ . Thus we have

		$\displaystyle\quad\sum_{N\geq 2}N^{-3/4}\sum_{N_{1}\sim N_{2}\gg N}\\|\eta P_{N}\mathscr{B}(\mathscr{A}_{\lambda}(P_{N_{1}}uP_{1}v)P_{N_{2}}\bar{w})\\|_{U^{2}_{K}}$		(2.8)
		$\displaystyle\lesssim\lambda^{23/60}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}\\|w\\|_{X_{\lambda}}.$		(2.8)

By (2.5), for $M_{2}^{2}\nsim\lambda M_{1}$ we have

\displaystyle\|\eta P_{N_{1}}\mathscr{A}_{\lambda}(P_{M_{1}}uP_{M_{2}}v)\|_{U^{2}_{S_{\lambda}}}\lesssim\frac{\lambda^{3/4}M_{1}^{3/16}M_{2}^{1/4}}{\max\{M_{2}^{2},\lambda M_{1}\}^{1/2}}\|u\|_{X_{\lambda}}\|v\|_{Z}.

By (2.6), for $M_{2}^{2}\sim\lambda M_{1}$ we have

\displaystyle\|\eta P_{N_{1}}\mathscr{A}_{\lambda}(P_{M_{1}}uP_{M_{2}}v)\|_{U^{2}_{S_{\lambda}}}\lesssim\lambda^{5/6}M_{2}^{-1/4}M_{1}^{3/16}(\log(M_{2}/\lambda))^{3}\|u\|_{X_{\lambda}}\|v\|_{Z}.

Thus,

		$\displaystyle\quad\\|\eta P_{N_{1}}(\mathscr{A}_{\lambda}(uP_{>1}v))\\|_{U^{2}_{S_{\lambda}}}$		(2.9)
		$\displaystyle\lesssim\sum_{M_{2}^{2}\nsim\lambda M_{1}}\frac{\lambda^{3/4}M_{1}^{3/16}M_{2}^{1/4}}{\max\{M_{2}^{2},\lambda M_{1}\}^{1/2}}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}$
		$\displaystyle\quad+\sum_{M_{2}^{2}\sim\lambda M_{1}\sim\lambda N_{1}}\lambda^{5/6}M_{2}^{-1/4}M_{1}^{3/16}(\log(M_{2}/\lambda))^{3}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}$
		$\displaystyle\lesssim(\lambda^{9/16}+\lambda^{17/24}N_{1}^{1/16}(\log N_{1})^{3})\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}.$

Let $\tilde{u}=\eta\mathscr{A}_{\lambda}(uP_{>1}v)$ . Now we estimate $\|\eta P_{N}\mathscr{B}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\|_{U^{2}_{K}}$ .

If $N^{2}\ll\lambda N_{1}$ by Lemma 4.12, (4.7) in [2] one has

\left|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})P_{N}\bar{\tilde{v}}~{}dxdt\right|\lesssim\lambda^{-1}N_{1}^{-1/2}\|\tilde{u}\|_{U^{2}_{S_{\lambda}}}\|w\|_{U^{2}_{S_{\lambda}}}\|v\|_{U^{2}_{K}}.

On the other hand by the Hölder inequality we also have

	$\displaystyle\left\|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})P_{N}\bar{\tilde{v}}~{}dxdt\right\|$	$\displaystyle\lesssim N\\|P_{N}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\\|_{L^{2}_{t,x}}\\|\tilde{\eta}P_{N}\tilde{v}\\|_{L^{2}_{t,x}}$
		$\displaystyle\lesssim\lambda^{-1/2}N^{1/2}\\|\tilde{u}\\|_{U^{2}_{S_{\lambda}}}\\|w\\|_{U^{2}_{S_{\lambda}}}\\|v\\|_{U^{p}_{K}},\quad p>2.$

Then by the interpolation theorem in [7] one has

\left|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})P_{N}\bar{\tilde{v}}~{}dxdt\right|\lesssim\lambda^{-1}N_{1}^{-1/2}\log(N_{1}/\lambda)\|\tilde{u}\|_{U^{2}_{S_{\lambda}}}\|w\|_{U^{2}_{S_{\lambda}}}\|v\|_{V^{2}_{K}}.

Combining Lemma 2.2 we have

		$\displaystyle\quad\sum_{N\geq 2}N^{-3/4}\sum_{N_{1}\sim N_{2}\gg N^{2}/\lambda}\\|\eta P_{N}\mathscr{B}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\\|_{U^{2}_{K}}$		(2.10)
		$\displaystyle\lesssim\sum_{N\geq 2}N^{-3/4}\sum_{N_{1}\sim N_{2}\gg N^{2}/\lambda}\lambda^{-1}N_{1}^{-1/2}\log(N_{1}/\lambda)\\|P_{N_{1}}\tilde{u}\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}$
		$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}\gg 1/\lambda}\lambda^{-1}N_{1}^{-1/2}\log(N_{1}/\lambda)\\|P_{N_{1}}\tilde{u}\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}$
		$\displaystyle\lesssim\lambda^{-7/8}\\|\tilde{u}\\|_{X_{\lambda}}\\|w\\|_{X_{\lambda}}\lesssim\lambda^{-3/8}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}\\|w\\|_{X_{\lambda}}.$

Similarly if $N^{2}\gg\lambda N_{1}$ by Lemma 4.12, (4.6) in [2] and the interpolation one has

\|\eta P_{N}\mathscr{B}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\|_{U^{2}_{K}}\lesssim\lambda^{-1/2}N^{-1}\log(N/\lambda)\|\tilde{u}\|_{U^{2}_{S_{\lambda}}}\|w\|_{U^{2}_{S_{\lambda}}}.

Combining Lemma 2.2 we have

		$\displaystyle\quad\sum_{N\geq 2}N^{-3/4}\sum_{N_{1},N_{2}\ll N^{2}/\lambda}\\|\eta P_{N}\mathscr{B}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\\|_{U^{2}_{K}}$		(2.11)
		$\displaystyle\lesssim\sum_{N\geq 2}N^{-3/4}\sum_{N_{1},N_{2}\ll N^{2}/\lambda}\lambda^{-1/2}N^{-1}\log(N/\lambda)\\|P_{N_{1}}\tilde{u}\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}$
		$\displaystyle\lesssim\sum_{N\geq 2}N^{-3/4}\sum_{N_{1},N_{2}\ll N^{2}/\lambda}\lambda^{-1/2}N^{-1}\log(N/\lambda)(N_{1}N_{2})^{3/16}\\|\tilde{u}\\|_{X_{\lambda}}\\|w\\|_{X_{\lambda}}$
		$\displaystyle\lesssim\lambda^{-7/8}\log(1/\lambda)\\|\tilde{u}\\|_{X_{\lambda}}\\|w\\|_{X_{\lambda}}\lesssim\lambda^{-3/8}\log(1/\lambda)\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}\\|w\\|_{X_{\lambda}}.$

If $N^{2}\sim\lambda N_{1}$ similar to the former argument by Lemma 4.12, (4.8) one has

\|\eta P_{N}\mathscr{B}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\|_{U^{2}_{K}}\lesssim\log(N/\lambda)\|\tilde{u}\|_{U^{2}_{S_{\lambda}}}\|w\|_{U^{2}_{S_{\lambda}}}.

Then by (2.9) we have

		$\displaystyle\quad\sum_{N\geq 2}N^{-3/4}\sum_{N_{1}\sim N_{2}\sim N^{2}/\lambda}\\|\eta P_{N}\mathscr{B}(P_{N_{1}}\tilde{u}P_{N_{2}}\bar{w})\\|_{U^{2}_{K}}$		(2.12)
		$\displaystyle\lesssim\sum_{N\geq 2}N^{-3/4}\sum_{N_{1}\sim N_{2}\sim N^{2}/\lambda}\log(N/\lambda)\\|P_{N_{1}}\tilde{u}\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}w\\|_{U^{2}_{S_{\lambda}}}$
		$\displaystyle\lesssim\sum_{N\geq 2}N^{-3/4}\sum_{N_{1}\sim N_{2}\sim N^{2}/\lambda}\log(N/\lambda)N_{2}^{3/16}$
		$\displaystyle\hskip 80.0pt\cdot(\lambda^{9/16}+\lambda^{17/24}N_{1}^{1/16}(\log N_{1})^{3})\\|u\\|_{X_{\lambda}}\\|v\\|_{Y}\\|w\\|_{X_{\lambda}}$
		$\displaystyle\lesssim\lambda^{3/8}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}\\|w\\|_{X_{\lambda}}.$

Combining (2.10), (2.11), and (2.12) we conclude the proof. ∎

3 The estimates for $F[u_{0}]$

In this section, we show some estimates for $F[u_{0}]$ . For the nonresonant case, we show that it belongs to $U^{2}$ type space. See Lemma 3.1. Also we show $F[u_{0}]\in Z$ .

Lemma 3.1.

Recall the definition of $F[u_{0}]$ by (2.1). We have

\|N^{-3/4}P_{N}(F[u_{0}]-F[P_{\sim N^{2}/\lambda}u_{0}])\|_{l^{2}_{N}U^{2}_{K}}\lesssim\lambda^{-1}\|u_{0}\|_{H^{-3/16}}^{2}.

Proof.

If $N\lesssim 1$ , we claim that

\left|\int_{\mathbb{R}^{2}}S_{\lambda}(t)P_{N_{1}}u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}}P_{N}\bar{v}~{}dxdt\right|\lesssim N_{1}^{-3/8}\|u_{0}\|_{L^{2}}^{2}\|v\|_{V^{2}_{K}}

(3.1)

where $v$ is supported on $[-1/2,1/2]\times\mathbb{R}$ . Let $q>2$ . By the Hölder inequality, local smoothing, maximal function, and Strichartz estimates one has

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}S_{\lambda}(t)P_{N_{1}}u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}}P_{N}\bar{v}~{}dxdt\right\|$
	$\displaystyle\lesssim\\|\tilde{\eta}S_{\lambda}(t)P_{N_{1}}u_{0}\\|_{L_{x}^{2q/(q-2)}L_{t}^{2}}\\|\tilde{\eta}S_{\lambda}(t)P_{N_{2}}u_{0}\\|_{L_{t,x}^{2}}\\|P_{N}v\\|_{L_{x}^{q}L_{t}^{\infty}}$
	$\displaystyle\lesssim\\|\tilde{\eta}S_{\lambda}(t)P_{N_{1}}u_{0}\\|_{L_{x}^{\infty}L_{t}^{2}}^{2/q}\\|u_{0}\\|^{1+(q-2)/q}_{L_{x}^{2}}\\|v\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim(\lambda N_{1})^{-1/q}\\|u_{0}\\|_{L^{2}}^{2}\\|v\\|_{V^{2}_{K}}.$

By choosing $q=8/3$ we obtain (3.1). Then by the duality one has

	$\displaystyle\quad\sum_{N\lesssim 1}\sum_{N_{1}\sim N_{2}}N^{-3/4}\left\\|\eta P_{N}\mathscr{B}(S_{\lambda}(t)P_{N_{1}}u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})\right\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}}(\lambda N_{1})^{-3/8}\\|P_{N_{1}}u_{0}\\|_{L^{2}}\\|P_{N_{2}}u_{0}\\|_{L^{2}}\lesssim\lambda^{-3/8}\\|u_{0}\\|_{H^{3/16}}^{2}.$

If $\lambda N_{1}\ll N^{2}$ , by Lemma 4.12 in [2] one has

\left|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})P_{N}\bar{v}~{}dxdt\right|\lesssim\lambda^{-1/2}N^{-1}\|u_{0}\|_{L^{2}}^{2}\|v\|_{U^{2}_{K}}.

By the Hölder inequality and the Strichartz estimate one also has

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})P_{N}\bar{v}~{}dxdt\right\|$
	$\displaystyle\lesssim\\|\tilde{\eta}P_{N_{1}}S_{\lambda}(t)u_{0}\\|_{L^{3}_{t,x}}\\|\tilde{\eta}P_{N_{2}}S_{\lambda}(t)u_{0}\\|_{L^{2}_{t,x}}\\|\partial_{x}P_{N}v\\|_{L^{6}_{t,x}}$
	$\displaystyle\lesssim\\|P_{N_{1}}S_{\lambda}(t)u_{0}\\|^{1/2}_{L^{6}_{t,x}}\\|u_{0}\\|^{3/2}_{L^{2}}N^{5/6}\\|v\\|_{U^{6}_{K}}$
	$\displaystyle\lesssim\lambda^{-1/12}N^{5/6}\\|u_{0}\\|_{L^{2}}^{2}\\|v\\|_{U^{6}_{K}}$

where $v$ is supported on $[-1/2,1/2]\times\mathbb{R}$ . Then by the interpolation in [7] we have

\left\|\eta P_{N}\mathscr{B}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})\right\|_{U^{2}_{K}}\lesssim\lambda^{-1/2}N^{-1}\log N\|u_{0}\|_{L^{2}}^{2}.

Since $\max\{N_{1},N_{2},N\}\sim\mathrm{med}\{N_{1},N_{2},N\}$ , thus

	$\displaystyle\quad\sum_{N\gg 1}\sum_{\lambda N_{1}\ll N^{2}}N^{-3/4}\left\\|\eta P_{N}\mathscr{B}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})\right\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N\gg 1}\sum_{\lambda\max\{N_{1},N_{2}\}\ll N^{2}}N^{-3/4}\lambda^{-1/2}N^{-1}\log N\\|P_{N_{1}}u_{0}\\|_{L^{2}}\\|P_{N_{2}}u_{0}\\|_{L^{2}}$
	$\displaystyle\lesssim\lambda^{-7/8}\\|u_{0}\\|_{H^{-3/16}}^{2}.$

If $\lambda N_{1}\gg N^{2}$ , by Lemma 4.12 in [2] one has

\left|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})P_{N}\bar{v}~{}dxdt\right|\lesssim\lambda^{-1}N_{1}^{-1/2}\|u_{0}\|_{L^{2}}^{2}\|v\|_{U^{2}_{K}}.

By similar argument as $\lambda N_{1}\ll N^{2}$ and the interpolation in [7] we have

\left\|\eta P_{N}\mathscr{B}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})\right\|_{U^{2}_{K}}\lesssim\lambda^{-1}N_{1}^{-1/2}\log N_{1}\|u_{0}\|_{L^{2}}^{2}.

Thus,

	$\displaystyle\quad\sum_{N\gg 1}\sum_{\lambda N_{1}\gg N^{2}}N^{-3/4}\left\\|\eta P_{N}\mathscr{B}(P_{N_{1}}S_{\lambda}(t)u_{0}\overline{S_{\lambda}(t)P_{N_{2}}u_{0}})\right\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N\gg 1}\sum_{N_{1}\sim N_{2}}N^{-3/4}\lambda^{-1}N_{1}^{-1/2}\log N_{1}\\|P_{N_{1}}u_{0}\\|_{L^{2}}\\|P_{N_{2}}u_{0}\\|_{L^{2}}$
	$\displaystyle\lesssim\lambda^{-1}\\|u_{0}\\|_{H^{-3/16}}^{2}.$

We conclude the proof. ∎

Although we can not show the desired estimate for $F[P_{\sim N^{2}/\lambda}u_{0}]$ in $U^{2}_{K}$ , one has the following estimate.

Lemma 3.2.

$\|N^{-3/4}P_{N}F[P_{\sim N^{2}/\lambda}u_{0}]\|_{l^{2}_{N}V^{2}_{K}}\lesssim\lambda^{-3/8}\|u_{0}\|_{H^{-3/16}}^{2}$ .

Proof.

By Lemma 4.12 in [2] one has

\left|\int_{\mathbb{R}^{2}}\partial_{x}(|S_{\lambda}(t)P_{\sim N^{2}/\lambda}u_{0}|^{2})P_{N}\bar{v}~{}dxdt\right|\lesssim\|u_{0}\|_{L^{2}}^{2}\|v\|_{U^{2}_{K}}

where $v$ is supported on $[-1,1]\times\mathbb{R}$ . Then by the duality one has

	$\displaystyle\\|N^{-3/4}P_{N}F[P_{\sim N^{2}/\lambda}u_{0}]\\|_{l^{2}_{N}V^{2}_{K}}$	$\displaystyle\lesssim\\|N^{-3/4}\\|P_{\sim N^{2}/\lambda}u_{0}\\|_{L^{2}}^{2}\\|_{l^{2}_{N}}$
		$\displaystyle\lesssim\\|(\lambda M)^{-3/16}P_{M}u_{0}\\|^{2}_{l^{2}_{M}L^{2}}$
		$\displaystyle\lesssim\lambda^{-3/8}\\|u_{0}\\|_{H^{-3/16}}^{2}.$

To show the second inequality in the former argument we use that the carnality of $\{N\in 2^{\mathbb{N}_{0}}:N^{2}\sim\lambda M\}$ is uniformly bounded for $M$ . ∎

We give some refinement for the case $\lambda N_{1}\sim\lambda N_{2}\sim N^{2}$ in following two lemmas.

Lemma 3.3.

Let $N\gg 1$ , $0<c\ll 1\ll C$ and $I_{1},I_{2}$ be intervals included in $[-CN^{2},-cN^{2}]\cup[cN^{2},CN^{2}]$ with length less than $cN$ . We assume $J:=I_{1}-I_{2}\subset[-CN,cN]\cup[cN,CN]$ and $|\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2}|\gtrsim N$ for any $\xi\in J,\xi_{j}\in I_{j}$ , $\xi_{1}+\xi_{2}=\xi$ . Then we have

\displaystyle\left\|\eta P_{J}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right\|_{U^{2}_{K}}\lesssim\lambda^{-1/2}\|u_{0}\|_{L^{2}}^{2}.

Proof.

Let $L=cN$ . Since $|\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2}|>L/2$ for $\xi_{j}\in I_{j},\xi\in J,\xi_{1}+\xi_{2}=\xi$ , we have

\int_{\mathbb{R}^{2}}\partial_{x}(S_{\lambda}(t)P_{I_{1}}u_{0}S_{\lambda}(t)P_{I_{2}}u_{0})Q^{K}_{\leq L}P_{J}\bar{v}~{}dxdt=0.

By the high modulation estimate and Lemma 4.2, (4.2) in [2] one has

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})Q_{>L}^{K}P_{J}\bar{v}~{}dxdt\right\|$
	$\displaystyle\lesssim\\|P_{J}(S_{\lambda}(s)P_{I_{1}}u_{0}\overline{S_{\lambda}(s)P_{I_{2}}u_{0}})\\|_{L^{2}_{t,x}}\\|\partial_{x}Q^{K}_{>L}P_{J}v\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim(\lambda N)^{-1/2}\\|u_{0}\\|_{L^{2}}^{2}L^{-1/2}N\\|v\\|_{V^{2}_{K}}\sim\lambda^{-1/2}\\|u_{0}\\|_{L^{2}}^{2}\\|v\\|_{V^{2}_{K}}.$

By the duality we conclude the proof. ∎

Lemma 3.4.

Let $N\gg 1$ , $0<c\ll 1\ll C$ and $I_{1},I_{2}$ be intervals included in $[-CN^{2},-cN^{2}]\cup[cN^{2},CN^{2}]$ with length $1/N$ . We assume $J$ included in $[-CN,-cN]\cup[cN,CN]$ with length $1/N^{2}$ and $|\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2}|\lesssim 1$ for any $\xi\in J,\xi_{j}\in I_{j}$ , $\xi_{1}+\xi_{2}=\xi$ . Then we have

\displaystyle\left\|\eta P_{J}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right\|_{U^{2}_{K}}\lesssim\|u_{0}\|_{L^{2}}^{2}.

Proof.

Let $P_{I_{1}}u_{0}=u_{1},P_{I_{2}}u_{0}=u_{2}$ . By the definition of $U^{2}_{K}$ and the Taylor expansion one has

	$\displaystyle\quad\left\\|\eta P_{J}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right\\|_{U^{2}_{K}}$
	$\displaystyle=\left\\|\eta P_{J}\partial_{x}\int_{0}^{t}K(-s)(S_{\lambda}(s)u_{1}\overline{S_{\lambda}(s)u_{2})}~{}ds\right\\|_{U^{2}_{t}(L^{2}_{x})}$
	$\displaystyle\sim\left\\|\eta(t)\xi\chi_{\xi\in J}\int_{0}^{t}e^{-is\xi^{3}}\int_{\xi_{1}+\xi_{2}=\xi}e^{-i\lambda s\xi_{1}^{2}+i\lambda s\xi_{2}^{2}}\widehat{u_{1}}(\xi_{1})\overline{\widehat{u_{2}}}(-\xi_{2})~{}ds\right\\|_{U^{2}_{t}(L^{2}_{\xi})}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{1}{k!}\left\\|\eta(t)\xi\chi_{\xi\in J}\int_{0}^{t}\int_{\xi_{1}+\xi_{2}=\xi}\widehat{u_{1}}(\xi_{1})\overline{\widehat{u_{2}}}(-\xi_{2})s^{k}(\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2})^{k}~{}ds\right\\|_{U^{2}_{t}L^{2}_{\xi}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{C^{k}N}{k!}\left\\|\chi_{\xi\in J}\int_{\xi_{1}+\xi_{2}=\xi}\|\widehat{u_{1}}(\xi_{1})\widehat{u_{2}}(-\xi_{2})\|~{}ds\right\\|_{L^{2}_{\xi}}\left\\|\eta(t)\frac{t^{k+1}-1}{k+1}\right\\|_{U^{2}_{t}}$
	$\displaystyle\lesssim N\\|u_{1}\\|_{L^{2}}\\|u_{2}\\|_{L^{2}}\|J\|^{1/2}\lesssim\\|u_{0}\\|_{L^{2}}^{2}.$

Here we use $\|\eta(t)t^{k+1}\|_{U^{2}_{t}}\lesssim\|\eta(t)t^{k+1}\|_{H^{1}}\lesssim k$ . ∎

Lemma 3.5.

$\|N^{1/4}P_{N}F[u_{0}]\|_{l^{2}_{N}L_{x}^{\infty}L_{t}^{2}}\lesssim\lambda^{-1}\|u_{0}\|_{H^{-3/16}}^{2}$ .

Proof.

Firstly by Lemma 3.1 and the local smoothing estimate one has

\|N^{1/4}P_{N}(F[u_{0}]-F[P_{\sim N^{2}/\lambda}u_{0}])\|_{l^{2}_{N}L_{x}^{\infty}L_{t}^{2}}\lesssim\lambda^{-1}\|u_{0}\|_{H^{-3/16}}^{2}

Thus we only need to show

\displaystyle\|N^{1/4}P_{N}F[P_{\sim N^{2}/\lambda}u_{0}]\|_{l^{2}_{N}L^{\infty}_{x}L^{2}_{t}}\lesssim\lambda^{-1}\|u_{0}\|_{H^{-3/16}}^{2}.

If $N\lesssim 1$ , by the Sobolev inequality, then

\displaystyle N^{1/4}\|P_{N}F[P_{\sim N^{2}/\lambda}u_{0}]\|_{L^{\infty}_{x}L^{2}_{t}}\lesssim\|P_{\sim N^{2}/\lambda}u_{0}\|_{L^{4}_{x}}^{2}\lesssim\lambda^{-7/8}\|u_{0}\|_{H^{-3/16}}^{2}.

If $N\gg 1$ , we claim

\left\|P_{N}F[P_{\sim N^{2}/\lambda}u_{0}]\right\|_{L_{x}^{\infty}L_{t}^{2}}\lesssim\lambda^{-1/2}N^{-1}\|u_{0}\|_{L^{2}}^{2}.

(3.2)

Let $I_{j}$ , $j=1,2$ be intervals included in $[-CN^{2}/\lambda,cN^{2}/\lambda]\cup[cN^{2}/\lambda,CN^{2}/\lambda]$ with length $cN$ . Also we assume for any $\xi_{j}\in I_{j}$ , $j=1,2$ one has $|\eta_{1}-\eta_{2}|\sim N$ . We only need to show

\left\|\eta P_{N}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}v_{0}})\right\|_{L_{x}^{\infty}L_{t}^{2}}\lesssim\lambda^{-1/2}N^{-1}\|u_{0}\|_{L^{2}}\|v_{0}\|_{L^{2}}.

There exists a interval $J$ included in $[-N,-N/2]\cup[N/2,N]$ with length $1$ such that $|\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2}|\gtrsim N^{2}$ for any $\xi\in[-N,-N/2]\cup[N/2,N]\setminus J:=\hat{J}$ , $\xi_{j}\in I_{j}$ , $\xi=\xi_{1}+\xi_{2}$ . Then by Lemma 3.3, the local smoothing estimate, and the orthogonality we only need to show

\left\|\eta P_{J}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}v_{0}})\right\|_{L_{x}^{\infty}L_{t}^{2}}\lesssim\lambda^{-1/2}N^{-1}\|u_{0}\|_{L^{2}}\|v_{0}\|_{L^{2}}

where $I_{1},I_{2},J$ are intervals with length $1$ , $|\xi_{j}|\sim N^{2}$ for any $\xi_{j}\in I_{j}$ , $|\xi|\sim N$ for any $\xi\in J$ .

There exists a interval $J_{1}$ included in $J$ with length $1/N$ such that $|\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2}|\gtrsim N$ for any $\xi\in J\setminus J_{1}$ , $\xi_{j}\in I_{j}$ , $\xi=\xi_{1}+\xi_{2}$ . Then again by Lemma 3.3, the local smoothing estimate, and the orthogonality we only need to show

\left\|\eta P_{J_{1}}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}v_{0}})\right\|_{L_{x}^{\infty}L_{t}^{2}}\lesssim\lambda^{-1/2}N^{-1}\|u_{0}\|_{L^{2}}\|v_{0}\|_{L^{2}}

where $I_{1},I_{2},J_{1}$ are intervals with length $1/N$ .

There exists a interval $J_{2}$ included in $J_{1}$ with length $1/N^{2}$ such that one has $|\xi^{3}+\lambda\xi_{1}^{2}-\lambda\xi_{2}^{2}|\lesssim 1$ for any $\xi\in J_{2}$ , $\xi_{j}\in I_{j}$ , $\xi=\xi_{1}+\xi_{2}$ . Then by Lemma 3.4 and the local smoothing estimate one has

	$\displaystyle\quad\left\\|\eta P_{J_{2}}\mathscr{B}(S_{\lambda}(t)P_{I}u_{0}\overline{S_{\lambda}(t)P_{J}v_{0}})\right\\|_{L_{x}^{\infty}L_{t}^{2}}$
	$\displaystyle\lesssim N^{-1}\left\\|\eta P_{J_{2}}\mathscr{B}(S_{\lambda}(t)P_{I}u_{0}\overline{S_{\lambda}(t)P_{J}v_{0}})\right\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim N^{-1}\\|u_{0}\\|_{L^{2}}\\|v_{0}\\|_{L^{2}}.$

Note that

	$\displaystyle\quad\mathscr{F}_{x}\left(\eta(t)P_{J_{1}\setminus J_{2}}\mathscr{B}(S_{\lambda}(t)P_{I}u_{0}\overline{S_{\lambda}(t)P_{J}v_{0}})\right)$
	$\displaystyle\sim\eta(t)\xi\int_{0}^{t}\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}e^{i(t-s)\xi^{3}}e^{-i\lambda s\xi_{1}^{2}+i\lambda s\xi_{2}^{2}}\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})$
	$\displaystyle\sim\eta(t)\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}\frac{e^{it\xi^{3}}-e^{i\lambda t\xi(\xi_{2}-\xi_{1})}}{\xi^{2}+\lambda\xi_{1}-\lambda\xi_{2}}\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})$
	$\displaystyle:=\Lambda_{1}+\Lambda_{2}.$

By the local smoothing estimate we have

	$\displaystyle\\|\mathscr{F}_{\xi}^{-1}\Lambda_{1}\\|_{L_{x}^{\infty}L_{t}^{2}}$	$\displaystyle\lesssim N^{-1}\left\\|\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}\frac{\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})}{\xi^{2}+\lambda\xi_{1}-\lambda\xi_{2}}\right\\|_{L^{2}_{\xi}}$
		$\displaystyle\lesssim N^{-1}\\|u_{0}\\|_{L^{2}}\\|v_{0}\\|_{L^{2}}\left\\|\frac{\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{1}\in I_{1},\xi-\xi_{1}\in I_{2}}}{\xi^{2}+2\lambda\xi_{1}-\lambda\xi}\right\\|_{L^{\infty}_{\xi_{1}}L^{2}_{\xi}}$
		$\displaystyle\lesssim N^{-1}\\|u_{0}\\|_{L^{2}}\\|v_{0}\\|_{L^{2}}.$

Let $\xi_{j}^{0}$ be the center of $I_{j}$ respectively. By the Taylor expansion we have

	$\displaystyle\quad\\|\mathscr{F}^{-1}_{\xi}\Lambda_{2}\\|_{L^{\infty}_{x}L^{2}_{t}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{1}{k!}\Bigg{\\|}\eta(t)\mathscr{F}_{\xi}^{-1}\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}\frac{e^{i\lambda t\xi(\xi_{2}^{0}-\xi_{1}^{0})}}{\xi^{2}+\lambda\xi_{1}-\lambda\xi_{2}}\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})$
	$\displaystyle\qquad\qquad\cdot(i\lambda t\xi(\xi_{2}-\xi_{2}^{0}-\xi_{1}+\xi_{1}^{0}))^{k}\Bigg{\\|}_{L_{x}^{\infty}L_{t}^{2}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{1}{k!}\Bigg{\\|}\mathscr{F}^{-1}_{\xi}\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}\frac{e^{i\lambda t\xi(\xi_{2}^{0}-\xi_{1}^{0})}}{\xi^{2}+\lambda\xi_{1}-\lambda\xi_{2}}\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})$
	$\displaystyle\qquad\qquad\cdot\xi^{k}(\xi_{2}-\xi_{2}^{0}-\xi_{1}+\xi_{1}^{0})^{k}\Bigg{\\|}_{L_{x}^{\infty}L_{t}^{2}}.$

Note that $|\xi_{2}^{0}-\xi_{1}^{0}|\sim N^{2}$ . By the local smoothing estimate one has

	$\displaystyle\\|\mathscr{F}^{-1}_{\xi}\Lambda_{2}\\|_{L^{\infty}_{x}L^{2}_{t}}$	$\displaystyle\lesssim N^{-1}\sum_{k=0}^{\infty}\frac{1}{k!}\Bigg{\\|}\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}\frac{\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})}{\xi^{2}+\lambda\xi_{1}-\lambda\xi_{2}}$
		$\displaystyle\qquad\qquad\cdot\xi^{k}(\xi_{2}-\xi_{2}^{0}-\xi_{1}+\xi_{1}^{0})^{k}\Bigg{\\|}_{L_{\xi}^{2}}$
		$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{C^{k}}{k!}N^{-1}\Bigg{\\|}\int_{\xi_{1}+\xi_{2}=\xi}\chi_{\|\xi\|\sim J_{1}\setminus J_{2},\xi_{j}\in I_{j}}\frac{\widehat{u_{0}}(\xi_{1})\overline{\widehat{v_{0}}}(-\xi_{2})}{\xi^{2}+\lambda\xi_{1}-\lambda\xi_{2}}\Bigg{\\|}_{L_{\xi}^{2}}$
		$\displaystyle\lesssim N^{-1}\\|u_{0}\\|_{L^{2}}\\|v_{0}\\|_{L^{2}}.$

We conclude the proof of (3.2). Then

	$\displaystyle\\|N^{1/4}P_{N}F[P_{\sim N^{2}/\lambda}u_{0}]\\|_{l^{2}_{N\gg 1}L_{x}^{\infty}L_{t}^{2}}$	$\displaystyle\lesssim\lambda^{-1/2}\\|N^{1/4}N^{-1}\\|P_{\sim N^{2}/\lambda}u_{0}\\|_{L^{2}}^{2}\\|_{l^{2}_{N}}$
		$\displaystyle\lesssim\lambda^{-7/8}\\|u_{0}\\|_{H^{-3/16}}^{2}.$

Combining the nonresonant and low frequency cases, we finish the proof. ∎

Recall the definition for the norm $Z$ by (2.3). We have:

Corollary 3.6.

Combining Lemmas 3.1, 3.2, and 3.5 then

\|F[u_{0}]\|_{Z}\lesssim\lambda^{-1}\|u_{0}\|_{H^{-3/16}}^{2}.

4 Bilinear estimates for KdV equation

We mainly use the argument in [10], §5.3.

Lemma 4.1.

Let $N_{1},N_{2}\geq 2$ , $N_{1}\sim N_{2}$ . Then

\displaystyle\|\eta P_{\lesssim 1}\mathscr{B}(P_{N_{1}}uP_{N_{2}}v)\|_{L_{x}^{2}L_{t}^{\infty}}\lesssim N_{1}^{-3/2}\|u\|_{V^{2}_{K}}\|v\|_{V^{2}_{K}}.

Proof.

If $N_{1}\lesssim 1$ we have

	$\displaystyle\\|\eta P_{\lesssim 1}\mathscr{B}(P_{N_{1}}uP_{N_{2}}v)\\|_{L_{x}^{2}L_{t}^{\infty}}$	$\displaystyle\lesssim\\|\tilde{\eta}P_{N_{1}}uP_{N_{2}}v\\|_{L_{t}^{1}L_{x}^{2}}$
		$\displaystyle\lesssim\\|P_{N_{1}}u\\|_{L_{t}^{\infty}L^{4}_{x}}\\|P_{N_{2}}u\\|_{L^{\infty}_{t}L^{4}_{x}}$
		$\displaystyle\lesssim N_{1}^{-3/2}\\|u\\|_{L^{\infty}_{t}L_{x}^{2}}\\|v\\|_{L^{\infty}_{t}L^{2}_{x}}.$

Thus we consider the case $N_{1}\sim N_{2}\gg 1$ . Let $L_{\min}=\min\{L_{1},L_{2}\},L_{\max}=\max\{L_{1},L_{2}\}$ . We decompose $P_{\lesssim 1}$ into $\sum_{K\lesssim 1}P_{K}$ . By maximal function estimate, Lemma 5.7, (5.37) and (5.38) in [10] one has

	$\displaystyle\quad\sum_{K\lesssim N_{1}^{-2/3}}\\|\eta P_{K}(\mathscr{B}(P_{N_{1}}uP_{N_{2}}v))\\|_{L_{x}^{2}L_{t}^{\infty}}$
	$\displaystyle\lesssim\sum_{K\lesssim N_{1}^{-2/3}}\sum_{L_{1},L_{2},L}KL^{-1/2}\\|Q_{L_{1}}^{K}P_{N_{1}}uQ_{L_{2}}^{K}P_{N_{2}}v\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim\sum_{K\lesssim N_{1}^{-2/3}}K\sum_{L_{1},L_{2},L}L^{-1/2}\min\{L_{\min}^{1/2}K^{1/2},L_{\min}^{1/2}L^{1/2}N_{1}^{-1}\}$
	$\displaystyle\hskip 80.0pt\cdot\\|P_{N_{1}}Q_{L_{1}}^{K}u\\|_{L^{2}_{t,x}}\\|P_{N_{2}}Q_{L_{2}}^{K}v\\|_{L_{t,x}^{2}}$
	$\displaystyle\lesssim\sum_{K\lesssim N_{1}^{-2/3}}K\sum_{L_{1},L_{2},L}(LL_{\max})^{-1/2}\min\{K^{1/2},L^{1/2}N_{1}^{-1}\}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim\sum_{K\lesssim N_{1}^{-2/3}}K\sum_{L}L^{-1/2}\min\{K^{1/2},L^{1/2}N_{1}^{-1}\}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim N_{1}^{-3/2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}.$

If $K\geq N_{1}^{-2/3}$ , we firstly manipulate the case $L_{\max}\geq N_{1}^{5/4}$ . Similar to the former argument one has

		$\displaystyle\quad\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}\geq N_{1}^{5/4}}\\|\eta P_{K}(\mathscr{B}(P_{N_{1}}Q^{K}_{L_{1}}uP_{N_{2}}Q^{K}_{L_{2}}v))\\|_{L_{x}^{2}L_{t}^{\infty}}$		(4.1)
		$\displaystyle\lesssim\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}\geq N_{1}^{5/4},L}K(LL_{\max})^{-1/2}\min\{K^{1/2},L^{1/2}N_{1}^{-1}\}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim\sum_{L_{\max}\geq N_{1}^{5/4}}L_{\max}^{-1/2}N_{1}^{-1}\log(N_{1})\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim N_{1}^{-3/2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}.$

Thus we only need to estimate $L_{\max}<N_{1}^{5/4}$ for $N_{1}^{-2/3}\leq K\lesssim 1$ . Let

	$\displaystyle f_{N_{1},L_{1}}(\tau,\xi)$	$\displaystyle=\mathscr{F}_{t,x}(P_{N_{1}}Q^{K}_{L_{1}}u)(\tau+\xi^{3},\xi),$
	$\displaystyle g_{N_{2},L_{2}}(\tau,\xi)$	$\displaystyle=\mathscr{F}_{t,x}(P_{N_{2}}Q^{K}_{L_{2}}u)(\tau+\xi^{3},\xi).$

Then

	$\displaystyle\quad\mathscr{F}_{x}\left(\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}P_{K}(\eta\mathscr{B}(P_{N_{1}}Q^{K}_{L_{1}}uP_{N_{2}}Q^{K}_{L_{2}}v))\right)$
	$\displaystyle\sim\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\eta(t)\int_{0}^{t}\psi(\xi/K)\xi e^{i(t-s)\xi^{3}}$
	$\displaystyle\quad\cdot\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\int_{\xi=\xi_{1}+\xi_{2}}f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2})e^{it(\tau_{1}+\xi_{1}^{3})+it(\tau_{2}+\xi_{2}^{3})}$
	$\displaystyle\sim\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\eta(t)\psi(\xi/K)\xi\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\int_{\xi=\xi_{1}+\xi_{2}}$
	$\displaystyle\quad\cdot\frac{e^{it(\tau_{1}+\tau_{2}+\xi_{1}^{3}+\xi_{2}^{3})}-e^{it\xi^{3}}}{\tau_{1}+\tau_{2}+\xi_{1}^{3}+\xi_{2}^{3}-\xi^{3}}f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2})$
	$\displaystyle:=\mathscr{F}_{x}I-\mathscr{F}_{x}II.$

Note that on the support set of $f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2})$ one has $|\tau_{1}+\tau_{2}+\xi_{1}^{3}+\xi_{2}^{3}-\xi^{3}|\sim N_{1}^{2}|\xi|$ since $L_{\max}<N_{1}^{5/4}$ , $|\xi|\sim K\geq N_{1}^{-2/3}$ . By the maximal function estimate and the Plancherel identity one has

	$\displaystyle\quad\\|II\\|_{L_{x}^{2}L_{t}^{\infty}}$
	$\displaystyle\sim\left\\|\int_{\begin{subarray}{c}\xi=\xi_{1}+\xi_{2},\\ \tau_{1},\tau_{2}\end{subarray}}\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\frac{\psi(\xi/K)\xi f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2})}{\tau_{1}+\tau_{2}+\xi_{1}^{3}+\xi_{2}^{3}-\xi^{3}}\right\\|_{L^{2}_{\xi}}$
	$\displaystyle\lesssim N_{1}^{-2}\int_{\mathbb{R}_{\tau_{1},\tau_{2}}^{2}}\sum_{L_{\max}<N_{1}^{5/4}}\\|f_{N_{1},L_{1}}(\tau_{1},\cdot)\\|_{L^{2}}\\|g_{N_{2},L_{2}}(\tau_{2},\cdot)\\|_{L^{2}}$
	$\displaystyle\lesssim N_{1}^{-2}\sum_{L_{\max}<N_{1}^{5/4}}L_{1}^{1/2}L_{2}^{1/2}\\|f_{N_{1},L_{1}}(\tau,\xi)\\|_{L^{2}_{\tau,\xi}}\\|f_{N_{2},L_{2}}\\|_{L^{2}_{\tau,\xi}}$
	$\displaystyle\lesssim N_{1}^{-2}\sum_{L_{\max}<N_{1}^{5/4}}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}\lesssim N_{1}^{-3/2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}.$

Define

	$\displaystyle\mathscr{F}_{x}I^{\prime}$	$\displaystyle:=\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\eta(t)\psi(\xi/K)\xi\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\int_{\xi=\xi_{1}+\xi_{2}}$
		$\displaystyle\hskip 80.0pt\frac{e^{it(\tau_{1}+\tau_{2}+\xi_{1}^{3}+\xi_{2}^{3})}}{\xi_{1}^{3}+\xi_{2}^{3}-\xi^{3}}f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2})$
		$\displaystyle=\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\eta(t)\psi(\xi/K)\int_{\xi=\xi_{1}+\xi_{2}}$
		$\displaystyle\hskip 80.0pt\frac{1}{-3\xi_{1}\xi_{2}}\mathscr{F}_{x}(P_{N_{1}}Q^{K}_{L_{1}}u)(t,\xi_{1})\mathscr{F}_{x}(P_{N_{2}}Q^{K}_{L_{2}}v)(t,\xi_{2}).$

Let $f:=\sum_{L_{1}<N_{1}^{5/4}}f_{N_{1},L_{2}}$ , $g:=\sum_{L_{2}<N_{1}^{5/4}}g_{N_{2},L_{2}}$ . Then

	$\displaystyle\mathscr{F}_{x}I-\mathscr{F}_{x}I^{\prime}$	$\displaystyle=\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\sum_{k=1}^{\infty}\eta(t)\psi(\xi/K)\xi\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\int_{\xi=\xi_{1}+\xi_{2}}$
		$\displaystyle\quad\frac{(-1)^{k}(\tau_{1}+\tau_{2})^{k}e^{it(\tau_{1}+\tau_{2}+\xi_{1}^{3}+\xi_{2}^{3})}}{(\xi_{1}^{3}+\xi_{2}^{3}-\xi^{3})^{k+1}}f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2}).$

Thus by the Strichartz estimate [8] one has

	$\displaystyle\quad\\|I-I^{\prime}\\|_{L_{x}^{2}L_{t}^{\infty}}$
	$\displaystyle\lesssim\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\sum_{k=1}^{\infty}\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}\|\tau_{1}+\tau_{2}\|^{k}K^{-k}$
	$\displaystyle\cdot\left\\|\eta(t)\mathscr{F}^{-1}_{\xi}\int_{\xi=\xi_{1}+\xi_{2}}\left(\frac{e^{it(\xi_{1}^{3}+\xi_{2}^{3})}}{\xi_{1}^{k+1}\xi_{2}^{k+1}}f_{N_{1},L_{1}}(\tau_{1},\xi_{1})g_{N_{2},L_{2}}(\tau_{2},\xi_{2})\right)\right\\|_{L^{2}_{x}L_{t}^{\infty}}$
	$\displaystyle\lesssim\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\sum_{k=1}^{\infty}\sum_{N_{1}^{-2/3}\leq K\lesssim 1}\sum_{L_{\max}<N_{1}^{5/4}}N_{1}^{5k/4}K^{-k}$
	$\displaystyle\cdot\\|\mathscr{F}^{-1}_{\xi}(e^{it\xi^{3}}f_{N_{1},L_{1}}(\tau_{1},\xi)/\xi^{k+1})\\|_{L_{x}^{4}L_{t}^{\infty}}\\|\mathscr{F}^{-1}_{\xi}(e^{it\xi^{3}}f_{N_{2},L_{2}}(\tau_{2},\xi)/\xi^{k+1})\\|_{L_{x}^{4}L_{t}^{\infty}}$
	$\displaystyle\lesssim\int_{\mathbb{R}^{2}_{\tau_{1},\tau_{2}}}\sum_{k=1}^{\infty}\sum_{L_{\max}<N_{1}^{5/4}}N_{1}^{23k/12-2(k+1)+1/2}\\|f_{N_{1},L_{1}}(\tau_{1},\cdot)\\|_{L^{2}}\\|f_{N_{2},L_{2}}(\tau_{2},\cdot)\\|_{L^{2}}$
	$\displaystyle\lesssim\sum_{L_{\max}<N_{1}^{5/4}}N_{1}^{-37/12}(L_{1}L_{2})^{1/2}\\|Q_{L_{1}}^{K}P_{N_{1}}u\\|_{L^{2}_{t,x}}\\|Q_{L_{2}}^{K}P_{N_{2}}v\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim\sum_{L_{\max}<N_{1}^{5/4}}N_{1}^{-37/12}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}\lesssim N^{-3/2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}.$

Note that there exists constant $\tilde{c}$ such that

I^{\prime}=\tilde{c}\eta(t)P_{N^{-2/3}\leq\cdot\lesssim 1}((\partial_{x}^{-1}P_{N_{1}}Q^{K}_{<N_{1}^{5/4}}u)(\partial_{x}^{-1}P_{N_{1}}Q^{K}_{<N_{1}^{5/4}}v)).

By the Hölder inequality we have

	$\displaystyle\\|I^{\prime}\\|_{L_{x}^{2}L_{t}^{\infty}}$	$\displaystyle\lesssim\\|\partial_{x}^{-1}P_{N_{1}}Q^{K}_{<N_{1}^{5/4}}u\\|_{L_{x}^{4}L_{t}^{\infty}}\\|\partial_{x}^{-1}P_{N_{1}}Q^{K}_{<N_{1}^{5/4}}v\\|_{L_{x}^{4}L_{t}^{\infty}}$
		$\displaystyle\lesssim N^{-2}\\|P_{N_{1}}Q^{K}_{<N_{1}^{5/4}}u\\|_{L_{x}^{4}L_{t}^{\infty}}\\|P_{N_{2}}Q^{K}_{<N_{1}^{5/4}}v\\|_{L_{x}^{4}L_{t}^{\infty}}$
		$\displaystyle\lesssim N^{-3/2}\\|Q^{K}_{<N_{1}^{5/4}}u\\|_{U^{4}_{K}}\\|Q^{K}_{<N_{1}^{5/4}}v\\|_{U^{4}_{K}}$
		$\displaystyle\lesssim N^{-3/2}\\|Q^{K}_{<N_{1}^{5/4}}u\\|_{V^{2}_{K}}\\|Q^{K}_{<N_{1}^{5/4}}v\\|_{V^{2}_{K}}\lesssim N^{-3/2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}.$

We conclude the proof. ∎

Proposition 4.2.

$\|\eta\mathscr{B}(uv)\|_{Y}\lesssim\|u\|_{Y}\|v\|_{Y}$ .

Proof.

By Lemma 4.1 we have

	$\displaystyle\quad\\|\eta P_{1}\mathscr{B}(uv)\\|_{L_{x}^{2}L_{t}^{\infty}}$
	$\displaystyle\leq\\|\eta P_{1}\mathscr{B}(P_{\lesssim 1}uP_{\lesssim 1}v)\\|_{L_{x}^{2}L_{t}^{\infty}}+\sum_{N_{1}\sim N_{2}\gg 1}\\|\eta P_{1}\mathscr{B}(P_{N_{1}}uP_{N_{2}}v)\\|_{L_{x}^{2}L_{t}^{\infty}}$
	$\displaystyle\lesssim\\|\tilde{\eta}P_{\lesssim 1}uP_{\lesssim 1}v\\|_{L_{t}^{1}L_{x}^{2}}+\sum_{N_{1}\sim N_{2}\gg 1}N_{1}^{-3/2}\\|P_{N_{1}}u\\|_{V^{2}_{K}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim\\|P_{\lesssim 1}u\\|_{L_{t}^{\infty}L_{x}^{4}}\\|P_{\lesssim 1}v\\|_{L_{t}^{\infty}L_{x}^{4}}+\\|u\\|_{Z}\\|v\\|_{Z}$
	$\displaystyle\lesssim\\|u\\|_{Z}\\|v\\|_{Z}.$

Let $N\geq 2$ . By the triangle inequality we have

	$\displaystyle\quad\\|\eta P_{N}\mathscr{B}(uv)\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N_{1}\sim N}\\|\eta P_{N}\mathscr{B}(P_{N_{1}}uP_{\lesssim 1}v)\\|_{U^{2}_{K}}+\sum_{N_{2}\sim N}\\|\eta P_{N}\mathscr{B}(P_{\lesssim 1}uP_{N_{2}}v)\\|_{U^{2}_{K}}$
	$\displaystyle\quad+\sum_{N_{1},N_{2}\gg 1}\\|\eta P_{N}\mathscr{B}(P_{N_{1}}uP_{N_{2}}v)\\|_{U^{2}_{K}}:=I_{N}+I^{\prime}_{N}+II_{N}.$

For the term $I$ we use Proposition 5.12 in [10].

	$\displaystyle\\|I_{N}\\|_{U^{2}_{K}}$	$\displaystyle\lesssim\sum_{N_{1}\sim N}\\|\tilde{\eta}^{2}\partial_{x}(P_{N_{1}}uP_{\lesssim 1}v)\\|_{L^{2}_{t,x}}$
		$\displaystyle\lesssim\sum_{N_{1}\sim N}N\\|\tilde{\eta}P_{N_{1}}u\\|_{L_{x}^{\infty}L_{t}^{2}}\\|\tilde{\eta}P_{\lesssim 1}v\\|_{L_{x}^{2}L_{t}^{\infty}}.$

Then by the local smoothing estimate one has

	$\displaystyle\\|N^{-3/4}I_{N}\\|_{l^{2}_{N\geq 2}U^{2}_{K}}$	$\displaystyle\lesssim\\|\tilde{\eta}N^{1/4}P_{N}u\\|_{l^{2}_{N}L_{x}^{\infty}L_{t}^{2}}\\|\tilde{\eta}P_{\lesssim 1}v\\|_{L_{x}^{2}L_{t}^{\infty}}$
		$\displaystyle\lesssim\\|u\\|_{Z}\\|v\\|_{Z}$

By symmetry we control $I^{\prime}_{N}$ similarly.

For $II_{N}$ we decompose the summation into four parts.

	$\displaystyle II_{N}$	$\displaystyle=\sum_{N_{1}\sim N_{2}\sim N\gg 1}+\sum_{N_{1}\sim N\gg N_{2}\gg 1}+\sum_{N_{2}\sim N\gg N_{1}\gg 1}+\sum_{N_{1}\sim N_{2}\gg N}\cdots$
		$\displaystyle:=II_{N}^{(1)}+II_{N}^{(2)}+II_{N}^{(3)}+II_{N}^{(4)}.$

For $N_{1}\sim N_{2}\sim N\gg 1$ we show

\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}uP_{N_{2}}v)P_{N}\bar{w}~{}dxdt\right|\lesssim N^{-3/4}\|u\|_{V^{2}_{K}}\|v\|_{V^{2}_{K}}\|w\|_{V^{2}_{K}}

where $u,v,w$ are supported on $[-1,1]\times\mathbb{R}$ . Let $L_{\min},L_{\mathrm{med}},L_{\max}$ be the maximum, medium and minimum among $L_{1},L_{2},L$ . By Lemma 5.7 in [10] we have

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}uP_{N_{2}}v)P_{N}\bar{w}~{}dxdt\right\|$
	$\displaystyle\leq\sum_{L_{\max}\gtrsim cN^{3}}\left\|\int_{\mathbb{R}^{2}}\partial_{x}(Q_{L_{1}}^{K}P_{N_{1}}uQ^{K}_{L_{2}}P_{N_{2}}v)\overline{P_{N}Q_{L}^{K}w}~{}dxdt\right\|$
	$\displaystyle\lesssim\sum_{L_{\max}\gtrsim cN^{3}}NL_{\min}^{1/2}L_{\mathrm{med}}^{1/4}N^{-1/4}\\|Q_{L_{1}}^{K}P_{N_{1}}u\\|_{L_{t,x}^{2}}\\|Q_{L_{2}}^{K}P_{N_{2}}v\\|_{L_{t,x}^{2}}\\|Q_{L}^{K}P_{N}w\\|_{L_{t,x}^{2}}$
	$\displaystyle\lesssim\sum_{L_{\max}\gtrsim cN^{3}}L_{\min}^{1/2}L_{\mathrm{med}}^{1/4}N^{3/4}(L_{1}L_{2}L)^{-1/2}\\|u\\|_{V_{K}^{2}}\\|v\\|_{V_{K}^{2}}\\|w\\|_{V_{K}^{2}}$
	$\displaystyle\lesssim N^{-3/4}\\|u\\|_{V_{K}^{2}}\\|v\\|_{V_{K}^{2}}\\|w\\|_{V_{K}^{2}}$

Then by the duality we have

	$\displaystyle\\|N^{-3/4}II_{N}^{(1)}\\|_{l^{2}_{N\geq 2}U^{2}_{K}}$	$\displaystyle\lesssim\left\\|N^{-3/2}\sum_{N_{1}\sim N_{2}\sim N\gg 1}\\|P_{N_{1}}u\\|_{V^{2}_{K}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}\right\\|_{l^{2}_{N\geq 2}}$
		$\displaystyle\lesssim\\|N^{-3/4}P_{N}u\\|_{l^{2}_{N\geq 2}V^{2}_{K}}\\|N^{-3/4}P_{N}v\\|_{l^{2}_{N\geq 2}V^{2}_{K}}$
		$\displaystyle\lesssim\\|u\\|_{Z}\\|v\\|_{Z}.$

For $N_{1}\sim N\gg N_{2}\gg 1$ we show

\displaystyle\left|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}uP_{N_{2}}v)P_{N}\bar{w}~{}dxdt\right|\lesssim N^{-1/2}N_{2}^{1/4}\|u\|_{V^{2}_{K}}\|v\|_{V^{2}_{K}}\|w\|_{V^{2}_{K}}.

In fact by Lemma 5.7 (5.38) in [10] we have

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}uP_{N_{2}}v)P_{N}\bar{w}~{}dxdt\right\|$
	$\displaystyle\leq\sum_{L_{1},L_{2},L}\left\|\int_{\mathbb{R}^{2}}\partial_{x}(Q_{L_{1}}^{K}P_{N_{1}}uQ^{K}_{L_{2}}P_{N_{2}}v)\overline{P_{N}Q_{L}^{K}w}~{}dxdt\right\|$
	$\displaystyle\lesssim\sum_{L_{1},L_{2},L}N(L_{1}L_{2}L)^{1/3}N^{-1/2}(N^{2}N_{2})^{-1/2}$
	$\displaystyle\quad\cdot\\|Q_{L_{1}}^{K}P_{N_{1}}u\\|_{L_{t,x}^{2}}\\|Q_{L_{2}}^{K}P_{N_{2}}v\\|_{L_{t,x}^{2}}\\|Q_{L}^{K}P_{N}w\\|_{L_{t,x}^{2}}$
	$\displaystyle\lesssim\sum_{L_{1},L_{2},L}(L_{1}L_{2}L)^{1/3}N^{-1/2}N_{2}^{-1/2}(L_{1}L_{2}L)^{-1/2}\\|u\\|_{V_{K}^{2}}\\|v\\|_{V_{K}^{2}}\\|w\\|_{V_{K}^{2}}$
	$\displaystyle\lesssim N^{-1/2}N_{2}^{-1/2}\\|u\\|_{V_{K}^{2}}\\|v\\|_{V_{K}^{2}}\\|w\\|_{V_{K}^{2}}.$

Then by the duality we have

	$\displaystyle\\|N^{-3/4}II_{N}^{(2)}\\|_{l^{2}_{N\geq 2}U^{2}_{K}}$	$\displaystyle\lesssim\sum_{N_{1}\sim N\gg N_{2}\gg 1}N^{-5/4}N_{2}^{-1/2}\\|P_{N_{1}}u\\|_{V^{2}_{K}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim\\|N^{-3/4}P_{N}u\\|_{l^{2}_{N\geq 2}V^{2}_{K}}\\|N^{-3/4}P_{N}v\\|_{l^{2}_{N\geq 2}V^{2}_{K}}$
		$\displaystyle\lesssim\\|u\\|_{Z}\\|v\\|_{Z}.$

By the symmetry we can control $II_{N}^{(3)}$ similarly.

Now, we estimate the term $II_{N}^{(4)}$ . Let $N_{1}\sim N_{2}\gg N\geq 2$ , $L=cN_{1}^{2}N$ . Firstly we estimate

\displaystyle\|\eta P_{N}\mathscr{B}(Q_{\leq L}^{K}(P_{N_{1}}uP_{N_{2}}v))\|_{U^{2}_{K}}.

Typically we control

\displaystyle\int_{\mathbb{R}^{2}}\partial_{x}(Q_{>L}^{K}P_{N_{1}}uP_{N_{2}}v)\overline{Q_{\leq L}^{K}P_{N}w}~{}dxdt.

By high modulation and transversal estimates, we have

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(Q_{>L}^{K}P_{N_{1}}uP_{N_{2}}v)\overline{Q_{\leq L}P_{N}w}~{}dxdt\right\|$
	$\displaystyle\lesssim\\|Q_{>L}^{K}P_{N_{1}}u\\|_{L_{t,x}^{2}}\\|P_{N_{2}}v\partial_{x}Q^{K}_{\leq L}P_{N}w\\|_{L_{t,x}^{2}}$
	$\displaystyle\lesssim L^{-1/2}\\|u\\|_{V^{2}_{K}}N_{2}^{-1}N\\|v\\|_{U^{2}_{K}}\\|w\\|_{U^{2}_{K}}$
	$\displaystyle\sim N_{1}^{-2}N^{1/2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{U^{2}_{K}}\\|w\\|_{U^{2}_{K}}.$

On the other hand by the Strichartz estimate we have

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(Q^{K}_{>L}P_{N_{1}}uP_{N_{2}}v)\overline{Q^{K}_{\leq L}P_{N}w}~{}dxdt\right\|$
	$\displaystyle\lesssim\\|Q^{K}_{>L}P_{N_{1}}u\\|_{L_{t,x}^{4}}\\|P_{N_{2}}v\\|_{L_{t,x}^{4}}\\|\partial_{x}Q^{K}_{\leq L}P_{N}w\\|_{L_{t,x}^{2}}$
	$\displaystyle\lesssim\\|Q_{>L}^{K}P_{N_{1}}u\\|_{L_{t,x}^{2}}^{1/4}\\|Q_{>L}^{K}P_{N_{1}}u\\|_{L_{t,x}^{6}}^{3/4}\\|P_{N_{2}}v\\|_{L_{t,x}^{2}}^{1/4}\\|P_{N_{2}}v\\|_{L_{t,x}^{6}}^{3/4}N\\|w\\|_{U^{6}_{K}}$
	$\displaystyle\sim N_{1}^{-1/4}N\\|u\\|_{V^{2}_{K}}\\|v\\|_{U^{6}_{K}}\\|w\\|_{U^{6}_{K}}.$

Then, by the interpolation [7], one has

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(Q_{>L}^{K}P_{N_{1}}uP_{N_{2}}v)\overline{Q^{K}_{\leq L}P_{N}w}~{}dxdt\right\|$
	$\displaystyle\lesssim N_{1}^{-2}N^{1/2}(\log N_{1})^{2}\\|u\\|_{V^{2}_{K}}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}.$

Thus,

	$\displaystyle\quad\sum_{N_{1}\sim N_{2}\gg N\geq 2}N^{-3/4}\\|\eta P_{N}\mathscr{B}(Q^{K}_{\leq L}(P_{N_{1}}uP_{N_{2}}v))\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}\gg N\geq 2}N^{-3/4}N_{1}^{-2}N^{1/2}(\log N_{1})^{2}\\|P_{N_{1}}u\\|_{V^{2}_{K}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim\\|u\\|_{Z}\\|v\\|_{Z}.$

Secondly we estimate $\|\eta P_{N}\mathscr{B}(Q^{K}_{\leq L}(P_{N_{1}}uP_{N_{2}}v))\|_{U^{2}_{K}}$ . By high modulation and transversal estimates, we have

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}\partial_{x}(P_{N_{1}}uP_{N_{2}}v)\overline{Q^{K}_{>L}P_{N}w}~{}dxdt\right\|$
	$\displaystyle\lesssim\\|P_{N}(P_{N_{1}}uP_{N_{2}}v)\\|_{L_{t,x}^{2}}\\|\partial_{x}Q^{K}_{>L}P_{N}w\\|_{L_{t,x}^{2}}$
	$\displaystyle\lesssim(NN_{1})^{-1/2}\\|u\\|_{U^{2}_{K}}\\|v\\|_{U^{2}_{K}}L^{-1/2}N\\|w\\|_{V^{2}_{K}}$
	$\displaystyle\sim N_{1}^{-3/2}\\|u\\|_{U^{2}_{K}}\\|v\\|_{U^{2}_{K}}\\|w\\|_{V^{2}_{K}}.$

Then by the duality one has

	$\displaystyle\quad\sum_{N_{1}\sim N_{2}\gg N\geq 2}\\|N^{-3/4}\eta P_{N}\mathscr{B}(Q^{K}_{\leq L}(P_{N_{1}}uP_{N_{2}}v))\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\sum_{N_{1}\sim N_{2}\gg N\geq 2}N^{-3/4}N_{1}^{-3/2}\\|P_{N_{1}}u\\|_{U^{2}_{K}}\\|P_{N_{2}}v\\|_{U^{2}_{K}}$
	$\displaystyle\lesssim\\|u\\|_{Y}\\|v\\|_{Y}.$

We finish the proof of this proposition. ∎

In the proof of Proposition 4.2, we control many parts by $\|u\|_{Z}\|v\|_{Z}$ except $II_{N}^{(4)}$ . Fortunately, we have the following lemma which can help us to control this case when $v=F[u_{0}]$ .

Lemma 4.3.

$2\leq N\ll N_{1}\sim N_{2}$ .

	$\displaystyle\\|\eta P_{N}\mathscr{B}(P_{N_{1}}vP_{N_{2}}F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\\|_{U^{2}_{K}}\lesssim(\lambda N_{1}^{3}/N)^{-1/2}\\|v\\|_{U^{2}_{K}}\\|u_{0}\\|_{L^{2}}^{2},$
	$\displaystyle\\|\eta P_{N}\mathscr{B}(P_{N_{1}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\\|_{U^{2}_{K}}\lesssim(\lambda^{2}N_{1}^{3}/N)^{-1/2}\\|u_{0}\\|_{L^{2}}^{4}.$

Proof.

By the duality, we only need to show

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\partial_{x}P_{N}w~{}dxdt\right\|$
	$\displaystyle\lesssim(\lambda N_{1}^{3}/N)^{-1/2}\\|v\\|_{U^{2}_{K}}\\|u_{0}\\|_{L^{2}}^{2}\\|w\\|_{V^{2}_{K}}$

and

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\partial_{x}P_{N}w~{}dxdt\right\|$
	$\displaystyle\lesssim(\lambda^{2}N_{1}^{3}/N)^{-1/2}\\|u_{0}\\|_{L^{2}}^{4}\\|w\\|_{V^{2}_{K}}$

where $v,w$ are supported on $[-1,1]\times\mathbb{R}$ . Let $L=cNN_{1}N_{2}$ . Firstly, we control

\displaystyle\left|\int_{\mathbb{R}^{2}}P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])Q_{\leq L}^{K}\partial_{x}P_{N}w~{}dxdt\right|.

Since for $|\xi_{1}|\sim N_{1}$ , $|\xi_{1}|\sim N_{2}$ , $|\xi|\sim N$ , $\xi_{1}+\xi_{2}=\xi$ , one has $|\xi_{1}^{3}+\xi_{2}^{3}-(\xi_{1}+\xi_{2})^{3}|\gtrsim L$ . Typically, we need to control

\left|\int_{\mathbb{R}^{2}}P_{N_{1}}vQ^{K}_{>L}P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])Q_{\leq L}^{K}\partial_{x}P_{N}w~{}dxdt\right|.

By transversal and high modulation estimates we can control this term by

	$\displaystyle\quad\\|P_{N_{1}}vQ_{\leq L}^{K}\partial_{x}P_{N}w\\|_{L^{2}_{t,x}}\\|Q^{K}_{>L}P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim N_{1}^{-1}N\\|v\\|_{U^{2}_{K}}\\|w\\|_{U^{2}_{K}}L^{-1/2}\\|P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\\|_{V^{2}_{K}}$
	$\displaystyle\sim N_{1}^{-2}N^{1/2}\\|v\\|_{U^{2}_{K}}\\|w\\|_{U^{2}_{K}}\\|P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\\|_{V^{2}_{K}}.$

Recall that $v$ is supported on $[-1,1]\times\mathbb{R}$ . Thus for any $p\geq 1$ , one has $\|v\|_{L_{t,x}^{2}}\lesssim\|v\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim\|v\|_{U^{p}}$ . By the Hölder inequality and Strichartz estimates, we can also control this term by

	$\displaystyle\quad\\|P_{N_{1}}v\\|_{L_{t,x}^{3}}\\|Q_{\leq L}^{K}\partial_{x}P_{N}w\\|_{L_{t,x}^{6}}\\|Q_{>L}^{K}P_{N_{2}}(F[P_{\sim N_{2}^{2}}u_{0}])\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim\\|P_{N_{1}}v\\|_{L^{2}_{t,x}}^{1/2}\\|P_{N_{1}}v\\|_{L^{6}_{t,x}}^{1/2}N\\|P_{N}w\\|_{U^{6}_{K}}L^{-1/2}\\|P_{N_{2}}(F[P_{\sim N_{2}^{2}}u_{0}])\\|_{V^{2}_{K}}$
	$\displaystyle\sim N_{1}^{-13/12}N^{1/2}\\|v\\|_{U^{6}_{K}}\\|w\\|_{U^{6}_{K}}\\|P_{N_{2}}(F[P_{\sim N_{2}^{2}}u_{0}])\\|_{V^{2}_{K}}.$

Then, by the interpolation [7] and Lemma 3.2, one has

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])Q_{\leq L}^{K}\partial_{x}P_{N}w~{}dxdt\right\|$
	$\displaystyle\lesssim N_{1}^{-2}N^{1/2}(\log N_{1})^{2}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}\\|P_{N_{2}}(F[P_{\sim N_{2}^{2}}u_{0}])\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim\lambda^{-3/8}N_{1}^{-2}N^{1/2}(\log N_{1})^{2}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}\\|P_{N_{2}}(F[P_{\sim N_{2}^{2}}u_{0}])\\|_{V^{2}_{K}}.$

Similarly, one has

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])Q_{\leq L}^{K}\partial_{x}P_{N}w~{}dxdt\right\|$
	$\displaystyle\lesssim N_{1}^{-2}N^{1/2}(\log N_{1})^{2}\\|P_{N_{1}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])\\|_{V^{2}_{K}}\\|P_{N_{2}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}$
	$\displaystyle\lesssim\lambda^{-3/4}N_{1}^{-2}N^{1/2}(\log N_{1})^{2}\\|u_{0}\\|_{L^{2}}^{4}\\|w\\|_{V^{2}_{K}}.$

For the other part, by the Hölder inequality,

	$\displaystyle\quad\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}])Q_{>L}^{K}\partial_{x}P_{N}w~{}dxdt\right\|$
	$\displaystyle\lesssim\\|P_{N}(P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\\|_{L^{2}_{t,x}}\\|Q_{>L}^{K}\partial_{x}P_{N}w\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim NL^{-1/2}\\|P_{N}(P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\\|_{L^{2}_{t,x}}\\|w\\|_{V^{2}_{K}}.$

To conclude the proof of this lemma, we show

	$\displaystyle\\|P_{N}(P_{N_{1}}vP_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\\|_{L^{2}_{t,x}}\lesssim(\lambda NN_{1})^{-1/2}\\|v\\|_{U^{2}_{K}}\\|u_{0}\\|_{L^{2}}^{2},$
	$\displaystyle\\|P_{N}(P_{N_{1}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\\|_{L^{2}_{t,x}}\lesssim(NN_{1})^{-1/2}\\|u_{0}\\|_{L^{2}}^{4}.$

To show the first one, we only need to show

\|P_{N}(P_{N_{1}}K(t)v_{0}P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\|_{L^{2}_{t,x}}\lesssim(\lambda NN_{1})^{-1/2}\|v_{0}\|_{L^{2}}\|u_{0}\|_{L^{2}}^{2}.

Let $I_{1},I_{2}$ be intervals included in $[-CN_{2}^{2}/\lambda,-cN_{2}^{2}/\lambda]\cup[cN_{2}^{2}/\lambda,CN_{2}^{2}/\lambda]$ with length $cN_{2}$ . Also, we can assume that for any $\eta_{1}\in I_{1}$ , $\eta_{2}\in I_{2}$ one has $|\eta_{1}-\eta_{2}|\sim N_{2}$ . It is equivalent to

	$\displaystyle\quad\left\\|P_{N}\left(P_{N_{1}}K(t)v_{0}\tilde{\eta}P_{N_{2}}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right)\right\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim(\lambda NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}.$

There exists a interval $J$ included in $[-CN_{2},-cN_{2}]\cup[cN_{2},CN_{2}]$ with length $1$ such that one has $|\xi_{2}^{3}+\lambda\eta_{1}^{2}-\lambda\eta_{2}^{2}|\gtrsim N_{2}^{2}$ for any $\xi_{2}\in[-CN_{2},-cN_{2}]\cup[cN_{2},CN_{2}]\setminus J:=\hat{J}$ , $\eta_{1}\in I_{1},\eta_{2}\in I_{2},\eta_{1}+\eta_{2}=\xi_{2}$ . Then by Lemma 3.3 and

\|P_{N}(P_{N_{1}}K(t)v_{0}P_{N_{2}}K(t)v_{1})\|_{L^{2}_{t,x}}\lesssim(NN_{1})^{-1/2}\|v_{0}\|_{L^{2}}\|v_{1}\|_{L^{2}},

(4.2)

we can control this part. Thus we only need to show

	$\displaystyle\quad\left\\|P_{N}\left(P_{N_{1}}K(t)v_{0}\tilde{\eta}P_{J}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right)\right\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim(\lambda NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}.$

By the orthogonality, we can assume that $I_{1},I_{2}$ are intervals with length $1$ .

There exists a interval $J_{1}$ included in $J$ with length at most $1/N_{2}$ such that one has $|\xi_{2}^{3}+\lambda\eta_{1}^{2}-\lambda\eta_{2}^{2}|\gtrsim N_{2}$ for any $\eta\in J\setminus J_{1}$ , $\eta_{1}\in I_{1},\eta_{2}\in I_{2},\eta_{1}+\eta_{2}=\eta$ . By Lemma 3.3 and the former argument we only need to show

	$\displaystyle\quad\left\\|P_{N}\left(P_{N_{1}}K(t)v_{0}\tilde{\eta}P_{J_{1}}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right)\right\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim(\lambda NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}$

where $I_{1},I_{2}$ are intervals with length $1/N_{2}$ .

There exists a interval $J_{2}$ included in $J_{1}$ with length $1/N_{2}^{2}$ such that one has $|\xi_{2}^{3}+\lambda\eta_{1}^{2}-\lambda\eta_{2}^{2}|\lesssim 1$ for any $\xi_{2}\in J_{2}$ , $\eta_{1}\in I_{1},\eta_{2}\in I_{2},\eta_{1}+\eta_{2}=\xi_{2}$ . By Lemma 3.4 and following the former argument we only need to show

	$\displaystyle\quad\left\\|P_{N}\left(P_{N_{1}}K(t)v_{0}\tilde{\eta}P_{J_{1}\setminus J_{2}}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right)\right\\|_{L^{2}_{t,x}}$
	$\displaystyle\lesssim(\lambda NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}$

where $I_{1},I_{2}$ are intervals with length $1/N_{2}$ . Note that

	$\displaystyle\quad\mathscr{F}_{x}\left(P_{N}\left(P_{N_{1}}K(t)v_{0}\tilde{\eta}P_{J_{1}\setminus J_{2}}\mathscr{B}(S_{\lambda}(t)P_{I_{1}}u_{0}\overline{S_{\lambda}(t)P_{I_{2}}u_{0}})\right)\right)$
	$\displaystyle\sim\int_{\begin{subarray}{c}\xi_{1}+\xi_{2}=\xi,\\ \eta_{1}+\eta_{2}=\xi_{2}\end{subarray}}M(\xi,\xi_{1},\eta_{1})\widehat{v_{0}}(\xi_{1})\widehat{u_{0}}(\eta_{1})\overline{\widehat{u_{0}}}(-\eta_{2})\tilde{\eta}(e^{it(\xi_{1}^{3}+\xi_{2}^{3})}-e^{it(\xi_{1}^{3}+\lambda\xi_{2}(\eta_{2}-\eta_{1}))})$
	$\displaystyle:=\Lambda_{1}+\Lambda_{2}.$

where $M(\xi,\xi_{1},\eta_{1})=\chi_{|\xi|\sim N,|\xi_{1}|\sim N_{1},\xi_{2}\in J_{1}\setminus J_{2},\eta_{1}\in I_{1},\eta_{2}\in I_{2}}/(\xi_{2}^{2}+\lambda\eta_{1}-\lambda\eta_{2})$ . Note that $M(\xi,\xi_{2},\eta_{1})\lesssim N_{2}$ . For $\Lambda_{1}$ , by (4.2) and the Plancherel identity we have

	$\displaystyle\\|\Lambda_{1}\\|_{L_{t}^{2}L_{\xi}^{2}}$	$\displaystyle\lesssim(NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}$
		$\displaystyle\quad\cdot\left\\|\int_{\eta_{1}+\eta_{2}=\xi_{2}}\frac{\widehat{u_{0}}(\eta_{1})\overline{\widehat{u_{0}}}(-\eta_{2})}{\xi_{2}^{2}+\lambda\eta_{1}-\lambda\eta_{2}}\chi_{\xi_{2}\in J_{1}\setminus J_{2},\eta_{1}\in I_{1},\eta_{2}\in I_{2}}\right\\|_{L^{2}_{\xi_{2}}}$
		$\displaystyle\lesssim(NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}\left\\|\frac{\chi_{\xi_{2}\in J_{1}\setminus J_{2},\eta_{1}\in I_{1},\eta_{2}\in I_{2}}}{\xi_{2}^{2}+2\lambda\eta_{1}-\lambda\xi_{2}}\right\\|_{L^{\infty}_{\eta_{1}}L^{2}_{\xi_{2}}}$
		$\displaystyle\lesssim(NN_{1})^{-1/2}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}.$

Let $\eta_{j}^{0}$ be the center of $I_{j}$ , $j=1,2$ . Note that

	$\displaystyle\partial_{\xi_{1}}(\xi_{1}^{3}+\lambda(\xi-\xi_{1})(\eta_{2}^{0}-\eta_{1}^{0}))$	$\displaystyle=3\xi_{1}^{2}-\lambda(\eta_{2}^{0}-\eta_{1}^{0})$
		$\displaystyle=3\xi_{1}^{2}-\xi_{2}^{2}+\xi_{2}^{2}-\lambda(\eta_{2}^{0}-\eta_{1}^{0})$
		$\displaystyle=2\xi_{1}^{2}+\xi(\xi_{1}-\xi_{2})+\xi_{2}^{2}-\lambda(\eta_{2}^{0}-\eta_{1}^{0})\sim N_{1}^{2}.$

Thus by the Taylor expansion and the transversal estimate one has

	$\displaystyle\quad\\|\Lambda_{2}\\|_{L_{t}^{2}L_{\xi}^{2}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\Bigg{\\|}\int_{\begin{subarray}{c}\xi_{1}+\xi_{2}=\xi,\\ \eta_{1}+\eta_{2}=\xi_{2}\end{subarray}}M(\xi,\xi_{1},\eta_{1})\widehat{v_{0}}(\xi_{1})\widehat{u_{0}}(\eta_{1})\overline{\widehat{u_{0}}}(-\eta_{2})e^{it(\xi_{1}^{3}+\lambda\xi_{2}(\eta_{2}^{0}-\eta_{1}^{0}))}$
	$\displaystyle\qquad\qquad\cdot\tilde{\eta}\frac{(i\lambda t\xi_{2}(\eta_{2}-\eta_{2}^{0}-\eta_{1}+\eta_{1}^{0}))^{k}}{k!}\Bigg{\\|}_{L_{t}^{2}L_{\xi}^{2}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{1}{k!}\Bigg{\\|}\int_{\begin{subarray}{c}\xi_{1}+\xi_{2}=\xi,\\ \eta_{1}+\eta_{2}=\xi_{2}\end{subarray}}M(\xi,\xi_{1},\eta_{1})\widehat{v_{0}}(\xi_{1})\widehat{u_{0}}(\eta_{1})\overline{\widehat{u_{0}}}(-\eta_{2})e^{it(\xi_{1}^{3}+\lambda\xi_{2}(\eta_{2}^{0}-\eta_{1}^{0}))}$
	$\displaystyle\qquad\qquad\cdot(\xi_{2}(\eta_{2}-\eta_{2}^{0}-\eta_{1}+\eta_{1}^{0}))^{k}\Bigg{\\|}_{L_{t}^{2}L_{\xi}^{2}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{1}{k!}N_{1}^{-1}\\|v_{0}\\|_{L^{2}}\Bigg{\\|}\int_{\eta_{1}+\eta_{2}=\xi_{2}}\frac{\widehat{u_{0}}(\eta_{1})\overline{\widehat{u_{0}}}(-\eta_{2})}{\xi_{2}^{2}+\lambda\eta_{1}-\lambda\eta_{2}}$
	$\displaystyle\qquad\qquad\cdot\chi_{\xi_{2}\in J_{1}\setminus J_{2},\eta_{1}\in I_{1},\eta_{2}\in I_{2}}(\xi_{2}(\eta_{2}-\eta_{2}^{0}-\eta_{1}+\eta_{1}^{0}))^{k}\Bigg{\\|}_{L_{\xi_{2}}^{2}}$
	$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{C^{k}}{k!}N_{1}^{-1}\\|v_{0}\\|_{L^{2}}\Bigg{\\|}\int_{\eta_{1}+\eta_{2}=\xi_{2}}\frac{\|\widehat{u_{0}}(\eta_{1})\widehat{u_{0}}(-\eta_{2})\|}{\xi_{2}^{2}+\lambda\eta_{1}-\lambda\eta_{2}}\chi_{\xi_{2}\in J_{1}\setminus J_{2},\eta_{1}\in I_{1},\eta_{2}\in I_{2}}\Bigg{\\|}_{L_{\xi_{2}}^{2}}$
	$\displaystyle\lesssim N_{1}^{-1}\\|v_{0}\\|_{L^{2}}\\|u_{0}\\|_{L^{2}}^{2}.$

Thus we conclude the proof of the first inequality.

For the second one, similar to the former argument, we reduce it to

	$\displaystyle\Bigg{\\|}\int_{\zeta_{1}+\zeta_{2}+\eta_{1}+\eta_{2}=\xi}\frac{\tilde{\eta}\chi_{\|\xi\|\sim N}M_{1}(\zeta_{1},\zeta_{2})M_{2}(\eta_{1},\eta_{2})\widehat{u_{0}}(\zeta_{1})\widehat{u_{0}}(\zeta_{2})\widehat{u_{0}}(\eta_{1})\widehat{u_{0}}(\eta_{2})}{((\zeta_{1}+\zeta_{2})^{2}+\lambda\zeta_{1}-\lambda\zeta_{2})((\eta_{1}+\eta_{2})^{2}+\lambda\eta_{1}-\lambda\eta_{2})}$
	$\displaystyle\qquad\cdot(e^{it(\zeta_{1}+\zeta_{2})^{2}}-e^{i\lambda t(\zeta_{1}+\zeta_{2})(\zeta_{2}-\zeta_{1})})(e^{it(\eta_{1}+\eta_{2})^{2}}-e^{i\lambda t(\eta_{1}+\eta_{2})(\eta_{2}-\eta_{1})})\Bigg{\\|}_{L_{t}^{2}L_{\xi}^{2}}$
	$\displaystyle\lesssim(NN_{1})^{-1/2}\\|u_{0}\\|_{L^{2}}^{4}$

where $M_{1}(\zeta_{1},\zeta_{2})=\chi_{\zeta_{1}+\zeta_{2}\in J,\zeta_{1}\in I_{1},\zeta_{2}\in I_{2}},M_{2}(\eta_{1},\eta_{2})=\chi_{\eta_{1}+\eta_{2}\in\tilde{J},\eta_{1}\in\tilde{I}_{1},\eta_{2}\in\tilde{I}_{2}}$ , $I_{1},I_{2},\tilde{I}_{1},\tilde{I}_{2}$ are intervals included in $[-CN_{1}^{2},-cN_{1}^{2}]\cup[cN_{1}^{2},CN_{1}^{2}]$ , $J,\tilde{J}$ are intervals included in $[-CN_{1},cN_{1}]\cup[cN_{1},CN_{1}]$ and

N_{1}^{-1}\leq|(\zeta_{1}+\zeta_{2})^{2}+\lambda\zeta_{1}-\lambda\zeta_{2}|,|(\eta_{1}+\eta_{2})^{2}+\lambda\eta_{1}-\lambda\eta_{2}|\ll N_{1}.

(4.3)

Recall the former argument. Let $\zeta_{1}+\zeta_{2}=\zeta,\eta_{1}+\eta_{2}=\eta$ , we only need to show

	$\displaystyle\Bigg{\\|}\int_{\zeta+\eta=\xi}\frac{\tilde{\eta}\chi_{\|\xi\|\sim N}M_{1}(\zeta_{1},\zeta_{2})M_{2}(\eta_{1},\eta_{2})\widehat{u_{0}}(\zeta_{1})\widehat{u_{0}}(\zeta_{2})\widehat{u_{0}}(\eta_{1})\widehat{u_{0}}(\eta_{2})}{(\zeta^{2}+\lambda\zeta_{1}-\lambda\zeta_{2})(\eta^{2}+\lambda\eta_{1}-\lambda\eta_{2})}$
	$\displaystyle\qquad\cdot e^{i\lambda t(\zeta(\zeta_{2}-\zeta_{1})+\eta(\eta_{2}-\eta_{1}))}\Bigg{\\|}_{L_{t}^{2}L_{\xi}^{2}}$
	$\displaystyle\lesssim(NN_{1})^{-1/2}\\|u_{0}\\|_{L^{2}}^{4}$

Let $\zeta_{j}^{0},\eta_{j}^{0}$ be the center of $I_{j},\tilde{I}_{j}$ respectively. Note that

	$\displaystyle\quad\partial_{\zeta}(\lambda\zeta(\zeta_{2}^{0}-\zeta_{1}^{0})+\lambda(\xi-\zeta)(\eta_{2}^{0}-\eta_{1}^{0}))$
	$\displaystyle=\lambda\zeta_{2}^{0}-\lambda\zeta_{1}^{0}-\lambda(\eta_{2}^{0}-\eta_{1}^{0})$
	$\displaystyle=\zeta^{2}-\eta^{2}+\lambda\zeta_{2}^{0}-\lambda\zeta_{1}^{0}-\zeta^{2}+\eta^{2}-\lambda(\eta_{2}^{0}-\eta_{1}^{0})$
	$\displaystyle=\xi(\zeta-\eta)-(\zeta^{2}+\lambda\zeta_{1}^{0}-\lambda\zeta_{2}^{0})+(\eta^{2}-\lambda\eta_{2}^{0}+\eta_{1}^{0}).$

By (4.3), $|\xi|\sim N,|\zeta-\eta|\sim N_{1}$ , we obtain

|\partial_{\zeta}(\lambda\zeta(\zeta_{2}^{0}-\zeta_{1}^{0})+\lambda(\xi-\zeta)(\eta_{2}^{0}-\eta_{1}^{0}))|\sim NN_{1}.

Following the same argument for the first inequality, by the Taylor expansion and the transversal estimate, we conclude the proof. ∎

Proposition 4.4.

$\|\eta\mathscr{B}(uF[u_{0}])\|_{Y}\lesssim\lambda^{-1}\|u\|_{Y}\|u_{0}\|_{H^{-3/16}}^{2}$ .

Proof.

By the proof of Proposition 4.2 and Lemma 3.6, we only need to control the following term

\sum_{N_{1}\sim N_{2}\gg N\geq 2}\|\eta P_{N}\mathscr{B}(P_{N_{1}}uP_{N_{2}}F[P_{\sim N_{2}^{2}/\lambda}u_{0}])\|_{U^{2}_{K}}.

By Lemma 4.3, we can control it by

	$\displaystyle\quad\sum_{N_{1}\sim N_{2}\gg N\geq 2}N^{-3/4}N_{1}^{-3/2}N^{1/2}\lambda^{-1/2}\\|P_{N_{1}}u\\|_{U^{2}_{K}}\\|P_{\sim N_{2}^{2}/\lambda}u_{0}\\|_{L^{2}}^{2}$
	$\displaystyle\lesssim\lambda^{-7/8}\\|u\\|_{Y}\\|u_{0}\\|_{H^{-3/16}}^{2}.$

Combining Corollary 3.6 and Proposition 4.2, we conclude the proof. ∎

Proposition 4.5.

$\|\eta\mathscr{B}(F[u_{0}]^{2})\|_{Y}\lesssim\lambda^{-2}\|u_{0}\|_{H^{-3/16}}^{4}$ .

Proof.

Similar to the argument for Proposition 4.4, we only need to control the term

\sum_{N_{1}\sim N_{2}\gg N\geq 2}N^{-3/4}\|\eta P_{N}\mathscr{B}(P_{N_{1}}(F[P_{\sim N_{1}^{2}/\lambda}u_{0}])P_{N_{2}}(F[P_{\sim N_{2}^{2}/\lambda}u_{0}]))\|_{U^{2}_{K}}.

By Lemma 4.3 we can control it by

\displaystyle\sum_{N_{1}\sim N_{2}\gg N\geq 2}N^{-3/4}N_{1}^{-3/2}N^{1/2}\lambda^{-1}\|P_{\sim N_{2}^{2}/\lambda}u_{0}\|_{L^{2}}^{4}\lesssim\lambda^{-7/4}\|u_{0}\|_{H^{-3/16}}^{4}.

By Corollary 3.6 and Proposition 4.2, we conclude the proof. ∎

Combining Lemmas 2.1–2.3, Corollary 3.6, Propositions 4.2, 4.4, and 4.5, one can construct the solution $(u,w)$ of (2.2). Let $v=w+F[u_{0}]$ . By rescaling we obtain the local solution $(u,v)\in C([0,T];H^{-3/16}\times H^{-3/4})$ of initial system (S-KdV) with $(s_{1},s_{2})=(-3/16,-3/4)$ . For general $(s_{1},s_{2})$ , see the argument in [2]. One can use the method in this paper to manipulate the case $s_{1},s_{2}<0$ with some modification. However there exists easier argument. For example if $s_{2}>-3/4$ , we do not need rescaling. One can use the norms

\|u\|_{X_{\lambda}^{s_{1}}}:=\|N^{s_{1}}P_{N}u\|_{l^{2}_{N}U^{2}_{S_{\lambda}}},\quad\|v\|_{Y^{s_{2}}}:=\|P_{1}v\|_{L_{x}^{2}L_{t}^{\infty}}+\|N^{s_{2}}P_{N}v\|_{l^{2}_{N\geq 2}V^{2}_{K}}.

Then we can show the contraction for the iteration of the initial integral equation. If $s_{2}=-3/4$ , $s_{1}>-3/16$ , we can show $\|\eta\mathscr{B}(\partial_{x}|u|^{2})\|_{Y}\lesssim\|u\|_{X_{\lambda}^{s_{1}}}$ . The argument is also much easier than the case $(s_{1},s_{2})=(-3/16,-3/4)$ . See also [11]. We omit the details.

Acknowledgments: The first author is supported in part by the NSFC, grant 12301116. The authors would like to thank Professors Boling Guo and Baoxiang Wang for their invaluable support and encouragement.

References

[1] K. Appert and J. Vaclavik, Dynamics of coupled solitons, Phys. Fluids 20 (1977), no. 11, 1845–1849.
[2] Y. Ban, J. Chen, and Y. Zhang, Local well-posedness for Schrödinger-KdV system in $H^{s_{1}}\times H^{s_{2}}$ , submitted (2024).
[3] D. Bekiranov, T. Ogawa, and G. Ponce, Weak solvability and well-posedness of a coupled Schrödinger-Korteweg-de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2907–2919.
[4] A. Corcho and F. Linares, Well-posedness for the Schrödinger-Korteweg-de Vries system, Tran. Amer. Math. Soc. 359 (2007), no. 9, 4089–4106.
[5] B. Guo and C. Miao, Well-posedness of the Cauchy problem for the coupled system of the Schrödinger-KdV equations, Acta Math. Sin. 15 (1999), 215–224.
[6] Z. Guo and Y. Wang, On the well-posedness of the Schrödinger-Korteweg-de Vries system, J. Differ. Equ. 249 (2010), no. 10, 2500–2520.
[7] M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 3, 917–941.
[8] C. E. Kenig, G. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69.
[9] M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Math. Sci. Appl 2 (1993), 513–528.
[10] B. Wang, Z. Huo, C. Hao, and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific, 2011.
[11] H. Wang and S. Cui, The Cauchy problem for the Schrödinger–KdV system, J. Differ. Equ. 250 (2011), no. 9, 3559–3583.

Yingzhe Ban: The Graduate School of China Academy of Engineering Physics, Beijing, 100088, P.R. China

E-mail address: [email protected]
Jie Chen: School of Sciences, Jimei University, Xiamen 361021, P.R. China

E-mail address: [email protected]
Ying Zhang: The Graduate School of China Academy of Engineering Physics, Beijing, 100088, P.R. China

E-mail address: [email protected]

	$\displaystyle\left\|\int_{\mathbb{R}^{2}}P_{N}(uP_{1}v)\tilde{\eta}^{2}\bar{w}~{}dxdt\right\|$	$\displaystyle\lesssim\sum_{N_{1}\sim N}\\|\tilde{\eta}P_{N_{1}}u\\|_{L_{x}^{\infty}L_{t}^{2}}\\|P_{1}v\\|_{L_{x}^{2}L_{t}^{\infty}}\\|\tilde{\eta}w\\|_{L_{t,x}^{2}}$		(2.4)
		$\displaystyle\lesssim(\lambda N)^{-1/2}\sum_{N_{1}\sim N}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|v\\|_{Z}\\|w\\|_{V^{2}_{S_{\lambda}}}.$		(2.4)

	$\displaystyle\|I\|$	$\displaystyle\lesssim\\|Q_{>L}^{S_{\lambda}}P_{N_{1}}u\\|_{L_{t,x}^{2}}\\|P_{N_{2}}v\\|_{L_{t,x}^{4}}\\|P_{N}w\\|_{L_{t,x}^{4}}$
		$\displaystyle\lesssim L^{-1/2}\lambda^{-1/8}\\|u\\|_{V^{2}_{S_{\lambda}}}N_{2}^{-1/8}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim\lambda^{-1/8}N_{2}^{-5/8}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\\|u\\|_{U^{2}_{S_{\lambda}}}\\|v\\|_{V^{2}_{K}}\\|w\\|_{V^{2}_{K}}.$

		$\displaystyle\quad\\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\\|_{U^{2}_{S_{\lambda}}}$		(2.5)
		$\displaystyle\lesssim\lambda\lambda^{-1/4}N_{2}^{-1/2}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$
		$\displaystyle\lesssim\lambda^{3/4}N_{1}^{3/16}N_{2}^{1/4}\max\{N_{2}^{2},\lambda N_{1}\}^{-1/2}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}.$

	$\displaystyle\left\|\int_{\mathbb{R}^{2}}P_{N_{1}}uP_{N_{2}}vP_{N}\bar{w}~{}dxdt\right\|$	$\displaystyle\lesssim\\|u\\|_{L_{t,x}^{3}}\\|P_{N_{2}}v\\|_{L_{t,x}^{3}}\\|w\\|_{L_{t,x}^{3}}$
		$\displaystyle\lesssim\\|u\\|_{L_{t}^{12}L_{x}^{3}}\\|P_{N_{2}}v\\|_{L_{t}^{12}L_{x}^{3}}\\|w\\|_{L_{t}^{12}L_{x}^{3}}$
		$\displaystyle\lesssim\lambda^{-1/6}N_{2}^{-1/12}\\|u\\|_{U^{12}_{S_{\lambda}}}\\|v\\|_{U^{12}_{K}}\\|w\\|_{U^{12}_{S_{\lambda}}}.$

	$\displaystyle\\|\eta P_{N}\mathscr{A}_{\lambda}(P_{N_{1}}uP_{N_{2}}v)\\|_{U^{2}_{S_{\lambda}}}$	$\displaystyle\lesssim\lambda N_{2}^{-1}\lambda^{-1/6}(\log N_{2})^{3}\\|P_{N_{1}}u\\|_{U^{2}_{S_{\lambda}}}\\|P_{N_{2}}v\\|_{V^{2}_{K}}$		(2.6)
		$\displaystyle\lesssim\lambda^{5/6}N_{2}^{-1/4}N_{1}^{3/16}(\log N_{2})^{3}\\|u\\|_{X_{\lambda}}\\|v\\|_{Z}.$		(2.6)

Local well-posedness for the Schrödinger-KdV system in Hs1×Hs2H^{s_{1}}\times H^{s_{2}}, II

Abstract

1 Introduction

Theorem 1.1.

Corollary 1.2.

Theorem 1.3.

2 Function spaces and multilinear estimates

Lemma 2.1.

Lemma 2.2.

Proof..

Lemma 2.3.

Proof.

3 The estimates for F​[u0]F[u_{0}]

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Corollary 3.6.

4 Bilinear estimates for KdV equation

Lemma 4.1.

Proof.

Proposition 4.2.

Proof.

Lemma 4.3.

Proof.

Proposition 4.4.

Proof.

Proposition 4.5.

Proof.

References

Local well-posedness for the Schrödinger-KdV system in $H^{s_{1}}\times H^{s_{2}}$ , II

3 The estimates for $F[u_{0}]$