The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity

Abstract

The Zakharov–Kuznetsov equation in spatial dimension $d\ge 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces, and it is proved that solutions scatter to free solutions as $t \rightarrow \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $d=4$.

1 Introduction

This paper is concerned with the Zakharov–Kuznetsov equation

$$\begin{aligned} \begin{array}{l@{\quad }l} \partial _t u+\partial _{x_1}\Delta u=\partial _{x_1}u^2 &{}\text { in}{\mathbb {R}}\times {\mathbb {R}}^d\\ u(0,\cdot )=u_0 &{}\text { on } {\mathbb {R}}^d \end{array} \end{aligned}$$

(1.1)

where $d\ge 2$, $u=u(t,x)$, $(t,x) =(t,x_1,\ldots , x_d) \in {\mathbb {R}}\times {\mathbb {R}}^d$, u is real-valued, and $\Delta$ denotes the Laplacian with respect to x.

The Zakharov–Kuznetsov equation was introduced in [13] as a model for propagation of ion-sound waves in magnetic fields. The Zakharov–Kuznetsov equation can be seen as a multidimensional extension of the well-known Korteweg–de Vries (KdV) equation. In contrast to the KdV equation, the Zakharov–Kuznetsov equation is not completely integrable, but possesses two invariants,

$$\begin{aligned} M(u) := \int _{{\mathbb {R}}^d} u^2 dx, \quad E(u) := \int _{{\mathbb {R}}^d} \frac{1}{2} |\nabla u|^2 + \frac{1}{3} u^3 dx. \end{aligned}$$

In the following, $H^s({\mathbb {R}}^d)$ denotes the standard $L^2$-based inhomogeneous Sobolev space and $B^s_{2,1}({\mathbb {R}}^d)$ is the Besov refinement, and the dotted versions their homogeneous counterparts, see below for definitions. The scale-invariant regularity threshold for (1.1) is $s_c=\frac{d-4}{2}$.

Before we state our main results, let us briefly summarize the progress which has been made regarding the well-posedness problem associated to (1.1). In the two-dimensional case, Faminskiĭ [3] established global well-posedness in the energy space $H^1({\mathbb {R}}^2)$. Later, Linares and Pastor [9] proved local well-posedness in $H^s({\mathbb {R}}^2)$ for $s>3/4$; before Grünrock and Herr [4] and Molinet and Pilod [11] showed local well-posedness for $s > 1/2$. Recently, the second author [7] proved local well-posedness for $s>-1/4$. In dimension $d=3$, Linares and Saut [10] obtained local well-posedness in $H^s({\mathbb {R}}^3)$ for $s > 9/8$. Ribaud and Vento [12] proved local well-posedness for $s > 1$ and in $B_2^{1,1}({\mathbb {R}}^3)$. The global well-posedness in $H^s({\mathbb {R}}^3)$ for $s>1$ was obtained by Molinet and Pilod in [11]. Recently, in dimensions $d \ge 3$, local well-posedness in $H^s({\mathbb {R}}^d)$ in the full subcritical range $s>s_c$ was proved in [5], which implies global well-posedness in $H^1({\mathbb {R}}^d)$ if $3 \le d \le 5$ and in $L^2({\mathbb {R}}^3)$. We refer the reader to these papers for a more thorough account on the Zakharov–Kuznetsov equation and more references.

In the present paper, we address the problem of global well-posedness and scattering for small initial data in critical spaces. By well-posedness we mean existence of a (mild) solution, uniqueness of solutions (in some subspace) and (locally Lipschitz) continuous dependence of solutions on the initial data. We say that a global solution $u \in C({\mathbb {R}},H^s({\mathbb {R}}^d))$ of (1.1) scatters as $t \rightarrow \pm \infty$, if there exist $u_\pm \in H^s({\mathbb {R}}^d)$ such that

$$\begin{aligned} \Vert u(t)-e^{tS}u_\pm \Vert _{H^s({\mathbb {R}}^d)}\rightarrow 0 \quad \left( t \rightarrow \pm \infty \right) . \end{aligned}$$

Here, $e^{tS}$ denotes the unitary group generated by the skew-adjoint linear operator $S = -\partial _{x_1} \Delta$, so that $e^{tS}u_\pm$ solves the linear homogeneous equation.

Our first main result covers small data in dimension $d=5$.

Theorem 1.1

For $d =5$, the Cauchy problem (1.1) is globally well-posed for small initial data in $B^{s_c}_{2,1}({\mathbb {R}}^5)$, and solutions scatter as $t\rightarrow \pm \infty$. The same result holds in $\dot{B}^{s_c}_{2,1}({\mathbb {R}}^d)$.

In dimensions $d\ge 6$, we can extend this to Sobolev regularity.

Theorem 1.2

For $d \ge 6$, the Cauchy problem (1.1) is globally well-posed for small initial data in $H^{s_c}({\mathbb {R}}^d)$, and solutions scatter as $t\rightarrow \pm \infty$. The same result holds in $\dot{H}^{s_c}({\mathbb {R}}^d)$.

Note that in $d=6$ this result includes the energy space $\dot{H}^1({\mathbb {R}}^6)$.

If we restrict to initial data which is radial in the last $(d-1)$ variables (see below for definitions), we obtain small data global well-posedness and scattering in the critical Sobolev spaces for any dimension $d\ge 4$.

Theorem 1.3

For $d \ge 4$, the Cauchy problem (1.1) is globally well-posed for small data in $H^{s_c}_{\mathrm {rad}}({\mathbb {R}}^d)$, and solutions scatter as $t\rightarrow \pm \infty$. The same result holds for radial data in $\dot{H}^{s_c}_{\mathrm {rad}}({\mathbb {R}}^d)$.

As the proof shows, the radiality assumption can be weakened to an angular regularity assumption, but we do not pursue this. One of the most interesting special cases here is $d=4$, when $s_c=0$; hence, the result covers the radial $L^2$ space.

The main idea of this paper is to combine a new set of non-isotropic Strichartz estimates with the bilinear transversal estimate and an interpolation argument in critical function spaces.

The paper is structured as follows: In Sect. 1.1 we introduce notation. In Sect. 2 we derive Strichartz-type estimates which are based on the well-known Strichartz estimates for the $(d-1)$-dimensional Schrödinger equation and allow us to treat the case $d=5$. In Sect. 3 we combine this with the bilinear transversal estimate and an interpolation argument, which leads to a proof of Theorem 1.2. Finally, in Sect. 4 we discuss an variation of these ideas under the additional radiality assumption and a proof of Theorem 1.3.

1.1 Notation

We write $x'=(x_2,\ldots ,x_d)$, $D_{x_j}=-i \partial _j$, $D=(-\Delta )^\frac{1}{2}$, $|\nabla _{x'}|^s=\mathcal {F}_{x'}^{-1} |\xi '|^s \mathcal {F}_{x'}$, and $\langle \nabla _{x'}\rangle ^s=\mathcal {F}_{x'}^{-1} \langle \xi '\rangle ^s \mathcal {F}_{x'}$. Here and in the sequel, we denote the Fourier transform of u in time, space, and the first spatial variable, by $\mathcal {F}_t u$, $\mathcal {F}_{x} u$, $\mathcal {F}_{x_1} u$, and $\mathcal {F}_{x'}$, respectively. $\mathcal {F}_{t, x} u = \widehat{u}$ denotes the Fourier transform of u in time and space. Choose a nonnegative bump function $\psi \in C_c^\infty ({\mathbb {R}})$ supported in the interval (1/2, 2) with the property that $\sum _{N \in 2^{\mathbb Z}} \psi (r/N)=1$ for $r>0$, and set $\psi _N=\psi (\cdot /N)$. For N, $\lambda , M \in 2^{{\mathbb Z}}$, we define (spatial) frequency projections $P_N$, $Q_{\lambda }$, and $R_M$ as the Fourier multipliers with symbols $\psi _{N}(|\xi | )$, $\psi _{\lambda }(|\xi _1| )$, and $\psi _{M}(|\xi '| )$, respectively, where $(\tau ,\xi )=(\tau ,\xi _1,\xi ')=(\tau ,\xi _1,\ldots ,\xi _d) \in {\mathbb {R}}\times {\mathbb {R}}^d$ are temporal and spatial frequencies. In addition, we define

$$\begin{aligned} P_{\le 1}=\sum _{1\ge N \in 2^{\mathbb Z}}P_N, \quad Q_{\le 1}:= \sum _{1\ge \lambda \in 2^{\mathbb Z}} Q_\lambda , \text { and } R_{\le 1}:= \sum _{1\ge M \in 2^{\mathbb Z}} R_M. \end{aligned}$$

As usual, the Sobolev space $H^s({\mathbb {R}}^d)$ is defined as the completion of the Schwartz functions with respect to the norm

$$\begin{aligned} \Vert f\Vert _{H^s({\mathbb {R}}^d)}=\left( \int _{{\mathbb {R}}^d} {\langle {\xi } \rangle }^{2s} |\widehat{f}(\xi )|^2 d\xi \right) ^{\frac{1}{2}}, \end{aligned}$$

and the (smaller) Besov space $B^s_{2,1}({\mathbb {R}}^d)$ as the completion of the Schwartz functions $\mathcal {S}({\mathbb {R}}^{d})$ with respect to the norm

$$\begin{aligned} \Vert f\Vert _{B^s_{2,1}({\mathbb {R}}^d)}=\Vert P_{\le 1}f\Vert _{L^2} +\sum _{N\in 2^{{\mathbb N}} }N^s\Vert P_Nf\Vert _{L^2}. \end{aligned}$$

Similarly, for $s\ge 0$, the homogeneous Sobolev space $\dot{H}^s({\mathbb {R}}^d)$ is defined as the completion of the Schwartz functions with respect to the norm

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s({\mathbb {R}}^d)}=\Big (\int _{{\mathbb {R}}^d} |\xi |^{2s} |\widehat{f}(\xi )|^2 d\xi \Big )^{\frac{1}{2}}, \end{aligned}$$

and the homogeneous Besov space $\dot{B}^s_{2,1}({\mathbb {R}}^d)$ as the completion of the Schwartz functions with respect to the norm

$$\begin{aligned} \Vert f\Vert _{\dot{B}^s_{2,1}({\mathbb {R}}^d)}=\sum _{N\in 2^{{\mathbb Z}} }N^s\Vert P_Nf\Vert _{L^2}. \end{aligned}$$

The radial subspaces $H_{\mathrm {rad}}^s({\mathbb {R}}^d)$ and $\dot{H}^s_{\mathrm {rad}}({\mathbb {R}}^d)$ are defined by the requirement that $f(x_1,x') = f (x_1, y')$ if $|x'|=|y'|$, i.e., for fixed $x_1$, the functions are radial in $x'$.

Finally, the Duhamel operator is denoted by

$$\begin{aligned} \mathcal {I}(F)(t) := \int _0^t e^{(t-t')S} F(t') d t'. \end{aligned}$$

2 Strichartz estimates and the proof of Theorem 1.1

For $d \ge 2$, we say (q, r) is $(d-1)$-admissible if

$$\begin{aligned} 2 \le q,r \le \infty , \; 2/q = (d-1)(1/2-1/r), \; (d, q, r) \not = (3,2,\infty ). \end{aligned}$$

Theorem 2.1

Let $d \ge 2$ and $(q_1, r_1)$, $(q_2,r_2)$ be $(d-1)$-admissible. Then, we have

$$\begin{aligned}&\Vert D_{x_1}^{\frac{1}{q}} e^{t S} f \Vert _{L_t^{q_1} L_{x'}^{r_1} {{L}_{x_1}^2}} \lesssim \Vert f \Vert _{L_x^2}. \end{aligned}$$

(2.1)

$$\begin{aligned}&\Vert D_{x_1}^{\frac{1}{q_1}+\frac{1}{q_2}}\mathcal {I} F\Vert _{L_t^{q_1} L_{x'}^{r_1} {{L}_{x_1}^2}} \lesssim \Vert F\Vert _{L_t^{q_2'} L_{x'}^{r_2'} {{L}_{x_1}^2}}, \end{aligned}$$

(2.2)

where $1/q_2'=1-1/q_2$ and $1/r_2'=1-1/r_2$.

Proof

Let $\Delta _{x'} = \sum _{j=2}^d \partial _{x_j}^2$. For fixed $\xi _1 \in {\mathbb {R}}$, define $V_{\xi _1}(t) f(x') := (e^{-i t \xi _1 \Delta _{x'}} f)(x')$. Since $V_{\xi _1}(t/\xi _1) = e^{it \Delta _{x'}}$, for $f \in \mathcal {S}({\mathbb {R}}^{d-1})$ and $F \in \mathcal {S}({\mathbb {R}}\times {\mathbb {R}}^{d-1})$, the Strichartz estimates of Schrödinger equations in ${\mathbb {R}}^{d-1}$ imply

$$\begin{aligned} \Vert |\xi _1|^{\frac{1}{q_1}} V_{\xi _1}(t) f \Vert _{L_t^{q_1} L_{x'}^{r_1}} \lesssim {}&\Vert f \Vert _{L_{x'}^2}, \end{aligned}$$

(2.3)

$$\begin{aligned} \Bigl \Vert \int _0^t |\xi _1|^{\frac{1}{q_1}+\frac{1}{q_2}} V_{\xi _1}(t-t') F(t') d t' \Bigr \Vert _{L_t^{q_1} L_{x'}^{r_1}} \lesssim {}&\Vert F\Vert _{L_t^{q_2'} L_{x'}^{r_2'}}, \end{aligned}$$

(2.4)

see [6, Theorem 1.2] for details. We deduce from Plancherel’s theorem, Minkowski’s inequality and (2.3) that

$$\begin{aligned} \Vert D_{x_1}^{\frac{1}{q}} e^{t S} f \Vert _{L_t^q L_{x'}^r L_{x_1}^2} = \left( \int _{{\mathbb {R}}} \Vert |\xi _1|^{\frac{1}{q}} V_{\xi _1} \mathcal {F}_{x_1} f \Vert _{L_t^q L_{x'}^r}^{2} d \xi _1 \right) ^{\frac{1}{2}} \lesssim \Vert f \Vert _{L_x^2}, \end{aligned}$$

which is (2.1). Similarly, by (2.4),

$$\begin{aligned}&\Vert D_{x_1}^{\frac{1}{q_1}+\frac{1}{q_2}} \mathcal {I} (F)\Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \\&\quad = \left( \int _{{\mathbb {R}}} \Bigl \Vert \int _0^t |\xi _1|^{\frac{1}{q_1}+\frac{1}{q_2}} V_{\xi _1}(t-t') \mathcal {F}_{x_1}(F)(t') d t' \Bigr \Vert _{L_t^{q_1} L_{x'}^{r_1}}^2 d \xi _1 \right) ^{\frac{1}{2}}\\&\quad \lesssim \Vert F\Vert _{L_t^{q_2'} L_{x'}^{r_2'} L_{x_1}^2}, \end{aligned}$$

which is (2.2). $\square$

Now, we can complete the proof of Theorem 1.1. Recall that $d=5$ implies $s_c = 1/2$.

Definition 2.2

We define

$$\begin{aligned} \Vert u \Vert _{Z^{\frac{1}{2}}} :={}&\Vert P_{\le 1}u \Vert _{L_t^{\infty } L_x^2} + \Vert P_{\le 1} D_{x_1}^{\frac{1}{2}} u\Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \\ {}&+ \sum _{N \in 2^{{\mathbb N}}}N^{\frac{1}{2}} \left( \Vert P_{N}u \Vert _{L_t^{\infty } L_x^2} + \Vert P_{N} D_{x_1}^{\frac{1}{2}} u\Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \right) , \\ \Vert u \Vert _{\dot{Z}^{\frac{1}{2}}} :={}&\sum _{N \in 2^{{\mathbb Z}}}N^{\frac{1}{2}} \left( \Vert P_{N}u \Vert _{L_t^{\infty } L_x^2} + \Vert P_{N} D_{x_1}^{\frac{1}{2}} u\Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \right) , \end{aligned}$$

and the corresponding Banach spaces.

By the standard argument involving the contraction mapping principle, it suffices to prove the following:

Proposition 2.3

Let $d=5$. Then, we have

$$\begin{aligned} \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{Z^{\frac{1}{2}}} \lesssim \Vert u_1 \Vert _{Z^{\frac{1}{2}}} \Vert u_2 \Vert _{Z^{\frac{1}{2}}}, \quad \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{\dot{Z}^{\frac{1}{2}}} \lesssim \Vert u_1 \Vert _{\dot{Z}^{\frac{1}{2}}} \Vert u_2 \Vert _{\dot{Z}^{\frac{1}{2}}}. \end{aligned}$$

Proof

Let $N_{\max } = \max (N_1,N_2,N_3)$ and $N_{\min } = \min (N_1,N_2, N_3)$. Theorem 2.1 gives

$$\begin{aligned}&\Vert P_{N_3}\mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert |_{L_t^{\infty } L_x^2} + \Vert P_{N_3}D_{x_1}^{\frac{1}{2}} \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{L_t^2 L_{x'}^4 L_{x_1}^2}\\&\quad \lesssim \Vert P_{N_3} D_{x_1}^{\frac{1}{2}} \left( u_1 \, u_2\right) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2}. \end{aligned}$$

We observe that $P_{N_3}=P_{N_3}Q_{\le N_3}$. Since the kernel of the operator $P_{N_3}$ is uniformly bounded in $L^1_x$, we obtain

$$\begin{aligned} \Vert P_{N_3} D_{x_1}^{\frac{1}{2}} (u_1 \, u_2) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2} \lesssim \Vert Q_{\le N_3} D_{x_1}^{\frac{1}{2}} (u_1 \, u_2) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2}. \end{aligned}$$

Now, by freezing $(t,x')$, we can use Plancherel’s theorem and Bernstein’s inequality in $x_1$ to compute

$$\begin{aligned} \Vert Q_{\le N_3} D_{x_1}^{\frac{1}{2}} (u_{N_1} \, u_{N_2}) \Vert _{L_{x_1}^2}\lesssim N_{\min }^{\frac{1}{2}}(\Vert D_{x_1}^{\frac{1}{2}} u_{N_1}\Vert _{L_{x_1}^2}\Vert u_{N_2} \Vert _{L_{x_1}^2}+\Vert u_{N_1} \Vert _{L_{x_1}^2}\Vert D_{x_1}^{\frac{1}{2}} u_{N_2}\Vert _{L_{x_1}^2}) \end{aligned}$$

Finally, Hölder’s inequality with respect to $(t,x')$ implies

$$\begin{aligned}&\Vert P_{N_3} D_{x_1}^{\frac{1}{2}} (u_{N_1} \, u_{N_2}) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2} \\&\quad \lesssim N_{\min }^{\frac{1}{2}}\Vert D_{x_1}^{\frac{1}{2}} u_{N_1} \Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \Vert u_{N_2} \Vert _{L_t^{\infty } L_x^2} + N_{\min }^{\frac{1}{2}} \Vert u_{N_1} \Vert _{L_t^{\infty } L_x^2} \Vert D_{x_1}^{\frac{1}{2}} u_{N_2} \Vert _{L_t^2 L_{x'}^4 L_{x_1}^2}. \end{aligned}$$

This can be summed up both in the homogeneous and in the inhomogeneous version.

$\square$

This argument also implies the scattering claim, since it implies that the Duhamel integral converges to a free solutions as $t\rightarrow \pm \infty$. We omit the details of this standard argument.

3 Transversal estimates and the proof of Theorem 1.2

Lemma 3.1

Let $d\ge 2$ and $f_{N_1,\lambda _1} = Q_{\lambda _1} P_{N_1} f$, $g_{N_2,\lambda _2} = Q_{\lambda _2} P_{N_2} g$. For all $\lambda _j,N_j\in 2^{\mathbb Z}$ such that

$$\begin{aligned} |\nabla \varphi (\xi )-\nabla \varphi (\eta )| > rsim \max \{\lambda _1,\lambda _2\} N_{\max }, \end{aligned}$$

for all $\xi \in {\text {supp}}\widehat{f}_{N_1,\lambda _1}$, $\eta \in {\text {supp}}\widehat{g}_{N_2,\lambda _2}$, it holds that

$$\begin{aligned} \Vert P_{N_3} (e^{tS}f_{N_1,\lambda _1} \, e^{tS}g_{N_2,\lambda _2}) \Vert _{L_t^2 L_x^2} \lesssim \left( \frac{N_{\min }^{d-1}}{ \max \{\lambda _1,\lambda _2\} N_{\max }}\right) ^{\frac{1}{2}} \Vert f_{N_1,\lambda _1}\Vert _{L^2} \Vert g_{N_2,\lambda _2} \Vert _{L^2}. \end{aligned}$$

(3.1)

This is an instance of the well-known bilinear transversal estimate, e.g., a special case of [1, Lemma 2.6], where a proof can be found.

Next, we recall the definitions of $U^p$ and $V^p$ spaces, which have been introduced in [8] the dispersive PDE context. We refer the reader to [1] and the references therein for further details. For $1\le p <\infty$, we call a function $a: {\mathbb {R}}\rightarrow L^2({\mathbb {R}}^d)$ a $p-$atom, if there exists a finite partition $\mathcal {J}=\{(-\infty ,t_1),[t_2,t_3), \ldots , [t_K,\infty )\}$ of the real line such that

$$\begin{aligned} a(t)=\sum _{J \in \mathcal {J}}\mathbf {1}(t)f_J, \quad \sum _{J \in \mathcal {J}}\Vert f_J\Vert _{L^2}^p\le 1. \end{aligned}$$

Now, $U^p$ is defined as the space of all $u: {\mathbb {R}}\rightarrow L^2({\mathbb {R}}^d)$, such that there exists an atomic decomposition $u=\sum _{j=1 }^\infty c_j a_j$, where $(c_j)\in \ell ^1({\mathbb N})$ and the $a_j$’s are $p-$atoms. Then, $\Vert u\Vert _{U^p}=\inf \sum _{j=1}^\infty |c_j|$ is a norm (the infimum is taken with respect to all possible atomic decompositions), so that $U^p$ is a Banach space. Further, let $V^p$ denote the space of all right-continuous functions $v: {\mathbb {R}}\rightarrow L^2({\mathbb {R}}^d)$, such that

$$\begin{aligned} \Vert v\Vert _{V^p}=\Vert v\Vert _{L^\infty _t L^2_x}+\sup \Big (\sum _{j \in {\mathbb Z}}\Vert v(t_j)-v(t_{j-1})\Vert _{L^2_x}^p\Big )^{\frac{1}{p}}<\infty , \end{aligned}$$

where the supremum is taken over all increasing sequences $(t_j)$. Now, we define the atomic space $U_{S}^p = e^{\cdot \, S} U^p$ with norm $\Vert u \Vert _{U_S^p} = \Vert e^{- \, \cdot \, S} u\Vert _{U^p}$, and $V_{S}^p = e^{\cdot \, S} V^p$ with norm $\Vert u \Vert _{V_S^p} = \Vert e^{- \, \cdot \, S} u\Vert _{V^p}$.

There is the embedding $V_{S}^p\subset U_{S}^q$ if $p<q$, see [8, Lemma 6.4]. Due to the atomic structure of $U_{S}^q$ and the Strichartz estimate (2.1), we have

$$\begin{aligned} \Vert D_{x_1}^{\frac{1}{q}} u\Vert _{L_t^{q} L_{x'}^{r} L_{x_1}^2} \lesssim \Vert u \Vert _{U^{q}_S} \end{aligned}$$

(3.2)

for $(d-1)-$admissible pairs, and $\Vert u \Vert _{U^{q}_S}$ may be replaced by $\Vert u \Vert _{V^{2}_S}$ for non-endpoint pairs, i.e., when $q>2$.

Let $\lambda _{\max } := \max (\lambda _1,\lambda _2,\lambda _3)$ and $\lambda _{\min } := \min (\lambda _1,\lambda _2,\lambda _3)$. We use the shorthand notation $u_N := P_{N} u$, $u_{N, \lambda } := Q_{\lambda } P_{N} u$, etc.

Proposition 3.2

Let $d\ge 6$ and the pair (q, r) be $(d-1)$-admissible with $2<q<\frac{2(d-3)}{d-5}$, and let $\varepsilon >0$. Suppose

$$\begin{aligned} |\nabla \varphi (\xi )-\nabla \varphi (\eta )| > rsim \max \{\lambda _1,\lambda _2\} N_{\max }, \end{aligned}$$

for all $\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1,\lambda _1}$, $\eta \in {\text {supp}}_{\xi } \widehat{v}_{N_2,\lambda _2}$. Then, for all $\lambda _j,N_j \in 2^{\mathbb Z}$,

$$\begin{aligned} \begin{aligned}&\Vert P_{N_3} Q_{\lambda _3} ( u_{N_1,\lambda _1} v_{N_2,\lambda _2}) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \\&\quad \lesssim \lambda _{\max }^{-\frac{1}{2}+ \frac{2 \varepsilon }{d-3}} \lambda _{\min }^{-\frac{1}{2}+\frac{1}{d-3}+\frac{1}{q}-\frac{2 \varepsilon }{d-3}} N_{\max }^{-\frac{1}{2} + \frac{1}{d-3}} N_{\min }^{\frac{d-3}{2} - \frac{2}{d-3}} \Vert u_{N_1,\lambda _1} \Vert _{V^2_S} \Vert v_{N_2,\lambda _2}\Vert _{V^2_S}. \end{aligned} \end{aligned}$$

(3.3)

Proof

By symmetry, we may assume that $\lambda _1 \sim \lambda _{\max }$. For a sufficiently small $\varepsilon >0$, we define the $(d-1)$-admissible pairs $(q_1,r_1)$ and $(q_2,r_2)$ by

$$\begin{aligned} \left( \frac{1}{q_1}, \ \frac{1}{r_1} \right)= & {} \left( \frac{1}{2} -\varepsilon , \frac{1}{2}-\frac{1- 2\varepsilon }{d-1} \right) , \ \left( \frac{1}{q_2}, \ \frac{1}{r_2}\right) \\= & {} \left( \frac{d-3}{4}- \frac{d-3}{2 q} + \varepsilon , \ \frac{d-3}{q(d-1)} + \frac{1- 2 \varepsilon }{d-1} \right) . \end{aligned}$$

In addition, letting

$$\begin{aligned} \frac{1}{\alpha } = \frac{1}{q_1} + \frac{1}{q_2} , \qquad \frac{1}{\beta } = \frac{1}{r_1} + \frac{1}{r_2}, \end{aligned}$$

by using (3.2), we have

$$\begin{aligned} \Vert P_{N_3}Q_{\lambda _3}( u_{N_1,\lambda _1} v_{N_2,\lambda _2}) \Vert _{L_t^{\alpha } L_{x'}^{\beta } L_{x_1}^2}&\lesssim \lambda _{\min }^{\frac{1}{2}} \Vert u_{N_1,\lambda _1} \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert v_{N_2,\lambda _2}\Vert _{L_t^{q_2} L_{x'}^{r_2} L_{x_1}^2}\nonumber \\&\lesssim \lambda _{\min }^{\frac{1}{2}-\frac{1}{q_2}} \lambda _{\max }^{-\frac{1}{q_1}} \Vert u_{N_1,\lambda _1} \Vert _{U^{q_1}_S} \Vert v_{N_2,\lambda _2}\Vert _{U_S^{q_2}}. \end{aligned}$$

(3.4)

Lemma 3.1 immediately extends from free solutions to $2-$atoms. Therefore, the atomic structure of $U^2$ implies

$$\begin{aligned} \Vert P_{N_3}Q_{\lambda _3}\left( u_{N_1,\lambda _1} v_{N_2,\lambda _2}\right) \Vert _{L_t^2 L_x^2} \lesssim \left( \frac{N_{\min }^{d-1}}{ \lambda _{\max } N_{\max }}\right) ^{\frac{1}{2}} \Vert u_{N_1,\lambda _1} \Vert _{U^{2}_S} \Vert v_{N_2,\lambda _2}\Vert _{U_S^{2}}. \end{aligned}$$

(3.5)

For $\theta = \frac{2}{d-3}$, it is observed that

$$\begin{aligned} \frac{\theta }{\alpha } + \frac{1- \theta }{2} = 1 - \frac{1}{q} \Bigl ( =: \frac{1}{q'} \Bigr ), \quad \frac{\theta }{\beta } + \frac{1- \theta }{2} = 1 - \frac{1}{r} \Bigl (=: \frac{1}{r'} \Bigr ). \end{aligned}$$

Fig. 1

Choice of parameters in the interpolation argument

Now, we interpolate (3.4) and (3.5) to obtain (3.3). More precisely, we follow the argument in [1, p. 1203]: For brevity, we set $u:=u_{N_1,\lambda _1}$, $v:=v_{N_2,\lambda _2}$. Then, [8, Lemma 6.4] implies that there exist decompositions $u=\sum _{k=1}^\infty u_{k}$, such that $\widehat{u_{k}}\subset {\text {supp}}\widehat{u}$, and for any $q \ge 2$ we have $\Vert u_{k}\Vert _{U^{q}_S}\lesssim 2^{k(\frac{2}{q}-1)}\Vert u\Vert _{V^2_S},$ and the analogous decomposition for v. Then, by convexity, we obtain

$$\begin{aligned} \Vert P_{N_3} Q_{\lambda _3} ( u v) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \lesssim {}&\sum _{k,k'\in {\mathbb N}}\Vert P_{N_3} Q_{\lambda _3} \left( u_k v_{k'}\right) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \\ \lesssim {}&\sum _{k,k'\in {\mathbb N}}\Vert P_{N_3} Q_{\lambda _3} \left( u_k v_{k'}\right) \Vert _{L_t^{\alpha } L_{x'}^{\beta } L_{x_1}^2}^\theta \Vert P_{N_3} Q_{\lambda _3} \left( u_k v_{k'}\right) \Vert _{L^2_{t,x}}^{1-\theta }. \end{aligned}$$

Estimates (3.4) and (3.5) further imply

$$\begin{aligned} \Vert P_{N_3} Q_{\lambda _3} ( u v) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \lesssim \Big (\sum _{k,k'\in {\mathbb N}} 2^{k\theta (\frac{2}{q_1}-1)}2^{k'\theta (\frac{2}{q_2}-1)}\Big ) \frac{\lambda _{\min }^{\frac{\theta }{2}-\frac{\theta }{q_2}} N_{\min }^{\frac{d-1}{2}(1-\theta )}}{ \lambda _{\max }^{\frac{\theta }{q_1}+\frac{1-\theta }{2}} N_{\max }^{\frac{1-\theta }{2}}} \Vert u\Vert _{V^2_S}\Vert v\Vert _{V^2_S} . \end{aligned}$$

Since $q_1,q_2>2$, the sums converge, and the proof of (3.3) is complete. $\square$

Define

$$\begin{aligned} \mathcal {R}_{\lambda _{\max },N_{\max }} = \{ (\xi _1, \xi ') \in {\mathbb {R}}\times {\mathbb {R}}^{d-1} \, | \, |\xi _1|\ll \lambda _{\max }, \ |\xi '| \ll N_{\max } \}. \end{aligned}$$

Lemma 3.3

Assume that there exist $\gamma _1$, $\gamma _2$, $\gamma _3 \in {\mathbb {R}}^{d}$ such that $\gamma _1 + \gamma _2 -\gamma _3 \in \mathcal {R}_{4 \lambda _{\max }, 4N_{\max }}$ and

$$\begin{aligned} {\text {supp}}_{\xi } \widehat{u}_{N_i, \lambda _i} \subset \mathcal {R}_{\lambda _{\max },N_{\max }} + \gamma _i := \{\xi \in {\mathbb {R}}^d \, | \, \xi - \gamma _i \in \mathcal {R}_{\lambda _{\max },N_{\max }} \} . \end{aligned}$$

(3.6)

Then, we have either

$$\begin{aligned} |\nabla \varphi (\xi ) - \nabla \varphi (\eta )| > rsim \lambda _{\max }N_{\max }, \end{aligned}$$

(3.7)

for all $\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}$, or

$$\begin{aligned} |\nabla \varphi (\eta )-\nabla \varphi (\zeta )| > rsim \lambda _{\max }N_{\max }, \end{aligned}$$

(3.8)

for all $\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}$, $\zeta \in {\text {supp}}_{\xi } \widehat{u}_{N_3, \lambda _3}$.

Proof

Firstly, we consider the case $\max (|\xi '|, |\eta '|, |\zeta '|) \ll N_{\max }$. We deduce from $\partial _1 \varphi (\xi )= 3 \xi _1^2 + |\xi '|^2$ that

$$\begin{aligned}&|\partial _1 \varphi (\xi ) - \partial _1 \varphi (\eta )| + | \partial _1 \varphi (\eta ) - \partial _1 \varphi (\xi + \eta )|\\&\quad \ge 3|\xi _1^2 - \eta _1^2| + 3|\xi _1\left( \xi _1+2\eta _1\right) | - |\xi '|^2-|\eta '|^2-|\xi '+\eta '|^2 > rsim N_{\max }^2, \end{aligned}$$

which implies the claim since $|\nabla ^2 \varphi (\xi )| \lesssim |\xi |$.

Next, we assume $\max (|\xi '|, |\eta '|, |\zeta '|) \sim N_{\max }$. For all $\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}$, $\xi + \eta \in {\text {supp}}_{\xi } \widehat{u}_{N_3, \lambda _3}$, we will show

$$\begin{aligned} \sum _{j=2}^d \left( |\partial _j \varphi (\xi ) - \partial _j \varphi (\eta )| + |\partial _j \varphi (\eta )-\partial _j \varphi (\xi + \eta )| \right) > rsim \lambda _{\max }N_{\max }. \end{aligned}$$

(3.9)

We may assume $|\xi '| \sim N_{\max }$, $\lambda _1 \sim \lambda _{\max }$. For $2 \le j \le d$, it is observed that $\partial _j \varphi (\xi ) = 2 \xi _1 \xi _j$. Then, for (3.9), it suffices to show

$$\begin{aligned} |\xi _1 \xi ' - \eta _1 \eta '| + |\eta _1 \eta ' - \left( \xi _1+\eta _1\right) \left( \xi ' + \eta '\right) | > rsim \lambda _1 N_1. \end{aligned}$$

(3.10)

Since $|\xi '| \sim N_{\max }$, $\lambda _1 \sim \lambda _{\max }$, if either $\lambda _{\min } \ll \lambda _{\max }$ or $\min (|\eta '|, |\xi '+\eta '|) \ll N_{\max }$ holds, we easily verify (3.10). Then, we assume $\lambda _1 \sim \lambda _2 \sim \lambda _3$ and $|\xi '| \sim |\eta '| \sim |\xi '+\eta '|$. We observe

$$\begin{aligned}&|\eta _1 \eta ' - \left( \xi _1+\eta _1\right) \left( \xi ' + \eta '\right) |\\&\quad = \bigl |\eta _1 \eta ' -\left( \xi _1 + \eta _1\right) \left( \xi ' - \frac{\eta _1}{\xi _1} \eta ' + \frac{\eta _1}{\xi _1} \eta ' + \eta ' \right) \bigr |\\&\quad \ge \bigl | \eta _1 \eta ' - (\xi _1 + \eta _1) \left( 1 + \frac{\eta _1}{\xi _1}\right) \eta ' \bigr | - \bigl | \left( 1+\frac{\eta _1}{\xi _1}\right) \left( \xi _1 \xi ' - \eta _1 \eta '\right) \bigr |\\&\quad = \bigl | 1+\frac{\eta _1}{\xi _1}+ \frac{\xi _1}{\eta _1} \, \bigr | |\eta _1 \eta '| - \bigl | \left( 1+\frac{\eta _1}{\xi _1}\right) \left( \xi _1 \xi ' - \eta _1 \eta '\right) \bigr |. \end{aligned}$$

Since $|\alpha + \alpha ^{-1}| \ge 2$ for any $\alpha \in {\mathbb {R}}$, this completes the proof of (3.10).

From (3.9), without loss of generality, we can assume that there exist $\xi _0 \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}$, $\eta _0 \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}$ such that

$$\begin{aligned} \sum _{j=2}^d |\partial _j \varphi \left( \xi _0\right) - \partial _j \varphi \left( \eta _0\right) | > rsim \lambda _{\max }N_{\max }. \end{aligned}$$

(3.11)

For $2 \le j, k \le d$ and all $\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}$, since $|\partial _1 \partial _j \varphi (\xi )| + |\partial _1 \partial _j \varphi (\eta )| \lesssim N_{\max }$ and $|\partial _k \partial _j \varphi (\xi )| +|\partial _k \partial _j \varphi (\eta )| \lesssim \lambda _{\max }$, we get

$$\begin{aligned} | \partial _j \varphi (\xi ) - \partial _j \varphi \left( \xi _0\right) | + | \partial _j \varphi (\eta ) - \partial _j \varphi \left( \eta _0\right) | \ll \lambda _{\max } N_{\max }, \end{aligned}$$

for all $\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}$. This estimate and (3.11) yield the claim (3.7). $\square$

Now, we define the solution spaces as $Y^s:=C({\mathbb {R}}; H^s({\mathbb {R}}^d)) \cap {\langle {\nabla _x} \rangle }^{-s} V^2_S$ and $\dot{Y}^s:=C({\mathbb {R}}; \dot{H}^s({\mathbb {R}}^d)) \cap {|\nabla _x|}^{-s} V^2_S$, with norms

$$\begin{aligned} \Vert u \Vert _{Y^s} :=&\left( \sum _{N \in 2^{{\mathbb Z}}} \langle N\rangle ^{2 s} \Vert P_N u \Vert _{V^2_S}^2 \right) ^{1/2} , \\ \Vert u \Vert _{\dot{Y}^s} :=&\left( \sum _{N \in 2^{{\mathbb Z}}} N^{2 s} \Vert P_N u \Vert _{V^2_S}^2 \right) ^{1/2}, \end{aligned}$$

respectively.

Proposition 3.4

Let $d \ge 6$. Then, we have

$$\begin{aligned} \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{Y^{s_c}} \lesssim \Vert u_1 \Vert _{Y^{s_c}} \Vert u_2 \Vert _{Y^{s_c}}, \quad \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{\dot{Y}^{s_c}} \lesssim \Vert u_1 \Vert _{\dot{Y}^{s_c}} \Vert u_2 \Vert _{\dot{Y}^{s_c}}. \end{aligned}$$

Proof

We show first that there exists $\varepsilon >0$ such that for any $N_1,N_2,N_3\in 2^{\mathbb Z}$ we have

$$\begin{aligned} \Bigl | \iint P_{N_1} u_1 P_{N_2} u_2 \partial _{x_1} P_{N_3} u_3 dx dt \Bigr | \lesssim N_{\min }^{s_c + \varepsilon } N_{\max }^{- \varepsilon } \prod _{i=1}^3 \Vert P_{N_i}u_{i} \Vert _{V^2_S}. \end{aligned}$$

(3.12)

As before, we use the shorthand notation $u_{N_j} := P_{N_j} u_j$, $u_{N_j, \lambda _j} := Q_{\lambda _j} P_{N_j} u_j$, etc. Obviously, (3.12) is implied by

$$\begin{aligned} \sum _{\lambda _1, \lambda _2, \lambda _{3} \in 2^{{\mathbb Z}}}\lambda _3 \Bigl | \iint u_{N_1,\lambda _1} u_{N_2,\lambda _2} u_{N_3,\lambda _3} dx dt \Bigr | \lesssim N_{\min }^{s_c + \varepsilon } N_{\max }^{- \varepsilon } \prod _{i=1}^3 \Vert u_{N_i} \Vert _{V^2_S}. \end{aligned}$$

(3.13)

Now we show (3.13). After harmless decompositions, we may assume that there exist $\gamma _1$, $\gamma _2$, $\gamma _3 \in {\mathbb {R}}^{d}$ such that $\gamma _1 + \gamma _2 -\gamma _3 \in \mathcal {R}_{4 \lambda _{\max }, 4N_{\max }}$ and (3.6). Lemma 3.3 provides either $|\nabla \varphi (\xi )-\nabla \varphi (\eta )| > rsim \lambda _{\max }N_{\max }$ for all $\xi \in {\text {supp}}_{\xi } u_{N_1,\lambda _1}$, $\eta \in {\text {supp}}_{\xi } u_{N_2,\lambda _2}$ or $|\nabla \varphi (\eta )-\nabla \varphi (\zeta )| > rsim \lambda _{\max }N_{\max }$ for all $\eta \in {\text {supp}}_{\xi } u_{N_2,\lambda _2}$ and $\zeta \in {\text {supp}}_{\xi } u_{N_3,\lambda _3}$. For the former case, it follows from the Hölder’s inequality, the Strichartz estimate (3.2), and the bilinear estimate (3.3) that

$$\begin{aligned}&\sum _{\lambda _1, \lambda _2, \lambda _{3} \in 2^{{\mathbb Z}}}\lambda _3 \Bigl | \iint u_{N_1,\lambda _1} u_{N_2,\lambda _2} u_{N_3,\lambda _3} dx dt \Bigr |\\&\quad \le \sum _{\lambda _i \le N_i (i=1,2,3)}\lambda _3 \Vert P_{N_3} Q_{\lambda _3} \left( u_{N_1,\lambda _1} u_{N_2,\lambda _2}\right) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \Vert u_{N_3,\lambda _3}\Vert _{L_t^{q} L_{x'}^{r} L_{x_1}^2}\\&\quad \lesssim \sum _{\lambda _i \le N_i (i=1,2,3)} \lambda _{\max }^{\frac{d-1}{d-3}\varepsilon } \lambda _{\min }^{\frac{1}{d-3}-\frac{d-1}{d-3} \varepsilon } N_{\max }^{-\frac{1}{2} + \frac{1}{d-3}} N_{\min }^{\frac{d-3}{2} - \frac{2}{d-3}} \Vert u_{N_1,\lambda _1} \Vert _{V^2_S} \Vert u_{N_2,\lambda _2}\Vert _{V_S^{2}} \Vert u_{N_3,\lambda _3}\Vert _{V^2_S}\\&\quad \le N_{\min }^{s_c + \frac{1}{2} -\frac{1}{d-3}- \frac{d-1}{d-3} \varepsilon }N_{\max }^{- \frac{1}{2} + \frac{1}{d-3}+\frac{d-1}{d-3} \varepsilon } \Vert u_{N_1} \Vert _{V^2_S} \Vert u_{N_2}\Vert _{V_S^{2}} \Vert u_{N_3}\Vert _{V^2_S}. \end{aligned}$$

Here, the pair (q, r) should satisfy the hypothesis of Proposition 3.2, and we have used $\lambda _{\max } \le N_{\max }$ and $\lambda _{\min }\le N_{\min }$. In the similar way, the latter case is treated as follows:

$$\begin{aligned}&\sum _{\lambda _1, \lambda _2, \lambda _{3} \in 2^{{\mathbb Z}}}\lambda _3 \Bigl | \iint u_{N_1,\lambda _1} u_{N_2,\lambda _2} u_{N_3,\lambda _3} dx dt \Bigr |\\&\quad \le \sum _{\lambda _i \le N_i (i=1,2,3)} \lambda _3 \Vert P_{N_1} Q_{\lambda _1} \left( u_{N_2,\lambda _2} u_{N_3,\lambda _3}\right) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \Vert u_{N_1,\lambda _1}\Vert _{L_t^{q} L_{x'}^{r} L_{x_1}^2}\\&\quad \le N_{\min }^{s_c + \frac{1}{2} -\frac{1}{d-3}- \frac{d-1}{d-3} \varepsilon }N_{\max }^{- \frac{1}{2} + \frac{1}{d-3}+\frac{d-1}{d-3} \varepsilon } \Vert u_{N_1} \Vert _{V^2_S} \Vert u_{N_2}\Vert _{V_S^{2}} \Vert u_{N_3}\Vert _{V^2_S}. \end{aligned}$$

Finally, we explain why (3.12) implies Proposition 3.4. By duality, see, e.g., [1, Lemma 7.3], we obtain

$$\begin{aligned} \Vert P_{N_3} \mathcal {I}\left( \partial _{x_1}\left( P_{N_1} u_1 P_{N_2} u_2\right) \right) \Vert _{V^2_S} \lesssim N_{\min }^{s_c}\left( \frac{N_{\min }}{N_{\max }}\right) ^{ \varepsilon } \Vert P_{N_1}u_{1} \Vert _{V^2_S}\Vert P_{N_2}u_{2} \Vert _{V^2_S}. \end{aligned}$$

This can be easily summed up. $\square$

Again, the proof of Theorem 1.2 is a straightforward application of the contraction mapping principle. The scattering claim follows from the well-known fact that functions in $V^2$ have limits at $\pm \infty$.

4 Radial Strichartz estimates and the proof of Theorem 1.3

We first prove a variant of the Strichartz estimates in 2.1 for functions which, for fixed $x_1$, are radial in $x'$.

Theorem 4.1

Let $d \ge 3$ and $2 \le q$,$r \le \infty$ satisfy

$$\begin{aligned} \frac{2}{q} \le (2 d -3) \left( \frac{1}{2} - \frac{1}{r} \right) , \quad (d,q,r)\not =(3,2,\infty ), \quad (q,r)\not =\left( 2,\frac{2(2d-3)}{2d-5}\right) , \end{aligned}$$

and let $\sigma =-\frac{d-1}{2}+\frac{d-1}{r} +\frac{2}{q}$. Then, for all functions $f \in L^2_{\text {rad}}({\mathbb {R}}^d)$, we have

$$\begin{aligned} \Vert D_{x_1}^{\frac{1}{q}} |\nabla _{x'}|^\sigma e^{t S} f \Vert _{L_t^{q} L_{x'}^{r} {{L}_{x_1}^2}} \lesssim \Vert f \Vert _{L_x^2}. \end{aligned}$$

(4.1)

The proof follows the exact same lines as the proof of Theorem 2.1, but with the Strichartz estimates for the $(d-1)$-dimensional Schrödinger equation from [6] replaced by the radial version obtained in [2, Theorem 1.1].

Lemma 4.2

Let $d\ge 2$ and $f_{N_1,\lambda _1,M_1} = R_{M_1} Q_{\lambda _1} P_{N_1} f$, $g_{N_2,\lambda _2,M_2} = R_{M_2} Q_{\lambda _2} P_{N_2} g$. (i) Suppose that there exists $\ell \in \{2,\ldots ,d\}$ such that

$$\begin{aligned} |\partial _{\ell } \varphi (\xi )-\partial _{\ell } \varphi (\eta )| > rsim N_{\max }^2, \end{aligned}$$

for all $\xi \in {\text {supp}}\widehat{f}_{N_1,\lambda _1,M_1}$, $\eta \in {\text {supp}}\widehat{g}_{N_2,\lambda _2,M_2}$. Then, it holds that

$$\begin{aligned} \begin{aligned}&\Vert P_{N_3}\left( e^{tS}f_{\lambda _1,N_1,M_1} \, e^{tS}g_{\lambda _2,N_2,M_2}\right) \Vert _{L_t^2 L_x^2} \\&\quad \lesssim \left( \frac{\min \left\{ \lambda _1,\lambda _2 \right\} \min \left\{ M_{1},M_2\right\} ^{d-2}}{ N_{\max }^2}\right) ^{\frac{1}{2}} \Vert f_{M_1,\lambda _1,N_1}\Vert _{L^2} \Vert g_{M_2,\lambda _2,N_2} \Vert _{L^2}. \end{aligned} \end{aligned}$$

(4.2)

(ii) Suppose that

$$\begin{aligned} |\partial _{1} \varphi (\xi )-\partial _{1} \varphi (\eta )| > rsim N_{\max }^2, \end{aligned}$$

for all $\xi \in {\text {supp}}\widehat{f}_{N_1,\lambda _1,M_1}$, $\eta \in {\text {supp}}\widehat{g}_{N_2,\lambda _2,M_2}$. Then, it holds that

$$\begin{aligned} \begin{aligned}&\Vert P_{N_3}(e^{tS}f_{\lambda _1,N_1,M_1} \, e^{tS}g_{\lambda _2,N_2,M_2}) \Vert _{L_t^2 L_x^2} \\&\quad \lesssim \left( \frac{ \min \{M_{1},M_2\}^{d-1}}{ N_{\max }^2}\right) ^{\frac{1}{2}} \Vert f_{M_1,\lambda _1,N_1}\Vert _{L^2} \Vert g_{M_2,\lambda _2,N_2} \Vert _{L^2}. \end{aligned} \end{aligned}$$

(4.3)

As above, the proof of this lemma follows from [1, Lemma 2.6]. As above, it immediately extends to $U^2_S$-functions.

Let $Y^{s}_{\text {rad}}$ and $\dot{Y}^{s}_{\text {rad}}$ be the subspaces of $Y^{s}$ and $\dot{Y}^{s}$ of functions which, for fixed $x_1$, are radial in $x'$, with the same norms. Then, the key for the proof of Theorem 1.3 is the following

Proposition 4.3

Let $d \ge 4$. Then, we have

$$\begin{aligned} \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{Y^{s_c}_{\text {rad}}} \lesssim \Vert u_1 \Vert _{Y^{s_c}_{\text {rad}}} \Vert u_2 \Vert _{Y^{s_c}_{\text {rad}}}, \quad \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{\dot{Y}^{s_c}_{\text {rad}}} \lesssim \Vert u_1 \Vert _{\dot{Y}^{s_c}_{\text {rad}}} \Vert u_2 \Vert _{\dot{Y}^{s_c}_{\text {rad}}}. \end{aligned}$$

Proof

For $i=1,2,3$, we use $u_{i} := R_{M_i} Q_{\lambda _i} P_{N_i} u$. As in the proof of Proposition 3.4, it suffices to show

$$\begin{aligned} \sum _{\lambda _i, M_i} \lambda _{\max } \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr | \lesssim N_{\min }^{s_c + \varepsilon } N_{\max }^{- \varepsilon } \prod _{i=1}^3 \Vert u_{N_i} \Vert _{V^2_S}. \end{aligned}$$

(4.4)

Here and in the sequel, all functions are implicitly assumed to satisfy the radiality hypothesis. Let

$$\begin{aligned} \left( \frac{1}{q_1}, \frac{1}{r_1} \right) =&\left( \frac{1}{2} - \varepsilon , \frac{2d-5}{2(2d-3)} \right) ,\\ \left( \frac{1}{q_2}, \frac{1}{r_2} \right) =&\left( \frac{(d-1)(2 d-3)}{2 \left( d-1+ 2 (2d-3) \varepsilon \right) } \varepsilon , \frac{d-1}{2 \left( d-1+ 2 (2d-3) \varepsilon \right) } \right) ,\\ \left( \frac{1}{q_3}, \frac{1}{r_3} \right) =&\left( 2 \varepsilon , \frac{2}{2d-3}\right) . \end{aligned}$$

Then, we have

$$\begin{aligned}&\Vert R_{M} Q_{\lambda } P_{N} u \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \lesssim \lambda ^{- \frac{1}{q_1}} M^{-\frac{d-2}{2 d-3} + 2 \varepsilon } \Vert R_{M} Q_{\lambda } P_{N} u\Vert _{U^{q_1}_S},\end{aligned}$$

(4.5)

$$\begin{aligned}&\Vert R_{M} Q_{\lambda } P_{N} u\Vert _{L_t^{q_2} L_{x'}^{r_2} L_{x_1}^2} \lesssim \lambda ^{- \frac{1}{q_2}} \Vert R_{M} Q_{\lambda } P_{N} u\Vert _{U^{q_2}_S}, \end{aligned}$$

(4.6)

$$\begin{aligned}&\Vert R_{M} Q_{\lambda } P_{N} u\Vert _{L_t^{q_3} L_{x'}^{r_3} L_{x_1}^2} \lesssim \lambda ^{- \frac{1}{q_3}} M^{\frac{(d-1)(2 d-7)}{2 (2 d-3)} - 4 \varepsilon }\Vert R_{M} Q_{\lambda } P_{N} u\Vert _{U^{q_3}_S}. \end{aligned}$$

(4.7)

By symmetry of (4.4), we may assume $N_3 \lesssim N_1 \sim N_2$, $\lambda _2 \lesssim \lambda _1$, and then, it is enough to consider the following three cases:

• (1) $M_1 \sim N_1$, $M_2 \sim N_2$. (2) $M_1 \sim N_1$, $M_2 \ll N_1$, (3) $M_1 \ll N_1$, $M_2 \ll N_1$.

  • (1) First, we assume $M_1 \sim N_1$, $M_2 \sim N_2$. By using (4.5) and (4.7), we obtain

    $$\begin{aligned}&\Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |\\&\quad \lesssim \lambda _{\min }^{\frac{1}{2}} \Vert u_1 \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_{2} \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_{3} \Vert _{L_t^{q_3} L_{x'}^{r_3} L_{x_1}^2} \\&\quad \lesssim \lambda _{\min }^{\frac{1}{2}} \lambda _1^{-\frac{1}{q_1}} \lambda _2^{-\frac{1}{q_1}} \lambda _3^{-2 \varepsilon } M_3^{\frac{(d-1)(2 d-7)}{2 (2 d-3)} - 4 \varepsilon } N_1^{-\frac{2(d-2)}{2d-3}+4 \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}\\&\quad \lesssim \lambda _{\min }^{\varepsilon } \lambda _{\max }^{-1}M_3^{s_c+\varepsilon }N_1^{-2 \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}, \end{aligned}$$

    which completes (4.4).

• (2) In the case $M_1 \sim N_1$, $M_2 \ll N_1$, it is observed that $\lambda _2 \sim M_3 \sim N_1$. Then, without loss of generality, we may assume $\lambda _3 \lesssim \lambda _1 \sim N_1$. In the case $\lambda _3 \ll \lambda _1$, for all $\xi \in {\text {supp}}_{\xi } \widehat{u}_{1}$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_{2}$ such that $\xi +\eta \in {\text {supp}}_{\xi } \widehat{u}_3$, we observe

  $$\begin{aligned}&|\tau _1- \varphi (\xi )|+|\tau _2 - \varphi (\eta )|+ |\tau _1+\tau _2 - \varphi (\xi +\eta )|\\&\quad > rsim {} |\varphi (\xi +\eta ) - \varphi (\xi ) - \varphi (\eta )| \\&\quad > rsim {} \bigl | \xi _1|\xi '|^2 \bigr | - |\xi _1+\eta _1| \, \bigl | |\xi + \eta |^2 + \xi _1^2 - \xi _1 \eta _1 + \eta _1^2 \bigr | - \bigl |\eta _1 | \eta '|^2 \bigr | > rsim N_1^3. \end{aligned}$$

  Thus, we can assume that at least one of $u_1$, $u_2$, $u_3$ satisfies ${\text {supp}}\widehat{u}_i \subset \{ (\tau ,\xi )\, |\, |\tau -\varphi (\xi )| > rsim N_1^3\}$. We easily see that this condition verifies the claim by utilizing Theorem 2.1 and (3.2). For example, if ${\text {supp}}\widehat{u}_1 \subset \{ (\tau ,\xi )\, |\, |\tau -\varphi (\xi )| > rsim N_1^3\}$, using Bernstein’s inequality and Theorem 4.1 we obtain

  $$\begin{aligned} \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |&\lesssim \Vert u_1 \Vert _{L_t^2 L_x^2} \Vert u_2 \Vert _{L_t^4 L_{x'}^{2(d-1)} L_{x_1}^2} \Vert u_3\Vert _{L_t^4 L_{x'}^{\frac{2(d-1)}{d-2}} L_{x_1}^{\infty }}\\&\lesssim N_1^{-\frac{3}{2}} \Vert u_1 \Vert _{V_S^2} M_2^{\frac{d-3}{2}} \Vert u_2 \Vert _{L_t^4 L_{x'}^{\frac{2(d-1)}{d-2}} L_{x_1}^2} \lambda _3^{\frac{1}{2}} \Vert u_3 \Vert _{L_t^4 L_{x'}^{\frac{2(d-1)}{d-2}} L_{x_1}^2}\\&\lesssim \lambda _3^{\frac{1}{4}} M_2^{s_c + \frac{1}{2}} N_1^{- \frac{7}{4}} \prod _{i=1,2,3}\Vert u_i \Vert _{V_S^2}. \end{aligned}$$

  Next we consider the case $\lambda _1 \sim \lambda _2 \sim \lambda _3 \sim N_1$. Since $M_2 \ll M_1 \sim \lambda _1$, we may assume that there exists $\ell \in \{2, \ldots , d\}$ such that $|\partial _{\ell } \varphi (\xi )-\partial _{\ell } \varphi (\eta )| > rsim N_1^2$ for $\xi \in {\text {supp}}_{\xi } \widehat{u}_1$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_2$. Then, from (4.2) we get

  $$\begin{aligned} \Vert u_1 u_2 \Vert _{L_t^2 L_x^2} \lesssim M_2^{\frac{d-2}{2}} N_1^{- \frac{1}{2}} \Vert u_1\Vert _{U^2_S} \Vert u_2\Vert _{U_S^2}. \end{aligned}$$

  (4.8)

  On the other hand, for $(\frac{1}{\alpha }, \frac{1}{\beta }) = (\frac{1}{q_1}+\frac{1}{q_2}, \frac{1}{r_1} + \frac{1}{r_2})$, we have

  $$\begin{aligned} \Vert u_1 u_2 \Vert _{L_t^{\alpha } L_{x'}^{\beta }L_{x_1}^2}&\lesssim \lambda _1^{\frac{1}{2}} \Vert u_1\Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_2\Vert _{L_t^{q_2} L_{x'}^{r_2} L_{x_1}^2} \end{aligned}$$

  (4.9)
  $$\begin{aligned}&\lesssim \lambda _1^{\frac{1}{2} -\frac{1}{q_1}-\frac{1}{q_2}} N_1^{-\frac{d-2}{2d-3}+2 \varepsilon } \Vert u_1\Vert _{U_S^{q_1}} \Vert u_2\Vert _{U^{q_2}_S}\end{aligned}$$

  (4.10)
  $$\begin{aligned}&\sim N_1^{-\frac{1}{q_2}-\frac{d-2}{2d-3}+3 \varepsilon } \Vert u_1\Vert _{U_S^{q_1}} \Vert u_2\Vert _{U^{q_2}_S}. \end{aligned}$$

  (4.11)

  We notice that $\alpha , \beta \ge 1$ and $q_1,q_2>2$ if $\varepsilon >0$ is chosen sufficiently small. Let $\theta = \frac{2(d-1)+ 4 (2 d-3)\varepsilon }{(d-1)(2 d -5) - 4 (2 d-3)\varepsilon }$. Then, since

  $$\begin{aligned} \frac{\theta }{\alpha } + \frac{1- \theta }{2} = \frac{1}{q_1'}, \quad \frac{\theta }{\beta } + \frac{1- \theta }{2} = \frac{1}{r_1'}, \end{aligned}$$

  by interpolating the above two estimates (with a similar argument as in the proof of Proposition 3.4), we have

  $$\begin{aligned} \Vert u_1 u_2 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2} \lesssim M_2^{\frac{d-2}{2}(1-\theta )} N_1^{-\frac{1}{2} - \frac{\theta }{q_2} + \frac{\theta }{2 (2 d-3)}+ 3 \varepsilon \theta }\Vert u_1\Vert _{V^2_S} \Vert u_2\Vert _{V_S^2}. \end{aligned}$$

  (4.12)

  This and (4.5) yield

  $$\begin{aligned} \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |&\lesssim \Vert u_1 u_2 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2} \Vert u_{3} \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \\&\lesssim M_2^{\frac{d-2}{2}(1-\theta )} N_1^{-\frac{3 d-5}{2 d-3}- \frac{\theta }{q_2} + \frac{\theta }{2 (2 d-3)}+ 3 \varepsilon (1+\theta )}\prod _{i=1,2,3}\Vert u_i\Vert _{V^2_S}\\&\lesssim M_2^{s_c+\varepsilon }N_1^{-1 - \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}. \end{aligned}$$

• (3) We deal with the last case $M_1 \ll N_1$, $M_2 \ll N_1$. By symmetry, we assume $M_2 \le M_1$. Assume first that $M_1 > rsim (\lambda _3 N_1)^{\frac{1}{2}}$ which implies $\lambda _3 \ll \lambda _1 \sim \lambda _2 \sim N_1$. Thus, we observe that $|\partial _1 \varphi (\xi )-\partial _1 \varphi (\eta )| > rsim N_1^2$ for $\xi \in {\text {supp}}_{\xi } \widehat{u}_2$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_3$. (4.3) implies

  $$\begin{aligned} \Vert u_2 u_3 \Vert _{L_t^2 L_x^2} \lesssim M_{\min }^{\frac{d-1}{2}} N_1^{-1} \Vert u_2\Vert _{U^2_S} \Vert u_3\Vert _{U_S^2}. \end{aligned}$$

  While, similarly to the above observation, we get

  $$\begin{aligned} \Vert u_2 u_3 \Vert _{L_t^{\alpha } L_{x'}^{\beta }L_{x_1}^2} \lesssim \lambda _2^{-\frac{1}{q_1}} \lambda _3^{\frac{1}{2}-\frac{1}{q_2}} M_{\min }^{-\frac{d-2}{2d-3}+2 \varepsilon } \Vert u_2\Vert _{U_S^{q_1}} \Vert u_3\Vert _{U^{q_2}_S}. \end{aligned}$$

  Interpolating the above two, we get

  $$\begin{aligned} \Vert u_2 u_3 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2} \lesssim \lambda _2^{-\frac{\theta }{q_1}} \lambda _3^{\frac{\theta }{2}-\frac{\theta }{q_2}} M_{\min }^{\frac{d-1}{2}(1-\theta )-\frac{d-2}{2 d-3} \theta + 2 \varepsilon \theta } N_1^{-1+\theta } \Vert u_1\Vert _{V^2_S} \Vert u_2\Vert _{V_S^2}. \end{aligned}$$

  Consequently, it follows from $M_1 > rsim \max \{M_{\min },(\lambda _3 N_1)^{\frac{1}{2}}\}$ that

  $$\begin{aligned}&\Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |\\&\quad \lesssim \Vert u_1 \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_2 u_3 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2}\\&\quad \lesssim \lambda _1^{-\frac{1+\theta }{q_1}} \lambda _3^{\frac{\theta }{2}-\frac{\theta }{q_2}} M_1^{-\frac{d-2}{2d-3}+2 \varepsilon } M_{\min }^{\frac{d-1}{2}(1-\theta )-\frac{d-2}{2 d-3} \theta + 2 \varepsilon \theta } N_1^{-1+\theta } \prod _{i=1,2,3}\Vert u_i\Vert _{V^2_S}\\&\quad \lesssim \lambda _1^{-1}\lambda _3^{\varepsilon } M_{\min }^{s_c+\varepsilon }N_1^{-2 \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}. \end{aligned}$$

  In the case $M_1 \ll (\lambda _3 N_1)^{\frac{1}{2}}$, we easily observe that at least one of $u_1$, $u_2$, $u_3$ satisfies ${\text {supp}}\widehat{u}_i \subset \{ (\tau ,\xi ) \, |\, |\tau -\varphi (\xi )| > rsim \lambda _3 N_1^2\}$. Indeed, $M_2 \lesssim M_1 \ll (\lambda _3 N_1)^{\frac{1}{2}}$ yields

  $$\begin{aligned}&|\tau _1- \varphi (\xi )|+|\tau _2 - \varphi (\eta )|+ |\tau _1+\tau _2 - \varphi (\xi +\eta )|\\&\quad > rsim {} |\varphi (\xi +\eta ) - \varphi (\xi ) - \varphi (\eta )| \\&\quad \ge {} x 3| \xi _1 \eta _1\left( \xi _1+\eta _1\right) | - 10\left( |\xi _1|+|\eta _1|\right) \left( |\xi '|^2 + |\eta '|^2\right) > rsim \lambda _3 N_1^2, \end{aligned}$$

  for all $\xi \in {\text {supp}}_{\xi } \widehat{u}_1$, $\eta \in {\text {supp}}_{\xi } \widehat{u}_2$ which satisfy $\xi +\eta \in {\text {supp}}_{\xi } \widehat{u}_3$. In the case ${\text {supp}}\widehat{u}_1 \subset \{ (\tau ,\xi )\, |\, |\tau -\varphi (\xi )| > rsim \lambda _3 N_1^2\}$ and $M_2 \lesssim M_3$, since $M_2 \lesssim M_1 \ll (\lambda _3 N_1)^{\frac{1}{2}}$, it follows from the Strichartz estimates (3.2) and Bernstein’s inequality that

  $$\begin{aligned} \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |&\lesssim \Vert u_1 \Vert _{L_t^2 L_x^2} \Vert u_2 \Vert _{L_t^3 L_{x'}^{3(d-1)} L_{x_1}^2} \Vert u_3\Vert _{L_t^{6} L_{x'}^{\frac{6(d-1)}{3 d-5}} L_{x_1}^{\infty }}\\&\lesssim \lambda _3^{-\frac{1}{2}} N_1^{-1} \Vert u_1 \Vert _{V_S^2} M_{2}^{\frac{d-3}{2}} \Vert u_2 \Vert _{L_t^3 L_{x'}^{\frac{6 (d-1)}{3 d -7}} L_{x_1}^2} \lambda _3^{\frac{1}{2}} \Vert u_3\Vert _{L_t^{6} L_{x'}^{\frac{6(d-1)}{3 d-5}} L_{x_1}^{2}}\\&\lesssim \lambda _3^{-\frac{1}{6}} M_1^{\frac{1}{3} + 2 \varepsilon }M_2^{s_c + \frac{1}{6}-2 \varepsilon } N_1^{-\frac{4}{3}} \prod _{i=1,2,3} \Vert u_i \Vert _{V_S^2}\\&\lesssim \lambda _3^{\varepsilon } M_2^{s_c + \frac{1}{6}-2 \varepsilon } N_1^{-\frac{7}{6}+ \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V_S^2}. \end{aligned}$$

  The other cases are treated similarly. $\square$

As above, Theorem 1.3 follows by the standard argument.