Analytic Smoothing Effect for the Nonlinear Schrödinger Equations Without Square Integrability

Abstract

In this study we consider the Cauchy problem for the nonlinear Schrödinger equations with data which belong to $L^p,$$1<p<2.$ In particular, we discuss analytic smoothing effect with data which satisfy exponentially decaying condition at spatial infinity in $L^p,$$1<p<2.$ We construct solutions in the function space of analytic vectors for the Galilei generator and the analytic Hardy space with the phase modulation operator based on $L^{p}$.

1 Introduction

We consider the local Cauchy problem for the nonlinear Schrödinger equations in setting of the Lebesgue space $L^p,$$1<p<2$:

$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _tu+\frac{1}{2}\Delta u=f(u),\ (t,x)\in I_T\times {\mathbb {R}}^n, \\ u(0)=\phi \in L^p \end{array}\right. } \end{aligned}$$

(1.1)

where $I_T=[0,T],$$T>0,$$i=\sqrt{-1},$$u:I_T\times {\mathbb {R}}^{n}\rightarrow {\mathbb {C}},$$\partial _t=\partial /\partial t,$$\Delta =\sum _{j=1}^n\partial _j^2/\partial x_j^2,$ and $n\ge 1.$ The nonlinearity f satisfies the gauge condition

$$\begin{aligned} e^{i\theta }f(u)=f(e^{i\theta }u),\ \theta \in {\mathbb {R}}. \end{aligned}$$

To be specific, we assume the following two types of nonlinearity in this study

$$\begin{aligned} f(u)&=\lambda |u|^2 u,\ \end{aligned}$$

(1.2)

$$\begin{aligned} f(u)&=\lambda \left( |x|^{-\gamma }*|u|^2\right) u \end{aligned}$$

(1.3)

where $\lambda \in {\mathbb {C}},$$0<\gamma <n$ and $\varphi *\psi$ denotes the convolution

$$\begin{aligned} (\varphi *\psi )(x)=\displaystyle \int _{{\mathbb {R}}^n}\varphi (x-y)\psi (y)dy, \ x\in {\mathbb {R}}^n. \end{aligned}$$

There is a large literature on the Cauchy problem for the nonlinear Schrödinger equations in the $L^2$-based framework (see for instance [3, 4, 6, 21, 27,28,29] and reference therein).

Analyticity and analytic smoothing effect for solutions to nonlinear evolution equations have been studied in many papers ([5, 8,9,10,11, 13,14,15, 20, 22,23,24]).

In particular, analytic smoothing effect for the nonlinear Schrödinger equations and the Hartree equations in the $L^2$ setting with data which satisfy exponentially decaying condition has been studied in [8,9,10,11, 13,14,15, 23,24,25] (see also reference therein).

On the other hand, as far as the authors know, there are no results on analytic smoothing effect for the nonlinear Schrödinger equations in the $L^p$-framework with $p\not =2.$ In the present paper we discuss the problem of analytic smoothing effect for solutions to (1.1) in the $L^p$-framework. We believe this problem is interesting since exponentially decaying $L^p$-functions for $1<p<2$ do not necessarily belong to $L^2.$ In fact, if $1<p<2,$ there exists

$$\begin{aligned} \phi \in L^p{\setminus } L^2 \end{aligned}$$

such that

$$\begin{aligned} \sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{L^p}<\infty . \end{aligned}$$

See Proposition 1 below for details. Furthermore, it is sufficient to consider the case $1<p<2,$ since any exponentially decaying $L^q$-function with $2<q<\infty$ is an $L^2$-function. To be more precise, let $D\subset {\mathbb {R}}^n$ be a domain with $0\in D$ and $2<q<\infty .$ If $\phi \in L^{q}$ satisfying

$$\begin{aligned} \sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{L^q}<\infty , \end{aligned}$$

then

$$\begin{aligned} \sup _{\delta \in D^\prime }\left\| e^{\delta \cdot x}\phi \right\| _{L^1}\le C \sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{L^q} \end{aligned}$$

for $D^\prime \Subset D$ (see Appendix of [15] and Chapter III of [26]). We put

$$\begin{aligned} \widehat{L^q}= \left\{ \phi \in \mathcal {S}^\prime ;\ {\mathcal {F}}[\phi ]\in L^{q^\prime } \right\} . \end{aligned}$$

If $\phi \in \widehat{L^q}$ satisfying

$$\begin{aligned} \sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{\widehat{L^q}}<\infty , \end{aligned}$$

then by the boundedness of the Fourier transform, we have

$$\begin{aligned} \sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{L^q}\le C \sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{\widehat{L^q} }<\infty . \end{aligned}$$

Hence if data satisfy exponentially decaying condition in $L^q$ or $\widehat{L^q},$ then data decay exponentially in $L^1 \cap L^q\subset L^2.$

Analytic smoothing effect for the cubic nonlinear Schrödinger equations in minimal regularity Sobolev setting based on $L^2$ has been studied in [23, 24]. Analytic smoothing effect for the nonlinear Schrödinger equations with non-gauge invariant quadratic nonlinearity in terms of the generator of dilations $P=2t\partial _t+x\cdot \nabla$ in the framework of negative exponent Sobolev space $H^s,$$s>-3/4$ has been studied in [20].

In the previous papers [7, 12, 17,18,19, 30, 31], the authors have attempted to construct solutions and proved the existence of solutions to the nonlinear Schrödinger equations in the framework of $L^p$ and $\widehat{L^p}.$ Especially, in the $L^p$-setting, in [31] the author proved the existence of local solutions to the 1D cubic nonlinear Schrödinger equations for data in $L^p$ with $1<p<2$. Later, in [12] we exploited his approach to show similar local well-posedness results for the Hartree equation for data in the Bessel potential spaces $H^{s,p}$ under suitable conditions on s, p, n. In particular, for $0<\gamma <\min (2,n)$, we obtained the local well-posedness for data in $L^p$ with $\max (\frac{2n}{n-\gamma +2},\frac{2n}{n+\gamma })<p<2$. In this paper, based on these local existence results, we investigate analytic smoothing effect for the 1D cubic NLS and the Hatree equation for data in $L^p$ when p is in the range stated above. To be more precise, our main purpose of this study is to show analytic smoothing effect for the Cauchy problem (1.1) in the framework of $L^{p}$-based spaces of functions as analytic vector for the Galilei generator $J=e^{i\frac{t}{2}\Delta }xe^{-i\frac{t}{2}\Delta }$ and the $L^{p}$-based analytic Hardy space characterized by the operator $A_\delta =e^{i\frac{t}{2}\Delta }e^{\delta \cdot x}e^{-i\frac{t}{2}\Delta }.$

Finally, the condition $p>\frac{2n}{n+\gamma }$ for the local well-posedness for the Hartree equation may seem unusual, since it is stronger than the condition for the cubic NLS. However, it is conjectured that this condition is optimal. See Remark 3 below.

We use the following notation throughout this paper. $L^p=L^p({\mathbb {R}}^n),$$1\le p\le \infty$ is the usual Lebesgue space. The Fourier transform ${\mathcal {F}}:\psi \mapsto \widehat{\psi }$ is defined by

$$\begin{aligned} \widehat{\psi }(\xi )=\frac{1}{(2\pi )^{n/2}}\int _{{\mathbb {R}}^n}e^{-i\xi \cdot x}\psi (x)dx,\ \xi \in {\mathbb {R}}^n, \end{aligned}$$

where $x\cdot y=\sum _{j=1}^nx_jy_j$ is the usual scaler product in ${\mathbb {R}}^n,$ and ${\mathcal {F}}^{-1}$ is the inverse Fourier transform. The Schrödinger propagator is defined by $U(t)\psi =\left( U\psi \right) (t)=e^{i\frac{t}{2}\Delta }\psi ={\mathcal {F}}^{-1}\left[ e^{-i\frac{t}{2}|\xi |^2}{\mathcal {F}}\left[ \psi \right] \right] ,$$t\in {\mathbb {R}},$ we often use the notation $U^{-1}(t)\psi =\left( U^{-1}\psi \right) (t)=U(-t)\psi ,$$t\in {\mathbb {R}}.$ As well known, U and $U^{-1}$ have the following factorization formula (see Chapter 2 of [3]):

$$\begin{aligned} U(t)\psi ={\mathcal {M}}(t)D(t){\mathcal {F}}{\mathcal {M}}(t)\psi ,\ U^{-1}(t)\psi ={\mathcal {M}}(-t){\mathcal {F}}^{-1}D^{-1}(t){\mathcal {M}}(-t)\psi \end{aligned}$$

for $t\not =0,$ where the phase modulation operator ${\mathcal {M}}(t):\psi \mapsto e^{i\frac{|x|^2}{2t}}\psi ,$ dilations $D(t):\psi \mapsto (it)^{-n/2}\psi \left( \frac{\cdot }{t}\right)$ and its inverse $D^{-1}(t)=i^nD\left( t^{-1}\right) ,$$t\not =0.$ We put the linear Schrödinger operator by

$$\begin{aligned} {\mathcal {L}}=i\partial _t+\frac{1}{2}\Delta =Ui\partial _tU^{-1}. \end{aligned}$$

The Duhamel integral operator is defined by $S\left[ f\right] (t)=\left( \int _0^{\cdot }U(\cdot -s)f(\tau )d\tau \right) (t)=\int _0^tU(t-\tau )f(\tau )d\tau$, $t\in {\mathbb {R}}.$ The Galilei generator is defined by

$$\begin{aligned} J(t)=U(t)xU(-t)=x+it\nabla ,\ t\in {\mathbb {R}}. \end{aligned}$$

We introduce an operator which gives analytic continuation (see also [11, 15, 23])

$$\begin{aligned} A_\delta (t)=U(t)e^{\delta \cdot x}U(-t),\ \delta \in {\mathbb {R}}^n \end{aligned}$$

for $t\in {\mathbb {R}}.$ We see that $A_\delta$ has another representation such as

$$\begin{aligned} A_\delta (t)={\mathcal {M}}(t)e^{it\delta \cdot \nabla }{\mathcal {M}}(-t),\ t\not =0, \end{aligned}$$

where $e^{it\delta \cdot \nabla }\psi ={\mathcal {F}}^{-1}\left[ e^{-t\delta \cdot \xi }{\widehat{\psi }}\right] ,$$\delta \in {\mathbb {R}}^n.$ The following commutation relation

$$\begin{aligned} \left[ J,{\mathcal {L}}\right] =U[x,i\partial _t]U^{-1}=0,\ \left[ A_\delta ,{\mathcal {L}}\right] =U[e^{\delta \cdot x},i\partial _t]U^{-1}=0 \end{aligned}$$

holds. The analytic Hardy space is defined by (see Chapter III of [26]):

$$\begin{aligned} \mathcal {H}^p(\Omega )=\left\{ \psi : \text{ analytic } \text{ on } {\mathbb {R}}^n+i\Omega ;\ \left\| \psi \right\| _{\mathcal {H}^p(\Omega )}=\sup _{y\in \Omega }\left\| \psi (\cdot +iy)\right\| _{L^p}<\infty \right\} \end{aligned}$$

with domain $\Omega \subset {\mathbb {R}}^n.$

2 Function Spaces

We introduce the following basic function spaces

$$\begin{aligned}&{\mathscr {X}}(I_T)\\ {}&\quad =\left\{ u:I_T\times {\mathbb {R}}^n \rightarrow {\mathbb {C}};\ U^{-1}u\in C(I_T;L^p),\ \left\| u\right\| _{{\mathscr {X}}(I_T)}=\left\| U^{-1}u \right\| _{L^\infty (I_T;L^p)}<\infty \right\} ,\\&X^{p}_{q,\theta }(I_T)\\ {}&\quad =\left\{ u:\ I_T\times {\mathbb {R}}^n\rightarrow {\mathbb {C}};\ \left\| u\right\| _{X^{p}_{q,\theta }(I_T)}=\left\{ \int _{I_T} \tau ^{\theta q}\left\| \left( U^{-1}{\mathcal {L}}u\right) (\tau ) \right\| _{L^p}^qd\tau \right\} ^{1/q}<\infty \right\} \end{aligned}$$

for $1\le p,q \le \infty ,$$\theta >0$ and we put the function spaces associated with $X^{p}_{q,\theta }(I_T)$ by

$$\begin{aligned} \widetilde{X}^{p}_{q,\theta }(I_T)=\left\{ u\in X^{p}_{q,\theta }(I_T);\ u(0)\in L^p,\ \left\| u\right\| _{\widetilde{X}^{p}_{q,\theta }(I_T)}=\left\| u(0)\right\| _{L^p}+\left\| u\right\| _{X^{p}_{q,\theta }(I_T)}<\infty \right\} . \end{aligned}$$

The function spaces such as above are firstly introduced in [31].

We introduce the following weighted function spaces:

$$\begin{aligned} G^{a}_{p}&=\left\{ \phi \in L^p;\ \left\| \phi \right\| _{G^{a}_{p}}=\sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| x^\alpha \phi \right\| _{L^p}<\infty \right\} ,\\ G^{D}_{p}&=\left\{ \phi \in L^p;\ \left\| \phi \right\| _{G^{D}_{p}}=\sup _{\delta \in D}\left\| e^{\delta \cdot x}\phi \right\| _{L^p}<\infty \right\} . \end{aligned}$$

The function space of analytic vectors for J is defined by

$$\begin{aligned}&G^{a}_{p,q,\theta }(I_T)=\left\{ u\in \widetilde{X}^p_{q,\theta }(I_T) ;\ \left\| u\right\| _{G^{a}_{p,q,\theta }(I_T)} =\left\| u(0)\right\| _{G^a_p}+\sum _{\alpha \ge 0}\frac{a^{\alpha }}{\alpha !}\left\| J^\alpha u\right\| _{X^p_{q,\theta }(I_T) }<\infty \right\} , \end{aligned}$$

for $a\in (0,\infty )^n$ and the analytic Hardy space with respect to $A_\delta$ is defined by

$$\begin{aligned} G^{D}_{p,q,\theta }(I_T)=\left\{ u\in \widetilde{X}^p_{q,\theta }(I_T);\ \left\| u\right\| _{G^{D}_{p,q,\theta }(I_T)} =\left\| u(0)\right\| _{G^D_p}+\sup _{\delta \in D}\left\| A_\delta u\right\| _{X^p_{q,\theta }(I_T) }<\infty \right\} \end{aligned}$$

for domain $D\subset {\mathbb {R}}^n.$

Our motivation of this study is based on the following proposition:

Proposition 1

Let $1<p<2$ and bounded domain $D\subset {\mathbb {R}}^n.$ Then

$$\begin{aligned} G^D_p {\setminus } L^2\not = \varnothing . \end{aligned}$$

Proof

The following function belongs to $G^D_p{\setminus } L^2,$

$$\begin{aligned} \phi (x)= {\left\{ \begin{array}{ll} |x|^{-n/(2-\varepsilon )}, &{}|x|\le 1,\\ e^{-r|x|}, &{}|x|>1 \end{array}\right. } \end{aligned}$$

where $0<\varepsilon <2-p$ and sufficiently large $r>0.$$\square$

The analyticity of functions which belong to $G^{a}_{p,q,\theta }(I_T)$ or $G^{D}_{p,q,\theta }(I_T)$ is shown by the following proposition:

Proposition 2

Let $1\le p < 2$ and $\frac{1}{q}+\frac{1}{\theta }>1.$

• (1) Let $u\in G^{a}_{p,q,\theta }(I_T).$ Then $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{p^\prime }\left( \prod _{j=1}^n(-a_jt,a_j t)\right)$$t\not =0$, where $a\in (0,\infty )^n$.

• (2) Let $u\in G^{D}_{p,q,\theta }(I_T).$ Then $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{p^\prime }\left( tD\right) ,$$t\not =0,$ where $D\subset {\mathbb {R}}^n$ is a domain with $0\in D$.

Proof

Because

$$\begin{aligned} \widetilde{X}^p_{q,\theta }(I_T)\subset {\mathscr {X}}(I_T) \end{aligned}$$

with

$$\begin{aligned} \left\| u\right\| _{{\mathscr {X}}(I_T)}&\le \left\| u(0)\right\| _{L^p}+T^{1-\theta \left( 1-\frac{1}{q} \right) }\left\| u\right\| _{X^p_{q,\theta }(I_T)}\\&\le \max \left\{ 1,T^{1-\theta \left( 1-\frac{1}{q}\right) }\right\} \left\| u\right\| _{\widetilde{X}^p_{q,\theta }(I_T)}, \end{aligned}$$

we see that

$$\begin{aligned} \sup _{\delta \in \prod _{j=1}^n(-a_j,a_j) }\left\| e^{\delta \cdot x}U^{-1}u\right\| _{L^\infty (I_T;L^p)} \le \sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| x^\alpha U^{-1}u\right\| _{L^\infty (I_T;L^p)}<\infty . \end{aligned}$$

Hence, it is sufficient to show real analyticity of $u\in G^\Omega _{p,q,\theta }$ with $\Omega =\prod _{j=1}^n(-a_j,a_j)$ in the case (1) and with $\Omega =D$ in the case (2). We see that

$$\begin{aligned} \left\| e^{\delta \cdot x}U(-t)u(t)\right\| _{L^1}&=\left\| e^{\delta \cdot x}{\mathcal {F}}^{-1}D^{-1}(t){\mathcal {M}}(-t)u(t) \right\| _{L^1}\\&= \left\| e^{\delta \cdot x}D(t){\mathcal {F}}^{-1} {\mathcal {M}}(-t)u(t)\right\| _{L^1}\\&= \left\| D(t)e^{t\delta \cdot x}{\mathcal {F}}^{-1} {\mathcal {M}}(-t)u(t)\right\| _{L^1}\\&=|t|^{n/2} \left\| e^{t\delta \cdot x}{\mathcal {F}}^{-1} {\mathcal {M}}(-t)u(t)\right\| _{L^1}\\&=|t|^{n/2} \left\| e^{-t\delta \cdot \xi }{\mathcal {F}} \left[ ({\mathcal {M}}^{-1}u)(t)\right] \right\| _{L^1}<\infty \end{aligned}$$

for all $t\not =0,\ \delta \in \Omega .$ Therefore, $({\mathcal {M}}^{-1}u)(t)$ is real analytic and has an analytic continuation

$$\begin{aligned} \left( {\mathcal {M}}^{-1}A_\delta u\right) (t,x)&= e^{it\delta \cdot \nabla }{\mathcal {M}}^{-1}u(t,x)\\&=\frac{1}{(2\pi )^{n/2}}\int _{{\mathbb {R}}^n}e^{i(x+it\delta )\cdot \xi }{\mathcal {F}}\left[ \left( {\mathcal {M}}^{-1}u\right) (t)\right] (\xi )d\xi \end{aligned}$$

for all $x+it\delta \in {\mathbb {R}}^n+it\Omega$ (see Appendix of [15]). Also we have

$$\begin{aligned} \sup _{\delta \in D}\left\| \left( {\mathcal {M}}^{-1}A_\delta u\right) (t)\right\| _{L^{p^\prime }}\le C |t|^{-n\left( \frac{1}{2}-\frac{1}{p^\prime }\right) }\sup _{\delta \in D}\left\| e^{\delta \cdot x}\left( U^{-1}u\right) (t)\right\| _{L^p}<\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{\delta \in \prod _{j=1}^n(-a_j,a_j)}\left\| \left( {\mathcal {M}}^{-1}A_\delta u\right) (t)\right\| _{L^{p^\prime }}&\le \sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| \left( J^\alpha u\right) (t)\right\| _{L^{p^\prime }}\\&\le C |t|^{-n\left( \frac{1}{2}-\frac{1}{p^\prime }\right) }\sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| x^\alpha \left( U^{-1}u\right) (t)\right\| _{L^p}\\&<\infty . \end{aligned}$$

$\square$

By the relation $e^{\delta \cdot x}U^{-1}U\phi =e^{\delta \cdot x}\phi ,$ we immediately have the following fact for the free solutions:

Corollary 1

Let $1\le p< 2.$

• (1) If $\phi \in G^a_p,$ then $U(t)\phi ,$$t\in {\mathbb {R}}{\setminus }\{0\},$ is real analytic and has an analytic continuation to ${\mathbb {R}}^n+it\prod _{j=1}^n(-a_j,a_j).$

• (2) If $\phi \in G^D_p,$ then $U(t)\phi ,$$t\in {\mathbb {R}}{\setminus }\{0\},$ is real analytic and has an analytic continuation to ${\mathbb {R}}^n+itD$.

3 Main Results

We put the interval $I_T=[0,T].$

Theorem 1

Let $n=1,$$1<p<2$ and $a\in (0,\infty ).$ Then for any $\eta >0$ there exists $T=T(\eta )>0$ such that; for any $\phi \in G^{a}_{p},$ satisfying $\left\| \phi \right\| _{G^{a}_{p}}\le \eta$ then the Cauchy problem (1.1)–(1.2) has a unique solution $u\in G^{a}_{p,p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T).$ Furthermore, $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{p^\prime }((-at,at)),$$t\in I_T{\setminus }\{0\}.$

Theorem 2

Let $n=1,$$1<p<2.$ Let a domain $D\subset {\mathbb {R}}$ satisfying $0\in D$ and $-D=D.$ Then for any $\eta >0$ there exists $T=T(\eta )>0$ such that; for any $\phi \in G^{D}_{p},$ satisfying $\left\| \phi \right\| _{G^{D}_{p}}\le \eta$ then the Cauchy problem (1.1)–(1.2) has a unique solution $u\in G^{D}_{p,p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T).$ Furthermore, $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{p^\prime }(tD),$$t\in I_T{\setminus }\{0\}.$

Theorem 3

Let $n\ge 1,$$0<\gamma <\min \left( n,2\right) ,$$\max \left( \frac{2n}{n+\gamma },\frac{2n}{n-\gamma +2}\right)<p<2$ and $a\in (0,\infty )^n.$ Then for any $\eta >0$ there exists $T=T(\eta )>0$ such that; for any $\phi \in G^{a}_{p},$ satisfying $\left\| \phi \right\| _{G^{a}_{p}}\le \eta$ then the Cauchy problem (1.1)–(1.3) has a unique solution $u\in G^{a}_{p,q,2n\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T),$ with $q=\frac{2p}{(n+\gamma )p-2n}.$ Furthermore, $\left( {\mathcal {M}}^{-1}u\right) (t) \in \mathcal {H}^{p^\prime }\left( \prod _{j=1}^n(-a_jt,a_jt)\right) ,$$t\in I_T{\setminus }\{0\}.$

Theorem 4

Let $n\ge 1,$$0<\gamma < \min (n,2)$ and $\max \left( \frac{2n}{n+\gamma },\frac{2n}{n-\gamma +2}\right)<p<2.$ Let a domain $D\subset {\mathbb {R}}^n$ satisfying $0\in D$ and $-D=D.$ Then for any $\eta >0$ there exists $T=T(\eta )>0$ such that; for any $\phi \in G^{D}_{p},$ satisfying $\left\| \phi \right\| _{G^{D}_{p}}\le \eta$ then the Cauchy problem (1.1)–(1.3) has a unique solution $u\in G^{D}_{p,q,2n\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T),$ with $q=\frac{2p}{(n+\gamma )p-2n}.$ Furthermore, $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{p^\prime }(tD),$$t\in I_T{\setminus }\{0\}.$

Remark 1

$G^a_p\subset G^\Omega _p\subset G^b_p,$ with $\Omega =\prod _{j=1}^n(-a_j,a_j),$ and $0<b_j<a_j,$$j=1,2,\cdots ,n$ (see Theorem 2 in [15]).

Remark 2

Let $1\le p\le \infty .$ We see that

$$\begin{aligned}&G^a_{p}\subset G^b_1,\ 0<b_j<a_j,\ G^{D_1}_{p}\subset G^{D_2}_1,\ D_2\Subset D_1 ,\\&\mathcal {H}^{p}(D_1)\subset \mathcal {H}^\infty (D_2),\ D_2\Subset D_1 \end{aligned}$$

where $D_1,\ D_2 \subset {\mathbb {R}}^n$ are domain (see Appendix of [15] and Chapter III of [26]). Therefore, the Cauchy data $\phi \in G^\Omega _p,$ satisfy $\phi \in G^{\Omega ^\prime }_1$ and solutions obtained in Theorems 1–4, $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{p^\prime }(t\Omega ),$$t\not =0,$ satisfy $\left( {\mathcal {M}}^{-1}u\right) (t)\in \mathcal {H}^{\infty }(t{\Omega }^\prime ),$$t\not =0,$ where $\Omega ^\prime \Subset \Omega$ with $\Omega =\prod _{j=1}^n(-a_j,a_j)$ or $\Omega =D.$

Remark 3

In Theorems 3 and 4, we need

$$\begin{aligned} p>\frac{2n}{n-\gamma +2}\ \ \text{ and } \ \ p>\frac{2n}{n+\gamma }. \end{aligned}$$

The exponent $\frac{2n}{n-\gamma +2}$ appearing in the first condition is called a scaling limit which is well known and is considered as one candidate of the thresholds for the local well-posedness of (1.1)–(1.3). Thus our local result can reach almost critical $L^p$ spaces if $n\ge 2$ and $\gamma >1$. The exponent in the second condition, on the other hand, seems unfamiliar and one may wonder if the local result still holds for p below this exponent. However, it is conjectured that the Cauchy problem is ill posed for $p<\frac{2n}{n+\gamma }$, because of the singularity at zero frequency. This is deduced from the recent works [2] and [16] which study the well-posendess of (1.1)–(1.3) in $\widehat{L^p}$. For details, see the introduction in [16]. Note that $\frac{2n}{n+\gamma }\rightarrow 1$ as $\gamma \rightarrow n$ and thus the limit coincides with the lower threshold of the local results for the cubic NLS (Theorems 1 and 2). Note also that $p=\frac{2n}{n+\gamma }$ is the exponent such that the trilinear operator

$$\begin{aligned} (u_1,u_2,u_3) \mapsto {\mathcal {F}}\left[ \left( |x|^{-\gamma } *(u_1 \overline{u_2})\right) u_3 \right] \end{aligned}$$

is defined and is continuous from $L^p\times L^p \times L^p$ to $L^p$.

4 Key Lemmas

We introduce the following two types of trilinear form ${\mathcal {T}}_0$ and ${\mathcal {T}}_\gamma$ by

$$\begin{aligned} {\mathcal {T}}_0(u_1,u_2,u_3)= u_1\overline{u_2}u_3 \end{aligned}$$

and

$$\begin{aligned} {\mathcal {T}}_\gamma (u_1,u_2,u_3)=\left( |x|^{-\gamma }*u_1\overline{u_2}\right) u_3, \end{aligned}$$

respectively. Then, we see that

$$\begin{aligned}&U(-t){\mathcal {T}}_0(u_1,u_2,u_3)\\&\quad ={\mathcal {M}}(-t){\mathcal {F}}^{-1}_{\xi \rightarrow x}i^nD(t^{-1})({\mathcal {M}}(-t)u_1\overline{{\mathcal {M}}(-t)u_2}) {\mathcal {M}}(-t)u_3\\&\quad =Ct^{-n}{\mathcal {M}}(-t){\mathcal {F}}^{-1}_{\xi \rightarrow x}(D(t^{-1}){\mathcal {M}}(-t)u_1\overline{D(t^{-1}){\mathcal {M}}(-t)u_2}) D(t^{-1}){\mathcal {M}}(-t)u_3\\&\quad =Ct^{-n}\left( {\mathcal {M}}(t)U(-t)u_1*\left( \overline{{\mathcal {M}}(t)U(-t)u_2}(-\cdot )\right) \right) *U(-t)u_3 \end{aligned}$$

and

$$\begin{aligned}&U(-t){\mathcal {T}}_\gamma (u_1,u_2,u_3)\\&\quad ={\mathcal {M}}(-t){\mathcal {F}}^{-1}_{\xi \rightarrow x}i^nD(t^{-1})(|\xi |^{-\gamma }*{\mathcal {M}}(-t)u_1\overline{{\mathcal {M}}(-t)u_2}) {\mathcal {M}}(-t)u_3\\&\quad =C|t|^{-\gamma }{\mathcal {M}}(-t){\mathcal {F}}^{-1}_{\xi \rightarrow x}(|\xi |^{-\gamma }*D(t^{-1}){\mathcal {M}}(-t)u_1\overline{D(t^{-1}){\mathcal {M}}(-t)u_2})\\&\qquad \times D(t^{-1}){\mathcal {M}}(-t)u_3\\&\quad =C|t|^{-\gamma }\left( |x|^{-(n-\gamma )}{\mathcal {M}} (t)U(-t)u_1*\left( \overline{{\mathcal {M}}(t)U(-t)u_2} (-\cdot )\right) \right) *U(-t)u_3, \end{aligned}$$

for $t\not =0.$

Lemma 1

([31]) Let $n=1.$ We have

$$\begin{aligned} \left\| \left( U^{-1}{\mathcal {T}}_0(u_1,u_2,u_3)\right) (t) \right\| _{L^1}\le C |t|^{-1}\prod _{j=1}^3\left\| \left( U^{-1}u_j\right) (t)\right\| _{L^1}, \end{aligned}$$

for $t\not =0,$

$$\begin{aligned}&\sup _{\tau \in I_T}\left( \tau \left\| \left( U^{-1}{\mathcal {T}}_0 \left( u_1,u_2,u_3\right) \right) (\tau )\right\| _{L^1}\right) \\&\quad \le C \prod _{j=1}^3\left\{ \left\| u_j(0)\right\| _{L^1} +\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}u_j\right) (\tau ) \right\| _{L^1}d\tau \right\} , \end{aligned}$$

and

$$\begin{aligned}&\left\{ \int _{ I_T}\left\| \left( U^{-1}{\mathcal {T}}_0(u_1,u_2,u_3) \right) (\tau )\right\| ^{2}_{L^2}d\tau \right\} ^{1/2}\\&\quad \le C \prod _{j=1}^3\left\{ \left\| u_j(0)\right\| _{L^2}+\int _{I_T} \left\| \left( U^{-1} {\mathcal {L}}u_j\right) (\tau )\right\| _{L^2}d\tau \right\} . \end{aligned}$$

Lemma 2

Let $n\ge 1$ and $0<\gamma <n.$ We have

$$\begin{aligned} \left\| \left( U^{-1}{\mathcal {T}}_\gamma (u_1,u_2,u_3)\right) (t) \right\| _{L^{\frac{2n}{n+\gamma }}}\le C |t|^{-\gamma }\prod _{j=1}^3 \left\| \left( U^{-1}u_j\right) (t)\right\| _{L^{\frac{2n}{n+\gamma }}} \end{aligned}$$

for all $t\not =0,$

$$\begin{aligned}&\sup _{\tau \in I_T }\left( \tau ^{\gamma }\left\| \left( U^{-1} {\mathcal {T}}_\gamma (u_1,u_2,u_3)\right) (\tau )\right\| _{L^{\frac{2n}{n+\gamma }}}\right) \\&\quad \le C\prod _{j=1}^3\left( \left\| u_j(0)\right\| _{L^{\frac{2n}{n+\gamma }}}+\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}u_j \right) (\tau )\right\| _{L^{\frac{2n}{n+\gamma }}}d\tau \right) \end{aligned}$$

and if $0<\gamma <\min (n,2),$ then

$$\begin{aligned}&\left\{ \int _{I_T}\left\| \left( U^{-1}{\mathcal {T}}_\gamma (u_1,u_2,u_3)\right) (\tau )\right\| _{L^2}^{2/\gamma } \right\} ^{\gamma /2}\\&\quad \le C \prod _{j=1}^3\left\{ \left\| u_j(0) \right\| _{L^2}+\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}u_j \right) (\tau )\right\| _{L^2}d\tau \right\} . \end{aligned}$$

Proof

$$\begin{aligned}&|t|^{\gamma }\left\| U(-t){\mathcal {T}}_\gamma (u_1,u_2,u_3)\right\| _{L^{p_0}}\\&\quad =C\left\| |x|^{-(n-\gamma )}{\mathcal {M}}(t)U(-t)u_1 *\left( \overline{{\mathcal {M}}(t)U(-t)u_2}(-\cdot )\right) *U(-t)u_3\right\| _{L^{p_0}}\\&\quad \le C \left\| |x|^{-(n-\gamma )}{\mathcal {M}}(t)U(-t)u_1 *\left( \overline{{\mathcal {M}}(t)U(-t)u_2}(-\cdot )\right) \right\| _{L^{\rho _1}}\left\| U(-t)u_3\right\| _{L^{p_3}}\\&\quad =C \left\| {\mathcal {F}}^{-1}\left[ |\xi |^{-\gamma }*\widehat{{\mathcal {M}}(t)U(-t)u_1}\widehat{ \left( \overline{{\mathcal {M}}(t) U(-t)u_2}(-\cdot )\right) }\right] \right\| _{L^{\rho _1}}\left\| U(-t) u_3\right\| _{L^{p_3}}\\&\quad \le C\left\| |\xi |^{-\gamma }*\widehat{{\mathcal {M}}(t)U(-t)u_1} \widehat{ \left( \overline{{\mathcal {M}}(t)U(-t)u_2}(-\cdot ) \right) }\right\| _{L^{\rho ^\prime _1}}\left\| U(-t)u_3\right\| _{L^{p_3}} \end{aligned}$$

for $\frac{1}{p_0}=\frac{1}{\rho _1}+\frac{1}{p_3}-1,$ with $2\le \rho _1\le \infty .$ By the Hardy–Littlewood–Sobolev inequality, with $\frac{1}{\rho ^\prime _1}=\frac{1}{\rho _2}+\frac{\gamma }{n}-1,$$2\le \rho _2 \le \infty$ and $n-\gamma <\frac{n}{\rho _2},$ we have

$$\begin{aligned}&|t|^\gamma \left\| U(-t){\mathcal {T}}_\gamma (u_1,u_2,u_3)\right\| _{L^{p_0}}\\&\quad \le C\left\| |\xi |^{-\gamma }*\widehat{{\mathcal {M}}(t) U(-t)u_1}\widehat{ \left( \overline{{\mathcal {M}}(t)U(-t)u_2}(-\cdot ) \right) }\right\| _{L^{\rho ^\prime _1}}\left\| U(-t)u_3\right\| _{L^{p_3}}\\&\quad \le C \left\| {\mathcal {F}}\left[ {\mathcal {M}}(t)U(-t)u_1 *\left( \overline{{\mathcal {M}}(t)U(-t)u_2}\right) (-\cdot ) \right] \right\| _{L^{\rho _2}}\left\| U(-t)u_3\right\| _{L^{p_3}}\\&\quad \le C \left\| {\mathcal {M}}(t)U(-t)u_1*\left( \overline{{\mathcal {M}}(t)U(-t)u_2}\right) (-\cdot ) \right\| _{L^{\rho _2^\prime }}\left\| U(-t)u_3\right\| _{L^{p_3}}\\&\quad \le C\prod _{j=1}^3 \left\| U(-t)u_j\right\| _{L^{p_j}} \end{aligned}$$

for $\frac{1}{\rho _2^\prime }=\frac{1}{p_1}+\frac{1}{p_2}-1$ and $\frac{1}{p_0}=\sum _{j=1}^3\frac{1}{p_j}+\frac{\gamma }{n}-1.$ In particular, $p_j=\frac{2n}{n+\gamma },$$j=0,1,2,3,$ satisfies these conditions. By

$$\begin{aligned} u_j=Uu_j(0)-iS\left[ {\mathcal {L}}u_j\right] ,\ U^{-1}u_j=u_j(0)-iU^{-1}S\left[ {\mathcal {L}}u_j\right] , \end{aligned}$$

we obtain the first and second inequalities. Finally, by the Hardy–Littlewood–Sobolev inequality with $\frac{\gamma }{3n}=\frac{3n-2\gamma }{3n}+\frac{\gamma }{n}-1$ and the Hölder inequality with $\frac{1}{2}=\frac{\gamma }{3n}+\frac{3n-2\gamma }{6n},$ we have

$$\begin{aligned} \left\{ \int _{I_T}\left\| {\mathcal {T}}_\gamma (u_1,u_2,u_3)\right\| ^{2/\gamma }_{L^2}\right\} ^{\gamma /2}&\le C\prod _{j=1}^3\left\| u_j\right\| _{L^{6/\gamma }\left( I_T;L^{\frac{6n}{3n-2\gamma }}\right) }, \end{aligned}$$

where $\left( \frac{6}{\gamma },\frac{6n}{3n-2\gamma }\right)$ is an admissible pair and by the Strichartz estimate

$$\begin{aligned}&\left\{ \int _{I_T}\left\| {\mathcal {T}}_\gamma (u_1,u_2,u_3)(\tau ) \right\| ^{2/\gamma }_{L^2}d\tau \right\} ^{\gamma /2} \\&\quad \le C \prod _{j=1}^3\left\{ \left\| u_j(0)\right\| _{L^2} +\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}u_j\right) (\tau ) \right\| _{L^2}d\tau \right\} . \end{aligned}$$

This completes the proof. $\square$

We obtain the following two inequalities by the multi-linear interpolation between $\tau ^{-\alpha }L^\infty \left( \tau ^{-2}d\tau ,I_T;L^1\right)$ and $\tau ^{-\alpha }L^q(\tau ^{-2}d\tau ,I_T;L^2),$ for $\alpha =1,\gamma$ respectively (see Chapter 4 of [1] and [31]), where

$$\begin{aligned} \left\| u\right\| _{\tau ^{\alpha } L^r(\tau ^{-2}d\tau ,I_T;L^p)}=\displaystyle \left\{ \int _{I_T}\tau ^{\alpha r}\left\| u(\tau ,\cdot )\right\| ^r_{L^p}\tau ^{-2}d\tau \right\} ^{1/r},\ 1\le r<\infty ,\ 1\le p\le \infty \end{aligned}$$

and

$$\begin{aligned} \left\| u\right\| _{\tau ^{\alpha } L^\infty (\tau ^{-2}d\tau ,I_T;L^p)}=\displaystyle \sup _{\tau \in I_T}\tau ^{\alpha }\left\| u(\tau ,\cdot )\right\| _{L^p},\ 1\le p\le \infty . \end{aligned}$$

Lemma 3

([12, 31]) We have

$$\begin{aligned}&\left\{ \int _{I_T}\tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) q} \left\| \left( U^{-1}{\mathcal {T}}_0(u_1,u_2,u_3)\right) (\tau ) \right\| _{L^p}^{q}d\tau \right\} ^{1/q}\\&\quad \le C\prod _{j=1}^3\left\{ \left\| u_j(0)\right\| _{L^p} +\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}u_j\right) (\tau ) \right\| _{L^p}d\tau \right\} , \end{aligned}$$

for $n=1,$$1<p<2,$$q=p^\prime$ and

$$\begin{aligned}&\left\{ \int _{I_T}\tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) q}\left\| \left( U^{-1}{\mathcal {T}}_\gamma (u_1,u_2,u_3)\right) (\tau )\right\| _{L^p}^qd\tau \right\} ^{1/q}\\&\quad \le C\prod _{j=1}^3\left\{ \left\| u_j(0)\right\| _{L^p}+\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}u_j\right) (\tau )\right\| _{L^p}d\tau \right\} , \end{aligned}$$

for $0<\gamma <\min (n, 2),$$\frac{2n}{n+\gamma }<p<2,$$q=\frac{2p}{(n+\gamma )p-2n}.$

Lemma 4

• (1) Let $1\le p<2,$$1\le q\le \infty ,$$\theta >0$ and let $u\in G^D_{p,q,\theta }(I_T).$ Then

  $$\begin{aligned} A_\delta (|u|^2u)=A_\delta u\overline{A_{-\delta }u}A_\delta u \end{aligned}$$

  for all $\delta \in D.$

• (2) Let $\frac{2n}{n+\gamma }\le p<2,$$1\le q\le \infty ,$$\theta >0$ and let $u\in G^D_{p,q,\theta }(I_T).$ Then

  $$\begin{aligned} A_\delta \left( (|x|^{-\gamma }*|u|^2)u\right) =\left( |x|^{-\gamma }*(A_\delta u\overline{A_{-\delta }u})\right) A_\delta u \end{aligned}$$

  for all $\delta \in D.$

Proof

Let $t\not =0.$ It is sufficient to show

$$\begin{aligned} e^{\delta \cdot x}U(-t)\left[ (|u|^2u)(t)\right] \in L^1 \end{aligned}$$

for all $\delta \in D,$ by Proposition 1 above. Indeed, we have

$$\begin{aligned}&e^{\delta \cdot x}U(-t)\left[ (|u|^2u)(t)\right] \\&\quad =Ct^{-n}\left( e^{\delta \cdot x}{\mathcal {M}}(t)U(-t)u(t) *\left( e^{\delta \cdot x}\overline{{\mathcal {M}}(t)U(-t) u(t)}(-\cdot )\right) \right) \\&\qquad *e^{\delta \cdot x}U(-t)u(t) \in L^1 \end{aligned}$$

for all $\delta \in D,$ because $e^{\delta \cdot x}U(-t)u(t)\in L^1 \cap L^p$ for all $\delta \in D.$ Hence $({\mathcal {M}}^{-1}|u|^2u)(t)$ is analytic on ${\mathbb {R}}^n+itD$ and its analytic continuation is represented as

$$\begin{aligned} ({\mathcal {M}}^{-1}|u|^2u)(t,x+it\delta )=e^{it\delta \cdot \nabla }({\mathcal {M}}^{-1}|u|^2u)(t,x),\ x+it\delta \in {\mathbb {R}}^n+itD, \end{aligned}$$

and

$$\begin{aligned} A_\delta (|u|^2u)(t)={\mathcal {M}}(t)e^{it\delta \cdot \nabla }{\mathcal {M}}(-t)(|u|^2u)(t)=\left( A_\delta u\right) (t)\overline{\left( A_{-\delta }u\right) (t)}\left( A_\delta u\right) (t) \end{aligned}$$

for all $\delta \in D.$ Similarly, we have

$$\begin{aligned}&e^{\delta \cdot x}U(-t)[((|x|^{-\gamma }*|u|^2)u)(t)]\\&\quad =C|t|^{-\gamma }\left( |x|^{-(n-\gamma )}e^{\delta \cdot x}{\mathcal {M}}(t)U(-t)u(t)*\left( e^{\delta \cdot x}\overline{{\mathcal {M}}(t)U(-t)u(t)}(-\cdot )\right) \right) \\&\qquad *e^{\delta \cdot x} U(-t)u(t) \in L^{\frac{2n}{n+\gamma }} \end{aligned}$$

for all $\delta \in D$ by Lemma 2 above and hence

$$\begin{aligned}&e^{\delta \cdot x}U(-t)[((|x|^{-\gamma }*|u|^2)u)(t)]\\&\quad =C|t|^{-\gamma }\left( |x|^{-(n-\gamma )}e^{\delta \cdot x}{\mathcal {M}}(t)U(-t)u(t)*\left( e^{\delta \cdot x}\overline{{\mathcal {M}}(t)U(-t)u(t)}(-\cdot )\right) \right) \\&\qquad *e^{\delta \cdot x} U(-t)u(t) \in L^1 \end{aligned}$$

for all $\delta \in D.$$\square$

5 Proof of Theorem 1

We define a complete metric space $\left( B^a_T(R),d\right)$ by

$$\begin{aligned} B^a_T(R)&=\left\{ u\in G^a_{p,p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T);\ u(0)=\phi ,\ \sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| J^\alpha u\right\| _{X^p_{p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}\le R\right\} ,\\ d(u,v)&=\sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !} \left\| J^\alpha (u-v)\right\| _{X^p_{p^\prime ,2\left( \frac{1}{p} -\frac{1}{2}\right) }(I_T)}. \end{aligned}$$

We show the map $\varPhi :u\mapsto \varPhi u,$$\varPhi u=U\phi -i\lambda S[|u|^2u],$ is a contraction mapping in $(B^a_T(R),d).$ We have

$$\begin{aligned} J^\alpha \varPhi u=U x^\alpha \phi -i\lambda \sum _{\beta +\gamma +\delta =\alpha }\frac{\alpha !(-1)^{|\gamma |}}{\beta !\gamma !\delta !}S\left[ J^\beta u \overline{J^{\gamma }u } J^\delta u\right] \end{aligned}$$

and

$$\begin{aligned} U^{-1}{\mathcal {L}}J^\alpha \varPhi u&=\lambda U^{-1}\sum _{\alpha _1 +\alpha _2+\alpha _3=\alpha }\frac{\alpha !(-1)^{|\alpha _2|}}{\alpha _1!\alpha _2!\alpha _3!}J^{\alpha _1} u \overline{J^{\alpha _2}u } J^{\alpha _3} u\\&=\lambda \sum _{\alpha _1+\alpha _2+\alpha _3=\alpha } \frac{\alpha !(-1)^{|\alpha _2|}}{\alpha _1!\alpha _2!\alpha _3!} U^{-1}{\mathcal {T}}_0\left( J^{\alpha _1} u, J^{\alpha _2}u , J^{\alpha _3} u\right) . \end{aligned}$$

Therefore,

$$\begin{aligned}&\left\{ \int _{I_T}\tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) p^\prime }\left\| \left( U^{-1}{\mathcal {L}} J^\alpha \varPhi u \right) (\tau )\right\| _{L^p}^{p^\prime }d\tau \right\} ^{1/p^\prime }\\&\quad \le C\sum _{\alpha _1+\alpha _2+\alpha _3=\alpha } \frac{\alpha !}{\alpha _1!\alpha _2!\alpha _3!}\prod _{j=1}^3\left\{ \left\| x^{\alpha _j}\phi \right\| _{L^p}+\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u\right) (\tau )\right\| _{L^p}d\tau \right\} \\&\quad =C \sum _{\alpha _1+\alpha _2+\alpha _3=\alpha }\alpha ! \prod _{j=1}^3\left\{ \frac{1}{\alpha _j!}\left\| x^{\alpha _j}\phi \right\| _{L^p}+\frac{1}{\alpha _j!}\int _{I_T}\left\| \left( U^{-1} {\mathcal {L}}J^{\alpha _j}u\right) (\tau )\right\| _{L^p}d\tau \right\} \end{aligned}$$

and

$$\begin{aligned}&\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u \right) (\tau )\right\| _{L^p}d\tau \\&\quad =\int _{I_T}\tau ^{-2\left( \frac{1}{p}-\frac{1}{2}\right) } \tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) }\left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u\right) (\tau )\right\| _{L^p}d\tau \\&\quad \le \left\{ \int _{I_T}\tau ^{p-2}d\tau \right\} ^{1/p} \left\{ \int _{I_T}\tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) p^\prime } \left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u \right) (\tau ) \right\| _{L^p}^{p^{\prime }}d\tau \right\} ^{1/p^\prime }\\&\quad = T^{\frac{1}{p^\prime }} \left\{ \int _{I_T}\tau ^{2\left( \frac{1}{p} -\frac{1}{2}\right) p^\prime }\left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j} u \right) (\tau )\right\| _{L^p}^{p^\prime }d\tau \right\} ^{1/p^\prime }. \end{aligned}$$

Also we have the difference term

$$\begin{aligned}&U^{-1}{\mathcal {L}}\left( \varPhi J^\alpha u-\varPhi J^\alpha v\right) \\&\quad =\lambda \sum _{\alpha _1+\alpha _2+\alpha _3=\alpha } \frac{\alpha !(-1)^{|\alpha _2|}}{\alpha _1!\alpha _2!\alpha _3!} U^{-1}\Big [{\mathcal {T}}_0\left( J^{\alpha _1}u,J^{\alpha _2}u, J^{\alpha _3}(u-v)\right) \\&\qquad +{\mathcal {T}}_0\left( J^{\alpha _1}v,J^{\alpha _2}v, J^{\alpha _3}(u-v)\right) +{\mathcal {T}}_0\left( J^{\alpha _1}u, J^{\alpha _2}(u-v),J^{\alpha _3}v\right) \Big ]. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| J^{\alpha } \varPhi u\right\| _{X^p_{p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}&\le C \left( \eta +T^{\frac{1}{p^\prime }}R\right) ^3,\\ d(\varPhi u,\varPhi v)&\le CT^{\frac{1}{p^\prime }} \left( \eta +T^{\frac{1}{p^\prime }}R\right) ^2d(u,v) \end{aligned}$$

and $\varPhi$ is a contraction mapping with $R>0$ and $T>0$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} T^{\frac{1}{p^\prime }}<\min \left( \frac{1}{2^{2/3}C\eta ^2}, \frac{2^{1/3}-1}{2C\eta ^2}\right) ,\\ R=2C\eta ^3. \end{array}\right. } \end{aligned}$$

6 Proof of Theorem 2

We define a complete metric space $\left( B^D_T(R),d\right)$ by

$$\begin{aligned} B^D_T(R)&=\left\{ u\in G^D_{p,p^\prime ,2\left( \frac{1}{p} -\frac{1}{2}\right) }(I_T);\ u(0)=\phi ,\ \sup _{\delta \in D} \left\| A_\delta u\right\| _{X^p_{p^\prime ,2\left( \frac{1}{p} -\frac{1}{2}\right) }(I_T)}\le R\right\} ,\\ d(u,v)&=\sup _{\delta \in D}\left\| A_\delta (u-v) \right\| _{X^p_{p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}. \end{aligned}$$

We show the map $\varPhi :u\mapsto \varPhi u,$$\varPhi u=U\phi -i\lambda S[|u|^2u],$ is a contraction mapping in $(B^D_T(R),d).$ We have

$$\begin{aligned} A_\delta \varPhi u=Ue^{\delta \cdot x}\phi -i\lambda S\left[ A_\delta u \overline{A_{-\delta }u } A_\delta u\right] , \end{aligned}$$

and

$$\begin{aligned} U^{-1}{\mathcal {L}}A_\delta \varPhi u=\lambda U^{-1}A_\delta u\overline{A_{-\delta }u}A_\delta u=\lambda U^{-1} {\mathcal {T}}_0(A_\delta u,A_{-\delta }u,A_\delta u). \end{aligned}$$

Therefore,

$$\begin{aligned}&\left\{ \int _{I_T}\tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) p^\prime }\left\| \left( U^{-1}{\mathcal {L}} A_\delta \varPhi u \right) (\tau )\right\| _{L^p}^{q}d\tau \right\} ^{1/q}\\&\quad \le C\prod _{j=1}^3\left\{ \left\| e^{(-1)^{j+1} \delta \cdot x}\phi \right\| _{L^p}+\int _{I_T}\left\| \left( U^{-1} {\mathcal {L}}A_{(-1)^{j+1}\delta }u\right) (\tau )\right\| _{L^p}d\tau \right\} \end{aligned}$$

and

$$\begin{aligned}&\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}A_{(-1)^{j+1}\delta } u\right) (\tau )\right\| _{L^p}d\tau \\&\quad =\int _{I_T}\tau ^{-2\left( \frac{1}{p}-\frac{1}{2}\right) } \tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) }\left\| \left( U^{-1}{\mathcal {L}}A_{(-1)^{j+1}\delta }u\right) (\tau ) \right\| _{L^p}d\tau \\&\quad \le \left\{ \int _{I_T}\tau ^{p-2}d\tau \right\} ^{1/p} \left\{ \int _{I_T}\tau ^{2\left( \frac{1}{p}-\frac{1}{2}\right) p^\prime } \left\| \left( U^{-1}{\mathcal {L}}A_{(-1)^{j+1}\delta }u \right) (\tau ) \right\| _{L^{p}}^{p^\prime }d\tau \right\} ^{1/p^\prime }\\&\quad = T^{\frac{1}{p^\prime }} \left\{ \int _{I_T}\tau ^{2 \left( \frac{1}{p}-\frac{1}{2}\right) p^\prime }\left\| \left( U^{-1} {\mathcal {L}}A_{(-1)^{j+1}\delta }u \right) (\tau )\right\| _{L^p}^{p^\prime } d\tau \right\} ^{1/p^\prime }. \end{aligned}$$

Also we have the difference term

$$\begin{aligned}&U^{-1}{\mathcal {L}}(\varPhi u-\varPhi v)\\&\quad =\lambda U^{-1}\left[ {\mathcal {T}}_0\left( A_\delta u,A_{-\delta }u,A_\delta (u-v)\right) +{\mathcal {T}}_0\left( A_\delta v,A_{-\delta } v,A_\delta (u-v)\right) \right. \\&\qquad \left. +{\mathcal {T}}_0\left( A_\delta u,A_{-\delta } (u-v),A_\delta v\right) \right] . \end{aligned}$$

Therefore,

$$\begin{aligned} \sup _{\delta \in D}\left\| A_\delta \varPhi u\right\| _{X^p_{p^\prime ,2\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}&\le C \left( \eta +T^{\frac{1}{p^\prime }}R\right) ^3,\\ d(\varPhi u,\varPhi v)&\le CT^{\frac{1}{p^\prime }} \left( \eta +T^{\frac{1}{p^\prime }}R\right) ^2d(u,v) \end{aligned}$$

and $\varPhi$ is a contraction mapping with $R>0$ and $T>0$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} T^{\frac{1}{p^\prime }}<\min \left( \frac{1}{2^{2/3}C\eta ^2}, \frac{2^{1/3}-1}{2C\eta ^2}\right) ,\\ R=2C\eta ^3. \end{array}\right. } \end{aligned}$$

7 Proof of Theorem 3

We define a complete metric space $\left( B^a_T(R),d\right)$ by

$$\begin{aligned} B^a_T(R)&=\left\{ u\in G^a_{p,q,2n\left( \frac{1}{p} -\frac{1}{2}\right) }(I_T);\ u(0)=\phi ,\ \sum _{\alpha \ge 0} \frac{a^\alpha }{\alpha !}\left\| J^\alpha u\right\| _{X^p_{q,2n \left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}\le R\right\} ,\\ d(u,v)&=\sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !} \left\| J^\alpha (u-v)\right\| _{X^p_{q,2n\left( \frac{1}{p} -\frac{1}{2}\right) }(I_T)}. \end{aligned}$$

We show the map $\varPhi :u\mapsto \varPhi u,$$\varPhi u=U\phi -i\lambda S[|u|^2u],$ is a contraction mapping in $(B^a_T(R),d).$ We have

$$\begin{aligned} J^\alpha \varPhi u=U x^\alpha \phi -i\lambda \sum _{\beta +\gamma +\delta =\alpha }\frac{\alpha !(-1)^{|\gamma |}}{\beta !\gamma !\delta !}S\left[ J^\beta u \overline{J^{\gamma }u } J^\delta u\right] \end{aligned}$$

and

$$\begin{aligned} U^{-1}{\mathcal {L}}J^\alpha \varPhi u&=\lambda U^{-1}\sum _{\alpha _1 +\alpha _2+\alpha _3=\alpha }\frac{\alpha !(-1)^{|\alpha _2|}}{\alpha _1!\alpha _2!\alpha _3!}\left( |x|^{-\gamma }*J^{\alpha _1} u \overline{J^{\alpha _2}u } \right) J^{\alpha _3} u\\&=\lambda \sum _{\alpha _1+\alpha _2+\alpha _3=\alpha } \frac{\alpha !(-1)^{|\alpha _2|}}{\alpha _1!\alpha _2!\alpha _3!} U^{-1}{\mathcal {T}}_\gamma \left( J^{\alpha _1} u, J^{\alpha _2}u , J^{\alpha _3} u\right) . \end{aligned}$$

Therefore,

$$\begin{aligned}&\left\{ \int _{I_T}\tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) q} \left\| \left( U^{-1} {\mathcal {L}}J^\alpha \varPhi u \right) (\tau )\right\| _{L^p}^{q}d\tau \right\} ^{1/q}\\&\quad \le C\sum _{\alpha _1+\alpha _2+\alpha _3=\alpha }\frac{\alpha !}{\alpha _1!\alpha _2!\alpha _3!}\prod _{j=1}^3\left\{ \left\| x^{\alpha _j}\phi \right\| _{L^p}+\int _{I_T} \left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u\right) (\tau ) \right\| _{L^p}d\tau \right\} \\&\quad =C \sum _{\alpha _1+\alpha _2+\alpha _3=\alpha }\alpha ! \prod _{j=1}^3\left\{ \frac{1}{\alpha _j!}\left\| x^{\alpha _j}\phi \right\| _{L^p}+\frac{1}{\alpha _j!}\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u\right) (\tau )\right\| _{L^p}d \tau \right\} \end{aligned}$$

and

$$\begin{aligned}&\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u \right) (\tau )\right\| _{L^p}d\tau \\&\quad =\int _{I_T}\tau ^{-2n\left( \frac{1}{p}-\frac{1}{2}\right) } \tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) }\left\| \left( \partial _\tau U^{-1}J^{\alpha _j}u\right) (\tau )\right\| _{L^p}d\tau \\&\quad \le \left\{ \int _{I_T}\tau ^{-2n\left( \frac{1}{p} -\frac{1}{2}\right) q^\prime }d\tau \right\} ^{1/q^\prime } \left\{ \int _{I_T}\tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) q} \left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u \right) (\tau ) \right\| _{L^p}^{q}d\tau \right\} ^{1/q}\\&\quad = T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n}} \left\{ \int _{I_T}\tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) q} \left\| \left( U^{-1}{\mathcal {L}}J^{\alpha _j}u \right) (\tau ) \right\| _{L^p}^{q}d\tau \right\} ^{1/q}. \end{aligned}$$

Also we have the difference term

$$\begin{aligned}&U^{-1}{\mathcal {L}}\left( \varPhi J^\alpha u-\varPhi J^\alpha v\right) \\&\quad =\lambda \sum _{\alpha _1+\alpha _2+\alpha _3=\alpha } \frac{\alpha !(-1)^{|\alpha _2|}}{\alpha _1!\alpha _2!\alpha _3!} U^{-1}\Big [{\mathcal {T}}_\gamma \left( J^{\alpha _1}(u-v),J^{\alpha _2} u,J^{\alpha _3}u\right) \\&\qquad +{\mathcal {T}}_\gamma \left( J^{\alpha _1}v,J^{\alpha _2}v, J^{\alpha _3}(u-v)\right) +{\mathcal {T}}_\gamma \left( J^{\alpha _1}v, J^{\alpha _2}(u-v),J^{\alpha _3}u\right) \Big ]. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{\alpha \ge 0}\frac{a^\alpha }{\alpha !}\left\| J^{\alpha } \varPhi u\right\| _{X^p_{q,2n\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}&\le C \left( \eta +T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p +2n}\frac{1}{q^\prime }}R\right) ^3,\\ d(\varPhi u,\varPhi v)&d\le CT^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n}\frac{1}{q^\prime }}\left( \eta +T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n} \frac{1}{q^\prime }}R\right) ^2d(u,v) \end{aligned}$$

and $\varPhi$ is a contraction mapping with $R>0$ and $T>0$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n}\frac{1}{q^\prime }} <\min \left( \frac{1}{2^{2/3}C\eta ^2},\frac{2^{1/3}-1}{2C\eta ^2}\right) ,\\ R=2C\eta ^3. \end{array}\right. } \end{aligned}$$

8 Proof of Theorem 4

We define a complete metric space $\left( B^D_T(R),d\right)$ by

$$\begin{aligned} B^D_T(R)&=\left\{ u\in G^D_{p,q,2n\left( \frac{1}{p}-\frac{1}{2} \right) }(I_T);\ u(0)=\phi ,\ \sup _{\delta \in D}\left\| A_\delta u \right\| _{X^p_{q,2n\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}\le R\right\} ,\\ d(u,v)&=\sup _{\delta \in D}\left\| A_\delta (u-v)\right\| _{X^p_{q,2n \left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}. \end{aligned}$$

We show the map $\varPhi :u\mapsto \varPhi u,$$\varPhi u=U\phi -i\lambda S[|u|^2u],$ is a contraction mapping in $(B^D_T(R),d).$ We have

$$\begin{aligned} A_\delta \varPhi u=Ue^{\delta \cdot x}\phi -i\lambda S\left[ A_\delta u \overline{A_{-\delta }u } A_\delta u\right] , \end{aligned}$$

and

$$\begin{aligned} U^{-1}{\mathcal {L}}A_\delta \varPhi u=\lambda U^{-1}\left( |x|^{-\gamma }*A_\delta u\overline{A_{-\delta }u}\right) A_\delta u=\lambda U^{-1} {\mathcal {T}}_\gamma (A_\delta u,A_{-\delta }u,A_\delta u). \end{aligned}$$

Therefore,

$$\begin{aligned}&\left\{ \int _{I_T}\tau ^{2n\left( \frac{1}{p}-\frac{1}{2} \right) q}\left\| \left( U^{-1}{\mathcal {L}} A_\delta \varPhi u\right) (\tau )\right\| _{L^p}^{q}d\tau \right\} ^{1/q}\\&\quad \le C\prod _{j=1}^3\left\{ \left\| e^{(-1)^{j+1}\delta \cdot x}\phi \right\| _{L^p}+\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}A_{(-1)^{j+1}\delta }u\right) (\tau ) \right\| _{L^p}d\tau \right\} \end{aligned}$$

and

$$\begin{aligned}&\int _{I_T}\left\| \left( U^{-1}{\mathcal {L}}A_{(-1)^{j+1}\delta }u\right) (\tau )\right\| _{L^p}d\tau \\&\quad =\int _{I_T}\tau ^{-2n\left( \frac{1}{p}-\frac{1}{2}\right) } \tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) }\left\| \left( U^{-1} {\mathcal {L}}A_{(-1)^{j+1}\delta }u\right) (\tau )\right\| _{L^p}d\tau \\&\quad \le \left\{ \int _{I_T}\tau ^{-2n\left( \frac{1}{p}-\frac{1}{2} \right) q^\prime }d\tau \right\} ^{1/q^\prime } \left\{ \int _{I_T}\tau ^{2n \left( \frac{1}{p}-\frac{1}{2}\right) q}\left\| \left( U^{-1}{\mathcal {L}} A_{(-1)^{j+1}\delta }u \right) (\tau )\right\| _{L^{p}}^qd\tau \right\} ^{1/q}\\&\quad = T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n}\frac{1}{q^\prime }} \left\{ \int _{I_T}\tau ^{2n\left( \frac{1}{p}-\frac{1}{2}\right) q} \left\| \left( U^{-1}{\mathcal {L}}A_{(-1)^{j+1}\delta }u \right) (\tau ) \right\| _{L^p}^{q}d\tau \right\} ^{1/q}. \end{aligned}$$

Also we have the difference term

$$\begin{aligned}&U^{-1}{\mathcal {L}}(\varPhi u-\varPhi v)\\&\quad =\lambda U^{-1}\left[ {\mathcal {T}}_\gamma \left( A_\delta (u-v),A_{-\delta }u,A_\delta u\right) +{\mathcal {T}}_\gamma \left( A_\delta v,A_{-\delta } v,A_\delta (u-v)\right) \right. \\&\qquad \left. +{\mathcal {T}}_\gamma \left( A_\delta v,A_{-\delta } (u-v),A_\delta u\right) \right] . \end{aligned}$$

Therefore,

$$\begin{aligned} \sup _{\delta \in D}\left\| A_\delta \varPhi u\right\| _{X^p_{p^\prime ,2n\left( \frac{1}{p}-\frac{1}{2}\right) }(I_T)}\le & {} C \left( \eta +T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n} \frac{1}{q^\prime }}R\right) ^3,\\ d(\varPhi u,\varPhi v)\le & {} CT^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p +2n}\frac{1}{q^\prime }}\left( \eta +T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p +2n}\frac{1}{q^\prime }}R\right) ^2d(u,v) \end{aligned}$$

and $\varPhi$ is a contraction mapping with $R>0$ and $T>0$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} T^{\frac{(2+n-\gamma )p-2n}{(2-n-\gamma )p+2n}\frac{1}{q^\prime }} <\min \left( \frac{1}{2^{2/3}C\eta ^2},\frac{2^{1/3}-1}{2C\eta ^2}\right) ,\\ R=2C\eta ^3. \end{array}\right. } \end{aligned}$$