Modified scattering operator for nonlinear Schrödinger equations with time-
decaying harmonic potentials

Masaki KAWAMOTO
Department of Engineering for Production, Graduate School of Science and
Engineering, Ehime University, 3 Bunkyo-cho Matsuyama, Ehime, 790-0826, Japan

Email<EMAIL_ADDRESS>
Hayato MIYAZAKI
Teacher Training Courses, Faculty of Education, Kagawa University, Takamatsu,
Kagawa 760-8522, Japan

E-mail<EMAIL_ADDRESS>
###### Abstract

This paper is concerned with nonlinear Schrödinger equations with a time-
decaying harmonic potential. The nonlinearity is gauge-invariant of the long-
range critical order. In [KM21] and [Ka21], it is proved that the equation
admits a nontrivial solution that behaves like a free solution with a
logarithmic phase correction in the frameworks of both the final state problem
and the initial value problem. Furthermore, a modified scattering operator has
been established in the case without the potential in [HN06]. In this paper,
we construct a modified scattering operator for our equation by utilizing a
generator of the Galilean transformation. Moreover, we remove a restriction
for the coefficient of the potential which is required in [Ka21].

2020 Mathematics Subject Classification: Primary: 35Q55, Secondly: 35B40,
35P25.

Key words and phrases: nonlinear Schrödinger equation; time-decaying harmonic
potential; long-range scattering; modified scattering operator

##  1 Introduction

###  1.1 Setting and Backgounrd

In this paper, we consider the nonlinear Schrödinger equation

$\displaystyle i\partial_{t}u-H(t)u=F(u),$ (NLS)

where $(t,x)\in{\bf R}\times{\bf R}^{d}$, $d=1,2,3$, and $u=u(t,x)$ is a
complex-valued unknown function. The operator $H(t)$ is defined by

$\displaystyle
H(t)=-\frac{1}{2}\Delta+\sigma(t)\frac{|x|^{2}}{2},\quad\sigma(t)\in
L^{\infty}({\bf R};{\bf R}).$

The coefficient of the potential $\sigma(t)$ satisfies the time-decay
condition
$\lim_{t\to\infty}t^{2}\sigma(t)\eqqcolon\sigma_{1}\in(-\infty,1/4)$, more
precisely see Assumption 1.1. The nonlinearity $F$ is gauge invariant of
degree $1+2/d(1-\lambda)$; that is, it is the form of

$\displaystyle F(u)=\eta|u|^{p_{c}}u,\quad
p_{c}\coloneqq\frac{2}{d(1-\lambda)},\quad\eta\in{\bf R},$ (1.1)

where $\lambda=(1-\sqrt{1-4\sigma_{1}})/2<1/2$ with $-1/3<\lambda$ if $d=3$.

The aim of this paper is to deal with a scattering operator to (NLS). As for
(NLS) with $\sigma(t)\equiv 0$, that is,

$\displaystyle i\partial_{t}u+\frac{1}{2}\Delta u=G(u),\quad
G(u)=\eta|u|^{p}u,$ (1.2)

then (1.2) admits a nontrivial solution that behaves like a free solution
$U_{0}(t)u_{\pm}$ as $t\rightarrow\pm\infty$, when $p>2/d$, where
$U_{0}(t)=e^{it\Delta/2}$ and $u_{\pm}$ is called the _final date_. In this
sense, we say the case $p>2/d$ is the short-range. On the other hand, if
$p\leq 2/d$, then (1.2) has no solutions that behave like the free solution in
$L^{2}$ (e.g., [S74, B84, TY84]), hence we say the case $p\leq d/2$ is the
long-range. Hence we can regard the exponent $p_{c}^{\prime}\coloneqq 2/d$ as
the threshold in the sense of scattering problems. In particular, when
$p=p_{c}^{\prime}$, there is the solution that behaves like a free solution
with a logarithmic phase correction
$U_{N}(t)u_{\pm}:=U_{0}(t)\mathcal{F}^{-1}\exp\left(-i\eta\left|\mathcal{F}u_{\pm}\right|^{p_{c}}\log|t|\right)\mathcal{F}u_{\pm}$
for a suitable given $u_{\pm}$ as $t\to\pm\infty$ (cf. [O91, GO93, HN98]),
where $\mathcal{F}$ is the usual Fourier transform. A scattering problem for
nonlinear Schrödinger equations has been considered in two directions. One
direction is the final state problem. We attempt to construct a solution $u$
to (1.2) that behaves like a prescribed asymptotics $U_{N}(t)u_{-}$ as
$t\rightarrow-\infty$ in the problem. The second is the initial value problem
that is dealt with a global-in-time solution and asymptotics $U_{N}(t)u_{+}$
for large time, specified by given initial data. If we find the solution under
the final state problem, then the wave operator $\mathcal{W}_{-}$ which is a
map from $u_{-}$ into $u(0)$ can be defined. On the other hand, we can define
the inverse wave operator $\mathcal{W}_{+}^{-1}$ from $u(0)$ into $u_{+}$,
once the initial value problem is solved. Furthermore, if the range of
$\mathcal{W}_{-}$ included in the domain of $\mathcal{W}_{+}^{-1}$, then the
scattering operator $\mathcal{S}\coloneqq\mathcal{W}_{+}^{-1}\mathcal{W}_{-}$
can be defined.

The scattering operator on the critical case $p=p_{c}^{\prime}$ is called the
_modified scattering operator_. In [Ca01], when $d=1$, Carles constructs a
modified scattering operator for (1.2) on $H^{0,3}\cap H^{1,2}$ if
$\norm{u_{-}}_{H^{1}\cap H^{0,1}}$ is small. Later on, the operator on the
neighborhood centered at the origin in $H^{\alpha,0}\cap H^{0,\alpha}$ has
been defined in Hayashi and Naumkin [HN06] under
$d/2<\alpha<\min(1+p_{c}^{\prime},d)$. This situation such that the operator
can be defined on the neighborhood centered at the origin is called _low-
energy scattering_ (cf. [Stra81a, Stra81b]). This kind of modified scattering
operator has been treated on other types of nonlinearity in some literature
(e.g., [Na02a, Na02b]). In terms of (NLS), Carles [Ca03] proves the existence
of the scattering operator, provided $\sigma(t)\equiv-\omega^{2}$ ($\omega>0$)
and $F(u)=\eta|u|^{2\sigma}u$, $\sigma>0$ when $d=1,2$ and $0<\sigma<2/(d-2)$
when $d\geq 3$.

Now we back to the case $\sigma(t)\neq 0$. Under the existence of
$\sigma(t)|x|^{2}$, the dispersive effect for free solution ((NLS) with
$F\equiv 0$) becomes weaker than the case of $\sigma(t)\equiv 0$ and a
threshold in the sense of scattering is changed from $p_{c}^{\prime}$ to
$p_{c}=2/(d(1-\lambda))$, (see below). Hence the construction of the
scattering operator for (NLS) is equivalent to that of the modified scattering
operator. Kawamoto and Muramatsu [KM21] study the initial value problem for
(NLS) with $F(u)\sim\eta|u|^{p_{c}}u$, and they succeed to construct the
inverse modified wave operator ${\cal W}^{-1}_{+}$ from $u_{0}\in
H^{0,\alpha}$ to $u_{+}\in L^{2}$. Also, Kawamoto [Ka21] proves that the
existence of modified wave operators ${\cal W}_{-}$ from $H^{0,\alpha}$ to
$u_{0}\in L^{2}$. Clearly, these results do NOT guarantee $\mathrm{Ran}({\cal
W}_{-})\subset{\mathscr{D}}({\cal W}_{+}^{-1})$ and hence to construct the
modified wave operator is difficult only with employing previous results.
Hence the first aim of this paper is to bridge this gap. Moreover, a strong
restriction for $\lambda$ has been imposed in [KM21] because of a use of the
time-weighted Strichartz estimate developed by [KY18] and [Ka20]. Hence the
second aim of this paper is to remove the restriction. The modified scattering
operator for (NLS) has not been discussed and so we attempt to define the
operator in this paper under the some smallness conditions which are
associated to the low energy condition in the case of $\sigma(t)=0$.

To state the reason why $p\leq p_{c}$ becomes long-range, we first let
$U(t,s)$ be a propagator for $H(t)$, defined by the family of unitary
operators $\\{U(t,s)\\}_{(t,s)\in{\bf R}^{2}}$ on $L^{2}({\bf R}^{n})$ such
that for all $t,$ $s$, $\tau\in{\bf R}$,

$\displaystyle i\partial_{t}U(t,s)=H(t)U(t,s),\quad
i\partial_{s}U(t,s)=-U(t,s)H(s),$ $\displaystyle
U(t,\tau)U(\tau,s)=U(t,s),\quad U(s,s)=\mathrm{Id}_{L^{2}({\bf R}^{n})},\quad
U(t,s){\mathscr{D}}(H(s))\subset{\mathscr{D}}(H(s))$

hold on ${\mathscr{D}}(H(s))$, i.e., $u_{0}(t):=U(t,0)u_{0}$ solves
$i\partial_{t}u_{0}(t)=H(t)u_{0}(t)$ with $u(0)=u_{0}\in{\mathscr{D}}(H(0))$.
We here mention that $U(t,s)$ has the following factorization formula

$\displaystyle U(t,0)={\cal
M}\left(\frac{\zeta_{2}(t)}{\zeta_{2}^{\prime}(t)}\right){\cal
D}(\zeta_{2}(t))\mathcal{F}{\cal
M}\left(\frac{\zeta_{2}(t)}{\zeta_{1}(t)}\right),$ (1.3)

where ${\cal M}(t)=e^{\frac{i|x|^{2}}{2t}}$, $\left({\cal
D}(t)\phi\right)(x)=(it)^{-\frac{d}{2}}\phi(x/t)$, $\mathcal{F}$ is a usual
Fourier transform and $\zeta_{j}(t)$, $j=1,2$ are solutions to

$\displaystyle\zeta_{j}^{\prime\prime}(t)+\sigma(t)\zeta_{j}(t)=0,\quad\begin{cases}\zeta_{1}(0)=1,\\\
\zeta_{1}^{\prime}(0)=0,\end{cases}\quad\begin{cases}\zeta_{2}(0)=0,\\\
\zeta_{2}^{\prime}(0)=1.\end{cases}$ (1.4)

Korotyaev [Ko89] first discovered similar factorization to (1.3) and Kawamoto
and Muramatsu [KM21] rewrites Korotyaev’s factorization to (1.3) to deal with
a nonlinear scattering problem on (NLS) motivated by Hayashi and Namunkin
[HN98] (see also [Ca03, Ca11, Ka21]). This factorization formula leads to a
kind of dispersive estimate

$\displaystyle\norm{U(t,0)\phi}_{\infty}\lesssim|\zeta_{2}(t)|^{-\frac{d}{2}}\norm{\phi}_{1}.$
(1.5)

The typical example of $\sigma(t)$ in our mind is

$\displaystyle\sigma(t)=\begin{cases}\sigma_{0}\quad(|t|<r_{1}),\\\
\sigma_{1}|t|^{-2}\quad(|t|\geq r_{1}),\end{cases}\quad r_{1},\sigma_{0}>0.$
(1.6)

For this case, $\zeta_{j}(t)$, $|t|\geq r_{1}$ can be written as a linear
combination of $|t|^{\lambda}$ and $|t|^{1-\lambda}$ with
$\lambda=(1-\sqrt{1-4\sigma_{1}})/2$. Namely, those solutions are

$\displaystyle\zeta_{1}(t)=b_{1}|t|^{1-\lambda}+b_{1}^{\prime}|t|^{\lambda},\quad\zeta_{2}(t)=b_{2}|t|^{1-\lambda}+b_{2}^{\prime}|t|^{\lambda}.$

Hence the time decay estimate becomes
$\|U(t,0)\phi\|_{\infty}\lesssim|t|^{-\frac{d}{2}(1-\lambda)}$ holds whenever
$b_{2}^{\prime}\neq 0$, and this decay is slower than the free case
$\sigma(t)\equiv 0$. The fact changes a threshold that the asymptotics
involves nonlinear effects coming from $G(u)=\eta|u|^{p}u$ with $p>0$.

###  1.2 Main results

In what follows, we assume the following implicit condition on $\sigma(t)$:

######  Assumption 1.1.

There exist solutions $\zeta(t)$ to (1.4) satisfying the following condition:
There exist $r_{0}\geq 1$ such that

  1. (A1)

there exists $\delta_{0}>0$ such that $|\zeta_{2}(t)|\geq\delta_{0}$ for any
$|t|\geq r_{0}$,

  2. (A2)

there exist $a_{j}\in{\bf R}$, $j=1,2$ with $a_{2}\neq 0$ and $\lambda<1/2$
with $-1/3<\lambda$ if $d=3$ such that

$\displaystyle\absolutevalue{\frac{\zeta_{2}(t)}{|t|^{1-\lambda}}-a_{2}}+\absolutevalue{\frac{\zeta_{1}(t)}{\zeta_{2}(t)}-\frac{a_{1}}{a_{2}}}\lesssim{}$
$\displaystyle\begin{cases}|t|^{-(1-2\lambda)}\quad(\lambda\geq 0),\\\
|t|^{-1}\quad(\lambda<0),\end{cases}$ (1.7)
$\displaystyle\absolutevalue{\frac{\zeta_{2}(t)}{|t|^{1-\lambda}}-a_{2}}\leq\frac{|a_{2}|}{2}$
(1.8)

for any $|t|\geq r_{0}$.

We remark that the condition (1.7) are required in order that
$\mathcal{M}\left(\zeta_{2}(t)/\zeta_{1}(t)\right)$ is approximated
$\mathcal{M}(a_{2}/a_{1})\eqqcolon\mathcal{M}_{+}$ as $t\rightarrow\pm\infty$.
More precisely, we obtain

$\displaystyle\absolutevalue{\mathcal{M}\left(\frac{\zeta_{2}(t)}{\zeta_{1}(t)}\right)-\mathcal{M}_{+}}=\absolutevalue{{\cal
M}\left(\frac{a_{2}\zeta_{2}(t)}{a_{2}\zeta_{1}(t)-a_{1}\zeta_{2}(t)}\right)-1}\lesssim|x|^{2\theta}\absolutevalue{\frac{\zeta_{1}(t)}{\zeta_{2}(t)}-\frac{a_{1}}{a_{2}}}^{\theta}$
(1.9)

for any $\theta\in[0,1]$. By choosing suitable $\sigma_{0}$ and $r_{1}$, we
can easily check that (1.6) satisfies Assumption 1.1. One also frequently
utilizes $\absolutevalue{\zeta_{2}(t)}\lesssim|t|^{1-\lambda}$ which arises
from (1.7). The condition (1.8) is only utilized in Lemma 3.2. Further, it is
shown that there exists $\sigma(t)$ satisfying Assumption 1.1 in general (see
[GMT93, Na84, W69]).

Based on the Duhamel principle, we give the definition of solutions to (NLS)
as follows:

######  Definition 1.1 ($\mathcal{F}H^{\beta}$-solution).

Let $I\subset{\bf R}$ be an interval and fix $t_{0}\in I$. Set $\beta>0$. We
say a function $u(t,x):I\times{\bf R}^{d}\to{\bf C}$ is a
$\mathcal{F}H^{\beta}$-solution to (NLS) on $I$ if $u\in C(I;L^{2}({\bf
R}^{d}))$ satisfies

$u(t)=U(t,t_{0})u(t_{0})-i\int_{t_{0}}^{t}U(t,s)F(u(s))\,ds$

in $L^{2}({\bf R}^{d})$ for any $t\in I$ and $U(0,\cdot)u\in
C(I;H^{0,\beta})$, where $H^{0,\beta}$ denotes the weighted $L^{2}$ space
equipped with the norm
$\norm{f}_{H^{0,\beta}}=\norm{(1+|x|^{2})^{\beta/2}f}_{2}$.

We now state the main theorems in this paper. Here we remark that the
factorization (1.3) is only justified for the case $|t|\geq r_{0}$. Indeed,
for the case of (1.6), there may exists $t_{0}\in[-r_{0},r_{0}]$ such that
$\zeta_{j}(t_{0})=0$. Hence we construct the modified scattering operator by
the following three steps:

1\. Construct the solution $u(t)$, $t\in(-\infty,-r_{0}]$ from $u_{-}$.

2\. Construct the $u(0)$ from $u(-r_{0})$ and $u(r_{0})$ from $u(0)$ by the
persistency argument.

3\. Construct the $u_{+}$ and the solution $u(t)$, $t\in[r_{0},\infty)$ from
$u(r_{0})$.

The first main result is concerned with the modified wave operator as follows:

######  Theorem 1.1 (Final state problem).

Assume $d/2<\beta<\alpha<\min(1+p_{c},d)$. Then there exists
$\varepsilon_{1}>0$ such that the following assertion holds: For any
$\varepsilon\in(0,\varepsilon_{1}]$ and for any $u_{-}\in H^{0,\alpha}$ with
$\norm{u_{-}}_{H^{0,\alpha}}\leq\varepsilon$, there exists a unique
$\mathcal{F}H^{\beta}$-solution to (NLS) on $(-\infty,-r_{0}]$. Moreover the
estimate

$\norm{U(0,t)\left(u(t)-u_{p}(t)\right)}_{H^{0,\beta}}\lesssim\varepsilon|t|^{-\mu}$

is valid for all $t\leq-r_{0}$, where $\mu>0$ is small enough and

$\displaystyle
u_{p}(t)=\mathcal{M}\left(\frac{\zeta_{2}(t)}{\zeta_{2}^{\prime}(t)}\right)\mathcal{D}(\zeta_{2}(t))\widehat{u_{-}}\exp\left(i\frac{\eta}{c_{+}}\left|\widehat{u_{-}}\right|^{p_{c}}\log|t|\right),\quad
c_{+}=|a_{2}|^{\frac{1}{1-\lambda}}.$ (1.10)

######  Remark 1.1.

If we replace the smallness assumption of $\norm{u_{-}}_{H^{0,\alpha}}$ by
$\norm{\widehat{u_{-}}}_{\infty}\leq\varepsilon$, then there exists $T\geq
r_{0}$ such that (NLS) admits a unique $\mathcal{F}H^{\beta}$-solution on
$(-\infty,-T]$ that behaves like $u_{p}$ as $t\rightarrow-\infty$ in the same
topology. Although the replacement is the weaker condition, we need in the
sequel the smallness in $H^{0,\alpha}$ to define the modified scattering
operator. Hence we have already required the smallness.

######  Remark 1.2.

In [Ka21] and [KaMi23], the time-weighted Strichartz estimate has been applied
to discuss the final state problem. However, a use of the Strichartz estimate
imposes the strong restriction of $\lambda$. We remove the restriction by
considering a solution in the space associated with a generator of the
Galilean transformation $J(t)\coloneqq U(t,0)xU(0,t)$.

######  Remark 1.3.

In Theorem 1.1, the same assertion holds if we replace $u_{-}$ by
$\mathcal{M}_{+}u_{-}$ and $u_{p}(t)$ behaves like
$U(t,0)\mathcal{F}^{-1}\exp\left(-i\eta/c_{+}\left|\widehat{u_{-}}\right|^{p_{c}}\log|t|\right)\widehat{u_{-}}$
as $t\rightarrow-\infty$ under suitable $u_{-}$.

We here define $B_{\varepsilon}^{\alpha}=\\{\phi\in
H^{0,\alpha}\mid\norm{\phi}_{H^{0,\alpha}}<\varepsilon\\}$ for any $\alpha$,
$\varepsilon>0$. Thanks to the persistence of regularity of solutions, see
Lemma 2.6, and the well-posedness in $L^{2}$ for (NLS), we can extend the
stopping time from $t=-r_{0}$ to $t=0$ when the solution propagates. Thus one
can define the modified wave operator as follows:

######  Corollary 1.1.

Under the assumption as in Theorem 1.1, the modified wave operator $W_{-}$ on
$B_{\varepsilon}$ is defined for any $\varepsilon\in(0,\varepsilon_{1}]$.
Namely, for some $c_{0}>0$, there exists a map

$\displaystyle\mathcal{W}_{-}\colon B^{\alpha}_{\varepsilon}\ni u_{-}\mapsto
u(0)\in B^{\beta}_{c_{0}\varepsilon},$

where $\varepsilon_{1}$ is given in Theorem 1.1.

Again using the persistence of regularity, for any
$u(0)\in{B}_{c_{0}\varepsilon}^{\beta}$, there uniquely exists
$\mathcal{F}H^{\beta}$-solution to (NLS) on $[0,r_{0})$ such that for some
$c_{1}>0$ and any $t\in[0,r_{0}]$,
$U(0,t)u(t)\in{B}_{c_{1}\varepsilon}^{\beta}$ holds.

Noting this fact, we state the second main result which is related to the
existence of modified inverse wave operator.

######  Theorem 1.2 (Modified scattering for the initial value problem).

Let $d/2<\delta<\beta<\min(1+p_{c},d)$. Then there exists $\varepsilon_{2}>0$
such that the following assertion holds: For any
$\varepsilon\in(0,\varepsilon_{2}]$ and $u_{0}\in L^{2}$ satisfying
$U(0,r_{0})u_{0}\in H^{0,\beta}$ with
$\norm{U(0,r_{0})u_{0}}_{H^{0,\beta}}\leq\varepsilon$, there exists a unique
$\mathcal{F}H^{\beta}$-solution to (NLS) on $[r_{0},\infty)$ under
$u(r_{0})=u_{0}$. Moreover, there exists a unique $u_{+}\in H^{0,\delta}$ such
that

$\displaystyle\norm{\mathcal{F}^{-1}\mathcal{P}(t)\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)-u_{+}}_{H^{0,\delta}}\lesssim{}$
$\displaystyle\varepsilon t^{-\mu}$ (1.11)

for any $t>r_{0}$, where

$\displaystyle\mathcal{P}(t)=\exp\left(i\eta\int_{r_{0}}^{t}|\mathcal{F}\mathcal{M}_{+}U(0,\tau)u(\tau)|^{p_{c}}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau\right)$

if $d=1$, otherwise

$\displaystyle\mathcal{P}(t)=\exp\left(i\eta\int_{r_{0}}^{t}\left(|\tau|^{-\chi}+|\mathcal{F}\mathcal{M}_{+}U(0,\tau)u(\tau)|^{2}\right)^{\frac{p_{c}}{2}}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau\right)$

for some small $\chi>0$ and $\mu>0$.

######  Remark 1.4.

The factor $\mathcal{M}_{+}$ appears in the phase correction $\mathcal{P}(t)$,
because we do not assume $a_{1}=0$ unlike [Ka21] and [KM21].

Similarly to the modified wave operator, we can construct the inverse wave
operator as consequence of Theorem 1.2.

######  Corollary 1.2.

Under the assumption as in Theorem 1.2, there exists
$\widetilde{\varepsilon}_{2}>0$ such that for any
$\varepsilon\in(0,\widetilde{\varepsilon}_{2}]$, the modified inverse wave
operator ${\cal W}_{+}^{-1}$ on $B_{\varepsilon}^{\beta}$ can be defined.
Namely, there exists a map

$\displaystyle\mathcal{W}_{+}^{-1}\colon B^{\beta}_{\varepsilon}\ni
u(0)\mapsto u_{+}\in H^{0,\gamma}.$

Combining Corollary 1.1 with Corollary 1.2, the following holds:

######  Corollary 1.3.

Let $d/2<\delta<\alpha<\min\left(d,1+p_{c}\right)$ Then there exists
$\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0}]$, the
modified scattering operator ${\cal
S}_{+}\coloneqq\mathcal{W}_{+}^{-1}\mathcal{W}_{-}$ on
$B_{\varepsilon}^{\alpha}$ can be defined. Namely, there exists a map

$\displaystyle{\cal S}_{+}\colon B^{\alpha}_{\varepsilon}\ni u_{-}\mapsto
u_{+}\in H^{0,\delta}.$

### Notations

We introduce some notations. For any $p\geq 1$, $L^{p}=L^{p}({\bf R}^{d})$
denotes the usual Lebesgue space on ${\bf R}^{d}$ equipped with the norm
$\norm{\,\cdot\,}_{p}\coloneqq\norm{\,\cdot\,}_{L^{p}}$. Set $\left\langle
a\right\rangle=(1+|a|^{2})^{1/2}$ for $a\in{\bf C}$ or ${\bf R}^{d}$.
${\mathscr{F}}[u]=\widehat{u}$ is the usual Fourier transform of a function
$u$ on ${\bf R}^{d}$ and ${\mathscr{F}}^{-1}[u]=\check{u}$ is its inverse. Let
$m$, $s\in{\bf R}$. The weighted Sobolev space and the homogeneous Sobolev
space on ${\bf R}^{d}$ are defined by $H^{m,s}=H^{m,s}({\bf
R}^{d})=\\{u\in\mathscr{S}^{\prime}\mid\left\langle
x\right\rangle^{s}\left\langle i\nabla\right\rangle^{m}u\in L^{2}\\}$ and
$\dot{H}^{s}=\dot{H}^{s}({\bf
R}^{d})=\\{u\in\mathscr{S}^{\prime}\mid|\nabla|^{s}u\in L^{2}\\}$,
respectively. Here $\mathscr{S}^{\prime}$ is the space of tempered
distributions. We simply write $H^{m}=H^{m,0}$. One denotes the smallest
integer $n_{0}$ such that $n_{0}\geq\delta$ by $\lceil\delta\rceil$ for any
$\delta\in{\bf R}$. $A\lesssim B$ denotes $A\leq CB$ for some constants $C>0$.

###  1.3 Strategy on the proof of main results

Before stating the strategy on the proof of main results, let us introduce
several notations associated with the factorization (1.3), which is the key
tool of our analysis. We denote

$\displaystyle{\cal M}\left(\frac{\zeta_{2}(t)}{\zeta_{1}(t)}\right)={\cal
M}_{+}{\cal M}_{2}(t),\quad{\cal M}_{+}={\cal
M}\left(\frac{a_{2}}{a_{1}}\right),\quad{\cal M}_{2}(t)={\cal
M}\left(\frac{a_{2}\zeta_{2}(t)}{a_{2}\zeta_{1}(t)-a_{1}\zeta_{2}(t)}\right).$

Then by (1.7) and (1.9), we notice

$\displaystyle\left|\left({\cal M}_{2}(t)\right)^{\pm
1}-1\right|\lesssim\begin{cases}|x|^{2\theta}t^{-\theta(1-2\lambda)},&\lambda>0,\\\
|x|^{2\theta}t^{-\theta},&\lambda\leq 0.\end{cases}$ (1.12)

Hereafter one also uses the notation

$\displaystyle{\cal M}_{1}(t)={\cal
M}\left(\frac{\zeta_{2}(t)}{\zeta_{2}^{\prime}(t)}\right),\quad\mathcal{D}_{1}(t)=\mathcal{D}(\zeta_{2}(t)).$

Then (1.3) is rewritten as

$\displaystyle U(t,0)={\cal M}_{1}(t){\cal D}_{1}(t)\mathcal{F}{\cal
M}_{+}{\cal M}_{2}(t).$ (1.13)

####  1.3.1 Strategy on the proof of Theorem 1.1

The strategy on the proof of Theorem 1.1 is to solve an integral equation
associated with (NLS) around a prescribed asymptotic profile $u_{p}$, where

$\displaystyle u_{p}(t)={}$ $\displaystyle{\cal M}_{1}(t){\cal
D}_{1}(t)\widehat{w}(t),\quad\widehat{w}(t)=\widehat{u_{-}}\exp\left(i\frac{\eta}{c_{+}}|\widehat{u_{-}}|^{p_{c}}\log|t|\right).$
(1.14)

We denote a new unknown $v$ by $v\coloneqq u-u_{p}$. Then one obtains the
equation

$\displaystyle i\partial_{t}v-H(t)v={}$ $\displaystyle
F(u_{p}+v)-F(u_{p})-\\{\left(i\partial_{t}-H(t)\right)u_{p}-F(u_{p})\\},$

which implies

$\displaystyle i\partial_{t}\left(U(0,t)v\right)={}$ $\displaystyle
U(0,t)\\{F(u_{p}+v)-F(u_{p})\\}-\\{i\partial_{t}\left(U(0,t)u_{p}\right)-U(0,t)F(u_{p})\\}.$
(1.15)

Since $u_{p}(t)$ is an expected asymptotic profile of $u(t)$, we may assume
that $\norm{v(t)}_{2}\rightarrow 0$ as $t\rightarrow-\infty$. Therefore,
applying (1.13), it follows from the Duhamel principle that $v(t)$ solves the
integral equation

$\displaystyle
v(t)=-i\int_{-\infty}^{t}U(t,s)(F(u_{p}(s)+v(s))-F(u_{p}(s)))\,ds+\mathcal{E}(t),$
(1.16)

where $\mathcal{E}(t)=\mathcal{A}(t)+\mathcal{E}_{r}(t)$ with

$\displaystyle\begin{aligned} {\cal A}(t)={}&-i\int_{-\infty}^{t}U(t,0){\cal
M}_{2}(s)^{-1}{\cal
M}_{+}^{-1}\mathcal{F}^{-1}\left(\frac{c_{+}|s|}{|\zeta_{2}(s)|^{1/(1-\lambda)}}-1\right)F(\widehat{w}(s))\,\frac{ds}{c_{+}|s|},\\\\[5.0pt]
{\cal
E}_{\mathrm{r}}(t)={}&R(t)\widehat{w}(t)+i\int_{-\infty}^{t}U(t,s)R(s)F(\widehat{w}(s))\frac{ds}{c_{+}|s|},\\\
R(t)={}&{\cal M}_{1}(t){\cal D}_{1}(t)\left(\mathcal{F}{\cal
M}_{2}(t)\mathcal{F}^{-1}-1\right).\end{aligned}$ (1.17)

Note that we give the derivation of (1.16) in Appendix B. Conversely, if we
find a solution to (1.16), then $u\coloneqq v+w$ is the solution to (NLS),
that is $u(t)$ solves

$\displaystyle
u(t)=u_{p}(t)+i\int_{-\infty}^{t}U(t,s)(F(u(s))-F(u_{p}(s)))\,ds+\mathcal{E}(t).$
(1.18)

Hence we attempt to find a unique solution to (1.16), based on the contraction
mapping argument in the complete metric space

$\displaystyle\begin{aligned} X_{b,R}={}&\left\\{\phi\in
C\left((-\infty,-r_{0}];L^{2}({\bf
R}^{d})\right)\mathrel{}\middle|\mathrel{}\left\|\phi\right\|_{X_{b}}\leq
R\right\\},\\\ \norm{\phi}_{X_{b}}={}&\sup_{t\leq-
r_{0}}\left(|t|^{\frac{\beta}{2}(1-2\lambda)+b}\norm{\phi(t)}_{L^{2}({\bf
R}^{d})}+|t|^{b}\norm{U(0,t)\phi(t)}_{H^{0,\beta}({\bf R}^{d})}\right),\\\
d(\phi,\psi)={}&\norm{\phi-\psi}_{L^{\infty}(-\infty,-r_{0};L^{2}({\bf
R}^{d}))}\end{aligned}$ (1.19)

for some $R>0$, where $b=b(\lambda,\beta,d)>0$ is small enough. If
$\norm{u_{-}}_{H^{0,\alpha}}$ is small enough and if the external term
$\mathcal{E}(t)$ satisfies $\norm{\mathcal{E}(t)}_{X_{b}}\leq
C_{0}\norm{u_{-}}_{\dot{H}^{0,\alpha}}$ for some $C_{0}>0$, then it enables us
to show that (1.16) admits a unique solution $u\in X_{b,R}$ with
$R=2C_{0}\norm{u_{-}}_{H^{0,\alpha}}$. To this end, we need to estimate
$F(u_{p}+v)-F(u_{p})$ in $X_{b,R}$ to show $\Psi(X_{b,R})\subset X_{b,R}$,
where $\Psi$ is a map associated with (1.16). Further, when $d=2$, $3$, the
smallness of the exponent $p_{c}$ causes the lack of differentiability of the
nonlinearity. This means that if $X_{b,R}$ is equipped with the distance
$d(\phi,\psi)=\norm{\phi-\psi}_{X_{b}}$, then it is difficult to show that
$\Psi$ is the contraction map in $X_{b,R}$. To overcome this difficultly, we
prove the completeness of $X_{b,R}$ equipped with the distance in (1.19)
consisting of the weaker norm than $X_{b}$-norm. This kind of technique is
originally introduced by Kato [TK87] (cf. [C03, p94]).

####  1.3.2 Strategy on the proof of Theorem 1.2

The proof of Theorem 1.2 consists of two parts. The first part is to show that
(NLS) admits a unique global $\mathcal{F}H^{\beta}$-solution that decays in
time of the order $-d(1-\lambda)/2$ in $L^{\infty}$ (Theorem A.1 below), and
the second part is to specify the final data $u_{+}$ in $H^{0,\delta}$. As
mentioned above, under $u(0)=u_{0}\in H^{\beta}\cap H^{0,\beta}$ with
$\beta>\max(1,d/2)$, the existence of such a solution and $u_{+}$ in $L^{2}$
are shown in [KM21]. To explain the strategy on their proof of the first part,
we focus attention on the following inequality given by (1.13):

$\displaystyle\norm{u(t)}_{\infty}\lesssim|t|^{-\frac{d}{2}(1-\lambda)}\norm{\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)}_{\infty}+o(|t|^{-\frac{d}{2}(1-\lambda)})$

for any $t\geq r_{0}$. According to the above inequality, we have to show the
boundedness of $\norm{\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)}_{\infty}$ to get
the desired solution. This claim can be proven by constructing the solution
$u$ that $|J(t)|^{\beta}u(t)$ increase in time of the small order depending on
the size of $u_{0}$. This kind of technique is originally developed by [HN98].
In [KM21], $u_{0}\in H^{\beta}$ is required to control $\norm{u(t)}_{\infty}$
near $t=0$, since they construct the solution $u(t)$ on $[0,\infty)$. To avoid
a use of the condition, we construct the solution on $[r_{0},\infty)$. They
also employ (1.3) instead of (1.13), because the factor $\mathcal{M}_{+}$ does
not appear due to the assumption $a_{1}=0$ in [KM21]. Hence, for the reader’s
convenience, we give a sketch of the proof of Theorem A.1 in Appendix A,
although it has been essentially proven in [KM21].

The novelty of the paper is to find the final data $u_{+}$ in $H^{0,\delta}$
with $d/2<\delta<\beta$. To state our technique, we only handle $d=2$, $3$.
Following [HN98], from (NLS), we derive

$\displaystyle i\partial_{t}(\mathcal{F}\mathcal{M}_{+}U(0,t)u)={}$
$\displaystyle\eta|\zeta_{2}(t)|^{-\frac{1}{1-\lambda}}|\mathcal{F}\mathcal{M}_{+}U(0,t)u|^{p_{c}}\mathcal{F}\mathcal{M}_{+}U(0,t)u+o(t^{-1}).$

Roughly speaking, the first term in the above right-hand side is
$\mathcal{O}(t^{-1})$, that is, it is non-integrable with respect to time on
$[r_{0},\infty)$. Hence we introduce a phase modification

$\displaystyle v_{p}\coloneqq{}$
$\displaystyle\mathcal{F}^{-1}\mathcal{P}(t)\mathcal{F}\mathcal{M}_{+}U(0,t)u,$
$\displaystyle\mathcal{P}(t)\coloneqq{}$
$\displaystyle\exp\left(i\eta\int_{r_{0}}^{t}\left(|\tau|^{-\chi}+|\mathcal{F}\mathcal{M}_{+}U(0,\tau)u(\tau)|^{2}\right)^{\frac{p_{c}}{2}}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau\right)$

for some $\chi>0$. By the modification, we obtain

$\displaystyle i\partial_{t}(\mathcal{F}v_{p})={}$
$\displaystyle\eta|\zeta_{2}(t)|^{-\frac{1}{1-\lambda}}\mathcal{P}(t)\left(|\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)|^{p_{c}}-\left(|t|^{-\chi}+|\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)|^{2}\right)^{\frac{p_{c}}{2}}\right)$
$\displaystyle{}\times\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)+o(t^{-1}).$

This implies that there exists $u_{+}\in H^{0,\delta}$ such that
$v_{p}\rightarrow u_{+}$ in $H^{0,\delta}$ as $t\rightarrow\infty$. Owing to
the lack of differentiability of the nonlinearity, the argument in [HN98] is
not directly applicable, since we need to find the final data in
$H^{0,\delta}$. To overcome the difficulty, we take the above phase
modification with the factor $|\tau|^{-\chi}$, which is firstly used by
Ginibre-Ozawa [GO93]. The key of the proof is to see that the first term of
the above right-hand side, which comes from the modification, is negligible
for large time. To deal with this issue, we develop fundamental inequalities
necessary to extract the time decay of the first term.

###  1.4 Organization of this paper

The rest of the paper is organized as follows: In Section 2, we collect
several ingredients to construct the modified scattering operator. One first
states properties for the Galilean operator $J(t)$ and estimates for the
fractional calculus. We also prove a persistent property for the regularity of
solutions necessary to connect the solution near $t=0$. Nonlinear estimates
involving fractional derivatives are discussed at the end of this section. We
next give the proof of Theorem 1.1 and construct the modified wave operator in
Section 3. In Section 4, key inequalities for modified phase corrections to
get rid of the lack of differentiability of the nonlinearity are developed,
and the proof of Theorem 1.2 will be given. Moreover, we shall construct the
modified inverse wave operator and finally define the modified scattering
operator at the end of this section. Appendix A provides us to prove Theorem
A.1, which is the former part of the proof of Theorem 1.2. The derivation of
the integral equation (1.16) is also treated in Appendix B. For the sake of
simplicity, we give all of statements in the both case $p_{c}\geq 1$ and
$p_{c}<1$, but usually deal with $p_{c}<1$ only because the other case are
easier or similar. Remark that $p_{c}\geq 1$ holds when $d=1$, $2$ or $d=3$
and $\lambda\in[1/3,1/2)$ and also if $d=3$ and $\lambda\in(-1/3,1/3)$, then
$p_{c}<1$. Because $1+p_{c}\leq d/2$ holds in the case $d=3$ and
$\lambda\leq-1/3$, we can not rely on the approach in the use of the Sobolev
inequality and hence we omit to discuss this case. Furthermore, the case

$\lambda\geq 0$

is only treated afterward since the case $\lambda<0$ is same as $\lambda=0$.

##  2 Preliminaries

###  2.1 Some tools

We summarize several tools frequently used throughout this paper. Let us first
introduce the operator $J(t)=U(t,0)xU(0,t)$ which is often called the Galilean
operator. Besides, define $|J(t)|^{\gamma}=U(t,0)|x|^{\gamma}U(0,t)$ for any
$\gamma\geq 0$. The operator plays a important role in this paper. Noting that

$\displaystyle|J(t)|^{\gamma}={}$ $\displaystyle{\cal M}_{1}(t){\cal
D}_{1}(t){\cal F}{\cal M}_{+}{\cal M}_{2}(t)|x|^{\gamma}{\cal
M}_{2}(t)^{-1}{\cal M}_{+}^{-1}{\cal F}^{-1}{\cal D}_{1}(t)^{-1}{\cal
M}_{1}(t)^{-1},$ $\displaystyle|J(t)|^{\gamma}R(t)={}$ $\displaystyle
U(t,0)|x|^{\gamma}U(0,t)\cdot{\cal M}_{1}(t){\cal D}_{1}(t)\left({\cal F}{\cal
M}_{2}(t){\cal F}^{-1}-1\right),$

we have the following properties of the operator:

######  Lemma 2.1.

For all $t\in{\bf R}$ and $\gamma>0$, it holds that

$\displaystyle|J(t)|^{\gamma}={}$
$\displaystyle\left|\zeta_{2}(t)\right|^{\gamma}\mathcal{M}_{1}(t)|\nabla|^{\gamma}\mathcal{M}_{1}^{-1}(t),$
(2.1) $\displaystyle|J(t)|^{\gamma}R(t)={}$ $\displaystyle
R(t)|\nabla|^{\gamma}.$ (2.2)

We often employ the following well-known estimates:

######  Lemma 2.2 (e.g. [KaMi23, Lemma 4.6]).

Let $\alpha>0$ and $n\in{\bf Z}$. If $\alpha\geq 1$, then

$\displaystyle\left||z|^{\alpha-n}z^{n}-|w|^{\alpha-n}w^{n}\right|\leq
C|n|\left(|z|^{\alpha-1}+|w|^{\alpha-1}\right)|z-w|$ (2.3)

for any $z$, $w\in{\bf C}$. Further, when $\alpha\in(0,1)$, the following
holds:

$\displaystyle\left||z|^{\alpha-n}z^{n}-|w|^{\alpha-n}w^{n}\right|\leq
C|z-w|^{\alpha}.$ (2.4)

The subsequent estimates are the fractional Leibniz rule and chain rule.

######  Lemma 2.3 ([KaP88, GraO14]).

Let $s>0$, $1<r<\infty$, $1<p_{1},p_{2},q_{1},q_{2}\leq\infty$ and
$1/r=1/p_{1}+1/q_{1}=1/p_{2}+1/q_{2}$. Then we have the following fractional
Leibniz rule:

$\norm{|\nabla|^{s}(fg)}_{r}\lesssim\left(\norm{|\nabla|^{s}f}_{p_{1}}\norm{g}_{q_{1}}+\norm{f}_{p_{2}}\norm{|\nabla|^{s}g}_{q_{2}}\right).$

######  Lemma 2.4 ([Vi07]).

Let $F$ be a Hölder continuous function of order $0<\rho<1$. Suppose that
$0<\sigma<\rho$, $1<p<\infty$ and $\sigma/\rho<s<1$. Then

$\displaystyle\norm{|\nabla|^{\sigma}F(u)}_{L^{p}}\lesssim\norm{|u|^{\rho-\sigma/s}}_{L^{p_{1}}}\norm{|\nabla|^{s}u}_{L^{p_{2}\sigma/s}}^{\sigma/s}$

provided $1/p=1/p_{1}+1/p_{2}$ and $(1-\sigma/(\rho s))p_{1}>1$.

###  2.2 Persistence of regularity of solutions

In Section 3, we construct the ${\cal F}H^{\beta}$-solution $u(t)$,
$t\in(-\infty,-r_{0}]$ from $u_{-}$, while the construction scheme strongly
rely on (1.3) which is the factorization of $U(t,s)$. In order to construct
the modified wave operator, we need to construct the nonlinear propagator
$u(-r_{0})\mapsto u(0)$ and in $[-r_{0},0]$, we can not rely on the approach
due to factorization. Similarly, in order to define the inverse wave operator
in Section 4, one needs the nonlinear propagator $u(0)\mapsto u(r_{0})$. In
this subsection, these propagators are established using the persistence of
regularity. We here show the persistence result in a generalized form, but, in
our case, it is enough to consider the case where $I=[-r_{0},0]$,
$t_{0}=-r_{0}$ and $u_{0}=u(-r_{0})$, (resp. $I=[0,r_{0}]$, $t_{0}=0$ and
$u_{0}=u(0)$) in Proposition 2.1 as follows:

######  Proposition 2.1.

Let $I$ be a bounded interval and fix $t_{0}\in I$ and
$(Q_{0},R_{0})=(4(2+p_{c})/(dp_{c}),2+p_{c})$. Assume $U(0,t_{0})u_{0}\in
H^{0,\beta}$ with $\|U(0,t_{0})u_{0}\|_{H^{0,\beta}}$ small enough compared to
$|I|$. Then (NLS) has a unique solution $u\in C(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}}),|J(\cdot)|^{\beta}u\in C(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}})$ with

$\displaystyle\norm{u}_{L^{\infty}(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}})}+\norm{|J(\cdot)|^{\beta}u}_{L^{\infty}(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}})}\leq{}$ $\displaystyle
A_{0}\norm{U(0,t_{0})u_{0}}_{H^{0,\beta}},$ (2.5)

where $A_{0}>0$ depends on $d$, $\lambda$, $I$ and $\norm{u_{0}}_{2}$ and
monotonically increases for $\norm{u_{0}}_{2}$.

######  Remark 2.1.

We only employ Proposition 2.1 to construct the propagators on
$[-r_{0},r_{0}]$.

Let us divide several steps into the construction of the nonlinear
propagators. Following the pioneer work of Tsutsumi [YT87], Carles [Ca11]
proves the global well-posedness in $L^{2}$ by the local-in time Strichartz
estimate and the conservation of $\norm{u(t)}_{2}$.

######  Lemma 2.5 (cf. [Ca11, Proposition 1.5]).

Let $I$ be a bounded interval and fix $t_{0}\in I$ and
$(Q_{0},R_{0})=(4(2+p_{c})/(dp_{c}),2+p_{c})$. Assume $u_{0}\in L^{2}$. Then
(NLS) has a unique global solution $u\in C(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}})$ with $u(t_{0})=u_{0}$ such that

$\displaystyle\norm{u}_{L^{\infty}(I;L^{2})\cap L^{Q_{0}}(I;L^{R_{0}})}\leq{}$
$\displaystyle A_{1}\norm{u_{0}}_{2},$ (2.6)

where $A_{1}>0$ depends on $d$, $\lambda$, $I$ and $\norm{u_{0}}_{2}$ and
monotonically increases for $\norm{u_{0}}_{2}$.

By means of the nonlinear estimates proven by the fractional calculus, we can
obtain the following persistent property for the regularity of solutions. We
here recall $|J(t)|^{\beta}=U(t,0)|x|^{\beta}U(0,t)$ in $L^{2}$. Remark that a
similar property has been discussed in [NP09].

######  Lemma 2.6 (Persistence of regularity of solutions).

Let $u$ be the solution as in Lemma 2.5 with the same interval $I$ under
$u(t_{0})=u_{0}$ and suppose $U(0,t_{0})u_{0}\in H^{0,\beta}$. Then
$|J(\cdot)|^{\beta}u\in C(I;L^{2})\cap L^{Q_{0}}(I;L^{R_{0}})$ and it holds
that

$\displaystyle\norm{|J(\cdot)|^{\beta}u}_{L^{\infty}(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}})}\leq{}$ $\displaystyle
A_{2}\norm{U(0,t_{0})u_{0}}_{H^{0,\beta}},$ (2.7)

where $A_{2}>0$ depends on $d$, $\lambda$, $I$ and $\norm{u_{0}}_{2}$ and
monotonically increases for $\norm{u_{0}}_{2}$.

In order to prove Lemma 2.6, we need the following nonlinear estimate:

######  Lemma 2.7.

Let $(Q_{0},R_{0})=(4(2+p_{c})/(dp_{c}),2+p_{c})$ as in Lemma 2.5. Then the
estimate

$\displaystyle\norm{|J(\cdot)|^{\beta}|u|^{p_{c}}u}_{L^{Q_{0}^{\prime}}(I;L^{R_{0}^{\prime}})}\lesssim{}$
$\displaystyle|I|^{\theta}\norm{u}_{L^{Q_{0}}(I;L^{R_{0}})}^{p_{c}}\norm{|J(t)|^{\beta}u}_{L^{Q_{0}}(I;L^{R_{0}})}$

holds, where $\theta=1-dp_{c}/4=1-1/2(1-\lambda)\geq 0$.

###### Proof of Lemma 2.7.

We only consider the case $p_{c}<1$ since the other cases are similar by a use
of the fractional chain rule (e.g. [CW91, K95]). Set
$\phi=\mathcal{M}_{1}^{-1}(t)u$. By (2.1), we have

$\displaystyle\left\||J(t)|^{\beta}|u(t)|^{p_{c}}u(t)\right\|_{R_{0}}=|\zeta_{2}(t)|^{\beta}\left\||\nabla|^{\beta}|\phi(t)|^{p_{c}}\phi(t)\right\|_{R_{0}}.$

Straightforward calculation gives

$\displaystyle\nabla\left(|\phi|^{p_{c}}\phi\right)={}$
$\displaystyle\left(\frac{p_{c}}{2}+1\right)|\phi|^{p_{c}}\nabla\phi+\frac{p_{c}}{2}|\phi|^{p_{c}-2}\phi^{2}\overline{\nabla\phi}\eqqcolon
B_{1}(\phi)\nabla\phi+B_{2}(\phi)\overline{\nabla\phi}.$

By Lemma 2.3, we obtain

$\displaystyle\norm{|\nabla|^{\beta-1}B_{1}(\phi)\nabla\phi}_{R_{0}^{\prime}}\lesssim{}$
$\displaystyle\norm{|\nabla|^{\beta-1}B_{1}(\phi)}_{R_{0}/p_{c}}\norm{\nabla\phi}_{R_{0}}+\norm{\phi}_{R_{0}}^{p_{c}}\norm{|\nabla|^{\beta}\phi}_{R_{0}}.$

Note that $1/R_{0}^{\prime}=p_{c}/R_{0}+1/R_{0}$. Applying Lemma 2.4, one has

$\displaystyle\norm{|\nabla|^{\beta-1}B_{1}(\phi)}_{R_{0}/p_{c}}\lesssim{}$
$\displaystyle\norm{|\phi|^{p_{c}-\frac{\beta-1}{s}}}_{R_{1}}\norm{|\nabla|^{s}\phi}_{\frac{\beta-1}{s}R_{2}}^{\frac{\beta-1}{s}}$

for any $s\in\left((\beta-1)/p_{c},1\right)$, where

$p_{c}/R_{0}=1/R_{1}+1/R_{2},\quad 1/R_{1}=(p_{c}-(\beta-1)/s)/R_{0},\quad
1/R_{2}=(\beta-1)/s\cdot 1/R_{0}.$

Hence, we see from the Gagliardo-Nirenberg inequality (see [BL76, p168]) that

$\displaystyle\norm{\nabla\phi}_{R_{0}}\lesssim\norm{\phi}_{R_{0}}^{1-\theta_{1}}\norm{|\nabla|^{\beta}\phi}_{R_{0}}^{\theta_{1}},\quad\norm{|\nabla|^{s}\phi}_{R_{0}}\lesssim\norm{\phi}_{R_{0}}^{1-\theta_{2}}\norm{|\nabla|^{\beta}\phi}_{R_{0}}^{\theta_{2}},$

where $\theta_{1}=1/\beta\in(0,1)$ and $\theta_{2}=s/\beta\in(0,1)$.
Collecting the above estimates,

$\displaystyle\norm{|\nabla|^{\beta-1}B_{1}(\phi)}_{p_{c}/R_{0}}\norm{\nabla\phi}_{R_{0}}\lesssim{}$
$\displaystyle\norm{\phi}_{R_{0}}^{p_{c}}\norm{|\nabla|^{\beta}\phi}_{R_{0}}.$

holds. Since the estimate $\norm{B_{2}(\phi)\overline{\nabla\phi}}_{R_{0}}$ is
the same, we obtain

$\displaystyle\norm{|J(t)|^{\beta}|u(t)|^{p_{c}}u(t)}_{R_{0}^{\prime}}\lesssim\norm{u(t)}_{R_{0}}^{p_{c}}\norm{|J(t)|^{\beta}u(t)}_{R_{0}}.$

By the Hölder inequality, one has

$\displaystyle\norm{|J(\cdot)|^{\beta}|u|^{p_{c}}u}_{L^{Q_{0}^{\prime}}(I,L^{R_{0}^{\prime}})}\lesssim{}$
$\displaystyle|I|^{\theta}\norm{u}_{L^{Q_{0}}(I;L^{R_{0}})}^{p_{c}}\norm{|J(t)|^{\beta}u}_{L^{Q_{0}}(I;L^{R_{0}})},$

where $\theta=1/Q_{0}^{\prime}-(p_{c}+1)/Q_{0}=1-dp_{c}/4$, as desired. ∎

###### Proof of Lemma 2.6.

Let us only consider the forward direction in time, because the backward
direction is same. Set $T_{0}=\sup I-t_{0}$. From the standard well-posedness
theory, there exists a unique local $\mathcal{F}H^{\beta}$-solution $u$,
$J(\cdot)u\in C([t_{0},t_{0}+T];L^{2})\cap L^{Q_{0}}(t_{0},t_{0}+T;L^{R_{0}})$
to (NLS) with $u(t_{0})=u_{0}$ for some $T\leq T_{0}$. We here denote the
maximal existence time of its solution by $T_{{\rm max}}$. Let us prove
$T_{\max}=T_{0}$ by a contradiction. Suppose that $T_{\max}<T_{0}$. Take
arbitrarily small $\varepsilon$, $\delta>0$ with $\delta>\varepsilon$. We set
$T_{\tau}=T_{\rm{max}}-\tau<T_{\rm{max}}$ and
$I_{\tau}=[t_{0},t_{0}+T_{\tau}]$ for any small $\tau>0$. By means of the
local-in-time Strichartz estimate, we see from Lemma 2.7 and (2.6) that

$\displaystyle{}\norm{|J(\cdot)|^{\beta}u}_{L^{\infty}(I_{\varepsilon};L^{2})}+\norm{|J(\cdot)|^{\beta}u}_{L^{Q_{0}}(I_{\varepsilon};L^{R_{0}})}$
$\displaystyle\leq{}$ $\displaystyle 2C_{0}\norm{|J(t_{0})|^{\beta}u_{0}}_{2}$
$\displaystyle{}+C\norm{u_{0}}_{2}^{p_{c}}\left(|T_{\delta}|^{\theta}\norm{|J(\cdot)|^{\beta}u}_{L^{Q_{0}}(I_{\delta};L^{R_{0}})}+|\delta-\varepsilon|^{\theta}\norm{|J(\cdot)|^{\beta}u}_{L^{Q_{0}}(t_{0}+T_{\delta},t_{0}+T_{\varepsilon};L^{R_{0}})}\right)$
$\displaystyle\leq{}$ $\displaystyle
C_{\delta}(u_{0})+a_{1}\delta^{\theta}\norm{u_{0}}_{2}^{p_{c}}\norm{|J(\cdot)|^{\beta}u}_{L^{Q_{0}}(I_{\varepsilon};L^{R_{0}})},$

where $\theta=1-dp_{c}/4$. Hence taking $\delta_{0}>0$ such that
$a_{1}|\delta_{0}|^{\theta}\norm{u_{0}}_{2}^{p_{c}}\leq 1/2$, one has

$\displaystyle\norm{|J(\cdot)|^{\beta}u}_{L^{\infty}(I_{\varepsilon};L^{2})}+\norm{|J(\cdot)|^{\beta}u}_{L^{Q_{0}}(I_{\varepsilon};L^{R_{0}})}\leq
C_{\delta_{0}}(u_{0}).$

Letting $\varepsilon\downarrow 0$, we have the contradiction and thus
$T_{\max}=T_{0}$. This yields $|J(\cdot)|^{\beta}u\in C(I;L^{2})\cap
L^{Q_{0}}(I;L^{R_{0}})$.

As for the estimate (2.7), similarly to the above, there exists $\tau_{0}>0$
such that

$\displaystyle\norm{|J(\cdot)|^{\beta}u}_{L^{\infty}(I^{\prime}_{\tau_{0}};L^{2})}+\norm{|J(\cdot)|^{\beta}u}_{L^{Q_{0}}(I^{\prime}_{\tau_{0}};L^{R_{0}})}\leq
2C_{0}\norm{|J(t_{0})|^{\beta}u_{0}}_{2},$

where $I^{\prime}_{\tau_{0}}=[t_{0},t_{0}+\tau_{0}]\subset I$. Since
$\tau_{0}$ only depends on $\norm{u_{0}}_{2}$, thanks to the conservation of
$\norm{u(t)}_{2}$, we can iterate the above argument with the same $\tau_{0}$
and hence the desired assertion holds. This completes the proof. ∎

###### Proof of Proposition 2.1.

It is the consequence of Lemma 2.5 and Lemma 2.6. ∎

###  2.3 Nonlinear estimates

In this section, we shall deal with the following estimates with regard to the
difference of the nonlinearity:

######  Lemma 2.8.

Let $d/2<\beta<\min\left(d,1+p_{c}\right)$. Then the followings hold: If
$d=1$, then

$\displaystyle\norm{F(\phi)-F(\psi)}_{\dot{H}^{\beta}}\lesssim{}$
$\displaystyle\left(\norm{\phi-\psi}_{\infty}+\norm{\psi}_{\infty}\right)^{p_{c}}\norm{\phi-\psi}_{\dot{H}^{\beta}}$
$\displaystyle{}+\left(\norm{\phi-\psi}_{\infty}+\norm{\psi}_{\infty}\right)^{p_{c}-1}\norm{\phi-\psi}_{\infty}\norm{\psi}_{\dot{H}^{\beta}}.$

When $d=2$, $3$, the estimate

$\displaystyle{}\norm{F(\phi)-F(\psi)}_{\dot{H}^{\beta}}$
$\displaystyle\lesssim{}$
$\displaystyle\norm{\phi}_{\infty}^{p_{c}+1-\beta}\norm{\nabla\phi}_{2\beta}^{\beta-1}\norm{\nabla(\phi-\psi)}_{2\beta}+\norm{\phi}_{\infty}^{p_{c}}\norm{\phi-\psi}_{\dot{H}^{\beta}}+\norm{\phi-\psi}_{\infty}^{p_{c}}\norm{\psi}_{\dot{H}^{\beta}}$
$\displaystyle{}+\norm{\phi-\psi}_{\infty}^{p_{c}-\gamma(\beta-1)}\left(\norm{\phi}_{\infty}^{\gamma-1}\norm{\nabla\phi}_{2\beta}+\norm{\psi}_{\infty}^{\gamma-1}\norm{\nabla\psi}_{2\beta}\right)^{\beta-1}\norm{\nabla\psi}_{2\beta}$

holds for any $\gamma\in(1,p_{c}/(\beta-1))$ if $p_{c}<1$, otherwise

$\displaystyle\norm{F(\phi)-F(\psi)}_{\dot{H}^{\beta}}\lesssim{}$
$\displaystyle\norm{\phi}_{\infty}^{p_{c}+1-\beta}\norm{\nabla\phi}_{2\beta}^{\beta-1}\norm{\nabla(\phi-\psi)}_{2\beta}+\norm{\phi}_{\infty}^{p_{c}}\norm{\phi-\psi}_{\dot{H}^{\beta}}$
$\displaystyle{}+\left(\norm{\phi}_{\infty}+\norm{\psi}_{\infty}\right)^{p_{c}-1}\norm{\phi-\psi}_{\infty}\norm{\psi}_{\dot{H}^{\beta}}$
$\displaystyle{}+\left(\norm{\phi}_{\infty}+\norm{\psi}_{\infty}\right)^{(p_{c}-1)(2-\beta)}\norm{\phi-\psi}_{\infty}^{2-\beta}$
$\displaystyle{}\times\left(\norm{\phi}_{\infty}^{p_{c}-1}\norm{\nabla\phi}_{2\beta}+\norm{\psi}_{\infty}^{p_{c}-1}\norm{\nabla\psi}_{2\beta}\right)^{\beta-1}\norm{\nabla\psi}_{2\beta}.$

######  Remark 2.2.

Connecting Lemma 2.8 with $\psi=0$ and the Gagliardo-Nirenberg inequality, we
easily verify
$\norm{F(\phi)}_{\dot{H}^{\beta}}\lesssim\norm{\phi}_{\infty}^{p_{c}}\norm{\phi}_{\dot{H}^{\beta}}$.

###### Proof.

The estimate in $d=1$ is immediate from a use of the representation of
$\dot{H}^{\beta}$-norm via the Gagliardo semi-norm (see [DPV12, Proposition
3.4]), after the calculation as follows:

$\displaystyle F(\phi)-F(\psi)={}$
$\displaystyle(\phi-\psi)\int_{0}^{1}\diffp{F}{z}(\psi+\theta(\phi-\psi))\,d\theta+(\overline{\phi-\psi})\int_{0}^{1}\diffp{F}{\overline{z}}(\psi+\theta(\phi-\psi))\,d\theta$

Let us consider the case $d=2$, $3$. Remark that

$\displaystyle\begin{aligned}
\nabla\left(|\phi|^{p_{c}}\phi-|\psi|^{p_{c}}\psi\right)={}&\left(\frac{p_{c}}{2}+1\right)\left(|\phi|^{p_{c}}\nabla(\phi-\psi)+\left(|\phi|^{p_{c}}-|\psi|^{p_{c}}\right)\nabla\psi\right)\\\
&{}+\frac{p_{c}}{2}\left(|\phi|^{p_{c}-2}\phi^{2}\nabla(\overline{\phi-\psi})+\left(|\phi|^{p_{c}-2}\phi^{2}-|\psi|^{p_{c}-2}\psi^{2}\right)\overline{\nabla\psi}\right).\end{aligned}$
(2.8)

We here mimic the argument of the proof of [KaMi23, Proposition 4.3] to
estimate the first term of the right-hand side of (2.8). Remark that it comes
from [KaMi23, Lemma 4.9] that

$\displaystyle\begin{aligned}
\norm{|\nabla|^{\beta-1}\left(|\varphi|^{\gamma-n}\varphi^{n}\right)}_{\frac{2\beta}{\beta-1}}\lesssim{}&\norm{\varphi}^{\gamma+1-\beta}_{\infty}\norm{\nabla\varphi}_{2\beta}^{\beta-1}\end{aligned}$
(2.9)

for any $n\in{\bf Z}$ and all $\gamma>\beta-1$. Applying Lemma 2.3 and (2.9),
we reach to

$\displaystyle\norm{|\phi|^{p_{c}}\nabla(\phi-\psi)}_{\dot{H}^{\beta-1}}\lesssim{}$
$\displaystyle\norm{|\nabla|^{\beta-1}|\phi|^{p_{c}}}_{\frac{2\beta}{\beta-1}}\norm{\nabla(\phi-\psi)}_{2\beta}+\norm{|\phi|^{p_{c}}}_{\infty}\norm{|\nabla|^{\beta}(\phi-\psi)}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\norm{\phi}_{\infty}^{p_{c}+1-\beta}\norm{\nabla\phi}_{2\beta}^{\beta-1}\norm{\nabla(\phi-\psi)}_{2\beta}+\norm{\phi}_{\infty}^{p_{c}}\norm{\phi-\psi}_{\dot{H}^{\beta}}.$

Let us next go to the second term of (2.8). we see from Lemma 2.3 that

$\displaystyle\begin{aligned}
&{}\norm{\left(|\phi|^{p_{c}}-|\psi|^{p_{c}}\right)\nabla\psi}_{\dot{H}^{\beta-1}}\\\
\lesssim{}&\norm{|\phi|^{p_{c}}-|\psi|^{p_{c}}}_{\infty}\norm{\psi}_{\dot{H}^{\beta}}+\norm{|\nabla|^{\beta-1}\left(|\phi|^{p_{c}}-|\psi|^{p_{c}}\right)}_{\frac{2\beta}{\beta-1}}\norm{\nabla\psi}_{2\beta}.\end{aligned}$
(2.10)

Hereafter, one shall treat the case $p_{c}<1$. In view of the concavity of
$|z|^{p_{c}}$,

$\displaystyle\norm{|\phi|^{p_{c}}-|\psi|^{p_{c}}}_{\infty}\leq\norm{\phi-\psi}_{\infty}^{p_{c}}$

holds. Hence, the Gagliardo-Nirenberg inequality leads to

$\displaystyle\begin{aligned}
\norm{|\nabla|^{\beta-1}\left(|\phi|^{p_{c}}-|\psi|^{p_{c}}\right)}_{\frac{2\beta}{\beta-1}}\lesssim{}&\norm{\phi-\psi}_{\infty}^{p_{c}\left(1-\frac{\beta-1}{s_{0}}\right)}\\\
&{}\times\left(\norm{|\nabla|^{s_{0}}|\phi|^{p_{c}}}_{\frac{2\beta}{s_{0}}}+\norm{|\nabla|^{s_{0}}|\psi|^{p_{c}}}_{\frac{2\beta}{s_{0}}}\right)^{\frac{\beta-1}{s_{0}}},\end{aligned}$
(2.11)

where $s_{0}=p_{c}-\kappa$ for any $\kappa\in(0,p_{c}-\beta+1)$. By Lemma 2.4,
we deduce that

$\displaystyle\norm{|\nabla|^{s_{0}}|\varphi|^{p_{c}}}_{\frac{2\beta}{s_{0}}}\lesssim{}$
$\displaystyle\norm{\varphi}_{\infty}^{p_{c}-\frac{s_{0}}{s}}\norm{|\nabla|^{s}\varphi}_{\frac{2\beta}{s}}^{\frac{s_{0}}{s}}$

for any $s\in\left(s_{0}/p_{c},1\right)$. We then see from the Gagliaro-
Nirenberg inequality that

$\displaystyle\norm{|\nabla|^{s_{0}}|\varphi|^{p_{c}}}_{\frac{2\beta}{s_{0}}}\lesssim{}$
$\displaystyle\norm{\varphi}_{\infty}^{p_{c}-s_{0}}\norm{\nabla\varphi}_{2\beta}^{s_{0}}.$

Plugging the above into (2.11), one has

$\displaystyle\norm{|\nabla|^{\beta-1}\left(|\phi|^{p_{c}}-|\psi|^{p_{c}}\right)}_{\frac{2\beta}{\beta-1}}\lesssim{}$
$\displaystyle\norm{\phi-\psi}_{\infty}^{p_{c}\left(1-\frac{\beta-1}{s_{0}}\right)}\left(\norm{\phi}_{\infty}^{\frac{p_{c}}{s_{0}}-1}\norm{\nabla\phi}_{2\beta}+\norm{\psi}_{\infty}^{\frac{p_{c}}{s_{0}}-1}\norm{\nabla\psi}_{2\beta}\right)^{\beta-1}.$

Taking $p_{c}/s_{0}=\gamma\in\left(1,p_{c}/(\beta-1)\right)$, the proof is
completed in $p_{c}<1$, because the third and last terms of (2.8) are the
same. The case $p_{c}\geq 1$ is easier, so we omit the proof. ∎

##  3 Construction of the modified wave operators

In this section, we shall prove Theorem 1.1. By the integral equation (1.18),
it is easy to notice that

$\displaystyle\left\|u-u_{p}\right\|_{X_{b}}\lesssim\left\|\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)ds\right\|_{X_{b}}+\left\|{\cal
K}(t)\right\|_{X_{b}}+\left\|{\cal E}_{r}(t)\right\|_{X_{b}},$ (3.1)

where $\norm{\phi}_{X_{b}}$ is defined by (1.19). Here, in order to deduce the
integrability in $(-\infty,r_{0}]$ for integrands of reminder terms ${\cal K}$
and ${\cal E}_{r}$, we take $\beta<\alpha$ in spite of $u_{-}\in
H^{0,\alpha}$. This gap yields an additional $s$ decay, see Lemma 3.3. As for
the first term of the RHS of (3.1), we employ the nonlinear estimates in
Section 2.3 and herein we give the time decay estimate for this term; however,
it is, especially in $d=3$, quite difficult to obtain the bound
$\|\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)ds\|_{X_{b}}\leq(\mbox{small})\times\|u-u_{p}\|_{X_{b}}$
since the lack of the derivativeness of nonlinearity. Indeed, as for $f,g\in
H^{\alpha}$, it is difficult to show
$\|e^{i|f|^{\alpha-1}}f-e^{i|g|^{\alpha-1}}g\|_{H^{\beta}}\lesssim\|f-g\|_{H^{\beta}}$
for $\alpha-1<\beta\leq\alpha$ . Because of this reason, the contraction
mapping argument on $(X_{b,R},{d}_{0})$ with
${d}_{0}(\phi,\psi)=\|\phi-\psi\|_{X_{b}}$ does not work. To avoid this
difficulty, we alternatively use weak norm distance $d$ and show that even if
such weak norm, $(X_{b,R},d)$ is a complete metric space.

###  3.1 Estimates for the external term

We first give estimates associated with the prescribed asymptotic profile.

######  Lemma 3.1.

Define $\widehat{w}$ as in (1.14). For any $|s|\geq 1$, the followings are
valid:

$\displaystyle\norm{\widehat{w}(s)}_{H^{\alpha}}\lesssim{}$
$\displaystyle\left\langle\log|s|\right\rangle^{\lceil\alpha\rceil}\norm{u_{-}}_{H^{0,\alpha}}\left(1+\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}}\right)^{\lceil\alpha\rceil},$
(3.2)
$\displaystyle\norm{F\left(\widehat{w}(s)\right)}_{H^{\alpha}}\lesssim{}$
$\displaystyle\left\langle\log|s|\right\rangle^{\lceil\alpha\rceil}\norm{u_{-}}_{H^{0,\alpha}}^{1+p_{c}}\left(1+\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}}\right)^{\lceil\alpha\rceil}.$
(3.3)

###### Proof.

Noting $\norm{\widehat{u_{-}}}_{\infty}\lesssim\norm{u_{-}}_{H^{0,\alpha}}$
given by the Sobolev embedding, the estimates have been already shown in
[KaMi23, Proposition 3.1 and Proposition 3.2]. ∎

Recall that $X_{b}$\- norm is defined by (1.19). To prove Theorem 1.1, we have
to control the $X_{b}$-norm for ${A}(t)$ and $\mathcal{E}_{r}(t)$ defined as
in (1.17).

######  Lemma 3.2.

Take $0<b<\left(1-\beta/2\right)(1-2\lambda)$. Then it holds that

$\displaystyle\norm{\mathcal{A}(t)}_{X_{b}}\lesssim{}$
$\displaystyle\norm{u_{-}}_{H^{0,\beta}}^{1+p_{c}}\left(1+\norm{u_{-}}_{H^{0,\beta}}^{p_{c}}\right)^{\lceil\beta\rceil}.$

###### Proof.

The estimate of $\norm{\mathcal{A}(t)}_{2}$ is easier, so we only treat
$\norm{U(0,t)\mathcal{A}(t)}_{H^{0,\beta}}$. Set
$f(s)=\zeta_{2}(s)a_{2}^{-1}|s|^{\lambda-1}$. Noting that $|f(s)|\geq 1/2$
follows form (1.8), using (1.7), we get

$\displaystyle\absolutevalue{|f(s)|^{-\frac{1}{1-\lambda}}-1}\lesssim\absolutevalue{|f(s)|-1}\int_{0}^{1}\frac{d\theta}{\left(1+\theta(|f(s)|-1)\right)^{\frac{1}{1-\lambda}+1}}\lesssim\absolutevalue{\frac{\zeta_{2}(s)}{|s|^{1-\lambda}}-a_{2}}\lesssim|s|^{-(1-2\lambda)}$

for any $s\leq-r_{0}$. Hence it follows from (3.3) that

$\displaystyle\norm{U(0,t)\mathcal{A}(t)}_{H^{0,\beta}}\leq{}$
$\displaystyle\norm{\int_{-\infty}^{t}|x|^{\beta}\mathcal{F}^{-1}\left(\frac{c_{+}s}{|\zeta_{2}(s)|^{1/(1-\lambda)}}-1\right)F(\widehat{w}(s))\frac{ds}{c_{+}|s|}}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\int_{-\infty}^{t}\absolutevalue{|f(s)|^{-\frac{1}{1-\lambda}}-1}\norm{F(\widehat{w}(s))}_{\dot{H}^{\beta}}\frac{ds}{c_{+}|s|}$
$\displaystyle\lesssim{}$
$\displaystyle|t|^{-(1-2\lambda)}\left\langle\log|t|\right\rangle^{\lceil\beta\rceil}\norm{u_{-}}_{H^{0,\alpha}}^{1+p_{c}}\left(1+\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}}\right)^{\lceil\alpha\rceil}$

holds for any $t\leq-r_{0}$. This completes the proof. ∎

The next statement is the estimates for $\mathcal{E}_{r}(t)$.

######  Lemma 3.3.

Let $0<b<(\alpha-\beta)(1-2\lambda)/2$. Then these hold that

$\displaystyle\norm{R(t)\widehat{w}(t)}_{X_{b}}\lesssim{}$
$\displaystyle\norm{u_{-}}_{H^{0,\alpha}}\left(1+\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}}\right)^{\lceil\alpha\rceil},$
$\displaystyle\norm{\int_{-\infty}^{t}U(t,s)R(s)F(\widehat{w}(s))\frac{ds}{c_{+}|s|}}_{X_{b}}\lesssim{}$
$\displaystyle\norm{u_{-}}_{H^{0,\alpha}}^{1+p_{c}}\left(1+\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}}\right)^{\lceil\alpha\rceil}.$

###### Proof.

Applying (1.7) and (1.9), we have

$\displaystyle\norm{R(t)\widehat{w}(t)}_{2}={}$
$\displaystyle\norm{(\mathcal{M}_{2}(t)-1)\mathcal{F}^{-1}\widehat{w}(t)}_{2}\lesssim|t|^{-\theta(1-2\lambda)}\norm{\widehat{w}(t)}_{\dot{H}^{2\theta}}$

for any $t\leq-r_{0}$. Taking $\theta=\alpha/2\in[0,1]$, one sees from (3.2)
that

$\displaystyle\norm{R(t)\widehat{w}(t)}_{2}\lesssim{}$
$\displaystyle|t|^{-\frac{\alpha}{2}(1-2\lambda)}\left\langle\log|t|\right\rangle^{\lceil\alpha\rceil}\norm{u_{+}}_{H^{0,\alpha}}\left(1+\norm{\widehat{u_{+}}}_{\infty}^{p_{c}}\right)^{\lceil\alpha\rceil}$

for any $t\leq-r_{0}$. Arguing as in the above, by (2.2), it holds that

$\displaystyle\norm{|J(t)|^{\beta}R(t)\widehat{w}(t)}_{2}\lesssim{}$
$\displaystyle\norm{(\mathcal{M}_{2}(t)-1)\mathcal{F}^{-1}|\nabla|^{\beta}\widehat{w}(t)}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle|t|^{-\frac{\alpha-\beta}{2}(1-2\lambda)}\left\langle\log|t|\right\rangle^{\lceil\alpha\rceil}\norm{u_{+}}_{H^{0,\alpha}}\left(1+\norm{\widehat{u_{+}}}_{\infty}^{p_{c}}\right)^{\lceil\alpha\rceil}$

for any $t\leq-r_{0}$. Here we take $\theta=\alpha-\beta$. Hence the first
estimate is valid. Combining (3.3) with (2.2), the second estimate similarly
follows. ∎

###  3.2 Estimation for leading terms

The subsequent estimate is relevant to $F(u(s))-F(u_{p}(s)$.

######  Lemma 3.4.

Assume $u_{-}\in H^{0,\beta}$. Let $b>0$. Then it holds that

$\displaystyle\sup_{t\leq-
r_{0}}|t|^{\frac{\beta}{2}(1-2\lambda)+b}\norm{\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\left(\norm{u_{-}}_{H^{0,\beta}}+\norm{u-u_{p}}_{X_{b}}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}.$

###### Proof.

The nonlinearity can be decomposed as $F(u)-F(u_{p})=F^{(1)}(u)+F^{(2)}(u)$,
where

$\displaystyle
F^{(1)}(u)=\chi_{\\{|u_{p}|\leq|u-u_{p}|\\}}\left(F(u)-F(u_{p})\right),\quad
F^{(2)}(u)=\chi_{\\{|u_{p}|\geq|u-u_{p}|\\}}\left(F(u)-F(u_{p})\right),$ (3.4)

and $\chi_{A}$ is a characteristic function on $A\subset{\bf R}^{1+d}$. Since

$|F(u)-F(u_{p})|\lesssim\left(|u-u_{p}|^{p_{c}+1}+|u_{p}|^{p_{c}}|u-u_{p}|\right),$

we have

$|F^{(1)}(u)|\lesssim|u-u_{p}|^{p_{c}+1},\quad|F^{(2)}(u)|\lesssim|u_{p}|^{p_{c}}|u-u_{p}|.$

As for $F^{(2)}(u)$, it holds that

$\displaystyle\norm{F^{(2)}(u)}_{2}\lesssim{}$
$\displaystyle\norm{u_{p}}_{\infty}^{p_{c}}\norm{u-u_{p}}_{2}\lesssim|s|^{-1-\frac{\beta}{2}(1-2\lambda)-b}\norm{\widehat{u_{-}}}_{\infty}^{p_{c}}\norm{u-u_{p}}_{X_{b}}.$

This implies that

$\displaystyle\norm{\int_{-\infty}^{t}U(t,s)F^{(2)}(u(s))\,ds}_{2}\lesssim{}$
$\displaystyle|t|^{-\frac{\beta}{2}(1-2\lambda)-b}\norm{\widehat{u_{-}}}_{\infty}^{p_{c}}\norm{u-u_{p}}_{X_{b}}$
(3.5)

for any $t\leq-r_{0}$. Let us handle the estimate for $F^{(1)}(u)$. By the
Gagliardo-Nirenberg inequality and (2.1), we obtain

$\displaystyle\begin{aligned}
\norm{u-u_{p}}_{\infty}\lesssim{}&\norm{u-u_{p}}_{2}^{1-\theta}\norm{\mathcal{M}_{1}(s)|\nabla|\mathcal{M}_{1}(-s)(u-u_{p})}_{2}^{\theta}\\\
\lesssim{}&|s|^{-\frac{1}{p_{c}}-\frac{\beta}{2}(1-2\lambda)(1-\theta)-b}\norm{u-u_{p}}_{X_{b}}\end{aligned}$
(3.6)

for any $s\leq-r_{0}$, where $\theta=d/(2\beta)\in(0,1)$. Hence since

$\displaystyle\norm{F^{(1)}(u)}_{2}\lesssim{}$
$\displaystyle\norm{u-u_{p}}_{\infty}^{p_{c}}\norm{u-u_{p}}_{2}\lesssim|s|^{-1-\frac{\beta}{2}(1-2\lambda)-b-\frac{\beta}{2}(1-2\lambda)(1-\theta)p_{c}-bp_{c}}\norm{u-u_{p}}_{X_{b}}^{1+p_{c}},$

it is established that

$\displaystyle\norm{\int_{-\infty}^{t}U(t,s)F^{(1)}(u(s))\,ds}_{2}\lesssim{}$
$\displaystyle\norm{u-u_{p}}_{X_{b}}^{1+p_{c}}\int_{-\infty}^{t}|s|^{-1-\frac{\beta}{2}(1-2\lambda)-b-\frac{\beta}{2}(1-2\lambda)(1-\theta)p_{c}-bp_{c}}\,ds$
$\displaystyle\lesssim{}$
$\displaystyle|t|^{-\frac{\beta}{2}(1-2\lambda)-b-\frac{\beta}{2}(1-2\lambda)(1-\theta)p_{c}-bp_{c}}\norm{u-u_{p}}_{X_{b}}^{1+p_{c}}$

for any $t\leq-r_{0}$. Thus there exists
$b_{0}=b_{0}(\lambda,d,\beta,\alpha)>0$ such that

$\displaystyle|t|^{\frac{\beta}{2}(1-2\lambda)+b}\norm{\int_{-\infty}^{t}U(t,s)F^{(1)}(u(s))\,ds}_{2}\lesssim{}$
$\displaystyle|t|^{-b_{0}}\norm{u-u_{p}}_{X_{b}}^{1+p_{c}}$ (3.7)

for any $t\leq-r_{0}$. Combining (3.5) with (3.7), we conclude the desired
estimate. ∎

######  Proposition 3.1.

Assume $u_{-}\in H^{0,\beta}$. Then there exists $b=b(\lambda,d,\beta)>0$ such
that

$\displaystyle{}\sup_{t\leq-
r_{0}}|t|^{b}\norm{|J(t)|^{\beta}\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\left(\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}$

is valid if $d=1$. When $d=2$, $3$, the estimate

$\displaystyle{}\sup_{t\leq-
r_{0}}|t|^{b}\norm{|J(t)|^{\beta}\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\left(\norm{u-u_{p}}_{X_{b}}+\norm{u_{-}}_{H^{0,\beta}}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}+\norm{u_{-}}_{H^{0,\beta}}\norm{u-u_{p}}_{X_{b}}^{p_{c}}$
$\displaystyle{}+\left(\norm{u-u_{p}}_{X_{b}}+\norm{u_{-}}_{H^{0,\beta}}\right)^{\gamma(\beta-1)+1}\norm{u-u_{p}}_{X_{b}}^{p_{c}-\gamma(\beta-1)}$

holds for any $\gamma\in(1,p_{c}/(\beta-1))$ if $p_{c}<1$, otherwise

$\displaystyle{}\sup_{t\leq-
r_{0}}|t|^{b}\norm{|J(t)|^{\beta}\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\left(\norm{u-u_{p}}_{X_{b}}+\norm{u_{-}}_{H^{0,\beta}}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}$
$\displaystyle{}+\left(\norm{u-u_{p}}_{X_{b}}+\norm{u_{-}}_{H^{0,\beta}}\right)^{p_{c}-1+\beta}\norm{u-u_{p}}_{X_{b}}^{2-\beta}.$

###### Proof.

Set $\phi=\mathcal{M}^{-1}_{1}(t)u$ and $\psi=\mathcal{M}_{1}^{-1}(t)u_{p}$.
we see from (2.1) that

$\displaystyle\begin{aligned}
&{}\norm{|J(t)|^{\beta}\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}\\\
\leq{}&\int_{-\infty}^{t}|\zeta_{2}(s)|^{\beta}\norm{|\nabla|^{\beta}\left(F(\phi(s))-F(\psi(s))\right)}_{2}\,ds.\end{aligned}$
(3.8)

Arguing as in (3.6), it is obtained that

$\displaystyle\begin{aligned}
\norm{\phi}_{\infty}\lesssim{}&|s|^{-\frac{1}{p_{c}}}\left(|s|^{-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right),\\\
\norm{\phi-\psi}_{\infty}\lesssim{}&|s|^{-\frac{1}{p_{c}}-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}\end{aligned}$
(3.9)

for any $s\leq-r_{0}$, where
$\beta_{\lambda}=2^{-1}\beta(1-2\lambda)(1-\theta)$ and
$\theta=d/(2\beta)\in(0,1)$. Also we easily verify from (3.2) that for any
$s\leq-r_{0}$. We see from
$\norm{\nabla\varphi}_{2\beta}\lesssim\norm{\varphi}_{\infty}^{1-\frac{1}{\beta}}\norm{\varphi}_{\dot{H}^{\beta}}^{\frac{1}{\beta}}$
and (LABEL:p31:3) that

$\displaystyle\norm{\nabla\psi}_{2\beta}\lesssim|s|^{-\left(1-\frac{1}{\beta}\right)\frac{1}{p_{c}}-(1-\lambda)}\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}\norm{\widehat{u_{-}}}_{\infty}^{1-\frac{1}{\beta}}\norm{u_{-}}_{H^{0,\beta}}^{\frac{1}{\beta}}.$
(3.10)

Let us deal with the case $d=1$. By means of (3.9) and (LABEL:p31:3), we
deduce from Lemma 2.8 that for any $s\leq-r_{0}$,

$\displaystyle{}|\zeta_{2}(s)|^{\beta}\norm{|\nabla|^{\beta}\left(F(\phi(s))-F(\psi(s))\right)}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\left(\norm{\phi-\psi}_{\infty}+\norm{\psi}_{\infty}\right)^{p_{c}}\norm{|\alpha(s)|^{\beta}(u-u_{p})(s)}_{2}$
$\displaystyle{}+\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}\left(\norm{\phi-\psi}_{\infty}+\norm{\psi}_{\infty}\right)^{p_{c}-1}\norm{\phi-\psi}_{\infty}\norm{u_{-}}_{H^{0,\beta}}$
$\displaystyle\lesssim{}$
$\displaystyle|s|^{-1-b}\left(|s|^{-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}$
$\displaystyle{}+\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}|s|^{-1-b-\beta_{\lambda}}\left(s^{-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}-1}\norm{u_{-}}_{H^{0,\beta}}\norm{u-u_{p}}_{X_{b}}.$

Hence it follows from (3.8) that there exists $b_{1}=b_{1}(d,\lambda,\beta)>0$
such that

$\displaystyle{}|t|^{b}\norm{|J(t)|^{\beta}\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle\left(|t|^{-b_{1}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}$
$\displaystyle{}+|t|^{-b_{1}}\left(|t|^{-b_{1}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}-1}\norm{u_{-}}_{H^{0,\beta}}\norm{u-u_{p}}_{X_{b}}$

for any $t\leq-r_{0}$, from which the desired assertion follows. We shall
estimate the case $d=2$, $3$. Firstly the Gagliardo-Nirenberg inequality and
(3.2) tell us that

$\displaystyle\begin{aligned}
\norm{\phi}_{\dot{H}^{\beta}}\lesssim{}&|s|^{-\beta(1-\lambda)}\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}\left(|s|^{-b}\norm{u-u_{p}}_{X_{b}}+\norm{u_{-}}_{H^{0,\beta}}\right),\\\
\norm{\phi-\psi}_{\dot{H}^{\beta}}\lesssim{}&|s|^{-\beta(1-\lambda)-b}\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}\norm{u-u_{p}}_{X_{b}}.\end{aligned}$
(3.11)

Further it comes from (3.10) that

$\displaystyle\begin{aligned}
\norm{\nabla\phi}_{2\beta}\lesssim{}&|s|^{-\left(1-\frac{1}{\beta}\right)\frac{1}{p_{c}}-(1-\lambda)}\left\langle\log
s\right\rangle^{\lceil\beta\rceil}\\\
&{}\times\left(|s|^{-\left(1-\frac{1}{\beta}\right)\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}^{1-\frac{1}{\beta}}\norm{u_{-}}_{H^{0,\beta}}^{\frac{1}{\beta}}\right),\\\
\norm{\nabla\left(\phi-\psi\right)}_{2\beta}\lesssim{}&|s|^{-\left(1-\frac{1}{\beta}\right)\left(\frac{1}{p_{c}}+\beta_{\lambda}\right)-(1-\lambda)-b}\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}\norm{u-u_{p}}_{X_{b}}\end{aligned}$
(3.12)

Arguing as in the case $d=1$, in terms of the first term, we estimate

$\displaystyle{}|\zeta_{2}(s)|^{\beta}\norm{\phi}_{\infty}^{p_{c}+1-\beta}\norm{\nabla\phi}_{2\beta}^{\beta-1}\norm{\nabla(\phi-\psi)}_{2\beta}$
$\displaystyle\lesssim{}$
$\displaystyle\left\langle\log|s|\right\rangle^{\beta\lceil\beta\rceil}|s|^{-1-b-\left(1-\frac{1}{\beta}\right)\beta_{\lambda}}\left(|s|^{-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}+1-\beta}$
$\displaystyle{}\times\left(|s|^{-\left(1-\frac{1}{\beta}\right)\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}^{1-\frac{1}{\beta}}\norm{u_{-}}_{H^{0,\beta}}^{\frac{1}{\beta}}\right)^{\beta-1}\norm{u-u_{p}}_{X_{b}}$

for any $s\leq-r_{0}$. The second term are led to

$\displaystyle|\zeta_{2}(s)|^{\beta}\norm{\phi}_{\infty}^{p_{c}}\norm{\phi-\psi}_{\dot{H}^{\beta}}\lesssim{}$
$\displaystyle|s|^{-1-b}\left(|s|^{-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}.$

From now on, we shall only deal with the case $p_{c}<1$, because the case
$p_{c}\geq 1$ is similar. In terms of the third term, one obtains

$\displaystyle|\zeta_{2}(s)|^{\beta}\norm{\phi-\psi}_{\infty}^{p_{c}}\norm{\psi}_{\dot{H}^{\beta}}\lesssim{}$
$\displaystyle\left\langle\log|s|\right\rangle^{\lceil\beta\rceil}|s|^{-1-p_{c}(\beta_{\lambda}+b)}\norm{u_{-}}_{H^{0,\beta}}\norm{u-u_{p}}_{X_{b}}^{p_{c}}$

for any $s\leq-r_{0}$. Further, as for the last term, it is calculated that

$\displaystyle{}|\zeta_{2}(s)|^{\beta}\norm{\phi-\psi}_{\infty}^{p_{c}-\gamma(\beta-1)}\left(\norm{\phi}_{\infty}^{\gamma-1}\norm{\nabla\phi}_{2\beta}+\norm{\psi}_{\infty}^{\gamma-1}\norm{\nabla\psi}_{2\beta}\right)^{\beta-1}\norm{\nabla\psi}_{2\beta}$
$\displaystyle\lesssim{}$
$\displaystyle\left\langle\log|s|\right\rangle^{\beta\lceil\beta\rceil}|s|^{-1-(\beta_{\lambda}+b)(p_{c}-\gamma(\beta-1))}\norm{u_{-}}_{H^{0,\beta}}\left(|s|^{-\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{(\gamma-1)(\beta-1)}$
$\displaystyle{}\quad\times\left(|s|^{-\left(1-\frac{1}{\beta}\right)\beta_{\lambda}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}^{1-\frac{1}{\beta}}\norm{u_{-}}_{H^{0,\beta}}^{\frac{1}{\beta}}\right)^{\beta-1}\norm{u-u_{p}}_{X_{b}}^{p_{c}-\gamma(\beta-1)}$

for any $s\leq-r_{0}$, as long as
$0<b<\beta_{\lambda}\frac{p_{c}-\gamma(\beta-1)}{1-p_{c}+\gamma(\beta-1)}$.
Connecting these above, we conclude that there exists
$b_{1}=b_{1}(d,\lambda,\beta)>0$ such that

$\displaystyle{}|t|^{b}\norm{|J(t)|^{\beta}\int_{-\infty}^{t}U(t,s)\left(F(u(s))-F(u_{p}(s))\right)\,ds}_{2}$
$\displaystyle\lesssim{}$
$\displaystyle|t|^{-b_{1}}\left(|t|^{-b_{1}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}+\norm{u_{-}}_{H^{0,\beta}}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}$
$\displaystyle{}+\left(|t|^{-b_{1}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{p_{c}}\norm{u-u_{p}}_{X_{b}}$
$\displaystyle{}+|t|^{-b_{1}}\norm{u_{-}}_{H^{0,\beta}}\norm{u-u_{p}}_{X_{b}}^{p_{c}}$
$\displaystyle{}+|t|^{-b_{1}}\left(|t|^{-b_{1}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}\right)^{(\gamma-1)(\beta-1)}$
$\displaystyle{}\quad\times\left(|t|^{-b_{1}-b}\norm{u-u_{p}}_{X_{b}}+\norm{\widehat{u_{-}}}_{\infty}+\norm{u_{-}}_{H^{0,\beta}}\right)^{\beta-1}\norm{u_{-}}_{H^{0,\beta}}\norm{u-u_{p}}_{X_{b}}^{p_{c}-\gamma(\beta-1)}$

for any $t\leq-r_{0}$. By the Sobolev embedding, this yields the desired
estimate. ∎

###  3.3 Proof of Theorem 1.1

We are now in position to prove Theorem 1.1.

Take $b>0$ given by Proposition 3.1. From the integral equation (1.16), we
define the map

$\displaystyle\Psi(v(t))=i\int_{-\infty}^{t}U(t,s)\left(F(v(t)+u_{p}(s))-F(u_{p}(s))\right)ds+{\cal
A}(t)+{\cal E}_{\mathrm{r}}(t)$

for any $v\in X_{b,R}$. We remark that $(X_{b,R},d)$ is the complete metric
space. Indeed, any Cauchy sequence $\\{v_{n}\\}\subset X_{b,R}$ has a strong
limit $v\in L^{\infty}(-\infty,-r_{0};L^{2})$ and the boundedness of $X_{b,R}$
provides us a subsequence $\\{|J(t)|^{\beta}v_{n_{k}}(t)\\}$ and $w(t)\in
L^{2}$ such that $|J(t)|^{\beta}v_{n_{k}}(t)\rightarrow w(t)$ in $L^{2}$ for
each $t\geq T$. Since $|J(t)|^{\beta}$ is the closed operator in
${\mathscr{D}}(|J(t)|^{\beta})\coloneqq\\{u\in
L^{2}\mid\||J(t)|^{\beta}u\|_{2}<\infty\\}$, in view of the uniqueness of the
limit in $\mathscr{S}^{\prime}$, it holds that $w(t)=|J(t)|^{\beta}v(t)$ in
$\mathscr{S}^{\prime}$ and hence

$\displaystyle{}t^{\frac{\beta}{2}(1-2\lambda)+b}\norm{v(t)}_{2}+t^{b}\norm{|J(t)|^{\beta}v(t)}_{2}$
$\displaystyle\leq{}$
$\displaystyle\liminf_{k\rightarrow\infty}\\{t^{\frac{\beta}{2}(1-2\lambda)+b}\norm{v_{n_{k}}(t)}_{2}+t^{b}\norm{|J(t)|^{\beta}v_{n_{k}}(t)}_{2}\\}\leq
R$

for any $t\geq T$. Thus $v\in X_{b,R}$. As for the external term, by Lemma 3.2
and Lemma 3.3,

$\displaystyle\norm{{\cal A}(\cdot)+{\cal E}_{\mathrm{r}}(\cdot)}_{X_{b}}\leq
C\left(1+\norm{u_{-}}_{H^{0,\alpha}}\right)^{3p_{c}}\norm{u_{-}}_{H^{0,\alpha}}\leq
C_{0}\norm{u_{-}}_{H^{0,\alpha}}$

if $\varepsilon_{1}\leq 1$. Let us here show that $\Psi\colon
X_{b,R}\rightarrow X_{b,R}$ is a contraction map with
$R=2C_{0}\norm{u_{-}}_{H^{0,\alpha}}$. We only deal with the case $p_{c}<1$
because the other case is similar. Recall $d/2<\beta<\alpha<\min(d,1+p_{c})$.
Combining Lemma 3.4 with Proposition 3.1, we see that there exists $C>0$ such
that

$\displaystyle\norm{\Psi(v)}_{X_{b}}\leq{}$ $\displaystyle
C\left(R+\norm{u_{-}}_{H^{0,\alpha}}\right)^{p_{c}}R+C\norm{u_{-}}_{H^{0,\alpha}}R^{p_{c}}+C\left(R+\norm{u_{-}}_{H^{0,\alpha}}\right)^{\gamma(\beta-1)+1}R^{p_{c}-\gamma(\beta-1)}$
$\displaystyle{}+C_{0}\norm{u_{-}}_{H^{0,\alpha}}$ $\displaystyle\leq{}$
$\displaystyle
b_{1}\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}+1}+C_{0}\norm{u_{-}}_{H^{0,\alpha}}$

for any $v\in X_{T,b,R}$, as long as $\varepsilon_{1}\leq 1$. Hence, we have
$\norm{\Psi(v)}_{X_{b}}\leq R$ if $b_{1}\epsilon_{1}^{p_{c}}\leq C_{0}$. Thus
$\Psi(v)\in X_{b,R}$ holds. In the same manner, it holds that

$\displaystyle d\left(\Psi(v_{1}),\Psi(v_{2})\right)\leq{}$ $\displaystyle
a_{1}\norm{u_{-}}_{H^{0,\alpha}}^{p_{c}}\norm{v_{1}-v_{2}}_{L^{\infty}(-\infty,-r_{0};L^{2})}\leq\frac{1}{2}d(v_{1},v_{2})$

for any $v_{1}$, $v_{2}\in X_{b,R}$ whenever $a_{1}\varepsilon_{1}^{p_{c}}\leq
1/2$. These imply that $\Psi\colon X_{b,R}\to X_{b,R}$ is the contraction map.
Thus, we have a unique solution in $X_{b,R}$. This completes the proof.

###  3.4 Proof of Corollary 1.1

We finish this section by proving the existence of the wave operator. Fix
$\varepsilon\in(0,\varepsilon_{1}]$, where $\varepsilon_{1}$ is given in
Theorem 1.1. We see from Theorem 1.1 that for any $u_{-}\in H^{0,\alpha}$ with
$\norm{u_{-}}_{H^{0,\alpha}}\leq\varepsilon$, (NLS) admits a unique
$\mathcal{F}H^{\beta}$-solution $u$ on $(-\infty,-r_{0}]$ satisfying

$\norm{U(0,t)\left(u(t)-u_{p}(t)\right)}_{H^{0,\beta}}\leq
C\varepsilon|t|^{-\mu}$

for all $t\leq-r_{0}$. By means of (3.2) and the Sobolev embedding, this
implies

$\displaystyle\norm{U(0,-r_{0})u(-r_{0})}_{H^{0,\beta}}\leq{}$
$\displaystyle\norm{U(0,-r_{0})u_{p}(-r_{0})}_{H^{0,\beta}}+C\varepsilon|r_{0}|^{-\mu}$
$\displaystyle\leq{}$
$\displaystyle\norm{u_{-}}_{2}+C(\eta,a_{2},r_{0})\norm{u_{-}}_{H^{0,\beta}}\left(1+\norm{u_{-}}_{H^{0,\beta}}^{p_{c}}\right)^{\lceil\alpha\rceil}+C(r_{0},\mu)\varepsilon$
$\displaystyle\leq{}$ $\displaystyle b_{1}\varepsilon$

and thus $U(0,-r_{0})u(-r_{0})\in H^{0,\beta}$. By Proposition 2.1, we have
the solution on $(-\infty,0]$ satisfying

$\displaystyle\norm{u(0)}_{H^{0,\beta}}\leq{}$
$\displaystyle\norm{|J(\cdot)|^{\beta}u}_{L^{\infty}(-r_{0},0;L^{2})}\leq
A_{0}\norm{U(0,-r_{0})u(-r_{0})}_{H^{0,\beta}}\leq A_{0}b_{1}\varepsilon$

where $A_{0}>0$ depends on $r_{0}$ and $\varepsilon_{1}$. Hence, setting
$c_{0}=A_{0}b_{1}$, we can define a map by

$\displaystyle\mathcal{W}_{-}\colon B^{\alpha}_{\varepsilon}\ni u_{-}\mapsto
u(0)\in B^{\beta}_{c_{0}\varepsilon}$

for any $\varepsilon\in(0,\varepsilon_{1}]$. This is the so-called modified
wave operator.

##  4 Construction of the modified inverse wave operator

Thanks to Corollary 1.1 and Proposition 2.1, we have the propagator from
$u_{-}$ to $u(r_{0})$, and in this section we construct the modified inverse
operator which propagates from $u(r_{0})$ to $u_{+}$. Let us first introduce
the function space

$\displaystyle\mathscr{X}^{\varepsilon,A}\coloneqq\\{u\in
C([r_{0},\infty);L^{2}\cap L^{\infty}({\bf
R}^{d}))\mid\norm{u}_{\mathscr{X}^{\varepsilon,A}}<\infty\\},$
$\displaystyle\norm{u}_{\mathscr{X}^{\varepsilon,A}}\coloneqq{}\sup_{t\geq
r_{0}}|t|^{-A\varepsilon^{\frac{1}{d(1-\lambda)}}}\norm{\left\langle
J(t)\right\rangle^{\beta}u}_{2}+\sup_{t\geq
r_{0}}|t|^{\frac{d}{2}(1-\lambda)}\norm{u(t)}_{\infty}$

for some small $\varepsilon>0$ and a certain $A>0$. The smallness of
$\varepsilon$ are needed to construct the global solution on $[r_{0},\infty)$.
As for the detail, see the beginning of Appendix A. As for the constant $A$,
we refer to comments above Proposition A.2 in Appendix A. Here we set

$\displaystyle\begin{aligned} v(t)&{}={\cal M}_{+}U(0,t)u(t),\\\
{\mathscr{A}}(t)&{}=\begin{cases}0,&d=1,\\\
|\widehat{v}(t,x)|^{p_{c}}-\left(|t|^{-\chi}+|\widehat{v}(t,x)|^{2}\right)^{\frac{p_{c}}{2}},&d=2,3,\end{cases}\\\
I_{1}(t)&{}=\mathcal{F}\left(\mathcal{M}_{2}^{-1}-1\right)\mathcal{F}^{-1}\left(|\mathcal{F}\mathcal{M}_{2}v|^{p_{c}}\mathcal{F}\mathcal{M}_{2}v\right),\\\\[5.69054pt]
I_{2}(t)&{}=|\mathcal{F}\mathcal{M}_{2}v|^{p_{c}}\mathcal{F}\mathcal{M}_{2}v-|\mathcal{F}v|^{p_{c}}\mathcal{F}v,\end{aligned}$
(4.1)

and

$\theta(t)=\theta(t,x)=\begin{cases}\vspace{0.1cm}{\displaystyle\eta\int_{r_{0}}^{t}|\widehat{v}(s,x)|^{p_{c}}|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}\,ds},&d=1,\\\
{\displaystyle\eta\int_{r_{0}}^{t}\left(|s|^{-\chi}+|\widehat{v}(s,x)|^{2}\right)^{\frac{p_{c}}{2}}|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}\,ds,}&d=2,3.\end{cases}$

We further define

$\displaystyle\mathcal{P}(t){}:=\exp(i\theta(t)),\quad v_{p}\coloneqq{}$
$\displaystyle\mathcal{F}^{-1}\mathcal{P}(t)\mathcal{F}v=\mathcal{F}^{-1}e^{i\theta(t)}\mathcal{F}v.$

Then by the similar arguments in Section 4 of [HN06], we have

$\displaystyle\begin{aligned}
\widehat{v_{p}}(t)-\widehat{v_{p}}(s)={}&-i\eta\int_{s}^{t}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)\,d\tau\\\
&{}-i\eta\int_{s}^{t}\mathcal{P}(\tau)\left(I_{1}(\tau)+I_{2}(\tau)\right)|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau.\end{aligned}$
(4.2)

In this section, we focus on the existence of inverse modified wave operators
${\cal W}_{+}^{-1}:u(0)\mapsto u_{+}$, namely, we establish that
$\widehat{v_{p}}(t)\in H^{\delta}$ has a limit by proving
$\widehat{v_{p}}(t)-\widehat{v_{p}}(s)\rightarrow 0$ as $t$,
$s\rightarrow\infty$ on $H^{\delta}$ under global solutions
$u\in\mathscr{X}^{\varepsilon,A}$ to (NLS). We first need to show that (NLS)
with $u_{0}=u(r_{0})$ is global well-posed. However, as mentioned in Section
1.3.2, we introduce the result on GWP as Theorem A.1 and give a sketch of the
proof in Appendix A, since result on GWP has been essentially proven in
[KM21]. Hence we only treat that $\widehat{v_{p}}(t)$ has a limit in
$H^{\delta}$ in this section.

In the following, we handle the RHS of (4.2) in $H^{\delta}$. If one can
obtain the integrand of which are integrable in $[r_{0},\infty)$, we
immediately find that ${v_{p}}(t)$ has the limit in $H^{0,\delta}$, that is
exactly the $u_{+}$. Hence the main purpose of this section is to seek the
decay estimate of the integrand of the RHS of (4.2), namely we show the
following Propositions:

######  Proposition 4.1.

Let $d=2$, $3$. Assume $d/2<\delta<\beta<\min(1+p_{c},d)$. Let $u$ be the
solution given by Theorem A.1. Set
$\epsilon_{\lambda}=A\varepsilon^{\frac{1}{d(1-\lambda)}}$. Then it holds that

$\displaystyle\norm{\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)}_{H^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\chi}{2}\kappa+2\epsilon_{\lambda}},$

where $\kappa=\kappa(d,\lambda,\delta)$ satisfies $\kappa=1+p_{c}-\delta$ if
$p_{c}\geq 1$ and
$\kappa=\min\left(3p_{c}-\delta,(1-p_{c})(1+p_{c}-\delta)+p_{c}^{2}\right)$ if
$p_{c}<1$.

######  Proposition 4.2.

Let $d\leq 3$. Assume $d/2<\delta<\beta<\min(1+p_{c},d)$. Let $u$ be the
solution given by Theorem A.1. Set
$\epsilon_{\lambda}=A\varepsilon^{\frac{1}{d(1-\lambda)}}$. If $d=1$, then

$\displaystyle\norm{\mathcal{P}(\tau)\left(I_{1}(\tau)+I_{2}(\tau)\right)}_{H^{\delta}}\lesssim\varepsilon\tau^{-\frac{\beta-\delta}{2}(1-2\lambda)+\left(p_{c}+1\right)\epsilon_{\lambda}}$

holds. When $d=2$, $3$, the following is valid:

$\displaystyle\norm{\mathcal{P}(\tau)\left(I_{1}(\tau)+I_{2}(\tau)\right)}_{H^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\beta-\delta}{2}(1+\min(1,p_{c})-\delta)(1-2\lambda)+\frac{\chi}{2}+(2p_{c}+1)\epsilon_{\lambda}},$

where $I_{j}(\tau)$ are defined by (4.1).

With these propositions in place, let us initially give the proof of Theorem
1.2. We postpone the proof of the propositions to the end of this section.

###  4.1 Proof of Theorem 1.2

Let us prove Theorem 1.2 by Proposition 4.1 and Proposition 4.2. We shall only
treat the case $p_{c}<1$, because the others are similar. Set

$0<\chi<(\beta-\delta)(1+p_{c}-\delta)(1-2\lambda).$

We see from Proposition 4.1 and Proposition 4.2 that there exists $\mu_{0}>0$
satisfying

$\displaystyle
0<\mu_{0}<\min\left(\frac{\chi}{2}\kappa,\frac{\beta-\delta}{2}(1+p_{c}-\delta)(1-2\lambda)-\frac{\chi}{2}\right),$

such that

$\displaystyle\norm{\widehat{v_{p}}(t)-\widehat{v_{p}}(s)}_{H^{\delta}}\lesssim{}$
$\displaystyle\int_{s}^{t}\norm{\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)}_{H^{\delta}}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau$
$\displaystyle{}+\int_{s}^{t}\norm{\mathcal{P}(\tau)\left(I_{1}(\tau)+I_{2}(\tau)\right)}_{H^{\delta}}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau$
$\displaystyle\lesssim{}$
$\displaystyle\varepsilon\left(t^{-\mu_{0}+(2p_{c}+1)\epsilon_{\lambda}}-s^{-\mu_{0}+(2p_{c}+1)\epsilon_{\lambda}}\right)$

for any $r_{0}<s<t$. Then there exists a unique limit $u_{+}\in H^{0,\delta}$
such that

$\displaystyle\norm{{v_{p}}(t)-u_{+}}_{H^{0,\delta}}\lesssim{}$
$\displaystyle\varepsilon t^{-\mu_{0}+(2p_{c}+1)\epsilon_{\lambda}}$

for any $t>r_{0}$. This completes the proof of Theorem 1.2.

###  4.2 Proof of Proposition 4.1 and Proposition 4.2.

Before showing these propositions, we introduce three important lemmata.

By the Gagliardo-Nirenberg inequality and (1.9), we easily verify the
followings for $\mathcal{M}_{+}U(0,t)u$ frequently used afterward.

######  Lemma 4.1.

Let $u$ be the solution given by Theorem A.1 and $v=\mathcal{M}_{+}U(0,t)u$.
Set $\epsilon_{\lambda}=A\varepsilon^{\frac{1}{d(1-\lambda)}}$. Fix
$\alpha\in[0,\beta]$. Then the following hold: For any $t\geq r_{0}$,

  * •

$\norm{\widehat{v}(t)}_{\dot{H}^{\alpha}}+\norm{\mathcal{F}\mathcal{M}_{2}v}_{\dot{H}^{\alpha}}\lesssim\varepsilon|t|^{\frac{\alpha}{\beta}\epsilon_{\lambda}}$,
$\norm{\mathcal{F}\mathcal{M}_{2}v}_{\infty}+\norm{\widehat{v}}_{\infty}\lesssim\varepsilon|t|^{\frac{d}{2\beta}\epsilon_{\lambda}}$,
$\norm{\nabla\widehat{v}(t)}_{2\delta}\lesssim\varepsilon
t^{\epsilon_{\lambda}}$.

  * •

$\norm{\mathcal{F}\left(\mathcal{M}_{2}-1\right)v}_{\infty}\lesssim\varepsilon|t|^{-\alpha(1-2\lambda)+\epsilon_{\lambda}}$,
$\norm{\mathcal{F}\left(\mathcal{M}_{2}-1\right)v}_{\dot{H}^{\alpha}}\lesssim\varepsilon|t|^{-\frac{\beta-\alpha}{2}(1-2\lambda)+\epsilon_{\lambda}}$,

where $\alpha\in[0,(\beta-d/2)/2)$.

In the modified wave operator, we need a modification of the phase correction
$e^{i\theta(t)}$ to remove a non-integrable part for $t$. Moreover, in $d=2$,
$3$, one employs a further modification such as $\mathcal{P}(t)$ which is
firstly used in [GO93], due to the lack of differentiability of the
nonlinearity. Hence the following sharp inequalities is needed to extract
necessary time decay for a term arising from the modification. Set $\chi>0$,
$n\in{\bf N}\cup\\{0\\}$ and $t\geq r_{0}$. We denote

$\displaystyle\mathscr{A}_{n}(t)={}$
$\displaystyle\mathscr{A}_{n}(t;z)=\left(|z|^{p_{c}-n}-\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{p_{c}-n}{2}}\right)z^{n},$
(4.3)

for any $z\in{\bf C}$. Further, define

$\displaystyle\mathscr{B}_{n}(t;z)=\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{p_{c}-n}{2}}z^{n},\quad\widetilde{\mathscr{B}_{n}}(t;z)=\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{p_{c}-n}{2}}|z|^{n}$

for any $z\in{\bf C}$, where we remark that

$\mathscr{A}(t)=\mathscr{A}_{0}(t;\hat{v}(t)),\quad d=2,3.$

######  Lemma 4.2.

The following estimates hold:

  * •

If $p_{c}\in(0,2)$, then

$\displaystyle{}\absolutevalue{\mathscr{A}_{n}(t;z)}\lesssim|t|^{-\frac{p_{c}}{2}\chi}.$
(4.4)

  * •

When $p_{c}\in(0,1)$,

$\displaystyle{}\absolutevalue{\widetilde{\mathscr{B}_{n}}(t;z)-\widetilde{\mathscr{B}_{n}}(t;w)}+\absolutevalue{\mathscr{B}_{n}(t;z)-\mathscr{B}_{n}(t;w)}\lesssim|z-w{}|^{p_{c}},$
(4.5)
$\displaystyle{}\absolutevalue{\mathscr{A}_{0}(t;z)z}\leq|t|^{-p_{c}\chi}|z|^{1-p_{c}}.$
(4.6)

###### Proof.

We shall prove (4.4) by an induction argument, with reference to the proof of
[AIMMU, Lemma 2.4]. The case $n=0$ immediately follows from the concavity of
$|z|^{p_{c}/2}$. When $n\in{\bf N}$, one computes

$\displaystyle\mathscr{A}_{n+1}(t;z)={}$
$\displaystyle\left\\{\left(|z|^{p_{c}-n}-(|\tau|^{-\chi}+|z|^{2})^{\frac{p_{c}-n}{2}}\right)z^{n}\right\\}|z|^{-1}z$
$\displaystyle{}+(|\tau|^{-\chi}+|z|^{2})^{\frac{p_{c}-(n+1)}{2}}z^{n+1}|z|^{-1}\left\\{(|\tau|^{-\chi}+|z|^{2})^{\frac{1}{2}}-|z|\right\\}.$

Thanks to the hypothesis of the induction, the first term is bounded by
$C|t|^{-\frac{p_{c}}{2}\chi}$. The second term is also dominated by
$C|t|^{-\frac{p_{c}}{2}\chi}$ from the direct calculation. This completes the
proof of (4.4). We shall next prove (4.5) by the induction. We only deal with
$\absolutevalue{\mathscr{B}_{n}(t;z)-\mathscr{B}_{n}(t;w)}\lesssim|z-w|^{p_{c}}$,
because the other part is similar. By the concavity of $|z|^{p_{c}}$, one has

$\displaystyle\absolutevalue{\mathscr{B}_{0}(t;z)-\mathscr{B}_{0}(t;w)}\leq{}$
$\displaystyle\absolutevalue{\frac{|z|^{2}-|w|^{2}}{(|\tau|^{-\chi}+|z|^{2})^{\frac{1}{2}}+(|\tau|^{-\chi}+|w|^{2})^{\frac{1}{2}}}}^{p_{c}}\leq\absolutevalue{z-w}^{p_{c}},$

which is the case $n=0$. When $n\in{\bf N}$, we compute

$\displaystyle{}\mathscr{B}_{n+1}(t;z)-\mathscr{B}_{n+1}(t;w)$
$\displaystyle={}$
$\displaystyle\left\\{\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{p_{c}-n}{2}}z^{n}-\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{p_{c}-n}{2}}w^{n}\right\\}\left(|t|^{-\chi}+|z|^{2}\right)^{-\frac{1}{2}}z$
$\displaystyle{}+\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{p_{c}-n}{2}}w^{n}\left\\{\left(|t|^{-\chi}+|z|^{2}\right)^{-\frac{1}{2}}z-\left(|t|^{-\chi}+|w|^{2}\right)^{-\frac{1}{2}}w\right\\}.$

The first term is bounded by $C|z-w|^{p_{c}}$ from the hypothesis of the
induction. If $|z-w|\geq\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{1}{2}}$, then
the second term is also dominated by $C|z-w|^{p_{c}}$. When
$|z-w|\leq\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{1}{2}}$, by a further
calculation,

$\displaystyle{}\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{p_{c}-n}{2}}w^{n}\left\\{\left(|t|^{-\chi}+|z|^{2}\right)^{-\frac{1}{2}}z-\left(|t|^{-\chi}+|w|^{2}\right)^{-\frac{1}{2}}w\right\\}$
$\displaystyle={}$
$\displaystyle\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{p_{c}-(n+1)}{2}}w^{n}\left(|t|^{-\chi}+|z|^{2}\right)^{-\frac{1}{2}}\left\\{\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{1}{2}}-\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{1}{2}}\right\\}z$
$\displaystyle{}+\left(|t|^{-\chi}+|w|^{2}\right)^{\frac{p_{c}-(n+1)}{2}}w^{n}(z-w).$

Hence, the second term is bounded by $C|z-w|^{p_{c}}$. The proof of (4.5) is
completed. Let us finally show (4.6). By means of the concavity of
$|z|^{p_{c}}$, we obtain

$\displaystyle\absolutevalue{\mathscr{A}_{0}(t;z)z}\leq{}$
$\displaystyle\absolutevalue{|z|-\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{1}{2}}}^{p_{c}}|z|\leq\absolutevalue{\frac{|t|^{-\chi}}{|z|+\left(|t|^{-\chi}+|z|^{2}\right)^{\frac{1}{2}}}}^{p_{c}}|z|,$

as desired. ∎

######  Lemma 4.3.

Let $d\leq 3$. Assume $d/2<\delta<\beta<\min(1+p_{c},d)$. Let $u$ be the
solution given by Theorem A.1. Set
$\epsilon_{\lambda}=A\varepsilon^{\frac{1}{d(1-\lambda)}}$. Then when $d=1$,
it holds that

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\tau^{p_{c}\epsilon_{\lambda}}.$

Under $d=2$, $3$, if $p_{c}<1$, then

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)+\delta\epsilon_{\lambda}}$

holds, otherwise,

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\lesssim\varepsilon\tau^{\frac{\chi}{2}(2-p_{c})(\delta-1)+p_{c}\epsilon_{\lambda}}.$

###### Proof of Lemma 4.3.

When $d=1$, noting
$|\mathcal{P}(\tau,x)-\mathcal{P}(\tau,y)|\lesssim|\theta(\tau,x)-\theta(\tau,y)|$
and $p_{c}\geq 2$, we see from a use of the representation via the Gagliardo
semi-norm that

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\norm{\theta(\tau)}_{\dot{H}^{\delta}}\lesssim\int_{r_{0}}^{\tau}\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}}_{L^{\infty}}\norm{\widehat{v}(s)}_{\dot{H}^{\delta}}|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}\,ds\lesssim\varepsilon\tau^{p_{c}\epsilon_{\lambda}}.$

Let us turn to the case $d=2$, $3$. We only handle the case $p_{c}<1$, because
the other case is easier. In view of
$\nabla\mathcal{P}(\tau)=i\nabla\theta(\tau)\mathcal{P}(\tau)$, we see from
Lemma 2.3 that

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}=\norm{\nabla\mathcal{P}(\tau)}_{\dot{H}^{\delta-1}}\lesssim{}$
$\displaystyle\norm{\theta(\tau)}_{\dot{H}^{\delta}}+\norm{\nabla\theta(\tau)}_{2\delta}\norm{|\nabla|^{\delta-1}\mathcal{P}(\tau)}_{\frac{2\delta}{\delta-1}}$

The Gagliardo-Nirenberg inequality gives us

$\displaystyle\norm{|\nabla|^{\delta-1}\mathcal{P}(\tau)}_{\frac{2\delta}{\delta-1}}\lesssim{}$
$\displaystyle\norm{\mathcal{P}(\tau)}_{\infty}^{2-\delta}\norm{\nabla\mathcal{P}(\tau)}_{2\delta}^{\delta-1}=\norm{\nabla\theta(\tau)}_{2\delta}^{\delta-1},$
(4.7)

which yields

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\norm{\theta(\tau)}_{\dot{H}^{\delta}}+\norm{\nabla\theta(\tau)}_{2\delta}^{\delta}.$
(4.8)

Since

$\displaystyle\nabla\theta(\tau)=\eta\int_{r_{0}}^{\tau}p_{c}\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\operatorname{Re}\left(\overline{\widehat{v}(s)}\nabla\widehat{v}(s)\right)|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}\,ds,$

we see from Lemma 4.1 that

$\displaystyle\begin{aligned}
\norm{\nabla\theta(\tau)}_{2\delta}\lesssim{}&\int_{r_{0}}^{\tau}\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-1}{2}}}_{\infty}\norm{\nabla\widehat{v}(s)}_{2\delta}|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}\,ds\\\
\lesssim{}&\int_{r_{0}}^{\tau}|s|^{-1+\frac{\chi}{2}(1-p_{c})}\norm{\nabla\widehat{v}(s)}_{2\delta}\,ds\lesssim\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})+\epsilon_{\lambda}}.\end{aligned}$
(4.9)

Further, applying Lemma 2.3, one has

$\displaystyle\begin{aligned}
&{}\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}\nabla\widehat{v}(s)}_{\dot{H}^{\delta-1}}\\\
\lesssim{}&\norm{|\nabla|^{\delta-1}\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}}_{\frac{2\delta}{\delta-1}}\norm{\nabla\widehat{v}(s)}_{2\delta}\\\
&{}+\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}}_{\infty}\norm{\widehat{v}(s)}_{\dot{H}^{\delta}}.\end{aligned}$
(4.10)

By means of the Gagliardo-Nirenberg inequality, we see that

$\displaystyle{}\norm{|\nabla|^{\delta-1}\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}}_{\frac{2\delta}{\delta-1}}$
$\displaystyle\lesssim{}$
$\displaystyle\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}}_{\infty}^{2-\delta}\norm{|\nabla|\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}}_{2\delta}^{\delta-1}$
$\displaystyle\lesssim{}$ $\displaystyle
s^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)}\norm{\nabla\widehat{v}(s)}_{2\delta}^{\delta-1}.$

Hence by Lemma 4.1, one obtains

$\displaystyle\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}\nabla\widehat{v}(s)}_{\dot{H}^{\delta-1}}\lesssim{}$
$\displaystyle
s^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)}\norm{\nabla\widehat{v}(s)}_{2\delta}^{\delta}+\varepsilon
s^{\frac{\chi}{2}(1-p_{c})+\epsilon_{\lambda}}$ $\displaystyle\lesssim{}$
$\displaystyle\varepsilon
s^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)+\delta\epsilon_{\lambda}}.$

This yields

$\displaystyle\begin{aligned}
\norm{\theta(\tau)}_{\dot{H}^{\delta}}\lesssim{}&\int_{r_{0}}^{\tau}\norm{\left(|s|^{-\chi}+|\widehat{v}(s)|^{2}\right)^{\frac{p_{c}-2}{2}}\overline{\widehat{v}(s)}\nabla\widehat{v}(s)}_{\dot{H}^{\delta-1}}|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}\,ds\\\
\lesssim{}&\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)+\delta\epsilon_{\lambda}}.\end{aligned}$
(4.11)

Plugging (4.11) and (4.9) into (4.8), we conclude

$\displaystyle\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)+\delta\epsilon_{\lambda}}+\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})\delta+\delta\epsilon_{\lambda}}\lesssim\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)+\delta\epsilon_{\lambda}},$

as desired. This completes the proof. ∎

Using these lemmata, we show Proposition 4.1 and Proposition 4.2.

###### Proof of Proposition 4.1.

From (4.4) and Lemma 4.1, we easily show

$\displaystyle\norm{\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)}_{2}\leq\norm{\mathscr{A}(\tau)}_{\infty}\norm{\widehat{v}(\tau)}_{2}\lesssim\varepsilon\tau^{-\frac{p_{c}}{2}+\epsilon_{\lambda}}.$

Also, the direct computation shows

$\displaystyle\nabla\left(\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)\right)={}$
$\displaystyle\nabla\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)+\frac{p_{c}}{2}\mathcal{P}(\tau)|\mathscr{A}_{2}(\tau)|\nabla\widehat{v}(\tau)$
$\displaystyle{}+\frac{p_{c}}{2}\mathcal{P}(\tau)\mathscr{A}_{2}(\tau)\nabla\widehat{v}(\tau)+\mathcal{P}(\tau)\mathscr{A}(\tau)\nabla\widehat{v}(\tau),$

where recall that $\mathscr{A}_{2}(\tau)$ is defined by (4.3). This implies

$\displaystyle\begin{aligned}
\norm{\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)}_{\dot{H}^{\delta}}\lesssim{}&\norm{\nabla\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)}_{\dot{H}^{\delta-1}}+\norm{\mathcal{P}(\tau)|\mathscr{A}_{2}(\tau)|\nabla\widehat{v}(\tau)}_{\dot{H}^{\delta-1}}\\\
&{}+\norm{\mathcal{P}(\tau)\mathscr{A}_{2}(\tau)\nabla\widehat{v}(\tau)}_{\dot{H}^{\delta-1}}+\norm{\mathcal{P}(\tau)\mathscr{A}(\tau)\nabla\widehat{v}(\tau)}_{\dot{H}^{\delta-1}}\\\
=:&{}J_{1}(\tau)+J_{2}(\tau)+J_{3}(\tau)+J_{4}(\tau).\end{aligned}$ (4.12)

In view of $\nabla\mathcal{P}(\tau)=i\nabla\theta(\tau)\mathcal{P}(\tau)$, we
see from Lemma 2.3 that

$\displaystyle\begin{aligned}
J_{1}(\tau)\lesssim{}&\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\norm{\mathscr{A}(\tau)\widehat{v}(\tau)}_{\infty}+\norm{\nabla\theta(\tau)}_{2\delta}\norm{|\nabla|^{\delta-1}\mathscr{A}(\tau)\widehat{v}(\tau)}_{\frac{2\delta}{\delta-1}}.\end{aligned}$
(4.13)

Hereafter, one shall address the case $p_{c}<1$. Collecting (4.4) and (4.6),
the Gagliardo-Nirenberg inequality gives us

$\displaystyle\norm{|\nabla|^{\delta-1}\mathscr{A}(\tau)\widehat{v}(\tau)}_{\frac{2\delta}{\delta-1}}\lesssim{}$
$\displaystyle\norm{\mathscr{A}(\tau)\widehat{v}(\tau)}_{\infty}^{2-\delta}\norm{\nabla\left(\mathscr{A}(\tau)\widehat{v}(\tau)\right)}_{2\delta}^{\delta-1}$
$\displaystyle\lesssim{}$
$\displaystyle\norm{\mathscr{A}(\tau)\widehat{v}(\tau)}_{\infty}^{2-\delta}\left(\norm{\mathscr{A}_{2}(\tau)}_{\infty}+\norm{\mathscr{A}(\tau)}_{\infty}\right)^{\delta-1}\norm{\nabla\widehat{v}(\tau)}_{2\delta}^{\delta-1}$
$\displaystyle\lesssim{}$
$\displaystyle\tau^{-\frac{\chi}{2}p_{c}(3-\delta)}\norm{\widehat{v}}_{\infty}^{(1-p_{c})(2-\delta)}\norm{\nabla\widehat{v}(\tau)}_{2\delta}^{\delta-1}.$

Noting (4.6), substituting these above into (4.13), applying Lemma 4.3, we
establish

$\displaystyle\begin{aligned}
J_{1}(\tau)\lesssim{}&\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})+\frac{\chi}{2}(\delta-1)+\delta\epsilon_{\lambda}}(\varepsilon\tau^{\epsilon_{\lambda}})^{(1-p_{c})}\tau^{-p_{c}\chi}+\varepsilon\tau^{\frac{1-p_{c}}{2}\chi-\frac{3-\delta}{2}p_{c}\chi+2\epsilon_{\lambda}}\\\
\lesssim{}&\varepsilon\tau^{-\frac{\chi}{2}(3p_{c}-\delta)+2\epsilon_{\lambda}}.\end{aligned}$
(4.14)

In terms of $J_{2}$, arguing as in $J_{1}$, together with Lemma 4.1, it
follows from (4.9) and (4.7) that

$\displaystyle J_{2}(\tau)\lesssim{}$
$\displaystyle\norm{|\nabla|^{\delta-1}\mathcal{P}(\tau)}_{\frac{2\delta}{\delta-1}}\norm{\mathscr{A}_{2}(\tau)}_{\infty}\norm{\nabla\widehat{v}(\tau)}_{2\delta}$
$\displaystyle{}+\norm{|\nabla|^{\delta-1}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{\delta-1}}\norm{\nabla\widehat{v}(\tau)}_{2\delta}+\norm{\mathscr{A}_{2}(\tau)}_{\infty}\norm{\widehat{v}(\tau)}_{\dot{H}^{\delta}}$
$\displaystyle\lesssim{}$
$\displaystyle\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})(\delta-1)-\frac{\chi}{2}p_{c}+\delta\epsilon_{\lambda}}+\norm{|\nabla|^{\delta-1}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{\delta-1}}\varepsilon\tau^{\epsilon_{\lambda}}+\varepsilon\tau^{-\frac{\chi}{2}p_{c}+\epsilon_{\lambda}}.$

Note that by (2.4) and (4.5), $|\mathscr{A}_{2}(\tau)|$ is a $p_{c}$-Hölder
continous function, namely

$\absolutevalue{|\mathscr{A}_{2}(\tau;z_{1})|-|\mathscr{A}_{2}(\tau;z_{2})|}\lesssim|z_{1}-z_{2}|^{p_{c}}.$

The Gagliardo-Nirenberg inequality leads to

$\displaystyle\begin{aligned}
\norm{|\nabla|^{\delta-1}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{\delta-1}}\lesssim{}&\norm{\mathscr{A}_{2}(\tau)}_{\infty}^{1-\frac{\delta-1}{s_{0}}}\norm{|\nabla|^{s_{0}}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{s_{0}}}^{\frac{\delta-1}{s_{0}}},\end{aligned}$
(4.15)

where $s_{0}=p_{c}-\varepsilon_{1}$ for any
$\varepsilon_{1}\in(0,p_{c}-\delta+1)$. Applying Lemma 2.4, we deduce that

$\displaystyle\norm{|\nabla|^{s_{0}}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{s_{0}}}\lesssim{}$
$\displaystyle\norm{\widehat{v}(\tau)}_{\infty}^{p_{c}-\frac{s_{0}}{s}}\norm{|\nabla|^{s}\widehat{v}(\tau)}_{\frac{2\delta}{s}}^{\frac{s_{0}}{s}}$

for any $s\in\left(s_{0}/p_{c},1\right)$. One then sees from the Gagliaro-
Nirenberg inequality that

$\norm{|\nabla|^{s}\widehat{v}(\tau)}_{\frac{2\delta}{s}}\lesssim\norm{\widehat{v}(\tau)}_{\infty}^{1-s}\norm{\nabla\widehat{v}(\tau)}_{2\delta}^{s},$

which implies

$\displaystyle\norm{|\nabla|^{s_{0}}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{s_{0}}}\lesssim{}$
$\displaystyle\norm{\widehat{v}(\tau)}_{\infty}^{p_{c}-s_{0}}\norm{\nabla\widehat{v}(\tau)}_{2\delta}^{s_{0}}.$

Plugging the above into (4.15), we have

$\displaystyle\norm{|\nabla|^{\delta-1}|\mathscr{A}_{2}(\tau)|}_{\frac{2\delta}{\delta-1}}\lesssim{}$
$\displaystyle\tau^{-\frac{\chi}{2}p_{c}\left(1-\frac{\delta-1}{s_{0}}\right)}\norm{\widehat{v}(\tau)}_{\infty}^{\left(\frac{p_{c}}{s_{0}}-1\right)(\delta-1)}\norm{\nabla\widehat{v}(\tau)}_{2\delta}^{\delta-1}$
$\displaystyle\lesssim{}$
$\displaystyle\varepsilon^{\frac{p_{c}}{s_{0}}(\delta-1)}\tau^{-\frac{\chi}{2}\left(p_{c}-\frac{p_{c}}{s_{0}}(\delta-1)\right)+\epsilon_{\lambda}},$

which implies

$\displaystyle\begin{aligned}
J_{2}(\tau)\lesssim{}&\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})(\delta-1)-\frac{\chi}{2}p_{c}+\delta\epsilon_{\lambda}}+\varepsilon^{1+\frac{p_{c}}{s_{0}}(\delta-1)}\tau^{-\frac{\chi}{2}\left(p_{c}-\frac{p_{c}}{s_{0}}(\delta-1)\right)+2\epsilon_{\lambda}}+\varepsilon\tau^{-\frac{\chi}{2}p_{c}+\epsilon_{\lambda}}\\\
\lesssim{}&\varepsilon\tau^{-\frac{\chi}{2}(1-p_{c})(1+p_{c}-\delta)-\frac{\chi}{2}p_{c}^{2}+2\epsilon_{\lambda}}.\end{aligned}$
(4.16)

Similarly to $J_{2}$, it holds that

$\displaystyle J_{3}(\tau)+J_{4}(\tau)\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\chi}{2}(1-p_{c})(1+p_{c}-\delta)-\frac{\chi}{2}p_{c}^{2}+2\epsilon_{\lambda}}.$
(4.17)

Plugging (4.14), (4.16) and (4.17) into (4.12), we conclude

$\displaystyle\norm{\mathcal{P}(\tau)\mathscr{A}(\tau)\widehat{v}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\chi}{2}(3p_{c}-\delta)+2\epsilon_{\lambda}}+\varepsilon\tau^{-\frac{\chi}{2}(1-p_{c})(1+p_{c}-\delta)-\frac{\chi}{2}p_{c}^{2}+2\epsilon_{\lambda}}\lesssim\varepsilon\tau^{-\frac{\chi}{2}\mu+2\epsilon_{\lambda}}$

as long as $\kappa=\kappa(d,\lambda,\delta)$ satisfies
$0<\kappa<\min\left(3p_{c}-\delta,(1-p_{c})(1+p_{c}-\delta)+p_{c}^{2}\right)$,
as desired in $p_{c}<1$. The case $p_{c}\geq 1$ is similar, so we omit the
proof. ∎

###### Proof of Proposition 4.2.

As in the preceding proposition, we will only treat the case $p_{c}<1$ in
$d=3$. Let us firstly handle $I_{1}(\tau)$. Together with
$\absolutevalue{\mathcal{M}^{-1}_{2}(t)-1}\lesssim|x|^{2\theta}|t|^{-\theta(1-2\lambda)}$
for any $\theta\in[0,1]$, it comes from Lemma 2.8 and Lemma 4.1 that

$\displaystyle\begin{aligned}
\norm{I_{1}(\tau)}_{\dot{H}^{\delta}}\leq{}&\norm{|x|^{\delta}\left(\mathcal{M}_{2}^{-1}-1\right)\mathcal{F}^{-1}\left(|\mathcal{F}\mathcal{M}_{2}v|^{p_{c}}\mathcal{F}\mathcal{M}_{2}v\right)}_{2}\\\
\lesssim{}&\tau^{-\frac{\beta-\delta}{2}(1-2\lambda)}\norm{\mathcal{F}\mathcal{M}_{2}v}_{\infty}^{p_{c}}\norm{\mathcal{F}\mathcal{M}_{2}v}_{\dot{H}^{\beta}}\\\
\lesssim{}&\varepsilon\tau^{-\frac{\beta-\delta}{2}(1-2\lambda)+\left(p_{c}+1\right)\epsilon_{\lambda}}.\end{aligned}$
(4.18)

Also, the estimate

$\displaystyle\norm{\mathcal{P}(\tau)I_{1}(\tau)}_{2}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\beta}{2}(1-2\lambda)+\left(p_{c}+1\right)\epsilon_{\lambda}}$
(4.19)

is valid in the same manner. Further, arguing as in (4.18), we see from the
Sobolev embedding that

$\displaystyle\begin{aligned}
\norm{I_{1}(\tau)}_{\infty}\lesssim{}&\tau^{-\alpha(1-2\lambda)}\norm{\mathcal{F}\mathcal{M}_{2}v}_{\infty}^{p_{c}}\norm{\mathcal{F}\mathcal{M}_{2}v}_{\dot{H}^{\beta}}\\\
\lesssim{}&\varepsilon\tau^{-\frac{1}{2}\left(\beta-\frac{d}{2}\right)(1-2\lambda)+\left(p_{c}+1\right)\epsilon_{\lambda}}.\end{aligned}$
(4.20)

for any $\alpha\in[0,(\beta-d/2)/2)$. By Lemma 2.3, we obtain

$\displaystyle\norm{\mathcal{P}(\tau)I_{1}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\norm{I_{1}(\tau)}_{\dot{H}^{\delta}}+\norm{|\nabla|^{\delta-1}\mathcal{P}(\tau)}_{\frac{2\delta}{\delta-1}}\norm{\nabla
I_{1}(\tau)}_{2\delta}$
$\displaystyle{}+\norm{\nabla\mathcal{P}(\tau)}_{2\delta}\norm{|\nabla|^{\delta-1}I_{1}(\tau)}_{\frac{2\delta}{\delta-1}}+\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\norm{I_{1}(\tau)}_{\infty}.$

By the Gagliardo-Nireberg inequality, it follows from (4.18) and (4.20) that

$\displaystyle\norm{\nabla I_{1}(\tau)}_{2\delta}\lesssim{}$
$\displaystyle\norm{I_{1}(\tau)}_{\infty}^{1-\frac{1}{\delta}}\norm{I_{1}(\tau)}_{\dot{H}^{\delta}}^{\frac{1}{\delta}}\lesssim\varepsilon\tau^{-\frac{\beta-\delta}{2}(1-2\lambda)+\left(p_{c}+1\right)\epsilon_{\lambda}},$

and

$\displaystyle\norm{|\nabla|^{\delta-1}I_{1}(\tau)}_{L^{\frac{2\delta}{\delta-1}}}\lesssim{}$
$\displaystyle\norm{I_{1}(\tau)}_{\infty}^{\frac{1}{\delta}}\norm{I_{1}(\tau)}_{\dot{H}^{\delta}}^{1-\frac{1}{\delta}}\lesssim\varepsilon\tau^{-\frac{\beta-\delta}{2}(1-2\lambda)+\left(p_{c}+1\right)\epsilon_{\lambda}}.$

Also, in view of (4.7) and (4.9), we obtain

$\displaystyle\norm{|\nabla|^{\delta-1}\mathcal{P}(\tau)}_{L^{\frac{2\delta}{\delta-1}}}\lesssim\varepsilon\tau^{\frac{\chi}{2}(1-p_{c})(\delta-1)+(\delta-1)\epsilon_{\lambda}}.$

Hence, choosing $\chi\in(0,(\beta-\delta)(1-2\lambda))$, we establish

$\displaystyle\norm{\mathcal{P}(\tau)I_{1}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\beta-\delta}{2}(1-2\lambda)+\frac{\chi}{2}+4\epsilon_{\lambda}}.$

Let us next consider $I_{2}(\tau)$. We see from Lemma 4.1 and (2.3) that

$\displaystyle\begin{aligned}
\norm{\mathcal{P}(\tau)I_{2}(\tau)}_{2}\lesssim{}&\norm{|\mathcal{F}\mathcal{M}_{2}v|^{p_{c}}\mathcal{F}\mathcal{M}_{2}v-|\mathcal{F}v|^{p_{c}}\mathcal{F}v}_{2}\\\
\lesssim{}&\left(\norm{\mathcal{F}\mathcal{M}_{2}v}_{\infty}+\norm{\mathcal{F}v}_{\infty}\right)^{p_{c}}\norm{\mathcal{F}\left(\mathcal{M}_{2}-1\right)v}_{2}\\\
\lesssim{}&\varepsilon\tau^{-\frac{\beta}{2}(1-2\lambda)+(p_{c}+1)\epsilon_{\lambda}}.\end{aligned}$
(4.21)

Also, the similar calculation gives us

$\displaystyle\begin{aligned}
\norm{I_{2}(\tau)}_{\infty}\lesssim{}&\left(\norm{\mathcal{F}\mathcal{M}_{2}v}_{\infty}+\norm{\mathcal{F}v}_{\infty}\right)^{p_{c}}\norm{\mathcal{F}\left(\mathcal{M}_{2}-1\right)v}_{\infty}\\\
\lesssim{}&\varepsilon\tau^{-\frac{1}{2}\left(\beta-\frac{d}{2}\right)(1-2\lambda)+(p_{c}+1)\epsilon_{\lambda}}.\end{aligned}$
(4.22)

We deduce from Lemma 2.3 that

$\displaystyle\norm{\mathcal{P}(\tau)I_{2}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\norm{I_{2}(\tau)}_{\dot{H}^{\delta}}+\norm{|\nabla|^{\delta-1}\mathcal{P}(\tau)}_{\frac{2\delta}{\delta-1}}\norm{\nabla
I_{2}(\tau)}_{2\delta}$
$\displaystyle{}+\norm{\nabla\mathcal{P}(\tau)}_{2\delta}\norm{|\nabla|^{\delta-1}I_{2}(\tau)}_{\frac{2\delta}{\delta-1}}+\norm{\mathcal{P}(\tau)}_{\dot{H}^{\delta}}\norm{I_{2}(\tau)}_{\infty}.$

Hereafter we only handle the case $p_{c}<1$. By Lemma 2.8 and Lemma 4.1, one
obtains

$\displaystyle\norm{I_{2}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\beta-\delta}{2}(1+p_{c}-\delta)(1-2\lambda)+(p_{c}+1)\epsilon_{\lambda}},$

where $\gamma\in(1,p_{c}/(\delta-1))$. By the Gagliardo-Nireberg inequality,
we have

$\displaystyle\norm{|\nabla|^{\delta-1}I_{2}(\tau)}_{L^{\frac{2\delta}{\delta-1}}}+\norm{\nabla
I_{2}(\tau)}_{2\delta}\lesssim\varepsilon\tau^{-\frac{\beta-\delta}{2}(1+p_{c}-\delta)(1-2\lambda)+(p_{c}+1)\epsilon_{\lambda}}.$

Combining these above, we conclude that

$\displaystyle\norm{\mathcal{P}(\tau)I_{2}(\tau)}_{\dot{H}^{\delta}}\lesssim{}$
$\displaystyle\varepsilon\tau^{-\frac{\beta-\delta}{2}(1+p_{c}-\delta)(1-2\lambda)+\frac{\chi}{2}+(2p_{c}+1)\epsilon_{\lambda}}.$

This completes the proof. ∎

###  4.3 Proof of Corollary 1.2

We herein construct the inverse modified wave operator. Let $u_{0}\in
B_{\varepsilon}^{\beta}$ for any
$\varepsilon\in(0,\widetilde{\varepsilon}_{2}]$, where
$\widetilde{\varepsilon}_{2}$ is chosen later. Proposition 2.1 tells us that
(NLS) admits a unique solution $u\in C([0,r_{0}];L^{2})$ with $u(0)=u_{0}$
satisfying

$\displaystyle\norm{U(0,t)u(t)}_{H^{0,\beta}}\leq{}$ $\displaystyle
A_{0}\norm{u_{0}}_{H^{0,\beta}}\leq A_{0}\varepsilon$

for any $t\in[0,r_{0}]$, where $A_{0}>0$ depends on $r_{0}$ and
$\widetilde{\varepsilon}_{2}$. This yields

$\displaystyle\norm{U(0,r_{0})u(r_{0})}_{H^{0,\beta}}\leq A_{0}\varepsilon.$

Hence, taking $\widetilde{\varepsilon}_{2}>0$ such that
$A_{0}\widetilde{\varepsilon}_{2}\leq\varepsilon_{2}$, we can apply Theorem
1.2 to construct a unique solution $u\in C([r_{0},\infty);L^{2})$ with
$U(0,\cdot)u\in C([r_{0},\infty;H^{0,\beta})$, where $\varepsilon_{1}$ is
given in Theorem 1.2. Moreover, there exists a unique $u_{+}\in H^{0,\delta}$
such that (1.11) holds. This implies that a map

$\displaystyle\mathcal{W}_{+}^{-1}\colon B^{\beta}_{\varepsilon}\ni
u(0)\mapsto u_{+}\in H^{0,\gamma}$

can be defined for any $\varepsilon\in(0,\widetilde{\varepsilon}_{2}]$. The
map is the so-called inverse wave operator.

###  4.4 Construction of the modified scattering operator

We are finally ready to define the modified scattering operator.

###### Proof of Corollary 1.3.

Take $u_{-}\in B^{\alpha}_{\varepsilon}$ for any
$\varepsilon\in(0,\varepsilon_{0}]$, where
$\varepsilon_{0}\in(0,\epsilon_{1})$ is chosen later. By Corollary 1.1, we
have the modified wave operator

$\displaystyle\mathcal{W}_{-}\colon B^{\alpha}_{\varepsilon}\ni u_{-}\mapsto
u(0)\in B^{\beta}_{c_{0}\varepsilon},$

and thus $u(0)=W_{-}u_{-}\in B^{\beta}_{c_{0}\varepsilon}$. Choosing
$\varepsilon_{0}>0$ as $c_{0}\varepsilon_{0}<\widetilde{\varepsilon}_{2}$, one
sees from Corollary 1.2 that there exists $u_{+}\in H^{0,\gamma}$ such that
the modified inverse wave operator $\mathcal{W}_{+}^{-1}$ maps $u(0)$ into
$u_{+}$. We then conclude $u_{+}=\mathcal{W}_{+}^{-1}W_{-}u_{-}$. Therefore,
the modified scattering operator $S_{+}\coloneqq\mathcal{W}_{+}^{-1}W_{-}$ can
be defined on $B^{\alpha}_{\varepsilon}$. This completes the proof. ∎

## Appendix

##  Appendix A Global-in-time well-posedness on $[r_{0},\infty)$

In this appendix, we shall show the following GWP result on $[r_{0},\infty)$.

######  Theorem A.1 (Global existence for initial value problem).

Let $d/2<\beta<\min(1+p_{c},d)$. There exists $\varepsilon_{2}>0$ such that
the following assertion holds: For any $\varepsilon\in(0,\varepsilon_{2}]$ and
$u_{0}\in L^{2}$ satisfying $U(0,r_{0})u_{0}\in H^{0,\beta}$ with
$\norm{U(0,r_{0})u_{0}}_{H^{0,\beta}}\leq\varepsilon$, there exists a unique
$\mathcal{F}H^{\beta}$-solution to (NLS) on $[r_{0},\infty)$ under
$u(r_{0})=u_{0}$ satisfying
$\norm{u(t)}_{\infty}\lesssim\varepsilon|t|^{-\frac{d}{2}(1-\lambda)}$ for any
$t\geq r_{0}$.

To show this theorem we introduce the function space

$\displaystyle\mathscr{X}_{T}^{\varepsilon,A}=\\{u\in C(I_{T};L^{2}\cap
L^{\infty}({\bf
R}^{d}))\mid\norm{u}_{\mathscr{X}_{T}^{\varepsilon,A}}<\infty\\}$

equipped with the distance function
$d(u,v)=\norm{u-v}_{L^{\infty}(I_{T},L^{2})}$, where
$\norm{\,\cdot\,}_{\mathscr{X}}$ is defined by

$\displaystyle\norm{u}_{\mathscr{X}}={}$
$\displaystyle\norm{u}_{\mathscr{X}_{T}^{\varepsilon,A}}\coloneqq{}\sup_{t\in
I_{T}}|t|^{-A\varepsilon^{\frac{1}{d(1-\lambda)}}}\norm{\left\langle
J(t)\right\rangle^{\beta}u}_{2}+\sup_{t\in
I_{T}}|t|^{\frac{d}{2}(1-\lambda)}\norm{u(t)}_{\infty}$

for any $\varepsilon>0$ and a certain $A>0$, where $I_{T}=[r_{0},r_{0}+T]$.
Afterward, the smallness of $\varepsilon$ will be imposed to extend the
solution globally in-time (see Corollary A.1 and Proposition A.2). As for the
constant $A$, we refer to comments above Proposition A.2. Arguing as in the
proof of Theorem 1.1, it can be proven that the metric space
$(\mathscr{X}_{T}^{\varepsilon,A},d)$ is complete.

We will begin with the local existence of solutions to (NLS) in
$\mathscr{X}_{T}^{\varepsilon,A}$.

######  Proposition A.1.

Let $\varepsilon>0$, $A>0$, and $K>0$. Assume that $u_{0}\in L^{2}({\bf
R}^{d})$ satisfies $|\alpha(r_{0})|^{\beta}u_{0}\in L^{2}$ with
$\norm{u_{0}}_{2}+\norm{|J(t_{0})|^{\beta}u_{0}}_{2}\leq K$ . Then there
exists $T=T(K)>0$ not depending on $\varepsilon$ and $A$ such that (NLS) has a
unique solution $u\in\mathscr{X}_{T}^{\varepsilon,A}$ with $u(r_{0})=u_{0}$.
Moreover the solution satisfies

$\displaystyle\norm{u}_{\mathscr{X}_{T}^{\varepsilon,A}}\leq B_{0}K$ (A.1)

for some constants $B_{0}>0$ not depending on $r_{0}$, $T$, $A$ and
$\varepsilon$.

###### Proof.

The proof is based on the standard contraction mapping argument from the
completeness of the metric space $(\mathscr{X}_{T}^{\varepsilon,A},d)$. So we
omit the proof. ∎

With Proposition A.1 in place, we have the following:

######  Corollary A.1.

Let $A>0$. Then, there exists $\varepsilon_{1}>0$ such that the following
assertion holds: For any $\varepsilon\in(0,\varepsilon_{1}]$ and $u_{0}\in
L^{2}$ satisfying $|\alpha(r_{0})|^{\beta}u_{0}\in L^{2}$ with
$\norm{u_{0}}_{2}+\norm{|\alpha(r_{0})|^{\beta}u_{0}}_{2}\leq\varepsilon$,
there exists $T=T(\varepsilon)>0$ not depending on $A$ such that (NLS) has a
unique solution $u\in\mathscr{X}_{T}^{\varepsilon,A}$ with $u(r_{0})=u_{0}$.
Moreover the solution satisfies

$\displaystyle\norm{u}_{\mathscr{X}_{T}^{\varepsilon,A}}<\varepsilon^{\frac{1}{2}}.$

###### Proof.

Taking $\varepsilon_{1}>0$ such that $B_{0}<\varepsilon_{1}^{-1/2}$, we have
$B_{0}\varepsilon<\varepsilon^{1/2}$ for any
$\varepsilon\in(0,\varepsilon_{1}]$, where $B_{0}$ is the constant as in
Proposition A.1. Applying Proposition A.1 as $K=\varepsilon$, we obtain the
desired assertion. ∎

In what follows, we fix $A=c_{0}$, where $c_{0}$ is given in Proposition A.3.
We show the following estimate.

######  Proposition A.2 (Bootstrap estimate).

There exists $\varepsilon_{1}>0$ such that the following holds: Let $T>0$. If
$\varepsilon\in(0,\varepsilon_{1}]$, $u_{0}\in L^{2}$ satisfies
$|J(r_{0})|^{\beta}u_{0}\in L^{2}$ with
$\norm{u_{0}}_{2}+\norm{|J(r_{0})|^{\beta}u_{0}}_{2}\leq\varepsilon$, and the
solution $u\in\mathscr{X}_{T}^{\varepsilon,A}$ of (NLS) with $u(r_{0})=u_{0}$
satisfies

$\displaystyle\norm{u}_{\mathscr{X}_{T}^{\varepsilon,A}}\leq\varepsilon^{\frac{1}{2}},$
(A.2)

then there exists a constant $b_{1}$ independent of $T$ and $\varepsilon$ such
that

$\displaystyle\norm{u}_{\mathscr{X}_{T}^{\varepsilon,A}}\leq
b_{1}\varepsilon.$

###### Proof of Theorem A.1.

Once we obtain Proposition A.2, Theorem A.1 can be proven by the standard
continuation argument (e.g. [AIMMU22, Theorem2.2]). ∎

To prove Proposition A.2, we need the following lemma immediately proven by
(1.13):

######  Lemma A.1.

Let $u\in C([r_{0},\infty);L^{2})$ satisfy $|J(\cdot)|^{\beta}u\in
C([r_{0},\infty);L^{2})$ and $\alpha\in[0,(\beta-d/2)/2)$. Then, it holds that

$\displaystyle\norm{u(t)}_{\infty}\lesssim|t|^{-\frac{d}{2}(1-\lambda)}\norm{\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)}_{\infty}+|t|^{-\frac{d}{2}(1-\lambda)-\alpha(1-2\lambda)}\left(\norm{u(t)}_{2}+\norm{|J(t)|^{\beta}u(t)}_{2}\right)$

for any $t\geq r_{0}$.

###### Proof.

The proof is similar to that of [HN98, Lemma 2.1] and [KM21, Lemma 2.11] and
hence it is omitted. ∎

We give the proof of Proposition A.2 dividing into two parts as follows:

######  Proposition A.3.

There exists $\varepsilon_{1}>0$ such that the following assertion holds: If
$\varepsilon$, $u_{0}$, and $u$ satisfy the assumption in Proposition A.2,
then the estimate

$\displaystyle{}|t|^{-A\varepsilon^{\frac{1}{d(1-\lambda)}}}\left(\norm{u(t)}_{2}+\norm{|J(t)|^{\beta}u(t)}_{2}\right)\leq\varepsilon$

is valid for any $t\in I_{T}$.

###### Proof.

Multiplying the integral equation associated with (NLS) with $u(r_{0})=u_{0}$
by the operator $|J(t)|^{\beta}$, we see from Lemma 2.1 that

$\displaystyle|J(t)|^{\beta}u(t)=U(t,r_{0})|J(r_{0})|^{\beta}u_{0}+i\int_{r_{0}}^{t}U(t,s)|J(s)|^{\beta}F(u)(s))\,ds.$

We see from (A.2) and Lemma 2.8 that

$\displaystyle\norm{|J(t)|^{\beta}u(t)}_{2}\leq{}$
$\displaystyle\norm{U(t,r_{0})|J(r_{0})|^{\beta}u_{0}}_{2}+\int_{r_{0}}^{t}\norm{|J(s)|^{\beta}F(u(s))}_{2}ds$
$\displaystyle\leq{}$
$\displaystyle\norm{|J(r_{0})|^{\beta}u_{0}}_{2}+c_{0}\varepsilon^{\frac{1}{d(1-\lambda)}}\int_{r_{0}}^{t}|s|^{-1}\norm{|J(s)|^{\beta}u(s)}_{2}\,ds.$

We here note $A=c_{0}$. Hence, the Gronwall inequality gives us

$\displaystyle|t|^{-A\varepsilon^{\frac{1}{d(1-\lambda)}}}\norm{|J(t)|^{\beta}u(t)}_{2}\leq\norm{|J(r_{0})|^{\beta}u_{0}}_{2}$
(A.3)

for any $t\in I_{T}$. Arguing as in the above, since

$\displaystyle\norm{u(t)}_{2}\leq{}$
$\displaystyle\norm{u_{0}}_{2}+c_{0}\varepsilon^{\frac{1}{d(1-\lambda)}}\int_{r_{0}}^{t}|s|^{-1}\norm{u(s)}_{2}\,ds,$

we get

$\displaystyle|t|^{-A\varepsilon^{\frac{1}{d(1-\lambda)}}}\norm{u(t)}_{2}\leq\norm{u_{0}}_{2}.$
(A.4)

Collecting (A.3) and (A.4), the desired estimate is valid. ∎

######  Proposition A.4.

There exists $\varepsilon_{1}\in(0,1)$ such that the following assertion
holds: If $\varepsilon$, $u_{0}$, and $u$ satisfy the assumption in
Proposition A.2, then the estimate

$\displaystyle{}|t|^{\frac{d}{2}(1-\lambda)}\norm{u(t)}_{\infty}\leq
C\varepsilon$

is valid for any $t\in I_{T}$.

###### Proof.

Take $\varepsilon_{1}<\min\\{1,\alpha(1-2\lambda)((1+p_{c})A)^{-1}\\}$ for
some $\alpha\in(0,(\beta-d/2)/2)$. Combining Lemma A.1 with Proposition A.3,
we obtain

$\displaystyle\norm{u(t)}_{\infty}\leq{}$ $\displaystyle
C|t|^{-\frac{d}{2}(1-\lambda)}\norm{\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)}_{\infty}+C\varepsilon|t|^{-\frac{d}{2}(1-\lambda)-\alpha(1-2\lambda)+A\varepsilon^{\frac{1}{d(1-\lambda)}}}$
(A.5)

for any $t\in I_{T}$. Let us estimate the first term in the above last line.
Let $v$ satisfy $U(t,0)\mathcal{M}_{+}^{-1}v=u$, namely,
$v=\mathcal{M}_{+}U(0,t)u$. Let us here introduce a phase modification

$\displaystyle
w_{p}\coloneqq\mathcal{F}^{-1}(\widetilde{\mathcal{P}}(t)\mathcal{F}v),\qquad\widetilde{\mathcal{P}}(t)\coloneqq\exp\left(i\eta\int_{r_{0}}^{t}|\mathcal{F}v(\tau)|^{p_{c}}|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau\right).$

Arguing as in the derivation of (3.14) in [HN98], we obtain

$\displaystyle
i\partial_{t}(\mathcal{F}w_{p})=\eta|\zeta_{2}(t)|^{-\frac{1}{1-\lambda}}\widetilde{\mathcal{P}}(t)\left(I_{1}(t)+I_{2}(t)\right),$

where $I_{j}(t)$ are defined by (4.1). Integrating the above on $[r_{0},t]$,
noting that
$\mathcal{F}\widetilde{w}(r_{0})=\mathcal{F}v(r_{0})=\mathcal{F}\mathcal{M}_{+}U(0,r_{0})u_{0}$,
one has

$\displaystyle\mathcal{F}w_{p}(t)=\mathcal{F}\mathcal{M}_{+}U(0,r_{0})u_{0}-i\eta\int_{r_{0}}^{t}\widetilde{\mathcal{P}}(\tau)\left(I_{1}(\tau)+I_{2}(\tau)\right)|\zeta_{2}(\tau)|^{-\frac{1}{1-\lambda}}\,d\tau.$
(A.6)

Arguing as in (4.20) and (4.22), one has

$\displaystyle\begin{aligned}
\norm{I_{1}(t)}_{\infty}+\norm{I_{2}(t)}_{\infty}\lesssim{}&|t|^{-\alpha(1-2\lambda)}\left(\norm{u(t)}_{2}+\norm{|J(t)|^{\beta}u(t)}_{2}\right)^{p_{c}+1}\end{aligned}$
(A.7)

for any $\alpha\in[0,(\beta-d/2)/2)$. Moreover, by the assumption, we have

$\displaystyle\norm{\mathcal{F}\mathcal{M}_{+}U(0,r_{0})u(r_{0})}_{\infty}\leq\norm{U(0,r_{0})u(r_{0})}_{1}\leq
C\left(\norm{u_{0}}_{2}+\norm{|J(r_{0})|^{\beta}u_{0}}_{2}\right)\leq
C\varepsilon.$

Combining (A.7) with Proposition A.3, we reach to

$\displaystyle\begin{aligned}
\norm{\mathcal{F}\mathcal{M}_{+}U(0,t)u(t)}_{\infty}\leq{}&C\varepsilon+C\varepsilon^{p_{c}+1}\int_{r_{0}}^{t}\tau^{-\alpha(1-2\lambda)-1+(p_{c}+1)A\varepsilon^{\frac{1}{d(1-\lambda)}}}\,d\tau\\\
\leq{}&C(\varepsilon_{1},\alpha,\lambda)\varepsilon\end{aligned}$ (A.8)

for any $t\in T_{T}$. Therefore collecting (A.5) and (A.8), we conclude

$\displaystyle|t|^{\frac{d}{2}(1-\lambda)}\norm{u(t)}_{\infty}\leq
C(1+t^{-\alpha(1-2\lambda)+A\varepsilon^{\frac{1}{d(1-\lambda)}}})\varepsilon\leq
C\varepsilon$

for any $t\in I_{T}$. This completes the proof. ∎

###### Proof of Proposition A.2.

The consequence immediately follows from Proposition A.4 and Proposition A.3.
∎

##  Appendix B Derivation of the integral equation (1.16)

In this appendix, we derive (1.16). Let us consider the term
$i\partial_{t}\left(U(0,t)u_{p}\right)-U(0,t)F(u_{p})$ in (1.15). Note that
$i\partial_{t}\widehat{w}=(c_{+}|s|)^{-1}F\left(\widehat{w}\right)$. By using
(1.13), we have

$\displaystyle U(0,t)F(u_{p})={}$
$\displaystyle\mathcal{M}_{2}^{-1}\mathcal{M}_{+}^{-1}\mathcal{F}^{-1}|\zeta_{2}(t)|^{-\frac{1}{1-\lambda}}F\left(\widehat{w}\right)$
$\displaystyle{}+\mathcal{M}_{+}^{-1}\left(\mathcal{M}_{2}^{-1}-1\right)\mathcal{F}^{-1}(c_{+}|t|)^{-1}F\left(\widehat{w}\right)+\mathcal{M}_{+}^{-1}\mathcal{F}^{-1}i\partial_{t}\widehat{w}.$

A direct computation shows

$\displaystyle
i\partial_{t}\left(U(0,t)u_{p}\right)=\mathcal{M}_{+}^{-1}\left(\mathcal{M}_{2}^{-1}-1\right)\mathcal{F}^{-1}i\partial_{t}\widehat{w}+\mathcal{M}_{+}^{-1}\mathcal{F}^{-1}i\partial_{t}\widehat{w}.$

Combining these above, we obtain

$\displaystyle{}i\partial_{t}\left(U(0,t)u_{p}(t)\right)-U(0,t)F(u_{p}(t))$
$\displaystyle={}$
$\displaystyle-\mathcal{M}_{+}^{-1}\mathcal{M}_{2}^{-1}(t)\mathcal{F}^{-1}\left(|\zeta_{2}(t)|^{-\frac{1}{1-\lambda}}-(c_{+}|t|)^{-1}\right)F\left(\widehat{w}(t)\right)$
$\displaystyle{}+\mathcal{M}_{+}^{-1}\left(1-\mathcal{M}_{2}^{-1}(t)\right)\mathcal{F}^{-1}(c_{+}|t|)^{-1}F\left(\widehat{w}(t)\right)$
$\displaystyle{}-\mathcal{M}_{+}^{-1}\left(1-\mathcal{M}_{2}^{-1}(t)\right)\mathcal{F}^{-1}i\partial_{t}\widehat{w}(t).$

Since we may assume that $\norm{v(t)}_{2}\rightarrow 0$ as
$t\rightarrow-\infty$, substituting the above into (1.15) and integrating with
respect to $t$, one sees from (1.13) that

$\displaystyle v(t)={}$
$\displaystyle-i\int_{-\infty}^{t}U(t,s)\\{F(u_{p}(s)+v(s))-F(u_{p}(s))\\}\,ds$
$\displaystyle{}-i\int_{-\infty}^{t}U(t,0)\mathcal{M}_{+}^{-1}\mathcal{M}_{2}^{-1}(s)\mathcal{F}^{-1}\left(|\zeta_{2}(s)|^{-\frac{1}{1-\lambda}}-(c_{+}|s|)^{-1}\right)F\left(\widehat{w}\right)\,ds$
$\displaystyle{}+i\int_{-\infty}^{t}U(t,s)\mathcal{M}_{1}(s)\mathcal{D}_{1}(s)\left(\mathcal{F}\mathcal{M}_{2}(s)\mathcal{F}^{-1}-1\right)F\left(\widehat{w}(s)\right)\,\frac{ds}{c_{+}|s|}$
$\displaystyle{}+\mathcal{M}_{1}(t)\mathcal{D}_{1}(t)\left(\mathcal{F}\mathcal{M}_{2}(t)\mathcal{F}^{-1}-1\right)\widehat{w}(t)$

and hence this is (1.16).

## Acknowledgments

M.K. was supported by JSPS KAKENHI Grant Numbers 20K14328. H.M. was supported
by JSPS KAKENHI Grant Numbers 22K13941. The authors express their gratitude to
Professor Satoshi Masaki for his valuable suggestion regarding the Dollard
type decomposition (1.13). The authors also would like to thank Professors
Kota Uriya and Yuta Wakasugi for their helpful advice on the proof of Theorem
1.1.

## References