\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 78, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/78\hfil Rigidity for rough solutions]
{On the rigidity of minimal mass solutions to the focusing
 mass-critical NLS for rough initial data}

\author[D. Li, X. Zhang\hfil EJDE-2009/78\hfilneg]
{Dong Li, Xiaoyi Zhang}  % in alphabetical order

\address{Dong Li \newline
Institute for Advanced Study, Princeton, NJ, 08544, USA}
\email{dongli@ias.edu}

\address{Xiaoyi Zhang \newline
Academy of Mathematics and System Sciences, Beijing,
China. \newline
Institute for Advanced Study, Princeton, NJ, 08544, USA}
\email{xiaoyi@ias.edu}

\thanks{Submitted April 15, 2009. Published June 16, 2009.}
\subjclass[2000]{35Q55}
\keywords{Mass-critical; nonlinear Schrodinger equation}

\begin{abstract}
 For the focusing mass-critical nonlinear Schr\"odinger equation
 $iu_t+\Delta u=-|u|^{4/d}u$,  an important problem is to
 establish Liouville type results for solutions with ground state mass.
 Here the ground state is the positive solution to elliptic equation
 $\Delta Q-Q+Q^{1+\frac 4d}=0$.
 Previous results in this direction were established in
 \cite{klvz,lz:2d,merle_duke,weinstein:charact} and they all require
 $u_0\in H_x^1(\mathbb{R}^d)$. In this paper, we consider the rigidity 
 results for rough initial data $u_0 \in H_x^s(\mathbb{R}^d)$ for any $s>0$.
 We show that in dimensions $d\ge 4$ and under the radial assumption,
 the only solution that does not scatter in both time directions
 (including the finite time blowup case) must be global and coincide
 with the solitary wave $e^{it}Q$ up to symmetries of the equation.
 The proof relies on a non-uniform local iteration scheme, the refined
 estimate involving the $P^{\pm}$ operator and a new smoothing estimate
 for spherically symmetric solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

\subsection{Background and main results}
We consider the focusing mass-critical nonlinear Schr\"odinger equation
\begin{equation}\label{nls}
iu_t+\Delta u=-|u|^{4/d}u
\end{equation}
in dimensions $d\geq 4$; here $u(t,x)$ is a complex-valued function
on $\mathbb R\times \mathbb{R}^d$. The name ``mass critical" refers to the
fact that the scaling symmetry
\begin{equation}
u(t,x) \mapsto  \lambda^{d/2}u(\lambda^{2}t,\lambda
x), \quad \forall\lambda>0 \label{scaling}
\end{equation}
leaves both the equation and the mass invariant. Here the mass is
defined as
\[
\mbox{Mass: } M(u(t))=\int_{\mathbb{R}^d} |u(t,x)|^2 dx=M(u_0).
\]
For the initial value problem of \eqref{nls}, the local theory was
established by Cazenave and Weissler in \cite{cwI}. To summarize,
for any initial data $u_0\in L_x^2(\mathbb{R}^d)$, they constructed the
unique local solution $u(t,x)\in C_t([-T,T];L_x^2)\cap
L_{t,x}^{2(d+2)/d}([-T,T]\times\mathbb{R}^d)$. Moreover, when the mass
of the initial data is small enough, the solution is global and
obeys the global spacetime estimate
\[
\|u\|_{L_{t,x}^{2(d+2)/d}(\mathbb{R}\times\mathbb{R}^d)} \le C(\|u_0\|_{L_x^2}).
\]
This estimate implies that the solution scatters in both time
directions asymptotically: there exist $u_{\pm}\in L_x^2(\mathbb{R}^d)$
such that
\[
\lim_{t\to \pm \infty}\|u(t)-e^{it\Delta}u_{\pm}\|_{L_x^2}=0.
\]

When the solution has large mass, blowup may occur at finite time. The
existence of finite blowup solutions was proved by Glassey
\cite{glassey}, basing on the virial argument. On the other hand,
the equation \eqref{nls} also admits solitary wave solutions of the
form $e^{it}R$, where $R$ solves the elliptic equation
\begin{equation}
\Delta R-R+|R|^{4/d}R=0.\label{elliptic}
\end{equation}
There are infinite many solutions to this equation, but only one
positive solution which is spherically symmetric and whose mass is
minimal among all these $R's$. This solution is usually called the

\begin{definition}[Ground state]\label{ground} \rm
The ground state $Q$ refers to the unique positive solution to the
equation \eqref{elliptic}. According to \cite{blions,kwong},
$Q$ is spherically symmetric and decays exponentially fast as
$|x|\to \infty$.
\end{definition}

It is believed that the mass of $Q$ is the minimal mass among all
the non-scattering solutions. The precise statement of this general
belief is the following scattering conjecture:

\begin{conjecture}
 Let $u_0\in L_x^2(\mathbb{R}^d)$ be such that $M(u_0)<M(Q)$.
Then the corresponding solution exists globally and scatters in both
time directions.
\end{conjecture}

So far, this conjecture has been proved in dimensions $d\ge 2$ when
the initial data $u_0$ is spherically symmetric,
see \cite{ktv:2d,kvz:blowup}.

At the level of minimal mass, there are two explicit examples of
non-scattering solutions: the solitary wave $SW$ and the
pseudo-conformal ground state $Pc(Q)$.
\begin{gather*}
SW =e^{it}Q(x),\\
Pc(Q)=|t|^{-\frac d2}e^{\frac{i|x|^2-4}{4t}}Q(\frac x{t}).
\end{gather*}
It is conjectured that these are the only two threshold solutions
for scattering at the level of minimal mass. The associated is the
following rigidity conjecture which identify the solutions with
ground state mass as either $SW$ or $Pc(Q)$ if they do not scatter.
Since both mass and the equation are invariant under a couple of
symmetries, the coincidence of the solutions with the examples hold
after quotienting out these symmetries. Specifically, the symmetries
are: translation, phase rotation, scaling and the Galilean boost.

\begin{conjecture}
Let $u_0\in L_x^2(\mathbb{R}^d)$ satisfy $M(u_0)=M(Q)$. Then only the
three scenarios can occur
\begin{enumerate}
\item The solution $u(t,x)$ scatters in both time directions;

\item $u$ blows up at finite time, then $u$ must coincide with $Pc(Q)$ up
to symmetries of the equation.

\item $u$ is a global solution and $u$ coincide with $SW$ up to symmetries of the
equation.
\end{enumerate}
\end{conjecture}

Equivalently to say, the rigidity result identifies all the
non-scattering solutions with minimal mass as either the
pseudo-conformal ground state or the solitary wave. Therefore, the
proof of the rigidity conjecture is divided into the two parts: one
is concerned about the behavior of finite blowup solutions, and the other is
concerned with the asymptotics of global solutions.

The first result toward the rigidity conjecture is about finite
time blowup solutions, and they were established by Weinstein and
Merle. In \cite{weinstein:charact}, Weinstein showed that if an
$H_x^1$ solution blows up at finite time with minimal mass, then there
exist $\theta(t), x(t), \lambda(t)$ such that
\[
e^{i\theta(t)} \lambda(t)^{d/2}u(t,\lambda(t) x+x(t))\to Q\quad \mbox{in }
H_x^1.
\]
Later, Merle extended this result to show that if an $H_x^1$
solution with minimal mass blows up at finite time, then it must be
equal to $Pc(Q)$ up to symmetries. One can also see \cite{hmidi-keraani}
for a simpler proof of this result. The requirement that $u_0\in H_x^1$ is
essentially needed since it is the natural space to carry out the
spectral analysis and to well define the energy:
\[
\mbox{Energy:}\ E(u(t))=\frac 12\|\nabla u(t)\|_{L_x^2}^2-\frac d{2(d+2)}
\|u(t)\|_{L_{x}^{2(d+2)/d}}^{2(d+2)/d}=E(u_0).
\]
It is also worth
pointing out that their results work for all dimensions $d\ge 1$ and
there is no symmetry assumption on the initial data. But the proof
relies heavily on the finiteness of the blowup time.

From Merle's result and the pseudo-conformal
invariance for mass critical NLS, one easily sees that if $u_0\in
\Sigma=\{v\in H_x^1, xv\in L_x^2\}$ and the corresponding solution
exists globally but does not scatter in at least one time direction,
then it must be the solitary wave $SW$ up to symmetries. This is the first
result toward the rigidity result if the solution is global.

Without the additional decay assumption, it is not obvious at all if
the conjecture still holds. Recently in \cite{klvz}, \cite{lz:2d},
we established the rigidity result for global solutions under the
radial assumptions.

\begin{theorem}[\cite{klvz,lz:2d}]\label{h1_result}
 Let $d\ge 2$. Let $u_0\in H_x^1(\mathbb{R}^d)$ be spherically symmetric and
satisfy $M(u_0)=M(Q)$. Then only the following two scenarios can occur:
\begin{enumerate}
\item The solution exists globally and scatters in both time directions;

\item There exist $\theta_0, \lambda_0$ such that
\[
u(t,x)=e^{i\theta_0}\lambda_0^{d/2}e^{i\lambda_0^2 t}Q( {\lambda_0}x).
\]
\end{enumerate}
\end{theorem}

In dimensions $d\ge 4$, we can
relax the spherical symmetry to certain splitting-spherical
symmetry, see \cite{lz:2d} for more details.

As indicated by the statement of the theorem, the rigidity
conjecture concerning the global solution holds under several
additional conditions: the spherical symmetry( or the splitting-spherical
symmetry) on the initial data; the dimension $d\ge 2$; and the
$H_x^1$ regularity on the initial data. Each of them is used heavily in the proof.
 To give a brief explanation, the symmetry assumption
is used to freeze the center of mass and provide enough decay as
$|x|\to \infty$. The one dimension function does not have decay as
$|x|\to \infty$, this is why the restriction on the dimension comes
in. Finally, since the proof relies on the virial argument, the
$H_x^1$ regularity is naturally needed to define the energy.
Meanwhile, in low dimensions $d=2,3$, the $H_x^1$ regularity is also
used to get the weak compactness for the kinetic energy, see
\cite{lz:2d} for more details.


Therefore, removing the reliance on any of these conditions makes a
very challenging problem. In this paper, we try to remove the
reliance on the $H_x^1$ regularity in the rigidity conjecture. For
some technical reasons which will be clear in the proof, we shall
consider solutions which have the minimal mass and do not
scatter in both time directions. Our result is the following:

\begin{theorem}[Rigidity of SW for rough solutions]\label{main_thm}
Let $d\ge 4$, $s>0$. Let $u_0\in H_x^s(\mathbb{R}^d)$ be spherically symmetric
and such that $M(u_0)=M(Q)$. Suppose that the corresponding maximal
lifespan solution $u(t,x):(-T_*, T^*)\times \mathbb{R}^d\to \mathbb{C}$ does not
scatter in both time directions:
\[
\|u\|_{L_{t,x}^{2(d+2)/d}((-T_*,0]\times\mathbb{R}^d)}=
\|u\|_{L_{t,x}^{2(d+2)/d}([0,T^*)\times\mathbb{R}^d)}=\infty.
\]
Then the solution must be global
\[
T_*=T^*=+\infty.
\]
And there exist $\lambda_0,\theta_0$ such that
\[
u(t,x)=e^{i\theta_0}e^{i\lambda_0^2t}\lambda_0^{d/2}Q(\lambda_0
x).
\]
\end{theorem}

As expected, the main part of the proof is devoted to upgrading the
$H_x^s$ regularity of the solution to $H_x^1$, when the result for
$H_x^1$ solution Theorem \ref{h1_result} can be applied. The
possibility that we can upgrade the regularity of the solution comes
ultimately from the fact that the solution we are considering has
the \emph{minimal mass} and does not scatter \emph{in both time
directions}.

Our strategy for upgrading the regularity is the
following: Firstly, since $u$ has the minimal mass and does not
scatter on both sides, applying the same argument as in \cite{klvz},
one easily gets that
\[
\|\phi_{>1}\nabla u(t)\|_{L_x^2}\lesssim 1.
\]
This means that away from the origin, the solution is regular uniformly
in time, thus it suffices for us to examine the solution near the origin.
There we carefully design a local iteration scheme enabling us to go
from $H_x^t$ to $H_x^{t+\epsilon}$ for any $t<1$ and an $\epsilon$ increasing
in $t$. After finite many times of iteration, we get the desired
$H_x^1$ regularity. Here by "local", we mean that the scheme is
designed to upgrade the regularity of the solution at some fixed
time $t$, for example $t=0$, not uniformly in time.  More
precisely, the quantity we will look at is
\begin{equation}
\|\phi_{\le 1} P_N u_0\|_{L_x^2}, \ N\ge 1.\label{small1}
\end{equation}
(Not that the piece $\|\phi_{>1} P_N u_0\|_{L_x^2}$ already gives us
$N^{-1-\epsilon(d)}$ decay following the argument in \cite{klvz}, which
is already very good). Now we split \eqref{small1} into two parts by
introducing a spatial cutoff
\begin{gather}
\| \phi_{\le N^{-1-\gamma}} P_N u_0 \|_{L_x^2},\label{first}\\
\| \phi_{N^{-1-\gamma}<\cdot\le 1} P_N u_0 \|_{L_x^2}.\label{second}
\end{gather}

By H\"older and Bernstein, the first quantity gives us the bound:
$N^{-s-\frac d2\gamma}$ which is good for the iteration. To estimate the
second piece, we project it into the incoming and outgoing wave, for
the incoming wave, we use the Duhamel formula backward in time; for
the outgoing wave, we use the Duhamel formula forward in time. The
assumption that the solution does not scatter on both sides forbids the
scattering wave, for which there is no hope to upgrade the
regularity, to participate in the estimates.

The first issue when we estimate these two pieces comes from the
fact that in \eqref{second}, the spatial cutoff and the frequency
cutoff does not obey the scaling like in \cite{ktv:2d},
\cite{kvz:blowup}, there we have good estimates for
$P_N^{\pm}$ with a natural spatial cutoff $\phi_{>\frac 1N}$.
Indeed, when approaching the origin,
the operator $P^{\pm}$ have strong singularities. To get around this
problem, we refine the estimates for the operator
$P_N^{\pm}$ with spatial cutoff of the form
$\phi_{N^{-1-\gamma}<\cdot\le 1}$. It turns out that there will be a loss
of $N$ to some power related to $\gamma$. This loss of power is not
too harmful for us if we make a judicious choice of $\gamma$ and
other relevant parameters in the iteration scheme.
We give the detailed discussion on the properties of the operators
in Section 3.

Having the operator estimate in hand, we then estimate the contribution
from the in-out wave by chopping the $t$-integration into different
pieces. Since the stationary phase point moves with time $t$ at speed $N$,
the contribution from the large time piece is presumably fine due to
the decay property of radial functions.

It turns out that the most troublesome term is the following local piece
\begin{equation}\label{trouble}
\Big\|\phi_{N^{-1-\gamma}<\cdot\le 1}\int_0^{\frac 1{N^{2-\sigma}}}
P_N^+ e^{-i\tau\Delta} \phi_{\le 1}F(u)(\tau)d\tau\Big\|_{L_x^2}.
\end{equation}
Here $0<\sigma<2$ is a small constant to be specified later in the proof. One observation from
the expression \eqref{trouble} is that it is spatially localized, which
suggests that the additional regularity should come from some sort of smoothing estimate.

The classical smoothing estimate
\cite{sjolin,kpv:smoothing93,constantin_saut}
asserts that the linear propagation gain half derivative locally. This is
crucial to study the NLS containing first order derivatives.
In our setting, since we are considering the spherically symmetric
functions, we develop the following global
smoothing estimate:
\begin{equation}\label{smoothing}
\||x|^{(d-1)/2}|\nabla|^{1/2}
e^{it\Delta}u_0\|_{L_x^{\infty}L_t^2(\mathbb{R}^d\times\mathbb{R})}\lesssim
\|u_0\|_{L_x^2}.
\end{equation}
Using the dual form of this estimate, we can successfully control the
term \eqref{trouble} and close the argument.
The proof of the estimate \eqref{smoothing} can be found in Section 3.

We make two remarks here. First of all, it is worth pointing out
that the strategy here for upgrading the regularity is quite different
from the one in \cite{ktv:2d, kvz:blowup}. There the
argument relies on the fact that the solution is almost periodic
modulo scaling, and the solution is uniformly flat. Namely, there exists $N(t)>0$ such that
$\{N(t)^{-\frac d2}u(t,\frac x{N(t)})\}$ is precompact in
$L_x^2(\mathbb{R}^d)$ \emph{and $N(t)\le 1$}. In our setting, the solution
also enjoys such compactness, but there is no a priori control on
$N(t)$. Actually, the most difficult case is that $N(t)$ can
fluctuate out of any control. Secondly, like in \cite{klvz}, our
proof needs the assumption $d\ge 4$ since the nonlinearity
$|u|^{4/d} u$ can easily be controlled without knowing other information
than $M(u)$ being finite. It is certainly an interesting problem
to prove the theorem in lower dimensions.

\section{Preliminaries}

\subsection{Some notation}
We write $X \lesssim Y$ or $Y \gtrsim X$ to indicate $X \leq CY$ for
some constant $C>0$.  We use $O(Y)$ to denote any quantity $X$ such
that $|X| \lesssim Y$.  We use the notation $X \sim Y$ whenever $X
\lesssim Y \lesssim X$.  The fact that these constants depend upon
the dimension $d$ will be suppressed.  If $C$ depends upon some
additional parameters, we will indicate this with subscripts; for
example, $X \lesssim_u Y$ denotes the assertion that $X \leq C_u Y$
for some $C_u$ depending on $u$.  We denote by $X\pm$ any quantity of the form
$X\pm \epsilon$ for any $\epsilon>0$.

We use the `Japanese bracket' convention $\langle x \rangle := (1
+|x|^2)^{1/2}$.

We write $L^q_t L^r_{x}$ to denote the Banach space with norm
$$ \| u \|_{L^q_t L^r_x(\mathbb{R} \times \mathbb{R}^d)} := \Bigl(\int_\mathbb{R} \Bigl(\int_{\mathbb{R}^d} |u(t,x)|^r\ dx\Bigr)^{q/r}\ dt\Bigr)^{1/q},$$
with the usual modifications when $q$ or $r$ are equal to infinity,
or when the domain $\mathbb{R} \times \mathbb{R}^d$ is replaced by a smaller region
of spacetime such as $I \times \mathbb{R}^d$.  When $q=r$ we abbreviate
$L^q_t L^q_x$ as $L^q_{t,x}$.

Throughout this paper, we will use $\phi\in C^\infty(\mathbb{R}^d)$ be a
radial bump function supported in the ball $\{ x \in \mathbb{R}^d: |x| \leq
\frac{25} {24} \}$ and equal to one on the ball $\{ x \in \mathbb{R}^d: |x|
\leq 1 \}$.  For any constant $C>0$, we denote $\phi_{\le C}(x):=
\phi \bigl( \tfrac{x}{C}\bigr)$ and $\phi_{> C}:=1-\phi_{\le C}$.

\subsection{Basic harmonic analysis}\label{ss:basic}

For each number $N > 0$, we define the Fourier multipliers
\begin{gather*}
\widehat{P_{\leq N} f}(\xi) := \phi_{\leq N}(\xi) \hat f(\xi)\\
\widehat{P_{> N} f}(\xi) := \phi_{> N}(\xi) \hat f(\xi)\\
\widehat{P_N f}(\xi) := (\phi_{\leq N} - \phi_{\leq N/2})(\xi) \hat
f(\xi)
\end{gather*}
and similarly $P_{<N}$ and $P_{\geq N}$.  We also define
$$
P_{M < \cdot \leq N} := P_{\leq N} - P_{\leq M}
 = \sum_{M < N' \leq N} P_{N'}
$$
whenever $M < N$.  We will usually use these multipliers when $M$
and $N$ are \emph{dyadic numbers} (that is, of the form $2^n$ for
some integer $n$); in particular, all summations over $N$ or $M$ are
understood to be over dyadic numbers.  Nevertheless, it will
occasionally be convenient to allow $M$ and $N$ to not be a power of
$2$.  As $P_N$ is not truly a projection, $P_N^2\neq P_N$, we will
occasionally need to use fattened Littlewood-Paley operators:
\begin{equation}\label{PMtilde}
\tilde P_N := P_{N/2} + P_N +P_{2N}.
\end{equation}
These obey $P_N \tilde P_N = \tilde P_N P_N= P_N$.

Like all Fourier multipliers, the Littlewood-Paley operators commute
with the propagator $e^{it\Delta}$, as well as with differential
operators such as $i\partial_t + \Delta$. We will use basic
properties of these operators many many times, including

\begin{lemma}[Bernstein estimates]\label{Bernstein}
 For $1 \leq p \leq q \leq \infty$,
\begin{gather*}
\bigl\| |\nabla|^{\pm s} P_N f\bigr\|_{L^p_x}
 \sim N^{\pm s} \| P_N f \|_{L^p_x},\\
\|P_{\leq N} f\|_{L^q_x}
  \lesssim N^{\frac{d}{p}-\frac{d}{q}} \|P_{\leq N} f\|_{L^p_x},\\
\|P_N f\|_{L^q_x} \lesssim N^{\frac{d}{p}-\frac{d}{q}} \| P_N
f\|_{L^p_x}.
\end{gather*}
\end{lemma}

While it is true that spatial cutoffs do not commute with
Littlewood-Paley operators, we still have the following result.

\begin{lemma}[Mismatch estimates in real space]\label{L:mismatch_real}
Let $R,N>0$.  Then
\begin{gather*}
\bigl\| \phi_{> R} \nabla P_{\le N} \phi_{\le\frac R2} f
\bigr\|_{L_x^p}
\lesssim_m N^{1-m} R^{-m} \|f\|_{L_x^p} \\
\bigl\| \phi_{> R}  P_{\leq N} \phi_{\le\frac R2} f
\bigr\|_{L_x^p} \lesssim _m N^{-m} R^{-m} \|f\|_{L_x^p}
\end{gather*}
for any $1\le p\le \infty$ and $m\geq 0$.
\end{lemma}

\begin{proof}
We will only prove the first inequality; the second follows
similarly.

It is not hard to obtain kernel estimates for the operator $\phi_{>
R}\nabla P_{\le N}\phi_{\le\frac R2}$. Indeed, an exercise in
non-stationary phase shows
\[
\bigl|\phi_{> R}\nabla P_{\le N}\phi_{\le R/2}(x,y)\bigr|
\lesssim N^{d+1-2k} |x-y|^{-2k}\phi_{|x-y|>\frac R2}
\]
for any $k\geq 0$.  An application of Young's inequality yields the
claim.
\end{proof}

Similar estimates hold when the roles of the frequency and physical
spaces are interchanged.  The proof is easiest when working on
$L_x^2$, which is the case we will need; nevertheless, the following
statement holds on $L_x^p$ for any $1\leq p\leq \infty$.

\begin{lemma}[Mismatch estimates in frequency space]
\label{L:mismatch_fre}
For $R>0$ and $N,M>0$ such that $\max\{N,M\}\geq 4\min\{N,M\}$,
\begin{gather*}
\bigl\|  P_N \phi_{\le{R}} P_M f \bigr\|_{L_x^2}
 \lesssim_m \max\{N,M\}^{-m} R^{-m} \|f\|_{L_x^2} \\
\bigl\|  P_N \phi_{\le {R}} \nabla P_M f \bigr\|_{L_x^2}
 \lesssim_m M \max\{N,M\}^{-m} R^{-m} \|f\|_{L_x^2}.
\end{gather*}
for any $m\geq 0$.  The same estimates hold if we replace
$\phi_{\le R}$ by $\phi_{>R}$.
\end{lemma}

\begin{proof}
The first claim follows from Plancherel's Theorem and
Lemma~\ref{L:mismatch_real} and its adjoint.  To obtain the second
claim from this, we write
$$
P_N \phi_{\le {R}} \nabla P_M = P_N \phi_{\le {R}} P_M \nabla \tilde
P_M
$$
and note that $\|\nabla \tilde P_M\|_{L_x^2\to L_x^2}\lesssim M$.
\end{proof}

We will need the following radial Sobolev embedding to exploit the
decay property of a radial function. For the proof and the more
complete version, one refers to see \cite{tvz:hd}.

\begin{lemma}[Radial Sobolev embedding, \cite{tvz:hd}]
 \label{L:radial_embed}
Let dimension $d\ge 2$. Let $s>0$, $\alpha>0$, $1<p,q<\infty$ obeys
the scaling restriction: $\alpha+s=d(\frac 1q-\frac 1p)$. Then the
following holds:
\[
\||x|^{\alpha} f\|_{L_x^p}\lesssim \||\nabla|^s f\|_{L^q_x},
\]
where the implicit constant depends on $s,\alpha,p,q$. When
$p=\infty$, we have
\[
\||x|^{(d-1)/2}P_N f\|_{L^\infty_x}\lesssim N^{1/2}\|P_N
f\|_{L_x^2}.
\]
\end{lemma}

We will need the following fractional chain rule lemma.

\begin{lemma}[Fractional chain rule for a $C^1$ function,
\cite{chris:weinstein}\cite{staf}\cite{taylor}]\label{lem_chain}
Let $F\in C^1(\mathbb C)$, $\sigma \in (0,1)$, and
$1<r,r_1,r_2<\infty$ such that $\frac 1r=\frac 1{r_1}+\frac 1{r_2}$.
Then we have
\[
\||\nabla|^{\sigma}F(u)\|_{L_x^r}\lesssim
\|F'(u)\|_{L_x^{r_1}}\||\nabla|^{\sigma}u\|_{L_x^{r_2}}.
\]
\end{lemma}

For a proof of the above lemma, see \cite{chris:weinstein,staf}
and \cite{taylor}.

\subsection{Strichartz estimates}

The free Schr\"odinger flow has the explicit expression
\[
e^{it\Delta } f(x)=\frac 1{(4\pi t)^{d/2}}\int_{\mathbb{R}^d}
e^{i|x-y|^2/4t}f(y)dy,
\]
We will frequently use the standard Strichartz estimate.

\begin{lemma}[Strichartz]\label{L:strichartz}
Let $d\ge 2$. Let $I$ be an interval, $t_0 \in I$, and let
$u_0 \in L^2_x(\mathbb{R}^d)$
and $F \in L^{2(d+2)/(d+4)}_{t,x}(I \times \mathbb{R}^d)$.  Then, the
function $u$ defined by
$$
u(t) := e^{i(t-t_0)\Delta} u_0 - i \int_{t_0}^t e^{i(t-t')
\Delta} F(t')\ dt'
$$
obeys the estimate
$$
\|u \|_{L^\infty_t L^2_x} + \| u \|_{L^{2(d+2)/d}_{t,x}}
    \lesssim \| u_0 \|_{L^2_x} + \|F\|_{L^{\frac{2(d+2)}{d+4}}_{t,x}},
$$
where all spacetime norms are over $I\times\mathbb{R}^d$.
\end{lemma}

\begin{proof}
See, for example, \cite{gv:strichartz, strichartz}.  For the
endpoint see \cite{tao:keel}.
\end{proof}

We will also need a weighted Strichartz estimate, which exploits
heavily the spherical symmetry in order to obtain spatial decay.

\begin{lemma}[Weighted Strichartz, \cite{ktv:2d, kvz:blowup}]\label{L:wes}
Let $d\ge 2$. Let $I$ be an interval, $t_0 \in I$, and let
$F:I\times\mathbb{R}^d\to \mathbb{C}$ be spherically symmetric.  Then,
$$
\Big\|\int_{t_0}^t e^{i(t-t')\Delta} F(t')\, dt' \Big\|_{L_x^2}
\lesssim \bigl\||x|^{-\frac{2(d-1)}q}F
\bigr\|_{L_t^{\frac{q}{q-1}}L_x^{\frac{2q}{q+4}}(I \times \mathbb{R}^d)}
$$
for $4\leq q\leq \infty$.
\end{lemma}

\section{Smoothing estimate and the refined operator estimates}

\subsection{Kato smoothing for radial solutions}

Kato smoothing estimate \cite{sjolin,kpv:smoothing93,constantin_saut} plays an important
role in studying the wellposedness for nonlinear Schr\"odinger equation with derivative.
In one spatial dimension, the typical Kato smoothing takes the form:
\begin{equation}\label{kato_1d}
\||\nabla|^{1/2} e^{it\partial_{xx}}u_0\|_{L_x^\infty L_t^2(\mathbb{R}\times\mathbb{R})}
\lesssim \|u_0\|_{L_x^2}.
\end{equation}
The smoothing estimate in high dimensions involves the spatial
localization or a decay weight.
We will not discuss in detail here. In this paper, we will need the
following smoothing estimate for radial functions, which can be
viewed as an extension of the one dimensional estimate
\eqref{kato_1d}.

\begin{lemma}[Kato smoothing for radial functions with $d\ge 2$]
\label{lem_hom}
Let the dimension $d\ge 2$. Then for any radial function $f$ we have
\[
 \big\| |x|^{\frac {d-1} 2} |\nabla |^{1/2} e^{it \Delta} f
 \big\|_{L_x^\infty L_t^2 (\mathbb{R}^d\times \mathbb{R})}
\lesssim \|f\|_2.
\]
\end{lemma}

\begin{proof}
By passing to radial coordinates, we can write
\begin{equation} \label{eq0}
 |x|^{\frac {d-1} 2} |\nabla|^{1/2} e^{it\Delta} f
=  |x|^{\frac {d-1} 2} \int_0^\infty k^{1/2} e^{-itk^2} \hat
f(k) k^{d-1} \Big( \int_{|\omega|=1} e^{ik|x| \omega_1} d
\sigma(\omega) \Big) dk,
\end{equation}
where $\hat f$ is the Fourier transform of the function $f$ and
$d\sigma(\omega)$ is the the surface measure on $S^{d-1}$. Since the
Fourier transform of a radial function is still radial, we can
slightly abuse the notation $\hat f(k)$ to denote the Fourier
transform of $f$.  Now consider the function
\[
 h(\rho) := \int_{|\omega|=1} e^{i\rho \omega_1} d\sigma (\omega).
\]
It is clear that
\[
 \rho^{\frac{d-2} 2} h(\rho)= J_{(d-2)/2} (\rho),
\]
where $J_{(d-2)/2}$ is the usual Bessel function of order $\frac
{d-2}2 $. Then by using the asymptotics for Bessel functions, it is
not difficult to see that
\begin{equation} \label{eq1a}
 \sup_{\rho>0} \rho^{(d-1)/2} |h(\rho)| \le C_1<\infty,
\end{equation}
where $C_1$ is a constant depending only on the dimension $d$. By
Plancherel, one can show that for any one-dimensional function $F$,
we have
\begin{equation} \label{eq1b}
 \big\| \int_0^\infty e^{-itk^2} F(k) dk \big\|_{L_t^2}^2
= \frac 12 \int_0^\infty |F(k)|^2 \frac {dk} k.
\end{equation}
Now by \eqref{eq0}, \eqref{eq1a}, \eqref{eq1b}, we obtain
\begin{align*}
 \big\| |x|^{(d-1)/2} |\nabla|^{1/2} e^{it\Delta} f\big\|_{L_t^2}^2
& = \frac 12 \int_0^\infty |\hat f(k)|^2 \cdot |x|^{d-1} \cdot k^{2(d-1)} \cdot |h(k|x|)|^2 dk \\
& \le \frac 12 \int_0^\infty |\hat f(k)|^2 k^{d-1} dk \cdot
\Big(\sup_{\rho>0} \rho^{\frac {d-1}2} |h(\rho)| \Big)^2 \\
& \le C_1^2 \|f\|_2^2.
\end{align*}
The lemma is proved.
\end{proof}

\subsection{The in-out decomposition and refined operator estimates}

We will need an incoming/outgoing decomposition; we will use the one
developed in \cite{ktv:2d, kvz:blowup}. As there, we define
operators $P^{\pm}$ by
\[
[P^{\pm} f](r) :=\tfrac12 f(r)\pm \tfrac{i}{\pi} \int_0^\infty
\frac{r^{2-d}\,f(\rho)\,\rho^{d-1}\,d\rho}{r^2-\rho^2},
\]
where the radial function $f: \mathbb{R}^d\to \mathbb{C}$ is written as a function
of radius only. We will refer to $P^+$ is the projection onto
outgoing spherical waves; however, it is not a true projection as it
is neither idempotent nor self-adjoint.  Similarly, $P^-$ plays the
role of a projection onto incoming spherical waves; its kernel is
the complex conjugate of the kernel of $P^+$ as required by
time-reversal symmetry.

\subsection{The two-dimensional case}
For $N>0$ let $P_N^{\pm}$ denote the product $P^{\pm}P_N$ where
$P_N$ is the Littlewood-Paley projection. We record the following
properties of $P^{\pm}$ from \cite{ktv:2d, kvz:blowup}:

\begin{proposition}[Properties of $P^\pm$, \cite{ktv:2d, kvz:blowup}]
\label{P:P properties}\leavevmode
\begin{itemize}
\item[(i)] $P^+ + P^- $ represents the projection from $L^2$ onto
$L^2_{\rm rad}$.

\item[(ii)]  $P^{\pm}$ are bounded on $L^2(\mathbb{R}^2)$.
\item[(iii)] For $|x|\gtrsim N^{-1}$ and $t\gtrsim N^{-2}$, the
integral kernel obeys
\[
\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim
\begin{cases}
    (|x||y|)^{-1/2}|t|^{-1/2}  & |y|-|x|\sim  Nt \\[1ex]
     \frac{N^2}{(N|x|)^{1/2}\langle N|y|\rangle^{1/2}}
     \bigl\langle N^2t + N|x| - N|y| \bigr\rangle^{-m}
            &\text{otherwise}\end{cases}
\]
for all $m\geq 0$.

\item[(iv)] For $|x|\gtrsim N^{-1}$ and $|t|\lesssim N^{-2}$, the
integral kernel obeys
\begin{equation*}
\bigl| [P^\pm_N e^{\mp it\Delta}](x,y) \bigr|
    \lesssim   \frac{N^2}{(N|x|)^{1/2}\langle N|y|\rangle^{1/2}}
     \bigl\langle N|x| - N|y| \bigr\rangle^{-m}
\end{equation*}
for any $m\geq 0$.
\end{itemize}
\end{proposition}

For a proof of the above proposition, see \cite{ktv:2d, kvz:blowup}.

We will also need the following Proposition concerning the
properties of $P^{\pm}$ in the small $x$ regime (i.e. $|x| \lesssim
N^{-1}$) where Bessel functions have logarithmic singularities. More
precisely, we have the following result.

\begin{proposition}[Properties of $P^\pm$, small $x$ regime,
\cite{lz:2d}] \label{Psmall_properties}
Let  dimension $d=2$.
 \begin{itemize}
 \item[(i)] For $t \gtrsim N^{-2}$, $N^{-3} \lesssim |x| \lesssim N^{-1}$,
$|y|\ll Nt$  or $|y| \gg Nt$, the integral
kernel satisfies
\[
 \bigl|  [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^2 \log N}{\langle N|y| \rangle^{1/2} }
\langle N^2 t + N|y| \rangle^{-m}, \quad \forall\, m\ge 0.
\]

\item[(ii)] For $t \gtrsim N^{-2}$, $N^{-3} \lesssim |x| \lesssim N^{-1}$,
$|y|\sim Nt$, the integral kernel satisfies
\[
 \bigl|  [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim
 \frac {N^2 \log N}{\langle N|y| \rangle^{1/2} }.
\]
 \end{itemize}
\end{proposition}

For a proof of the above proposition, see \cite{lz:2d}.

\subsection{The case $d\ge 3$}
The next lemma allows us to bound the operator $P_N^{\pm}$ slightly
below the $|x| \sim 1/N$ barrier, i.e. in the regime
$\frac 1 {N^{1+\gamma}} \le |x| \le \frac 1N$ for some $\gamma>0$.
The price to pay is a polynomial growth factor in $N$.

\begin{lemma} \label{L:bdd}
Let the dimension $d\ge 3$.  Fix $N \gtrsim 1$ and $\gamma>0$.
For any spherically symmetric function $f\in L_x^2(\mathbb{R}^d)$,
\[
\bigl\|P^\pm P_{ N} f \bigr\|_{L^2_x(\frac 1 {N^{1+\gamma} } \le |x|
\le  \frac 1N)} \lesssim
\begin{cases}
N^{\frac{(d-4)\gamma}2}  \cdot \bigl\| f \bigr\|_{L^2_x(\mathbb{R}^d)},
&\text{if $d\ge 5$}, \\
\langle \log N \rangle^{1/2} \cdot \bigl\| f \bigr\|_{L^2_x(\mathbb{R}^d)},
&\text{if $d=4$}, \\
\bigl\| f \bigr\|_{L^2_x(\mathbb{R}^d)}, &\text{if $d=3$},
\end{cases}
\]
where the implied constant depends only on $\gamma$ and $d$.
Here $\langle \cdot \rangle$ is the Japanese bracket.
\end{lemma}

\begin{proof}
 We shall only prove the inequality for $P^+$. The result for $P^{-}$
is similar (or one can use the fact $P^++P^-$ acts as an identity on
$L^2_{\rm rad}(\mathbb{R}^d)$). By the definition of $P^{+}$, we have
 \begin{equation}
 \bigl\|P^\pm P_{N} f \bigr\|_{L^2_x(\frac 1 {N^{1+\gamma} } \le |x|
 \le  \frac 1 N)}^2
 = \int_{\frac 1 {N^{1+\gamma}}}^{\frac 1 N}
 \Big| \int_0^\infty H^{(1)}_{\frac {d-2} 2} (kr) \hat f(k) k^{d/2}
 \psi(\tfrac k N) dk \Big|^2 r dr.
    \label{tmp_1047_a}
\end{equation}
Since $k\sim N$, $\frac 1 {N^{1+\gamma} } \le r \le \frac 1 N$,
$\frac 1 {N^{\gamma}} \le kr \lesssim 1$, we have
\[
 \big| H^{(1)}_{(d-2)/2} (kr) \big| \lesssim (kr)^{-(d-2)/2}.
\]
Therefore, by Cauchy-Schwartz, we obtain
\begin{equation}
\begin{aligned}
 \text{RHS of \eqref{tmp_1047_a} }
& \lesssim \int_{\frac 1 {N^{1+\gamma}}}^{\frac 1 N} r^{3-d} dr \cdot N^{2-d}
\cdot \int_0^\infty |\hat f(k)|^2 k^{d-1} dk \cdot
  \int_0^\infty |\psi(\frac kN)|^2 k dk  \\
& \lesssim \int_{\frac 1 {N^{1+\gamma}}}^{\frac 1N} r^{3-d} dr
\cdot N^{4-d} \cdot \|f\|_{L_x^2(\mathbb{R}^d)}^2.
\end{aligned} \label{tmp_1047_b}
\end{equation}
Now if $d \ge 5$, then
\[
 \int_{\frac 1{N^{1+\gamma}}}^{\frac 1 N} r^{3-d} dr \lesssim N^{(1+\gamma)(d-4)}
\]
and RHS of \eqref{tmp_1047_b} $\lesssim N^{(d-4)\gamma} \|f\|_{L_x^2(\mathbb{R}^d)}^2$.

If $d=4$, then
\[
 \int_{\frac 1{N^{1+\gamma}}}^{\frac 1 N} r^{3-d} dr \lesssim \log N
\]
and
RHS of \eqref{tmp_1047_b} $\lesssim (\log N)\cdot \|f\|_{L_x^2(\mathbb{R}^d)}^2$.

If $d=3$, then clearly
\[
 \text{RHS of \eqref{tmp_1047_b} } \lesssim \|f\|_{L_x^2(\mathbb{R}^d)}^2.
\]
The lemma is proved.
\end{proof}

In the next lemma we shall give bounds of some integrals needed later
in the kernel estimates. To fix notations, we assume $\tilde g_1$,
$\tilde g_2$ are one-dimensional functions such
that
\begin{equation} \label{eq_1236a}
 \big| \frac{d^m \tilde g_i (r)} { d r^m} \big| \lesssim 1, \quad
\forall 0<r\lesssim 1, \; m\ge 0, \; i=1,2,
\end{equation}
and $a(\cdot)$ is a one-dimensional function such that
\begin{equation} \label{eq_1236b}
 \big| \frac{d^m a(r)}{dr^m} \big| \lesssim \langle r \rangle^{-m}, \quad
\forall r\ge 0,\; m\ge 0,
\end{equation}
where $\langle \cdot \rangle$ is the Japanese bracket. We shall denote
by $\psi$ the multiplier function
from the Littlewood-Paley projection. With these notations, we state
the following lemma.

\begin{lemma} \label{lem_tmp12}
Let $N\gtrsim 1$ be a dyadic number. Assume $0<c_1, \,c_2\lesssim \frac 1 N$
are two fixed numbers. Then for any
$t \gtrsim N^{-2}$, we have
\begin{equation} \label{eq_1237a}
 \int_0^\infty \tilde g_1(kc_1) \tilde g_2(k c_2) e^{itk^2} \psi(\tfrac k N ) dk \lesssim N \cdot
\langle N^2 t + N c_2 \rangle^{-m}, \quad \forall\, m\ge 0.
\end{equation}
 If $c_3$ is a number such that $\frac 1 N \lesssim c_3 \ll Nt$, then
\begin{equation} \label{eq_1237b}
 \int_0^\infty \tilde g_1(kc_1) \frac {a(kc_3)} {\langle k c_3 \rangle^{1/2}} e^{i(tk^2  \pm c_3 k)} dk
\lesssim  \frac N {\langle Nc_3 \rangle^{1/2}} \cdot \langle N^2 t
+ N c_3 \rangle^{-m}, \quad \forall\, m\ge 0.
\end{equation}
Similarly if $c_4$ is such that $ c_4 \gg Nt$, then
\begin{equation} \label{eq_1237c}
 \int_0^\infty \tilde g_1(kc_1) \frac {a(kc_4)} {\langle k c_4 \rangle^{1/2}} e^{i(tk^2  \pm c_4 k)} dk
\lesssim  \frac N {\langle Nc_4 \rangle^{1/2}} \cdot \langle N^2 t
+ N c_4 \rangle^{-m}, \quad \forall\, m\ge 0.
\end{equation}
\end{lemma}

\begin{proof}
All the estimates \eqref{eq_1237a}--\eqref{eq_1237c} essentially follow
from integrating by parts. Let $k=N\tilde k$, then $\tilde k \sim 1$
due to the cut-off function $\psi$. Change $k$ to $N\tilde k$ in
\eqref{eq_1237a}--\eqref{eq_1237c}. Note by \eqref{eq_1236a} and the
fact that $0<c_1,\, c_2\lesssim \frac 1N$,
$N\gtrsim 1$, we have
\[
 \big| \frac{d^m}{d {\tilde k}^m} ( \tilde g_i (\tilde k c_i N) ) \big|
\lesssim 1, \quad \forall\, \tilde k\sim 1, \; m\ge 0,\; i=1,2.
\]
Also by \eqref{eq_1236b} and the assumptions on $c_3$, $c_4$, we have
\[
 \big| \frac {d^m}{d {\tilde k}^m} a(\tilde k c_3 N) \big|
+ \big| \frac {d^m}{d {\tilde k}^m} a(\tilde k c_4 N) \big| \lesssim 1,
\quad \forall\, \tilde k\sim 1,\, m\ge 0.
\]
The desired estimates \eqref{eq_1237a}--\eqref{eq_1237c} now follow
from integration by parts
and the above derivative estimates on $\tilde g_1$, $\tilde g_2$, $a$.
\end{proof}

\begin{proposition}[Properties of $P^\pm$, small $x$ regime]
\label{Psmall_properties_highD}

Let dimension $d \ge 3$ and assume $\gamma>0$. Let $N\gtrsim 1$ be a dyadic number.
 \begin{itemize}
 \item[(i)] For $t \gtrsim N^{-2}$, $\frac 1 {N^{1+\gamma}} \lesssim |x| \lesssim N^{-1}$, $|y|\ll Nt$  or $|y| \gg Nt$, the integral
kernel satisfies
\[
 \bigl|  [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^{(1+\gamma)(d-2)+2}}{\langle N|y| \rangle^{1/2} }
\langle N^2 t + N|y| \rangle^{-m}, \quad \forall\, m\ge 0.
\]

\item[(ii)] For $t \gtrsim N^{-2}$, $\frac 1 {N^{1+\gamma}} \lesssim |x| \lesssim N^{-1}$, $|y|\sim Nt$, the integral
kernel satisfies
\[
 \bigl|  [P^\pm_N e^{\mp it\Delta}](x,y) \bigr| \lesssim \frac {N^{(1+\gamma)(d-2)+2} \log N}{\langle N|y| \rangle^{1/2} }.
\]
 \end{itemize}
\end{proposition}


\begin{proof}
We shall only provide the proof for $P_N^{+} e^{-it \Delta}$ since the
other kernel is its complex conjugate.
The first claim is an exercise in stationary phase. By Fourier
transform we have the following formula for the kernel
\begin{equation}\label{kernel_highD}
\begin{aligned}
&[P^+_N e^{-it\Delta}](x,y)\\
&=
    \tfrac12 \bigl(|x||y|\bigr)^{-(d-2)/2}
 \int_0^\infty H^{(1)}_{(d-2)/2}(k |x|) J_{(d-2)/2}(k |y|)
    e^{itk^2} \psi\bigl(\tfrac kN\bigr)\,k\,dk
\end{aligned}
\end{equation}
where $\psi$ is the multiplier function from the Littlewood--Paley
projection. First note that
\begin{equation} \label{eq_tmp_1210_1}
 H^{(1)}_{(d-2)/2} (r) = J_{(d-2)/2} (r) + iY_{(d-2)/2}(r).
\end{equation}
Since $k\sim N$, $\frac 1{N^{1+\gamma}} \lesssim |x| \lesssim \frac 1N$,
we have $r=k|x|$ satisfies $\frac 1{N^{\gamma}} \lesssim r \lesssim 1$.
Now using the expansion
\[
 J_{(d-2)/2} (r) = \sum_{m=0}^\infty \frac {(-1)^m} {m! \Gamma(m+\tfrac d2)} \cdot (\tfrac r 2)^{2m+\frac {d-2}2},
\]
we can write
\begin{equation} \label{eq_tmp_1210_2}
 r^{-(d-2)/2} J_{(d-2)/2} (r) = \tilde g_1(r),
\end{equation}
where
\[
 \big| \frac {\partial^m \tilde g_1(r)} {\partial r^m} \big| \lesssim 1,
\quad \forall\, m \ge0, \, r\lesssim 1.
\]
Here the factor $r^{-(d-2)/2}$ in \eqref{eq_tmp_1210_2} is needed
since the dimension $d$ may possibly be a odd integer.
To treat the function $Y_{\frac {d-2} 2}$ in the regime
$\frac 1 {N^{\gamma}} \lesssim r \lesssim 1$, we discuss two
cases. If the dimension $d$ is even, then we use the series
\begin{align*}
 Y_{(d-2)/2} (r)
&= - \frac{ (\tfrac r 2)^{-(d-2)/2}}{\pi}
  \sum_{k=0}^{\frac{d-4}2} \frac{(\tfrac {d-4}2 -k)!}{k!}
 \cdot \big( \frac 14 r^2 \big)^k + \frac 2 {\pi}
\log (\tfrac r 2) J_{(d-2)/2} (r) \\
& \quad - \frac{ (\tfrac r 2)^{\frac{d-2}2}}{\pi}
\sum_{k=0}^\infty \bigl( \psi_0(k+1) + \psi_0(n+k+1) \bigr) \cdot
 \frac {(-\tfrac 14 r^2)^k} {k! (\tfrac{d-2}2 +k)!},
\end{align*}
where $\psi_0$ is the digamma function defined by
\[
 \psi_0(n) = - \gamma_0 + \sum_{k=1}^{n-1} \frac 1k,
\]
and $\gamma_0$ is the Euler-Masheroni constant. It follows easily that
\begin{equation} \label{eq_tmp_1210_3a}
 Y_{(d-2)/2} (r) = r^{-(d-2)/2} \tilde g_2(r) + \log r \cdot r^{\frac{d-2}2} \tilde g_3(r) +r^{\frac{d-2}2} \cdot\tilde g_4(r),
\end{equation}
where
\[
 \big| \frac{\partial^m \tilde g_j (r)}{\partial r^m} \big| \lesssim 1,
\quad \forall\, m\ge0,\,
r\lesssim 1,\; j=2,3,4.
\]
Now if the dimension $d$ is odd, then we use the formula
\[
 Y_{\frac {d-2} 2}(r) = Y_{\frac {d-3}2 +\frac 12} (r)
 = - \frac {2 \cdot (\tfrac r 2)^{\frac{d-2}2}} {\sqrt{ \pi \cdot (\tfrac{d-3}2)!}}
\cdot \Big( 1+\frac{d^2}{dr^2} \Big)^{\frac{d-3}2}
\big( \frac {\cos r}{r} \big).
\]
It follows that we can write
\begin{equation} \label{eq_tmp_1210_3b}
 r^{-(d-2)/2} \cdot Y_{(d-2)/2} (r) = r^{-(d-2)} \tilde g_5(r),
\end{equation}
where
\[
 \big| \frac{\partial^m \tilde g_5 (r)}{\partial r^m} \big| \lesssim 1,
\quad \forall\, m\ge0,\,
r\lesssim 1.
\]

Next we also use the following information
about Bessel functions in the regime $r\gtrsim 1$:
\begin{equation} \label{eq_tmp_1210_4}
J_{(d-2)/2}(r) = \frac{a(r) e^{ir}}{\langle r\rangle^{1/2}}
+ \frac{\bar a(r) e^{-ir}}{\langle r\rangle^{1/2}}\,,
\end{equation}
where $a(r)$ obeys the symbol estimates
\[
\Bigr| \frac{\partial^m a(r)}{\partial r^m} \Bigr| \lesssim \langle r \rangle^{-m}
    \quad \text{for all $m\geq0$, $r\gtrsim 1$}
\]
Finally substitute \eqref{eq_tmp_1210_1}, \eqref{eq_tmp_1210_2}, \eqref{eq_tmp_1210_3a} (when $d$ is even),
\eqref{eq_tmp_1210_3b} (when $d$ is odd), \eqref{eq_tmp_1210_4} into \eqref{kernel_highD}. Consider three regimes
of $y$: $1/N \lesssim |y| \ll N|t|$, $|y|\lesssim 1/N$, $|y| \gg N|t|$ and use different asymptotics of the Bessel
function in these regimes. Note also that the singular part of the Hankel function near $r=0$ adds only a power of $N$
due to our lower bound on $x$. It is then easy to see that a stationary phase point can only occur when $|y| \sim Nt$.
Since we assume $|y|\ll Nt$ or $|y| \gg Nt$, integrating by parts and using Lemma \ref{lem_tmp12} yield the first claim. The second claim follows
from a trivial $L^1$ estimate. We omit
the details.
\end{proof}

\section{The proof of Theorem \ref{main_thm}}

We first explain why it suffices for us to show that such
two way non-scattering solution with minimal mass must be regular: $u_0\in H_x^1$. Indeed, if
$u_0\in H_x^1$ and the corresponding solution blows up at finite time, according to
Merle's result \cite{merle_duke}, we know it must scatter one way which contradicts our
assumption. Then the solution must be global, here a direct application of Theorem
\ref{h1_result} immediately yields the coincidence of the solution with $SW$ up to symmetries.

Since the following proof of upgrading the regularity works for all two-way non-scattering solutions,
for the sake of simplicity, we assume the solution is global. The discussion of the
finite time blowup solutions is only notationally more complicated.

To begin with, we recall the following result.
The proof of this result is implicitly contained in \cite{klvz}.

\begin{lemma}[Regularity of solutions away from the origin,
\cite{klvz}]\label{weak_com}
Let $d\ge 4$. Let $u_0\in L_x^2(\mathbb{R}^d)$ be spherically symmetric and $M(u_0)=M(Q)$.
Let $u(t,x)$ be the corresponding solution such that it does not
scatter in both time directions:
\[
\|u\|_{L_{t,x}^{\frac {2(d+2)}d}((-\infty,0]\times\mathbb{R}^d)}
=\|u\|_{L_{t,x}^{\frac {2(d+2)}d}([0,\infty)\times\mathbb{R}^d)}=\infty.
\]
Then there exists $\epsilon=\epsilon(d)>0$ such that
\[
\|\phi_{>1}P_N u(t)\|_{L_x^2}\lesssim N^{-1-\epsilon}, \ \forall N\ge 1, \ t\in \mathbb{R}.
\]
In particular,
\[
\|\phi_{>1}\nabla u(t)\|_{L_x^2}\lesssim 1, \ \forall t\in \mathbb{R}.
\]
\end{lemma}

Now we use this information to upgrade the regularity of the initial data.
To this end, we seek for the refined decay estimate for single
frequency $P_N u_0$ with $N\ge 1$. Let $\gamma>0$ be a small parameter
to be chosen later, we use triangle inequality to bound
\begin{align}
\|P_N u_0\|_{L_x^2}
&\lesssim \|\phi_{\le N^{-1-\gamma}}  P_N u_0\|_{L_x^2}\label{low_cut}\\
&\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1} P_N u_0\|_{L_x^2}\label{mid_cut}\\
&\quad +\|\phi_{>1}P_N u_0\|_{L_x^2}.\label{hi_cut}
\end{align}
First of all, Lemma \ref{weak_com} yields that
$\eqref{hi_cut}\lesssim N^{-1-\epsilon}$.
Next, using H\"older and Bernstein, \eqref{low_cut} can be controlled
rather easily:
\[
\eqref{hi_cut} \lesssim N^{\frac d2(-1-\gamma)}\|P_N u_0\|_{L_x^{\infty}}
\lesssim N^{-s-\frac d2\gamma}\|u_0\|_{H_x^s}\lesssim N^{-s-\frac d2\gamma}.
\]
The task now is to estimate \eqref{mid_cut}, for which we will use
the in-out decomposition
and the improved Duhamel formula as we explain now. Since the solution
$u$ does not scatter
in both time directions and has minimal mass, according to
\cite{compact, kvz:blowup}
\footnote{The first reference established the improved Duhamel
 formula for minimal-mass non-scattering solution in which the scattering wave vanish when
 the $t$ approaches the maximal life time. The second one identifies $M(Q)$ as the minimal
 mass within all the spherically symmetric solutions.} we have
\begin{align}
u(t)&=\lim_{T\to \infty} -i\int_t^T e^{i(t-s)\Delta}F(u(s))ds\label{duh_for}\\
&=\lim_{T\to -\infty}i\int_T^t e^{i(t-s)\Delta}F(u(s))ds,\label{duh_bak}
\end{align}
where $F(u)=|u|^{4/d}u$ and the limit is understood in the weak
$L_x^2$ sense.
Using the in-out decomposition and \eqref{duh_for}, \eqref{duh_bak},
we estimate
\begin{align}
\eqref{mid_cut}
&\le \|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ u_0\|_{L_x^2}
+\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^- u_0\|_{L_x^2}\notag\\
&\lesssim \|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+\int_0^{\infty}
e^{-i\tau\Delta} F(u(\tau))d\tau
\|_{L_x^2}\label{6}\\
&\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^-\int_0^{\infty} e^{i\tau\Delta} F(u(-\tau)) d\tau\|
_{L_x^2}.\label{7}
\end{align}
Expression \eqref{6} and \eqref{7} will give the same contribution so
we only need to estimate one of
them. By splitting into different time pieces and introducing spatial
cutoffs, we
estimate \eqref{6} as follows
\begin{align}
\eqref{6}
&\lesssim \|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int_{\frac 1N}^\infty
e^{-i\tau\Delta} \phi_{>\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.1}\\
&\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int_{\frac 1N}^\infty
e^{-i\tau\Delta} \phi_{\le\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.2}\\
&\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int^{\frac 1N}_{\frac 1{N^{2-\sigma}}}
e^{-i\tau\Delta} \phi_{>\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.3}\\
&\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int^{\frac 1N}_{\frac 1{N^{2-\sigma}}}
e^{-i\tau\Delta} \phi_{\le\frac {N\tau}2}F(u(\tau))d\tau\|_{L_x^2}\label{6.4}\\
&\quad +\|\phi_{N^{-1-\gamma}<\cdot\le 1}P_N^+ \int_{0}^{\frac 1{N^{2-\sigma}}}
e^{-i\tau\Delta} F(u(\tau))d\tau\|_{L_x^2},\label{6.5}
\end{align}
where $0<\sigma<2$ is a small constant to be fixed later.
We first look at \eqref{6.2}, \eqref{6.4} where the desired decay in
$N$ comes from the kernel estimate Lemma \ref{Psmall_properties}.
Let $A=(1+\gamma)(d-2)+2$, then for any $\tau>\frac 1{N^2}$, $m>0$,
\begin{align*}
\Big| \Bigl( \phi_{N^{-1-\gamma}<\cdot\le 1}
 P_N^+ e^{-i\tau\Delta}
\phi_{\le \frac{N\tau}2} \Bigr) (x,y)\Big|
&\lesssim_m N^A\langle N^2\tau+N|x|+N|y|\rangle^{-2m}\\
&\lesssim_m N^A|N^2\tau|^{-m}\langle N|x-y|\rangle^{-m}.
\end{align*}
Using this and Young's inequality, \eqref{6.2}, \eqref{6.4} can be
bounded as follows
\begin{align*}
\eqref{6.2}
&\lesssim_m N^A \int_{\frac 1N}^{\infty} |N^2\tau|^{-m}\|\langle N|\cdot|\rangle*
F(u(\tau))\|_{L_x^2} d\tau\\
&\lesssim_m N^A N^{-2m}\int_{\frac 1N}^{\infty} \tau^{-m} d\tau \|\langle N|\cdot|\rangle\|
_{L_x^{\frac d{d-2}}}\|F(u)\|_{L_t^{\infty}L_x^{\frac{2d}{d+4}}}\\
&\lesssim_m N^{-m+1+A-d}\|u\|_{L_t^{\infty}L_x^2}^{1+\frac 4d}\\
&\lesssim_m N^{-m+1+A-d}.
\end{align*}
Thus, by taking $m$ large enough depending on $d$,
\[
\eqref{6.2}\lesssim N^{-10}.
\]
Expression \eqref{6.4} can be estimated in a similar way:
\begin{align*}
\eqref{6.4}
&\lesssim_m N^A\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N}|N^2\tau|^{-m}
\|\langle N|\cdot|\rangle* F(u(\tau))\|_{L_x^2} d\tau\\
&\lesssim_m N^{-2m+A+2-d}\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N} \tau^{-m}d\tau \\
&\lesssim_m N^{A-(m-1)\sigma-d}.
\end{align*}
For \eqref{6.1} and \eqref{6.3}, we will use the weighted Strichartz
estimate Lemma \ref{L:wes}.
In what follows we shall only present the details for $d\ge 5$.
The case $d=4$ is similar and will
be omitted. In dimension $d\ge 5$, from the $L_x^2$-boundedness of the operator $\phi_{>\frac 1{N^{1+\gamma}}}P_N^+$
Lemma \ref{L:bdd}, Lemma \ref{L:wes} and Lemma \ref{L:strichartz}, we have
\begin{align}
\eqref{6.1}
&\lesssim N^{\frac{\gamma}2(d-4)}
\Big\|\int_{\frac 1N}^{\infty}e^{-i\tau\Delta}\tilde{P}_N
 \phi_{>\frac{N\tau}2}F(u(\tau))d\tau\Big\|_{L_x^2}\notag\\
&\lesssim N^{\frac{\gamma}2(d-4)}\Big\|\int_{\frac 1N}^{\infty}
e^{-i\tau\Delta}\tilde{P}_N\phi_{>\frac {N\tau}2}F(u\phi_{>\frac{N\tau}4})(\tau)d\tau\Big\|_{L_x^2}\notag\\
&\lesssim N^{\frac{\gamma}2(d-4)}\Big(\Big\|\int_{\frac 1N}^{\infty}
e^{-i\tau\Delta}\tilde{P}_N\phi_{>\frac{N\tau}2}P_{\le N/8}F(u\phi_{>\frac{N\tau}4})(\tau)d\tau\Big\|_{L_x^2}\notag\\
&\quad+\Big\|\int_{\frac 1N}^{\infty}e^{-i\tau\Delta}\tilde{P}_N\phi_{>\frac{N\tau}2}P_{>N/8}
F(u\phi_{>\frac{N\tau}4})(\tau)d\tau\Big\|_{L_x^2}\Big)\notag\\
&\lesssim N^{\frac{\gamma}2(d-4)}\Big(\|\tilde{P}_N \phi_{>\frac{N\tau}2}P_{\le N/8}F(u\phi_{>\frac{N\tau}4})
\|_{L_{\tau}^1L_x^2([\frac 1N,\infty)\times\mathbb{R}^d)}\label{8}\\
&\quad+\|(N\tau)^{-\frac{2(d-1)}d}P_{>N/8}F(u\phi_{>\frac{N\tau}4})\|_{L_{\tau}^{\frac d{d-1}}
L_x^{\frac{2d}{d+4}}([\frac 1N,\infty)\times\mathbb{R}^d)}\Big).\label{9}
\end{align}
Using the mismatch estimate Lemma \ref{L:mismatch_fre}, we can
bound \eqref{8} as
\begin{align*}
\eqref{8}
&\lesssim \|(N^2\tau)^{-11}P_{\le N} F(u\phi_{>\frac{N\tau}4})\|_{L_{\tau}^1L_x^2
([\frac 1N,\infty)\times\mathbb{R}^d)}\\
&\lesssim N^{-20}\|\tau^{-11} F(u\phi_{>\frac{N\tau}4})\|_{L_{\tau}^1L_x^{\frac{2d}{d+4}}
([\frac 1N,\infty)\times\mathbb{R}^d)}\\
&\lesssim N^{-20}\|\tau^{-11}\|_{L_{\tau}^1([\frac 1N,\infty))}\lesssim N^{-10}.
\end{align*}
For \eqref{9}, we use Bernstein estimate and Lemma \ref{weak_com}
to get
\begin{align*}
\eqref{9}
&\lesssim N^{-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}N^{-1}\|
 \nabla F(u\phi_{>\frac{N\tau}4})
\|_{L_x^{\frac{2d}{d+4}}}\|_{L_{\tau}^{\frac d{d-1}}([\frac 1N,\infty))}\\
&\lesssim N^{-1-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}\|u
 \phi_{>\frac{N\tau}4}\|_{L_x^2}^{4/d}
\|\nabla (u\phi_{>\frac{N\tau}4})\|_{L_x^2}\|_{L_{\tau}^{\frac d{d-1}}
 ([\frac 1N,\infty))}\\
&\lesssim N^{-1-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}\|_{L_{\tau}
^{d/(d-1)}(\frac 1N,\infty)}\\
&\lesssim N^{-1-\frac {d-1}d}.
\end{align*}
Therefore, summarizing the two pieces together we have
\[
\eqref{6.1}\lesssim N^{\frac{\gamma}2(d-4)}N^{-1-\frac{d-1}d}.
\]

Now we look at the piece \eqref{6.3} where the uniform kinetic energy estimate Lemma
\ref{weak_com} is no longer available. Instead, we will use the fact $u_0\in H_x^s$, therefore
locally we have the bound
\begin{equation}
\|u\|_{L_{\tau}^{\infty}H_x^s([0,1]\times\mathbb{R}^d)}\lesssim_{u_0} 1.\label{local_hs}
\end{equation}
Using this information, Lemma \ref{L:bdd}, Lemma \ref{L:mismatch_fre} and
Lemma \ref{L:wes}, we control \eqref{6.3} as
\begin{align*}
\eqref{6.3}
&\lesssim N^{\frac{\gamma}2(d-4)}\Big\|\tilde{P}_N\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N}
e^{-i\tau\Delta}\phi_{>\frac{N\tau}2}F(u(\tau))d\tau\Big\|_{L_x^2}\\
&\lesssim N^{\frac{\gamma}2(d-4)}\Big(\Big\|\tilde{P}_N\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N}
e^{-i\tau\Delta}\phi_{>\frac{N\tau}2}P_{\le\frac N8}F(u(\tau))d\tau\Big\|_{L_x^2}\\
&\quad +\Big\|\tilde{P}_N\int_{\frac 1{N^{2-\sigma}}}^{\frac 1N}e^{-i\tau\Delta} \phi_{>\frac{N\tau}2}
P_{>N/8}F(u(\tau))d\tau\Big\|_{L_x^2}\Big)\\
&\lesssim N^{\frac{\gamma}2(d-4)}\Big(\|\tilde{P}_N\phi_{>\frac{N\tau}2}P_{\le N/8}F(u)\|
_{L_{\tau}^1L_x^2([\frac 1{N^{2-\sigma}},\frac 1N]\times\mathbb{R}^d)}\\
&\quad+\||N\tau|^{-\frac{2(d-1)}d}P_{>\frac N8}F(u(\tau))\|_{L_{\tau}^{\frac d{d-1}}
L_x^{\frac{2d}{d+4}}([\frac 1{N^{2-\sigma}},\frac 1N]\times\mathbb{R}^d)}\Big)\\
&\lesssim_m N^{\frac{\gamma}2(d-4)}
\Big(\||N^2\tau|^{-m}\|P_{\le N/8}F(u)\|_{L_x^2}\|_{L_{\tau}^1([\frac 1{N^{2-\sigma}},\frac 1N])}\\
&\quad+\||N\tau|^{-\frac{2(d-1)}d}N^{-s}\||\nabla|^s P_{> N/8}F(u)\|_{L_x^{\frac{2d}{d+4}}}
\|_{L_{\tau}^{\frac d{d-1}}([\frac 1{N^{2-\sigma}},\frac 1N])}\\
&\lesssim_{m,u_0} N^{\frac{\gamma}2(d-4)}(N^{2-2m}\|
 \tau^{-m}\|_{L_{\tau}^1([\frac 1{N^{2-\sigma}},\frac 1N])} \\
&\quad +N^{-s-\frac{2(d-1)}d}\|\tau^{-\frac{2(d-1)}d}\|_{L_{\tau}^{\frac d{d-1}}([\frac 1{N^{2-\sigma}},
\frac 1N])}\Big)\\
&\lesssim_{m,u_0} N^{\frac{\gamma}2(d-4)}(N^{-\sigma(m-1)}+N^{-s-\frac{d-1}d\sigma}).
\end{align*}
Finally, we give the estimate of \eqref{6.5} which can firstly be
 trivially bounded as
\begin{align}
\eqref{6.5}
&\lesssim \|\phi_{N^{-1-\gamma}<\cdot\le 1}\int_0^{\frac 1{N^{2-\sigma}}}
P_N^+ e^{-i\tau\Delta}\phi_{>1}\tilde{P}_N F(u(\tau))d\tau
\|_{L_x^2}\label{6.6}\\
&\quad+\|\phi_{N^{-1-\gamma}<\cdot\le 1}\int_0^{\frac 1{N^{2-\sigma}}}
P_N^+ e^{-i\tau\Delta}\phi_{\le 1}\tilde{P}_N F(u(\tau))
d\tau\|_{L_x^2}.
\label{6.7}
\end{align}

For \eqref{6.6}, we use the $L_x^2$ boundedness Lemma \ref{L:bdd},
weighted Strichartz Lemma \ref{L:wes}, Bernstein and
local estimate \eqref{local_hs} to get
\begin{align*}
\eqref{6.6}&\lesssim N^{\frac \gamma 2(d-4)}\|\int_0^{\frac 1{N^{2-\sigma}}}e^{-i\tau\Delta}\phi_{>1}\tilde{P}_N
F(u(\tau))d\tau\|_{L_x^2}\\
&\lesssim N^{\frac{\gamma}2(d-4)}\|\tilde{P}_N F(u)\|_{L_{\tau}^{\frac d{d-1}}L_x^{\frac{2d}{d+4}}([0,
\frac 1{N^{2-\sigma}}]\times\mathbb{R}^d)}\\
&\lesssim N^{\frac{\gamma}2(d-4)}N^{-(2-\sigma)\frac{d-1}d}N^{-s}
\||\nabla|^s F(u)\|_{L_{\tau}^{\infty}L_x^{\frac{2d}{d+4}}}\\
&\lesssim_{u_0} N^{\frac{\gamma}2(d-4)}N^{-\frac{(2-\sigma)(d-1)}d}N^{-s}.
\end{align*}
To estimate \eqref{6.7}, we will use the duality of the smoothing
estimate \eqref{smoothing} as follows:
\begin{equation}
\|\int_{\mathbb{R}} e^{-i\tau\Delta} |\nabla|^{1/2} f(\tau)d\tau\|_{L_x^2}\lesssim
\||x|^{-\frac{d-1}2}f\|_{L_x^1L_t^2(\mathbb{R}^d\times\mathbb{R})}.\label{dual}
\end{equation}
Let $\eta>0$ be a tiny number to be chosen later, using Lemma \ref{L:bdd}
and \eqref{dual} we have
\begin{align*}
\eqref{6.7}&\lesssim N^{\frac{\gamma}2(d-4)}N^{-1/2}\|\int_{\mathbb{R}}
e^{-i\tau\Delta}|\nabla|^{1/2}(\phi_{\le 1}\tilde{P}_N F(u(\tau))\chi_{0<\tau\le\frac 1{N^{2-\sigma}}})
d\tau\|_{L_x^2}\\
&\lesssim N^{\frac{\gamma}2(d-4)}N^{-1/2}\||x|^{-\frac{d-1} 2}\phi_{\le 1}\tilde{P}_N F(u)\|
_{L_x^1L_{\tau}^2(\mathbb{R}^d\times[0,\frac 1{N^{2-\sigma}}])}\\
&\lesssim N^{\frac{\gamma}2(d-4)}N^{-1/2}\||x|^{\frac 12-\eta}\tilde{P}_N F(u)\|_{L_{\tau,x}^2([0,
\frac 1{N^{2-\sigma}}]\times\mathbb{R}^d)}\||x|^{-\frac d2+\eta}\phi_{\le 1}\|_{L_x^2}.\\
&\lesssim  N^{\frac{\gamma}2(d-4)-\frac 12-\frac{2-\sigma}2}\||x|^{\frac 12-\eta}\tilde{P}_N
F(u)\|_{L_{\tau}^{\infty}L_x^2([0,\frac 1{N^{2-\sigma}}]\times\mathbb{R}^d)}.
\end{align*}
Now, using the radial Sobolev embedding Lemma \ref{L:radial_embed},
Bernstein,
and \eqref{local_hs}, we bound the $F(u)$ term as
\begin{align*}
\||x|^{\frac 12-\eta}\tilde{P}_N F(u)\|_{L_x^2}&\lesssim \||\nabla|^{\eta}\tilde{P}_N F(u)\|_
{L_x^{\frac{2d}{d+1}}}\\
&\lesssim N^{\eta}N^{d(\frac{d^2+4d-8s}{2d^2}-\frac{d+1}{2d})}\|\tilde{P}_N F(u)\|_{L_x^{\frac{2d^2}
{d^2+4d-8s}}}\\
&\lesssim N^{\eta+\frac{3d-8s}{2d}}N^{-s}\||\nabla|^s u\|_{L_x^2}\|u\|_{L_x^{\frac{2d}{d-2s}}}^{4/d}\\
&\lesssim_{u_0} N^{\eta+\frac 32-\frac{4s}d-s}.
\end{align*}
Plugging in this estimate back to the estimate of \eqref{6.7} we have
\[
\eqref{6.7}\lesssim_{u_0} N^{\frac{\gamma}2(d-4)+\eta
 +\frac{\sigma}2-\frac{4s}d-s}.
\]
Combining the estimate for \eqref{6.6} and \eqref{6.7} together gives
the final estimate of \eqref{6.5}:
\[
\eqref{6.5}\lesssim_{u_0} N^{\frac{\gamma}2(d-4)}(N^{-\frac{(2-\sigma)(d-1)}d -s}
+N^{\eta+\frac{\sigma}2-\frac{4s}d-s}).
\]
Now adding all the estimate for \eqref{6.1} through \eqref{6.5},
we finish estimating the term \eqref{6}. This together with the
estimate for \eqref{low_cut} and \eqref{hi_cut} finally gives that
\begin{align*}
\|P_N u_0\|_{L_x^2} &\lesssim_{m,u_0} N^{-1-\epsilon}+N^{-s-\frac d2\gamma}
+N^{(1+\gamma)(d-2)+2-d-\sigma(m-1)} \\
&\quad+N^{\frac{\gamma} 2(d-4)}(N^{-\frac{d-1}d\sigma-s}+N^{-\sigma(m-1)}
+N^{-\frac{d-1}d(2-\sigma) -s}
+N^{\eta+\frac{\sigma}2-\frac{4s}d-s}).
\end{align*}
For any $s>0$, choosing $\sigma=\frac s{100d}$, $\eta=\frac s{1000d}$, $\gamma=\frac s{1000d^2}$,
$m=1+\frac{200d}s$, we finally obtain
\[
\|P_N u_0\|_{L_x^2}\lesssim_{u_0}N^{-1-\epsilon}+N^{-s-\frac s{2000d}}.
\]
It is easy to see that after finite many times of iteration we obtain
\[
\|P_N u_0\|_{L_x^2}\lesssim_{u_0} N^{-1-},\ \forall N\ge 1.
\]
Therefore, $u_0\in H_x^1$. The proof of Theorem \ref{main_thm} is
complete.

\subsection*{Acknowledgements}
Both authors were supported by the National Science Foundation under
agreement No. DMS-0635607 and start-up funding from the Mathematics
Department of University of Iowa.
X.~Zhang was also supported by NSF grant No.~10601060
and project 973 in China.


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\end{document}