Next, we prove that for $y\gg 1$ and
$f,g\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ satisfying (4.37),

$\langle
f,g\rangle_{\dot{H}^{\sigma}}=\frac{-i}{2\pi}\int_{\mathbb{R}}\left\langle\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma+iy)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma-
iy)\right)f,g\right\rangle_{\dot{H}^{\sigma}}d\lambda.$ (4.41)

We first choose $y$ large enough such that
$\|\mathcal{S}_{b}^{\pm}(z+ib\sigma\pm
iy)\|_{\mathcal{L}(\dot{H}^{\sigma})}\ll 1$ by Lemma 3.2, leading to
$(I+V_{2}\mathcal{S}_{b}^{\pm}(\lambda+ib\sigma\pm iy)V_{1})^{-1}$ and hence
$\mathcal{S}_{b,V}^{\pm}(\lambda+ib\sigma\pm iy)$ uniformly bounded in
$\mathcal{L}(\dot{H}^{\sigma})$ for $\lambda\in\mathbb{R}$. The formula (4.41)
therefore makes sense. Increase $y$ if necessary to make
$y>1+\|D^{\sigma}VD^{-\sigma}\|_{\mathcal{L}(L^{2})}$. So for any
$f,g\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ satisfying (4.37), we consider
the function of $t$

$\langle
e^{it({\mathcal{H}}_{b}-ib\sigma+V)}e^{-y|t|}f,g\rangle_{\dot{H}^{\sigma}}=\langle
e^{it({\mathcal{H}}-ib(\sigma-s_{c})}e^{-y|t|}f,g\rangle_{\dot{H}^{\sigma}}$

which belongs to $L^{1}(\mathbb{R})$ due to its exponential decay at
$\pm\infty$. Using the integral representation of resolvents (like (4.10)),
the Fourier transform of this function is

$\int_{\mathbb{R}}e^{-it\lambda}\langle e^{it({\mathcal{H}}-ib(\sigma-
s_{c})}e^{-y|t|}f,g\rangle_{\dot{H}^{\sigma}}dt=-i\left\langle\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma+iy)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma-
iy)\right)f,g\right\rangle_{\dot{H}^{\sigma}}$ (4.42)

We claim that this is again in $L^{1}(\mathbb{R})$. Indeed, we first expand
$\mathcal{S}_{b,V}^{\pm}(z)$ with (4.24) to obtain

$\displaystyle\mathrm{RHS\,\,of\,\,\eqref{eqpropr31}}=-i\left\langle\left(\mathcal{S}_{b}^{+}(\lambda+iy)-\mathcal{S}_{b}^{-}(\lambda-
iy)\right)D^{\sigma}f,D^{\sigma}g\right\rangle_{L^{2}}$ $\displaystyle+$
$\displaystyle i\sum_{\pm}\pm\left\langle\mathcal{S}_{b}^{\pm}(\lambda\pm
iy)D^{\sigma}V_{1}(I+V_{2}\mathcal{S}_{b}^{\pm}(\lambda+ib\sigma\pm
iy)V_{1})^{-1}V_{2}D^{-\sigma}\mathcal{S}_{b}^{\pm}(\lambda\pm
iy)D^{\sigma}f,D^{\sigma}g\right\rangle_{L^{2}}.$

Now we apply Lemma 3.3 with $p=p_{0}$ to gain decay in $\lambda$. Apply (3.19)
to the first term and (3.18) to the last two terms after transposing the first
$\mathcal{S}_{b}^{\pm}(\lambda\pm iy)$ onto $D^{\sigma}g$. Together with the
uniform boundedness of $(I+V_{2}\mathcal{S}_{b}^{\pm}(\lambda+ib\sigma\pm
iy)V_{1})^{-1}$ in $\mathcal{L}(\dot{H}^{\sigma})$ and
$D^{\sigma}V_{i}D^{-\sigma}\in\mathcal{L}(L^{p_{0}}+L^{2}\to
L^{2})\cap\mathcal{L}(L^{2}\to L^{p_{0}^{\prime}}\cap L^{2})$, for any
$p\in(2,p_{0}]$, we have

$\displaystyle\left|\mathrm{RHS\,\,of\,\,\eqref{eqpropr31}}\right|\lesssim\langle\lambda\rangle^{-2}$
$\displaystyle\left(\|D^{\sigma}f\|_{L^{p}\cap
L^{p^{\prime}}}+\|{\mathcal{H}}_{b}D^{\sigma}f\|_{L^{p^{\prime}}}+\|{\mathcal{H}}_{b}^{2}D^{\sigma}f\|_{L^{p^{\prime}}}\right)$
(4.43) $\displaystyle\cdot\left(\|D^{\sigma}g\|_{L^{p}\cap
L^{p^{\prime}}}+\|{\mathcal{H}}_{b}D^{\sigma}g\|_{L^{p^{\prime}}}\right)\in
L^{1}(\mathbb{R})$

So (4.42) implies (4.41) through the Fourier inversion formula. Note that this
estimate also holds for $y\geq 0$ whenever $\\{\mathbb{R}+i(b\sigma-bs_{c}\pm
y)\\}$ does not intersect with $\tilde{{\mathcal{E}}}^{\pm}_{\sigma}$.

Finally, we will shift the integration contour in (4.41) toward the essential
spectrum $\\{\mathbb{R}+ib(\sigma-s_{c})\\}$. Lemma 4.4 (5) guaranteed the
analyticity of
$\left\langle\mathcal{S}_{b,V}^{\pm}(z+ibs_{c})f,g\right\rangle_{\dot{H}^{\sigma}}$
on the corresponding half-plane except $\tilde{{\mathcal{E}}}^{\pm}_{\sigma}$,
so we take contours consisting of boundaries of $[-R,R]\times[0,y]$,
$[-R,R]\times[-y,0]$ and small circular contours around all exceptional
values. Lemma 3.2 and (4.43) with $y=0$ guarantees the residual to vanish when
$R$ goes to $+\infty$, so we finally arrived at

$\displaystyle\langle f,g\rangle_{\dot{H}^{\sigma}}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\left\langle\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma)\right)f,g\right\rangle_{\dot{H}^{\sigma}}d\lambda+\sum_{\xi\in\tilde{{\mathcal{E}}}}\langle
P_{\xi}f,g\rangle_{\dot{H}^{\sigma}}$

which infers (4.38). ∎

At the end of this step, we use this representation formula to show $P_{\rm
ess}$ can remove the singularity of the resolvent
$\mathcal{S}_{b,V}^{-}(z+ibs_{c})$.

######  Corollary 4.7.

Assume 4.1 and take $\delta_{1}$ from Lemma 4.3 and $p_{1}$ from Proposition
4.5. For $\sigma\in(s_{c},s_{c}+\delta_{1})$, we have for any $p\in(2,p_{1}]$,
$\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm ess}$ is a well-defined bounded
operator from $\dot{W}^{\sigma,p^{\prime}}$ to $\dot{W}^{\sigma,p}$ for $\Im
z\leq b(\sigma-s_{c})$, and has the uniform bound

$\mathop{\rm sup}_{z:\Im z\leq b(\sigma-
s_{c})}\left\|\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm
ess}\right\|_{\dot{W}^{\sigma,p^{\prime}}\to\dot{W}^{\sigma,p}}<\infty.$
(4.44)

###### Proof.

Fix a $p\in(2,p_{1}]$. We will use the representation formula (4.38),
boundedness of $P_{\rm ess}\in\mathcal{L}(\dot{W}^{\sigma,p^{\prime}})$ and
resolvent identities (4.30) (4.31). Recalling the boundedness (4.26), we only
need to estimate for $z$ close to $\tilde{{\mathcal{E}}}^{-}_{\sigma}$. In
particular, we can also assume $\Im z-b(\sigma-s_{c})\leq-\delta_{2}$ for some
$\delta_{2}>0$.

Firstly, we check the validity of the representation formula

$\displaystyle\left\langle\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm
ess}f,g\right\rangle_{\dot{H}^{\sigma}}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\left\langle\mathcal{S}_{b,V}^{-}(z+ibs_{c})\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma)\right)f,g\right\rangle_{\dot{H}^{\sigma}}$
(4.45)

for $\Im z\leq b(\sigma-s_{c})-\delta_{2}$,
$z\notin\tilde{{\mathcal{E}}}^{-}_{\sigma}$ and $f,g\in\mathscr{S}$.
Since666Here the adjoint operator are in the sense of $L^{2}$ rather than
$\dot{H}^{\sigma}$.

$\displaystyle\left\langle\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm
ess}f,g\right\rangle_{\dot{H}^{\sigma}}=\left\langle
D^{\sigma}\mathcal{S}_{b,V}^{-}(z+ibs_{c})D^{-\sigma}D^{\sigma}P_{\rm
ess}f,D^{\sigma}g\right\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle\left\langle D^{\sigma}P_{\rm
ess}f,(D^{\sigma}\mathcal{S}_{b,V}^{-}(z+ibs_{c})D^{-\sigma})^{*}D^{\sigma}g\right\rangle_{L^{2}}$
$\displaystyle=$ $\displaystyle\left\langle P_{\rm
ess}f,D^{-\sigma}(D^{\sigma}\mathcal{S}_{b,V}^{-}(z+ibs_{c})D^{-\sigma})^{*}D^{\sigma}g\right\rangle_{\dot{H}^{\sigma}}$

It suffices to prove $f$ and
$G:=D^{-\sigma}(D^{\sigma}\mathcal{S}_{b,V}^{-}(z+ibs_{c})D^{-\sigma})^{*}D^{\sigma}g$
satisfying (4.37). From the resolvent property Lemma 4.4 (2), we have

$\displaystyle(D^{\sigma}\mathcal{S}_{b,V}^{-}(z+ibs_{c})D^{-\sigma})^{*}=\left[({\mathcal{H}}_{b}+ib(\sigma-
s_{c})-z+D^{\sigma}VD^{-\sigma})^{-1}\right]^{*}$ $\displaystyle=$
$\displaystyle({\mathcal{H}}_{b}-ib(\sigma-
s_{c})-{\overline{z}}+D^{-\sigma}{\overline{V}}^{T}D^{\sigma})^{-1}=:({\mathcal{H}}_{b}-\zeta+A)^{-1}\in\mathcal{L}(L^{2})$

where $A:=D^{-\sigma}{\overline{V}}^{T}D^{\sigma}$ and $\zeta:=ib(\sigma-
s_{c})+{\overline{z}}$ which satisfies $\Im\zeta\leq-\delta_{2}$. From (B.4),
we know for $0\leq\gamma\ll 1$, $A\in\mathcal{L}(L^{2}\to L^{2}_{\langle
x\rangle^{\gamma}})$. Now from resolvent identities

$({\mathcal{H}}_{b}-\zeta+A)^{-1}-\mathcal{S}_{b}(\zeta)=-({\mathcal{H}}_{b}-\zeta+A)^{-1}A\mathcal{S}_{b}(\zeta)=-\mathcal{S}_{b}(\zeta)A({\mathcal{H}}_{b}-\zeta+A)^{-1},$

we have

$({\mathcal{H}}_{b}-\zeta+A)^{-1}=\mathcal{S}_{b}(\zeta)-\mathcal{S}_{b}(\zeta)A\mathcal{S}_{b}(\zeta)+\mathcal{S}_{b}(\zeta)A({\mathcal{H}}_{b}-\zeta+A)^{-1}A\mathcal{S}_{b}(\zeta).$

Using Lemma 2.13, the boundedness of $A$ above and
$g\in\mathscr{S}(\mathbb{R}^{d})$, we can find $0<\tilde{\gamma}\ll 1$,
$0<\tilde{p}-2\ll 1$ such that
$D^{\sigma}G=({\mathcal{H}}_{b}-\zeta+A)^{-1}D^{\sigma}g\in L^{2}_{\langle
x\rangle^{\tilde{\gamma}}}\cap L^{\tilde{p}}$. Also noticing that
${\mathcal{H}}_{b}({\mathcal{H}}_{b}-\zeta+A)^{-1}=I+(\zeta-A)({\mathcal{H}}_{b}-\zeta+A)^{-1}$,
we know $G$ satisfies
$G\in\dot{W}^{\sigma,\tilde{p}}\cap\dot{W}^{\sigma,\tilde{p}^{\prime}}$ and
${\mathcal{H}}_{b}D^{\sigma}G\in L^{\tilde{p}^{\prime}}$. With
$f\in\mathscr{S}$, this confirms $f$ and $G$ satisfying (4.37) and therefore
the validity of (4.45).

Now for $z_{0}$ with $\Im z_{0}\leq b(\sigma-s_{c})-\delta_{2}$,
$z_{0}\notin\tilde{{\mathcal{E}}}^{-}_{\sigma}$, we apply (4.45) and the
resolvent property of $\mathcal{S}_{b,V}^{-}(z_{0}+ibs_{c})$ to get

$\displaystyle\left\langle\mathcal{S}_{b,V}^{-}(z_{0}+ibs_{c})P_{\rm
ess}f,g\right\rangle_{\dot{H}^{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{R}}\left\langle\mathcal{S}_{b,V}^{-}(z_{0}+ibs_{c})\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma)\right)f,g\right\rangle_{\dot{H}^{\sigma}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{R}}\left\langle\frac{1}{z_{0}-(\lambda+ib(\sigma-
s_{c}))}\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma)\right)f,g\right\rangle_{\dot{H}^{\sigma}}$

where the integral converges absolutely due to Proposition 4.6. Now fix an
arbitrary $z_{1}\in\\{\Im z\leq b(\sigma-
s_{c})-\delta_{2}\\}-\tilde{{\mathcal{E}}}^{-}_{\sigma}$ so that
$\mathcal{S}_{b,V}^{-}(z_{0}+ibs_{c})P_{\rm ess}$ is bounded from
$\dot{W}^{\sigma,p^{\prime}}$ to $\dot{W}^{\sigma,p}$. For any other
$z_{2}\in\\{\Im z\leq b(\sigma-s_{c})-\delta_{2}\\}$, we have

$\displaystyle\left\langle\mathcal{S}_{b,V}^{-}(z_{1}+ibs_{c})P_{\rm
ess}f,g\right\rangle_{\dot{H}^{\sigma}}-\left\langle\mathcal{S}_{b,V}^{-}(z_{2}+ibs_{c})P_{\rm
ess}f,g\right\rangle_{\dot{H}^{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{R}}\bigg{\langle}\frac{z_{2}-z_{1}}{[z_{1}-(\lambda+ib(\sigma-
s_{c}))][z_{2}-(\lambda+ib(\sigma-s_{c}))]}$
$\displaystyle\quad\quad\quad\left(\mathcal{S}_{b,V}^{+}(\lambda+ib\sigma)-\mathcal{S}_{b,V}^{-}(\lambda+ib\sigma)\right)f,g\bigg{\rangle}_{\dot{H}^{\sigma}}.$

The $\langle\lambda\rangle^{-2}$ decay and uniform boundedness of
$\mathcal{S}_{b,V}^{\pm}(\lambda+ib\sigma)\in\mathcal{L}(\dot{W}^{\sigma,p^{\prime}}\to\dot{W}^{\sigma,p})$
indicate the boundedness of $\mathcal{S}_{b,V}^{-}(z_{2}+ibs_{c})P_{\rm ess}$,
particularly when $z_{2}$ is close to $\tilde{{\mathcal{E}}}^{-}_{\sigma}$.
This enables us to extend the definition
$\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm ess}$ to
$z\in\tilde{{\mathcal{E}}_{\sigma}}$ and implies the uniform boundedness. ∎

##  5\. Abstract Wiener’s Theorem and Proof of Strichartz estimates

In this section, we will prove Theorem 1.2. We first recall the abstract
Wiener’s theorem of Beceanu from [6] (also [5]), and then present the proof of
our Strichartz estimates by verifying conditions for a modified Schrödinger
system.

###  5.1. Abstract Wiener’s Theorem

In this subsection, we introduce the setting and abstract tool in Beceanu’s
work [6, 5], and then present a version for our later use.

Let $H$ be a complex Hilbert space, $\mathcal{L}(H)$ be the space of bounded
linear operators from $H$ to itself, and $L^{1}_{t}\mathcal{L}(H)$ be the
Banach space of operator-valued $L^{1}$ functions on $\mathbb{R}$. We add a
convolutional unit

$I_{K}:=\delta_{t=0}I_{H}$

to define the normed vector space $K$

$\displaystyle K$ $\displaystyle:=$ $\displaystyle
L^{1}_{t}\mathcal{L}(H)\oplus\mathbb{C}\\{I_{K}\\};$ (5.1)
$\displaystyle\|\mu+aI_{K}\|_{K}$ $\displaystyle:=$
$\displaystyle\|\mu\|_{L^{1}_{t}\mathcal{L}(H)}+|a|,\qquad\forall
a\in\mathbb{C},\,\,\mu\in L^{1}_{t}\mathcal{L}(H).$ (5.2)

Then it satisfies the following properties:

######  Lemma 5.1 (Properties of $K$).

Let $K$ defined as (5.1). Then the following statements are true.

  1. (1)

$K$ is an unital Banach algebra.

  2. (2)

Multiplication by bounded continuous functions is bounded on $K$:

$\|fk\|_{K}\leq\|f\|_{C^{0}(\mathbb{R})}\|k\|_{K}.$ (5.3)

  3. (3)

If $k=\mu(t)+aI_{K}\in K$, then $k^{*}=\mu(t)^{*}+{\overline{a}}I_{K}\in K$ as
well.

  4. (4)

The Fourier transform is well-defined by

$\hat{k}(\lambda):=\int_{\mathbb{R}}e^{-it\lambda}k(t)dt,$ (5.4)

which satisfies $\|\hat{k}\|_{C^{0}_{\lambda}\mathcal{L}(H)}\leq\|k\|_{K}$.

###### Proof.

The algebra structure is given by $L^{1}_{t}$-convolution

$(\mu+aI_{K})*(\nu+bI_{K})=\mu*\nu+a\nu+b\mu+abI_{K}.$

Then all these properties are trivial. ∎

We stress that the invertibility of $k\in K$ means there exists $k^{-1}\in K$
such that $k*k^{-1}=k^{-1}*k=I_{K}$.

Now we are in place to state the abstract Wiener’s theorem, a sufficient
condition for $k\in K$ to be invertible characterized by invertibility of
$\hat{k}(\lambda)\in\mathcal{L}(H)$ for all $\lambda$. We also collect the
criterion for the upper triangular property (meaning that $A\in K$ is
supported on $[0,\infty)$) to be preserved under inversion.

######  Theorem 5.2 ([5, Theorem 3.3], [6, Theorem 2.3], ).

Let $K$ be the operator algebra of (5.1). If $A\in K$ is invertible, then
$\hat{A}(\lambda)\in\mathcal{L}(H)$ is invertible for every
$\lambda\in\mathbb{R}$. Conversely, assume $\hat{A}(\lambda)$ is invertible
for each $\lambda\in\mathbb{R}$, $A-I_{K}=L\in L^{1}_{t}(\mathcal{L}(H))$, and

$\mathop{\rm lim}_{\varepsilon\to
0}\|L(\cdot+\varepsilon)-L\|_{K}=0,\quad\mathop{\rm
lim}_{R\to\infty}\|1-\psi(t/R)L(t)\|_{K}=0$ (5.5)

where $\psi\in C^{\infty}_{c}([-2,2])$ satisfies $\psi(t)=1$ for $t\in[-1,1]$.
Then $A$ is invertible and $A^{-1}-I_{K}\in L^{1}_{t}(\mathcal{L}(H))$.

######  Lemma 5.3 ([5, Lemma 3.4], [6, Lemma 2.5]).

Given $A\in K$ upper triangular with $A^{-1}\in K$, $A^{-1}$ is upper
triangular if and only if $\hat{A}$ can be extended to a weakly analytic
family of invertible operators in the lower half-plane (namely it is
continuous up to the boundary, uniformly bounded, and has uniformly bounded
inverse).

The following lemma was also implicitly used in Beceanu’s work [6]. It bridges
such invertibility with our Strichartz estimates. We present a proof for
completeness.

######  Lemma 5.4.

If $A,A^{-1}\in K$ and both are upper triangular, then
$\int_{0}^{t}A(t-s)ds\in\mathcal{L}(L^{2}_{t}([0,\infty),H))$ has an inversion
$\int_{0}^{t}A^{-1}(t-s)ds\in\mathcal{L}(L^{2}_{t}([0,\infty),H))$.

###### Proof.

We first show the boundedness. Since $A=I_{K}+L$ with $L\in
L^{1}_{t}\mathcal{L}(H)$, for $u\in L^{2}_{t}([0,\infty),H)$,

$\displaystyle\left\|\int_{0}^{t}A(t-s)u(s)ds\right\|_{L^{2}_{t}([0,\infty),H)}$
$\displaystyle\leq$
$\displaystyle\left\|\int_{0}^{t}\|L(t-s)\|_{\mathcal{L}(H)}\|u(s)\|_{H}ds\right\|_{L^{2}_{t}(\mathbb{R})}+\|u\|_{L^{2}_{t}([0,\infty),H)}=\|A\|_{K}\|u\|_{L^{2}_{t}([0,\infty),H)}$

from Young’s inequality. And similar for $\int_{0}^{t}A^{-1}(t-s)ds$. That
$\int_{0}^{t}A^{-1}(t-s)ds$ is the inverse of $\int_{0}^{t}A(t-s)ds$ can be
checked using $A^{-1}*A=A*A^{-1}=I_{K}$. ∎

To conclude this subsection, we present the following version of abstract
Wiener’s theorem for operators with exponential decay in $t$.

######  Corollary 5.5 (Abstract Wiener’s theorem with exponential decay).

Let $K$ be the Banach algebra of (5.1) and $\varepsilon>0$. If $A=I_{K}+L\in
K$ is upper triangular with $L\in L^{1}_{t}\mathcal{L}(H)$ and satisfies

  1. (1)

$\|L\|_{L^{1}_{t}e^{\varepsilon
t}\mathcal{L}(H)}=\int_{0}^{\infty}\|L(t)\|_{\mathcal{L}(H)}e^{\varepsilon
t}dt<\infty$.

  2. (2)

$\mathop{\rm lim}_{\epsilon\to
0}\|L(\cdot+\epsilon)-L\|_{L^{1}_{t}e^{\varepsilon t}\mathcal{L}(H)}=0.$

  3. (3)

The Fourier transform $\hat{A}(\lambda)$ can be extended to a weakly analytic
family of invertible operators on $H$ for $\\{z\in\mathbb{C}:\Im
z\leq\varepsilon\\}$.

Then $\int_{0}^{t}A(t-s)ds\in\mathcal{L}(L^{2}_{t}([0,\infty),e^{\varepsilon
t}H))$ has inversion
$\int_{0}^{t}A^{-1}(t-s)ds\in\mathcal{L}(L^{2}_{t}([0,\infty),e^{\varepsilon
t}H))$.

###### Proof.

Let $B(t):=A(t)e^{\varepsilon t}$. Then $B=I_{K}+L_{B}$ is upper triangular
with $L_{B}\in L^{1}_{t}\mathcal{L}(H)$. Since

$\hat{B}(\lambda)=\int_{0}^{\infty}e^{-it\lambda}e^{\varepsilon
t}A(t)dt=\hat{A}(\lambda+i\varepsilon)$

is a weak analytic family in the lower-half plane, $B$ meets the conditions of
Theorem 5.2 (the second limit in (5.5) follows (1) here), Lemma 5.3 and Lemma
5.4. Hence there exists $B^{-1}\in K$. For $\mu,\nu\in K$, a computation as in
Lemma 5.4 indicates

$\displaystyle\left((e^{-\varepsilon(\cdot)}\mu)*(e^{-\varepsilon(\cdot)}\nu)\right)(t)=\int_{0}^{t}e^{-\varepsilon(t-\tau)}(\mu*\nu)(t-\tau)d\tau.$

Thus $B^{-1}*(e^{\varepsilon(\cdot)}A)=(e^{\varepsilon(\cdot)}A)*B^{-1}=I_{K}$
implies
$(e^{-\varepsilon(\cdot)}B^{-1})*A=A*(e^{-\varepsilon(\cdot)}B^{-1})=I_{K}$.
By the uniqueness of inversion in $K$, we know
$B^{-1}=e^{\varepsilon(\cdot)}A^{-1}\in L^{1}_{t}\mathcal{L}(H)$. Finally, for
the boundedness, by acting $\int_{0}^{t}e^{\varepsilon(t-s)}A(t-s)ds$ on
$e^{\varepsilon(\cdot)}u$, Lemma 5.4 indicates

$\left\|\int_{0}^{t}A(t-s)ds\right\|_{\mathcal{L}(L^{2}_{t}([0,\infty),e^{\varepsilon
t}H))}\leq\|e^{\varepsilon(\cdot)}A\|_{L^{1}_{t}\mathcal{L}(H)}<\infty.$

Replacing $A$ by $A^{-1}$, we obtain the same bound which concludes the proof.
∎

###  5.2. Modified Schrödinger system and Proof of Theorem 1.2

We consider the modified system

$i\partial_{t}\tilde{Z}+{\mathcal{H}}P_{\rm ess}\tilde{Z}+i\mu P_{\rm
disc}\tilde{Z}=\tilde{F}$ (5.6)

where $\mu>b(\sigma-s_{c})$ and $P_{\rm disc}$ and $P_{\rm ess}$ are spectral
projections of ${\mathcal{H}}$ defined in (4.36). Our aim of this section is
to apply Corollary 4.7 to verify the condition in Corollary 5.5 and prove the
Strichartz estimate.

As a preparation, since

${\mathcal{H}}P_{\rm ess}+i\mu P_{\rm disc}={\mathcal{H}}_{b}-ibs_{c}+V-P_{\rm
disc}({\mathcal{H}}-i\mu)=:{\mathcal{H}}_{b}-ibs_{c}+\tilde{V},$ (5.7)

we first derive a decomposition of the modified potential $\tilde{V}:=V-P_{\rm
disc}({\mathcal{H}}-i\mu)$.

######  Lemma 5.6 (Decomposition of modified potential).

Assume (4.1) and take $\delta_{1}$ from Lemma 4.3 and $p_{1}$ from Proposition
4.5. For $\sigma\in(s_{c},s_{c}+\delta_{1})$ and $\mu>b(\sigma-s_{c})$, there
exist $p_{2}\in(2,2^{*})$, $\gamma_{2}>0$ and a decomposition

$\tilde{V}=\tilde{V}_{1}\tilde{V}_{2}$ (5.8)

such that
$\tilde{V_{1}}\in\mathcal{L}(\dot{H}^{\sigma},\dot{W}^{\sigma,p_{2}^{\prime}}\cap\dot{H}^{\sigma}_{\langle
x\rangle^{\gamma_{2}}})$ and
$\tilde{V}_{2}\in\mathcal{L}(\dot{W}^{\sigma,p_{2}}+\dot{H}^{\sigma},\dot{H}^{\sigma})$.

###### Proof.

We will specify $0<\gamma_{1}<\min\\{\alpha,\gamma_{0}\\}$ with $\gamma_{0}$
from Proposition 4.5 and decompose trivially

$\tilde{V_{1}}=\langle x\rangle^{-\gamma_{1}},\quad\tilde{V_{2}}=\langle
x\rangle^{\gamma_{1}}\left(V-P_{\rm disc}({\mathcal{H}}-i\mu)\right).$ (5.9)

Lemma B.1 and (B.3) implies
$\tilde{V}_{1}\in\mathcal{L}(\dot{H}^{\sigma},\dot{W}^{\sigma,p_{2}^{\prime}}\cap\dot{H}^{\sigma}_{\langle
x\rangle^{\gamma_{2}}})$ and $\langle
x\rangle^{\gamma_{1}}V\in\mathcal{L}(\dot{W}^{\sigma,p_{2}}+\dot{H}^{\sigma},\dot{H}^{\sigma})$
for some $p_{2}>2$ and $\gamma_{2}>0$. Notice that for
$\xi\in\tilde{{\mathcal{E}}}_{\sigma}=\sigma_{disc}({\mathcal{H}})$,

$P_{\xi}^{1}:=P_{\xi}({\mathcal{H}}-\xi)=-\frac{1}{2\pi
i}\int_{|z-\xi|=\varepsilon}(z-\xi)\mathcal{S}_{b,V}(z+ibs_{c})dz$

satisfies $P_{\xi}^{1}P_{\xi}=P_{\xi}P_{\xi}^{1}=P_{\xi}^{1}$ by functional
calculus (see for example [60, Theorem V.8.1]) and similar for
$(P_{\xi}^{1})^{*}$ and $P_{\xi}^{*}$. In particular, we have
$\mathrm{Ran}P_{\xi}^{1}\subset\mathrm{Ran}P_{\xi}$ and
$\mathrm{Ran}(P_{\xi}^{1})^{*}\subset\mathrm{Ran}P_{\xi}^{*}$. Thus
Proposition 4.5 indicates that $P_{\rm disc}({\mathcal{H}}-i\mu)$ is bounded
from $\dot{W}^{\sigma,p_{1}}+\dot{H}^{\sigma}$ to $\dot{H}^{\sigma}_{\langle
x\rangle^{\gamma_{0}}}$ and the bound of $\tilde{V_{2}}$ follows (B.5). ∎

Then we can define the resolvent of (5.7) and derive its relation with
$\mathcal{S}_{b,V}(\lambda+ibs_{c})$ and $\mathcal{S}_{b}(\lambda+ibs_{c})$. A
formal computation suggests that

$(P_{\rm ess}{\mathcal{H}}+i\mu P_{\rm
disc}-z)\left(\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm ess}-(z-i\mu)^{-1}P_{\rm
disc}\right)=I.$ (5.10)

This motivates the following definition and lemma.

######  Lemma 5.7 (Extended resolvents for $\tilde{{\mathcal{H}}}$).

Assume (4.1) and take $\delta_{1}$ from Lemma 4.3. For
$\sigma\in(s_{c},s_{c}+\delta_{1})$, $\mu>b(\sigma-s_{c})$ and
$\tilde{V}_{1}$, $\tilde{V}_{2}$ obtained in Lemma 5.6, we define

$\tilde{\mathcal{S}}^{-}_{b,V}(z+ibs_{c}):=\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm
ess}-(z-i\mu)^{-1}P_{\rm disc}$ (5.11)

for $\Im z\leq b(\sigma-s_{c})$. It satisfies the following properties.

  1. (1)

Uniform boundedness: for some $p_{3}\in(2,2^{*})$,
$\tilde{S}_{b,V}^{-}(z+ibs_{c})$ is well-defined and uniformly bounded in
$\mathcal{L}(\dot{W}^{\sigma,p_{3}^{\prime}}\to\dot{W}^{\sigma,p_{3}})$ for
$\Im z\leq b(\sigma-s_{c})$.

  2. (2)

Relation with $\mathcal{S}_{b}^{-}$: the following identities hold for $\Im
z\leq b(\sigma-s_{c})$

$\displaystyle\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})=\mathcal{S}_{b}^{-}(z+ibs_{c})-$
$\displaystyle\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1}(I+\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1})^{-1}\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})$
(5.12) $\displaystyle
I-\tilde{V}_{2}\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})\tilde{V}_{1}$
$\displaystyle=(I+\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1})^{-1}.$
(5.13)

Moreover, $I+\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1}$ and
$I-\tilde{V}_{2}\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})\tilde{V}_{1}$ are
both analytic and uniformly bounded in $\mathcal{L}(\dot{H}^{\sigma})$ for
$\Im z\leq b(\sigma-s_{c})$.

###### Proof.

Firstly we discuss the commutability issue. Let
$z\in\mathbb{C}-\tilde{{\mathcal{E}}}_{\sigma}$ with $\Im z\leq b(\sigma-
s_{c})$. The resolvent identities (4.30) and (4.31) indicate that
$\mathcal{S}_{b,V}^{-}(z+ibs_{c})$ is commutable with all
$\mathcal{S}_{b,V}(z^{\prime}+ibs_{c})$ for $\Im z^{\prime}\neq b(\sigma-
s_{c})$ and $z^{\prime}\notin\tilde{{\mathcal{E}}}_{\sigma}$. This implies

$P_{\rm
disc}\mathcal{S}_{b,V}^{-}(z+ibs_{c})=\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm
disc}$

and hence

$P_{\rm
ess}\mathcal{S}_{b,V}^{-}(z+ibs_{c})=\mathcal{S}_{b,V}^{-}(z+ibs_{c})P_{\rm
ess}.$ (5.14)

From the uniform boundedness Corollary 4.7, the identity (5.14) makes sense
for all $\Im z\leq b(\sigma-s_{c})$.

From now on, we suppose $z\in\mathbb{C}$ only satisfying $\Im z\leq b(\sigma-
s_{c})$ and take $p_{3}=\min\\{p_{1},p_{2}\\}$ with $p_{1}$ from Proposition
4.5 and $p_{2}$ from Lemma 5.6. Then Lemma 4.4 and Corollary 4.7 imply (1).

For (2), we claim the following resolvent identities

$\displaystyle\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})-\mathcal{S}_{b}^{-}(z+ibs_{c})$
$\displaystyle=$
$\displaystyle-\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})$
(5.15) $\displaystyle=$
$\displaystyle-\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})\tilde{V}\mathcal{S}_{b}^{-}(z+ibs_{c}),$
(5.16)

which imply (5.12) and (5.13) easily. These two resolvent identities will
follow some computation involving the above commutability and inversion
property of $\mathcal{S}_{b}^{-}$. Take (5.15) as an example. From definitions
of $\tilde{\mathcal{S}}_{b,V}^{-}$, $\mathcal{S}_{b,V}^{-}$, $\tilde{V}$ and
identities (4.27), (4.28) and (5.14), we have (the variable $z+ibs_{c}$ of
$\mathcal{S}_{b}^{-}$, $\mathcal{S}_{b,V}^{-}$ or
$\tilde{\mathcal{S}}_{b,V}^{-}$ are omitted in the following computation for
convenience)

$\displaystyle\mathrm{LHS\,\,of\,\,\eqref{eqSbVt3}}$ $\displaystyle=$
$\displaystyle\left(\mathcal{S}_{b,V}^{-}-\mathcal{S}_{b}^{-}\right)P_{\rm
ess}-\left((z-i\mu)^{-1}+\mathcal{S}_{b}^{-}\right)P_{\rm disc},$
$\displaystyle\mathrm{RHS\,\,of\,\,\eqref{eqSbVt3}}$ $\displaystyle=$
$\displaystyle-\mathcal{S}_{b}^{-}\left(V-P_{\rm
disc}({\mathcal{H}}-i\mu)\right)\left(\mathcal{S}_{b,V}^{-}P_{\rm
ess}-(z-i\mu)^{-1}P_{\rm disc}\right)$ $\displaystyle=$
$\displaystyle-\mathcal{S}_{b}^{-}V\mathcal{S}_{b,V}^{-}P_{\rm
ess}-\mathcal{S}_{b}^{-}\left[-V(z-i\mu)^{-1}+({\mathcal{H}}-i\mu)(z-i\mu)^{-1}\right]P_{\rm
disc}$ $\displaystyle=$
$\displaystyle\left(\mathcal{S}_{b,V}^{-}-\mathcal{S}_{b}^{-}\right)P_{\rm
ess}-\mathcal{S}_{b}^{-}P_{\rm
disc}-(z-i\mu)^{-1}\mathcal{S}_{b}^{-}({\mathcal{H}}_{b}-z-ibs_{c})P_{\rm
disc}$

Noticing that for $u\in\dot{W}^{\sigma,p_{3}^{\prime}}$, we have $P_{\rm
disc}u\in\dot{W}^{\sigma,p_{3}^{\prime}}$ and ${\mathcal{H}}_{b}P_{\rm
disc}u\in\dot{W}^{\sigma,p_{3}^{\prime}}$ since $P_{\rm disc}u$ is a
generalized eigenfunction of ${\mathcal{H}}={\mathcal{H}}_{b}-ibs_{c}+V$. Then
the last term is just $(z-i\mu)^{-1}P_{\rm disc}$ by the inversion property
(2.24), and (5.15) follows. The identity (5.16) comes similarly with (2.23).

Finally, we prove the analyticity and uniform boundedness of
$I+\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1}$ and
$I-\tilde{V}_{2}\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})\tilde{V}_{1}$. We
first see the analyticity of
$\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1}$ by specifying a
$0<\gamma_{3}\leq\gamma_{2}$ small enough such that
$p_{\gamma_{3}}:=\frac{2d}{d-\gamma_{3}}\leq p_{3}$. Then differentiability of
$\mathcal{S}_{b}^{-}(z+ibs_{c})$ (Proposition 4.5 (5)) and boundedness of
$\tilde{V}_{1}$ and $\tilde{V}_{2}$ (Lemma 5.7) imply the desired analyticity.
Next, Lemma 3.1 (ii) and finite-rankness of $P_{\rm disc}$ easily indicate
that $\tilde{V}_{2}\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})\tilde{V}_{1}$ is a
compact operator on $\dot{H}^{\sigma}$. Hence we can use a similar argument as
in Lemma 4.1 plus the uniform boundedness of
$\tilde{\mathcal{S}}_{b,V}^{-}(z+ibs_{c})$ to finish the proof. ∎

Now we are ready to prove the Strichartz estimate.

###### Proof of Theorem 1.2.

Step 1. Application of abstract Wiener’s theorem. Let $H=\dot{H}^{\sigma}$ and
$K$ be the Banach algebra as in (5.1). Define

$L(t):=-i\tilde{V}_{2}e^{it({\mathcal{H}}_{b}-ibs_{c})}\tilde{V}_{1}$ (5.17)

for $t\geq 0$ and

$A=I_{K}+L.$ (5.18)

We will check the three conditions in Corollary 5.5 with
$\varepsilon=b(\sigma-s_{c})$ to apply it.

(1) Firstly, the dispersive estimate (2.10) and commutator relation (2.54)
indicate that $e^{it({\mathcal{H}}_{b}-ibs_{c})}\in L^{1}_{t}e^{b(\sigma-
s_{c})t}\mathcal{L}(\dot{W}^{\sigma,p_{2}^{\prime}}\to\dot{W}^{\sigma,p_{2}})$.
Combined with Lemma 5.6, we have $L\in L^{1}_{t}e^{b(\sigma-
s_{c})t}\mathcal{L}(\dot{H}^{\sigma})$ and hence $A\in K$ is upper triangular.

(2) Next, we show that for any $[a_{1},a_{2}]\in(0,\infty)$,

$\mathop{\rm
sup}_{t\in[a_{1},a_{2}]}\|L(\cdot+\epsilon)-L(\cdot)\|_{\mathcal{L}(\dot{H}^{\sigma})}=o_{\epsilon}(1).$
(5.19)

It implies the second condition thanks to the integrability $L\in
L^{1}_{t}e^{b(\sigma-s_{c})t}\mathcal{L}(\dot{H}^{\sigma})$. Recall (5.9) that

$\tilde{V}_{2}=\langle x\rangle^{\gamma_{1}}V+\left(-\langle
x\rangle^{\gamma_{1}}P_{\rm
disc}({\mathcal{H}}-i\mu)\right)=:\tilde{V}_{2,1}+\tilde{V}_{2,2}.$

We will treat these two parts separately.

_(2.a) The multiplication part $\tilde{V}_{2,1}:=\langle
x\rangle^{\gamma_{1}}V$._ We first approximate $\tilde{V}_{1}$ and
$\tilde{V}_{2,1}$ by $W_{1},W_{2}\in\mathscr{S}$ with
$\hat{W}_{1},\hat{W}_{2}$ compactly supported. Supposing
$\hat{W}_{1},\hat{W}_{2}\in C^{\infty}_{c}(B_{M})$ with $M>0$, we consider
$\epsilon<\min\\{(10b)^{-1},(2e^{b(a_{2}+1)}M)^{-4}\\}$ and let
$R=\epsilon^{-\frac{1}{4}}$. We decompose for any $t\in[a_{1},a_{2}]$ as

$\displaystyle\|W_{2}(e^{i(t+\epsilon)({\mathcal{H}}_{b}-ibs_{c})}-e^{it({\mathcal{H}}_{b}-ibs_{c})})W_{1}f\|_{\dot{H}^{\sigma}}$
(5.20) $\displaystyle\leq$
$\displaystyle\left\|W_{2}\varphi(R^{-1}D)(e^{i(t+\epsilon)({\mathcal{H}}_{b}-ibs_{c})}-e^{it({\mathcal{H}}_{b}-ibs_{c})})W_{1}f\right\|_{\dot{H}^{\sigma}}$
(5.21) $\displaystyle+$
$\displaystyle\sum_{\pm}\left\|W_{2}(1-\varphi(R^{-1}D))e^{i(t\pm\frac{\epsilon}{2})({\mathcal{H}}_{b}-ibs_{c})}W_{1}f\right\|_{\dot{H}^{\sigma}}$
(5.22)

where $\varphi$ is a smooth cutoff as in Lemma 2.12. For the low-frequency
part (5.21), notice that

$\displaystyle\mathscr{F}\left(\varphi(R^{-1}D)(e^{i(t+\epsilon)\Delta_{b}}-e^{it\Delta_{b}})W_{1}f\right)(\eta)$
$\displaystyle=$
$\displaystyle\varphi(R^{-1}\eta)\int_{\mathbb{R}^{d}}\bigg{[}\left(e^{b(t+\epsilon)\frac{d}{2}}e^{-i\frac{e^{2b(t+\epsilon)}-1}{2b}|\eta|^{2}}\hat{W}_{1}(e^{b(t+\epsilon)}\eta-\zeta)\right)$
$\displaystyle\quad\quad-\left(e^{bt\frac{d}{2}}e^{-i\frac{e^{2bt}-1}{2b}|\eta|^{2}}\hat{W}_{1}(e^{bt}\eta-\zeta)\right)\bigg{]}|\zeta|^{-\sigma}\widehat{D^{\sigma}f}(\zeta)d\zeta$

where the absolute value of the bracket can be bounded by

$2(b+b^{-1})\epsilon
R^{2}e^{b(a_{2}+1)\left(\frac{d}{2}+2\right)}\left[\int_{0}^{1}|\nabla\hat{W}_{1}(e^{b(t+\epsilon
s)}\eta-\zeta)|ds+\sum_{\pm}|\hat{W}_{1}(e^{b(t\pm\frac{\epsilon}{2})}\eta-\zeta)|\right]$

where we get smallness $\epsilon R^{2}=\epsilon^{\frac{1}{2}}$. The compact
support of $\hat{W}_{1}$ and $\hat{W}_{2}$ also indicates
$|\xi|\sim|\eta|\sim|\zeta|$ when $|\xi|\gg M$ so that $|\zeta|^{-\sigma}$
cancels $|\xi|^{\sigma}$ from the $\dot{H}^{\sigma}$ norm. Therefore, Young’s
inequality implies

$\eqref{eqinteract1}\lesssim_{d,b,a_{2},W_{1},W_{2}}\epsilon^{\frac{1}{2}}\|f\|_{\dot{H}^{\sigma}}.$

For the high-frequency part, we again estimate pointwisely in the Fourier
space. Let $\tilde{f}=D^{\sigma}f$ and $\tilde{t}$ to be either
$t\pm\frac{\epsilon}{2}$. Compute

$\displaystyle\mathscr{F}D^{\sigma}W_{2}D^{-\sigma}\left(1-\varphi(R^{-1}D)\right)e^{i\tilde{t}\Delta_{b}}D^{\sigma}W_{1}D^{-\sigma}\tilde{f}(\xi)$
$\displaystyle=$ $\displaystyle
C|\xi|^{\sigma}\int_{\mathbb{R}^{d}}\hat{W}_{2}(\xi-\eta)(1-\varphi(R^{-1}\eta))e^{b\tilde{t}\left(\sigma+\frac{d}{2}\right)}e^{-i\frac{e^{2b\tilde{t}}-1}{2b}|\eta|^{2}}\left(\widehat{W_{1}D^{-\sigma}\tilde{f}}\right)(e^{b\tilde{t}}\eta)d\eta$
$\displaystyle=$
$\displaystyle-C\frac{be^{b\tilde{t}\left(\sigma+\frac{d}{2}\right)}}{e^{2b\tilde{t}}-1}|\xi|^{\sigma}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-i\frac{e^{2b\tilde{t}}-1}{2b}|\eta|^{2}}$
$\displaystyle\qquad\nabla_{\eta}\cdot\left[\hat{W}_{2}(\xi-\eta)(1-\varphi(R^{-1}\eta))\frac{\eta}{|\eta|^{2}}\hat{W}_{1}(e^{b\tilde{t}}\eta-\zeta)\right]|\zeta|^{-\sigma}\hat{\tilde{f}}(\zeta)d\zeta
d\eta$

where we integrate by parts over
$e^{-i\frac{e^{2b\tilde{t}}-1}{2b}|\eta|^{2}}$. So from the support of
$1-\varphi(R^{-1}\eta)$ and $\hat{V}_{1}$, $\hat{V}_{2}$, we know
$|\xi|\sim|\zeta|\sim|\eta|\geq R$. Hence by Young’s inequality,

$\displaystyle|\eqref{eqfinitespeed}|\lesssim_{d,\sigma}R^{-1}\left|\frac{be^{b\tilde{t}\left(\sigma+1+\frac{d}{2}\right)}}{e^{2b\tilde{t}}-1}\right|\left\\{(|\hat{W}_{2}|+|\nabla\hat{W}_{2}|)*\left[(|\hat{W}_{1}|+|\nabla\hat{W}_{1}|)*|\hat{\tilde{f}}|\right](e^{b\tilde{t}}\cdot)\right\\}(\xi).$

Young’s inequality again implies

$\eqref{eqinteract2}\lesssim_{d,b,a_{1},a_{2},W_{1},W_{2}}\epsilon^{-\frac{1}{4}}\|f\|_{\dot{H}^{\sigma}}.$

So far, we have verified

$\mathop{\rm
sup}_{t\in[a_{1},a_{2}]}\eqref{eqinteract0}=o_{\epsilon}(1)\|f\|_{\dot{H}^{\sigma}}.$

_(2.b) The finite-rank part $\tilde{V}_{2,2}:=-\langle
x\rangle^{\gamma_{1}}P_{\rm disc}({\mathcal{H}}-i\mu)$._ We can assume
$\tilde{V}_{2,2}g=\sum_{i=1}^{N}(g,u_{i})_{\dot{H}^{\sigma}}\langle
x\rangle^{\gamma_{1}}v_{i}$ with $v_{i}\in\mathrm{Ran}P_{\rm disc}$ and
$u_{i}\in\mathrm{Ran}P_{\rm disc}^{*}$. Since
$u_{i}\in\dot{W}^{\sigma,p_{2}^{\prime}}$ and $\langle
x\rangle^{\gamma_{1}}v_{i}\in\dot{H}^{\sigma}$ (from Proposition 4.5 and
(B.5)), we can approximate $u_{i}$ with
$\tilde{u}_{i}\in\mathscr{S}(\mathbb{R}^{d})$ and obtain uniform smallness for
$t\in[a_{1},a_{2}]$ from the strongly continuity of the semigroup

$\left(e^{i(t+\epsilon)({\mathcal{H}}_{b}-ibs_{c})}-e^{it({\mathcal{H}}_{b}-ibs_{c})}\right)^{*}\tilde{u}_{i}\to
0\quad\mathrm{in}\,\,\dot{H}^{\sigma}\,\,\mathrm{as}\,\,\epsilon\to 0.$

So (5.19) and hence the second condition are confirmed by combining these two
estimates for $\tilde{V}_{2,1}$ and $\tilde{V}_{2,2}$.

(3) Recalling the definition of $\mathcal{S}_{b}^{-}$ in (4.10), we compute

$\hat{A}(z)=I-i\int_{0}^{\infty}\tilde{V}_{2}e^{it({\mathcal{H}}_{b}-ibs_{c})}\tilde{V}_{1}e^{-itz}dt=I+\tilde{V}_{2}\mathcal{S}_{b}^{-}(z+ibs_{c})\tilde{V}_{1}.$

The third follows Lemma 5.7 (2) directly.

Step 2. Concluding the proof. Now we define

$U_{b}:=\int_{0}^{t}e^{i(t-s)({\mathcal{H}}_{b}-ibs_{c})}ds.$

The conclusion of Corollary 5.5 implies

$\left(\int_{0}^{t}A(t-s)ds\right)^{-1}=\left(I-i\tilde{V}_{2}U_{b}\tilde{V}_{1}\right)^{-1}\in\mathcal{L}(L^{2}_{t}([0,\infty),e^{b(\sigma-
s_{c})t}\dot{H}^{\sigma}).$ (5.24)

Also from the Strichartz estimate for $e^{it\Delta_{b}}$ from (2.11) and the
commutator relation (2.54), we have

$U_{b}\in\mathcal{L}\left(L_{t}^{q_{2}^{\prime}}e^{tb(\sigma-
s_{c})}\dot{W}^{\sigma,p_{2}^{\prime}}\to L_{t}^{q_{1}}e^{tb(\sigma-
s_{c})}\dot{W}^{\sigma,p_{1}}\right)$ (5.25)

where $(q_{i},p_{i})$ for $i=1,2$ satisfies the admissible condition (1.24).

Finally, we will come back to the modified system (5.6) and prove Theorem 1.2.
The argument in Section 1.5 implies

$\tilde{Z}=\left[U_{b}+iU_{b}\tilde{V}_{1}(I-i\tilde{V}_{2}U_{b}\tilde{V}_{1})^{-1}\tilde{V}_{2}U_{b}\right](\delta_{t=0}\tilde{Z}(0)-i\tilde{F})$
(5.26)

The bounds (5.24), (5.25) and boundedness of $\tilde{V}_{1}$ and
$\tilde{V}_{2}$ from Lemma 5.6 indicate that

$\left\|e^{bt(\sigma-
s_{c})}\tilde{Z}\right\|_{L^{q_{1}}_{t}\dot{W}^{\sigma,p_{1}}_{x}}\lesssim\|\tilde{Z}(0)\|_{\dot{H}^{\sigma}}+\left\|e^{bt(\sigma-
s_{c})}\tilde{F}\right\|_{L_{t}^{q_{2}^{\prime}}\dot{W}^{\sigma,p_{2}^{\prime}}_{x}}.$
(5.27)

We emphasize that the enlarged admissible region of (5.24) and (5.25) is
necessary because $\tilde{V}_{1}$ and $\tilde{V}_{2}$ can only slightly change
the integrability.

Now consider $Z$ with $Z(0)=Z_{0}\in\dot{H}^{\sigma}$ satisfying the original
system (1.22) and let $\tilde{Z}(0)=P_{\rm ess}Z(0)\in\dot{H}^{\sigma}$ evolve
in the modified system (5.6). Then we will have $P_{\rm ess}Z(t)=\tilde{Z}(t)$
and the Strichartz estimate above implies (1.23) for $\nu=b(\sigma-s_{c})$.

For general $\nu\in[0,b(\sigma-s_{c})]$, it suffices to recover (5.24) and
(5.25) with $b(\sigma-s_{c})$ substituted by $\nu$. Noticing that the
condition of Corollary 5.5 for $\varepsilon$ is stronger than that for
$\tilde{\varepsilon}<\varepsilon$, the above arguments also imply (5.24) for
$\nu$. Since

$\left\|e^{it\Delta_{b}}e^{-[b(\sigma-s_{c})-\nu]t}\right\|_{L^{q^{\prime}}\to
L^{q}}\leq\left\|e^{it\Delta_{b}}\right\|_{L^{q^{\prime}}\to
L^{q}}\lesssim|t|^{-\left(\frac{d}{2}-\frac{d}{q}\right)},$

this dispersive estimate similarly yields the free Strichartz (5.25) with
exponential weight $e^{\nu t}$. These two elements conclude the proof of for
all $\nu\in[0,b(\sigma-s_{c})]$.

∎

##  6\. Finite-codimensional stability

In this section, we turn to the finite-codimensional stability of self-similar
blowup of (NLS). More precisely, we will construct a Lipschitz manifold of
initial data leading to the self-similar blowup plus a decaying perturbation
in $\dot{H}^{\sigma}$.

###  6.1. $\mathcal{J}$-invariance and spectrally decoupled system

In this subsection, we will transform the nonlinear Schrödinger equation into
a renormalized system with stable modes and unstable modes decoupled. The
extra degree of freedom in the system will be removed by an
$\mathbb{R}$-linear operator ${\mathcal{J}}$ (borrowed from [56]).

From (NLS) to the system (1.14). We begin by clarifying our setting of (1.14)

$i\partial_{\tau}Z+\mathcal{H}Z=F(Z).$ (1.14)

where ${\mathcal{H}}$ is as (1.19) with potential (1.20),
$Z=\left(\begin{array}[]{c}Z_{1}\\\ Z_{2}\end{array}\right)\in\mathbb{C}^{2}$
and $F=\left(\begin{array}[]{c}-R(Z_{1},Q_{b})\\\
\overline{R({\overline{Z}_{2}},Q_{b})}\end{array}\right)$ with

$\displaystyle
R(\varepsilon,v)=(\varepsilon+v)|\varepsilon+v|^{p-1}-v|v|^{p-1}-\frac{p+1}{2}|v|^{p-1}\varepsilon-\frac{p-1}{2}v^{2}|v|^{p-3}{\overline{\varepsilon}}.$
(6.1)

Then we define the $\mathbb{R}$-linear subspace

$X_{+}:=\left\\{\left(\begin{array}[]{c}\varepsilon\\\
{\overline{\varepsilon}}\end{array}\right):\varepsilon\in\dot{H}^{\sigma}(\mathbb{R}^{d})\right\\}\subset\dot{H}^{\sigma}(\mathbb{R}^{d})\times\dot{H}^{\sigma}(\mathbb{R}^{d})$
(6.2)

and the $\mathbb{R}$-linear transform

${\mathcal{J}}:\left(\begin{array}[]{c}Z_{1}\\\
Z_{2}\end{array}\right)\mapsto\overline{\left(\begin{array}[]{c}Z_{2}\\\
Z_{1}\end{array}\right)}$ (6.3)

Obviously, for
$Z\in\dot{H}^{\sigma}(\mathbb{R}^{d})\times\dot{H}^{\sigma}(\mathbb{R}^{d})$,
$Z\in X_{+}$ if and only if ${\mathcal{J}}Z=Z$. With the help of
${\mathcal{J}}$, we can show $X_{+}$ is invariant under the evolution (1.14).

######  Lemma 6.1 (${\mathcal{J}}$-invariance for (1.14)).

We have

$\displaystyle{\mathcal{J}}\circ i$ $\displaystyle=$
$\displaystyle-i\circ{\mathcal{J}},$
$\displaystyle{\mathcal{J}}\circ{\mathcal{H}}$ $\displaystyle=$
$\displaystyle-{\mathcal{H}}\circ{\mathcal{J}},$
$\displaystyle{\mathcal{J}}F(Z)$ $\displaystyle=$
$\displaystyle-F({\mathcal{J}}Z).$

Thus if $Z(0)\in X_{+}$, the solution $Z(\tau)$ of (1.14) will satisfies
$Z(\tau)\in X_{+}$ for all $\tau$ in the lifespan.

###### Proof.

Those identities follows the definition of ${\mathcal{J}}$ and $i$,
${\mathcal{H}}$, $F(Z)$. We only remark that
$\overline{\Delta_{b}}=\Delta_{-b}$.

For the invariance, suppose $Z(0)\in X_{+}$ generates a solution $Z(\tau)$.
Since ${\mathcal{J}}Z$ satisfies the same equation with initial data
${\mathcal{J}}(Z)(0)=Z(0)\in X_{+}$, the uniqueness of solution from
Proposition D.2 indicates $Z(\tau)={\mathcal{J}}Z(\tau)$ for all $\tau$, which
implies $Z(\tau)\in X_{+}$. ∎

Define the exponent

$\alpha_{c}:=d\left(\frac{1}{2}-\frac{1}{p+1}\right)=\frac{d}{d+2-2s_{c}}\in(s_{c},1)$
(6.4)

such that $\dot{H}^{\alpha_{c}}(\mathbb{R}^{d})\hookrightarrow
L^{p+1}(\mathbb{R}^{d})$. Recall the local-wellposedness of (NLS) and (1.14)
in $\dot{H}^{\sigma}$, $\sigma\in(s_{c},\alpha_{c})$ from Proposition D.1 and
Proposition D.2 respectively. The identification of these two systems via
${\mathcal{J}}$-invariance follows.

######  Lemma 6.2.

Let $d\geq 1$, $0<s_{c}<\min\left\\{\frac{d}{2},1\right\\}$,
$\sigma\in(s_{c},\alpha_{c})$. Let $u$ solves (NLS) with initial data
$u_{0}\in\dot{H}^{\sigma}$, and $Z$ solves (1.14) with initial data
$Z_{0}=\left(\begin{array}[]{c}\varepsilon_{0}\\\
{\overline{\varepsilon}_{0}}\end{array}\right)\in(\dot{H}^{\sigma})^{2}$. Take
any $T>0$ and $\lambda(t)$, $\tau(t)$ from (1.4). If the initial data are
related by

$u_{0}=\lambda(0)^{-\frac{2}{p-1}}(Q_{b}+\varepsilon_{0})(x/\lambda(0))e^{i\tau(0)},$

then these two solutions identify by
$Z(\tau,y)=\left(\begin{array}[]{c}\varepsilon(\tau,y)\\\
{\overline{\varepsilon}}(\tau,y)\end{array}\right)$ and

$u(t,x)=\lambda(t)^{-\frac{2}{p-1}}(Q_{b}+\varepsilon)\left(t,\frac{x}{\lambda(t)}\right)e^{i\left(\tau(t)\right)}.$
(6.5)

###### Proof.

The form $Z=\left(\begin{array}[]{c}\varepsilon\\\
{\overline{\varepsilon}}\end{array}\right)\in X_{+}$ follows the previous
lemma. Using the equation of $Q_{b}$ (1.6), we can check the $u$ defined in
(6.5) satisfies (NLS) with initial data $u_{0}$. Thus the local well-posedness
Proposition D.1 implies the correspondence. ∎

From (1.14) to spectrally decoupled system (6.12). Next, we decouple the
evolution of stable and unstable modes using spectral theory in Section 4 for
general ${\mathcal{H}}$. Those modes are also ${\mathcal{J}}$-invariant.

######  Lemma 6.3 (Spectrum and ${\mathcal{J}}$-invariance of
${\mathcal{H}}$).

Let $d\geq 1$, $0<s_{c}<\min\\{\frac{d}{2},1\\}$ and assume the existence of
$Q_{b}$ satisfying (1.6), (1.7) and (1.8). For the operator ${\mathcal{H}}$ in
(1.19) with potential (1.20), there exists $\delta_{1}>0$ such that for
$\sigma\in(s_{c},s_{c}+\delta_{1})$, ${\mathcal{H}}\big{|}_{\dot{H}^{\sigma}}$
has the spectrum decomposition

$\sigma_{\rm
ess}({\mathcal{H}}\big{|}_{\dot{H}^{\sigma}})=\\{\mathbb{R}+ib(\sigma-
s_{c})\\},\quad\sigma_{\rm
disc}({\mathcal{H}}\big{|}_{\dot{H}^{\sigma}})={\mathcal{E}}_{s}\cup{\mathcal{E}}_{u}$
(6.6)

with the finite discrete sets ${\mathcal{E}}_{s}$, ${\mathcal{E}}_{u}$
satisfying

${\mathcal{E}}_{s}\subset\\{z\in\mathbb{C}:\Im z\geq
b\delta_{1}\\},\quad{\mathcal{E}}_{u}\subset\\{z\in\mathbb{C}:\Im z\leq 0\\}.$
(6.7)

The corresponding spectral projections as $P_{c}:=P_{\rm ess}$777We replace
subscription $\rm{ess}$ by $c$ for simplicity., $P_{s}$ and $P_{u}$ will
satisfy

  1. (1)

$I=P_{\rm c}+P_{u}+P_{s}$.

  2. (2)

$P_{s},P_{u}\in\mathcal{L}((\dot{W}^{\sigma,p_{1}}+\dot{W}^{\sigma,(2^{*-})^{\prime}})^{2}\to(\dot{W}^{\sigma,2^{*-}}\cap\dot{W}^{\sigma,p_{1}^{\prime}})^{2})$
for some $p_{1}>2$.

  3. (3)

${\mathcal{J}}$-invariance:

${\mathcal{J}}\circ
P_{\alpha}=P_{\alpha}\circ{\mathcal{J}},\quad\alpha\in\\{s,u,c\\}.$ (6.8)

###### Proof.

The spectral decomposition is direct conclusion of Proposition 4.5 with the
$\delta_{1}$ and decoupling (6.7) coming from Lemma (4.3). The only statement
left is the ${\mathcal{J}}$-invariance property.

It suffices to prove (6.8) for $\alpha\in P_{s}$. From the boundedness and gap
of ${\mathcal{E}}_{s}$ and ${\mathcal{E}}_{u}$, we can pick contours
$\gamma_{s},\gamma_{u}\subset\mathbb{C}$ to be symmetric about $i\mathbb{R}$
such that

$P_{\alpha}=-\frac{1}{2\pi
i}\oint_{\gamma_{\alpha}}({\mathcal{H}}-z)^{-1}dz,\quad\alpha\in\\{u,s\\}.$

Thus $\gamma_{\alpha}=-{\overline{\gamma}}_{\alpha}$ as sets but with reversed
orientation as oriented curves. Denote $J:=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)$, then ${\mathcal{J}}Z=\overline{JZ}$. Direct
computation implies $J{\mathcal{H}}J=-\overline{{\mathcal{H}}}$. Thus

$\displaystyle{\mathcal{J}}P_{\alpha}Z$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\oint_{\gamma_{\alpha}}\overline{J({\mathcal{H}}-z)^{-1}J}d{\overline{z}}\overline{JZ}=\frac{1}{2\pi
i}\oint_{\gamma_{\alpha}}(-{\mathcal{H}}-{\overline{z}})^{-1}d{\overline{z}}\overline{JZ}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\oint_{-{\overline{\gamma}}_{\alpha}}({\mathcal{H}}-\omega)^{-1}d\omega\overline{JZ}=-\frac{1}{2\pi
i}\oint_{\gamma_{\alpha}}({\mathcal{H}}-\omega)^{-1}d\omega\overline{JZ}=P_{\alpha}{\mathcal{J}}Z$

for $\alpha\in\\{u,s\\}$. ∎

Applying these projections onto (1.14), we get the decoupled system

$\displaystyle i\partial_{\tau}Z_{c}+{\mathcal{H}}Z_{c}$ $\displaystyle=$
$\displaystyle F_{c}$ (6.9) $\displaystyle
i\partial_{\tau}Z_{s}+{\mathcal{H}}Z_{s}$ $\displaystyle=$ $\displaystyle
F_{s}$ (6.10) $\displaystyle i\partial_{\tau}Z_{u}+{\mathcal{H}}Z_{u}$
$\displaystyle=$ $\displaystyle F_{u}$ (6.11)

with $Z_{\alpha}=P_{\alpha}Z$, $F_{\alpha}=P_{\alpha}F$ for
$\alpha\in\\{u,s,c\\}$. We further parametrize the evolution in
$X_{s}:=P_{s}X_{+}$ and $X_{u}:=P_{u}X_{+}$ as $\mathbb{R}^{d_{s}}$ and
$\mathbb{R}^{d_{u}}$ respectively888We cannot view $X_{u}$ or $X_{s}$ as
complex-valued space since $X_{+}$ is only $\mathbb{R}$-linear.. Then the
system (6.9)-(6.11) finally turns into

$\left\\{\begin{array}[]{ll}i\partial_{\tau}Z_{c}+{\mathcal{H}}Z_{c}&=F_{c}\\\
\frac{d}{d\tau}z_{s}+A_{s}z_{s}&=f_{s},\\\
\frac{d}{d\tau}z_{u}+A_{u}z_{u}&=f_{u}.\end{array}\right.$ (6.12)

where

$\displaystyle
z_{\alpha}(\tau)=\phi_{\alpha}(Z_{\alpha}(\tau))\in\mathbb{R}^{d_{\alpha}},$
$\displaystyle\qquad
f_{\alpha}(\tau)=\phi_{\alpha}(-iF_{\alpha}(\tau))\in\mathbb{R}^{d_{\alpha}},$
$\displaystyle A_{\alpha}=\phi_{\alpha}^{-1}\circ$
$\displaystyle(-i{\mathcal{H}}P_{\alpha})\circ\phi_{\alpha}\in\mathbb{R}^{d_{\alpha}\times
d_{\alpha}},$

with $\phi_{\alpha}:P_{\alpha}X_{+}\to\mathbb{R}^{d_{\alpha}}$ a
$\mathbb{R}$-linear homeomorphism for $\alpha\in\\{u,s\\}$. Moreover, they
satisfy

$\displaystyle\Re\left(\mathrm{eig}(A_{s})\right)\geq b\delta_{1},$
$\displaystyle\Re\left(\mathrm{eig}(A_{u})\right)\leq 0,$ (6.13)
$\displaystyle|f_{\alpha}|\sim\|F_{\alpha}\|_{\dot{H}^{\sigma}}\lesssim\|F\|_{\dot{H}^{\sigma}+\dot{W}^{\sigma,(2^{*-})^{\prime}}},$
$\displaystyle|z_{\alpha}|\sim\|Z_{\alpha}\|_{\dot{W}^{\sigma,q}}\,\,\forall
q\in[2,2^{*-})$ (6.14)

from Lemma 6.3.

Now we state the local well-posedness of this spectrally decoupled system
(6.12) in $X:=P_{c}X_{+}\times\mathbb{R}^{d_{s}}\times\mathbb{R}^{d_{u}}$ and
identify with the original (NLS). We also denote the Strichartz norm as

$S([0,T]\times\mathbb{R}^{d}):=C_{\tau}^{0}([0,T],(\dot{H}_{y}^{\sigma})^{2}(\mathbb{R}^{d}))\cap
L_{\tau}^{2}([0,T],(\dot{W}_{y}^{\sigma,2^{*}})^{2}(\mathbb{R}^{d}))$ (6.15)

and in $d=2$ case the non-endpoint norm for $r\geq 2$ as

$S_{r}([0,T]\times\mathbb{R}^{2}):=C_{\tau}^{0}([0,T],(\dot{H}_{y}^{\sigma})^{2}(\mathbb{R}^{d}))\cap
L_{\tau}^{r^{\prime}}([0,T],(\dot{W}_{y}^{\sigma,r})^{2}(\mathbb{R}^{d})).$
(6.16)

######  Proposition 6.4 (Local well-posedness of (6.12) and identification
with (NLS)).

Let $d\geq 1$, $0<s_{c}<\min\left\\{\frac{d}{2},1\right\\}$, and

$s_{c}<\sigma<\min\left\\{s_{c}+\frac{1}{2}\delta_{1},\alpha_{c}\right\\}$
(6.17)

with $\delta_{1}$ from Lemma 6.3. For any initial data
$(Z_{c}(0),z_{s}(0),z_{u}(0))\in X$, if $d\neq 2$, there exists
$T_{0}=T_{0}(\|(Z_{c}(0),Z_{s}(0),Z_{u}(0))\|_{X})$ such that the Cauchy
problem for (6.12) has a unique solution

$(Z_{c},z_{s},z_{u})\in S([0,T_{0}]\times\mathbb{R}^{d})\times
C^{0}_{\tau}([0,T_{0}]\to\mathbb{R}^{d_{s}})\times
C^{0}_{\tau}([0,T_{0}]\to\mathbb{R}^{d_{u}})$

satisfying

$\|Z_{c}\|_{S([0,T_{0}])}+\|z_{s}\|_{C^{0}_{\tau}([0,T_{0}])}+\|z_{u}\|_{C^{0}_{\tau}([0,T_{0}])}\lesssim\|(Z_{c}(0),Z_{s}(0),Z_{u}(0))\|_{X},$
(6.18)

and the solution map $(Z_{c}(0),Z_{s}(0),Z_{u}(0))\mapsto(Z_{c},Z_{s},Z_{u})$
is Lipschitz. If $d=2$, the same conclusion holds after replacing $S$ by
$S_{r}$ for $r\geq 2$ and the time by
$T_{0}=T_{0}(\|(Z_{c}(0),Z_{s}(0),Z_{u}(0))\|_{X},r)$.

Besides, $Z:=Z_{c}+\phi_{s}^{-1}(z_{s})+\phi_{u}^{-1}(z_{u})\in X_{+}$ will
solve (1.14) and $u$ defined by (6.5) will solve (NLS).

###### Proof.

Using argument in Lemma 6.1, the ${\mathcal{J}}$-invariance from Lemma 6.3 and
local well-posedness Proposition D.2 indicate $P_{c}X_{+}\times
P_{s}X_{+}\times P_{u}X_{+}$ to be invariant in the evolution of (6.9)-(6.11).
Thus (6.9)-(6.11) is local well-posed in $P_{c}X_{+}\times P_{s}X_{+}\times
P_{u}X_{+}$, which immediately imply the above local well-posedness of (6.12)
in $X$ since $\varphi_{s}$, $\varphi_{u}$ are linear homeomorphisms. The
identification of $(Z_{c},z_{s},z_{u})$ with $Z$ follows this parametrization,
and Lemma 6.2 concludes their identification with $u$ of (NLS). ∎

###  6.2. A priori control of solution

In preparation for Theorem 1.1, we prove an a priori decay estimate of the
solution of (6.12) given a uniform control of unstable modes.

We first introduce the following exponents:

$\frac{1}{r_{1}}:=\frac{\sigma}{d}+\frac{1}{p+1}\frac{d-2\sigma}{d},\quad\frac{1}{r_{2}}=\frac{p}{p+1}\frac{d-2\sigma}{d},\quad\frac{1}{q_{1}}=\frac{(p-1)(d-2\sigma)}{4(p+1)}.$
(6.19)

Using $\alpha_{c}\in(0,1)$ from (6.4), we have
$\frac{1}{r_{1}}=\frac{1}{2}-\frac{\alpha_{c}}{d}\frac{d-2\sigma}{d}\in\left(\frac{1}{2^{*}},\frac{1}{2}\right)$,
$\frac{1}{q_{1}}=\frac{\alpha_{c}(d-2\sigma)}{2d}$. One can then check
$\frac{2}{q_{1}}+\frac{d}{r_{1}}=\frac{d}{2}$,
$\frac{1}{r_{1}^{\prime}}=\frac{1}{r_{1}}+\frac{p-1}{pr_{2}}$,
$\dot{W}^{\sigma,r_{1}^{\prime}}\hookrightarrow L^{r_{2}}$ and
$\dot{W}^{\sigma,r_{1}}\hookrightarrow L^{pr_{2}}$. With
$s_{c}<\sigma<\alpha_{c}$, we also define

$\beta:=\frac{1}{q_{1}^{\prime}}-\frac{p}{q_{1}}=\frac{\sigma-
s_{c}}{\frac{d}{2}-s_{c}}>0.$ (6.20)

We also recall the Strichartz norms $S$ and $S_{r}$ from (6.15), (6.16). Now
we come to the main result of this subsection.

######  Lemma 6.5 (A priori control with decaying unstable modes).

Under the assumption of Theorem 1.1 and (6.17), for $d\neq 2$ and
$0<\delta\leq b(\sigma-s_{c})$, there exists $\varepsilon_{0}>0$ and
$C_{0},C_{1},C_{2}>0$ such that the following holds: for any
$\varepsilon\leq\varepsilon_{0}$ and any solution $(Z_{c},z_{s},z_{u})\in
S([0,T_{0}]\times\mathbb{R}^{d})\times
C^{0}_{\tau}([0,T_{0}]\to\mathbb{R}^{d_{s}})\times
C^{0}_{\tau}([0,T_{0}]\to\mathbb{R}^{d_{u}})$ solving the system (6.12) for
$T_{0}\in(0,\infty]$, suppose

  * •

smallness of initial data:
$\|Z_{c}(0)\|_{(\dot{H}^{\sigma})^{2}}+|z_{s}(0)|\leq\varepsilon$,

  * •

uniform control of unstable modes: $|z_{u}(\tau)|\leq\varepsilon
e^{\frac{\tilde{p}+1}{2}\delta\tau}$ for all $\tau\in[0,T_{0}]$ where
$\tilde{p}:=\min\\{\frac{p+1}{2},\frac{3}{2}\\}$,

then we have

$\left\|e^{\delta\tau}Z_{c}\right\|_{S([0,T_{0}]\times\mathbb{R}^{d})}+\left\|e^{\frac{\tilde{p}+1}{2}\delta\tau}z_{s}\right\|_{L^{\infty}_{\tau}([0,T_{0}])}+\left\|e^{\frac{\tilde{p}+1}{2}\delta\tau}z_{u}\right\|_{L^{\infty}_{\tau}([0,T_{0}])}\leq
C_{0}\varepsilon$ (6.21)

and

$\left\|e^{\tilde{p}\delta\tau}F\right\|_{L^{q_{1}^{\prime}}_{\tau}([0,T_{0}],\dot{W}^{\sigma,r_{1}^{\prime}}_{y})}\leq
C_{1}\varepsilon^{p}$ (6.22)

where $r_{1},q_{1}$ are defined in (6.19).

For $d=2$, these inequalities hold after replacing $S$ by $S_{r}$ for any
$2\leq r<\infty$.

###### Proof.

The proof is a bootstrap argument based on nonlinear estimates. For
simplicity, we will omit the domain of a norm if it is $\mathbb{R}^{d}$ in
space and $[0,T_{0}]$ in time. We also assume $d\neq 2$ since $d=2$ case just
require replacing $S$ by $S_{r}$ and applying the non-endpoint Strichartz.

Case 1. $1<p\leq 2$. We first obtain a spatial nonlinear estimate from the
fractional difference estimate Lemma C.1. Note that $Q_{b}$ is non-vanishing
(1.7) and satisfies the condition (C.1) with $a=1$. Then we have

$\|F\|_{\dot{W}^{\sigma,r_{1}^{\prime}}}\lesssim\|Z\|_{L^{pr_{2}^{-}}}^{p}+\|Z\|_{L^{pr_{2}}}^{p-1}\|Z\|_{\dot{W}^{\sigma,r_{1}}}\lesssim\|Z\|_{\dot{W}^{\sigma,r_{1}}\cap\dot{W}^{\sigma,r_{1}^{-}}}^{p}$
(6.23)

for $1<p\leq 2$. Here $r^{-}_{i}$ with $i=1,2$ are fixed such that
$0<r_{i}-r_{i}^{-}\ll 1$ and $\dot{W}^{\sigma,r_{1}^{-}}\hookrightarrow
L^{pr_{2}^{-}}$. We also pick $q_{1}^{+}$ and $\beta^{+}$ such that
$\frac{2}{q_{1}^{+}}+\frac{d}{r_{1}^{-}}=\frac{d}{2}$,
$\beta^{+}=\frac{1}{q_{1}^{\prime}}-\frac{p}{q_{1}^{+}}.$

With the nonlinear estimate (6.23) at hand, we consider nonlinear estimates in
weighted Strichartz norms. Suppose $\nu,\mu>0$ with $\nu<p\mu$. Then for
$1<p\leq 2$,

$\displaystyle\|e^{\nu\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}\lesssim\left\|e^{\nu\tau}\left(\|Z\|_{\dot{W}^{\sigma,r_{1}}_{y}}^{p}+\|Z\|_{\dot{W}^{\sigma,r_{1}^{-}}_{y}}^{p}\right)\right\|_{L^{q_{1}^{\prime}}_{\tau}}$
(6.24) $\displaystyle\lesssim$
$\displaystyle\left\|e^{\nu\tau}\left(\|Z_{c}\|_{\dot{W}^{\sigma,r_{1}}_{y}}^{p}+\|Z_{c}\|_{\dot{W}^{\sigma,r_{1}^{-}}_{y}}^{p}+|z_{u}|^{p}+|z_{s}|^{p}\right)\right\|_{L^{q_{1}^{\prime}}_{\tau}}$
$\displaystyle\lesssim$
$\displaystyle\|e^{-(p\mu-\nu)\tau}\|_{L^{q_{1}^{\prime}}_{\tau}}\left(\|e^{\mu\tau}z_{s}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\mu\tau}z_{u}\|_{L^{\infty}_{\tau}}^{p}\right)+\|e^{-(p\mu-\nu)\tau}\|_{L^{\frac{1}{\beta}}_{\tau}}\|e^{\mu\tau}Z_{c}\|_{L^{q_{1}}_{\tau}\dot{W}^{\sigma,r_{1}}_{y}}^{p}$
$\displaystyle+$
$\displaystyle\|e^{-(p\mu-\nu)\tau}\|_{L^{\frac{1}{\beta^{+}}}_{\tau}}\|e^{\mu\tau}Z_{c}\|_{L^{q_{1}^{+}}_{\tau}\dot{W}^{\sigma,r_{1}^{-}}_{y}}^{p}$
$\displaystyle\lesssim$
$\displaystyle\|e^{\mu\tau}Z_{c}\|_{S}^{p}+\|e^{\mu\tau}z_{s}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\mu\tau}z_{u}\|_{L^{\infty}_{\tau}}^{p}$

where the second inequality follows (6.14). We emphasize that this estimate is
independent of the time region. That is, the constant is uniform for all
$T_{0}\in(0,\infty]$, and we can replace the domain $[0,T_{0}]$ by any
$[0,T]\subset[0,T_{0}]$.

Using the Strichartz estimate (1.23) from Theorem 1.2 with $\mu=\delta$ and
$\nu=\frac{p+1}{2}\delta$, we have

$\displaystyle\|e^{\delta\tau}Z_{c}\|_{S}$ $\displaystyle\lesssim$
$\displaystyle\|Z_{c}(0)\|_{\dot{H}^{\sigma}_{y}}+\|e^{\delta\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}\leq\|Z_{c}(0)\|_{\dot{H}^{\sigma}_{y}}+\|e^{\frac{p+1}{2}\delta\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}$
(6.25) $\displaystyle\lesssim$
$\displaystyle\|Z_{c}(0)\|_{\dot{H}^{\sigma}_{y}}+\|e^{\delta\tau}Z_{c}\|_{S}^{p}+\|e^{\delta\tau}z_{s}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\delta\tau}z_{u}\|_{L^{\infty}_{\tau}}^{p}$

Note that (6.13) and $\delta_{1}\geq 2(\sigma-s_{c})$ indicate
$\|e^{-A_{s}\tau}\|_{\mathbb{R}^{d_{s}}\to\mathbb{R}^{d_{s}}}\lesssim
e^{-\frac{p+3}{4}\delta\tau}$. By Duhamel’s formula for $z_{s}$-evolution in
(6.12) and (6.14), we have

$\displaystyle|z_{s}(\tau)|$ $\displaystyle\leq$
$\displaystyle\left|e^{-A_{s}\tau}z_{s}(0)\right|+\left|\int_{0}^{\tau}e^{-A_{s}(\tau-\tau^{\prime})}f_{s}(\tau^{\prime})d\tau^{\prime}\right|$
$\displaystyle\lesssim$ $\displaystyle
e^{-\frac{p+3}{4}\delta\tau}\left(|z_{s}(0)|+\|e^{\frac{p+3}{4}\delta\tau^{\prime}}\|F(\tau^{\prime})\|_{\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}\|_{L^{1}_{\tau^{\prime}}([0,\tau])}\right)$
$\displaystyle\lesssim$ $\displaystyle
e^{-\frac{p+3}{4}\delta\tau}\left(|z_{s}(0)|+\|e^{-\frac{p-1}{4}\delta\tau^{\prime}}\|_{L^{q_{1}}_{\tau^{\prime}}}\|e^{\frac{p+1}{2}\delta\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}\right).$

Thus

$\|e^{\frac{p+3}{4}\delta\tau}z_{s}\|_{L^{\infty}_{\tau}}\lesssim|z_{s}(0)|+\|e^{\delta\tau}Z_{c}\|_{S}^{p}+\|e^{\delta\tau}z_{s}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\delta\tau}z_{u}\|_{L^{\infty}_{\tau}}^{p}.$
(6.26)

Now define

$A(T):=\|e^{\delta\tau}Z_{c}\|_{S([0,T])}+\|e^{\frac{p+3}{4}\delta\tau}z_{s}\|_{L^{\infty}_{\tau}([0,T])}.$
(6.27)

We know $A\in C^{1}([0,T_{0}])$. Combine these two estimates and using
conditions on initial data and decay of $z_{u}(\tau)$, we obtain

$A(T)\leq C\varepsilon+CA(T)^{p},\quad T\in[0,T_{0}].$ (6.28)

Taking $C_{0}=4C$ and $\varepsilon_{0}=(4C)^{-\frac{p}{p-1}}$, we will have

$A(T)\leq C_{0}\varepsilon\Rightarrow A(T)\leq\frac{1}{2}C_{0}\varepsilon$

for $\varepsilon\leq\varepsilon_{0}$. A bootstrap argument then implies
$A(T)\leq\frac{1}{2}C_{0}\varepsilon$ for $T\in[0,T_{0}]$ and therefore the
exponential decay (6.21) when $1<p\leq 2$. The decay of nonlinearity follows
(6.24) with $\mu$ and $\nu$ as above.

Case 2. $p>2$.  We only sketch the proof since the argument is similar. Like
(6.23), we have a similar nonlinear estimate

$\|F\|_{\dot{W}^{\sigma,r_{1}^{\prime}}}\lesssim\|Z\|_{\dot{W}^{\sigma,r_{1}}\cap\dot{W}^{\sigma,r_{1}^{-}}}^{p}+\|Z\|_{\dot{W}^{\sigma,r_{1}}\cap\dot{W}^{\sigma,r_{1}^{-}}}^{2}$
(6.29)

for $p>2$. To deal with the additional quadratic terms, we introduce
$\beta_{1}:=\frac{1}{q_{1}^{\prime}}-\frac{2}{q_{1}}>\frac{1}{q_{1}}-\frac{p}{q_{1}}=\beta$,
and correspondingly
$\beta_{1}^{+}=\frac{1}{q_{1}^{\prime}}-\frac{2}{q_{1}^{+}}$. Then for
$\nu>\mu>0$ with $\nu<2\mu$, the weighted nonlinear estimate becomes

$\displaystyle\|e^{\nu\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}$
$\displaystyle\lesssim$
$\displaystyle\|e^{-(p\mu-\nu)\tau}\|_{L^{q_{1}^{\prime}}_{\tau}}\left(\|e^{\mu\tau}z_{s}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\mu\tau}z_{u}\|_{L^{\infty}_{\tau}}^{p}\right)+\|e^{-(p\mu-\nu)\tau}\|_{L^{\frac{1}{\beta}}_{\tau}}\|e^{\mu\tau}Z_{c}\|_{L^{q_{1}}_{\tau}\dot{W}^{\sigma,r_{1}}_{y}}^{p}$
$\displaystyle+$
$\displaystyle\|e^{-(p\mu-\nu)\tau}\|_{L^{\frac{1}{\beta^{+}}}_{\tau}}\|e^{\mu\tau}Z_{c}\|_{L^{q_{1}^{+}}_{\tau}\dot{W}^{\sigma,r_{1}^{-}}_{y}}^{p}$
$\displaystyle+$
$\displaystyle\|e^{-(2\mu-\nu)\tau}\|_{L^{q_{1}^{\prime}}_{\tau}}\left(\|e^{\mu\tau}z_{s}\|_{L^{\infty}_{\tau}}^{2}+\|e^{\mu\tau}z_{u}\|_{L^{\infty}_{\tau}}^{2}\right)+\|e^{-(2\mu-\nu)\tau}\|_{L^{\frac{1}{\beta_{1}}}_{\tau}}\|e^{\mu\tau}Z_{c}\|_{L^{q_{1}}_{\tau}\dot{W}^{\sigma,r_{1}}_{y}}^{2}$
$\displaystyle+$
$\displaystyle\|e^{-(2\mu-\nu)\tau}\|_{L^{\frac{1}{\beta_{1}^{+}}}_{\tau}}\|e^{\mu\tau}Z_{c}\|_{L^{q_{1}^{+}}_{\tau}\dot{W}^{\sigma,r_{1}^{-}}_{y}}^{2}$
$\displaystyle\lesssim$
$\displaystyle\|e^{\mu\tau}Z_{c}\|_{S}^{p}+\|e^{\mu\tau}z_{s}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\mu\tau}z_{u}\|_{L^{\infty}_{\tau}}^{p}+\|e^{\mu\tau}Z_{c}\|_{S}^{2}+\|e^{\mu\tau}z_{s}\|_{L^{\infty}_{\tau}}^{2}+\|e^{\mu\tau}z_{u}\|_{L^{\infty}_{\tau}}^{2}$

So let $\mu=\delta$ and $\nu=\frac{3}{2}\mu=\delta$ and $A(T)$ defined as
(6.27), we get from Strichartz estimate (1.23) and Duhamel’s principle that

$A(T)\leq C^{\prime}\varepsilon+C^{\prime}A(T)^{2}+C^{\prime}A(T)^{p},\quad
T\in[0,T_{0}].$

The uniform control is then concluded from the similar bootstrap argument with
$\varepsilon_{0}$ small enough. ∎

###  6.3. Construction of finite codimensional blowup manifold

In this subsection, we construct the stable blowup manifold as stated in
Theorem 1.1. More precisely, given initial data in the center-stable space
$(Z_{c}(0),z_{s}(0))\in P_{c}X_{+}\times\mathbb{R}^{d_{s}}$, we use a modified
Brouwer argument to find $z_{u}(0)\in\mathbb{R}^{d_{u}}$ exponential decay, so
that the a priori estimate Lemma 6.5 holds.

The main difficulty is the worse time regularity $L^{q_{1}^{\prime}}_{t}$ of
the nonlinearity, which has been traded for better spatial regularity via
Strichartz estimates. The direct construction of $z_{u}(0)\mapsto
x(\tau^{*}(z_{u}(0)))$ (with $\tau^{*}(z_{u}(0))$ the out-going time) can be
discontinuous. To pick a continuous section, we use instead a mollified
trajectory for Brouwer’s argument.

######  Lemma 6.6 (Existence of decaying trojectory).

Under the assumption of Theorem 1.1 and (6.17), there exists
$\varepsilon_{0}\ll 1$ and $C>0$ such that for any $(Z_{c}(0),z_{s}(0))\in
P_{c}X_{+}\times\mathbb{R}^{d_{s}}$,

$\|Z_{c}(0)\|_{(\dot{H}^{\sigma})^{2}}+|z_{s}(0)|=\varepsilon\leq\varepsilon_{0},$
(6.30)

we can find $z_{u}(0)\in\mathbb{R}^{d_{u}}$ such that

$|z_{u}(0)|\leq C\varepsilon$ (6.31)

and

$\|z_{u}(\tau)|\leq C\varepsilon e^{-\frac{\tilde{p}+1}{2}b(\sigma-
s_{c})\tau}\quad\forall\tau\geq 0.$ (6.32)

###### Proof.

Within this proof, we fix $\mu=b(\sigma-s_{c})>0$,
$\tilde{p}:=\min\\{\frac{p+1}{2},\frac{3}{2}\\}$ and denote $B_{c}^{Y}$ to be
open ball of radius $c$ in a Banach space $Y$.

Step 1. Diagonalized evolution $w$. Firstly, there exists
$Q\in\mathbb{C}^{d_{u}\times d_{u}}$ invertible such that

$Q\left(A_{u}-\frac{\tilde{p}+3}{4}\mu
I_{d_{u}}\right)Q^{-1}=:\Lambda=\mathrm{diag}\\{\lambda_{k}I_{d_{k}}+N_{d_{k}}\\}_{k=1}^{K}$

where $\Lambda$ is the Jordan normal form, $\lambda_{k}I_{d_{k}}+N_{d_{k}}$ is
a Jordan block of size $d_{k}$ with $\sum_{k=1}^{K}d_{i}=d_{u}$ and

$A_{+}\geq-\Re\lambda_{k}\geq A_{-}.$ (6.33)

Here $A_{-}=\frac{\tilde{p}+3}{4}\mu>0$ from (6.13), and $A_{+}<+\infty$ by
the finiteness of ${\mathcal{E}}_{u}$. We further define

$\Lambda_{0}:=\mathrm{diag}\\{\lambda_{k}I_{d_{k}}\\}_{k=1}^{K}$

and

$\displaystyle
w(\tau):=Qe^{(\Lambda-\Lambda_{0}+\frac{\tilde{p}+3}{4}\mu)\tau}z_{u}(\tau),\quad
g(\tau):=Qe^{(\Lambda-\Lambda_{0}+\frac{\tilde{p}+3}{4}\mu)\tau}f_{u}(\tau).$
(6.34)

Then $z_{u}$-evolution in (6.12) implies

$\frac{d}{d\tau}w+\Lambda_{0}w=g.$ (6.35)

Noticing that
$\|e^{(\Lambda-\Lambda_{0})\tau}\|_{\mathbb{C}^{d_{u}}\to\mathbb{C}^{d_{u}}}\lesssim(1+|\tau|)^{d_{u}}$,
$|Qv|_{\mathbb{C}^{d_{u}}}\sim|v|_{\mathbb{R}^{d_{u}}}$, we know

$|e^{\frac{\tilde{p}+1}{2}\mu\tau}z_{u}(\tau)|_{\mathbb{R}^{d_{u}}}\leq
C_{3}|w(\tau)|$ (6.36)

for some $C_{3}>0$.

From the local well-posedness Proposition 6.4 and the a priori control Lemma
6.5 with $\delta=\mu$, when $\varepsilon_{0}\ll 1$ small enough, for any
$\varepsilon\leq\varepsilon_{0}$, the solution of (6.12) with initial data
satisfying (6.30) and

$\displaystyle|w(0)|=|Qz_{u}(0)|$ $\displaystyle\leq$
$\displaystyle\varepsilon$ (6.37)

can be extended whenever $|w(\tau)|\leq\varepsilon$. Besides,
$Z_{c},z_{s},z_{u}$ and $F$ decay exponentially as in (6.21)-(6.22) within
this time period.

Hence, with $(Z_{c}(0),z_{s}(0))\in P_{c}X_{+}\times\mathbb{R}^{d_{s}}$
satisfying (6.30) for $\varepsilon\leq\varepsilon_{0}$, we can define

$\displaystyle\tau^{*}(z_{u}(0)):=\inf\left\\{T\geq
0:|w_{z_{u}(0)}(\tau)|\geq\varepsilon\right\\}\in[0,\infty]$

for any
$z_{u}(0)\in{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}$.
Here $w_{z_{u}(0)}$ is the trajectory defined by (6.34), where $z_{u}$ is the
solution of the system (6.12) with initial data
$(Z_{c}(0),z_{s}(0),z_{u}(0))$. Likewise, we use $g_{z_{u}(0)}$ to indicate
the dependence on initial data. Moreover, by continuity of $w_{z_{u}(0)}$, we
take $T_{0}\leq 1$ be the local lifespan from Proposition 6.4, unifrom for
$\|Z_{c}(0)\|+|z_{s}(0)|+|z_{u}(0)|\lesssim\varepsilon_{0}$ (invariant when we
further shrink $\varepsilon_{0}$). We can then extend the solution to
$\tau\in[-T_{0},\tau^{*}(z_{u}(0))+T_{0}]$ with the estimate

$\mathop{\rm
sup}_{\tau\in[-T_{0},\tau^{*}+T_{0}]}|w_{z_{u}(0)}(\tau)|\lesssim\varepsilon$
(6.38)

from (6.18). Decreasing $\varepsilon_{0}$ if necessary, we can have
exponential bounds (6.21) and (6.22) on $[-T_{0},\tau^{*}+T_{0}]$. The source
term $g_{z_{u}(0)}$ is now bounded by

$\|g_{z_{u}(0)}\|_{L^{q_{1}^{\prime}}_{\tau}([-T_{0},\tau^{*}+T_{0}])}\lesssim\|e^{\tilde{p}\mu\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}([-T_{0},\tau^{*}+T_{0}],\dot{W}^{\sigma,r_{1}^{\prime}}_{y})}\lesssim\varepsilon^{p}.$
(6.39)

Step 2. Continuity and outgoing property of $w$. We state and prove the
following two continuity properties for the evolution of $w_{z_{u}(0)}$.

  * •

_Continuous dependence on initial data_ : for any $z_{u0}\in
B^{\mathbb{R}^{d_{u}}}_{c_{0}\varepsilon}$, for any finite
$T\leq\tau^{*}(z_{u0})$, there exists $\varepsilon(z_{u0},T)$ and
$C(z_{u0},T)$, such that for
$|z_{u0}^{\prime}-z_{u0}|\leq\varepsilon(z_{u0},T)$,

$\|w_{z_{u0}^{\prime}}-w_{z_{u0}}\|_{C^{0}_{\tau}([-T_{0},T+T_{0}])}\leq
C(z_{u0},T)|z_{u0}^{\prime}-z_{u0}|.$ (6.40)

Indeed, proposition D.2 implies a short-time Lipschitz dependence on initial
data. Thanks to the uniform boundedness given by (6.38) and hence
$Z_{c},z_{s}$ and $z_{u}$ from Lemma 6.5, we can iterate the short-time
Lipschitz continuity finite times to obtain the long-time version as above.

  * •

_Continuity of trajectory_ : for any $z_{u0}\in
B^{\mathbb{R}^{d_{u}}}_{c_{0}\varepsilon}$ and any
$\tau_{1},\tau_{2}\in[0,\tau^{*}(z_{u0})+\tau_{0}]$ with
$|\tau_{1}-\tau_{2}|\leq 1$, there exists $C_{4}>0$ such that

$|w_{z_{u0}}(\tau_{2})-e^{-\Lambda_{0}(\tau_{2}-\tau_{1})}w_{z_{u0}}(\tau_{1})|\leq
C_{4}\varepsilon^{p}|\tau_{2}-\tau_{1}|^{\frac{1}{q_{1}}}.$ (6.41)

Indeed, by Duhamel’s formula for (6.35), we have

$\displaystyle
w_{z_{u0}}(\tau_{2})-e^{-\Lambda_{0}(\tau_{2}-\tau_{1})}w_{z_{u0}}(\tau_{1})=\int_{0}^{\tau_{2}-\tau_{1}}e^{-\Lambda_{0}(\tau_{2}-\tau_{1}-\tau^{\prime})}g_{z_{u}(0)}(\tau_{1}+\tau^{\prime})d\tau^{\prime}$

Then (6.41) follows Hölder’s inequality, (6.39) and boundedness of
$\Lambda_{0}$.

In particular, these facts imply the continuity of solution map
$(z_{u0},t)\mapsto w_{z_{u0}}(t)$ with
$(z_{u0},t)\in\\{(v,t)\in{\overline{B}}^{\mathbb{R}^{d_{u}}}_{\varepsilon}\times\mathbb{R}:t\in[-T_{0},\tau^{*}(v)+T_{0})\\}$.

  * •

_Boundedness near $\tau^{*}$:_ When $\varepsilon_{0}\ll 1$, there exists
$C_{5}=2e^{A_{+}}$ such that we have

$\frac{1}{2}\varepsilon\leq|w_{z_{u0}}(\tau)|\leq
C_{5}\varepsilon,\quad\forall\tau\in[\tau^{*}(z_{u0})-T_{0},\tau^{*}(z_{u0})+T_{0}].$
(6.42)

This easily follows (6.41), (6.33) and $T_{0}\leq 1$.

  * •

_Outgoing property with time gap:_ When $\varepsilon_{0}\ll 1$, there exists

$\tau_{0}=\varepsilon^{\frac{\tilde{p}-1}{2}q_{1}^{\prime}}\ll T_{0}$

such that for any
$[\tau,\tau+h]\in[\tau^{*}(z_{u0})-T_{0},\tau^{*}(z_{u0})+T_{0}]$ and
$h\geq\tau_{0}$, we have

$|w_{z_{u0}}(\tau+h)|-|w_{z_{u0}}(\tau)|\gtrsim\varepsilon h.$ (6.43)

Indeed, using

$\frac{d}{d\tau}|w_{z_{u0}}(\tau)|^{2}=2\Re(-\Lambda_{0}w_{z_{u0}}\cdot\overline{w_{z_{u0}}})+2\Re(g_{z_{u0}}\cdot\overline{w_{z_{u0}}})$

and the above boundedness (6.42), we get

$\frac{1}{2}\left(|w_{z_{u0}}(\tau+h)|^{2}-|w_{z_{u0}}(\tau)|^{2}\right)\geq\frac{1}{4}A_{-}\varepsilon^{2}h-C_{4}C_{5}\varepsilon^{\tilde{p}+1}h^{\frac{1}{q_{1}}}$

Hence taking $\varepsilon_{0}\ll 1$ depending on $A_{-},A_{+}$, we see
$\tau_{0}=\varepsilon^{\frac{\tilde{p}-1}{2}q_{1}^{\prime}}\ll\varepsilon^{(\tilde{p}-1)q_{1}^{\prime}}$
will ensure (6.43). The smallness of $\tau_{0}$ also follows
$\varepsilon_{0}\ll 1$.

Step 3. Modified evolution $\tilde{w}$. Take a $\rho\in
C^{\infty}_{c}(B_{1}^{\mathbb{R}})$, $\rho\geq 0$, $\int_{-1}^{1}\rho=1$, and
$\rho_{a}:=\rho(a^{-1}\cdot)a^{-1}$ with $a>0$. Using the small time scale
$\tau_{0}\ll T_{0}$ introduced from the above outgoing property, we now define
the modified evolution by mollifying $w$ and $g$ in time

$\tilde{w}(\tau):=w*\rho_{\tau_{0}}(\tau),\quad\tilde{g}(\tau):=g*\rho_{\tau_{0}}(\tau)$
(6.44)

for $\tau\in[-T_{0}+\tau_{0},\tau^{*}(z_{u}(0))+T_{0}-\tau_{0}]$. Then (6.35)
becomes

$\frac{d}{d\tau}\tilde{w}+\Lambda_{0}\tilde{w}=\tilde{g}$ (6.45)

Define

$\tilde{\tau}^{*}(z_{u}(0)):=\inf\left\\{\tau:|\tilde{w}_{z_{u}(0)}(\tau)|\geq\varepsilon\right\\}\in(\tau_{0},\infty].$
(6.46)

We claim that

$|\tilde{\tau}^{*}(z_{u0})-\tau^{*}(z_{u0})|\leq 2\tau_{0}.$ (6.47)

Indeed, $\tilde{\tau}^{*}(z_{u0})\geq\tau^{*}(z_{u0})-\tau_{0}$ follows
definition of $\tilde{w}$, and the outgoing property (6.43) indicates
$\tilde{w}_{z_{u0}}(\tau^{*}(z_{u0})+2\tau_{0})>\varepsilon$, which concludes
the other side.

As one last preparation, we check a stronger outgoing condition of
$\tilde{w}_{z_{u}(0)}$ at $\tilde{\tau}^{*}(z_{u}(0))$ (if it is finite). From
the definition of $\rho_{\tau_{0}}$ and (6.41),

$|\tilde{g}_{z_{u}(0)}(\tilde{\tau}^{*}(z_{u}(0)))|\leq\|g_{z_{u}(0)}\|_{L^{q_{1}^{\prime}}([0,\tau^{*}+\tau_{0}])}\|\rho_{\tau_{0}}\|_{L^{q_{1}}}\lesssim\varepsilon^{\tilde{p}}\tau_{0}^{-\frac{1}{q_{1}^{\prime}}}=\varepsilon^{\frac{\tilde{p}+1}{2}}\ll\varepsilon$

as $\varepsilon_{0}\ll 1$. Therefore,

$\frac{d}{d\tau}|\tilde{w}_{z_{u}(0)}(\tilde{\tau}^{*})|^{2}=2\Re(-\Lambda_{0}\tilde{w}_{z_{u}(0)}\cdot\overline{\tilde{w}_{z_{u}(0)}})+2\Re(\tilde{g}_{z_{u}(0)}\cdot\overline{\tilde{w}_{z_{u}(0)}})\gtrsim\varepsilon^{2}.$
(6.48)

Step 4. Brouwer’s argument and end of the proof. Picking $\varepsilon_{0}\ll
1$ small enough as required in the previous steps, for any
$\varepsilon\leq\varepsilon_{0}$ and $(Z_{c}(0),z_{s}(0))\in
P_{c}X_{+}\times\mathbb{R}^{d_{s}}$ satisfying (6.30), we will show by
contradiction that there exists
$z_{u}(0)=z_{u0}^{*}\in{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}$
such that $\tilde{\tau}^{*}(z_{u0}^{*})=+\infty$. Then (6.47) indicates
$\tau^{*}(z_{u0}^{*})=+\infty$ and thus (6.32) is proven.

Otherwise, we have $\tilde{\tau}^{*}(z_{u0})<+\infty$ for any
$z_{u0}\in{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}$.
Note that $\tilde{w}_{z_{u0}}(\tau)$ depends on $z_{u0}$ and $t$ continuously
from those continuity of $w$ and definition of $\tilde{w}$. Combined with the
outgoing condition (6.48), we know

$\tilde{\tau}^{*}:{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}\to(-T_{0},+\infty),\quad
z_{u0}\mapsto\tilde{\tau}^{*}(z_{u0})$

is continuous. Since $\tau^{*}(z_{u0})=0$ for boundary data
$|Qz_{u0}|=\varepsilon$, we know $\tilde{\tau}^{*}\big{|}_{\partial
B^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}}\in[-2\tau_{0},2\tau_{0}]$
from (6.47). Then by continuous extension, we can find a continuous function

$\triangle\tau^{*}:{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}\to[-2\tau_{0},2\tau_{0}]$

such that $\triangle\tau^{*}\big{|}_{\partial
B^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}}=\tilde{\tau}^{*}\big{|}_{\partial
B^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}}$. Then
we can define

$\Psi:{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}\to\partial
B^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon},\quad
z_{u0}\mapsto\varepsilon\frac{Q^{-1}w_{z_{u0}}\left(\tilde{\tau}^{*}(z_{u0})-\triangle\tau^{*}(z_{u0})\right)}{|w_{z_{u0}}\left(\tilde{\tau}^{*}(z_{u0})-\triangle\tau^{*}(z_{u0})\right)|_{\mathbb{C}^{d_{u}}}}$

which is continuous (by continuity of $w$, $\tilde{\tau}^{*}$ and
$\triangle\tau^{*}$) and satisfies $\Psi\big{|}_{\partial
B^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}}={\rm
Id}$. Here we also exploited
$e^{(\Lambda-\Lambda_{0}+\mu)\tau}\in\mathbb{R}^{d_{u}\times d_{u}}$ to ensure
$w(\tau)\in Q\mathbb{R}^{d_{u}}$, and the boundedness (6.42) to make the
normalization well-defined. By Brouwer’s argument, this map $\Psi$ cannot be
continuous. This contradiction concludes the proof. ∎

######  Remark 6.7.

Note that $\Psi$ is homotopic to the continuous map

$\Psi_{0}:{\overline{B}}^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon}\to\partial
B^{(\mathbb{R}^{d_{u}},|Q\cdot|_{\mathbb{C}^{d_{u}}})}_{\varepsilon},\quad
z_{u0}\mapsto Q^{-1}\tilde{w}_{z_{u0}}\left(\tilde{\tau}^{*}(z_{u0})\right).$

So the construction can be viewed as a homotopic transformation with
adjustment on the boundary, and Brouwer’s argument still work since $0$ never
touches the boundary during the homotopic transformation.

###  6.4. The Lipschitz dependence and Proof of Theorem 1.1

We show the procedure in Lemma 6.6 of finding $z_{u}(0)$ from
$(Z_{c}(0),z_{s}(0))$ is a well-defined and Lipschitz map, which is the last
piece towards Theorem 1.1.

######  Lemma 6.8 (Lipschitz dependence).

Under the assumption of Theorem 1.1 and (6.17), there exists
$\varepsilon_{1}\leq\varepsilon_{0}$ from Lemma 6.6 and $C^{\prime}>0$ such
that: for any two global solutions $(Z_{c},z_{s},z_{u})$ and
$(\tilde{Z}_{c},\tilde{z}_{s},\tilde{z}_{u})$ of (6.12) satisfying
(6.30)-(6.32) with $\varepsilon_{0}$ replaced by $\varepsilon_{1}$, we have

$|\triangle z_{u}(0)|\leq
C^{\prime}(\varepsilon+\tilde{\varepsilon})^{\min\\{p,2\\}-1}\left(\|\triangle
Z_{c}(0)\|_{(\dot{H}^{\sigma})^{2}}+|\triangle z_{s}(0)|\right).$ (6.49)

Here $\triangle Z_{c}:=\tilde{Z}_{c}-Z_{c}$ and $\triangle z_{u}$, $\triangle
z_{s}$ are defined likewise.

###### Proof.

The proof is mainly a nonlinear estimate for the difference flow of (6.12).
Without loss of generality, we assume $\varepsilon\geq\tilde{\varepsilon}$. We
also employ notations $\tilde{F}=F(\tilde{Z})$ and $\triangle F=\tilde{F}-F$.

From Lemma 6.5 and $\varepsilon_{0}\ll 1$, we can assume the nonlinearity $F$,
$\tilde{F}$ satisfying

$\|e^{\tilde{p}\delta\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}([0,\infty),\dot{W}^{\sigma,r_{1}^{\prime}}_{y})}+\|e^{\tilde{p}\delta\tau}\tilde{F}\|_{L^{q_{1}^{\prime}}_{\tau}([0,\infty),\dot{W}^{\sigma,r_{1}^{\prime}}_{y})}\leq
1.$

Since the unit ball in
$L^{q_{1}^{\prime}}_{\tau}([0,\infty),(\dot{W}^{\sigma,r_{1}^{\prime}}_{y})^{2})$
is closed in $L^{q_{1}}_{\tau}([0,\infty),(L^{r_{2}}_{y})^{2})$, the inverse
mapping theorem then implies

$\left\|e^{\tilde{p}\delta\tau}\triangle
F\right\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}\sim\left\|e^{\tilde{p}\delta\tau}\triangle
F\right\|_{L^{q_{1}^{\prime}}_{\tau}L^{r_{2}}_{y}}.$

Note that Lemma C.3 indicates

$\displaystyle\|\triangle F\|_{L^{r_{2}}}\lesssim\|\triangle
Z\|_{L^{pr_{2}}}\left(\|Z\|_{L^{pr_{2}}}^{p-1}+\|\tilde{Z}\|_{L^{pr_{2}}}^{p-1}\right)\lesssim\|\triangle
Z\|_{\dot{W}^{\sigma,r_{1}}}\left(\|Z\|_{\dot{W}^{\sigma,r_{1}}}^{p-1}+\|\tilde{Z}\|_{\dot{W}^{\sigma,r_{1}}}^{p-1}\right)$

for $1<p\leq 2$ and

$\displaystyle\|\triangle F\|_{L^{r_{2}}}\lesssim\|\triangle
Z\|_{\dot{W}^{\sigma,r_{1}}}\left(\|Z\|_{\dot{W}^{\sigma,r_{1}}}^{p-1}+\|\tilde{Z}\|_{\dot{W}^{\sigma,r_{1}}}^{p-1}+\|Z\|_{\dot{W}^{\sigma,r_{1}}}+\|\tilde{Z}\|_{\dot{W}^{\sigma,r_{1}}}\right)$

for $p>2$. For simplicity, we denote ${\overline{p}}:=\min\\{p,2\\}$.

We consider the difference of equations (6.12) for $Z$ and $\tilde{Z}$

$\displaystyle(i\partial_{\tau}+{\mathcal{H}})\triangle Z_{c}=\triangle
F_{c},\quad\left(\frac{d}{d\tau}+A_{s}\right)\triangle z_{s}=\triangle
f_{s},\quad\left(\frac{d}{d\tau}+A_{u}\right)\triangle z_{u}=\triangle f_{u}.$

From the evolution of $\triangle Z_{c}$ and $\triangle z_{s}$, we use
Strichartz estimates (1.23), Duhamel’s formula and the above nonlinear
estimates to obtain

$\displaystyle\|e^{\delta\tau}\triangle Z_{c}\|_{S}\lesssim\|\triangle
Z_{c}(0)\|_{\dot{H}^{\sigma}}+\left\|e^{\tilde{p}\delta\tau}\triangle
F\right\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}$
$\displaystyle\lesssim$ $\displaystyle\|\triangle
Z_{c}(0)\|_{\dot{H}^{\sigma}}+\varepsilon^{{\overline{p}}-1}\bigg{(}\left\|e^{\delta\tau}\triangle
Z_{c}\right\|_{L^{q_{1}}_{\tau}\dot{W}^{\sigma,r_{1}}_{y}}+\|e^{\delta\tau}\triangle
z_{s}\|_{L^{\infty}_{\tau}}+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\bigg{)}$

and

$\displaystyle\|e^{\delta\tau}\triangle z_{s}\|_{L^{\infty}_{\tau}}$
$\displaystyle\lesssim$ $\displaystyle|\triangle z_{s}(0)|$ $\displaystyle+$
$\displaystyle\varepsilon^{{\overline{p}}-1}\bigg{(}\left\|e^{\delta\tau}\triangle
Z_{c}\right\|_{L^{q_{1}}_{\tau}\dot{W}^{\sigma,r_{1}}_{y}}+\|e^{\delta\tau}\triangle
z_{s}\|_{L^{\infty}_{\tau}}+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\bigg{)}.$

Thus, by taking $\varepsilon_{1}$ small enough,

$\|e^{\delta\tau}\triangle Z_{c}\|_{S}+\|e^{\delta\tau}\triangle
z_{s}\|_{L^{\infty}_{\tau}}\lesssim\|\triangle
Z_{c}(0)\|_{\dot{H}^{\sigma}}+|\triangle
z_{s}(0)|+\varepsilon^{{\overline{p}}-1}\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}.$ (6.50)

Next, we use the evolution of $\triangle z_{u}$ to get

$e^{\tau A_{u}}\triangle z_{u}(\tau)=\triangle
z_{u}(0)+\int_{0}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}-\int_{\tau}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}$ (6.51)

Then use $\|e^{\tau A_{u}}\|\lesssim e^{\frac{\tilde{p}-1}{3}\tau}$ with
$\tilde{p}:=\min\\{\frac{p+3}{2},\frac{3}{2}\\}$ for all $\tau\geq 0$, ,

$\displaystyle\left|\int_{\tau}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}\right|$ $\displaystyle\lesssim$
$\displaystyle\left(\int_{\tau}^{\infty}\|e^{-\tau^{\prime}(\tilde{p}\delta-
A_{u})}\|^{q_{1}}d\tau^{\prime}\right)^{\frac{1}{q_{1}}}\|e^{\tilde{p}\delta\tau}\triangle
F\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}$
$\displaystyle\lesssim$
$\displaystyle\varepsilon^{{\overline{p}}-1}e^{-\tau\frac{2\tilde{p}+1}{3}\delta}\left(\|e^{\delta\tau}\triangle
Z_{c}\|_{S}+\|e^{\delta\tau}\triangle
z_{s}\|_{L^{\infty}_{\tau}}+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\right).$

Combined with the exponential decay of $z_{u}$ and $\tilde{z}_{u}$ (6.32),
this implies

$\displaystyle\left|\triangle
z_{u}(0)+\int_{0}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}\right|$ $\displaystyle\lesssim$
$\displaystyle\left|e^{\tau A_{u}}\triangle
z_{u}(\tau)+\int_{\tau}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}\right|$ $\displaystyle\lesssim$
$\displaystyle\varepsilon
e^{-\tau\frac{4-\tilde{p}}{3}}+\varepsilon^{{\overline{p}}}e^{-\tau\frac{2\tilde{p}+1}{3}\delta}$

Taking $\tau\to+\infty$, we know the LHS of the above inequality is zero.
Hence

$\displaystyle|\triangle z_{u}(0)|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}\right|$ $\displaystyle\lesssim$
$\displaystyle\varepsilon^{{\overline{p}}-1}\left(\|e^{\delta\tau}\triangle
Z_{c}\|_{S}+\|e^{\delta\tau}\triangle
z_{s}\|_{L^{\infty}_{\tau}}+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\right)$ $\displaystyle\lesssim$
$\displaystyle\varepsilon^{{\overline{p}}-1}\left(\|\triangle
Z_{c}(0)\|_{\dot{H}^{\sigma}}+|\triangle z_{s}(0)|+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\right)$

Using (6.51) again and applying (6.50), we get

$\displaystyle|e^{\tau\delta}\triangle z_{u}(\tau)|$ $\displaystyle=$
$\displaystyle\left|\int_{\tau}^{\infty}e^{\tau^{\prime}A_{u}}\triangle
f_{u}(\tau^{\prime})d\tau^{\prime}\right|$ $\displaystyle\lesssim$
$\displaystyle\varepsilon^{p-1}e^{-\tau\frac{\tilde{p}-1}{3}\delta}\left(\|e^{\delta\tau}\triangle
Z_{c}\|_{S}+\|e^{\delta\tau}\triangle
z_{s}\|_{L^{\infty}_{\tau}}+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\right)$ $\displaystyle\lesssim$
$\displaystyle\varepsilon^{p-1}\left(\|\triangle
Z_{c}(0)\|_{\dot{H}^{\sigma}}+|\triangle z_{s}(0)|+\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\right).$

Taking $\varepsilon_{1}$ small enough and taking the supremum of $\tau\geq 0$,
we can absorb the $\triangle z_{u}$ term on RHS to get

$\displaystyle\|e^{\delta\tau}\triangle
z_{u}\|_{L^{\infty}_{\tau}}\lesssim\varepsilon^{p-1}\left(\|\triangle
Z_{c}(0)\|_{(\dot{H}^{\sigma})^{2}}+|\triangle z_{s}(0)|\right),$

which implies (6.49). ∎

Now we are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Take $\sigma$ as (6.17). Then for $\varepsilon\leq\varepsilon_{1}$ from Lemma
6.8, Lemma 6.6 and Lemma 6.8 indicate that every $(Z_{c}(0),z_{s}(0))\in
P_{c}X_{+}\times\mathbb{R}^{d_{s}}$ satisfying (6.30) yields a unique
$z_{u}(0)$ satisfying (6.31)-(6.32), and the corresponding solution to (6.12)
is global. Moreover, this map is Lipschitz from (6.49).

Therefore, let $U:Z=\left(\begin{array}[]{l}\varepsilon\\\
{\overline{\varepsilon}}\end{array}\right)\mapsto\varepsilon$ is an real-
linear isometry between $X_{+}$ and $\dot{H}^{\sigma}$. We define

$X_{s}:=U(P_{c}X_{+}\oplus P_{s}X_{+}),\quad X_{cu}:=UP_{u}X_{+}.$ (6.52)

Then (1.11) holds and the Lipschitz mapping

$\Phi:X_{s}\cap B^{\dot{H}^{\sigma}}_{\varepsilon_{1}}\to X_{cu}\quad
U(Z_{c}(0)+Z_{s}(0))\mapsto UZ_{u}(0)$ (6.53)

generates a the Lipschitz manifold of initial data by (1.12). It suffices to
check (1.9) and (1.10) to conclude the proof.

For the decay of perturbation in renormalized coordinates (1.9), let
$\varepsilon$ be the first component of $Z$. Let $\mu=b(\sigma-s_{c})$ and
recall Lemma 6.5 and regularity of eigenspaces (6.14). Then

$\displaystyle\|e^{\mu\tau}Z\|_{L^{\infty}_{\tau}\dot{H}^{\sigma}\cap
L^{2^{+}}_{\tau}\dot{W}^{\sigma,2^{*-}}}\lesssim\|e^{\mu\tau}Z_{c}\|_{S}+\|e^{\mu\tau}z_{s}\|_{L^{2}_{\tau}}+\|e^{\mu\tau}z_{u}\|_{L^{2}_{\tau}}\lesssim\varepsilon.$

Noticing that $\tau(t)=-\frac{\ln(1-2bt)}{2b}$ and hence
$e^{\mu\tau}=(1-2bt)^{-\frac{\sigma-s_{c}}{2}}$, this is the boundedness of
$\varepsilon$ (1.9).

As for the asymptotics in the original coordinates (1.10), from

$i\partial_{\tau}Z+({\mathcal{H}}_{b}-ibs_{c})Z=F-VZ,$

we have

$\displaystyle e^{-i\tau({\mathcal{H}}_{b}-ibs_{c})}Z(\tau)$ $\displaystyle=$
$\displaystyle\left(Z(0)-i\int_{0}^{\infty}e^{-i\tau^{\prime}({\mathcal{H}}_{b}-ibs_{c})}(F-VZ)(\tau^{\prime})d\tau^{\prime}\right)$
$\displaystyle+$ $\displaystyle
i\int_{\tau}^{\infty}e^{-i\tau^{\prime}({\mathcal{H}}_{b}-ibs_{c})}(F-VZ)(\tau^{\prime})d\tau^{\prime}$
$\displaystyle=:$ $\displaystyle Z_{+}+\mathfrak{r}(\tau).$

Similarly, we bound the linear term by

$\|e^{\mu\tau}VZ\|_{L^{2}_{\tau}\dot{W}^{\sigma,\frac{4d}{2d+1}}_{y}}\lesssim\|e^{\mu\tau}VZ_{c}\|_{L^{2}_{\tau}\dot{W}^{\sigma,\frac{4d}{2d+1}}_{y}}+\|\varepsilon
e^{-\frac{\tilde{p}-1}{2}\mu\tau}\|_{L^{2}_{\tau}}\lesssim\varepsilon.$

Together with (6.22) and enhanced Strichartz estimate for
$e^{it{\mathcal{H}}_{b}}$ (2.11), we know

$\displaystyle\left\|\mathfrak{r}(\tau)\right\|_{\dot{H}^{\sigma}}\lesssim\|e^{\mu\tau}VZ\|_{L^{2}_{\tau}([\tau,\infty),\dot{W}^{\sigma,\frac{4d}{2d+1}}_{y})}+\|e^{\mu\tau}F\|_{L^{q_{1}^{\prime}}_{\tau}([\tau,\infty),\dot{W}^{\sigma,r_{1}^{\prime}}_{y})}\to
0$

as $\tau\to+\infty$ and $Z_{+}\in(\dot{H}^{\sigma})^{2}$. Thus

$e^{b(\sigma-
s_{c})\tau}Z(\tau)-e^{i\tau({\mathcal{H}}_{b}-ib\sigma)}Z_{+}=e^{i\tau({\mathcal{H}}_{b}-ib\sigma)}\mathfrak{r}(\tau)\to
0,\quad\mathrm{in}\,\,(\dot{H}^{\sigma})^{2}$

as $\tau\to+\infty$ since
$\|e^{i\tau({\mathcal{H}}_{b}-ib\sigma)}\mathfrak{r}(\tau)\|_{(\dot{H}^{\sigma})^{2}}=\|\mathfrak{r}(\tau)\|_{(\dot{H}^{\sigma})^{2}}$.
Denote $\varepsilon_{+}$ and $\varepsilon_{r}$ as the first component of
$Z_{+}$ and $\mathfrak{r}$ respectively. We invoke (2.9) to go back to the
original coordinates

$\displaystyle\frac{1}{\lambda(t)^{\frac{2}{p-1}}}\varepsilon\left(t,\frac{x}{\lambda(t)}\right)e^{i\tau(t)}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda(t)^{\frac{d}{2}}}\left(e^{i\tau(t)(\Delta_{b}-1)}(\varepsilon_{+}+\varepsilon_{r}(\tau(t))\right)\left(\frac{x}{\lambda(t)}\right)e^{i\tau(t)}$
$\displaystyle=$ $\displaystyle
e^{it\Delta}\varepsilon_{+}(x)+(e^{it\Delta}\varepsilon_{r}(\tau(t)))(x).$

This implies (1.10) with $u_{*}:=e^{i\frac{1}{2b}\Delta}\varepsilon_{+}$.

∎

###  6.5. Critical norm behavior

Finally, we discuss the critical norm $L^{p_{c}}$ of $u$ and show (1.13).

###### Proof of (1.13).

Recall that $p_{c}=\frac{d(p-1)}{2}$ such that $\dot{H}^{s_{c}}\hookrightarrow
L^{p_{c}}$ and denote $\tilde{Q_{b}}$ and $\tilde{\varepsilon}$ to be the
rescaled functions

$u=e^{i\tau(t)}\lambda(t)^{-\frac{2}{p-1}}(Q_{b}+\varepsilon)\left(t,\frac{\cdot}{\lambda(t)}\right)=:\tilde{Q}_{b}+\tilde{\varepsilon}.$

We now fix an $R>0$ and take a smooth cutoff
$\varphi_{R}:=\varphi(R^{-1}\cdot)$ with $\varphi\in
C^{\infty}_{c,rad}(\mathbb{R}^{d})$, ${\rm supp}~{}\varphi\subset B_{1}$, and
$\varphi=1$ in $B_{1/2}$. It suffices to prove the following four estimates

$\displaystyle\|\tilde{\varepsilon}(t)\varphi_{R}\|_{\dot{H}^{s_{c}}}$
$\displaystyle\lesssim$ $\displaystyle 1,$ (6.54)
$\displaystyle\|\tilde{Q}_{b}(t)\varphi_{R}\|_{L^{p_{c}}}$ $\displaystyle\sim$
$\displaystyle|{\rm log}(T-t)|^{\frac{1}{p_{c}}},$ (6.55)
$\displaystyle\|\tilde{Q}_{b}(t)\varphi_{R}\|_{\dot{H}^{s_{c}}}$
$\displaystyle\sim$ $\displaystyle|{\rm log}(T-t)|^{\frac{1}{2}},$ (6.56)
$\displaystyle\|u(t)(1-\varphi_{R})\|_{\dot{H}^{s_{c}}}$
$\displaystyle\lesssim$ $\displaystyle 1$ (6.57)

where the constant is independent of $t$ as $t\to T$.

For (6.54), we need the following estimate

$\|f\varphi_{M}\|_{\dot{H}^{s_{c}}}\lesssim M^{\sigma-
s_{c}}\|f\|_{\dot{H}^{\sigma}}.$ (6.58)

for any $M>0$, $0\leq s_{c}\leq\sigma<\min\\{\frac{d}{2},1\\}$. It can be
easily proven through high-low decomposition of $f$. Applying this inequality
and (1.9), we obtain

$\displaystyle\|\tilde{\varepsilon}(t)\varphi_{R}\|_{\dot{H}^{s_{c}}}=\|\varepsilon(t)\varphi_{R\lambda(t)^{-1}}\|_{\dot{H}^{s_{c}}}\lesssim\|\varepsilon(t)\|_{\dot{H}^{\sigma}}(R\lambda(t)^{-1})^{\sigma-
s_{c}}\lesssim 1$

Next, recall from (1.8) that

$|Q_{b}|\sim\langle
x\rangle^{-\frac{2}{p-1}},\quad|Q_{b}^{\prime}|\lesssim\langle
x\rangle^{-\frac{p+1}{p-1}}$ (6.59)

and for a large enough $R_{0}\gg 1$,

$|Q_{b}(x)-Q_{b}(x^{\prime})|\sim\langle
x\rangle^{-\frac{2}{p-1}}\qquad\forall|x|\geq R_{0},\,\,|x^{\prime}|\geq
2|x|.$ (6.60)

Hence the $L^{p_{c}}$ estimate (6.55) follows immediately

$\|\tilde{Q}_{b}\varphi_{R}\|_{L^{p_{c}}}\sim\left(\int_{B_{R\lambda(t)^{-1}}}\langle
x\rangle^{-\frac{2}{p-1}p_{c}}dx\right)^{\frac{1}{p_{c}}}\sim|{\rm
log}(T-t)|^{\frac{1}{p_{c}}}.$

Using

$\displaystyle\|\tilde{Q}_{b}(t)\varphi_{R}\|_{\dot{H}^{s_{c}}}^{2}=\|Q_{b}\varphi_{\tilde{R}}\|_{\dot{H}^{s_{c}}}^{2}$
$\displaystyle\sim$
$\displaystyle\iint_{\mathbb{R}^{2d}}\frac{\left|(Q_{b}\varphi_{\tilde{R}})(x-y)-(Q_{b}\varphi_{\tilde{R}})(x)\right|^{2}}{|y|^{d+2s_{c}}}dxdy$
$\displaystyle=:$ $\displaystyle\iint F(x,y)dxdy$

where $\tilde{R}:=R\lambda(t)^{-1}$. The $\dot{H}^{s_{c}}$ estimate comes from
a careful partition of $\mathbb{R}^{2d}$. Indeed, let $\tilde{R}=:2^{K}$, with
$K\gg 1$, we define $K^{*}$ the smallest integer greater than $K$. Define the
disjoint partition $\mathbb{R}^{2d}=\cup_{i=1}^{5}I_{i}$ with

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle\\{|x|\geq
2^{K^{*}+1}\\},$ $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\\{|x|<2^{K^{*}+1},|y|\geq 2^{K^{*}+1}\\},$ $\displaystyle
I_{3}$ $\displaystyle=$
$\displaystyle\\{|x|<2^{K^{*}+1},|y|\in[2^{K^{*}-2},2^{K^{*}+1})\\}$
$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\\{|x|<2^{K^{*}+1},|y|\in[2,2^{K^{*}-2})\\}$ $\displaystyle
I_{5}$ $\displaystyle=$ $\displaystyle\\{|x|<2^{K^{*}+1},|y|<4\\},$

and further decompose $I_{4}$ with $k\in[2,K^{*}-3]$ by
$I_{4}=\cup_{k=2}^{K^{*}-3}\cup_{l=1}^{4}I_{4,k,l}$

$\displaystyle I_{4,k,1}$ $\displaystyle=$ $\displaystyle
I_{4}\cap\\{|x|<2^{k-2}\\},$ $\displaystyle I_{4,k,2}$ $\displaystyle=$
$\displaystyle I_{4}\cap\\{|x|\geq 2^{k-2},|x-y|<2^{k-2}\\},$ $\displaystyle
I_{4,k,3}$ $\displaystyle=$ $\displaystyle
I_{4}\cap\\{|x|\in[2^{k-2},2^{k+2}),|x-y|\geq 2^{k-2}\\},$ $\displaystyle
I_{4,k,4}$ $\displaystyle=$ $\displaystyle I_{4}\cap\\{|x|\geq 2^{k+2}\\}.$

Brute force computations with the support of $Q_{b}\varphi_{\tilde{R}}$ and
(6.59) yield

$\displaystyle\iint_{I_{j}}F(x,y)dxdy\lesssim 1,$ $\displaystyle j=1,2,3,5;$
$\displaystyle\iint_{I_{4,k,l}}F(x,y)dxdy\lesssim 1,$ $\displaystyle
l=1,2,3,4;\quad k\in[2,K^{*}-3].$

And via (6.60), one can check the lower bound

$\iint_{I_{4,k,l}}F(x,y)dxdy\sim 1,\qquad l=1,2;\quad k\in[{\rm
log}R_{0}+5,K^{*}-3].$

Thus by summing them up and using $K^{*}\sim{\rm log}\tilde{R}\sim|{\rm
log}(T-t)|$, we obtain (6.56).

Finally we show (6.57). As a preparation, we first consider the subcritical
$\dot{H}^{\tilde{\sigma}}$ convergence with $0\leq\tilde{\sigma}<s_{c}$.
Consider the Strichartz pair

$r_{\tilde{\sigma}}=\frac{d(p+1)}{d+\tilde{\sigma}(p-1)},\quad\gamma_{\tilde{\sigma}}=\frac{4(p+1)}{(p-1)(d-2\tilde{\sigma})},\quad\frac{2}{\gamma_{\tilde{\sigma}}}+\frac{d}{r_{\tilde{\sigma}}}=\frac{d}{2}$

And let
$\frac{1}{r_{\sigma}}-\frac{1}{r_{\tilde{\sigma}}}=\frac{\sigma-\tilde{\sigma}}{d}$
such that
$\dot{W}^{\sigma,r_{\sigma}}\hookrightarrow\dot{W}^{\tilde{\sigma},r_{\tilde{\sigma}}}$.
Then by fractional Leibniz rule,

$\|D^{\tilde{\sigma}}(u|u|^{p-1})\|_{L^{r_{\tilde{\sigma}}^{\prime}}}\lesssim\|D^{\tilde{\sigma}}u\|_{L^{r_{\tilde{\sigma}}}}^{p}\lesssim\|D^{\sigma}u\|_{L^{r_{\sigma}}}^{p}.$

From Duhamel formula and Strichartz estimate of $e^{it\Delta}$, for
$0<t_{0}\leq t_{1}\leq t_{2}<T$

$\displaystyle\|u(t_{1})-u(t_{2})\|_{\dot{H}^{\tilde{\sigma}}}\lesssim\|e^{i(t_{1}-t_{0})\Delta}u(t_{0})-e^{i(t_{2}-t_{0})\Delta}u(t_{0})\|_{\dot{H}^{\tilde{\sigma}}}+\|D^{\sigma}u\|_{L^{p\gamma_{\tilde{\sigma}}^{\prime}}_{t}L^{r_{\sigma}}_{x}[t_{1},T)}^{p}$

Note that

$\|e^{i(t_{1}-t_{0})\Delta}u(t_{0})-e^{i(t_{2}-t_{0})\Delta}u(t_{0})\|_{\dot{H}^{\tilde{\sigma}}}=\|(e^{i(t_{2}-t_{1})\Delta}-I)u(t_{0})\|_{\dot{H}^{\tilde{\sigma}}}=o_{|t_{2}-t_{1}|\to
0}(1)$

with the speed depending on $t_{0}$, and

$\displaystyle\|D^{\sigma}u\|_{L^{p\gamma_{\tilde{\sigma}}^{\prime}}_{t}L^{r_{\sigma}}_{x}[t_{1},T)}^{p}\lesssim\left(\int_{t_{1}}^{T}\lambda(t)^{-2+\gamma_{\tilde{\sigma}}^{\prime}(s_{c}-\tilde{\sigma})}(\|Q_{b}\|_{\dot{W}^{\sigma,r_{\sigma}}}^{p\gamma_{\tilde{\sigma}}^{\prime}}+\|\varepsilon(t)\|_{\dot{W}^{\sigma,r_{\sigma}}}^{p\gamma_{\tilde{\sigma}}^{\prime}})dt\right)^{\frac{1}{\gamma_{\tilde{\sigma}}^{\prime}}}$
$\displaystyle\lesssim(T-t)^{s_{c}-\tilde{\sigma}}(\|Q_{b}\|_{\dot{W}^{\sigma,r_{\sigma}}}^{p}+\|\varepsilon\|_{L^{p\gamma_{\tilde{\sigma}}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{\sigma}}_{y}}^{p})=o_{t_{0}\to
T}(1)$

using $Q_{b}\in\dot{W}^{\sigma,r_{\sigma}}$ and (1.9) with
$\frac{2}{p\gamma_{\tilde{\sigma}}^{\prime}}+\frac{d}{r_{\sigma}}>\frac{d}{2}$.
We know $\\{u(t)\\}$ is a Cauchy sequence in $\dot{H}^{\tilde{\sigma}}$ as
$t\to T$. In particular,
$\|u(t)\|_{L^{\infty}_{t}\dot{H}^{\tilde{\sigma}}[0,T]}<\infty$.

By interpolation between $\dot{H}^{\tilde{\sigma}}$ and $\dot{H}^{\sigma}$,
(6.57) is reduced to bound $\|u(t)(1-\varphi_{R})\|_{\dot{H}^{\sigma}}$.
Decompose by

$D^{\sigma}(u(t)(1-\varphi_{R}))=(D^{\sigma}\tilde{Q}_{b})(1-\varphi_{R})-[D^{\sigma},\varphi_{R}]\tilde{Q}_{b}+D^{\sigma}(\tilde{\varepsilon}(1-\varphi_{R})).$

The last term involving $\varepsilon$ is bounded by $O(1)$ from (1.9). The
$O(1)$ bound of the first term follows a pointwise decay of $D^{\sigma}Q_{b}$

$|D^{\sigma}Q_{b}(x)|=C_{\sigma,d}\left|\mathop{\rm lim}_{\epsilon\to
0}\int_{\\{|y|\geq\epsilon\\}}\frac{Q_{b}(x-y)-Q_{b}(x)}{|y|^{d+\sigma}}dy\right|\lesssim_{\sigma}\langle
x\rangle^{-\frac{2}{p-1}-\sigma}$

which is proven by splitting the integral in two and applying pointwise bounds
of $Q_{b}$ and $\nabla Q_{b}$ (1.8) respectively. Finally, by the self-similar
scaling and decay, we can uniformly bound

$\mathop{\rm
sup}_{t\in[0,T)}|\tilde{Q}_{b}(t,x)|\lesssim|x|^{-\frac{2}{p-1}}.$

Thus the commutator has a pointwise bound

$\displaystyle|[D^{\sigma},\varphi_{R}]\tilde{Q}_{b}|$ $\displaystyle\lesssim$
$\displaystyle\left|\int\frac{\varphi_{R}(x-y)-\varphi_{R}(x)}{|y|^{d+\sigma}}\tilde{Q}_{b}(x-y)dy\right|$
$\displaystyle\lesssim$
$\displaystyle\mathbbm{1}_{B_{3R}}\left(|\cdot|^{-d-\sigma+1}\mathbbm{1}_{B_{\frac{R}{2}}}+|\cdot|^{-d-\sigma}\mathbbm{1}_{B_{\frac{R}{2}}^{c}}\right)*|\cdot|^{-\frac{2}{p-1}}$
$\displaystyle+\left(|\cdot|^{-d-\sigma}\mathbbm{1}_{B_{\frac{R}{2}}^{c}}\right)*(|\cdot|^{-\frac{2}{p-1}}\mathbbm{1}_{B_{2R}})$
$\displaystyle\lesssim_{R}$
$\displaystyle|x|^{-\frac{2}{p-1}}\mathbbm{1}_{B_{3R}}+\left(|\cdot|^{-d-\sigma}\mathbbm{1}_{B_{\frac{R}{2}}^{c}}\right)*(|\cdot|^{-\frac{2}{p-1}}\mathbbm{1}_{B_{2R}})$

by again splitting the integral. This implies
$\|[D^{\sigma},\varphi_{R}]\tilde{Q}_{b}\|_{L^{2}}\lesssim 1$ from Hölder and
Young’s inequality. This yields (6.57) and hence concludes (1.13). ∎

##  Appendix A Fractional Non-stationary Phase

In Lemma 2.12 and Lemma 2.13, we want to make use of the
$H^{\delta}$-regularity to integrate by parts and get decay from the quadratic
oscillatory phase. For $\delta<1$, we achieve this via mollification, with the
mollifier’s size adapted to the decay $r^{-1}$ we gain. The idea of this part
is credited to Xiaodong Li.

######  Lemma A.1.

Let smooth cutoff function $\psi\in C^{\infty}_{c,rad}(\mathbb{R}^{d})$, ${\rm
supp}~{}\psi\subset B_{2}$, non-negative and
$\int_{\mathbb{R}^{d}}\psi(x)dx=1$. Define the rescaling of $\psi$ (or general
$C^{\infty}_{c}$ functions) by
$\psi_{\lambda}(r):=\lambda^{-d}\psi(\lambda^{-1}r)$. For
$f\in\mathscr{S}(\mathbb{R}^{d})$ and $\delta\in(0,1)$, we have

$\displaystyle\left\|f*\psi_{|\cdot|^{-1}}\right\|_{L^{2}\\{|x|\geq 1/2\\}}$
$\displaystyle\lesssim_{\psi,d}$ $\displaystyle\|f\|_{L^{2}}$ (A.1)
$\displaystyle\left\|\partial_{r}(f*\psi_{|\cdot|^{-1}})|\cdot|^{-1+\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}$ $\displaystyle\lesssim_{\psi,d,\delta}$
$\displaystyle\|f\|_{H^{\delta}}$ (A.2)
$\displaystyle\left\|\left(f-f*\psi_{|\cdot|^{-1}}\right)|\cdot|^{\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}$ $\displaystyle\lesssim_{\psi,d,\delta}$
$\displaystyle\|f\|_{H^{\delta}}$ (A.3)

where (A.1) also holds for general $\psi\in C^{\infty}_{c}(B_{2})$ with
constant depending on $\psi$.

###### Proof.

(1) For (A.1), we have

$\displaystyle\left\|f*\psi_{|\cdot|^{-1}}\right\|_{L^{2}\\{|x|\geq
1/2\\}}^{2}=\int_{B_{1/2}^{c}}\left|\int_{\mathbb{R}^{d}}f(y)\psi_{|x|^{-1}}(x-y)dy\right|^{2}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{1/2}^{c}}\left(\int_{\mathbb{R}^{d}}|f(y)|^{2}|\psi_{|x|^{-1}}(x-y)|dy\right)\left(\int_{\mathbb{R}^{d}}|\psi_{|x|^{-1}}(x-y)|dy\right)dx$
$\displaystyle=$
$\displaystyle\|\psi\|_{L^{1}}\int_{\mathbb{R}^{d}}|f(y)|^{2}\int_{B_{1/2}^{c}}|\psi(|x|(x-y))||x|^{d}dxdy$
$\displaystyle\leq$
$\displaystyle\|f\|_{L^{2}}^{2}\|\psi\|_{L^{1}}\|\psi\|_{L^{\infty}}\mathop{\rm
sup}_{y\in\mathbb{R}^{d}}\int_{\left\\{x:|x|\geq\frac{1}{2},|x-y|\leq
2|x|^{-1}\right\\}}|x|^{d}dx.$

We claim that

$\mathop{\rm
sup}_{y\in\mathbb{R}^{d}}\int_{\left\\{x:|x|\geq\frac{1}{2},|x-y|\leq
2|x|^{-1}\right\\}}|x|^{d}dx\lesssim_{d}1.$ (A.4)

and (A.1) follows with general $\psi\in C^{\infty}_{c}(B^{2})$. Indeed, when
$|y|\geq 8$, noticing that

$\left\\{x:|x|\geq\frac{1}{2},|x-y|\leq
2|x|^{-1}\right\\}\subset\\{x:|x-y|\leq 4\\},$

we have $\frac{1}{2}|y|\leq|x|\leq 2|y|$. So

$\displaystyle\int_{\left\\{x:|x|\geq\frac{1}{2},|x-y|\leq
2|x|^{-1}\right\\}}|x|^{d}dx\leq\int_{|x-y|\leq
4|y|^{-1}}2^{d}|y|^{d}dx\lesssim_{d}1.$

Also, when $|y|\leq 8$,

$\displaystyle\int_{\left\\{x:|x|\geq\frac{1}{2},|x-y|\leq
2|x|^{-1}\right\\}}|x|^{d}dx\leq\int_{|x-y|\leq
4}|x|^{d}dx\leq\int_{B_{12}}|x|^{d}dx\lesssim_{d}1,$

which verifies (A.4) and hence (A.1).

(2) For (A.2), we first compute

$\displaystyle\partial_{r}(f*\psi_{|x|^{-1}})(x)$ $\displaystyle=$
$\displaystyle(f*\partial_{\lambda}\psi_{|x|^{-1}})(x)(r^{-1})^{\prime}+\frac{x}{|x|}\cdot(f*\nabla(\psi_{|x|^{-1}}))(x)$
$\displaystyle=$
$\displaystyle|x|^{-1}\left(f*[(d+(\cdot)\cdot\nabla)\psi]_{|x|^{-1}}\right)(x)+x\cdot(f*(\nabla\psi)_{|x|^{-1}})(x)$

Let $\tilde{\psi}:=(d+x\cdot\nabla)\psi\in C^{\infty}_{c}(B^{2})$, we have

$\displaystyle\left\|\partial_{r}(f*\psi_{|\cdot|^{-1}})|\cdot|^{-1+\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}$ $\displaystyle\leq$
$\displaystyle\left\|(f*\tilde{\psi}_{|\cdot|^{-1}})|\cdot|^{-2+\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}+\left\|(f*(\nabla\psi)_{|x|^{-1}})|\cdot|^{\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}.$

Using (A.1), the first term is bounded by $C_{\psi,d,\delta}\|f\|_{L^{2}}$. So
it suffices to prove

$\displaystyle\left\|\left(f*(\nabla\psi)_{|\cdot|^{-1}}\right)|\cdot|^{\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}$ $\displaystyle\lesssim_{\psi,d,\delta}$
$\displaystyle\|f\|_{\dot{H}^{\delta}}$ (A.5)

For (A.5),

$\displaystyle\left\|\left(f*(\nabla\psi)_{|\cdot|^{-1}}\right)|\cdot|^{\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}^{2}=\int_{B_{1/2}^{c}}\left|\int_{\mathbb{R}^{d}}f(x-y)(\nabla\psi)_{|x|^{-1}}(y)dy\right|^{2}|x|^{2\delta}dx$
$\displaystyle=$
$\displaystyle\int_{B_{1/2}^{c}}\left|\int_{\mathbb{R}^{d}}\left(f(x-y)-f(x)\right)(\nabla\psi)_{|x|^{-1}}(y)dy\right|^{2}|x|^{2\delta}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{1/2}^{c}}\left(\int_{\mathbb{R}^{d}}\left|f(x-y)-f(x)\right|\left(\frac{2|x|^{-1}}{|y|}\right)^{\frac{d+2\delta}{2}}|(\nabla\psi)_{|x|^{-1}}(y)|dy\right)^{2}|x|^{2\delta}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{1/2}^{c}}2^{d+2\delta}|x|^{-(d+2\delta)}\left(\int_{\mathbb{R}^{d}}\frac{\left|f(x-y)-f(x)\right|^{2}}{|y|^{d+2\delta}}dy\right)\left(\int_{\mathbb{R}^{d}}|(\nabla\psi)_{|x|^{-1}}(y)|^{2}dy\right)|x|^{2\delta}dx$
$\displaystyle\leq$
$\displaystyle\|\nabla\psi\|_{L^{2}}^{2}2^{d+2\delta}\int_{B^{c}_{1/2}}\int_{\mathbb{R}^{d}}\frac{\left|f(x-y)-f(x)\right|^{2}}{|y|^{d+2\delta}}dydx$

where we used the vanishing
$\int_{\mathbb{R}^{d}}(\nabla\psi)_{\lambda}(x)dx=\vec{0}$ for any $\lambda>0$
and its support and scaling. With the equivalent expression of
$\dot{H}^{\delta}$ norm (2.43), the above computation implies (A.5) and hence
(A.2).

(3) Finally, for (A.3), note that $\int\psi(x)dx=1$ indicates

$\displaystyle\left\|\left(f-f*\psi_{|\cdot|^{-1}}\right)|\cdot|^{\delta}\right\|_{L^{2}\\{|x|\geq
1/2\\}}^{2}$ $\displaystyle=$
$\displaystyle\int_{B_{1/2}^{c}}\left|\int_{\mathbb{R}^{d}}\left(f(x-y)-f(x)\right)\psi_{|x|^{-1}}(y)dy\right|^{2}|x|^{2\delta}dx$
$\displaystyle\leq$
$\displaystyle\|\psi\|_{L^{2}}^{2}2^{d+2\delta}\int_{B^{c}_{1/2}}\int_{\mathbb{R}^{d}}\frac{\left|f(x-y)-f(x)\right|^{2}}{|y|^{d+2\delta}}dydx.$

The last inequality follows almost the same computation as the above one for
(A.5). ∎

##  Appendix B Commutator estimates of multiplication in $\dot{H}^{\sigma}$

We consider several boundedness and commutator estimates for scalar potential
$V$ in $\dot{H}^{\sigma}$. Recall our polynomial decay assumption on potential

$|V(x)|\lesssim\langle x\rangle^{-\alpha},\quad|\nabla V(x)|\lesssim\langle
x\rangle^{-\alpha-1}$ (1.21)

for some $\alpha>0$.

######  Lemma B.1.

For $d\geq 1$, $0<\sigma<\min\left\\{1,\frac{d}{2}\right\\}$ and $V\in
C^{1}(\mathbb{R}^{d})$, for $2\leq p<\frac{d}{\sigma}$,

$\|D^{\sigma}VD^{-\sigma}\|_{L^{2}\to
L^{p^{\prime}}}+\|D^{\sigma}VD^{-\sigma}\|_{L^{p}\to
L^{2}}\lesssim_{d,p,\sigma}\|V\|_{\dot{W}^{\sigma,\left(\frac{1}{2}+\frac{\sigma}{d}-\frac{1}{p}\right)^{-1}}}$
(B.1)

In particular, for $V$ satisfying (1.21) with $\alpha>0$,
$D^{\sigma}VD^{-\sigma}\in\mathcal{L}(L^{2}\to
L^{p^{\prime}})\cap\mathcal{L}(L^{p}\to L^{2})$ for
$\frac{1}{2}\geq\frac{1}{p}>\max\\{\frac{1}{2}-\frac{\alpha}{d},\frac{\sigma}{d}\\}$.

###### Proof.

It is an easy application of Sobolev embedding and the fractional Lebnitz rule
(see for example [25, Theorem 1])

$\|D^{s}(f_{1}f_{2})\|_{L^{r}}\lesssim_{d,s,r,p_{1},p_{2},q_{1},q_{2}}\|D^{s}f_{1}\|_{L^{p_{1}}}\|f_{2}\|_{L^{q_{1}}}+\|f_{1}\|_{L^{p_{2}}}\|D^{s}f_{2}\|_{L^{q_{2}}}$
(B.2)

with $1>s>0$, $r\in(1,\infty)$, $p_{1},p_{2},q_{1},q_{2}\in(1,\infty]$
satisfying
$\frac{1}{r}=\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}$
and $f,g\in\mathscr{S}(\mathbb{R}^{d})$. ∎

Moreover, the shape of $V$ indicates a stronger bound: $\|\langle
x\rangle^{\alpha}Vf\|_{L^{2}}\lesssim\|f\|_{L^{2}}$ in $L^{2}$. This remains
almost true after conjugating $V$ with fractional derivatives.

######  Lemma B.2.

Let $d\geq 1$, $0<\sigma<\min\left\\{1,\frac{d}{2}\right\\}$ and $V\in
C^{1}(\mathbb{R}^{d})$ satisfying (1.21) with parameter $\alpha>0$. We have
the following statements:

  1. (i)

For $0\leq\beta<\min\left\\{\alpha,1\right\\}$ and any
$f\in\mathscr{S}(\mathbb{R}^{d})$, we have

$\|\langle
x\rangle^{\beta}D^{\sigma}VD^{-\sigma}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}.$ (B.3)

  2. (ii)

For $0\leq\beta<\min\\{\alpha,\frac{d}{2}-\sigma\\}$ and
$f\in\mathscr{S}(\mathbb{R}^{d})$, we have

$\|\langle
x\rangle^{\beta}D^{-\sigma}VD^{\sigma}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}$ (B.4)

  3. (iii)

Let $0<\alpha+\beta<\gamma<1$, we have

$\|\langle
x\rangle^{\alpha}D^{\sigma}\langle\cdot\rangle^{\beta}D^{-\sigma}\langle\cdot\rangle^{-\gamma}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}.$
(B.5)

Consequently, $\langle
x\rangle^{\beta}\in\mathcal{L}(\dot{H}^{\sigma}_{\langle
x\rangle^{\gamma}}\to\dot{H}^{\sigma}_{\langle x\rangle^{\alpha}})$.

###### Proof.

(i) Since $\langle x\rangle^{\beta}V$ satisfies the polynomial decay condition
(1.21) with parameter $\alpha-\beta>0$, Lemma B.1 suggests that it suffices to
estimate the commutator

$\left\|[D^{\sigma},\langle\cdot\rangle^{\beta}]VD^{-\sigma}f\right\|_{L^{2}}\lesssim\|f\|_{L^{2}}.$
(B.6)

Without less of generality, we assume $\alpha\in(\beta,1)$.

With the physical representation of $D^{\sigma}$ and the cancellation from the
commutator, we have the decomposition

$\displaystyle\left|[D^{\sigma},\langle\cdot\rangle^{\beta}]VD^{-\sigma}f\right|(x)=C\left|\int_{\mathbb{R}^{d}}\frac{\langle
x\rangle^{\beta}-\langle
x-y\rangle^{\beta}}{|y|^{d+\sigma}}(VD^{-\sigma}f)(x-y)dy\right|$
$\displaystyle\lesssim$ $\displaystyle\int_{|y|\geq 2\langle
x\rangle}|y|^{-(d+\sigma+\alpha-\beta)}\left(|\cdot|^{-(d-\sigma)}*|f|\right)(x-y)dy$
(B.7) $\displaystyle+$ $\displaystyle\int_{|y|\leq 2\langle
x\rangle}|y|^{-(d+\sigma-1)}\langle x\rangle^{\beta-1}\langle
x-y\rangle^{-\alpha}\left[\left(|\cdot|^{-(d-\sigma)}\mathbbm{1}_{B_{4\langle
x\rangle}}\right)*|f|\right](x-y)dy$ (B.8) $\displaystyle+$
$\displaystyle\int_{|y|\leq 2\langle x\rangle}|y|^{-(d+\sigma-1)}\langle
x\rangle^{\beta-1}\langle
x-y\rangle^{-\alpha}\left[\left(|\cdot|^{-(d-\sigma)}\mathbbm{1}_{B_{4\langle
x\rangle}^{c}}\right)*|f|\right](x-y)dy$ (B.9)

where $|V|$ is substituted by $\langle x\rangle^{-\alpha}$ and the fractional
weight is bounded by

$\left|\langle x\rangle^{\beta}-\langle x-y\rangle^{\beta}\right|\langle
x-y\rangle^{-\alpha}\lesssim\left\\{\begin{array}[]{ll}|y|^{\beta-\alpha}&|y|\geq
2\langle x\rangle,\\\ |y|\langle x\rangle^{\beta-1}\langle
x-y\rangle^{-\alpha}&|y|\leq 2\langle x\rangle.\end{array}\right.$

Now we will bound the $L^{2}$-norm of (B.7)-(B.9) with $\|f\|_{L^{2}}$.

For (B.7), using Hölder and Hardy-Littlewood-Sobolev inequality, we get

$\displaystyle\eqref{eqcom01}\lesssim\||\cdot|^{-(d+\sigma+\alpha-\beta)}\|_{L^{\frac{2d}{d+2\sigma}}(B_{2\langle
x\rangle}^{c})}\||\cdot|^{-(d-\sigma)}*f\|_{L^{\frac{2d}{d-2\sigma}}}\lesssim\langle
x\rangle^{-\frac{d}{2}-\alpha+\beta}\|f\|_{L^{2}}.$

Since $\alpha>\beta$, we get
$\|\eqref{eqcom01}\|_{L^{2}}\lesssim\|f\|_{L^{2}}$.

For (B.8), noticing that $|z|\geq 4\langle x\rangle$ and $|y|\leq 2\langle
x\rangle$ imply $|x-y-z|\sim|z|$, so

$\displaystyle\eqref{eqcom02}\lesssim\langle
x\rangle^{\beta-1}\int_{B_{2\langle x\rangle}}|y|^{-(d+\sigma-1)}\langle
x-y\rangle^{-\alpha}dy\int_{B_{4\langle
x\rangle}^{c}}\frac{|f(z)|}{|z|^{d-\sigma}}dz\lesssim\langle
x\rangle^{-\frac{d}{2}-\alpha+\beta}\|f\|_{L^{2}}$

where the first integral is bounded by $\langle x\rangle^{1-\sigma-\alpha}$
when $\alpha<d$ via elementary computation. Therefore this term has the
desired bound.

Finally, we write (B.9) as

$\displaystyle\eqref{eqcom03}=\langle
x\rangle^{\beta-1}\left(|\cdot|^{-(d+\sigma-1)}\mathbbm{1}_{B_{2\langle
x\rangle}}\right)*\left[\langle\cdot\rangle^{-\alpha}\mathbbm{1}_{B_{3\langle
x\rangle}}\left(|\cdot|^{-(d-\sigma)}\mathbbm{1}_{B_{4\langle
x\rangle}}*|f|\right)\right](x).$

Then we can estimate their $L^{2}$ norm on dyadic annulus with $k\geq 0$

$\displaystyle\|\eqref{eqcom03}\|_{L^{2}(B_{2^{k+1}}-B_{2^{k}})}$
$\displaystyle\lesssim$ $\displaystyle
2^{-k(1-\beta)}\||\cdot|^{-(d+\sigma-1)}\|_{L^{p_{1}}(B_{2^{k+3}})}\|\langle\cdot\rangle^{-\alpha}\|_{L^{p_{2}}(B_{2^{k+3}})}$
$\displaystyle\cdot\||\cdot|^{-(d-\sigma)}\|_{L^{p_{3}}(B_{2^{k+3}})}\|f\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle 2^{-k(\alpha-\beta)}\|f\|_{L^{2}}.$

where we choose $p_{1},p_{2},p_{3}\in(1,\infty)$ such that
$\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=2$,
$p_{1}<\frac{d}{d+\sigma-1}$, $p_{2}<\frac{d}{\alpha}$ and
$p_{3}<\frac{d}{d-\sigma}$ which is possible when $\alpha<1$. The contribution
on $B_{1}$ can be bounded similarly, and we conclude the estimate of (B.9) by
summing all these estimates up using $\alpha>\beta$.

Combining these three parts, we obtain (B.6) and hence (B.3).

(ii) Without loss of generality, we assume $\alpha<\frac{d}{2}-\sigma$.
Because $\langle x\rangle^{\beta}V\in L^{\infty}$, it suffices to show

$\left\|\langle
x\rangle^{\beta}D^{-\sigma}[D^{\sigma},V]f\right\|_{L^{2}}\lesssim\|f\|_{L^{2}}.$
(B.10)

Notice that

$\displaystyle\langle x\rangle^{\beta}D^{-\sigma}[D^{\sigma},V]f(x)$
$\displaystyle=$ $\displaystyle C\langle
x\rangle^{\beta}\int_{\mathbb{R}^{d}}\frac{1}{|y|^{d-\sigma}}\int_{\mathbb{R}^{d}}\frac{V(x-y)-V(x-y-z)}{|z|^{d+\sigma}}f(x-y-z)dzdy.$

Once again, we decompose the integral region and estimate the weights
correspondingly. Here the estimate for interaction is

$|V(x_{1})-V(x_{2})|\lesssim\left\\{\begin{array}[]{ll}\langle
x_{1}\rangle^{-\alpha-1}|x_{1}-x_{2}|&|x_{1}-x_{2}|\leq\frac{1}{2}\langle
x_{1}\rangle\\\ \langle
x_{2}\rangle^{-\alpha}&|x_{1}-x_{2}|\in[\frac{1}{8}\langle
x_{1}\rangle,8\langle x_{1}\rangle]\\\ \langle
x_{1}\rangle^{-\alpha}&|x_{1}-x_{2}|\geq 2\langle
x_{1}\rangle.\end{array}\right.$ (B.11)

So we will partition $y$ and $z$ into $3\times 3$ regions:

  1. (1)

$\left\\{|y|\leq\frac{1}{2}\langle x\rangle,|z|\leq\frac{1}{4}\langle
x\rangle\right\\},\left\\{|y|\leq\frac{1}{2}\langle
x\rangle,|z|\in\left[\frac{1}{4},4\right]\langle
x\rangle\right\\},\left\\{|y|\leq\frac{1}{2}\langle x\rangle,|z|\geq 4\langle
x\rangle\right\\};$

  2. (2)

$\left\\{|y|\in\left[\frac{1}{2},2\right]\langle
x\rangle,|z|\leq\frac{1}{4}\langle
x-y\rangle\right\\},\left\\{|y|\in\left[\frac{1}{2},2\right]\langle
x\rangle,|z|\in\left[\frac{1}{4},4\right]\langle x-y\rangle\right\\},\\\
\left\\{|y|\in\left[\frac{1}{2},2\right]\langle x\rangle,|z|\geq 4\langle
x-y\rangle\right\\};$

  3. (3)

$\left\\{|y|\geq 2\langle x\rangle,|z|\leq\frac{1}{4}\langle
y\rangle\right\\},\left\\{|y|\geq 2\langle
x\rangle,|z|\in\left[\frac{1}{4},4\right]\langle
y\rangle\right\\},\left\\{|y|\geq 2\langle x\rangle,|z|\geq 4\langle
y\rangle\right\\}$.

Therefore we have

$\displaystyle\left|\langle
x\rangle^{\beta}D^{-\sigma}[D^{\sigma},V]f(x)\right|$ (B.12)
$\displaystyle\lesssim$ $\displaystyle\langle
x\rangle^{-1-\alpha+\beta}\left(|\cdot|^{-(d-\sigma)}\mathbbm{1}_{B_{\frac{1}{2}\langle
x\rangle}}\right)*\left(|\cdot|^{-(d+\sigma-1)}\mathbbm{1}_{B_{\frac{1}{4}\langle
x\rangle}}\right)*|f|(x)$ $\displaystyle+$ $\displaystyle\langle
x\rangle^{-d-\sigma+\beta}\int_{B_{\frac{1}{2}\langle
x\rangle}}|y|^{-(d-\sigma)}\int_{B_{4\langle
x\rangle}}\frac{|f(x-y-z)|}{\langle x-y-z\rangle^{\alpha}}dzdy$ (B.13)
$\displaystyle+$ $\displaystyle\langle
x\rangle^{-\alpha+\beta}\left(|\cdot|^{-(d-\sigma)}\mathbbm{1}_{B_{\frac{1}{2}\langle
x\rangle}}\right)*\left(|\cdot|^{-(d+\sigma)}\mathbbm{1}_{B^{c}_{4\langle
x\rangle}}\right)*|f|(x)$ (B.14) $\displaystyle+$ $\displaystyle\langle
x\rangle^{-d+\sigma+\beta}\int_{B_{2\langle x\rangle}-B_{\frac{1}{2}\langle
x\rangle}}\langle x-y\rangle^{-\alpha-1}\int_{B_{\frac{1}{4}\langle
x-y\rangle}}\frac{|f(x-y-z)|}{|z|^{d+\sigma-1}}dzdy$ (B.15) $\displaystyle+$
$\displaystyle\langle x\rangle^{-d+\sigma+\beta}\int_{B_{2\langle
x\rangle}-B_{\frac{1}{2}\langle x\rangle}}\langle
x-y\rangle^{-d-\sigma}\int_{B_{4\langle x-y\rangle}-B_{\frac{1}{4}\langle
x-y\rangle}}\frac{|f(x-y-z)|}{\langle x-y-z\rangle^{\alpha}}dzdy$ (B.16)
$\displaystyle+$ $\displaystyle\langle
x\rangle^{-d+\sigma+\beta}\int_{B_{2\langle x\rangle}-B_{\frac{1}{2}\langle
x\rangle}}\langle x-y\rangle^{-\alpha}\int_{B^{c}_{4\langle
x-y\rangle}}\frac{|f(x-y-z)|}{|z|^{d+\sigma}}dzdy$ (B.17) $\displaystyle+$
$\displaystyle\langle x\rangle^{\beta}\int_{B_{2\langle
x\rangle}^{c}}|y|^{-(d-\sigma+\alpha+1)}\int_{B_{\frac{1}{4}\langle
y\rangle}}\frac{|f(x-y-z)|}{|z|^{d+\sigma-1}}dzdy$ (B.18) $\displaystyle+$
$\displaystyle\langle x\rangle^{\beta}\int_{B_{2\langle
x\rangle}^{c}}|y|^{-2d}\int_{B_{4\langle y\rangle}-B_{\frac{1}{4}\langle
y\rangle}}\frac{|f(x-y-z)|}{\langle x-y-z\rangle^{\alpha}}dzdy$ (B.19)
$\displaystyle+$ $\displaystyle\langle x\rangle^{\beta}\int_{B_{2\langle
x\rangle}^{c}}|y|^{-(d-\sigma+\alpha)}\int_{B_{4\langle
y\rangle}^{c}}\frac{|f(x-y-z)|}{|z|^{d+\sigma}}dzdy.$ (B.20)

We partition these 9 terms into four groups to estimate.

  1. (1)

For (B.12) and (B.14), we exploit the convolution structure to estimate the
$L^{2}$ norm on non-homogeneous dyadic annulus, resulting in
$\|\eqref{eqcom41}\|_{L^{2}(B_{2^{k+1}}-B_{2^{k}})}\lesssim
2^{-k(\alpha-\beta)}\|f\|_{L^{2}}$,
$\|\eqref{eqcom41}\|_{L^{2}(B_{1})}\lesssim\|f\|_{L^{2}}$, and similar for
(B.14).

  2. (2)

For (B.13), (B.16) and (B.19), a straightforward application of Hölder’s
inequality implies pointwise bound by $\langle
x\rangle^{-\frac{d}{2}-\alpha+\beta}\|f\|_{L^{2}}$.

  3. (3)

We address (B.18) and (B.20) similarly to (B.12) and (B.14), but with an
additional dyadic decomposition for the domain of $y$ to fix the integration
domain of $z$. For example,

$\displaystyle\|\eqref{eqcom47}\|_{L^{2}(B_{2^{k+1}}-B_{2^{k}})}$
$\displaystyle\lesssim$ $\displaystyle 2^{k\beta}\sum_{j\geq
k}\left\|\left(|\cdot|^{-(d-\sigma+\alpha+1)}\mathbbm{1}_{B_{2^{j+1}}-B_{2^{j}}}\right)*\left(|\cdot|^{-(d+\sigma-1)}\mathbbm{1}_{B_{2^{j}}}\right)*|f|\right\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle 2^{k\beta}\sum_{j\geq
k}2^{-k\alpha}\|f\|_{L^{2}}\lesssim 2^{-k(\alpha-\beta)}\|f\|_{L^{2}}.$

  4. (4)

For (B.15)-(B.17), we first apply a change of variable $y^{\prime}:=x-y\in
B_{4\langle x\rangle}$ so that $\langle x\rangle$ only affects the size of the
integration domain of $y^{\prime}$. By considering $x$ and $y$ dyadically, we
can finally arrive at the same pointwise bound $\langle
x\rangle^{-\frac{d}{2}+\beta-\alpha}\|f\|_{L^{2}}$. For instance, when $x\in
B_{2^{k+1}}-B_{2^{k}}$,

$\displaystyle\eqref{eqcom46}\lesssim\langle
x\rangle^{-d+\sigma+\beta}\int_{B_{2^{k+3}}}\langle
y^{\prime}\rangle^{-\alpha}\int_{B_{4\langle
y^{\prime}\rangle}^{c}}\frac{|f(y^{\prime}-z)|}{|z|^{d+\sigma}}dzdy$
$\displaystyle\lesssim$ $\displaystyle 2^{(-d+\sigma+\beta)k}\sum_{0\leq j\leq
k}\int_{B_{2^{j+3}}-B_{2^{j+2}}}2^{-j\alpha}\left[\left(|\cdot|^{-(d+\sigma)}\mathbbm{1}_{B_{2^{j+1}}^{c}}\right)*|f|\right](y^{\prime})dy^{\prime}$
$\displaystyle+2^{(-d+\sigma+\beta)k}\int_{B_{4}}\left[\left(|\cdot|^{-(d+\sigma)}\mathbbm{1}_{B_{8}^{c}}\right)*|f|\right](y^{\prime})dy^{\prime}$
$\displaystyle\lesssim$ $\displaystyle 2^{(-d+\sigma+\beta)k}\sum_{0\leq j\leq
k}2^{j(\frac{d}{2}-\sigma-\alpha)}\|f\|_{L^{2}}\lesssim
2^{\left(-\frac{d}{2}+\beta-\alpha\right)k}\|f\|_{L^{2}}.$

These bounds conclude the proof.

(iii) It suffices to estimate the commutator

$\|\langle
x\rangle^{\alpha}[D^{\sigma},\langle\cdot\rangle^{\beta}]D^{-\sigma}\langle\cdot\rangle^{-\gamma}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}.$
(B.21)

We decompose the domain of integration for $y$ into $\\{|y|\geq 2\langle
x\rangle\\}$ and $\\{|y|\leq 2\langle x\rangle\\}$, and further decompose the
domain of $z$ by comparing $|z|$ with $\frac{1}{4}|y|$ or $4\langle x\rangle$
respectively. Estimating $|\langle x\rangle^{\beta}-\langle
x-y\rangle^{\beta}|$, $|x-y-z|$ and $\langle z\rangle$ correspondingly, we
obtain

$\displaystyle\left|\langle
x\rangle^{\alpha}[D^{\sigma},\langle\cdot\rangle^{\beta}]D^{-\sigma}\langle\cdot\rangle^{-\gamma}f(x)\right|$
$\displaystyle=$ $\displaystyle C\left|\langle
x\rangle^{\alpha}\int_{\mathbb{R}^{d}}\frac{\langle x\rangle^{\beta}-\langle
x-y\rangle^{\beta}}{|y|^{d+\sigma}}\int_{\mathbb{R}^{d}}\frac{\langle
z\rangle^{-\gamma}f(z)}{|x-y-z|^{d-\sigma}}dzdy\right|$
$\displaystyle\lesssim$ $\displaystyle\langle
x\rangle^{\alpha}\int_{B_{2\langle
x\rangle}^{c}}|y|^{-(d+\sigma-\beta)}\int_{B_{\frac{1}{4}|y|}}|y|^{-(d-\sigma)}\langle
z\rangle^{-\gamma}|f(z)|dzdy$ (B.22) $\displaystyle+$ $\displaystyle\langle
x\rangle^{\alpha}\int_{B_{2\langle
x\rangle}^{c}}|y|^{-(d+\sigma-\beta+\gamma)}\left(|\cdot|^{-(d-\sigma)}*|f|\right)(x-y)dy$
(B.23) $\displaystyle+$ $\displaystyle\langle
x\rangle^{\alpha+\beta-1}\left(|\cdot|^{-(d+\sigma-1)}\mathbbm{1}_{B_{2\langle
x\rangle}}\right)*\left(|\cdot|^{-(d-\sigma)}\mathbbm{1}_{B_{8\langle
x\rangle}}\right)*\left(\langle\cdot\rangle^{-\gamma}\mathbbm{1}_{B_{4\langle
x\rangle}}|f|\right)(x)$ (B.24) $\displaystyle+$ $\displaystyle\langle
x\rangle^{\alpha+\beta-1}\int_{B_{2\langle
x\rangle}}|y|^{-(d+\sigma-1)}dy\int_{B_{4\langle
x\rangle}^{c}}\frac{|f(z)|}{|z|^{d-\sigma+\gamma}}dz$ (B.25)

For (B.22), (B.23) and (B.25), Hölder’s and Hardy-Littlewood-Sobolev
inequalities imply a pointwise bound by $\langle
x\rangle^{-(d/2+\gamma-\alpha-\beta)}\|f\|_{L^{2}}$, so their $L^{2}$-norm is
bounded by $\|f\|_{L^{2}}$. For (B.24), similar to (B.9), we can control their
$L^{2}$-norm on dyadic annulus $B_{2^{k+1}}-B_{2^{k}}$ by
$2^{-k(\gamma-\alpha-\beta)}\|f\|_{L^{2}}$ and on $B_{1}$ by $\|f\|_{L^{2}}$.
Combining these bounds, we get (B.21) and hence (B.5). ∎

##  Appendix C Nonlinear estimates in fractional Sobolev space

The following estimates in fractional Sobolev space are for the nonlinearity
in (6.1). We first state the main result and give some remarks.

######  Lemma C.1.

Given $d\geq 1$ and $W\in C^{1}(\mathbb{R}^{d}\to\mathbb{C}-\\{0\\})$
satisfying

$|\nabla W(x)|\leq C_{W}\langle
x\rangle^{-a}|W(x)|,\quad\forall\,x\in\mathbb{R}^{d}$ (C.1)

for some $C_{W}>0$, $a\in[0,1]$, then for $p>1$, $s\in(0,1)$, $r\in(1,\infty)$
and $\varepsilon\in\mathscr{S}(\mathbb{R}^{d}\to\mathbb{C})$, we have the
following estimates for

$R(\varepsilon):=\bigg{[}|W+\varepsilon|^{p-1}(W+\varepsilon)-|W|^{p-1}W-\frac{p+1}{2}|W|^{p-1}\varepsilon-\frac{p-1}{2}|W|^{p-3}W^{2}{\overline{\varepsilon}}\bigg{]}$
(C.2)

with $\tilde{r},r_{1},r_{2}\in(1,\infty)$ such that
$\frac{1}{r}-\frac{1}{\tilde{r}}\in\left[0,\frac{as}{d}\right)$,
$\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}$ and $(p-1)r_{1}>1$:

  * •

If $1<p\leq 2$,

$\|R(\varepsilon)\|_{\dot{W}^{s,r}}\lesssim_{d,p,s,r,\tilde{r},r_{1},r_{2},a,C_{W}}\||\varepsilon|^{p}\|_{L^{\tilde{r}}}+\|\varepsilon\|_{L^{(p-1)r_{1}}}^{p-1}\|\varepsilon\|_{\dot{W}^{s,r_{2}}}$
(C.3)

  * •

If $p>2$,

$\begin{split}\|R(\varepsilon)\|_{\dot{W}^{s,r}}\lesssim_{d,p,s,r,\tilde{r},r_{1},r_{2},a,C_{W}}&\||\varepsilon|^{p}\|_{L^{\tilde{r}}}+\||W|^{p-2}|\varepsilon|^{2}\|_{L^{\tilde{r}}}\\\
+&\left(\|\varepsilon\|_{L^{(p-1)r_{1}}}^{p-1}+\||W|^{p-2}|\varepsilon|\|_{L^{r_{1}}}\right)\|\varepsilon\|_{\dot{W}^{s,r_{2}}}.\end{split}$
(C.4)

######  Remark C.2.

When $p$ is large, these estimates easily follow the fractional chain rule for
$F^{\prime\prime}(W+t\varepsilon)$ with $F(z):=|z|^{p-1}z$. However, when
$p-1\ll 1$ (as $d\gg 1$), $F^{\prime\prime}(u)$ may not even be continuous for
generic $u\in C^{\infty}_{c}$, let alone estimates on $D^{s}$ derivatives.

Fortunately, we exploit the specialty of $F$ and $W$ here: the nonlinearity
$F$ only has a singularity at $0$ while $W$ does not vanish anywhere. Thus,
with an appropriate estimate for $\nabla W$, the Taylor expansion of $F$
around $W$ will be good enough.

###### Proof.

Denote $W=:\Sigma+i\Theta$, $\varepsilon=:\varepsilon_{1}+i\varepsilon_{2}$
and $R_{1}(\varepsilon):=\Re R(\varepsilon)$, $R_{2}(\varepsilon):=\Im
R(\varepsilon)$. Also let $F_{i}:\mathbb{R}^{2}\to\mathbb{R}$ with $i=1,2$ to
be

$F_{1}(a,b)=(a^{2}+b^{2})^{\frac{p-1}{2}}a,\quad
F_{2}(a,b)=(a^{2}+b^{2})^{\frac{p-1}{2}}b.$ (C.5)

Then for $i=1,2$,

$\displaystyle
R_{i}(\varepsilon)=F_{i}\left(\Sigma+\varepsilon_{1},\Theta+\varepsilon_{2}\right)-F_{i}(\Sigma,\Theta)-\partial_{1}F_{i}(\Sigma,\Theta)\varepsilon_{1}-\partial_{2}F_{i}(\Sigma,\Theta)\varepsilon_{2}$

From the expression of $F_{i}$, it is easy to see $F_{i}\in
C^{\infty}(\mathbb{R}^{2}-\\{0\\},\mathbb{R})$ and

$\left|\nabla^{n}F_{i}(a,b)\right|\lesssim_{n}(a^{2}+b^{2})^{\frac{p-n}{2}},\quad\forall\,n\geq
0,\,\forall\,(a,b)\neq(0,0).$ (C.6)

So if $(W+t\varepsilon)(x)\neq 0$ for any $t\in[0,1]$, the following
representation also holds

$R_{i}(\varepsilon)(x)=\sum_{j,k=1}^{2}\varepsilon_{j}(x)\varepsilon_{k}(x)\int_{0}^{1}\partial_{j}\partial_{k}F_{i}(\Sigma(x)+t\varepsilon_{1}(x),\Theta(x)+t\varepsilon_{2}(x))(1-t)dt$
(C.7)

We will prove (C.3)-(C.4) in two steps, following the strategy of [12] or
[61]. For simplicity, we only prove the case $1<p\leq 2$ while the $p>2$ case
follows in a similar way.

Step 1. Pointwise estimates. We first derive pointwise estimate for difference
of $R(\varepsilon)(x+h)-R(\varepsilon)(x)$ with $x,h\in\mathbb{R}^{d}$. For
notational simplicity, we define the difference operator

$\displaystyle\Delta_{h}f(x):=f(x+h)-f(x).$

Then we claim the following estimates:

$\begin{split}|\Delta_{h}R(\varepsilon)(x)|\lesssim_{p,C_{W},a}&\left(|\varepsilon(x+h)|^{p}+|\varepsilon(x)|^{p}\right)\min\\{1,\langle
x\rangle^{-a}|h|\\}\\\
+&\left(|\varepsilon(x+h)|^{p-1}+|\varepsilon(x)|^{p-1}\right)\left|\Delta_{h}\varepsilon(x)\right|;\end{split}$
(C.8)

We only consider estimates for $R_{1}(\varepsilon)$ as an example.

From (C.1), there exists a $c_{0}=c_{0}(C_{W},a)>0$ such that

$|W(x)-W(x+h)|\leq\frac{1}{3}|W(x)|,\quad\forall\,x\in\mathbb{R}^{d},\,\forall\,|h|\leq\langle
x\rangle^{a}c_{0}.$ (C.9)

Then we distinguish two cases by comparing $|h|$ with $\langle
x\rangle^{a}c_{0}$.

_Case 1. $|h|\geq\langle x\rangle^{a}c_{0}$._ It suffices to bound
$R_{1}(\varepsilon)(x)$ and $R_{1}(\varepsilon)(x+h)$ separately by
$|R(\varepsilon)(\cdot)|\lesssim_{p}|\varepsilon(\cdot)|^{p}$.

If $|\varepsilon(x)|\geq\frac{1}{2}|W(x)|$, the trivial bound
$|R_{1}(\varepsilon)(x)|\lesssim_{p}|W(x)|^{p}+|\varepsilon(x)|^{p}$ suffices.
If $|\varepsilon(x)|\leq\frac{1}{2}|W(x)|$, from (C.6) we know

$\mathop{\rm
sup}_{t\in[0,1]}\left|\nabla^{2}F_{1}(\Sigma(x)+t\varepsilon_{1}(x),\Theta(x)+t\varepsilon_{2}(x))\right|\lesssim_{p}|W(x)|^{p-2}.$

Thus (C.7) yields

$|R_{1}(\varepsilon)(x)|\lesssim|W(x)|^{p-2}|\varepsilon(x)|^{2}\lesssim_{p}|\varepsilon(x)|^{p}.$

_Case 2. $|h|\leq\langle x\rangle^{a}c_{0}$._ We consider the following three
subcases.

_Case 2.1. $|\varepsilon(x)|\geq\frac{1}{6}|W(x)|$._ Since
$|\varepsilon(x)|\gtrsim|W(x)|\sim|W(x+h)|$, we can estimate most interaction
terms in a direct fashion

$\displaystyle|\Delta_{h}(|W|^{p-1}\Sigma)|+|\Delta_{h}(|W|^{p-1}\varepsilon_{1})|+|\Delta_{h}(|W|^{p-3}(\Sigma^{2}-\Theta^{2})\varepsilon_{1})|$
$\displaystyle\qquad+|\Delta_{h}(|W|^{p-3}\Sigma\Theta\varepsilon_{2})|$
$\displaystyle\lesssim_{p,C_{W}}$ $\displaystyle|h|\langle
x\rangle^{-a}|W(x)|^{p-1}\left(|\varepsilon(x)|+|\varepsilon(x+h)|+|W(x)|\right)+|W(x)|^{p-1}|\Delta_{h}\varepsilon(x)|$
$\displaystyle\lesssim_{p}$ $\displaystyle|h|\langle
x\rangle^{-a}\left(|\varepsilon(x)|^{p}+|\varepsilon(x+h)|^{p}\right)+|\varepsilon(x)|^{p-1}|\Delta_{h}\varepsilon(x)|$

where we used $|\nabla W(x+th)|\leq C_{W}\langle
x+th\rangle^{-a}|W(x+th)|\sim\langle x\rangle^{-a}|W(x)|$ for $t\in[0,1]$ to
bound the difference of $W$. For the first term, we exploit the elementary
estimate

$\left|\frac{A^{\alpha}-B^{\alpha}}{A-B}\right|\sim_{\alpha}\max\\{A,B\\}^{\alpha-1},\quad\alpha>0$
(C.10)

for $A,B>0$. Thus

$\displaystyle\left|\Delta_{h}\left(|W+\varepsilon|^{p-1}(\Sigma+\varepsilon_{1})\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\Delta_{h}|W+\varepsilon|^{p-1}}{\Delta_{h}|W+\varepsilon|}\left(\Delta_{h}|W+\varepsilon|\right)(\Sigma+\varepsilon_{1})(x+h)\right|+|W+\varepsilon|^{p-1}(x)\left|\Delta_{h}(\Sigma+\varepsilon_{1})\right|$
$\displaystyle\lesssim_{p,C_{W}}$
$\displaystyle\left(|W+\varepsilon|^{p-1}(x+h)+|W+\varepsilon|^{p-1}(x)\right)\left(|h|\langle
x\rangle^{-a}|W(x)|+|\Delta_{h}\varepsilon|\right)$
$\displaystyle\lesssim_{p}$ $\displaystyle|h|\langle
x\rangle^{-a}\left(|\varepsilon(x)|^{p}+|\varepsilon(x+h)|^{p}\right)+\left(|\varepsilon(x)|^{p-1}+|\varepsilon(x+h)|^{p-1}\right)|\Delta_{h}\varepsilon(x)|.$

These estimates implies both (C.8).

_Case 2.2. $|\varepsilon(x)|\leq\frac{1}{6}|W(x)|\leq|\varepsilon(x+h)|$._
Noticing the symmetry between $x$ and $x+h$ term and that
$|\varepsilon(x+h)|\gtrsim|W(x)|\sim|W(x+h)|$, the bound in this case follows
exactly the same argument as Case 2.1 above.

_Case 2.3. $|\varepsilon(x)|\leq\frac{1}{6}|W(x)|$,
$|\varepsilon(x+h)|\leq\frac{1}{6}|W(x)|$._ In the following computation, we
slightly abuse notation to denote $W:=W(x)$, $W^{\prime}:=W(x+h)$ and
similarly for $\varepsilon$ and $\varepsilon^{\prime}$. Applying (C.7), we get

$\displaystyle R_{1}(\varepsilon)(x+h)-R_{1}(\varepsilon)(x)$ $\displaystyle=$
$\displaystyle\sum_{j,k=1}^{2}\Bigg{[}\varepsilon_{j}^{\prime}\varepsilon_{k}^{\prime}\int_{0}^{1}\partial_{j}\partial_{k}F_{1}(W^{\prime}+t\varepsilon^{\prime})(1-t)dt-\varepsilon_{j}\varepsilon_{k}\int_{0}^{1}\partial_{j}\partial_{k}F_{1}(W+t\varepsilon)(1-t)dt\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{j,k=1}^{2}\left(\Delta_{h}(\varepsilon_{j}\varepsilon_{k})\right)\int_{0}^{1}\partial_{j}\partial_{k}F_{1}(W^{\prime}+t\varepsilon^{\prime})(1-t)dt$
$\displaystyle+$
$\displaystyle\sum_{j,k=1}^{2}\varepsilon_{j}\varepsilon_{k}\int_{0}^{1}(1-t)\int_{0}^{1}[(W^{\prime}-W)+t(\varepsilon^{\prime}-\varepsilon)]\cdot\nabla\partial_{j}\partial_{k}F_{1}\big{(}(1-s)(W+t\varepsilon)+s(W^{\prime}+t\varepsilon^{\prime})\big{)}dsdt.$

Thanks to (C.9) and the smallness of $\varepsilon(x)$, $\varepsilon(x+h)$, we
have

$\left|(1-s)(W+t\varepsilon)+s(W^{\prime}+t\varepsilon^{\prime})-W\right|\leq\frac{1}{2}|W(x)|,\quad\forall\,t,s\in[0,1],$

and hence all the derivatives acting on $F_{1}$ are well-defined. This and
(C.6) for $k=3$ imply the boundedness

$\displaystyle|\Delta_{h}R_{1}(\varepsilon)(x)|\lesssim_{p,C_{W}}|\Delta_{h}\varepsilon|(|\varepsilon^{\prime}|+|\varepsilon|)|W|^{p-2}+|\varepsilon|^{2}\left(|h|\langle
x\rangle^{-a}|W|+|\Delta_{h}\varepsilon|\right)|W|^{p-3}.$

which leads to the bounds (C.8).

Step 2. Littlewood-Paley and maximal function estimates. Next, we apply the
argument in [61] to derive estimates in fractional Sobolev spaces. Denote the
homogeneous Littlewood-Paley decomposition as

$P_{k}f=\mathscr{F}^{-1}(\psi_{k}\cdot)\hat{f})=\check{\psi}_{k}*f,\quad
k\in\mathbb{Z},$

and the Hardy-Littlewood maximal operator as $M$. Since
$\psi_{k}=\psi(2^{-k}\cdot)$ with
$\psi\in\mathcal{C}^{\infty}_{c}(B_{2}-B_{\frac{1}{2}})$ for $k\in\mathbb{Z}$,
we have
$\check{\psi}_{k}=2^{kd}\check{\psi}(2^{k}\cdot)\in\mathscr{S}(\mathbb{R}^{d})$
and

$\displaystyle|\check{\psi}_{k}(x)|$ $\displaystyle\lesssim_{d,m}$
$\displaystyle 2^{dk}\langle 2^{k}x\rangle^{-m},\quad\forall\,m\geq 0,$ (C.11)
$\displaystyle\int_{\mathbb{R}^{d}}\psi_{k}(x)dx$ $\displaystyle=$
$\displaystyle 0.$ (C.12)

The Littlewood-Paley theory indicates

$\left\|f\right\|_{\dot{W}^{s,r}}\sim_{d,s,r}\left\|\sum_{j=-\infty}^{+\infty}2^{js}P_{j}f\right\|_{L^{r}}\sim_{d,s,r}\left\|\left(\sum_{j}2^{2js}|P_{j}f(\cdot)|^{2}\right)^{\frac{1}{2}}\right\|_{L^{r}}$
(C.13)

for $s\in(0,1)$ and $r\in(1,\infty)$. We recall the following estimates
involving maximal functions from [61, Chap. 2 Lemma 3.1-3.5]

$\displaystyle\begin{split}\int|P_{k}u(x)-P_{k}u(y)|\cdot|\check{\psi}_{j}(x-y)|dy\lesssim_{d}\min\\{2^{k-j},1\\}M(P_{k}u)(x),\end{split}$
(C.14)
$\displaystyle\begin{split}&\int|P_{k}u(x)-P_{k}u(y)|\cdot|\check{\psi}_{j}(x-y)|H(y)dy\\\
&\qquad\lesssim_{d}\min\\{2^{k-j},1\\}\left[M(P_{k}u)(x)MH(x)+M(|P_{k}u|H)(x)\right]\end{split}$
(C.15)

where $H$ is a positive function and $k,j\in\mathbb{Z}$.

Now we derive a pointwise estimate for the function in (C.13) and finish the
proof.

Using the vanishing condition (C.12) and the pointwise estimates (C.8), we
have

$\displaystyle\sum_{j}2^{2js}|P_{j}R(\varepsilon)(x)|^{2}\leq\sum_{j}2^{2js}\left[\int|R(\varepsilon)(y)-R(\varepsilon)(x)||\check{\psi}_{j}(x-y)|dy\right]^{2}$
(C.16) $\displaystyle\lesssim$
$\displaystyle\sum_{j}2^{2js}\left[\int\left(|\varepsilon(y)|^{p}+|\varepsilon(x)|^{p}\right)\min\\{1,\langle
x\rangle^{-a}|x-y|\\}|\check{\psi}_{j}(x-y)|dy\right]^{2}$ $\displaystyle+$
$\displaystyle\sum_{j}2^{2js}\left[\int\left(|\varepsilon(y)|^{p-1}+|\varepsilon(x)|^{p-1}\right)|\varepsilon(x)-\varepsilon(y)||\check{\psi}_{j}(x-y)|dy\right]^{2}$
(C.17)

For (C.16), we claim

$\eqref{eqdifterm1}\lesssim(M(|\varepsilon|^{p})(x))^{2}\langle
x\rangle^{-2as}.$ (C.18)

Indeed, the decay (C.11) implies

$\int|\check{\psi}_{j}(x-y)|\min\\{1,\langle
x\rangle^{-a}|x-y|\\}dy\lesssim\min\\{2^{-j}\langle x\rangle^{-a},1\\}.$

Then the bound on $|\varepsilon(x)|^{p}$ term follows

$\sum_{j\in\mathbb{Z}}2^{2js}\min\\{2^{-2j}\langle
x\rangle^{-2a},1\\}\sim\langle x\rangle^{-2as}$

thanks to $s\in(0,1)$. Likewise, for the other term involving
$|\varepsilon(y)|^{p}$, it suffices to show the maximal function bound

$A_{j}(x):=\int|\varepsilon(y)|^{p}|\check{\psi}_{j}(x-y)|\min\\{1,\langle
x\rangle^{-a}|x-y|\\}dy\lesssim\min\\{2^{-j}\langle
x\rangle^{-a},1\\}M(|\varepsilon|^{p})(x).$

Let $j_{0}\in\mathbb{Z}$ such that $2^{j_{0}}\leq\langle
x\rangle^{-a}<2^{j_{0}+1}$. Take $m=d+2$, then for $j\leq j_{0}$,

$\displaystyle A_{j}(x)$ $\displaystyle\lesssim$ $\displaystyle\int_{|z|\leq
2^{-j}}|\varepsilon(x-z)|^{p}2^{dj}dz+\sum_{k\geq-j}\int_{2^{k}\leq|z|\leq
2^{k+1}}|\varepsilon(x-z)|^{p}2^{dj}2^{-(j+k)m}dz$ $\displaystyle\lesssim$
$\displaystyle
M(|\varepsilon|^{p})(x)\left(1+\sum_{k\geq-j}2^{(d-m)(j+k)}\right)\lesssim
M(|\varepsilon|^{p})(x);$

and for $j>j_{0}$,

$\displaystyle A_{j}(x)$ $\displaystyle\lesssim$ $\displaystyle\int_{|z|\leq
2^{-j}}|\varepsilon(x-z)|^{p}2^{dj}2^{j_{0}-j}dz+\sum_{k=-j}^{-j_{0}}\int_{2^{k}\leq|z|\leq
2^{k+1}}|\varepsilon(x-z)|^{p}2^{dj}2^{j_{0}+k}2^{-m(j+k)}dz$ $\displaystyle+$
$\displaystyle\sum_{k\geq-j_{0}+1}\int_{2^{k}\leq|z|\leq
2^{k+1}}|\varepsilon(x-z)|^{p}2^{dj}2^{-m(j+k)}dz$ $\displaystyle\lesssim$
$\displaystyle 2^{j_{0}-j}M(|\varepsilon|^{p})(x).$

For (C.17), we apply Littlewood-Paley decomposition to
$\varepsilon(x)-\varepsilon(y)$ and then use (C.14)-(C.15) to get

$\displaystyle\eqref{eqdifterm2}$ $\displaystyle\lesssim$
$\displaystyle\sum_{j}2^{2js}\left|\sum_{k}\int\left(|\varepsilon(y)|^{p-1}+|\varepsilon(x)|^{p-1}\right)|P_{k}\varepsilon(x)-P_{k}\varepsilon(y)||\check{\psi}_{j}(x-y)|dy\right|^{2}$
$\displaystyle\lesssim$
$\displaystyle\sum_{j}2^{2js}\left[\sum_{k}\min\\{2^{k-j},1\\}M(P_{k}\varepsilon)(x)M(|\varepsilon|^{p-1})(x)\right]^{2}$
$\displaystyle+\sum_{j}2^{2js}\left[\sum_{k}\min\\{2^{k-j},1\\}M(|P_{k}\varepsilon|\cdot|\varepsilon|^{p-1})(x)\right]^{2}$
$\displaystyle\lesssim$
$\displaystyle\left[M(|\varepsilon|^{p-1})(x)\right]^{2}\sum_{k}2^{2ks}\left[M(P_{k}\varepsilon)(x)\right]^{2}+\sum_{k}2^{2ks}\left[M(|P_{k}\varepsilon|\cdot|\varepsilon|^{p-1})(x)\right]^{2}$

where the last inequality follows Young’s inequality.

From (C.13) and these estimates of (C.11)-(C.12), we bound the $\dot{W}^{s,r}$
norm by

$\displaystyle\|R(\varepsilon)\|_{\dot{W}^{s,r}}$ $\displaystyle\lesssim$
$\displaystyle\left\|M(|\varepsilon|^{p})\langle
x\rangle^{-as}\right\|_{L^{r}}+\left\|M(|\varepsilon|^{p-1})(x)\left\|2^{ks}M(P_{k}\varepsilon)(x)\right\|_{l^{2}_{k}}\right\|_{L^{r}}$
$\displaystyle+\left\|\left\|2^{ks}M(|P_{k}\varepsilon|\cdot|\varepsilon|^{p-1})(x)\right\|_{l^{2}_{k}}\right\|_{L^{r}}$
$\displaystyle\lesssim$
$\displaystyle\||\varepsilon|^{p}\|_{L^{\tilde{r}}}+\|\varepsilon\|_{L^{(p-1)r_{1}}}^{p-1}\left\|\left\|2^{ks}|P_{k}\varepsilon(x)|\right\|_{l^{2}_{k}}\right\|_{L^{r_{2}}}+\left\||\varepsilon|^{p-1}(x)\left\|2^{ks}|P_{k}\varepsilon|(x)\right\|_{l^{2}_{k}}\right\|_{L^{r}}$
$\displaystyle\lesssim$
$\displaystyle\||\varepsilon|^{p}\|_{L^{\tilde{r}}}+\|\varepsilon\|_{L^{(p-1)r_{1}}}^{p-1}\|\varepsilon\|_{\dot{W}^{s,r_{2}}}$

with $\frac{1}{r}-\frac{1}{\tilde{r}}\in\left[0,\frac{as}{d}\right)$,
$\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}$ and $(p-1)r_{1}>1$. In the
second-to-last inequality, we invoked the Fefferman-Stein maximal theorem [20]

$\left\|\|M(h_{k})(x)\|_{l^{2}_{k}}\right\|_{L^{q}_{x}}\lesssim\left\|\|h_{k}(x)\|_{l^{2}_{k}}\right\|_{L^{q}_{x}}$

for a sequence of functions $\\{h_{k}\\}$ and $q\in(1,\infty)$.

∎

Next, we present a difference estimate in $L^{p}$ space.

######  Lemma C.3.

Given $d\geq 1$ and $W\in C^{1}(\mathbb{R}^{d}\to\mathbb{C})$, $p>1$ and
$u,v\in\mathscr{S}(\mathbb{R}^{d}\to\mathbb{C})$, for $R(u)$ defined as in
(C.2), we have difference estimate for $r,r_{1},r_{2}\in[1,\infty]$ with
$\frac{1}{r}=\frac{1}{r_{1}}+\frac{p-1}{r_{2}}$:

  * •

If $1<p\leq 2$,

$\|R(u)-R(v)\|_{L^{r}}\lesssim_{d,p,r,r_{1},r_{2}}\|u-v\|_{L^{r_{1}}}\left(\|u\|_{L^{r_{2}}}^{p-1}+\|v\|_{L^{r_{2}}}^{p-1}\right);$
(C.19)

  * •

If $p>2$,

$\begin{split}&\|R(u)-R(v)\|_{L^{r}}\\\
\lesssim_{d,p,s,r,r_{1},r_{2}}&\|u-v\|_{L^{r_{1}}}\left(\|u\|_{L^{r_{2}}}^{p-1}+\|v\|_{L^{r_{2}}}^{p-1}+\left(\|u\|_{L^{r_{2}}}+\|v\|_{L^{r_{2}}}\right)\|W\|_{L^{r_{2}}}^{p-2}\right).\end{split}$
(C.20)

###### Proof.

It suffices to prove the corresponding pointwise estimate for any $x$

$\displaystyle|R(u)-R(v)|\lesssim_{p}\left\\{\begin{array}[]{ll}|u-v|(|u|^{p-1}+|v|^{p-1})&1<p\leq
2,\\\ |u-v|(|u|^{p-1}+|v|^{p-1}+(|u|+|v|)|W|^{p-2})&p\geq
2.\end{array}\right.$

The proof is similar to Step 1 in Lemma C.1 and thus is omited.

∎

##  Appendix D Local Well-posedness in $\dot{H}^{\sigma}$

In this section, we prove local well-posedness of the intercritical
Schrödinger equation (NLS)

$\displaystyle\left\\{\begin{array}[]{l}i\partial_{t}u+\Delta
u+u|u|^{p-1}=0\\\ u_{|t=0}=u_{0}\in\dot{H}^{\sigma}\end{array}\right.$

and the renormalized system (1.14) (we denote the time variable as $t$)

$\displaystyle\left\\{\begin{array}[]{l}i\partial_{t}Z+\mathcal{H}Z=F,\\\
Z_{|t=0}=Z_{0}\in\dot{H}^{\sigma}\times\dot{H}^{\sigma}.\end{array}\right.$

with ${\mathcal{H}}$, $F=\left(\begin{array}[]{c}-R(\varepsilon,Q_{b})\\\
\overline{R(\varepsilon,Q_{b})}\end{array}\right)$ defined by (1.19), (1.20)
and (6.1). Here we assume $s_{c}<\sigma\leq\alpha_{c}=\frac{d}{d+2-2s_{c}}$
above the critical scaling. We will also use the exponents
$r_{1},r_{2},q_{1},\beta$ as in (6.19)-(6.20). We begin with the original
system (NLS).

######  Proposition D.1.

Suppose $d\geq 1$ and $0<s_{c}<\min\left\\{1,\frac{d}{2}\right\\}$ and
consider the Cauchy problem (NLS). For any $s_{c}<\sigma\leq\alpha_{c}$, we