# Growth of Sobolev Norms for $2d$ NLS with harmonic potential

Fabrice Planchon  Sorbonne Université, CNRS, IMJ-PRG F-75005 Paris, France
<EMAIL_ADDRESS>,  Nikolay Tzvetkov  Département de
Mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302
Cergy-Pontoise Cedex, France and Institut Universitaire de France
<EMAIL_ADDRESS>and  Nicola Visciglia  Dipartimento di Matematica,
Università di Pisa, Largo Bruno Pontecorvo 5, 56100 Pisa, Italy
<EMAIL_ADDRESS>
###### Abstract.

We prove polynomial upper bounds on the growth of solutions to $2d$ cubic NLS
where the Laplacian is confined by the harmonic potential. Due to better
bilinear effects our bounds improve on those available for the $2d$ cubic NLS
in the periodic setting: our growth rate for a Sobolev norm of order $s=2k$,
$k\in\mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an
direct proof, based on integration by parts, of bilinear estimates associated
with the harmonic oscillator.

The first author was supported by ERC grant ANADEL no 757996, the second
author by ANR grant ODA (ANR-18-CE40-0020-01), the third author by PRIN grant
2020XB3EFL. N.V. acknowledge the Gruppo Nazionale per l’ Analisi Matematica,
la Probabilità e le loro Applicazioni (GNAMPA) of the Istituzione Nazionale di
Alta Matematica (INDAM)

Dedicated to Professor Vladimir Georgiev for his 65’s birthday

##  1\. Introduction

In recent years, growth of Sobolev norms for solutions to nonlinear dispersive
equations generated a huge interest, in relation with weak turbulence
phenomena. Concerning upper bounds, we quote the pioneering work of Bourgain
[3] and its extension in a series of subsequent papers ([7],[8], [9], [14],
[18], [19], [21] to quote only a few of them). On the other end, growth of
Sobolev norm cannot occur in settings where the dispersive effect is too
strong. For instance consider the translation invariant cubic defocusing NLS
on ${\mathbb{R}}^{2}$. Then [10] proved the long standing conjecture that
nonlinear solutions scatter to free waves when time goes to infinity and hence
no growth phenomena is possible in such setting.

We are interested in the growth of solutions to the following nonlinear
Schrödinger equation:

(1.1) $\begin{cases}i\partial_{t}u+Au\pm
u|u|^{2}=0,\quad(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{2}\\\
u(0,x)=\varphi(x)\in{\mathcal{H}}^{s}\end{cases}$

where $x=(x_{1},x_{2})$, the operator $A$ is the usual Laplacian with an
harmonic potential,

(1.2) $A=-\Delta+|x|^{2},\quad\hbox{ where
}\Delta=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2},\quad|x|^{2}=x_{1}^{2}+x_{2}^{2}$

and $\|\varphi\|_{{\mathcal{H}}^{s}}=\|A^{s/2}\varphi\|_{L^{2}}$, where in
general we use the notation $L^{p}=L^{p}({\mathbb{R}}^{2})$. We shall also
denote $L^{p}_{t,x}=L^{p}({\mathbb{R}}\times{\mathbb{R}}^{2})$ to emphasize
the Lebesgue space of space-time dependent functions.

Let us first comment briefly about the local Cauchy theory associated with
(1.1). By combining preservation of regularity for the linear flow,
$\|e^{itA}\varphi\|_{{\mathcal{H}}^{s}}=\|\varphi\|_{{\mathcal{H}}^{s}}$ and
that ${\mathcal{H}}^{s}$ is an algebra for $s>1$, one proves existence of a
local solution to (1.1) by fixed point; its local time of existence depends on
the ${\mathcal{H}}^{s}$ norm of the initial datum. Moreover the solution map
is Lipschitz continuous. In order to globalize our solution one can rely on
the Brezis-Gallouët inequality (see [4]) provided that

(1.3)
$\sup_{t\in(-T_{min}(\varphi),T_{max}(\varphi))}\|u(t,x)\|_{\mathcal{H}^{1}}<\infty$

where $(-T_{min}(\varphi),T_{max}(\varphi))$, with
$T_{min}(\varphi),T_{max}(\varphi)>0$, is the maximal time interval of
existence of the solution associated with (1.1). In particular, assuming
(1.3), $T_{max}(\varphi)=T_{min}(\varphi)=\infty$ and a double exponential
bound holds:

(1.4) $\|u(t,x)\|_{\mathcal{H}^{s}}\leq C\exp(C\exp(C|t|)).$

Solutions to (1.1) satisfy the conservation of the Hamiltonian

$\frac{1}{2}\|u(t,x)\|_{{\mathcal{H}}^{1}}^{2}\pm\frac{1}{4}\|u(t,x)\|_{L^{4}}^{4}=const\,,$

therefore, in the defocusing case, (1.3) is automatically satisfied, while in
the focusing case it is not granted for free. Of course, by using more
sophisticated tools, e.g. Bourgain’s spaces $X^{s,b}$ associated with
$i\partial_{t}+A$, one can deal with initial data at lower regularity than
${\mathcal{H}}^{1+\varepsilon}$. These $X^{s,b}$ spaces will play a key role
in our analysis as they allow us to exploit a bilinear effect associated with
the propagator $e^{itA}$. They will be defined in Section 3 where we also
provide more useful facts about Cauchy theory.

Our main goal is to improve (1.4) and prove polynomial upper bounds for the
quantity $\|u(t,x)\|_{{\mathcal{H}}^{s}}$ when $t\rightarrow\pm\infty$ with
$s>1$. Along the rest of the paper the following equivalence of norms will be
useful: for every $s\geq 0$ there exist $C>0$ such that

(1.5) $\frac{1}{C}(\|D^{s}u\|_{L^{2}}^{2}+\|\langle
x\rangle^{s}u\|_{L^{2}}^{2})\leq\|\varphi\|_{{\mathcal{H}}^{s}}^{2}\leq
C(\|D^{s}u\|_{L^{2}}^{2}+\|\langle x\rangle^{s}u\|_{L^{2}}^{2})$

where $D^{s}$ is the operator associated with the Fourier multiplier
$|\xi|^{s}$ and $\langle x\rangle=\sqrt{1+x_{1}^{2}+x_{2}^{2}}$. The proof of
the equivalence (1.5) is a special case of a more general result proved in [1,
11]. In particular establishing growth upper bounds on ${\mathcal{H}}^{s}$
norm of the solution is equivalent to establish polynomial bounds on the
classical Sobolev norms $H^{s}$ and the corresponding moment of order $s$. We
now state our main result.

######  Theorem 1.1.

Let $\epsilon>0$ and $k\in{\mathbb{N}}$. For every global solution $u$ to
(1.1) such that $u(t,x)\in{\mathcal{C}}({\mathbb{R}},{\mathcal{H}}^{2k})$ and

(1.6) $\sup_{t\in\mathbb{R}}\|u(t,x)\|_{\mathcal{H}^{1}}<\infty$

there exists a constant $C$ such that

$\|D^{2k}u(t,x)\|_{L^{2}}+\|\langle x\rangle^{2k}u(t,x)\|_{L^{2}}\leq C\langle
t\rangle^{\frac{2(2k-1)}{3}+\epsilon}.$

Our bound may be compared to the corresponding bound for solutions to NLS on a
generic compact $2-d$ manifold $M^{2}$ and more specifically on the torus
${\mathbb{T}}^{2}$. In fact at the best of our knowledge the best known upper
bound available on the growth of the classical Sobolev norm
$H^{2k}({\mathbb{T}}^{2})$ for solutions to cubic NLS on ${\mathbb{T}}^{2}$ is
$(1+t)^{2k-1+\epsilon}$, as proved in [21], [14]. Notice also that in our case
we control the growth of the moments as well (see also [20] for a different
perspective on the moments).

Theorem 1.1 may also be compared with [7, Theorem 2], where the same bound on
the growth of Sobolev norm was achieved for the translation invariant cubic
NLS posed on ${\mathbb{R}}^{2}$, at a time where Dodson’s definitive result
was not available. As already mentioned, unlike the situation considered in
Theorem 1.1, where in general scattering theory is not available, in the
euclidean setting one can deduce uniform boundedness of high order Sobolev
norms, at least in the defocusing situation. Nevertheless bounds provided in
Theorem 1.1 are still meaningful and non trivial in the flat case either, if
one considers solutions to the focusing NLS such that the $H^{1}$ norm is
uniformly bounded. In fact under this assumption it is not true in general
that the solutions scatter to a free wave and hence the uniform boundedness of
Sobolev norms is not granted.

It would be very interesting to construct solutions to the defocusing (1.1)
such that the $H^{k}$ norms do not remain bounded in time for some $k>1$.
Unfortunately such results are rare in the context of canonical dispersive
models (with the notable exception of [12]).

##  2\. $X^{s,b}$ framework and linear estimates

We first define $X^{s,b}$ spaces associated with the harmonic oscillator in
dimension two: the spectrum of the harmonic oscillator is given by the
following set of integers $\\{2n+2,n\in{\mathbb{N}}\\}$. For every
$n\in{\mathbb{N}}$ we shall denote by $\Pi_{n}$ the orthogonal projector on
the eigenspace associated with the eigenvalue $2n+2$. Then the $X^{s,b}$ norm
is given by the expression

$\|u\|_{X^{s,b}}^{2}=\sum_{n\in{\mathbb{N}}}(2n+2)^{s}\big{\|}\langle\tau+2n+2\rangle^{b}{\mathcal{F}}_{t\rightarrow\tau}(\Pi_{n}u(t,x))\big{\|}^{2}_{L^{2}_{\tau,x}}$

where $u(t,x)$ is a function globally defined on space-time and
${\mathcal{F}}_{t\rightarrow\tau}$ denotes the Fourier transform with respect
to the time variable. Along with the $X^{s,b}$ spaces, which are defined for
global space-time functions, we also introduce its localized version for every
$T>0$. More precisely for functions $v(t,x)$ on the strip
$(-T,T)\times{\mathbb{R}}^{2}$ we define:

$\|v\|_{X^{s,b}_{T}}=\inf_{\begin{subarray}{c}\tilde{v}\in X^{s,b}\\\
v(t,x)=\tilde{v}(t,x)_{|(-T,T)\times{\mathbb{R}}^{2}}\end{subarray}}\|\tilde{v}\|_{X^{s,b}}.$

The main result of this section is the continuity of suitable linear operators
in the Bourgain’s spaces $X^{s,b}_{T}$.

######  Proposition 2.1.

For every $\delta\in(0,\frac{1}{2})$, $b\in(0,1)$ there exists $C>0$ such that
we have the following estimate for every $T>0$:

(2.1) $\|Lu\|_{X^{-\frac{1}{2}+\delta,\frac{1}{2}-\delta+2\delta b}_{T}}\leq
C\|u\|_{X^{\frac{1}{2}+\delta,\frac{1}{2}-\delta+2\delta b}_{T}}$ (2.2)
$\|Lu\|_{X^{\delta,(1-\delta)b}_{T}}\leq
C\|u\|_{X^{1+\delta,(1-\delta)b}_{T}}$

where $L$ can be either $\partial_{x_{i}}$, $i=1,2$ or multiplication by
$\langle x\rangle$.

###### Proof.

We prove Proposition 2.1 without the time localization. The corresponding
version in localized Bourgain’ spaces is straightforward. We will prove the
following bounds:

(2.3) $\displaystyle\|Lu\|_{X^{0,b}}\leq C\|u\|_{X^{1,b}},\quad b\in[0,1]$
(2.4) $\displaystyle\|Lu\|_{X^{1,0}}\leq C\|u\|_{X^{2,0}}.$

Notice that (2.2) follows by interpolation between (2.3) and (2.4). Moreover
we get

(2.5) $\|Lu\|_{X^{-1,0}}\leq C\|u\|_{X^{0,0}}$

by duality from (2.3) for $b=0$, and we also get

(2.6) $\|Lu\|_{X^{-1/2,1/2}}\leq C\|u\|_{X^{1/2,1/2}}$

by interpolation between (2.5) and (2.3) for $b=1$. Then (2.1) follows,
interpolating (2.3) and (2.6). Hence we focus on (2.3) and (2.4). Since the
proof is slightly different depending from the operator $L$ that we consider,
we consider two cases.

First case: proof of (2.3) and (2.4) for $Lu=\partial_{x_{i}}u$

First we prove that, for space-time dependent functions $u(t,x)$ we have

(2.7) $\|\partial_{x_{i}}u\|_{X^{0,0}}\leq C\|u\|_{X^{1,0}}\,.$

This estimate is a consequence of the following one for time independent
functions $v(x)$:

$\|\partial_{x_{i}}v\|_{L^{2}}\leq C\|\sqrt{A}v\|_{L^{2}}$

that in turn follows by $\|\sqrt{A}v\|_{L^{2}}=\|v\|_{{\mathcal{H}}^{1}}$ and
by recalling (1.5) for $s=1$. Next we prove

(2.8) $\|\partial_{x_{i}}u\|_{X^{0,1}}\leq C\|u\|_{X^{1,1}}\,,$

and by interpolation with (2.7), (2.3) will follow for $L=\partial_{x_{i}}$.
As $\|w(t,x)\|_{X^{0,1}}$ is equivalent to
$\|(i\partial_{t}+A)w\|_{L^{2}_{t,x}}+\|w\|_{L^{2}_{t,x}}$, in order to get
(2.8) we estimate

(2.9)
$\displaystyle\|(i\partial_{t}+A)\partial_{x_{i}}u\|_{L^{2}_{t,x}}+\|\partial_{x_{i}}u\|_{L^{2}_{t,x}}$
$\displaystyle=\|\partial_{x_{i}}(i\partial_{t}+A)u+[|x|^{2},\partial_{x_{i}}]u\|_{L^{2}_{t,x}}+\|\partial_{x_{i}}u\|_{L^{2}_{t,x}}$
(2.10)
$\displaystyle\leq\|\partial_{x_{i}}(i\partial_{t}+A)u\|_{L^{2}_{t,x}}+2\||x|u\|_{L^{2}_{t,x}}+\|\partial_{x_{i}}u\|_{L^{2}_{t,x}}\,.$

By combining (2.7) with the following identity

(2.11)
$\|\sqrt{A}v\|_{L^{2}}^{2}=(Av,v)=\|\nabla_{x}v\|_{L^{2}}^{2}+\||x|v\|_{L^{2}}^{2}$

we can continue (2.9) as follows:

$(\dots)\leq\|(i\partial_{t}+A)u\|_{X^{1,0}}+3\|\sqrt{A}u\|_{L^{2}_{t,x}}\leq\|u\|_{X^{1,1}}+3\|u\|_{X^{1,0}}\leq
4\|u\|_{X^{1,1}}\,.$

and (2.8) for $L=\partial_{x_{i}}$ follows. Next we prove (2.4) (where
$L=\partial_{x_{i}}$), namely

(2.12) $\|\partial_{x_{i}}u\|_{X^{1,0}}\leq C\|u\|_{X^{2,0}}.$

This estimate is a consequence of the following one for time independent
functions $v(x)$:

$\|\sqrt{A}\partial_{x_{i}}v\|_{L^{2}}\leq C\|Av\|_{L^{2}},$

that in turn is equivalent to

(2.13) $(A\partial_{x_{i}}v,\partial_{x_{i}}v)\leq C(Av,Av).$

As on the r.h.s. we get $\|v\|_{{\mathcal{H}}^{2}}^{2}$, by (1.5) and
elementary considerations it is sufficient to prove

(2.14) $\int|x|^{2}|\partial_{x_{i}}v|^{2}\leq
C(\|D^{2}v\|_{L^{2}}^{2}+\|\langle x\rangle^{2}v\|_{L^{2}}^{2})\,.$

In turn this last inequality follows by combining integration by parts and the
Cauchy-Schwarz inequality:

$\displaystyle\int|x|^{2}|\partial_{x_{i}}v|^{2}=-\int|x|^{2}\partial_{x_{i}}^{2}v\bar{v}-2\int
x_{i}\partial_{x_{i}}v\bar{v}$
$\displaystyle\leq\|\partial_{x_{i}}^{2}v\|_{L^{2}}\||x|^{2}v\|_{L^{2}}+2\||x|\partial_{x_{i}}v\|_{L^{2}}\|v\|_{L^{2}}$
$\displaystyle\leq\frac{1}{2}\|D^{2}u\|_{L^{2}}^{2}+\frac{1}{2}\|\langle
x\rangle^{2}v\|_{L^{2}}^{2}+\frac{1}{2}\||x|\partial_{x_{i}}v\|_{L^{2}}^{2}+2\|\langle
x\rangle^{2}v\|_{L^{2}}^{2}\,,$

from which we easily conclude moving
$\frac{1}{2}\||x|\partial_{x_{i}}v\|_{L^{2}}^{2}$ to the left-hand side.

Second case: proof of (2.3) and (2.4) for $Lu=\langle x\rangle u$

The proof follows the same steps as in the case $L=\partial_{x_{i}}$, with
minor modifications. First notice that we have for space-time dependent
functions $u(t,x)$ the following estimate:

(2.15) $\|\langle x\rangle u\|_{X^{0,0}}\leq C\|u\|_{X^{1,0}}.$

This is a consequence of the following estimate for time independent functions
$v(x)$:

$\|\langle x\rangle v\|_{L^{2}}\leq C\|\sqrt{A}v\|_{L^{2}}$

that in turn follows by noticing that
$\|\sqrt{A}v\|_{L^{2}}=\|v\|_{{\mathcal{H}}^{1}}$ and recalling (1.5) for
$s=1$. Moreover we have

(2.16) $\|\langle x\rangle u\|_{X^{0,1}}\leq C\|u\|_{X^{1,1}}.$

that by interpolation with (2.15) implies (2.3) for $L=\langle x\rangle$. In
order to prove this estimate recall again that $\|w(t,x)\|_{X^{0,1}}$ is
equivalent to $\|i\partial_{t}w+Aw\|_{L^{2}_{t,x}}+\|w\|_{L^{2}_{t,x}}$ and
hence we compute

$\displaystyle\|(i\partial_{t}+A)(\langle x\rangle
u)\|_{L^{2}_{t,x}}+\|\langle x\rangle u\|_{L^{2}_{t,x}}$
$\displaystyle=\|\langle x\rangle(i\partial_{t}+A)u+[\Delta,\langle
x\rangle]u\|_{L^{2}_{t,x}}+\|\langle x\rangle u\|_{L^{2}_{t,x}}$
$\displaystyle\leq\|\langle
x\rangle(i\partial_{t}+A)u\|_{L^{2}_{t,x}}+\|2\nabla(\langle
x\rangle)\cdot\nabla u+\Delta(\langle x\rangle)u\|_{L^{2}_{t,x}}+\|\langle
x\rangle u\|_{L^{2}_{t,x}}$ $\displaystyle\leq C(\|\langle
x\rangle(i\partial_{t}+A)u\|_{L^{2}_{t,x}}+\|\nabla
u\|_{L^{2}_{t,x}}+\|\langle x\rangle u\|_{L^{2}_{t,x}})\,.$

By combining (2.15) with the identity
$\|\sqrt{A}u\|_{L^{2}_{t,x}}=\|u\|_{{\mathcal{H}}^{1}}$ and by recalling (1.5)
for $s=1$, we can proceed with our estimate above,

$(...)\leq
C(\|(i\partial_{t}+A)u\|_{X^{1,0}}+\|\sqrt{A}u\|_{L^{2}_{t,x}})=C(\|u\|_{X^{1,1}}+\|u\|_{X^{1,0}})\leq
C\|u\|_{X^{1,1}}\,.$

Next we prove (2.4) (where $L=\langle x\rangle$), namely

(2.17) $\|\langle x\rangle u\|_{X^{1,0}}\leq C\|u\|_{X^{2,0}}.$

This estimate is a consequence of the following one for time independent
functions $v(x)$:

$\|\sqrt{A}(\langle x\rangle v)\|_{L^{2}}\leq C\|Av\|_{L^{2}}$

that in turn is equivalent to

(2.18) $(A(\langle x\rangle v),\langle x\rangle v)\leq
C\|v\|_{{\mathcal{H}}^{2}}.$

By (1.5) it is equivalent to

$\|\nabla(\langle x\rangle v)\|_{L^{2}}^{2}+\|\langle
x\rangle|x|v\|_{L^{2}}^{2}\leq C(\|D^{2}v\|_{L^{2}}^{2}+\|\langle
x\rangle^{2}v\|_{L^{2}}^{2}).$

In turn, developing the gradient on the l.h.s. the estimate above follows from

$\int\langle x\rangle^{2}|\nabla v|^{2}\leq C(\|D^{2}v\|_{L^{2}}^{2}+\|\langle
x\rangle^{2}v\|_{L^{2}}^{2})$

whose proof proceeds by integration by parts and Cauchy-Schwarz inequality as
we did for (2.14). ∎

##  3\. The Cauchy theory in $X^{s,b}$ and consequences

We first obtain a trilinear estimate, whose proof heavily relies on the
analysis of [16] (also available as [17]); for the sake of completeness, we
provide a relatively elementary proof of the crucial bilinear estimate from
[16] in the appendix, using the bilinear virial techniques from [15]. The only
novelty in our trilinear estimate is that we prove a tame estimate, while such
an estimate was not needed for the low regularity analysis of [16]. We first
recall the following key bilinear estimate [16, Theorem 2.3.13]. There exists
$\delta_{0}\in(0,\frac{1}{2}]$ such that for every $\delta\in(0,\delta_{0}]$
there exists $b^{\prime}<\frac{1}{2}$ and $C>0$ such that:

(3.1) $\|\Delta_{N}(u)\,\Delta_{M}(v)\|_{L^{2}((0,T);L^{2})}\leq
C(\min(M,N))^{\delta}\Big{(}\frac{\min(M,N)}{\max(M,N)}\Big{)}^{\frac{1}{2}-\delta}\|\Delta_{N}(u)\|_{X^{0,b^{\prime}}_{T}}\|\Delta_{M}(v)\|_{X^{0,b^{\prime}}_{T}}$

where $\Delta_{N}$, $\Delta_{M}$ are the Littelwood-Paley localization
associated with $A$ and $N$, $M$ are dyadic integers.

######  Proposition 3.1.

Let $0<T<1$ and $\epsilon>0$ be fixed. Then there exist $C>0$, $b>1/2$ and
$\gamma>0$ such that for $s\geq\varepsilon$:

(3.2)
$\Big{\|}\int_{0}^{t}e^{i(t-\tau)A}(u_{1}(\tau)u_{2}(\tau)\bar{u}_{3}(\tau))d\tau\Big{\|}_{X^{s,b}_{T}}\leq
CT^{\gamma}\sum_{\sigma\in{\mathcal{S}}_{3}}\|u_{\sigma(1)}\|_{X^{s,b}_{T}}\|u_{\sigma(2)}\|_{X^{\varepsilon,b}_{T}}\|u_{\sigma(3)}\|_{X^{\varepsilon,b}_{T}}.$

###### Proof.

Using standard arguments (see for instance [5, Proposition 3.3]), it suffices
to prove that

$\|u_{1}u_{2}\bar{u}_{3}\|_{X^{s,-b^{\prime}}_{T}}\leq
C\sum_{\sigma\in{\mathcal{S}}_{3}}\|u_{\sigma(1)}\|_{X^{s,b}_{T}}\|u_{\sigma(2)}\|_{X^{\varepsilon,b}_{T}}\|u_{\sigma(3)}\|_{X^{\varepsilon,b}_{T}}$

for some $b>1/2$, $b^{\prime}<1/2$ such that $b+b^{\prime}<1$. Using duality,
the last estimate is equivalent to:

$\Big{|}\int\int u_{1}u_{2}\bar{u}_{3}\bar{u}_{0}\Big{|}\leq
C\|u_{0}\|_{X^{-s,b^{\prime}}_{T}}\sum_{\sigma\in{\mathcal{S}}_{3}}\|u_{\sigma(1)}\|_{X^{s,b}_{T}}\|u_{\sigma(2)}\|_{X^{\varepsilon,b}_{T}}\|u_{\sigma(3)}\|_{X^{\varepsilon,b}_{T}}\,.$

where $\int\int$ denotes a space-time integral on
${\mathbb{R}}^{2}\times{\mathbb{R}}$ with respect to the Lebesgue measure
$dxdt$. We now perform a Littlewood-Paley decomposition in the left-hand side
of the last inequality and using a symmetry argument, we are reduced to
obtaining a bound on

$\Big{|}\sum_{N_{1}\geq N_{2}\geq
N_{3}}\sum_{N_{0}}\int\int\Delta_{N_{0}}(\bar{u}_{0})\Delta_{N_{1}}(u_{1})\Delta_{N_{2}}(u_{2})\Delta_{N_{3}}(\bar{u}_{3})\Big{|},$

where the summation is meant over dyadic values of $N_{1}$, $N_{2}$, $N_{3}$
and $N_{0}$. The other possible orders of magnitudes of $N_{1}$, $N_{2}$ and
$N_{3}$ provide all permutations involved in the sum of the right hand-side of
(3.2).

First case: $N_{0}\geq N_{1}^{1+\delta}$ for some $\delta>0$

In this case, we can apply the $2d$ version of [16, Lemme 2.1.23] to obtain
that for every $K$ there is $C_{K}$ such that

$\Big{|}\int\int\Delta_{N_{0}}(\bar{u}_{0})\Delta_{N_{1}}(u_{1})\Delta_{N_{2}}(u_{2})\Delta_{N_{3}}(\bar{u}_{3})\Big{|}\leq
C_{K}N_{0}^{-K}\|\Delta_{N_{0}}u_{0}\|_{X^{0,b^{\prime}}_{T}}\|\Delta_{N_{1}}u_{1}\|_{X^{0,b^{\prime}}_{T}}\|\Delta_{N_{2}}u_{2}\|_{X^{0,b^{\prime}}_{T}}\|\Delta_{N_{3}}u_{3}\|_{X^{0,b^{\prime}}_{T}}\,.$

where $b^{\prime}<\frac{1}{2}$. Now we can readily perform the $N_{0}$,
$N_{1}$, $N_{2}$, $N_{3}$ summations thanks to the large negative power of
$N_{0}$.

Second case: $N_{0}\leq N_{1}^{1+\delta}$, with $\delta>0$ to be chosen later
depending on $\varepsilon$

Combining Cauchy-Schwarz and (3.1), we write

$\displaystyle\Big{|}\int\int\Delta_{N_{0}}(\bar{u}_{0})\Delta_{N_{1}}(u_{1})\Delta_{N_{2}}(u_{2})\Delta_{N_{3}}(\bar{u}_{3})\Big{|}\
\leq$
$\displaystyle\|\Delta_{N_{1}}(u_{1})\Delta_{N_{2}}(u_{2})\|_{L^{2}((0,T);L^{2})}\|\Delta_{N_{0}}(u_{0})\Delta_{N_{3}}(u_{3})\|_{L^{2}((0,T);L^{2})}$
$\displaystyle\leq$ $\displaystyle
C(N_{2}N_{3})^{\delta}\Big{(}\frac{N_{2}N_{3}}{N_{0}N_{1}}\Big{)}^{\frac{1}{2}-\delta}\prod_{j=1}^{4}\|\Delta_{N_{j}}(u_{j})\|_{X^{0,b^{\prime}}_{T}}\,.$

A normalization yields that it suffices to prove the following inequality:

$\sum_{N_{1}\geq N_{2}\geq N_{3}}\sum_{N_{0}\leq
N_{1}^{1+\delta}}(N_{2}N_{3})^{\delta}\Big{(}\frac{N_{2}N_{3}}{N_{0}N_{1}}\Big{)}^{\frac{1}{2}-\delta}N_{0}^{s}N_{1}^{-s}(N_{2}N_{3})^{-\varepsilon}\\\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\|\Delta_{N_{0}}(u_{0})\|_{X^{-s,b^{\prime}}_{T}}\|\Delta_{N_{1}}(u_{1})\|_{X^{s,b^{\prime}}_{T}}\prod_{j=3}^{4}\|\Delta_{N_{j}}(u_{j})\|_{X^{\epsilon,b^{\prime}}_{T}}\\\
\leq
C\Big{(}\sum_{N}\|\Delta_{N}(u_{0})\|_{X^{-s,b^{\prime}}_{T}}^{2}\Big{)}^{1/2}\Big{(}\sum_{N}\|\Delta_{N}(u_{1})\|_{X^{s,b^{\prime}}_{T}}^{2}\Big{)}^{1/2}\prod_{j=3}^{4}\Big{(}\sum_{N}\|\Delta_{N}(u_{j})\|_{X^{\epsilon,b^{\prime}}_{T}}^{2}\Big{)}^{1/2}\,.$

In the range of summation,

$(N_{2}N_{3})^{\delta}\Big{(}\frac{N_{2}N_{3}}{N_{0}N_{1}}\Big{)}^{\frac{1}{2}-\delta}N_{0}^{s}N_{1}^{-s}(N_{2}N_{3})^{-\varepsilon}=N_{0}^{s-\frac{1}{2}+\delta}(N_{2}N_{3})^{\frac{1}{2}-\varepsilon}N_{1}^{-s-\frac{1}{2}+\delta}\leq
N_{1}^{-\kappa}$

where at the last step we have chosen $\delta>0$ small enough enough in such a
way that $\kappa>0$, allowing us to sum over $N_{0}$, $N_{1}$, $N_{2}$,
$N_{3}$. This completes the proof of Proposition 3.1. ∎

As a standard consequence of Proposition 3.1 (see e.g. [5, Proposition 3.3]),
we can obtain the following well-posedness result.

######  Proposition 3.2.

Let $R>0$ and $s_{0}\geq 1$ be given. Then there exists $T>0$ and
$b>\frac{1}{2}$ such that (1.1) has a unique local solution in
$X^{s_{0},b}_{T}$ for every $\varphi\in{\mathcal{H}}^{s_{0}}$ with
$\|\varphi\|_{{\mathcal{H}}^{1}}<R$. Moreover,

(3.3) $\|u(t,x)\|_{X^{s_{0},b}_{T}}\leq
2\|\varphi\|_{{\mathcal{H}}^{s_{0}}}\,.$

######  Remark 3.1.

While the Proposition is stated above regularity ${\mathcal{H}}^{1}$, it can
be extended at lower regularity ${\mathcal{H}}^{\varepsilon}$ with
$\varepsilon>0$, however we do not need such a low regularity later on.

Our next proposition reduces studying the growth of the ${\mathcal{H}}^{2k}$
norm of the solution $u(t,x)$ to the analysis of the growth of
$\|\partial_{t}^{k}u(t,x)\|_{L^{2}}$. In fact this last quantity is easier to
handle, as $\partial_{t}$ has better commutation properties with the nonlinear
Schrödinger flow than the operator $A$.

######  Proposition 3.3.

Let $k,s\in{\mathbb{N}}$ and $R,\delta>0$ be given. Let $T>0$ be associated
with $R$ and $s_{0}=2k+s$ as in Proposition 3.2 and let $u(t,x)\in
X^{2k+s,b}_{T}$ be the unique local solution to (1.1) with initial condition
$\varphi\in{\mathcal{H}}^{2k+s}$. Assume moreover that
$\sup_{t\in(-T,T)}\|u(t,x)\|_{{\mathcal{H}}^{1}}<R$. Then there exists $C>0$
such that

(3.4) $\displaystyle\forall
t\in(-T,T)\,,\quad\|\partial_{t}^{k}u(t)-i^{k}A^{k}u(t)\|_{{\mathcal{H}}^{s}}\leq
C\|u(t)\|_{{\mathcal{H}}^{s+2k-1}}^{1+\delta}\,.$

###### Proof.

We temporarily drop dependence on $t$ since the estimates we prove are
pointwise in time. In the sequel we shall also use without further comment
that $\sup_{t\in(-T,T)}\|u(t)\|_{{\mathcal{H}}^{1}}<R$. We shall denote by
$\delta>0$ an arbitrary small number that can change from line to line. We
start from the identity

(3.5)
$\partial_{t}^{h}u=i^{h}A^{h}u+\sum_{j=0}^{h-1}c_{j}\partial_{t}^{j}A^{h-j-1}(u|u|^{2})\,,$

available for every integer $h\geq 1$ and for suitable coefficients
$c_{j}\in{\mathbb{C}}$. Its elementary proof follows by induction on $h$,
using the equation solved by $u(t,x)$.

Next we argue by induction on $k$ in order to establish (3.4). More precisely
by assuming (3.4) we shall prove that the same estimate is true if we replace
$k$ by $k+1$. Indeed by (3.5), where we choose $h=k+1$, the estimate (3.4) for
$k+1$ reduces to

(3.6) $\|\partial_{t}^{j}(u|u|^{2})\|_{{\mathcal{H}}^{2k-2j+s}}\leq
C\|u\|_{{\mathcal{H}}^{s+2k+1}}^{1+\delta},\quad j=0,\dots,k.$

Hence we prove (3.6), assuming (3.4). Recalling (1.5), we have to prove

(3.7) $\displaystyle\|D^{2k-2j+s}\partial_{t}^{j}(u|u|^{2})\|_{L^{2}}\leq$
$\displaystyle C\|u\|_{{\mathcal{H}}^{s+2k+1}}^{1+\delta},\quad
j=0,\dots,k\,,$ (3.8) $\displaystyle\|\langle
x\rangle^{2k-2j+s}\partial_{t}^{j}(u|u|^{2})\|_{L^{2}}\leq$ $\displaystyle
C\|u\|_{{\mathcal{H}}^{s+2k+1}}^{1+\delta},\quad j=0,\dots,k\,.$

To prove (3.7) we expand time and space derivatives on the left-hand side.
Since $s$ is an integer and we never work with $L^{1}$ and $L^{\infty}$ norms,
we may replace the operator $D$ by the usual gradient operator $\nabla$, and
in particular, use the Leibniz rule. Hence by expanding space-time derivatives
and by using Hölder, we can estimate as follows the l.h.s. in (3.7):

$\displaystyle\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
s_{1}+s_{2}+s_{3}=2k-2j+s\end{subarray}}\prod_{l=1,2,3}\|\partial_{t}^{j_{l}}u\|_{W^{s_{l},6}}$
$\displaystyle\leq C\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
s_{1}+s_{2}+s_{3}=2k-2j+s\end{subarray}}\prod_{l=1,2,3}\|\partial_{t}^{j_{l}}u\|_{{\mathcal{H}}^{s_{l}+1}}$
$\displaystyle\leq C\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
s_{1}+s_{2}+s_{3}=2k-2j+s\end{subarray}}\prod_{l=1,2,3}\|u\|_{{\mathcal{H}}^{2j_{l}+s_{l}+1}}^{1+\delta}$

where we used the Sobolev embedding and the induction hypothesis at the last
step. We proceed with a trivial interpolation argument,

$(\dots)\leq
C\|u\|_{{\mathcal{H}}^{s+2k+1}}^{\delta}\Big{(}\prod_{l=1,2,3}\|u\|_{{\mathcal{H}}^{s+2k+1}}^{\theta_{l}}\|u\|_{{\mathcal{H}}^{1}}^{(1-\theta_{l})}\Big{)},$

where $\theta_{l}(s+2k+1)+(1-\theta_{l})=2j_{l}+s_{l}+1$. We conclude to (3.7)
since by direct computation we have $\sum_{l=1}^{3}\theta_{l}=1$ for every
$j=0,\dots,k$.

We now turn to (3.8): by Leibniz rule and Hölder, we estimate the l.h.s.

$\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
j_{2},j_{3}<j\end{subarray}}\|\langle
x\rangle^{2k-2j+s}\partial_{t}^{j_{1}}u\|_{L^{2}}\|\partial_{t}^{j_{2}}u\|_{L^{\infty}}\|\partial_{t}^{j_{3}}u\|_{L^{\infty}}\leq
C\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
j_{2},j_{3}<j\end{subarray}}\|\partial_{t}^{j_{1}}u\|_{{\mathcal{H}}^{2k-2j+s}}\|\partial_{t}^{j_{2}}u\|_{{\mathcal{H}}^{1+\delta}}\|\partial_{t}^{j_{3}}u\|_{{\mathcal{H}}^{1+\delta}}$

where we used Sobolev embedding. By interpolation we proceed with

$\displaystyle(\dots)$ $\displaystyle\leq
C\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
j_{2},j_{3}<j\end{subarray}}\|\partial_{t}^{j_{1}}u\|_{{\mathcal{H}}^{2k-2j+s}}\|\partial_{t}^{j_{2}}u\|_{{\mathcal{H}}^{1}}^{1-\delta}\|\partial_{t}^{j_{2}}u\|_{{\mathcal{H}}^{2}}^{\delta}\|\partial_{t}^{j_{3}}u\|_{{\mathcal{H}}^{1}}^{1-\delta}\|\partial_{t}^{j_{3}}u\|_{{\mathcal{H}}^{2}}^{\delta}$
$\displaystyle\leq C\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}=j\\\
j_{2},j_{3}<j\end{subarray}}\|u\|_{{\mathcal{H}}^{2j_{1}+2k-2j+s}}^{1+\delta}\|u\|_{{\mathcal{H}}^{2j_{2}+1}}\|u\|_{{\mathcal{H}}^{2j_{2}+2}}^{\delta}\|u\|_{{\mathcal{H}}^{2j_{3}+1}}\|u\|_{{\mathcal{H}}^{2j_{3}+2}}^{\delta}$

where we used the inductive assumption (3.4) to estimate
$\|\partial_{t}^{j_{l}}u\|_{{\mathcal{H}}^{s}}$ for $l=2,3$ and $s=1,2$. By a
further interpolation step and using again (3.4) we get

$(\dots)\leq
C\|u\|_{{\mathcal{H}}^{2k+s}}^{\delta}\Big{(}\prod_{l=1,2,3}\|u\|_{{\mathcal{H}}^{s+2k+1}}^{\theta_{l}}\|u\|_{{\mathcal{H}}^{1}}^{(1-\theta_{l})}\Big{)}\,,$

where we have chosen

$\theta_{1}(s+2k+1)+(1-\theta_{1})=2j_{1}+2k-2j+s\,,\,\,\,\,\theta_{l}(s+2k+1)+(1-\theta_{l})=2j_{l}+1,\quad
l=2,3\,.$

We conclude to (3.8) since one can check $\sum_{l=1}^{3}\theta_{l}<1$ for
$j=0,\dots,k$. ∎

The next proposition will be crucial in the sequel. It allows to estimate the
norm of time derivatives of the solution in the localized $X^{s,b}_{T}$
spaces, by using suitable Sobolev norms of the initial datum.

######  Proposition 3.4.

Let $l\in{\mathbb{N}}$, $R,\delta>0$ and $s\in(0,2]$ be given. Let $T>0$ be
associated with $R$ and $s_{0}=2l+2$ as in Proposition 3.2, and $u(t,x)\in
X^{2l+2,b}_{T}$ be the unique local solution to (1.1) with initial condition
$\varphi\in{\mathcal{H}}^{2l+2}$ and $\|\varphi\|_{{\mathcal{H}}^{1}}<R$.
Assume moreover that $\sup_{t\in(-T,T)}\|u(t,x)\|_{{\mathcal{H}}^{1}}<R$, then
there exists $C>0$ such that:

(3.9) $\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\leq
C\|\varphi\|_{{\mathcal{H}}^{2l}}^{1-s}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta},\quad\hbox{
if }s\in(0,1]$

and

(3.10) $\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\leq
C\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{s-1+\delta},\quad\hbox{
if }s\in(1,2].$

###### Proof.

We shall prove separately (3.9) and (3.10) by induction on $l$.

Proof of (3.9)

We consider the integral formulation of the equation solved by
$\partial_{t}^{l}u$,

$\partial_{t}^{l}u(t)=e^{itA}\partial_{t}^{l}u(0)+\int_{0}^{t}e^{i(t-\tau)A}\partial_{\tau}^{l}(u(\tau)|u(\tau)|^{2})d\tau$

and then by standard properties of the $X^{s,b}$ spaces,

(3.11) $\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\\\ \leq
C\Big{(}\|\partial_{t}^{l}u(0)\|_{{\mathcal{H}}^{s}}+\Big{\|}\int_{0}^{t}e^{i(t-\tau)A}\partial_{\tau}^{l}(u(\tau)|u(\tau)|^{2})d\tau\Big{\|}_{X^{s,b}_{T}}\Big{)}\,.$

Expanding the time derivative and using Proposition 3.1 we get

(3.12) $\|\partial_{t}^{l}u(t)\|_{X^{s,b}_{T}}\leq
C\Big{(}\|\partial_{t}^{l}u(0)\|_{{\mathcal{H}}^{s}}+T^{\gamma}\sum_{l_{1}+l_{2}+l_{3}=l}\|\partial_{t}^{l_{1}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{2}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{s,b}_{T}}\Big{)}\,.$

By interpolation and Proposition 3.3 we also have

(3.13)
$\|\partial_{t}^{l}u(0)\|_{{\mathcal{H}}^{s}}\leq\|\partial_{t}^{l}u(0)\|_{{\mathcal{H}}^{1}}^{s}\|\partial_{t}^{l}u(0)\|_{L^{2}}^{1-s}\leq
C\|u(0)\|_{{\mathcal{H}}^{2l+1}}^{s}\|u(0)\|_{{\mathcal{H}}^{2l}}^{1-s}\|u(0)\|_{{\mathcal{H}}^{2l+2}}^{\delta}=C\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l}}^{1-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}\,.$

Therefore, estimating the second term on the r.h.s. in (3.12) is sufficient.
We split the proof in two cases.

First case: $0<\min\\{l_{1},l_{2},l_{3}\\}\leq\max\\{l_{1},l_{2},l_{3}\\}<l$

We use induction on $l$ and estimate

$\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=l\\\
\max\\{l_{1},l_{2},l_{3}\\}<l\end{subarray}}\|\partial_{t}^{l_{1}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{2}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{s,b}_{T}}\\\
\leq C\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=l\\\
\max\\{l_{1},l_{2},l_{3}\\}<l\end{subarray}}\|\varphi\|_{{\mathcal{H}}^{2l_{1}}}^{1-s}\|\varphi\|_{{\mathcal{H}}^{2l_{2}}}^{1-s}\|\varphi\|_{{\mathcal{H}}^{2l_{3}}}^{1-s}\|\varphi\|_{{\mathcal{H}}^{2l_{1}+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l_{2}+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l_{3}+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}\\\
\leq
C\|\varphi\|_{{\mathcal{H}}^{2l}}^{(1-s){\eta_{1}}}\|\varphi\|_{{\mathcal{H}}^{2l}}^{(1-s)\eta_{2}}\|\varphi\|_{{\mathcal{H}}^{2l}}^{(1-s)\eta_{3}}\|\varphi\|_{{\mathcal{H}}^{1}}^{(1-s)(1-\eta_{1})}\|\varphi\|_{{\mathcal{H}}^{1}}^{(1-s)(1-\eta_{2})}\|\varphi\|_{{\mathcal{H}}^{1}}^{(1-s)(1-\eta_{3})}\\\
\times\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{s\theta_{1}}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{s\theta_{2}}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{s\theta_{3}}\|\varphi\|_{{\mathcal{H}}^{1}}^{s(1-\theta_{1})}\|\varphi\|_{{\mathcal{H}}^{1}}^{s(1-\theta_{2})}\|u(t_{0})\|_{{\mathcal{H}}^{1}}^{s(1-\theta_{3})}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}$

where

$\begin{cases}2l\eta_{1}+1-\eta_{1}=2l_{1}\\\ 2l\eta_{2}+1-\eta_{2}=2l_{2}\\\
2l\eta_{3}+1-\eta_{3}=2l_{3}\end{cases}\,\,\,\,\,\text{and}\,\,\,\,\,\begin{cases}\theta_{1}(2l+1)+1-\theta_{1}=2l_{1}+1\\\
\theta_{2}(2l+1)+1-\theta_{2}=2l_{2}+1\\\
\theta_{3}(2l+1)+1-\theta_{3}=2l_{3}+1\end{cases}\,.$

Since $\theta_{1}+\theta_{2}+\theta_{3}=1$ and $\eta_{1}+\eta_{2}+\eta_{3}<1$
we may conclude with

(3.14) $\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=l\\\
\max\\{l_{1},l_{2},l_{3}\\}<l\end{subarray}}\|\partial_{t}^{l_{1}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{2}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{s,b}_{T}}\leq
C\|\varphi\|_{{\mathcal{H}}^{2l}}^{1-s}\|u(t_{0})\|_{{\mathcal{H}}^{2l+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}.$

Second case: $0=\min\\{l_{1},l_{2},l_{3}\\}\leq\max\\{l_{1},l_{2},l_{3}\\}<l$

We can assume $l_{1}=0$. Then we argue exactly as above except that, since
$\|\varphi\|_{{\mathcal{H}}^{2l_{1}}}=\|\varphi\|_{L^{2}}$ is bounded since we
assume a control on the ${\mathcal{H}}^{1}$ norm of the initial datum, it is
not necessary to introduce the parameter $\eta_{1}$. Hence we need only
$\eta_{2},\eta_{3},\theta_{1},\theta_{2},\theta_{3}$. The conclusion is the
same as above.

Third case: $\max\\{l_{1},l_{2},l_{3}\\}=l$

We estimate the terms in the sum at the r.h.s. of (3.12) as follows

$\|u\|_{X^{s,b}_{T}}^{2}\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\leq\|u\|_{X^{1,b}_{T}}^{2}\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\leq
C\|\varphi\|_{{\mathcal{H}}^{1}}^{2}\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}$

where we have used (3.3) for $s_{0}=1$.

Summarizing (3.12), (3.13) and (3.14) we get

$\|\partial_{t}^{l}u(t)\|_{X^{s,b}_{T}}\leq
C\Big{(}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{s}\|\varphi\|_{{\mathcal{H}}^{2l}}^{1-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}+T^{\gamma}\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\Big{)}\,.$

We conclude by choosing a time $\bar{T}>0$ small enough in such a way that the
second term on the r.h.s. can be absorbed by the l.h.s. Notice that the bound
that we get on the short time $\bar{T}$ can be iterated since the constants
depends only from the ${\mathcal{H}}^{1}$ norm of the solution and hence we
get the desired bound up to the fixed time $T$ provided by the proposition
after a finite iteration of the previous argument.

Proof of (3.10)

By Proposition (3.1) we get

(3.15) $\|\partial_{t}^{l}u(t)\|_{X^{s,b}_{T}}\leq
C\Big{(}\|\partial_{t}^{l}u(0)\|_{{\mathcal{H}}^{s}}+T^{\gamma}\sum_{l_{1}+l_{2}+l_{3}=l}\|\partial_{t}^{l_{1}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{2}}u\|_{X^{2-s,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{2-s,b}_{T}}\Big{)}.$

We first notice that by interpolation and Proposition 3.3 (see the proof of
(3.13)),

(3.16) $\|\partial_{t}^{l}u(0)\|_{{\mathcal{H}}^{s}}\leq
C\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{s-1+\delta}\,.$

Next we estimate the sum on the r.h.s. of (3.15) by considering two cases.

First case: $0<\min\\{l_{1},l_{2},l_{3}\\}\leq\max\\{l_{1},l_{2},l_{3}\\}<l$

By combining the inductive assumption on $l$ and (3.9) we get:

$\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=l\\\
\max\\{l_{1},l_{2},l_{3}\\}<l\end{subarray}}\|\partial_{t}^{l_{1}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{2}}u\|_{X^{2-s,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{2-s,b}_{T}}\\\
\leq C\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=l\\\
\max\\{l_{1},l_{2},l_{3}\\}<l\end{subarray}}\|\varphi\|_{{\mathcal{H}}^{2l_{1}+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l_{2}}}^{s-1}\|\varphi\|_{{\mathcal{H}}^{2l_{3}}}^{s-1}\|\varphi\|_{{\mathcal{H}}^{2l_{1}+2}}^{s-1}\|\varphi\|_{{\mathcal{H}}^{2l_{2}+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l_{3}+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}\\\
\leq
C\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{(2-s){\eta_{1}}}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{(s-1)\eta_{2}}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{(s-1)\eta_{3}}\|\varphi\|_{{\mathcal{H}}^{1}}^{(2-s)(1-\eta_{1})}\|\varphi\|_{{\mathcal{H}}^{1}}^{(s-1)(1-\eta_{2})}\|\varphi\|_{{\mathcal{H}}^{1}}^{(s-1)(1-\eta_{3})}\\\
\times\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{(s-1)\theta_{1}}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{(2-s)\theta_{2}}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{(2-s)\theta_{3}}\|\varphi\|_{{\mathcal{H}}^{1}}^{(s-1)(1-\theta_{1})}\|\varphi\|_{{\mathcal{H}}^{1}}^{(2-s)(1-\theta_{2})}\|\varphi\|_{{\mathcal{H}}^{1}}^{(2-s)(1-\theta_{3})}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}$

where:

$\begin{cases}\eta_{1}(2l+1)+1-\eta_{1}=2l_{1}+1\\\
\eta_{2}(2l+2)+1-\eta_{2}=2l_{2}\\\
\eta_{3}(2l+2)+1-\eta_{3}=2l_{3}\end{cases}\,\,\,\,\,\text{and}\,\,\,\,\,\,\begin{cases}\theta_{1}(2l+2)+1-\theta_{1}=2l_{1}+2\\\
\theta_{2}(2l+1)+1-\theta_{2}=2l_{2}+1\\\
\theta_{3}(2l+1)+1-\theta_{3}=2l_{3}+1\end{cases}.$

By direct computation $\eta_{1}+\theta_{2}+\theta_{3}=1$ and
$\theta_{1}+\eta_{2}+\eta_{3}<1$. Hence the estimate above implies

(3.17) $\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=l\\\
\max\\{l_{1},l_{2},l_{3}\\}<l\end{subarray}}\|\partial_{t}^{l_{1}}u\|_{X^{s,b}_{T}}\|\partial_{t}^{l_{2}}u\|_{X^{2-s,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{2-s,b}_{T}}\leq
C\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{s-1+\delta}.$

Second case: $0=\min\\{l_{1},l_{2},l_{3}\\}\leq\max\\{l_{1},l_{2},l_{3}\\}<l$

If $l_{1}=0$, then our previous proof is valid since we have to deal with the
norm $\|\varphi\|_{{\mathcal{H}}^{2l_{1}+1}}$ and hence we have regularity
${\mathcal{H}}^{1}$ and the interpolation argument above can be applied.
However, in the cases $l_{2}=0$ or $l_{3}=0$ the proof needs to be slightly
modified. We can assume $l_{2}=0$ then in this case
$\|\varphi\|_{{\mathcal{H}}^{2l_{2}}}=\|\varphi\|_{L^{2}}$ is bounded since we
assume a control on the ${\mathcal{H}}^{1}$ norm of the initial datum, hence
it is not necessary to introduce the parameter $\eta_{2}$ in the interpolation
step. The conclusion is the same as above.

Third case: $\max\\{l_{1},l_{2},l_{3}\\}=l$

We have to consider three cases $(l_{1},l_{2},l_{3})=(l,0,0)$,
$(l_{1},l_{2},l_{3})=(0,l,0)$ and $(l_{1},l_{2},l_{3})=(0,0,l)$ (the last two
cases are similar). In the first case we have

(3.18)
$\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}\|u\|_{X^{2-s,b}_{T}}\|u\|_{X^{2-s,b}_{T}}\leq
C\|\partial_{t}^{l}u\|_{X^{s,b}_{T}}$

where we used the estimate

(3.19) $\|u\|_{X^{2-s,b}_{T}}\leq C\|\varphi\|_{{\mathcal{H}}^{1}},$

which is a consequence of (3.3) where we choose $s_{0}=1$. We conclude this
case by choosing $T$ small enough, exactly as we did along the proof of (3.9).
Finally, for $(l_{1},l_{2},l_{3})=(0,l,0)$ by Proposition 3.2, (3.19) and
(3.9),

(3.20)
$\|u\|_{X^{s,b}_{T}}\|\partial_{t}^{l}u\|_{X^{2-s,b}_{T}}\|u\|_{X^{2-s,b}_{T}}\\\
\leq C\|\varphi\|_{{\mathcal{H}}^{s}}\|\partial_{t}^{l}u\|_{X^{2-s,b}_{T}}\leq
C\|\varphi\|_{{\mathcal{H}}^{s}}\|\varphi\|_{{\mathcal{H}}^{2l}}^{s-1}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}\\\
\leq
C\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\frac{s-1}{2l+1}}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\frac{(s-1)(2l-1)}{1+2l}}\|\varphi\|_{{\mathcal{H}}^{2l+1}}^{2-s}\|\varphi\|_{{\mathcal{H}}^{2l+2}}^{\delta}$

where we used interpolation and the a priori bound on the $\mathcal{H}^{1}$
norm of the initial datum. This concludes the proof of (3.10). ∎

##  4\. Modified energies and proof of Theorem 1.1

The aim of this section is to introduce suitable energies and to measure how
far they are from being exact conservation laws. Those energies are the key
tool in order to achieve the growth estimate provided in Theorem 1.1. Along
this section we denote by $\int$ the integral on ${\mathbb{R}}^{2}$ with
respect to the Lebesgue measure $dx$, and $\int\int$ the integral on
${\mathbb{R}}^{2}\times{\mathbb{R}}$ with respect to the Lebesgue measure
$dxdt$.

######  Proposition 4.1.

Let $u(t,x)\in\mathcal{C}((-T,T);{\mathcal{H}}^{2k+2})$ be a local solution to
(1.1) with initial datum $\varphi\in{\mathcal{H}}^{2k+2}$. Then we have:

(4.1)
$\frac{d}{dt}\Big{(}\frac{1}{2}\|\partial_{t}^{k}Au(t,x)\|_{L^{2}}^{2}+{\mathcal{S}}_{2k+2}(u(t,x))\Big{)}={\mathcal{R}}_{2k+2}(u(t,x))$

where ${\mathcal{S}}_{2k+2}(u(t,x))$ is a linear combination of terms of the
following type:

$\int\partial_{t}^{k}Lu_{0}\partial_{t}^{m_{1}}Lu_{1}\partial_{t}^{m_{2}}u_{2}\partial_{t}^{m_{3}}u_{3},\\\
\quad m_{1}+m_{2}+m_{3}=k,\quad m_{1}<k.$

and ${\mathcal{R}}_{2k+2}(u(t,x))$ is a linear combination of terms of the
following type:

(4.2)
$\int\partial_{t}^{k}Lu_{0}\partial_{t}^{l_{1}}Lu_{1}\partial_{t}^{l_{2}}u_{2}\partial_{t}^{l_{3}}u_{3},\quad
l_{1}+l_{2}+l_{3}=k+1,\quad l_{1}\leq k.$

where in (4.1) and (4.2) we have $u_{0},u_{1},u_{2},u_{3}\in\\{u,\bar{u}\\}$
and $L$ can be any of the following operators:

$Lu=\partial_{x_{i}}u\hbox{ for }i=1,2,\quad Lu=\langle x\rangle u,\quad
Lu=u.$

###### Proof.

We have

$i\partial_{t}(\partial_{t}^{k}\sqrt{A}u)+A(\partial_{t}^{k}\sqrt{A}u)\pm\partial_{t}^{k}\sqrt{A}(u|u|^{2})=0\,.$

Next we multiply the equation above by $\partial_{t}^{k+1}\sqrt{A}\bar{u}$ and
we take the real part,

$\frac{1}{2}\frac{d}{dt}(\|\partial_{t}^{k}Au\|_{L^{2}}^{2})=\mp\mathrm{Re}\,\int\partial_{t}^{k}\sqrt{A}(u|u|^{2})\partial_{t}^{k+1}\sqrt{A}\bar{u}.$

By symmetry of the operator $\sqrt{A}$ we have

$\mathrm{Re}\,\int\partial_{t}^{k}\sqrt{A}(u|u|^{2})\partial_{t}^{k+1}\sqrt{A}\bar{u}=\mathrm{Re}\,\int\partial_{t}^{k}(u|u|^{2})\partial_{t}^{k+1}A\bar{u}\\\
=-\mathrm{Re}\,\int\partial_{t}^{k}(u|u|^{2})\partial_{t}^{k+1}\Delta\bar{u}+\mathrm{Re}\,\int\partial_{t}^{k}(u|u|^{2})\partial_{t}^{k+1}(|x|^{2}\bar{u})$

and we proceed by integration by parts

(4.3)
$(\dots)=\sum_{i=1}^{2}\mathrm{Re}\,\int\partial_{t}^{k}\partial_{x_{i}}(u|u|^{2})\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}+\mathrm{Re}\,\int|x|^{2}\partial_{t}^{k}(u|u|^{2})\partial_{t}^{k+1}\bar{u}.$

Next notice that the first term on the r.h.s. in (4.3) can be written as
follows

$\sum_{i=1}^{2}\mathrm{Re}\,\int\partial_{t}^{k}\partial_{x_{i}}(u|u|^{2})\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}\\\
=\sum_{i=1}^{2}\Big{(}2\mathrm{Re}\,\int|u|^{2}\partial_{t}^{k}\partial_{x_{i}}u\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}+\mathrm{Re}\,\int
u^{2}\partial_{t}^{k}\partial_{x_{i}}\bar{u}\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}\Big{)}\\\
+\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=k\\\ \\\
max\\{l_{1},l_{2},l_{3}\\}<k\end{subarray}}\mathrm{Re}\,\Big{(}a_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}\partial_{x_{i}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\bar{u}\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}+b_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}\partial_{x_{i}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\partial_{x_{i}}\bar{u}\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}\Big{)}$

where $a_{l_{1},l_{2},l_{3}},b_{l_{1},l_{2},l_{3}}$ are suitable real numbers.
Rewriting

$(\dots)=\frac{d}{dt}\sum_{i=1}^{2}\Big{(}\int|\partial_{t}^{k}\partial_{x_{i}}u|^{2}|u|^{2}+\frac{1}{2}\mathrm{Re}\,\int(\partial_{t}^{k}\partial_{x_{i}}\bar{u})^{2}u^{2}\Big{)}\\\
-\sum_{i=1}^{2}\Big{(}\int|\partial_{t}^{k}\partial_{x_{i}}u|^{2}\partial_{t}(|u|^{2})+\frac{1}{2}\mathrm{Re}\,\int(\partial_{t}^{k}\partial_{x_{i}}\bar{u})^{2}\partial_{t}(u^{2})\Big{)}\\\
+\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=k\\\ \\\
l_{1}<k\end{subarray}}\mathrm{Re}\,\Big{(}a_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}\partial_{x_{i}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\bar{u}\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}+b_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}\partial_{x_{i}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\partial_{x_{i}}\bar{u}\partial_{t}^{k+1}\partial_{x_{i}}\bar{u}\Big{)}\,,$

by elementary manipulations on the last two lines we get

$(\dots)=\frac{d}{dt}\sum_{i=1}^{2}\Big{(}\int|\partial_{t}^{k}\partial_{x_{i}}u|^{2}|u|^{2}+\frac{1}{2}\mathrm{Re}\,\int(\partial_{t}^{k}\partial_{x_{i}}\bar{u})^{2}u^{2}\Big{)}\\\
-\sum_{i=1}^{2}\Big{(}\int|\partial_{t}^{k}\partial_{x_{i}}u|^{2}\partial_{t}(|u|^{2})+\frac{1}{2}\mathrm{Re}\,\int(\partial_{t}^{k}\partial_{x_{i}}\bar{u})^{2}\partial_{t}(u^{2})\Big{)}\\\
+\frac{d}{dt}\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=k\\\ \\\
l_{1}<k\end{subarray}}\mathrm{Re}\,\Big{(}a_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}\partial_{x_{i}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\bar{u}\partial_{t}^{k}\partial_{x_{i}}\bar{u}+b_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\partial_{x_{i}}\bar{u}\partial_{t}^{k}\partial_{x_{i}}\bar{u}\Big{)}\\\
+\sum_{\begin{subarray}{c}l_{1}+l_{2}+l_{3}=k+1\\\ l_{1}\leq
k\end{subarray}}\mathrm{Re}\,\Big{(}\tilde{a}_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}\partial_{x_{i}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\bar{u}\partial_{t}^{k}\partial_{x_{i}}\bar{u}+\tilde{b}_{l_{1},l_{2},l_{3}}\partial_{t}^{l_{1}}u\partial_{t}^{l_{2}}u\partial_{t}^{l_{3}}\partial_{x_{i}}\bar{u}\partial_{t}^{k}\partial_{x_{i}}\bar{u}\Big{)}\,.$

This last expression is a sum of a linear combination of terms with structure
(4.1) with $Lu=\partial_{x_{i}}u$ plus a time derivative of a linear
combination of terms of type (4.2) with $Lu=\partial_{x_{i}}u$.

Notice that the second term on the r.h.s. in (4.3) rewrites

$\mathrm{Re}\,\int|x|^{2}\partial_{t}^{k}(u|u|^{2})\partial_{t}^{k+1}\bar{u}=\mathrm{Re}\,\int\partial_{t}^{k}(\langle
x\rangle u|u|^{2})\partial_{t}^{k+1}(\langle
x\rangle\bar{u})-\mathrm{Re}\,\int\partial_{t}^{k}(u|u|^{2})\partial_{t}^{k+1}\bar{u}$

and arguing as above one checks, by developing first the derivative of order
$k$ with respect to time, that this expression is a time derivative of terms
of type (4.1), where $Lu=\langle x\rangle u$ or $Lu=u$, plus a linear
combination of terms with structure (4.2) where $Lu=\langle x\rangle u$ or
$Lu=u$. ∎

Next we estimate the energy ${\mathcal{R}}_{2k+2}$ introduced in Proposition
4.1.

######  Proposition 4.2.

Let $k\in{\mathbb{N}}$, $R>0$ be given and $u(t,x)\in X^{2k+2,b}_{T}$ be the
unique local solution to (1.1) with initial condition
$\varphi\in{\mathcal{H}}^{2k+2}$ and $\|\varphi\|_{{\mathcal{H}}^{1}}<R$,
where $T>0$ is associated with $R$ and $s_{0}=2k+2$ as in Proposition 3.2.
Assume moreover that $\sup_{t\in(-T,T)}\|u(t,x)\|_{{\mathcal{H}}^{1}}<R$. Then
for every $\delta>0$ there exists $C>0$ such that:

$\big{|}\int_{0}^{T}{\mathcal{R}}_{2k+2}(u(\tau,x))d\tau\big{|}\leq
C\|\varphi\|_{\mathcal{H}^{2k+2}}^{\frac{8k+1}{4k+2}+\delta}.$

###### Proof.

We have to estimate integrals like (4.2). After a Littlewood-Paley
decomposition we are reduced to estimating

$\sum_{N_{0},N_{1},N_{2},N_{3}}\int\int\Delta_{N_{0}}(\partial_{t}^{k}Lu_{0})\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu_{1})\Delta_{N_{2}}(\partial_{t}^{l_{2}}u_{2})\Delta_{N_{3}}(\partial_{t}^{l_{3}}u_{3}).$

where

$l_{1}+l_{2}+l_{3}=k+1,\quad l_{1}\leq k.$

Here we have used the compact notation $\int\int$ to denote the space-time
integral on the strip $(-T,T)\times{\mathbb{R}}^{2}$ and $\Delta_{N}$ denotes
the Littlewood-Paley localization associated with the operator $A$ at dyadic
frequency $N$. We split the sum in several pieces depending on the frequencies
$N_{0},N_{1},N_{2},N_{3}$ and we shall make extensively use of the following
bilinear estimate (see [16, Proposition 2.3.15]). For every
$\delta\in(0,\frac{1}{2}]$, $b>\frac{1}{2}$ there exists $C>0$ such that:

(4.4) $\|(\Delta_{N}u)(\Delta_{M}v)\|_{L^{2}((0,T);L^{2})}\leq
C\left(\frac{\min\\{N,M\\}}{\max\\{N,M\\}}\right)^{\frac{1}{2}-\delta}\|\Delta_{N}u\|_{X^{0,b}_{T}}\|\Delta_{M}u\|_{X^{0,b}_{T}}.$

Using the equation solved by $u$ and noticing that, with the imposed
conditions on $l_{1},l_{2},l_{3}$ we may assume $l_{2}\geq 1$ (otherwise
$l_{3}\geq 1$ and it is symmetric) we get

(4.5)
$\Big{|}\int\int(\partial_{t}^{k}Lu_{0})(\partial_{t}^{l_{1}}Lu_{1})\partial_{t}^{l_{2}}u_{2}\partial_{t}^{l_{3}}u_{3}\Big{|}\leq
C\Big{(}\Big{|}\int\int(\partial_{t}^{k}Lu_{0})(\partial_{t}^{l_{1}}Lu_{1})(\partial_{t}^{l_{2}-1}Au_{2})\partial_{t}^{l_{3}}u_{3}\Big{|}\\\
+\int\int|(\partial_{t}^{k}Lu_{0})||(\partial_{t}^{l_{1}}Lu_{1})||(\partial_{t}^{l_{2}-1}(u|u|^{2})||\partial_{t}^{l_{3}}u|\Big{)}\,.$

The second term on the right hand side is estimated by Cauchy-Schwarz with

$\Big{(}\int_{0}^{T}\|\partial_{t}^{k}Lu_{0}(\tau)\|_{L^{2}}\|\partial_{t}^{l_{1}}Lu_{1}\|_{L^{2}}d\tau\Big{)}\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);L^{\infty})}\|\partial_{t}^{l_{3}}u\|_{L^{\infty}((0,T);L^{\infty})}\\\
\leq\Big{(}\int_{0}^{T}\|\partial_{t}^{k}Lu_{0}(\tau)\|_{L^{2}}\|\partial_{t}^{l_{1}}Lu_{1}\|_{L^{2}}d\tau\Big{)}\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\|\partial_{t}^{l_{3}}u\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}$

where we used Sobolev embedding. Using (3.9) we proceed with

$(\dots)\leq
C\|\varphi\|_{{\mathcal{H}}^{2k+1}}\|\varphi\|_{{\mathcal{H}}^{2l_{1}+1}}\|\varphi\|_{{\mathcal{H}}^{2l_{3}+1}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\\\
\leq
C\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\eta+\eta_{1}+\eta_{3}}\|\varphi\|_{{\mathcal{H}}^{1}}^{3-(\eta+\eta_{1}+\eta_{3})}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}$

where

$\begin{cases}\eta(2k+2)+(1-\eta)=2k+1\\\
\eta_{i}(2k+2)+(1-\eta_{i})=2l_{i}+1,\quad i=1,3\end{cases}$

and hence, by using the bound assumed on $\|\varphi\|_{{\mathcal{H}}^{1}}$ we
can continue the estimate above as follows

$\dots\leq
C\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\frac{2k+2l_{1}+2l_{3}}{2k+1}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\,.$

Expanding $\partial_{t}^{l_{2}-1}(u|u|^{2})$ and using that
${\mathcal{H}}^{1+\delta}$ is an algebra we get

$\displaystyle\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\leq$
$\displaystyle
C\sum_{j_{1}+j_{2}+j_{3}=l_{2}-1}\|\partial_{t}^{j_{1}}u\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\|\partial_{t}^{j_{2}}u\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\|\partial_{t}^{j_{3}}u\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j_{1}+j_{2}+j_{3}=l_{2}-1}\|\varphi\|_{{\mathcal{H}}^{2j_{1}+1}}\|u\|_{{\mathcal{H}}^{2j_{2}+1}}\|\varphi\|_{{\mathcal{H}}^{2j_{3}+1}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j_{1}+j_{2}+j_{3}=l_{2}-1}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\theta_{1}+\theta_{2}+\theta_{3}}\|\varphi\|_{{\mathcal{H}}^{1}}^{3-(\theta_{1}+\theta_{2}+\theta_{3})}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}$

where

$\theta_{i}(2k+2)+(1-\theta_{i})=2j_{i}+1,\quad i=1,2,3$

and recalling the a priori bound assumed on $\|\varphi\|_{{\mathcal{H}}^{1}}$
we conclude that

$\|(\partial_{t}^{l_{2}-1}(u|u|^{2})\|_{L^{\infty}((0,T);{\mathcal{H}}^{1+\delta})}\leq
C\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\frac{2l_{2}-2}{2k+1}}\,.$

By combining the estimates above we get that the second term on the right-hand
side in (4.5) can be estimated up to a constant by

$\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\frac{2k+2l_{1}+2l_{2}+2l_{3}-2}{2k+1}+\delta}=\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\frac{4k}{2k+1}+\delta}.$

We now focus on the first term on the right-hand side in (4.5) and by
Littlewood-Paley decomposition we are reduced to estimating

$\sum_{N_{0},N_{1},N_{2},N_{3}}\int\int\Delta_{N_{0}}(\partial_{t}^{k}Lu_{0})\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu_{1})\Delta_{N_{2}}(\partial_{t}^{l_{2}-1}Au_{2})\Delta_{N_{3}}(\partial_{t}^{l_{3}}u_{3}).$

Here we used again $\int\int$ to denote the space-time integral on the strip
$(-T,T)\times{\mathbb{R}}^{2}$ and $\Delta_{N}$ still denotes the Littlewood-
Paley localization associated with the operator $A$ at dyadic frequency $N$.
We split the sum in several pieces depending on the frequencies
$N_{0},N_{1},N_{2},N_{3}$ and we shall make again extensive use of the
bilinear estimate (4.4). Next we consider several subcases.

First subcase: $\min\\{N_{0},N_{2}\\}\geq\max\\{N_{1},N_{3}\\}$

By Cauchy-Schwarz,

$\Big{|}\int\int\Delta_{N_{0}}(\partial_{t}^{k}Lu_{0})\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu_{1})\Delta_{N_{2}}(\partial_{t}^{l_{2}-1}Au_{2})\Delta_{N_{3}}(\partial_{t}^{l_{3}}u_{3})\Big{|}\\\
\leq\|\Delta_{N_{0}}(\partial_{t}^{k}Lu_{0})\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu_{1})\|_{L^{2}((0,T);L^{2})}\|\Delta_{N_{2}}(\partial_{t}^{l_{2}-1}Au)\Delta_{N_{3}}(\partial_{t}^{l_{3}}u)\|_{L^{2}((0,T);L^{2})}$

and by (4.4) we can continue as follows

$\displaystyle(\dots)\leq$ $\displaystyle
C\frac{(N_{1}N_{3})^{\frac{1}{2}-\delta}}{(N_{0}N_{2})^{\frac{1}{2}-\delta}}\|\Delta_{N_{0}}(\partial_{t}^{k}Lu)\|_{X^{0,b}_{T}}\|\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu)\|_{X^{0,b}_{T}}\|\Delta_{N_{2}}(\partial_{t}^{l_{2}-1}Au_{2})\|_{X^{0,b}_{T}}\|\Delta_{N_{3}}(\partial_{t}^{l_{3}}u)\|_{X^{0,b}_{T}}$
$\displaystyle\leq$ $\displaystyle
C\frac{N_{3}^{\frac{1}{2}-\delta}}{N_{2}^{\frac{1}{2}-\delta}}\|\Delta_{N_{0}}(\partial_{t}^{k}Lu)\|_{X^{0,b}_{T}}\|\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu)\|_{X^{0,b}_{T}}\|\Delta_{N_{2}}(\partial_{t}^{l_{2}-1}Au_{2})\|_{X^{0,b}_{T}}\|\Delta_{N_{3}}(\partial_{t}^{l_{3}}u)\|_{X^{0,b}_{T}}$
$\displaystyle\leq$ $\displaystyle
C\|\Delta_{N_{0}}(\partial_{t}^{k}Lu)\|_{X^{0,b}_{T}}\|\Delta_{N_{1}}(\partial_{t}^{l_{1}}Lu)\|_{X^{0,b}_{T}}\|\Delta_{N_{2}}(\partial_{t}^{l_{2}-1}Au)\|_{X^{-\frac{1}{2}+\delta,b}_{T}}\|\Delta_{N_{3}}(\partial_{t}^{l_{3}}u)\|_{X^{\frac{1}{2}-\delta,b}_{T}}\,.$

Summarizing,

$\sum_{\begin{subarray}{c}N_{0},N_{1},N_{2},N_{3}\\\
\min\\{N_{0},N_{2}\\}\geq\max\\{N_{1},N_{3}\\}\end{subarray}}\Big{|}\int\int\Delta_{N_{0}}(Lu^{0})\Delta_{N_{1}}(Lu_{1})\Delta_{N_{2}}(Au_{2})\Delta_{N_{3}}(u_{3})\Big{|}\\\
\leq
C\|L\partial_{t}^{k}u\|_{X^{\delta,b}_{T}}\|L\partial_{t}^{l_{1}}u\|_{X^{\delta,b}_{T}}\|A\partial_{t}^{l_{2}-1}u\|_{X^{-\frac{1}{2}+\delta,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{\frac{1}{2}-\delta,b}_{T}}\\\
\leq
C\|\partial_{t}^{k}u\|_{X^{1+\delta,b}_{T}}\|\partial_{t}^{l_{1}}u\|_{X^{1+\delta,b}_{T}}\|\partial_{t}^{l_{2}-1}u\|_{X^{\frac{3}{2}+\delta,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{\frac{1}{2}+\delta,b}_{T}}$

where we used Lemma 2.1 at the last step, assuming we chose $b>\frac{1}{2}$ in
such a way the estimate at the last line fits with Lemma 2.1.

Second subcase: $\min\\{N_{1},N_{2}\\}\geq\max\\{N_{0},N_{3}\\}$

We can argue as above and we are reduced to the previous case by noticing that
we have the inequality $\frac{N_{0}N_{3}}{N_{1}N_{2}}\leq\frac{N_{3}}{N_{2}}$
since in this subcase $N_{0}\leq N_{1}$.

Third subcase: $\min\\{N_{3},N_{2}\\}\geq\max\\{N_{0},N_{1}\\}$

We can argue as above and we are reduced to the first subcase by noticing that
we have the inequality $N_{0}N_{1}\leq N_{3}^{2}$ and hence
$\frac{N_{0}N_{1}}{N_{2}N_{3}}\leq\frac{N_{3}}{N_{2}}$.

Fourth subcase: $\min\\{N_{1},N_{3}\\}\geq\max\\{N_{0},N_{2}\\}$

We can argue as above and we are reduced to the first subcase by noticing that
we have the inequality $\frac{N_{0}N_{2}}{N_{1}N_{3}}\leq\frac{N_{3}}{N_{2}}$.
In fact it is equivalent to $N_{0}N_{2}^{2}\leq N_{1}N_{3}^{2}$ which clearly
holds in that subcase.

Fifth subcase: $\min\\{N_{0},N_{3}\\}\geq\max\\{N_{1},N_{2}\\}$

We can argue as above and we are reduced to the first subcase by noticing that
we have the inequality $\frac{N_{1}N_{2}}{N_{0}N_{3}}\leq\frac{N_{3}}{N_{2}}$.
In fact it is equivalent to $N_{1}N_{2}^{2}\leq N_{0}N_{3}^{2}$ which clearly
holds in this subcase.

Sixth subcase: $\min\\{N_{0},N_{1}\\}\geq\max\\{N_{2},N_{3}\\}$

We can argue as above and we are reduced to the first subcase by noticing that
we have the inequality $\frac{N_{2}N_{3}}{N_{0}N_{1}}\leq\frac{N_{3}}{N_{2}}$
which in turn follows from the fact that in this subcase $N_{2}^{2}\leq
N_{0}N_{1}$.

We are therefore left with proving

$\|\partial_{t}^{k}u\|_{X^{1+\delta,b}_{T}}\|\partial_{t}^{l_{1}}u\|_{X^{1+\delta,b}_{T}}\|\partial_{t}^{l_{2}-1}u\|_{X^{\frac{3}{2}+\delta,b}_{T}}\|\partial_{t}^{l_{3}}u\|_{X^{\frac{1}{2}+\delta,b}_{T}}\leq
C\|\varphi\|_{\mathcal{H}^{2k+2}}^{\frac{8k+1}{4k+2}+\delta}.$

Using (3.9) and (3.10) we can control the right-hand side with

$\|\varphi\|_{{\mathcal{H}}^{2k+1}}\|\varphi\|_{{\mathcal{H}}^{2l_{1}+1}}\|\varphi\|_{{\mathcal{H}}^{2l_{2}-1}}^{\frac{1}{2}}\|\varphi\|_{{\mathcal{H}}^{2l_{2}}}^{\frac{1}{2}}\|\varphi\|_{{\mathcal{H}}^{2l_{3}}}^{\frac{1}{2}}\|\varphi\|_{{\mathcal{H}}^{2l_{3}+1}}^{\frac{1}{2}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}$

which, by interpolation and recalling the a priori bound on
${\mathcal{H}}^{1}$ norm, can be estimated with

$\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\theta}\|\varphi\|_{{\mathcal{H}}^{2l_{1}+1}}^{\theta_{1}}\|\varphi\|_{{\mathcal{H}}^{2l_{2}-1}}^{\frac{1}{2}\gamma_{2}}\|\varphi\|_{{\mathcal{H}}^{2l_{2}}}^{\frac{1}{2}\eta_{2}}\|\varphi\|_{{\mathcal{H}}^{2l_{3}}}^{\frac{1}{2}\eta_{3}}\|\varphi\|_{{\mathcal{H}}^{2l_{3}+1}}^{\frac{1}{2}\theta_{3}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}$

where

$\begin{cases}\theta(2k+2)+(1-\theta)=2k+1\\\
\theta_{1}(2k+2)+(1-\theta_{1})=2l_{1}+1\\\
\theta_{3}(2k+2)+(1-\theta_{3})=2l_{3}+1\\\
\gamma_{2}(2k+2)+(1-\gamma_{2})=2l_{2}-1\\\
\eta_{2}(2k+2)+(1-\eta_{2})=2l_{2}\\\
\eta_{3}(2k+2)+(1-\eta_{3})=2l_{3}.\end{cases}$

We conclude by computing
$\theta,\theta_{1},\theta_{3},\gamma_{2},\eta_{2},\eta_{3}$ and noticing that

$\theta+\theta_{1}+\frac{1}{2}\gamma_{2}+\frac{1}{2}\eta_{2}+\frac{1}{2}\eta_{3}+\frac{1}{2}\theta_{3}=\frac{8k+1}{4k+2}\,,$

and this concludes the proof. ∎

Next we estimate the terms involved in the expression of
${\mathcal{S}}_{2k+2}$ introduced in Proposition 4.1.

######  Proposition 4.3.

Let $k\in{\mathbb{N}}$, $R>0$ be given and $u(t,x)\in X^{2k+2,b}_{T}$ be the
unique local solution to (1.1) with initial condition
$\varphi\in{\mathcal{H}}^{2k+2}$ and $\|\varphi\|_{{\mathcal{H}}^{1}}<R$,
where $T>0$ is associated with $R$ and $s_{0}=2k+2$ as in Proposition 3.2.
Assume moreover that $\sup_{t\in(-T,T)}\|u(t,x)\|_{{\mathcal{H}}^{1}}<R$. Then
for every $\delta>0$ there exists $C>0$ such that

$\sup_{t\in(-T,T)}|{\mathcal{S}}_{2k+2}(u(t,x))|\leq
C\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\frac{4k}{2k+1}+\delta}\,.$

###### Proof.

We prove the desired estimate for every expression with type (4.1). Indeed by
Hölder we have for every fixed $t\in(-T,T)$

$\int\partial_{t}^{k}Lu_{0}\partial_{t}^{m_{1}}Lu_{1}\partial_{t}^{m_{2}}u_{2}\partial_{t}^{m_{3}}u_{3}\leq\|\partial_{t}^{k}Lu_{0}\|_{L^{2}}\|\partial_{t}^{m_{1}}Lu_{1}\|_{L^{2}}\|\partial_{t}^{m_{2}}u_{2}\|_{L^{\infty}}\|\partial_{t}^{m_{3}}u_{3}\|_{L^{\infty}}\,,$

and by Sobolev embedding and (1.5) we proceed with

$\displaystyle(\dots)\leq$ $\displaystyle
C\|\partial_{t}^{k}Lu\|_{L^{2}}\|\partial_{t}^{m_{1}}Lu\|_{L^{2}}\|\partial_{t}^{m_{2}}u\|_{{\mathcal{H}}^{1}}^{1-\delta}\|\partial_{t}^{m_{2}}u\|_{{\mathcal{H}}^{2}}^{\delta}\|\partial_{t}^{m_{3}}u\|_{{\mathcal{H}}^{1}}^{1-\delta}\|\partial_{t}^{m_{3}}u\|_{{\mathcal{H}}^{2}}^{\delta}$
$\displaystyle\leq$ $\displaystyle
C\|\partial_{t}^{k}u\|_{{\mathcal{H}}^{1}}\|\partial_{t}^{m_{1}}u\|_{{\mathcal{H}}^{1}}\|\partial_{t}^{m_{2}}u\|_{{\mathcal{H}}^{1}}^{1-\delta}\|\partial_{t}^{m_{2}}u\|_{{\mathcal{H}}^{2}}^{\delta}\|\partial_{t}^{m_{3}}u\|_{{\mathcal{H}}^{1}}^{1-\delta}\|\partial_{t}^{m_{3}}u\|_{{\mathcal{H}}^{2}}^{\delta}.$

Then we can use (3.4) to get

$\displaystyle(\dots)\leq$ $\displaystyle
C\|u\|_{{\mathcal{H}}^{2k+1}}\|u\|_{{\mathcal{H}}^{2m_{1}+1}}\|u\|_{{\mathcal{H}}^{2m_{2}+1}}\|u\|_{{\mathcal{H}}^{2m_{3}+1}}\|u\|_{{\mathcal{H}}^{2k+2}}^{\delta}$
$\displaystyle\leq$ $\displaystyle
C\|\varphi\|_{{\mathcal{H}}^{2k+1}}\|\varphi\|_{{\mathcal{H}}^{2m_{1}+1}}\|\varphi\|_{{\mathcal{H}}^{2m_{2}+1}}\|\varphi\|_{{\mathcal{H}}^{2m_{3}+1}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}$

where $\delta>0$ is a small constant that can change from line to line and the
last estimate follows from the embedding
$X^{s,b}_{T}\subset{\mathcal{C}}((-T,T);{\mathcal{H}}^{s})$ for
$b>\frac{1}{2}$ and (3.3). Next we choose
$\theta,\theta_{1},\theta_{2},\theta_{3}\in[0,1]$ such that

$\begin{cases}\theta(2k+2)+(1-\theta)=2k+1\\\
\theta_{i}(2k+2)+(1-\theta_{i})=2m_{i}+1,\quad i=1,2,3\end{cases}$

and by interpolation, we proceed with

$\int\partial_{t}^{k}Lu_{0}\partial_{t}^{m_{1}}Lu_{1}\partial_{t}^{m_{2}}u_{2}\partial_{t}^{m_{3}}u_{3}\leq
C\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\theta+\theta_{1}+\theta_{2}+\theta_{3}}\|\varphi\|_{{\mathcal{H}}^{2k+2}}^{\delta}$

and we conclude by computing explicitly
$\theta+\theta_{1}+\theta_{2}+\theta_{3}$. ∎

Proof of Theorem 1.1 It will follow as a consequence of Propositions 4.1, 4.2,
4.3. Let

(4.6) $\sup_{t\in(-\infty,\infty)}\|u(t,x)\|_{\mathcal{H}^{1}}=R$

then $R<\infty$ by (1.6). By integration on the strip $(0,T)$ of the identity
(4.1) we get

$\frac{1}{2}\|\partial_{t}^{k}Au(T,x)\|_{L^{2}}^{2}+{\mathcal{S}}_{2k+2}(u(T,x))=\frac{1}{2}\|\partial_{t}^{k}Au(0,x)\|_{L^{2}}^{2}+{\mathcal{S}}_{2k+2}(u(0,x))+\int_{0}^{T}{\mathcal{R}}_{2k+2}(u(\tau,x))d\tau$

where $T$ is the one defined in Propositions 4.2 and 4.3 with $R$ is defined
as above. Then we get

(4.7)
$\frac{1}{2}\|\partial_{t}^{k}Au(T,x)\|_{L^{2}}^{2}-\frac{1}{2}\|\partial_{t}^{k}Au(0,x)\|_{L^{2}}^{2}\leq
C\Big{(}\|u(0,x)\|_{{\mathcal{H}}^{2k+2}}^{\frac{4k}{2k+1}+\delta}+\|u(0,x)\|_{\mathcal{H}^{2k+2}}^{\frac{8k+1}{4k+2}+\delta}\Big{)}.$

Next notice that if we assume that $u(t,x)$ is the nontrivial solution
(otherwise the conclusion is trivial) then
$\|u(t,x)\|_{\mathcal{H}^{2k+2}}\geq\|u(t,x)\|_{{L}^{2}}=const>0$ and the
second term on the r.h.s. in (4.7) may be absorbed by the first one provided
that we modify the multiplicative constant; then

(4.8)
$\frac{1}{2}\|\partial_{t}^{k}Au(T,x)\|_{L^{2}}^{2}-\frac{1}{2}\|\partial_{t}^{k}Au(0,x)\|_{L^{2}}^{2}\leq
C\|u(0,x)\|_{\mathcal{H}^{2k+2}}^{\frac{8k+1}{4k+2}+\delta}.$

One easily checks that by (4.6) the bound above can be iterated with the same
constants, namely

(4.9)
$\frac{1}{2}\|\partial_{t}^{k}Au((n+1)T,x)\|_{L^{2}}^{2}-\frac{1}{2}\|\partial_{t}^{k}Au(nT,x)\|_{L^{2}}^{2}\leq
C\|u(nT,x)\|_{\mathcal{H}^{2k+2}}^{\frac{8k+1}{4k+2}+\delta}$

for every $n\in{\mathbb{N}}$. By summing up for $n\in[0,N-1]$ we obtain

$\|\partial_{t}^{k}Au(NT,x)\|_{L^{2}}^{2}\leq
C\sum_{n\in\\{0,\dots,N-1\\}}\|u(nT,x)\|_{\mathcal{H}^{2k+2}}^{\frac{8k+1}{4k+2}+\delta}$

and it implies

$\sup_{n\in[0,N]}\|\partial_{t}^{k}Au(nT,x)\|_{L^{2}}^{2}\leq
CN\Big{(}\sup_{n\in[0,N]}\|u(nT,x)\|_{\mathcal{H}^{2k+2}}\Big{)}^{\frac{8k+1}{4k+2}+\delta}$

which implies by (3.4) the estimate

$\sup_{n\in[0,N]}\|u(nT,x)\|_{\mathcal{H}^{2k+2}}\leq CN^{\frac{2(2k+1)}{3}}$

and $\forall N\in{\mathbb{N}}$,

$\|u(NT,x)\|_{\mathcal{H}^{2k+2}}\leq CN^{\frac{2(2k+1)}{3}}\,.$

By using (3.3) it is easy to deduce that the estimate above implies

$\sup_{t\in[NT,(N+1)T]}\|u(t,x)\|_{\mathcal{H}^{2k+2}}\leq
CN^{\frac{2(2k+1)}{3}}$

provided that we suitably modify the multiplicative constant $C$. Summarizing
we get that, for all $t>0$,

$\|u(t,x)\|_{\mathcal{H}^{2k+2}}\leq Ct^{\frac{2(2k+1)}{3}}\,.$

The same argument works for $t<0$.

## Appendix

We intend to provide a direct proof, based on integration by parts, of the
crucial bilinear estimate from [16], for solutions to

(4.10) $i\partial_{t}u-\Delta u+|x|^{2}u=0\,.$

######  Theorem 4.1.

Let $1\leq M\leq N$ be dyadic numbers ; for $T\in(0,\infty)$ there exists
$C_{T}$ such that

(4.11) $\|u_{N}v_{M}\|_{L^{2}((0,T);L^{2})}^{2}\leq
C_{T}MN^{-1}\|u_{N}(0)\|_{L^{2}}^{2}\|v_{N}(0)\|_{L^{2}}^{2}$

where $u_{N}$ and $v_{N}$ are spectrally localized solutions to (4.10) (namely
$\Delta_{N}u_{N}=u_{N}$, $\Delta_{M}v_{M}=v_{M}$) respectively with initial
datum $u_{N}(0)$, $v_{M}(0)$.

Such bilinear estimates were first obtained for solutions to the classical
linear Schrödinger equation in [2], using direct computations in Fourier
variables. In [6], so-called interaction Morawetz interaction estimates were
introduced for the 3D nonlinear Schrödinger equation, relying on a bilinear
version of the classical Morawetz estimate. Here, we rely on the bilinear
computation from [15] that non only extended such bilinear virial estimates to
low dimensions but also allowed to recover Bourgain’s estimates from [2]. We
will follow the strategy from [13] where bilinear estimates on bounded domains
were obtained, bypassing the need for Fourier localization. We split the proof
in several steps. First we prove that, for a given $T\in(0,\infty)$:

(4.12)
$\int_{0}^{T}\Big{(}\int\int_{|x-y|<\frac{1}{M}}M|u_{N}(x)\nabla_{y}\bar{v}_{M}(y)+\bar{v}_{M}(y)\nabla_{x}u_{N}(x)|^{2}\,dxdy\Big{)}dt\leq
C_{T}N\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}.$

Next we deduce from (4.12) that

(4.13) $\int_{0}^{T}\Big{(}\int|\nabla_{x}(v_{M}u_{N})|^{2}\,dx\Big{)}dt\leq
C_{T}MN\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}.$

Estimate (4.13), along with a companion easier estimate for
$\int_{0}^{T}\Big{(}\int|x|^{2}|v_{M}u_{N}|^{2}\,dx\Big{)}\,dt$, implies

(4.14) $\int_{0}^{T}\|v_{M}\bar{u}_{N}\|^{2}_{\mathcal{H}^{1}}\,dt\leq
C_{T}MN\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}\,.$

Finally, by a spectral localization argument, we prove that (4.14) implies
(4.11).

### Proof of (4.12)

We first remark for later use that once (4.12) will be established, then we
are allowed to replace $v_{N}$ by $Av_{N}$ (which is still a localized
solution to (4.10)) and we get:

(4.15)
$\int_{0}^{T}\Big{(}\int\int_{|x-y|<\frac{1}{M}}M|u_{N}(x)\nabla_{y}(A\bar{v}_{M})(y)+(A\bar{v}_{M})(y)\nabla_{x}u_{N}(x)|^{2}\,dxdy\Big{)}dt\\\
\leq C_{T}NM^{2}\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}.$

Next we focus on the proof of (4.12) and from now on, $T$ is fixed in
$(0,+\infty)$. Let $\rho:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ be a $C^{1}$
function whose derivative is piecewise differentiable, with $H_{\rho}$
denoting the bilinear form associated to its Hessian (as a distribution),
$H_{\rho}(a,b)=\sum_{k,l}(\partial^{2}_{kl}\rho)a_{k}b_{l}$; all
$\partial^{2}_{kl}\rho$ are actually piecewise continuous functions and under
such assumptions all subsequent integrations by parts are fully justified in
the classical sense. We claim that, for any couple of solutions $u,v$ of
(4.10),

(4.16) $\int_{0}^{T}\Big{(}\int\int
H_{\rho}(x-y)(\bar{v}(y)\nabla_{x}u(x)+u(x)\nabla_{y}\bar{v}(y),v(y)\nabla_{x}\bar{u}(x)+\bar{u}(x)\nabla_{y}v(y))\,dxdy\Big{)}dt\\\
\leq
C_{T}\|\nabla\rho\|_{L^{\infty}}(\|v(0)\|_{L^{2}}^{2}\|u(0)\|_{L^{2}}\|u(0)\|_{{\mathcal{H}}^{1}}+\|u(0)\|_{L^{2}}^{2}\|v(0)\|_{L^{2}}\|v(0)\|_{{\mathcal{H}}^{1}})$

where we dropped time dependence for notational simplicity. Following [13] we
define a convex function $\rho_{M}:{\mathbb{R}}\rightarrow{\mathbb{R}}$,

$\rho_{M}(z)=\begin{cases}\frac{M}{2}z^{2}+\frac{1}{2M},\quad
z\leq\frac{1}{M}\\\ z,\quad z>\frac{1}{M}\end{cases}$

and we use (4.16) with $\rho(x-y)=\rho_{M}(x_{1}-y_{1})$: we get, by direct
computation of the Hessian $H_{\rho}$,

(4.17)
$\int_{0}^{T}\Big{(}\int\int_{|x_{1}-y_{1}|<\frac{1}{M}}M|(\bar{v}(y)\partial_{x_{1}}u(x)+u(x)\partial_{y_{1}}\bar{v}(y)|^{2}\,dxdy\Big{)}dt\\\
\leq
C_{T}(\|v(0)\|_{L^{2}}^{2}\|u(0)\|_{L^{2}}\|u(0)\|_{{\mathcal{H}}^{1}}+\|u(0)\|_{L^{2}}^{2}\|v(0)\|_{L^{2}}\|v(0)\|_{{\mathcal{H}}^{1}})$

where there is no contribution in the region $|x_{1}-y_{1}|>\frac{1}{M}$ as
$H_{\rho}=0$ there, and we used that
$\|\rho_{M}^{\prime}(z)\|_{L^{\infty}}\leq 1$. Of course by choosing
$\rho(x-y)=\rho(x_{2}-y_{2})$ we get a similar estimate where $x_{1},y_{1}$
are replaced by $x_{2},y_{2}$ and by combining the two estimates we get (4.12)
where we noticed that
$|x-y|<\frac{1}{M}\subset\max\\{|x_{1}-y_{1}|,|x_{2}-y_{2}|\\}<\frac{1}{M}$.
Replacing $u$ and $v$ by $u_{N}$ and $v_{M}$ and using spectral localization
we get (4.12).

Next we focus on the proof of (4.16). We compute the second derivative w.r.t.
time of the functional

$I_{\rho}(t)=\int\int|u(x)|^{2}\rho(x-y)|v(y)|^{2}\,dxdy$

where for for simplicity we have dropped the time dependence of $u,v$. In
order to do so, recall that by the classical virial computation we get for a
solution $w(t,x)$ to (4.10) (we drop again time-dependence of $w$ and set
$\rho_{y}(x)=\rho(x-y)$ to emphasize that $y$ is a fixed base point here):

(4.18)
$\displaystyle\frac{d}{dt}\int\rho_{y}(x)|w(x)|^{2}\,dx=2\int\nabla\rho_{y}(x)\cdot\mathrm{Im}\,(\nabla\bar{w}(x)w(x))\,dx$
(4.19)
$\displaystyle\;\;\;\;\;\;\;\;\frac{d^{2}}{dt^{2}}\int\rho_{y}(x)|w|^{2}\,dx=4\int
H_{\rho_{y}}(\nabla
w(x),\nabla\bar{w}(x))-\int\Delta\rho_{y}(x)\Delta(|w(x)|^{2})\,dx-4\int
x\cdot\nabla\rho_{y}(x)|w(x)|^{2}dx\,,$

where we emphasize that we will not be using more than two derivatives on
$\rho_{y}$. Next, using (4.18) we get

(4.20)
$\frac{d}{dt}I_{\rho}(t)=2\int\int\nabla\rho(x-y)\cdot\mathrm{Im}\,(\nabla_{x}\bar{u}(x)u(x))|v(y)|^{2}\,dxdy-2\int\int\nabla\rho(x-y)\cdot\mathrm{Im}\,(\nabla_{y}\bar{v}(y)v(y))|u(x)|^{2}\,dxdy\,.$

Using that $\|\nabla w\|_{L^{2}}\leq C\|w\|_{{\mathcal{H}}^{1}}$ at fixed time
followed by conservation of mass and energy for (4.10), we get

(4.21) $\displaystyle|\frac{d}{dt}I_{\rho}(t)|\leq$ $\displaystyle
2\|\nabla\rho\|_{L^{\infty}}(\|v\|_{L^{2}}^{2}\|u\|_{L^{2}}\|\nabla
u\|_{L^{2}}+\|u\|_{L^{2}}^{2}\|v\|_{L^{2}}\|\nabla v\|_{L^{2}})$ (4.22)
$\displaystyle\leq$ $\displaystyle
C\|\nabla\rho\|_{L^{\infty}}(\|v(0)\|_{L^{2}}^{2}\|u(0)\|_{L^{2}}\|u(0)\|_{{\mathcal{H}}^{1}}+\|u(0)\|_{L^{2}}^{2}\|v(0)\|_{L^{2}}\|v(0)\|_{{\mathcal{H}}^{1}})\,.$

For later use notice also that using (4.18) on both mass densities,

(4.23)
$\displaystyle\int\Big{(}\int\rho(x-y)\frac{d}{dt}|u(x)|^{2}\,dx\Big{)}\frac{d}{dt}|v(y)|^{2}\,dy=$
$\displaystyle
2\int\Big{(}\int\nabla\rho(x-y)\cdot\mathrm{Im}\,(\nabla_{x}\bar{u}(x)u(x))\,dx\Big{)}\frac{d}{dt}|v(y)|^{2}\,dy$
(4.24) $\displaystyle=$ $\displaystyle-4\int\int
H_{\rho}(\mathrm{Im}\,(\nabla_{x}\bar{u}(x)u(x)),\mathrm{Im}\,(\nabla_{y}\bar{v}(y)v(y))\,dxdy\,.$

On the other hand by combining (4.19) and (4.23),

(4.25) $\displaystyle\frac{d^{2}}{dt^{2}}I_{\rho}(t)=$ $\displaystyle
4\int\int H_{\rho}(x-y)(\nabla u(x),\nabla\bar{u}(x))|v(y)|^{2}+4\int\int
H_{\rho}(x-y)(\nabla v(y),\nabla\bar{v}(y))|u(x)|^{2}\,dxdy$ (4.26)
$\displaystyle{}-\int\int\Delta\rho(x-y)\Delta(|u(x)|^{2})|v(y)|^{2}\,dxdy-\int\int\Delta\rho(x-y)|u(x)|^{2}\Delta(|u(y)|^{2})\,dxdy$
(4.27) $\displaystyle{}-8(\int
H_{\rho}(x-y)(\mathrm{Im}\,(\nabla\bar{u}(x)u(x)),\mathrm{Im}\,(\nabla\bar{v}(y)v(y)))\,dydx$
(4.28)
$\displaystyle{}-4\mathrm{Re}\,\int\int(x-y)\cdot\nabla\rho(x-y)|u(x)|^{2}|v(y)|^{2}\,dxdy=I+II+III+IV+V+VI\,.$

Following [15], we rewrite
$\Delta\rho(x-y)=-\nabla_{x}\cdot\nabla_{y}\rho(x-y)$ and integrate by parts
w.r.t. to $x$ and $y$:

$III+IV=8\int\int H_{\rho}(x-y)(\mathrm{Re}\,(\bar{u}(x)\nabla
u(x)),\mathrm{Re}\,(\bar{v}(y)\nabla v(y)))\,dxdy\,.$

Now, thinking about just one direction of derivation, we have the following
identity

$4|v|^{2}(y)|\partial u|^{2}(x)+4|u|^{2}(x)|\partial
v|^{2}(y)+8\frac{(v\partial\bar{v}+\bar{v}\partial
v)(y)}{2}\frac{(u\partial\bar{u}+\bar{u}\partial
u)(x)}{2}-8\frac{(v\partial\bar{v}-\bar{v}\partial
v)(y)}{2}\frac{(\bar{u}\partial u-u\partial\bar{u})(x)}{2}\\\
=4|v|^{2}(y)|\partial u|^{2}(x)+4|u|^{2}(x)|\partial
v|^{2}(y)+4v\partial\bar{v}(y)u\partial\bar{u}(x)+4\bar{v}\partial
v(y)\bar{u}\partial u(x)=4|\bar{v}(y)\partial
u(x)+u(x)\partial\bar{v}(y)|^{2}\,,$

which allows to recombine $I+II+III+IV+V$, to get

$\frac{d^{2}}{dt^{2}}I_{\rho}(t)=4\int\int H_{\rho}(x-y)(\bar{v}(y)\nabla
u(x)+u(x)\nabla\bar{v}(y),v(y)\nabla\bar{u}(x)+\bar{u}(x)\nabla v(y))\,dxdy\\\
-4\mathrm{Re}\,\int\int\nabla\rho(x-y)\cdot(x-y)|v(y)|^{2}|u(x)|^{2}\,dxdy.$

After integration in time of the identity above and by recalling (4.21) we get

$4\int_{0}^{T}\Big{(}\int\int
H_{\rho}(x-y)(\bar{v}(y)\nabla_{x}u(x)+u(x)\nabla_{y}\bar{v}(y),v(y)\nabla_{x}\bar{u}(x)+\bar{u}(x)\nabla_{y}v(y))\,dxdy\Big{)}dt\\\
\leq
C\|\nabla\rho\|_{L^{\infty}}(\|v(0)\|_{L^{2}}^{2}\|u(0)\|_{L^{2}}\|u(0)\|_{{\mathcal{H}}^{1}}+\|u(0)\|_{L^{2}}^{2}\|v(0)\|_{L^{2}}\|v(0)\|_{{\mathcal{H}}^{1}})\\\
+4\int_{0}^{T}\Big{(}\int\int|\nabla\rho(x-y)||y-x||v(y)|^{2}|u(x)|^{2}\,dxdy\Big{)}dt\\\
\leq
C\|\nabla\rho\|_{L^{\infty}}(\|v(0)\|_{L^{2}}^{2}\|u(0)\|_{L^{2}}\|u(0)\|_{{\mathcal{H}}^{1}}+\|u(0)\|_{L^{2}}^{2}\|v(0)\|_{L^{2}}\|v(0)\|_{{\mathcal{H}}^{1}})\\\
+C_{T}\|\nabla\rho\|_{L^{\infty}}\sup_{t\in(0,T)}(\|u(t)\|_{L^{2}}^{2}\|v(t)\|_{L^{2}}\|yv(t)\|_{L^{2}}+\|v(t)\|_{L^{2}}^{2}\|u(t)\|_{L^{2}}\|xu(t)\|_{L^{2}}^{2})\,.$

Using $\|yw\|_{L^{2}}^{2}\leq\|w\|_{{\mathcal{H}}^{1}}$ and, again,
conservation of mass and energy for (4.10), this estimate implies (4.16).

### Proof of (4.12) $\Rightarrow$ (4.13)

We need a suitable local elliptic estimate for our operator
$A=-\Delta+|x|^{2}$ to reproduce the computation from [13]. The next lemma is
a modification of Lemma 4.2 in [13].

######  Lemma 4.1.

There exists $C>0$ and $\lambda_{0}\geq 1$ such that, for any smooth function
$\phi$ in ${\mathbb{R}}^{2}$ and $\lambda\geq\lambda_{0}$, the following
pointwise estimate holds:

(4.29) $|\phi(x)|^{2}\leq
C\lambda^{-2}\int_{|x-y|<{\lambda^{-1}}}|A\phi|^{2}\,dy+C\lambda^{2}\int_{|x-y|<{\lambda^{-1}}}|\phi|^{2}\,dy,\quad\forall
x\in{\mathbb{R}}^{2}\,.$

###### Proof.

Without loss of generality we may restrict to real-valued $\phi$. The lemma is
proved in [13] if we replace in the r.h.s. the operator $A$ by $-\Delta$ and
the domain of integration by the smaller domain $|x-y|<(4\lambda)^{-1}$ (this
fact follows from classical elliptic theory and Sobolev embedding for
$\lambda=1$ and then any $\lambda>0$ by rescaling). Thus we conclude provided
that we prove

(4.30) $\lambda^{-2}\int_{|x-y|<{(4\lambda)^{-1}}}|\Delta\phi|^{2}\,dy\leq
C\lambda^{-2}\int_{|x-y|<{\lambda^{-1}}}|A\phi|^{2}\,dy+C\lambda^{2}\int_{|x-y|<{\lambda^{-1}}}|\phi|^{2}\,dy.$

In order to prove this estimate we expand the square $\int|\Delta
f|^{2}=\int|Af-|y|^{2}f|^{2}$ and after integrations by parts we get

(4.31) $\int(|\Delta f|^{2}+|y|^{4}|f|^{2}+2|y|^{2}|\nabla
f|^{2})\,dy=\int(|Af|^{2}+4|f|^{2})\,dy$

for any real-valued function $f\in C^{\infty}_{0}({\mathbb{R}}^{2})$. Next we
pick $f(y)=\chi_{\lambda}(y)\phi(y)$, where
$\chi_{\lambda}(y)=\chi(\lambda(y-x))$, with $\chi(|z|)=1$ on
$|z|<\frac{1}{4}$ and $\chi(|z|)=0$ on $|z|>\frac{1}{2}$. All subsequent
cutoffs w.r.t. $y$ will be centered at $x$. Expanding
$\int|\Delta(\chi_{\lambda}\phi)|^{2}$ and $\int|A(\chi_{\lambda}\phi)|^{2}$
and replacing in the previous identity we get:

(4.32)
$\int(|\chi_{\lambda}|^{2}|\Delta\phi|^{2}+|y|^{4}|\chi_{\lambda}|^{2}|\phi|^{2}+2|y|^{2}|\nabla(\chi_{\lambda}\phi)|^{2})\,dy\\\
=\int(|\chi_{\lambda}|^{2}|Av|^{2}+4|\chi_{\lambda}|^{2}|\phi|^{2})\,dy-2\int\chi_{\lambda}|y|^{2}\phi(2\nabla\chi_{\lambda}\cdot\nabla\phi+\Delta\chi_{\lambda}\phi)\,dy\,.$

By Cauchy-Schwarz and elementary manipulations we estimate the last term on
the r.h.s. as follows, for every $\mu>0$ and with a universal constant $C>0$:

$\Big{|}\int\chi_{\lambda}|y|^{2}\phi(2\nabla\chi_{\lambda}\cdot\nabla\phi+\phi\Delta\chi_{\lambda})\,dy\Big{|}^{2}\leq
C\mu\int|y|^{4}|\chi_{\lambda}|^{2}|\phi|^{2}\,dy+\frac{C}{\mu}\int(|\nabla\chi_{\lambda}\cdot\nabla\phi|^{2}+|\Delta\chi_{\lambda}|^{2}|\phi|^{2})\,dy.$

If we choose the constant $\mu$ small enough then we can absorb
$\int|y|^{4}|\chi_{\lambda}v|^{2}dx$ on the l.h.s. in (4.32) and by neglecting
some positive terms we get, abusing notation for the constant $C$,

(4.33) $\int|\chi_{\lambda}|^{2}|\Delta\phi|^{2}\,dy\leq
C\int(|\chi_{\lambda}|^{2}|A\phi|^{2}+4|\chi_{\lambda}|^{2}|\phi|^{2}+|\nabla\chi_{\lambda}\cdot\nabla\phi|^{2}+|\Delta\chi_{\lambda}|^{2})|\phi|^{2}\,dy$

and by elementary considerations

$\int_{|x-y|<(4\lambda)^{-1}}|\Delta\phi|^{2}\,dy\leq
C\int_{|x-y|<(2\lambda)^{-1}}|A\phi|^{2}\,dy+C\lambda^{2}\int\tilde{\chi}_{\lambda}|\nabla\phi|^{2}\,dy+C(1+\lambda^{2})\int_{|x-y|<(2\lambda)^{-1}}|\phi|^{2}\,dy$

where $\tilde{\chi}_{\lambda}$ is a suitable enlargement of $\chi_{\lambda}$,
namely
$\tilde{\chi}_{\lambda}(y)=\tilde{\chi}\big{(}\frac{y-x}{\lambda}\big{)}$,
with $\tilde{\chi}(|z|)=1$ on $|z|<\frac{1}{2}$ and $\tilde{\chi}(|z|)=0$ on
$|z|>1$. Then (4.30) follows provided that

$\int\tilde{\chi}_{\lambda}|\nabla\phi|^{2}dy\leq
C\lambda^{-2}\int_{|x-y|<{\lambda^{-1}}}|A\phi|^{2}\,dy+C\lambda^{2}\int_{|x-y|<{\lambda^{-1}}}|\phi|^{2}\,dy.$

In order to do that we write (either integrating by parts or replacing
$-\Delta$ by $A-|x|^{2}$)

$-2\int\tilde{\chi}_{\lambda}\phi\Delta\phi\,dy=2\int\tilde{\chi}_{\lambda}|\nabla\phi|^{2}\,dy-\int\Delta\tilde{\chi}_{\lambda}|\phi|^{2}\,dy=2\int\tilde{\chi}_{\lambda}\phi
A\phi\,dy-2\int|y|^{2}\tilde{\chi}_{\lambda}|\phi|^{2}\,dy$

and hence

$2\int\tilde{\chi}_{\lambda}|\nabla\phi|^{2}\,dy+2\int|y|^{2}\tilde{\chi}_{\lambda}|\phi|^{2}\,dy=-2\int\tilde{\chi}_{\lambda}\phi
A\phi\,dy+\int\Delta\tilde{\chi}_{\lambda}|\phi|^{2}\,dy\\\ \leq
C\lambda^{-2}\int_{|x-y|<\lambda^{-1}}|A\phi|^{2}\,dy+C(1+\lambda^{2})\int_{|x-y|<\lambda^{-1}}|\phi|^{2}\,dy$

where we used Cauchy-Schwarz at the last step. ∎

We now proceed to prove that (4.12) $\Rightarrow$ (4.13). The first term in
the square at the l.h.s. of (4.12) turns out to be lower order: we compute by
change of variable, Cauchy-Schwarz inequality and Strichartz estimate,

(4.34)
$\int_{0}^{T}\Big{(}\int\int_{|x-y|<\frac{1}{M}}|u_{N}(x)\nabla_{y}\bar{v}_{M}(y)|^{2}dxdy\Big{)}dt=\int_{0}^{T}\int_{|z|<\frac{1}{M}}|u_{N}(x)\nabla_{x}\bar{v}_{M}(x-z)|^{2}\,dxdzdt\\\
\leq\int_{|z|<\frac{1}{M}}\|u_{N}\|_{L^{4}((0,T);L^{4})}^{2}\|\nabla
v_{M}\|_{L^{4}((0,T);L^{4})}^{2}\,dz\leq
C_{T}\|u_{N}(0)\|^{2}_{2}\|v_{M}(0)\|^{2}_{L^{2}}$

where at the last step we used that $\|\nabla v_{M}\|_{L^{4}((0,T);L^{4})}\leq
C_{T}M\|v_{M}(0)\|_{L^{2}}$. In turn this bound follows by noticing that
$\nabla v_{M}$ is solution to the inhomogeneous equation associated with
(4.10) with forcing term $2xv_{M}$. Hence by the inhomogeneous Strichartz
estimate, placing the forcing term in $L^{1}((0,T);L^{2})$,

(4.35) $\|\nabla v_{M}\|_{L^{4}((0,T);L^{4})}\leq C\|\nabla
v_{M}(0)\|_{L^{2}}+C\||x|v_{M}\|_{L^{1}((0,T);L^{2})}\leq
C_{T}\|v_{M}(0)\|_{\mathcal{H}^{1}}\leq C_{T}M\|v_{M}(0)\|_{L^{2}}\,,$

where we used conservation of energy for (4.10) and the bound
$\||x|w\|_{L^{2}}\leq C\|w\|_{{\mathcal{H}}^{1}}$ for every time independent
function.
Recall that (4.34) holds with $v_{M}$ replaced by $Av_{M}$ (it is still a
solution to (4.10)), hence we get:

(4.36)
$\int_{0}^{T}\Big{(}\int\int_{|x-y|<\frac{1}{M}}|u_{N}(x)\nabla_{y}(A\bar{v}_{M})(y)|^{2}\,dxdy\Big{)}dt\leq
C_{T}M^{4}\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}\,.$

We now proceed using the Lemma 4.1 and we get

$\int_{0}^{T}\Big{(}\int|\bar{v}_{M}(x)\nabla_{x}u_{N}(x)|^{2}\,dx\Big{)}dt\leq
C\int_{0}^{T}\Big{(}\int\int_{|x-y|<\frac{1}{M}}M^{2}|\bar{v}_{M}(y)\nabla_{x}u_{N}(x)|^{2}\\\
{}+\frac{1}{M^{2}}|A\bar{v}_{M}(y)\nabla_{x}u_{N}(x)|^{2}\,dxdy\Big{)}dt$

that by (4.12) and (4.15) implies

$(\dots)\leq
C\int_{0}^{T}\Big{(}\int\int_{|x-y|<\frac{1}{M}}M^{2}|u_{N}(x)\nabla_{y}\bar{v}_{M}(y)|^{2}+\frac{1}{M^{2}}|u_{N}(x)\nabla_{y}(A\bar{v}_{M})(y)|^{2}\,dxdy\Big{)}dt\\\
+C_{T}NM\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}\,.$

Combining the above estimate with (4.34) and (4.36) we obtain

(4.37)
$\int_{0}^{T}\Big{(}\int|\bar{v}_{M}(x)\nabla_{x}u_{N}(x)|^{2}\,dx\Big{)}dt\leq
C_{T}(M^{2}+NM)\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}\\\ \leq
C_{T}NM\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}\,.$

On the other hand by Cauchy-Schwarz, Strichartz estimate and (4.35) we have

$\int_{0}^{T}\Big{(}\int|u_{N}(x)\nabla_{x}\bar{v}_{M}(x)|^{2}\,dx\Big{)}dt\leq\|u_{N}\|_{L^{4}((0,T);L^{4})}^{2}\|\nabla
v_{M}\|_{L^{4}((0,T);L^{4})}^{2}\leq
C_{T}M^{2}\|v_{M}(0)\|_{L^{2}}^{2}\|u_{N}(0)\|^{2}_{L^{2}}\,.$

Therefore combining this last estimate with (4.37) we get (4.13).

### Proof of (4.14)

Due to (4.13), it suffices to prove

(4.38) $\int_{0}^{T}\Big{(}\int|x|^{2}|v_{M}\bar{u}_{N}|^{2}dx\Big{)}dt\leq
C_{T}MN\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}.$

By Hölder inequality we have

(4.39)
$\int_{0}^{T}\Big{(}\int|x|^{2}|v_{M}\bar{u}_{N}|^{2}\,dx\Big{)}dt\leq\||x|v_{M}\|^{2}_{L^{4}((0,T);L^{4})}\|u_{N}\|^{2}_{L^{4}((0,T);L^{4})}\,.$

Next notice that $|x|^{2}v_{M}$ is solution to the inhomogeneous equation
associated with (4.10), with source term $-4v_{M}-2x\cdot\nabla v_{M}$. Again,
using Strichartz and placing the source term in $L^{1}((0,T);L^{2})$,

(4.40) $\displaystyle\||x|^{2}v_{M}\|_{L^{4}((0,T);L^{4})}\leq$ $\displaystyle
C\||x|^{2}v_{M}(0)\|_{L^{2}}+C\|v_{M}\|_{L^{1}((0,T);L^{2})}+C\|x\cdot\nabla
v_{M}\|_{L^{1}((0,T);L^{2})}$ (4.41) $\displaystyle\leq$ $\displaystyle
C_{T}\|v_{M}(0)\|_{\mathcal{H}^{2}}\leq C_{T}M^{2}\|v_{M}(0)\|_{L^{2}}$

where we used the time independent estimate $\|x\cdot\nabla w\|_{L^{2}}\leq
C\|w\|_{{\mathcal{H}}^{2}}$ (see (4.31)) and conservation of the
${\mathcal{H}}^{2}$ norm for (4.10). Interpolation between (4.40) and the
Strichartz estimate $\|v_{M}\|_{L^{4}((0,T);L^{4})}\leq C\|v_{M}(0)\|_{L^{2}}$
implies $\||x|v_{M}\|_{L^{4}((0,T);L^{4})}\leq CM\|v_{M}(0)\|_{L^{2}}$.
Combining this estimate, Strichartz for $u_{N}$ and (4.39) we obtain (4.38)
(actually, a stronger version of (4.38) as on the r.h.s. we get $M^{2}$).

### Proof of the implication (4.14) $\Rightarrow$ (4.11)

We can write

(4.42) $\|v_{M}u_{N}\|^{2}_{L^{2}((0,T);L^{2})}=\sum_{K\in
2^{\mathbb{N}}}\|\Delta_{K}(v_{M}u_{N})\|^{2}_{L^{2}((0,T);L^{2})}$

If $K>N$, we may forget about $\Delta_{K}$ and use (4.14) in order to get

(4.43) $\sum_{K>N}\|\Delta_{K}(v_{M}u_{N})\|^{2}_{L^{2}((0,T);L^{2})}\leq
C\sum_{K>N}(1+K)^{-2}\|v_{M}u_{N}\|_{L^{2}((0,T);{\mathcal{H}}^{1})}^{2}\leq
C_{T}MN^{-1}\|u_{N}(0)\|^{2}_{L^{2}}\|v_{M}(0)\|^{2}_{L^{2}}$

For $K\leq N$, denote $S_{N}=\sum_{K\leq N}\Delta_{K}$ and write directly

(4.44) $S_{N}(v_{M}u_{N})=S_{N}(v_{M}N^{-2}A\tilde{u}_{N})$

where $\tilde{u}_{N}=\tilde{\Delta}_{N}u_{N}$ and the localization operator
$\tilde{\Delta}_{N}$ was chosen so that $N^{-2}A\tilde{\Delta}_{N}$ is the
identity on the support of $\Delta_{N}$. We may now write
$v_{M}A\tilde{u}_{N}=A(v_{M}\tilde{u}_{N})+\tilde{u}_{N}\Delta v_{M}+2\nabla
v_{M}\cdot\nabla\tilde{u}_{N}$ and hence by (4.44), uniform boundedness of
$S_{N}$ and $N^{-1}\sqrt{A}S_{N}$ on $L^{2}$, we get

$\displaystyle\|S_{N}(v_{M}u_{N})\|_{L^{2}}^{2}\leq$ $\displaystyle
CN^{-2}\|N^{-1}\sqrt{A}S_{N}(\sqrt{A}(v_{M}\tilde{u}_{N})\|_{L^{2}}^{2}+CN^{-4}(\|\tilde{u}_{N}\Delta
v_{M}\|^{2}_{L^{2}}+\|\nabla v_{M}\cdot\nabla\tilde{u}_{N}\|^{2}_{L^{2}})$
$\displaystyle\leq$ $\displaystyle
CN^{-2}\|\sqrt{A}(v_{M}\tilde{u}_{N})\|_{L^{2}}^{2}+CN^{-4}\|\Delta
v_{M}\|^{2}_{L^{4}}\|\tilde{u}_{N}\|^{2}_{L^{4}}+CN^{-4}\|\nabla
v_{M}\|^{2}_{L^{4}}\|\nabla\tilde{u}_{N}\|^{2}_{L^{4}}\,.$

After integration in time, using Strichartz estimates to control $L^{4}$ norms
(use (4.35) to control $\|\nabla v_{M}\|_{L^{4}_{t,x}}$ and a similar argument
to control $\|\nabla\tilde{u}_{N}\|_{L^{4}_{t,x}}$) and (4.14), we get

$\displaystyle\int_{0}^{T}\|S_{N}(v_{M}u_{N})\|_{L^{2}}^{2}\,dt\leq$
$\displaystyle
CN^{-2}\int_{0}^{T}\|v_{M}\tilde{u}_{N}\|_{\mathcal{H}^{1}}^{2}\,dt+C_{T}M^{2}N^{-4}(M^{2}+N^{2})\|v_{M}(0)\|^{2}_{L^{2}}\|\tilde{u}_{N}(0)\|^{2}_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C_{T}MN^{-1}\|v_{M}(0)\|^{2}_{L^{2}}\|\tilde{u}_{N}(0)\|^{2}_{L^{2}}+C_{T}M^{2}N^{-2}\|v_{M}(0)\|^{2}_{L^{2}}\|\tilde{u}_{N}(0)\|^{2}_{L^{2}}\,,$

and we complete the proof with $\|\tilde{u}_{N}(0)\|_{L^{2}}\leq
C\|u_{N}(0)\|_{L^{2}}$.∎

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