# Inter-critical NLS:
critical $\dot{H}^{s}$-bounds imply scattering

Jason Murphy  Department of Mathematics, University of California at Los
Angeles, Los Angeles, CA 90095-1555, USA<EMAIL_ADDRESS>
###### Abstract.

We consider a class of power-type nonlinear Schrödinger equations for which
the power of the nonlinearity lies between the mass- and energy-critical
exponents. Following the concentration-compactness approach, we prove that if
a solution $u$ is bounded in the critical Sobolev space throughout its
lifespan, that is, $u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$, then $u$ is
global and scatters.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline of the proof of Theorem 1.3
  2. 2 Notation and useful lemmas
    1. 2.1 Some notation
    2. 2.2 Basic harmonic analysis
    3. 2.3 Strichartz estimates
    4. 2.4 Concentration-compactness
  3. 3 Local well-posedness
  4. 4 Reduction to almost periodic solutions
  5. 5 Long-time Strichartz estimates
  6. 6 The rapid frequency-cascade scenario
  7. 7 The frequency-localized interaction Morawetz inequality
  8. 8 The quasi-soliton scenario
  9. A Some basic estimates

##  1\. Introduction

We consider the initial-value problem for defocusing nonlinear Schrödinger
equations of the form

$\left\\{\begin{array}[]{ll}(i\partial_{t}+\Delta)u=|u|^{p}u\\\
u(0,x)=u_{0}(x),\end{array}\right.$ (1.1)

where $p>0$ is chosen to lie between the mass- and energy-critical exponents,
that is, $\tfrac{4}{d}<p<\tfrac{4}{d-2}$. Here
$u:\mathbb{R}_{t}\times\mathbb{R}_{x}^{d}\to\mathbb{C}$ is a complex-valued
function of time and space.

The class of solutions to (1.1) is left invariant by the scaling

$u(t,x)\mapsto\lambda^{\frac{2}{p}}u(\lambda^{2}t,\lambda x)$

for $\lambda>0$. This scaling defines a notion of _criticality_. In
particular, one can check that the only homogeneous $L_{x}^{2}$-based Sobolev
space that is left invariant by this scaling is
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, where the _critical regularity_ $s_{c}$
is given by $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}.$ If we take
$u_{0}\in\dot{H}_{x}^{s}(\mathbb{R}^{d})$ in (1.1), then for $s=s_{c}$, we
call the problem _critical_. For $s>s_{c}$, we call the problem _subcritical_
, while for $s<s_{c}$ the problem is _supercritical_.

We consider the critical problem for (1.1) in the _inter-critical_ regime,
that is, $0<s_{c}<1$. For $(d,s_{c})$ satisfying an appropriate set of
constraints, we prove that any maximal-lifespan solution that is uniformly
bounded (throughout its lifespan) in $\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$
must be global and scatter.

We begin by making the notion of a solution more precise.

######  Definition 1.1 (Solution).

A function $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ on a non-empty time interval
$I\ni 0$ is a _solution_ to (1.1) if it belongs to
$C_{t}\dot{H}_{x}^{s_{c}}(K\times\mathbb{R}^{d})\cap
L_{t,x}^{p(d+2)/2}(K\times\mathbb{R}^{d})$ for every compact $K\subset I$ and
obeys the Duhamel formula

$u(t)=e^{it\Delta}u_{0}-i\int_{0}^{t}e^{i(t-s)\Delta}(|u|^{p}u)(s)\,ds$

for all $t\in I$. We call $I$ the _lifespan_ of $u$; we say $u$ is a _maximal-
lifespan solution_ if it cannot be extended to any strictly larger interval.
If $I=\mathbb{R}$, we say $u$ is _global_.

We define the _scattering size_ of a solution $u$ to (1.1) on a time interval
$I$ by

$S_{I}(u):=\int_{I}\int_{\mathbb{R}^{d}}|u(t,x)|^{\frac{p(d+2)}{2}}\,dx\,dt.$
(1.2)

Standard arguments show that if a solution $u$ to (1.1) is global, with
$S_{\mathbb{R}}(u)<\infty$, then it _scatters_ ; that is, there exist unique
$u_{\pm}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ such that

$\lim_{t\to\pm\infty}\big{\|}u(t)-e^{it\Delta}u_{\pm}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}=0.$

The goal of this paper is to address some cases of the following

######  Conjecture 1.2.

Let $d\geq 1$ and $p>0$ such that $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}\geq 0$.
Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan solution to
(1.1) such that $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$. Then $u$ is global
and scatters, with

$S_{\mathbb{R}}(u)\leq
C(\big{\|}u\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(\mathbb{R}\times\mathbb{R}^{d})})$

for some function $C:[0,\infty)\to[0,\infty).$

For two special cases, it is unnecessary to take $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$ as an additional hypothesis, as this bound
follows from conservation laws. In particular, in the mass-critical case,
$s_{c}=0$ (i.e. $p=\tfrac{4}{d}$), the fact that $u\in
L_{t}^{\infty}L_{x}^{2}$ follows from the conservation of _mass_ , defined by

$M[u(t)]:=\int_{\mathbb{R}^{d}}|u(t,x)|^{2}\,dx,$

while in the energy-critical case, $s_{c}=1$ (i.e. $p=\tfrac{4}{d-2}$, $d\geq
3$), the fact that $u\in L_{t}^{\infty}\dot{H}_{x}^{1}$ follows from the
conservation of _energy_ , defined by

$E[u(t)]:=\int_{\mathbb{R}^{d}}\tfrac{1}{2}|\nabla
u(t,x)|^{2}+\tfrac{1}{p+2}|u(t,x)|^{p+2}\,dx.$

Due to the presence of conserved quantities at critical regularity, the mass-
and energy-critical equations have been the most widely studied; in fact,
Conjecture 1.2 has been settled in these cases. For $s_{c}\notin\\{0,1\\}$,
the hypothesis that $u$ stays bounded in $\dot{H}_{x}^{s_{c}}$ in Conjecture
1.2 plays the role of the ‘missing conservation law’ at critical regularity;
the aim of this paper is to show that with this extra assumption, the
techniques developed to treat the mass- and energy-critical NLS can be applied
in the inter-critical regime, $0<s_{c}<1.$

The defocusing energy-critical case was handled first by Bourgain [3],
Grillakis [23], and Tao [49] for radial data, and subsequently by
Colliander–Keel–Staffilani–Takaoka–Tao [15], Ryckman–Vişan [40], and Vişan
[55, 56] for arbitrary data. (See also [25, 32] for results in the focusing
case.) The primary obstacle to establishing these results was the lack of any
a priori estimates with critical scaling (besides the conservation of energy);
that is to say, none of the known monotonicity formulae for NLS (i.e. Morawetz
inequalities) scale like the energy (in contrast to the energy-critical
nonlinear wave equation, for example). It was ultimately the ‘induction on
energy’ technique of Bourgain that showed how one can move beyond this
difficulty: by finding a bubble of concentration inside a solution, one
introduces a characteristic length scale into a scale-invariant problem.
Having ‘broken’ the scaling symmetry in this way, the available Morawetz
inequalities come back into play, despite their non-critical scaling. All
subsequent work for NLS at critical regularity has built upon this fundamental
idea.

In the energy-critical case, the critical regularity associated to the
available Morawetz estimates is lower than the critical regularity of the
problem. Bourgain was able to make use of the Lin–Strauss Morawetz inequality
(appearing first in [38]), which scales like $\dot{H}_{x}^{\frac{1}{2}}$ and
is well-suited to the radial case; to remove the radial assumption,
Colliander–Keel–Staffilani–Takaoka–Tao introduced the interaction Morawetz
inequality (see [14]), which has the scaling of $\dot{H}_{x}^{\frac{1}{4}}$
(but still requires control over at least half of a derivative, cf. (1.3)
below). These considerations lead us to believe that the techniques developed
to handle the energy-critical problem should be applicable to resolve
Conjecture 1.2 in the case $s_{c}\geq\tfrac{1}{2}$.

In particular, we will be making use of the concentration-compactness (or
‘minimal counterexample’) approach to induction on energy. Minimal
counterexamples were introduced over the course of several papers in the
context of the mass-critical problem (see, for example, [1, 2, 7, 28, 29,
39]); however, the first application of minimal counterexamples to establish a
global well-posedness result was carried out by Kenig–Merle [25], who
developed the technique in the focusing, energy-critical setting.

Minimal counterexamples were also used to establish Conjecture 1.2 in the
mass-critical setting, first for spherically-symmetric data in dimensions
$d\geq 2$ (see [30, 37, 52]), and subsequently for arbitrary initial data in
all dimensions by Dodson [17, 18, 19]. (For results in the focusing case, see
[20, 30, 37].) Notice that in the mass-critical case, the critical regularity
of the problem is lower than that of the available Morawetz estimates; thus
one needs to prove additional regularity (instead of decay) to access these
estimates. We pause here to point out [36, 57], as well, which revisit the
defocusing energy-critical problem from the perspective of minimal
counterexamples.

The first case of Conjecture 1.2 at non-conserved critical regularity to be
addressed was the case $d=3$ and $s_{c}=\tfrac{1}{2}$, in which case the
nonlinearity is cubic. Kenig–Merle [26] were able to handle this case by using
their concentration-compactness technique (as in [25]), together with the
Lin–Strauss Morawetz inequality (which is scaling-critical in this case). As
we will see, this case also falls into the range of cases that we consider,
although we will opt to use the interaction Morawetz inequality instead.

Some cases of Conjecture 1.2 in the energy-supercritical regime (i.e.
$s_{c}>1$) have also been handled by Killip–Vişan [33], also through the use
of minimal counterexamples. In particular, they deal with the case of a cubic
nonlinearity for $d\geq 5$, along with some other cases for which $s_{c}>1$
and $d\geq 5$. Their restriction to high dimensions stems ultimately from
their use of the so-called ‘double Duhamel trick’; for more details, see [33]
and the references cited therein.

Before we discuss our contribution, we note that the analogous conjecture has
also been studied for the nonlinear wave equation. For progress in the energy-
supercritical case, one can refer to works of Kenig–Merle [27], Killip–Vişan
[34, 35], and Bulut [4, 5, 6]. For some results in the energy-subcritical case
with radial data, see [42, 43].

Finally, we are in a position to describe the cases of Conjecture 1.2 that we
will address in this paper. As mentioned above, we will work in the inter-
critical regime, $0<s_{c}<1$ (that is, $\tfrac{4}{d}<p<\tfrac{4}{d-2}$). Our
primary restriction is technical; namely, we only consider cases for which
$p\geq 1$. This restriction serves to simplify the analysis, which still
becomes a bit complicated. For example, when we need to estimate a fractional
number of derivatives of the function $G(z)=|z|^{p}$, things are quite a bit
simpler when $G$ is locally Lipschitz, rather than merely Hölder continuous.

Next, in order to make use of the interaction Morawetz inequality, we restrict
to the cases $d\geq 3$ and $s_{c}\geq\tfrac{1}{2}$ (cf. (1.3) below). For this
restriction to be compatible with $p\geq 1$, we must then restrict to $d\leq
5$. The use of the interaction Morawetz inequality ultimately leads to a
further (more severe) restriction: when $d=3$, we must exclude the cases
$\tfrac{3}{4}<s_{c}<1$. Let us briefly describe the reason for this additional
restriction.

The standard interaction Morawetz inequality may be written as follows: for
$u$ solving (1.1), we have

$\displaystyle-\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}$
$\displaystyle|u(t,x)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u(t,y)|^{2}\,dx\,dy\,dt$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}^{2}\big{\|}|\nabla|^{1/2}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}^{2}.$
(1.3)

As we are in the case $s_{c}\geq\tfrac{1}{2}$, we see that to guarantee that
the right-hand side is finite, we must truncate the solution $u$ to high
frequencies (that is, we must work with $u_{\geq N}$ for some $N>0$). In
Section 7, we do exactly this. However, $u_{\geq N}$ is no longer a solution
to (1.1); thus, the truncation results in error terms for (1.3) that must be
handled to arrive at a useful estimate. When $d=3$ and $s_{c}>\tfrac{3}{4}$,
we find that there is one error term that we cannot handle unless we also
impose a spatial truncation on the weight we use to derive (1.3); see, for
example, [15, 36], which address the case $d=3$, $s_{c}=1$. This spatial
truncation, however, results in even _more_ error terms; it turns out that
some of these additional error terms then require control over the
$\dot{H}_{x}^{1}$-regularity of the solution. Thus, in the energy-critical
case, one can push the argument through, while in our case we cannot succeed
without significant additional input. For further comments, see Remark 7.4.

Our final restriction is to exclude the case $(d,s_{c})=(5,\tfrac{1}{2})$, for
which $p=1$. In this case, the proof of Lemma 4.2, which is essential to the
reduction to almost periodic solutions in Section 4, breaks down. For further
discussion, see Remark 4.3.

To summarize, our main results will apply to (1.1) with $(d,s_{c})$ satisfying
the following:

$\left\\{\begin{array}[]{llllll}\tfrac{1}{2}\leq
s_{c}\leq\tfrac{3}{4}&\text{if }d=3,\vspace{1mm}\\\ \tfrac{1}{2}\leq
s_{c}<1&\text{if }d=4,\vspace{1mm}\\\ \tfrac{1}{2}<s_{c}<1&\text{if
}d=5.\end{array}\right.$ (1.4)

It is worth mentioning, however, that many of our results will apply to a less
restrictive set of $(d,s_{c})$. In particular, many of our results will apply
to

$d\in\\{3,4,5\\}\quad\text{and}\quad\tfrac{1}{2}\leq
s_{c}<1,\quad\text{excluding}\quad(d,s_{c})=(5,\tfrac{1}{2}).$ (1.5)

As we proceed, we will keep track of which restrictions are necessary for
which results.

Our main result is the following:

######  Theorem 1.3.

Suppose $(d,s_{c})$ satisfies (1.4). Let
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan solution to (1.1)
such that $u\in L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d}).$
Then $u$ is global and scatters, with

$S_{\mathbb{R}}(u)\leq
C(\big{\|}u\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(\mathbb{R}\times\mathbb{R}^{d})})$

for some function $C:[0,\infty)\to[0,\infty).$

To establish Theorem 1.3, we will model our approach after several sources,
including [31, 32, 33, 36, 57]. In particular, Section 3 follows the
presentation in [31, 33]; Section 4 draws heavily from [32]; and the
presentation of the remaining sections is inspired largely by [36, 57]. We
note also that we rely on [31] for several standard results regarding almost
periodic solutions in the outline below.

The first step towards a global-in-time theory for (1.1) is to develop a good
local-in-time theory for this equation. In particular, building off arguments
of Cazenave–Weissler [8], one can prove the following

######  Theorem 1.4 (Local well-posedness).

Let $(d,s_{c})$ satisfy (1.5). Then, given
$u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, there exists a unique maximal-
lifespan solution $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1). Moreover,
this solution satisfies the following:

$\bullet$ (Local existence) $I$ is an open neighborhood of $0$.

$\bullet$ (Blowup criterion) If $\sup I$ is finite, then $u$ blows up forward
in time, in the sense that $S_{[0,\sup I)}(u)=\infty$. If $\inf I$ is finite,
then $u$ blows up backwards in time, in the sense that $S_{(\inf
I,0]}(u)=\infty.$

$\bullet$ (Scattering) If $\sup I=+\infty$ and $u$ does not blow up forward in
time, then $u$ scatters forward in time; that is, there exists a unique
$u_{+}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ such that

$\lim_{t\to\infty}\big{\|}u(t)-e^{it\Delta}u_{+}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}=0.$
(1.6)

Conversely, for any $u_{+}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, there is a
unique solution $u$ to (1.1) so that (1.6) holds. The analogous statements
hold backward in time.

$\bullet$ (Small-data global existence) There exists $\eta_{0}=\eta_{0}(d,p)$
such that if

$\big{\|}u_{0}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{2}<\eta_{0},$

then $u$ is global and scatters, with
$S_{\mathbb{R}}(u)\lesssim\big{\|}u_{0}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{p(d+2)/2}.$

######  Remark 1.5.

We note here that the notion of blowup described above corresponds exactly to
the impossibility of extending the solution to a larger time interval in the
class described in Definition 1.1.

In Section 3, we will establish this theorem as a corollary of a local well-
posedness result of Cazenave–Weissler [8] (Theorem 3.1) and a stability result
(Theorem 3.4). This stability result plays an essential role in the argument
we present, specifically in the proof of Theorem 1.12.

###  1.1. Outline of the proof of Theorem 1.3

The proof is by contradiction. We first recall that Theorem 1.3 holds if we
restrict to sufficiently small initial data (cf. Theorem 1.4); thus, the
failure of Theorem 1.3 would imply the existence of a ‘threshold’ size, below
which Theorem 1.3 holds, but above which we can find (almost) counterexamples.
Using a limiting argument, we then find blowup solutions _at_ this threshold,
so-called ‘minimal blowup solutions’. By carefully analyzing such solutions,
we can show that they must have so many nice properties that in fact, they
cannot exist at all.

The main property of these special counterexamples is that of almost
periodicity modulo symmetries:

######  Definition 1.6 (Almost periodic solutions).

Let $s_{c}>0$. A solution $u$ to (1.1) with lifespan $I$ is said to be _almost
periodic (modulo symmetries)_ if $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$ and there exist
functions $N:I\to\mathbb{R}^{+},$ $x:I\to\mathbb{R}^{d}$, and
$C:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that for all $t\in I$ and all
$\eta>0$,

$\int_{|x-x(t)|\geq\frac{C(\eta)}{N(t)}}\big{|}|\nabla|^{s_{c}}u(t,x)\big{|}^{2}\,dx+\int_{|\xi|\geq
C(\eta)N(t)}|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta.$

We call $N$ the _frequency scale function_ , $x$ the _spatial center function_
, and $C$ the _compactness modulus function_.

######  Remark 1.7.

By the Arzelà–Ascoli theorem, a family of functions is precompact in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ if and only if it is norm-bounded and
there exists a compactness modulus function $C$ such that

$\int_{|x|\geq
C(\eta)}\big{|}|\nabla|^{s_{c}}f(x)\big{|}^{2}\,dx+\int_{|\xi|\geq
C(\eta)}|\xi|^{2s_{c}}|\widehat{f}(\xi)|^{2}\,d\xi\leq\eta$

uniformly for all functions $f$ in the family. Thus, an equivalent formulation
of Definition 1.6 is the following: $u$ is almost periodic (modulo symmetries)
if and only if

$\\{u(t):t\in
I\\}\subset\\{\lambda^{\frac{2}{p}}f(\lambda(x+x_{0})):\lambda\in(0,\infty),\
x_{0}\in\mathbb{R}^{d},\ \text{and}\ f\in K\\}$

for some compact $K\subset\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d}).$

Furthermore, Sobolev embedding gives that every compact set in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ is also compact in
$L_{x}^{\frac{dp}{2}}(\mathbb{R}^{d})$; thus, for any almost periodic solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$, we also have

$\int_{|x-x(t)|\geq\frac{C(\eta)}{N(t)}}|u(t,x)|^{\frac{dp}{2}}\,dx\leq\eta$

for all $t\in I$ and $\eta>0$.

######  Remark 1.8.

Another consequence of almost periodicity modulo symmetries is the existence
of a function $c:\mathbb{R}^{+}\to\mathbb{R}^{+}$ so that for all $t\in I$ and
all $\eta>0$,

$\int_{|x-x(t)|\leq\frac{c(\eta)}{N(t)}}\big{|}|\nabla|^{s_{c}}u(t,x)\big{|}^{2}\,dx+\int_{|\xi|\leq
c(\eta)N(t)}|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta.$

One can show (see [31, Lemma 5.18], for example) that the modulation
parameters of almost periodic solutions obey the following local constancy
property:

######  Lemma 1.9 (Local constancy).

Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan almost
periodic solution to (1.1). Then there exists $\delta=\delta(u)>0$ such that
if $t_{0}\in I$, then

$[t_{0}-\delta N(t_{0})^{-2},t_{0}+\delta N(t_{0})^{-2}]\subset I,$

with

$N(t)\sim_{u}N(t_{0})\indent\text{for}\quad|t-t_{0}|\leq\delta N(t_{0})^{-2}.$

Given a maximal-lifespan almost periodic solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1), Lemma 1.9 allows us to
subdivide $I$ into _characteristic subintervals_ $J_{k}$ on which $N(t)$ is
constant and equal to some $N_{k}$, with $|J_{k}|\sim_{u}N_{k}^{-2}$. To do
this, we need to modify the compactness modulus function by a time-independent
multiplicative factor.

The local constancy property also has the following consequence (see [31,
Corollary 5.19]):

######  Corollary 1.10 ($N(t)$ at blowup).

Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan almost
periodic solution to (1.1). If $T$ is any finite endpoint of $I$, then
$N(t)\gtrsim_{u}|T-t|^{-1/2}$; in particular, $\lim_{t\to T}N(t)=\infty.$ If
$I$ is infinite or semi-infinite, then for any $t_{0}\in I$, we have
$N(t)\gtrsim\langle t-t_{0}\rangle^{-1/2}.$

Finally, we need the following result, which relates the frequency scale
function of an almost periodic solution to its Strichartz norms.

######  Lemma 1.11 (Spacetime bounds).

Let $(d,s_{c})$ satisfy (1.5), and suppose $u$ is an almost periodic solution
to (1.1) on a time interval $I$. Then

$\int_{I}N(t)^{2}\,dt\lesssim_{u}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}^{2}\lesssim_{u}1+\int_{I}N(t)^{2}\,dt.$

One can prove this result by adapting the proof of [31, Lemma 5.21]; the key
is to notice that $\int_{I}N(t)^{2}\,dt$ is approximately the number of
characteristic subintervals inside $I$. The restriction on $(d,s_{c})$ is not
actually necessary, but it covers our cases of interest.

With these preliminaries established, we can now describe the first major step
in the proof of Theorem 1.3.

######  Theorem 1.12 (Reduction to almost periodic solutions).

Suppose that Theorem 1.3 fails for $(d,s_{c})$ satisfying (1.5). Then there
exists a maximal-lifespan solution $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to
(1.1) such that $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$, $u$ is almost
periodic modulo symmetries, and $u$ blows up both forward and backward in
time. Moreover, $u$ has minimal $L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$-norm among
all blowup solutions; i.e.,

$\sup_{t\in
I}\big{\|}u(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}\leq\sup_{t\in
J}\big{\|}v(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}$

for all maximal-lifespan solutions $v:J\times\mathbb{R}^{d}\to\mathbb{C}$ that
blow up in at least one time direction.

We sketch a proof of Theorem 1.12 in Section 4. By now, the reduction to
almost periodic solutions is a fairly standard technique in the study of
dispersive equations at critical regularity. Keraani [29] first proved the
existence of minimal blowup solutions (in the mass-critical setting), while
Kenig–Merle [25] were the first to use them as a tool to prove global well-
posedness (in the energy-critical setting). For many more examples of these
techniques, one can refer to [26, 27, 30, 32, 31, 33, 34, 35, 37, 53, 52], for
example.

Still, while the underlying ideas are well-established, we will see that to
carry out the reduction in the cases we consider will require some new ideas
and careful analysis; indeed, the resolution of this problem is the chief
novelty of this paper. One of the principal difficulties arises in the proof
of Lemma 4.2, in which we establish a decoupling of nonlinear profiles in
order to show that a sequence of approximate solutions to (1.1) converges in
some sense to a true solution.

For the mass- and energy-critical cases, one can use pointwise estimates and
well-known arguments of [28] to establish this decoupling; in our setting, the
nonlocal nature of $|\nabla|^{s_{c}}$ prevents the direct use of any pointwise
estimates. The authors of [33], who dealt with some cases in the energy-
supercritical regime, overcame this difficulty by establishing analogous
pointwise estimates for a square function of Strichartz that shares estimates
with $|\nabla|^{s_{c}}$ (see [46]). With such estimates in hand, the usual
arguments can then be pushed through. Their approach does not work in our
setting, however, as it strongly relies on the fact that $s_{c}>1$. In [26],
the authors treat a cubic nonlinearity in dimension $d=3$ (in which case
$s_{c}=\tfrac{1}{2}$); by exploiting the polynomial nature of the nonlinearity
and employing a paraproduct estimate, they too overcome the nonlocal nature of
$|\nabla|^{s_{c}}$ and put themselves in a position where the standard
arguments are applicable. In our setting, the combination of fractional
derivatives and non-polynomial nonlinearities presents a new technical
challenge. Ultimately, the resolution of this problem comes from a careful
reworking of the proof of the fractional chain rule, in which the
Littlewood–Paley square function allows us to work at the level of individual
frequencies. By making use of some tools from harmonic analysis (including
maximal and vector maximal inequalities), we are eventually able to adapt the
standard arguments to establish the decoupling in our setting. For further
discussion, see Section 4.

After establishing Theorem 1.12, we make some further refinements to the class
of solutions we consider. First, we can use a rescaling argument to restrict
our attention to almost periodic solutions that do not escape to arbitrarily
low frequencies on at least half of their maximal lifespan, say $[0,T_{max})$.
We will not include the details here; one can instead refer to Section 4 in
any of [30, 32, 52]. Next, following [17], we will divide these solutions into
two classes that depend on the control given by the interaction Morawetz
inequality; these classes will correspond to the ‘rapid frequency-cascade’ and
‘quasi-soliton’ scenarios. Finally, as described above, we use Lemma 1.9 to
subdivide $[0,T_{max})$ into characteristic subintervals $J_{k}$ and set
$N(t)$ to be constant and equal to $N_{k}$ on each $J_{k}$, with
$|J_{k}|\sim_{u}N_{k}^{-2}$. In this way, we arrive at

######  Theorem 1.13 (Two special scenarios for blowup).

Suppose that Theorem 1.3 fails for $(d,s_{c})$ satisfying (1.5). Then there
exists an almost periodic solution
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ that blows up forward in
time, with

$N(t)\equiv N_{k}\geq 1$

for each $t\in J_{k}$, where $[0,T_{max})=\cup_{k}J_{k}$. Furthermore,

$\text{either}\quad\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty\quad\text{or}\quad\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$

Thus, to establish Theorem 1.3, it remains to preclude the existence of the
almost periodic solutions described in Theorem 1.13. The main technical tool
we will use to achieve this will be a long-time Strichartz inequality,
Proposition 5.1. Such inequalities were originally developed by Dodson [17]
for almost periodic solutions in the mass-critical setting; for variants in
the energy-critical setting, see [36, 57]. In this paper, we establish a long-
time Strichartz estimate for the first time in the inter-critical regime. The
proof of Proposition 5.1 is by induction; the recurrence relation is derived
with the aid of Strichartz estimates, together with a paraproduct estimate
(Lemma 2.6) and a bilinear Strichartz inequality (Lemma 2.10).

In Section 6, we preclude the rapid frequency-cascade scenario, that is,
almost periodic solutions as in Theorem 1.13 for which
$\textstyle\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty.$ This proof requires
two ingredients. The first ingredient is the long-time Strichartz inequality
(Proposition 5.1), while the second is the following

######  Proposition 1.14 (No-waste Duhamel formula).

Let $u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}$. Then for all $t\in[0,T_{max}),$ we have

$u(t)=i\lim_{T\to T_{max}}\int_{t}^{T}e^{i(t-s)\Delta}(|u|^{p}u)(s)\,ds,$

where the limits are taken in the weak $\dot{H}_{x}^{s_{c}}$ topology.

To prove Proposition 1.14, one can adapt the proof of [31, Proposition 5.23];
we omit the details. Using Proposition 1.14 and Strichartz estimates, we can
upgrade the information given by Proposition 5.1 to see that a rapid
frequency-cascade solution must have finite mass. In fact, we can show that
the solution has zero mass, which contradicts that the solution blows up.

In Section 8, we preclude the quasi-soliton scenario, that is, almost periodic
solutions as in Theorem 1.13 for which
$\textstyle\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$ The main ingredient
is a frequency-localized interaction Morawetz inequality (Proposition 7.1),
which we prove in Section 7. To establish this estimate, we begin with the
usual interaction Morawetz inequality, truncate to high frequencies, and use
Proposition 5.1 to control the resulting error terms. (As described above, one
of these error terms eventually forces us to exclude the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ from Theorem 1.3; see Remark 7.4.)
To rule out the quasi-soliton scenario and thereby complete the proof of
Theorem 1.3, we notice that the frequency-localized interaction Morawetz
inequality provides uniform control over $\int_{I}N(t)^{3-4s_{c}}\,dt$ for all
compact time intervals $I\subset[0,T_{max})$; thus we can derive a
contradiction by taking $I$ to be sufficiently large inside of $[0,T_{max})$.

### Acknowledgements

I owe many thanks to my advisors, Rowan Killip and Monica Vişan, for all of
their guidance and support. I am very grateful to them, not only for bringing
this problem to my attention, but also for engaging in many helpful
discussions, and for a careful reading of the manuscript. This work was
supported in part by NSF grant DMS-1001531 (P.I. Rowan Killip).

##  2\. Notation and useful lemmas

###  2.1. Some notation

We write $X\lesssim Y$ or $Y\gtrsim X$ whenever $X\leq CY$ for some $C>0$. If
$X\lesssim Y\lesssim X$, we write $X\sim Y$. If the implicit constant $C$
depends on the dimension $d$ or the power $p$, this dependence will be
suppressed; dependence on additional parameters will be indicated with
subscripts. For example, $X\lesssim_{u}Y$ indicates that $X\leq CY$ for some
$C=C(u).$

For a spacetime slab $I\times\mathbb{R}^{d}$, we write
$L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})$ to denote the Banach space of
functions $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ equipped with the norm

$\big{\|}u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:=\left(\int_{I}\big{\|}u(t)\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}^{q}\,dt\right)^{1/q},$

with the usual conventions when $q$ or $r$ is infinity. If $q=r$, we
abbreviate $L_{t}^{q}L_{x}^{q}=L_{t,x}^{q}.$ At times we will also abbreviate
$\big{\|}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$ to
$\big{\|}f\big{\|}_{L_{x}^{r}}$ or $\big{\|}f\big{\|}_{r}.$

We define the Fourier transform on $\mathbb{R}^{d}$ by

$\widehat{f}(\xi):=(2\pi)^{-d/2}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)\,dx.$

For $s\in\mathbb{R}$, we can then define the fractional differentiation
operator $|\nabla|^{s}$ via

$\widehat{|\nabla|^{s}f}(\xi):=|\xi|^{s}\widehat{f}(\xi),$

which in turn defines the homogeneous Sobolev norm

$\big{\|}f\big{\|}_{\dot{H}_{x}^{s}(\mathbb{R}^{d})}:=\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}.$

###  2.2. Basic harmonic analysis

Let $\varphi$ be a radial bump function supported in the ball
$\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\tfrac{11}{10}\\}$ and equal to 1 on the
ball $\\{\xi\in\mathbb{R}^{d}:|\xi|\leq 1\\}.$ For $N\in 2^{\mathbb{Z}}$, we
define the Littlewood–Paley projection operators via

$\displaystyle\widehat{P_{\leq N}f}(\xi):=\widehat{f_{\leq
N}}(\xi):=\varphi(\xi/N)\widehat{f}(\xi),$
$\displaystyle\widehat{P_{>N}f}(\xi):=\widehat{f_{>N}}(\xi):=(1-\varphi(\xi/N))\widehat{f}(\xi),$
$\displaystyle\widehat{P_{N}f}(\xi):=\widehat{f_{N}}(\xi):=(\varphi(\xi/N)-\varphi(2\xi/N))\widehat{f}(\xi).$

We define $P_{<N}$ and $P_{\geq N}$ similarly. We also define

$P_{M<\cdot\leq N}:=P_{\leq N}-P_{\leq M}=\sum_{M<N^{\prime}\leq
N}P_{N^{\prime}}$

for $M<N.$ All such summations are understood to be over $N\in
2^{\mathbb{Z}}.$ Being Fourier multiplier operators, these Littlewood–Paley
projection operators commute with $e^{it\Delta}$, as well as differential
operators (for example, $i\partial_{t}+\Delta$). We will need the following
standard estimates for these operators:

######  Lemma 2.1 (Bernstein estimates).

For $1\leq r\leq q\leq\infty$ and $s\geq 0$,

$\displaystyle\big{\|}|\nabla|^{\pm
s}P_{N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$ $\displaystyle\sim N^{\pm
s}\big{\|}P_{N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}|\nabla|^{s}P_{\leq
N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$ $\displaystyle\lesssim
N^{s}\big{\|}P_{\leq N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}P_{\geq N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\lesssim N^{-s}\big{\|}|\nabla|^{s}P_{\geq
N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$ $\displaystyle\big{\|}P_{\leq
N}f\big{\|}_{L_{x}^{q}(\mathbb{R}^{d})}$ $\displaystyle\lesssim
N^{\frac{d}{r}-\frac{d}{q}}\big{\|}P_{\leq
N}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}.$

######  Lemma 2.2 (Littlewood–Paley square function estimates).

For $1<r<\infty$,

$\displaystyle\big{\|}\big{(}\sum|P_{N}f(x)|^{2}\big{)}^{1/2}\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\sim\big{\|}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}\big{(}\sum
N^{2s}|f_{N}(x)|^{2}\big{)}^{1/2}\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\sim\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}\quad\text{for
}s\in\mathbb{R},$ $\displaystyle\big{\|}\big{(}\sum
N^{2s}|f_{>N}(x)|^{2}\big{)}^{1/2}\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}$
$\displaystyle\sim\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}\quad\text{for
}s>0.$

We will also need the following general inequalities, which appear originally
in [10]. For a textbook treatment, one can refer to [48].

######  Lemma 2.3 (Fractional product rule, [10]).

Let $s\in(0,1]$ and $1<r,r_{1},r_{2},q_{1},q_{2}<\infty$ such that
$\tfrac{1}{r}=\tfrac{1}{r_{i}}+\tfrac{1}{q_{i}}$ for $i=1,2$. Then

$\big{\|}|\nabla|^{s}(fg)\big{\|}_{r}\lesssim\big{\|}f\big{\|}_{r_{1}}\big{\|}|\nabla|^{s}g\big{\|}_{q_{1}}+\big{\|}|\nabla|^{s}f\big{\|}_{r_{2}}\big{\|}g\big{\|}_{q_{2}}.$

######  Lemma 2.4 (Fractional chain rule, [10]).

Suppose $G\in C^{1}(\mathbb{C}),$ $s\in(0,1]$, and $1<r,r_{1},r_{2}<\infty$
are such that $\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{1}{r_{2}}.$ Then

$\big{\|}|\nabla|^{s}G(u)\big{\|}_{r}\lesssim\big{\|}G^{\prime}(u)\big{\|}_{r_{1}}\big{\|}|\nabla|^{s}u\big{\|}_{r_{2}}.$

We will also make use of the following refinement of the fractional chain
rule, which appears in [35]:

######  Lemma 2.5 (Derivatives of differences, [35]).

Fix $p>1$ and $0<s<1$. Then for $1<r,r_{1},r_{2}<\infty$ such that
$\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{p-1}{r_{2}},$ we have

$\big{\|}|\nabla|^{s}[|u+v|^{p}-|u|^{p}]\big{\|}_{r}\lesssim\big{\|}|\nabla|^{s}u\big{\|}_{r_{1}}\big{\|}v\big{\|}_{r_{2}}^{p-1}+\big{\|}|\nabla|^{s}v\big{\|}_{r_{1}}\big{\|}u+v\big{\|}_{r_{2}}^{p-1}.$

Finally, we prove a paraproduct estimate, very much in the spirit of Lemma 2.3
in [57].

######  Lemma 2.6 (Paraproduct estimate).

Fix $d\in\\{3,4,5\\}.$

(a) For $p>0$ such that $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}\in[\tfrac{1}{2},1)$,
we have

$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(fg)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}.$

(b) We also have

$\big{\|}|\nabla|^{-\frac{2}{5}}(fg)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}|\nabla|^{-\frac{2}{5}}f\big{\|}_{L_{x}^{\frac{10d}{5d-11}}}\big{\|}|\nabla|^{\frac{2}{5}}g\big{\|}_{L_{x}^{\frac{2d}{5}}}.$

###### Proof.

For (a), we will prove the equivalent estimate

$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\left(|\nabla|^{\frac{1}{2}s_{c}}f\,|\nabla|^{-\frac{1}{2}s_{c}}g\right)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}$

by decomposing the left-hand side into low-high and high-low frequency
interactions. More precisely, we introduce the projections $\pi_{l,h}$ and
$\pi_{h,l}$, defined for any pair of functions $\phi,\psi$ by

$\pi_{l,h}(\phi,\psi):=\sum_{N\lesssim
M}\phi_{N}\psi_{M}\quad\text{and}\quad\pi_{h,l}(\phi,\psi):=\sum_{N\gg
M}\phi_{N}\psi_{M}.$

We first consider the low-high interactions: by Sobolev embedding, we have

$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{4dp}{p(3d+4)-4}}}.$
(2.1)

We note here that when $d=3$, the assumption $s_{c}<1$ guarantees that

$\tfrac{4dp}{p(3d+4)-4}>1.$

If we now consider the multiplier of the operator given by

$T(f,g):=\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g),$

that is,

$\sum_{N\lesssim
M}|\xi_{1}|^{\frac{1}{2}s_{c}}\widehat{f_{N}}(\xi_{1})|\xi_{2}|^{-\frac{1}{2}s_{c}}\widehat{g_{M}}(\xi_{2}),$

then we see that this multiplier is a symbol of order zero with
$\xi=(\xi_{1},\xi_{2})$. Thus, continuing from (2.1), we can cite a theorem of
Coifman–Meyer (see [11, 13], for example) to conclude

$\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\pi_{l,h}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}.$

We now consider the high-low interactions: if we consider the multiplier of
the operator given by

$S(f,h):=|\nabla|^{-\frac{1}{2}s_{c}}\pi_{h,l}(|\nabla|^{\frac{1}{2}s_{c}}f,h),$

that is,

$\sum_{N\gg
M}|\xi_{1}+\xi_{2}|^{-\frac{1}{2}s_{c}}|\xi_{1}|^{\frac{1}{2}s_{c}}\widehat{f_{N}}(\xi_{1})\widehat{h_{M}}(\xi_{2}),$

then we see that this multiplier is also a symbol of order zero. Thus, using
the result cited above, along with Sobolev embedding, we can estimate

$\displaystyle\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}\pi_{h,l}(|\nabla|^{\frac{1}{2}s_{c}}f,|\nabla|^{-\frac{1}{2}s_{c}}g)\big{\|}_{L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}g\big{\|}_{L_{x}^{\frac{d}{2}}}$
$\displaystyle\lesssim\big{\|}f\big{\|}_{L_{x}^{\frac{2d}{d-2}}}\big{\|}g\big{\|}_{L_{x}^{\frac{4dp}{p(d+8)-4}}}.$

Combining the low-high and high-low interactions, we recover (a). Mutis
mutandis, the exact same proof gives (b). ∎

###  2.3. Strichartz estimates

Let $e^{it\Delta}$ be the free Schrödinger propagator,

$[e^{it\Delta}f](x)=\tfrac{1}{(4\pi
it)^{d/2}}\int_{\mathbb{R}^{d}}e^{i|x-y|^{2}/4t}f(y)\,dy$

for $t\neq 0$. This explicit formula immediately implies the dispersive
estimate

$\big{\|}e^{it\Delta}f\big{\|}_{L_{x}^{\infty}(\mathbb{R}^{d})}\lesssim|t|^{-\frac{d}{2}}\big{\|}f\big{\|}_{L_{x}^{1}(\mathbb{R}^{d})}$

for $t\neq 0$. Interpolating with
$\big{\|}e^{it\Delta}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}=\big{\|}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}$
(cf. Plancherel), one arrives at

$\big{\|}e^{it\Delta}f\big{\|}_{L_{x}^{r}(\mathbb{R}^{d})}\lesssim|t|^{-(\frac{d}{2}-\frac{d}{r})}\big{\|}f\big{\|}_{L_{x}^{r^{\prime}}(\mathbb{R}^{d})}$
(2.2)

for $t\neq 0$ and $2\leq r\leq\infty$, with
$\tfrac{1}{r}+\tfrac{1}{r^{\prime}}=1.$ This estimate can be used to prove the
standard Strichartz estimates, which we state below. First, we need the
following

######  Definition 2.7 (Admissible pairs).

For $d\geq 3$, we call a pair of exponents $(q,r)$ _Schrödinger admissible_ if

$\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}\indent\text{and}\indent 2\leq
q,r\leq\infty.$

For a spacetime slab $I\times\mathbb{R}^{d}$, we define

$\big{\|}u\big{\|}_{S^{0}(I)}:=\sup\left\\{\big{\|}u\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:(q,r)\text{
admissible}\right\\}.$

We define $S^{0}(I)$ to be the closure of the test functions under this norm,
and denote the dual of $S^{0}(I)$ by $N^{0}(I)$. We note

$\big{\|}u\big{\|}_{N^{0}(I)}\lesssim\big{\|}u\big{\|}_{L_{t}^{q^{\prime}}L_{x}^{r^{\prime}}(I\times\mathbb{R}^{d})}\quad\text{for
any admissible pair }(q,r).$

We now state the Strichartz estimates in the form we will need them.

######  Lemma 2.8 (Strichartz).

Let $s\geq 0$, let $I$ be a compact time interval, and let
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to the forced Schrödinger
equation

$(i\partial_{t}+\Delta)u=F.$

Then

$\big{\|}|\nabla|^{s}u\big{\|}_{S^{0}(I)}\lesssim\big{\|}|\nabla|^{s}u(t_{0})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s}F\big{\|}_{N^{0}(I)}$

for any $t_{0}\in I$.

###### Proof.

As mentioned, the key ingredient is (2.2). For the endpoint
$(q,r)=(2,\tfrac{2d}{d-2})$ in $d\geq 3$, see [24]. For the non-endpoint
cases, see [22, 47], for example. ∎

The free propagator also obeys some local smoothing estimates (see [16, 44,
54] for the original results). We will make use of the following, which
appears as Proposition 4.14 in [31]:

######  Lemma 2.9 (Local smoothing).

For any $f\in L_{x}^{2}(\mathbb{R}^{d})$ and any $\varepsilon>0$,

$\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\big{|}[|\nabla|^{\frac{1}{2}}e^{it\Delta}f](x)\big{|}^{2}\langle
x\rangle^{-1-\varepsilon}\,dx\,dt\lesssim_{\varepsilon}\big{\|}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}.$

Next, we record the following bilinear Strichartz estimates. The version we
need can be deduced from (the proof of) Corollary 4.19 in [31].

######  Lemma 2.10 (Bilinear Strichartz).

Let $0<s_{c}<\tfrac{d-1}{2}$. For any spacetime slab $I\times\mathbb{R}^{d}$
and any frequencies $M>0$ and $N>0$, we have

$\big{\|}u_{\leq M}v_{\geq
N}\big{\|}_{L_{t,x}^{2}(I\times\mathbb{R}^{d})}\lesssim
M^{(\frac{d-1}{2}-s_{c})}N^{-(\frac{1}{2}+s_{c})}\big{\|}|\nabla|^{s_{c}}u_{\leq
M}\big{\|}_{S_{0}^{*}(I)}\big{\|}|\nabla|^{s_{c}}v_{\geq
N}\big{\|}_{S_{0}^{*}(I)},$

where

$\big{\|}u\big{\|}_{S_{0}^{*}(I)}:=\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}+\big{\|}(i\partial_{t}+\Delta)u\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}.$

######  Remark 2.11.

We will use Lemma 2.10 in the proof of Proposition 5.1. In that context, we
will have $u=v$ an almost periodic solution to (1.1) and $I=J_{k}$, a
characteristic subinterval. In this case, interpolating between $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}$ and Lemma 1.11 gives

$\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(J_{k})}\lesssim_{u}1,$

so that we can use the fractional chain rule and Sobolev embedding to estimate

$\displaystyle\big{\|}|\nabla|^{s_{c}}(|u|^{p}u)\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}(J_{k}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(J_{k}\times\mathbb{R}^{d})}^{p}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}(J_{k}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(J_{k})}^{p+1}$
$\displaystyle\lesssim_{u}1.$

Thus, in this setting, an application of Lemma 2.10 gives

$\big{\|}u_{\leq M}u_{\geq
N}\big{\|}_{L_{t,x}^{2}(J_{k}\times\mathbb{R}^{d})}\lesssim_{u}M^{(\frac{d-1}{2}-s_{c})}N^{-(\frac{1}{2}+s_{c})}.$

###  2.4. Concentration-compactness

We record here the linear profile decomposition that we will use to prove the
reduction in Theorem 1.12. We begin with the following

######  Definition 2.12 (Symmetry group).

For any position $x_{0}\in\mathbb{R}^{d}$ and scaling parameter $\lambda>0$,
we define a unitary transformation
$g_{x_{0},\lambda}:\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})\to\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$
by

$[g_{x_{0},\lambda}f](x):=\lambda^{-\frac{2}{p}}f\left(\lambda^{-1}(x-x_{0})\right)$

(recall $s_{c}:=\tfrac{d}{2}-\tfrac{2}{p}$). We let $G$ denote the collection
of such transformations. For a function
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$, we define
$T_{g_{x_{0},\lambda}}u:\lambda^{2}I\times\mathbb{R}^{d}\to\mathbb{C}$ by the
formula

$[T_{g_{x_{0},\lambda}}u](t,x):=\lambda^{-\frac{2}{p}}u\left(\lambda^{-2}t,\lambda^{-1}(x-x_{0})\right),$

where $\lambda^{2}I:=\\{\lambda^{2}t:t\in I\\}.$ Note that if $u$ is a
solution to (1.1), then $T_{g}u$ is a solution to (1.1) with initial data
$gu_{0}$.

######  Remark 2.13.

It is easily seen that $G$ is a group under composition. The map $u\mapsto
T_{g}u$ maps solutions to (1.1) to solutions with the same scattering size,
that is, $S(T_{g}u)=S(u)$. Furthermore, $u$ is a maximal-lifespan solution if
and only if $T_{g}u$ is a maximal-lifespan solution.

We now state the linear profile decomposition in the form that we need. For
$s_{c}=0$, this result was originally proven in [1, 7, 39], while for
$s_{c}=1$ it was established in [28]. In the generality we need, a proof can
be found in [41].

######  Lemma 2.14 (Linear profile decomposition, [41]).

Fix $0<s_{c}<1$ and let $\\{u_{n}\\}_{n\geq 1}$ be bounded sequence in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d}).$ Then, after passing to a subsequence if
necessary, there exist functions $\\{\phi^{j}\\}_{j\geq
1}\subset\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d}),$ group elements $g_{n}^{j}\in
G$, and times $t_{n}^{j}\in\mathbb{R}$ such that for all $J\geq 1$, we have
the decomposition

$u_{n}=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}$

with the following properties:

$\bullet$ For all $n$ and all $J\geq 1$, we have
$w_{n}^{J}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, with

$\lim_{J\to\infty}\limsup_{n\to\infty}\
\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(\mathbb{R}\times\mathbb{R}^{d})}=0.$
(2.3)

$\bullet$ For any $j\neq k$, we have the following asymptotic orthogonality of
parameters:

$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}+\frac{\lambda_{n}^{k}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{k}|^{2}}{\lambda_{n}^{j}\lambda_{n}^{k}}+\frac{|t_{n}^{j}(\lambda_{n}^{j})^{2}-t_{n}^{k}(\lambda_{n}^{k})^{2}|}{\lambda_{n}^{j}\lambda_{n}^{k}}\to\infty\quad\text{as}\
n\to\infty.$ (2.4)

$\bullet$ For any $J\geq 1$, we have the decoupling properties:

$\lim_{n\to\infty}\bigg{[}\big{\|}|\nabla|^{s_{c}}u_{n}\big{\|}_{2}^{2}-\sum_{j=1}^{J}\big{\|}|\nabla|^{s_{c}}\phi^{j}\big{\|}_{2}^{2}-\big{\|}|\nabla|^{s_{c}}w_{n}^{J}\big{\|}_{2}^{2}\bigg{]}=0,$
(2.5)

and for any $1\leq j\leq J$,

$e^{-it_{n}^{j}\Delta}[(g_{n}^{j})^{-1}w_{n}^{J}]\rightharpoonup
0\quad\text{weakly in }\dot{H}_{x}^{s_{c}}\ \text{as }n\to\infty.$ (2.6)

######  Remark 2.15.

In this linear profile decomposition, we may always choose the scaling
parameters $\lambda_{n}^{k}$ so that they belong to $2^{\mathbb{Z}}$.

##  3\. Local well-posedness

In this section, we develop a local theory for (1.1). We begin by recording a
standard local well-posedness result proven by Cazenave–Weissler [8]; see also
[9, 31, 50]. We will also need to establish a stability result (appearing as
Theorem 3.4), which will be essential in the reduction to almost periodic
solutions in Section 4. For stability results in the mass- and energy-critical
settings, see [15, 40, 51, 53].

For the following local well-posedness result, one must assume that the
initial data belongs to the inhomogeneous Sobolev space
$H_{x}^{s_{c}}(\mathbb{R}^{d}).$ This assumption serves to simplify the proof
(allowing for a contraction mapping argument in mass-critical spaces); we can
remove it a posteriori by using the stability result we prove below.

######  Theorem 3.1 (Standard local well-posedness [8]).

Let $d\geq 1$, $0<s_{c}<1$, and $u_{0}\in H_{x}^{s_{c}}(\mathbb{R}^{d}).$ Then
there exists $\eta_{0}>0$ so that if $0<\eta\leq\eta_{0}$ and $I$ is an
interval containing zero such that

$\big{\|}|\nabla|^{s_{c}}e^{it\Delta}u_{0}\big{\|}_{L_{t}^{p+2}L_{x}^{\frac{2d(p+2)}{2(d-2)+dp}}(I\times\mathbb{R}^{d})}\leq\eta,$
(3.1)

then there exists a unique solution $u$ to (1.1) that obeys the following
bounds:

$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{p+2}L_{x}^{\frac{2d(p+2)}{2(d-2)+dp}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq 2\eta,$
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{0}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\eta^{p+1},$
$\displaystyle\big{\|}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\big{\|}u_{0}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}.$

######  Remark 3.2.

By Strichartz, we have

$\big{\|}|\nabla|^{s_{c}}e^{it\Delta}u_{0}\big{\|}_{L_{t}^{p+2}L_{x}^{\frac{2d(p+2)}{2(d-2)+dp}}(I\times\mathbb{R}^{d})}\lesssim\big{\|}|\nabla|^{s_{c}}u_{0}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})},$

so that (3.1) holds with $I=\mathbb{R}$ for sufficiently small initial data.
One can also guarantee that (3.1) holds by taking $|I|$ sufficiently small
(cf. monotone convergence).

We now turn to the question of stability for (1.1). We will prove a stability
result for $(d,s_{c})$ satisfying (1.5), in which case we always have $p\geq
1$. As we will see, this assumption allows for a very simple stability theory.
On the other hand, when $p<1$, developing a stability theory can become quite
delicate. For a discussion in the energy-critical case, see [31, Section 3.4]
and the references cited therein. See also [33] for a stability theory in the
energy-supercritical regime, as well as [21] for a stability theory in the
inter-critical regime in high dimensions.

Following the arguments in [31], we begin with the following

######  Lemma 3.3 (Short-time perturbations).

Fix $(d,s_{c})$ satisfying (1.5). Let $I$ be a compact interval and
$\tilde{u}:I\times\mathbb{R}^{d}\to\mathbb{C}$ a solution to

$(i\partial_{t}+\Delta)\tilde{u}=|\tilde{u}|^{p}\tilde{u}+e$

for some function $e$. Assume that

$\big{\|}\tilde{u}\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})}\leq
E.$ (3.2)

Let $t_{0}\in I$ and $u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$. Then there
exist $\varepsilon_{0},$ $\delta>0$ (depending on $E$) such that for all
$0<\varepsilon<\varepsilon_{0}$, if

$\displaystyle\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq\delta,$ (3.3)
$\displaystyle\big{\|}u_{0}-\tilde{u}(t_{0})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}$
$\displaystyle\leq\varepsilon,$ (3.4)
$\displaystyle\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\leq\varepsilon,$ (3.5)

then there exists $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ solving
$(i\partial_{t}+\Delta)u=|u|^{p}u$ with $u(t_{0})=u_{0}$ satisfying

$\displaystyle\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\varepsilon,$ (3.6)
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim E,$ (3.7)
$\displaystyle\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(I)}$
$\displaystyle\lesssim\varepsilon.$ (3.8)

###### Proof.

We prove the lemma under the additional hypothesis $u_{0}\in
L_{x}^{2}(\mathbb{R}^{d})$; this allows us (by Theorem 3.1) to find a solution
$u$, so that we are left to prove all of the estimates as a priori estimates.
Once the lemma is proven, we can use approximation by
$H_{x}^{s_{c}}(\mathbb{R}^{d})$ functions (along with the lemma itself) to see
that the lemma holds for $u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$.

Define $w=u-\tilde{u}$, so that
$(i\partial_{t}+\Delta)w=|u|^{p}u-|\tilde{u}|^{p}\tilde{u}-e$. Without loss of
generality, assume $t_{0}=\inf I$, and define

$A(t)=\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}([t_{0},t))}.$

We first note that by Duhamel, Strichartz, (3.4), and (3.5), we get

$\displaystyle\big{\|}$
$\displaystyle|\nabla|^{s_{c}}w\big{\|}_{S^{0}([t_{0},t))}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}w(t_{0})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}([t_{0},t))}+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\lesssim\varepsilon+A(t).$ (3.9)

Using this fact, together with Lemma 2.5, (3.3), and Sobolev embedding, we can
estimate (with all spacetime norms over $[t_{0},t)\times\mathbb{R}^{d}$)

$\displaystyle A(t)$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(|\tilde{u}+w|^{p}(\tilde{u}+w)-|\tilde{u}|^{p}\tilde{u})\big{\|}_{L_{t}^{\frac{p(d+2)}{2(p+1)}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+6dp-8}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}}\big{\|}w\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{p}$
$\displaystyle\ \ \ \ \ \ \
+\big{\|}|\nabla|^{s_{c}}w\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}}\big{\|}\tilde{u}+w\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{p}$
$\displaystyle\lesssim\delta[\varepsilon+A(t)]^{p}+[\varepsilon+A(t)][\delta^{p}+(\varepsilon+A(t))^{p}].$

Thus, recalling $p\geq 1$ and choosing $\delta,\varepsilon$ sufficiently
small, we conclude $A(t)\lesssim\varepsilon$ for all $t\in I$, which gives
(3.8). Combining (3.8) with (3.9), we also get (3.6). Finally, we can prove
(3.7) as follows: by Strichartz, (3.6), (3.2), (3.5), (3.3), the fractional
chain rule, and Sobolev embedding,

$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I)}+\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}$
$\displaystyle\lesssim\varepsilon+\big{\|}|\nabla|^{s_{c}}\tilde{u}(t_{0})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s_{c}}(|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(I)}+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\lesssim\varepsilon+E+\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}(I\times\mathbb{R}^{d})}\big{\|}\widetilde{u}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(I\times\mathbb{R}^{d})}^{p}$
$\displaystyle\lesssim E+\varepsilon+\delta^{p+1}$ $\displaystyle\lesssim E$

for $\varepsilon$ and $\delta$ sufficiently small depending on $E$. ∎

With Lemma 3.3 established, we now turn to

######  Theorem 3.4 (Stability).

Fix $(d,s_{c})$ satisfying (1.5). Let $I$ be a compact time interval and
$\tilde{u}:I\times\mathbb{R}^{d}\to\mathbb{C}$ a solution to

$(i\partial_{t}+\Delta)\tilde{u}=|\tilde{u}|^{p}\tilde{u}+e$

for some function $e$. Assume that

$\displaystyle\big{\|}\tilde{u}\big{\|}_{L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq E,$ (3.10) $\displaystyle S_{I}(\tilde{u})$
$\displaystyle\leq L.$ (3.11)

Let $t_{0}\in I$ and $u_{0}\in\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$. Then there
exists $\varepsilon_{1}=\varepsilon_{1}(E,L)$ such that if

$\displaystyle\big{\|}u_{0}-\tilde{u}(t_{0})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}$
$\displaystyle\leq\varepsilon,$ (3.12)
$\displaystyle\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(I)}$
$\displaystyle\leq\varepsilon$ (3.13)

for some $0<\varepsilon<\varepsilon_{1}$, then there exists a solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to $(i\partial_{t}+\Delta)u=|u|^{p}u$
with $u(t_{0})=u_{0}$ satisfying

$\displaystyle\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I)}$
$\displaystyle\leq C(E,L)\varepsilon,$ (3.14)
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I)}$ $\displaystyle\leq
C(E,L).$ (3.15)

###### Proof.

Once again, we may assume $t_{0}=\inf I$. To begin, we let $\eta>0$ be a small
parameter to be determined shortly. By (3.11), we may subdivide $I$ into
(finitely many, depending on $\eta$ and $L$) intervals $J_{k}=[t_{k},t_{k+1})$
so that

$\big{\|}\tilde{u}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(J_{k}\times\mathbb{R}^{d})}\sim\eta$

for each $k$. Then by Strichartz, (3.10), (3.13), and the fractional chain
rule, we have

$\displaystyle\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(J_{k})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}\tilde{u}(t_{k})\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+\big{\|}|\nabla|^{s_{c}}(|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(J_{k})}+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}(J_{k})}$
$\displaystyle\lesssim
E+\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}\big{\|}\tilde{u}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}(J_{k}\times\mathbb{R}^{d})}^{p}+\varepsilon$
$\displaystyle\lesssim
E+\varepsilon+\eta^{p}\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}.$

Thus for $\varepsilon\leq E$ and $\eta$ sufficiently small, we find

$\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(J_{k})}\lesssim E.$

Adding these bounds, we find

$\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{S^{0}(I)}\leq C(E,L).$ (3.16)

Now, we take $\delta>0$ as in Lemma 3.3 and subdivide $I$ into finitely many,
say $J_{0}=J_{0}(C(E,L),\delta)$ intervals $I_{j}=[t_{j},t_{j+1})$ so that

$\big{\|}|\nabla|^{s_{c}}\tilde{u}\big{\|}_{L_{t}^{\frac{p(d+2)}{2}}L_{x}^{\frac{2dp(d+2)}{d^{2}p+2dp-8}}(I_{j}\times\mathbb{R}^{d})}\leq\delta$

for each $j$. We now wish to proceed inductively. We may apply Lemma 3.3 on
each $I_{j}$, provided we can guarantee

$\big{\|}u(t_{j})-\tilde{u}(t_{j})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}\leq\varepsilon$
(3.17)

for some $0<\varepsilon<\varepsilon_{0}$ and each $j$ (where $\varepsilon_{0}$
is as in Lemma 3.3). In the event that (3.17) holds for some $j$, applying
Lemma 3.3 on $I_{j}=[t_{j},t_{j+1})$ gives

$\displaystyle\big{\|}|\nabla|^{s_{c}}(u-\tilde{u})\big{\|}_{S^{0}(I_{j})}$
$\displaystyle\leq C(j)\varepsilon,$ (3.18)
$\displaystyle\big{\|}|\nabla|^{s_{c}}u\big{\|}_{S^{0}(I_{j})}$
$\displaystyle\leq C(j)E,$ (3.19)
$\displaystyle\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}(I_{j})}$
$\displaystyle\leq C(j)\varepsilon.$ (3.20)

Now, we first note that (3.17) holds for $j=0$, provided we take
$\varepsilon_{1}<\varepsilon_{0}$. Next, assuming that (3.17) holds for $0\leq
k\leq j-1$, we can use Strichartz, (3.12), (3.13), and the inductive
hypothesis (3.20) to estimate

$\displaystyle\big{\|}u$
$\displaystyle(t_{j})-\tilde{u}(t_{j})\big{\|}_{\dot{H}^{s_{c}}_{x}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}u(t_{0})-\tilde{u}(t_{0})\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}\\!+\big{\|}|\nabla|^{s_{c}}(|u|^{p}u-|\tilde{u}|^{p}\tilde{u})\big{\|}_{N^{0}([t_{0},t_{j}))}\\!+\big{\|}|\nabla|^{s_{c}}e\big{\|}_{N^{0}([t_{0},t_{j}))}$
$\displaystyle\lesssim\varepsilon+\sum_{k=0}^{j-1}C(k)\varepsilon+\varepsilon$
$\displaystyle<\varepsilon_{0},$

provided $\varepsilon_{1}=\varepsilon_{1}(\varepsilon_{0},J_{0})$ is taken
sufficiently small. Thus, by induction, we get (3.18) and (3.19) on each
$I_{j}$. Adding these bounds over the $I_{j}$ yields (3.14) and (3.15). ∎

######  Remark 3.5.

As mentioned above, with this stability result in hand, we can see that
Theorem 3.1 holds without the assumption $u_{0}\in L_{x}^{2}(\mathbb{R}^{d})$.
Using this updated version of Theorem 3.1 (along with the original proof of
Theorem 3.1), one can then derive Theorem 1.4. We omit the standard arguments;
one can refer instead to [8, 9].

##  4\. Reduction to almost periodic solutions

The goal of this section is to sketch a proof of Theorem 1.12. We will follow
the argument presented in [32, Section 3]. By now, the general procedure is
fairly standard; see, for example, [25, 26, 27, 31, 53] for other instances in
different contexts. Thus, we will merely outline the main steps of the
argument, providing full details only when significant new difficulties arise
in our setting.

We suppose that Theorem 1.3 fails for some $(d,s_{c})$ satisfying (1.5). We
then define the function $L:[0,\infty)\to[0,\infty]$ by

$\displaystyle
L(E):=\sup\\{S_{I}(u):u:I\\!\times\\!\mathbb{R}^{d}\to\mathbb{C}\text{ solving
}\eqref{nls}\text{ with }\sup_{t\in
I}\big{\|}u(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{2}\leq E\\},$

where $S_{I}(u)$ is defined as in (1.2). We note that $L$ is a non-decreasing
function, and that Theorem 1.4 implies

$L(E)\lesssim E^{\frac{p(d+2)}{4}}\indent\text{for}\indent E<\eta_{0},$ (4.1)

where $\eta_{0}$ is the small-data threshold. Thus, there exists a unique
‘critical’ threshold $E_{c}\in(0,\infty]$ such that $L(E)<\infty$ for
$E<E_{c}$ and $L(E)=\infty$ for $E>E_{c}$. The failure of Theorem 1.3 implies
that $0<E_{c}<\infty.$

The key ingredient to proving Theorem 1.12 is a Palais–Smale condition modulo
the symmetries of the equation; indeed, once the following proposition is
proven, deriving Theorem 1.12 is standard (see [32]).

######  Proposition 4.1 (Palais–Smale condition modulo symmetries).

Let $(d,s_{c})$ satisfy (1.5). Let
$u_{n}:I_{n}\times\mathbb{R}^{d}\to\mathbb{C}$ be a sequence of solutions to
(1.1) such that

$\limsup_{n\to\infty}\sup_{t\in
I_{n}}\big{\|}u_{n}(t)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{2}=E_{c},$

and suppose $t_{n}\in I_{n}$ are such that

$\lim_{n\to\infty}S_{[t_{n},\sup I_{n})}(u_{n})=\lim_{n\to\infty}S_{(\inf
I_{n},t_{n}]}(u_{n})=\infty.$ (4.2)

Then the sequence $u_{n}(t_{n})$ has a subsequence that converges in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$ modulo symmetries; that is, there exist
$g_{n}\in G$ such that $g_{n}[u_{n}(t_{n})]$ converges along a subsequence in
$\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})$, where $G$ is as in Definition 2.12.

We now sketch the proof of this proposition, following the argument as it
appears in [32]. As in that setting, the main ingredients will be a linear
profile decomposition (Lemma 2.14 in our case) and a stability result (Theorem
3.4 in our case). However, as we will see, combining fractional derivatives
with non-polynomial nonlinearities will present some significant new
difficulties in our setting when it comes time to apply the stability result.

We begin by translating so that each $t_{n}=0$, and apply the linear profile
decomposition Lemma 2.14 to write

$u_{n}(0)=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}$ (4.3)

along some subsequence. Refining the subsequence for each $j$ and
diagonalizing, we may assume that for each $j$, we have $t_{n}^{j}\to
t^{j}\in[-\infty,\infty].$ If $t^{j}\in(-\infty,\infty)$, then we replace
$\phi^{j}$ by $e^{it^{j}\Delta}\phi^{j}$, so that we may take $t^{j}=0$.
Moreover, we can absorb the error $e^{it_{n}^{j}\Delta}\phi^{j}-\phi^{j}$ into
the error term $w_{n}^{J}$, and so we may take $t_{n}^{j}\equiv 0$. Thus,
without loss of generality, either $t_{n}^{j}\equiv 0$ or
$t_{n}^{j}\to\pm\infty$.

Next, appealing to Theorem 1.4, for each $j$ we define
$v^{j}:I^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ to be the maximal-lifespan
solution to (1.1) such that

$\left\\{\begin{array}[]{ll}v^{j}(0)=\phi^{j}&\text{if }t_{n}^{j}\equiv 0,\\\
\\\ v^{j}\text{ scatters to }\phi^{j}\text{ as }t\to\pm\infty&\text{if
}t_{n}^{j}\to\pm\infty.\end{array}\right.$

We now define the nonlinear profiles
$v_{n}^{j}:I_{n}^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ by

$v_{n}^{j}(t)=g_{n}^{j}v^{j}\left((\lambda_{n}^{j})^{-2}t+t_{n}^{j}\right),$

where $I_{n}^{j}=\\{t:(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\in I^{j}\\}.$ Now, the
$\dot{H}_{x}^{s_{c}}$ decoupling of the profiles $\phi^{j}$, (2.5),
immediately tells us that the $v_{n}^{j}$ are global and scatter for $j$
sufficiently large, say $j\geq J_{0}$; indeed, for large enough $j$, we are in
the small-data regime. We want to show that there exists some $1\leq
j_{0}<J_{0}$ such that

$\limsup_{n\to\infty}S_{[0,\sup I_{n}^{j_{0}})}(v_{n}^{j_{0}})=\infty.$ (4.4)

Once we obtain at least one such ‘bad’ nonlinear profile, we can show that in
fact, there can only be one profile. To see this, one needs to adapt the
argument in [32, Lemma 3.3] to see that the $\dot{H}_{x}^{s_{c}}$ decoupling
of the profiles persists in time (this does not follow immediately, as the
$\dot{H}_{x}^{s_{c}}$-norm is not a conserved quantity for (1.1)). Then, the
‘critical’ nature of $E_{c}$ can be used to rule out the possibility of
multiple profiles.

Comparing with (4.3), one sees that once we show that there is only one
profile $\phi^{j_{0}}$, the proof of Proposition 4.1 is nearly complete; one
needs only to rule out the cases $t_{n}^{j_{0}}\to\pm\infty$. This can be
easily done by applying the stability theory; we omit the details and instead
refer the reader to [32].

We turn now to proving that there is at least one bad profile. We suppose
towards a contradiction that there are no bad nonlinear profiles. In this
case, we can show

$\sum_{j\geq 1}S_{[0,\infty)}(v_{n}^{j})\lesssim_{E_{c}}1$ (4.5)

for $n$ sufficiently large (to control the tail of the sum, for example, we
recall that for $j\geq J_{0}$, we are in the small-data regime; thus we can
use (2.5) and (4.1) to bound the tail by $E_{c}$ for $n$ sufficiently large).
We would like to use (4.5) and the stability result (Theorem 3.4) to deduce a
bound on the scattering size of the $u_{n}$, thus contradicting (4.2).

To this end, we define the approximations

$u_{n}^{J}(t)=\sum_{j=1}^{J}v_{n}^{j}(t)+e^{it\Delta}w_{n}^{J}.$

By the construction of the $v_{n}^{j}$, it is easy to see that

$\limsup_{n\to\infty}\big{\|}u_{n}(0)-u_{n}^{J}(0)\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}=0.$
(4.6)

We also claim that we have

$\lim_{J\to\infty}\limsup_{n\to\infty}S_{[0,\infty)}(u_{n}^{J})\lesssim_{E_{c}}1.$
(4.7)

To see why (4.7) holds, first note that by (2.3) and (4.5), it suffices to
show

$\lim_{J\to\infty}\limsup_{n\to\infty}\bigg{|}S_{[0,\infty)}\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}-\sum_{j=1}^{J}S_{[0,\infty)}(v_{n}^{j})\bigg{|}=0.$
(4.8)

To establish (4.8), we can first use the pointwise inequality

$\bigg{|}\
\bigg{|}\sum_{j=1}^{J}v_{n}^{j}\bigg{|}^{\frac{p(d+2)}{2}}-\sum_{j=1}^{J}|v_{n}^{j}|^{\frac{p(d+2)}{2}}\bigg{|}\lesssim_{J}\sum_{j\neq
k}|v_{n}^{j}|^{\frac{p(d+2)}{2}-1}|v_{n}^{k}|$

along with Hölder’s inequality to see

$\displaystyle\text{LHS}\eqref{ok version3}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}([0,\infty)\times\mathbb{R}^{d})}^{\frac{p(d+2)}{2}-2}\big{\|}v_{n}^{j}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{4}}([0,\infty)\times\mathbb{R}^{d})}.$
(4.9)

Then, following an argument of Keraani (cf. [28, Lemma 2.7]), given $j\neq k$,
we can approximate $v^{j}$ and $v^{k}$ by compactly supported functions in
$\mathbb{R}\times\mathbb{R}^{d}$ and use the asymptotic orthogonality of
parameters (2.4) to show

$\limsup_{n\to\infty}\big{\|}v_{n}^{j}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{4}}([0,\infty)\times\mathbb{R}^{d})}=0.$
(4.10)

Thus, continuing from (4.9), we get that (4.8) (and therefore (4.7)) holds.

With (4.6) and (4.7) in place, we see that if we can show that the $u_{n}^{J}$
asymptotically solve (1.1), that is,

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{[}(i\partial_{t}+\Delta)u_{n}^{J}-|u_{n}^{J}|^{p}u_{n}^{J}\big{]}\big{\|}_{N^{0}([0,\infty))}=0,$

then we will be able to apply Theorem 3.4 to deduce bounds on the scattering
size of the $u_{n}$. Writing $F(z)=|z|^{p}z$, the proof of Proposition 4.1
therefore reduces to showing the following

######  Lemma 4.2 (Decoupling of nonlinear profiles).

$\lim_{J\to\infty}\limsup_{n\to\infty}\bigg{\|}|\nabla|^{s_{c}}\bigg{(}F\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}-\sum_{j=1}^{J}F(v_{n}^{j})\bigg{)}\bigg{\|}_{N^{0}([0,\infty))}=0,$
(4.11)

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\left(F(u_{n}^{J}-e^{it\Delta}w_{n}^{J})-F(u_{n}^{J})\right)\big{\|}_{N^{0}([0,\infty))}=0.$
(4.12)

While many of the ideas needed to establish this lemma may be found in [28],
we will see that new difficulties appear in our setting. Consider, for
example, (4.11). In the mass-critical setting, i.e. $s_{c}=0$, one has the
pointwise estimate

$\bigg{|}\
F\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}-\sum_{j=1}^{J}F(v_{n}^{j})\bigg{|}\lesssim_{J}\sum_{j\neq
k}|v_{n}^{j}|\,|v_{n}^{k}|^{p}.$ (4.13)

To see that the contribution of the terms on the the right-hand side of (4.13)
is acceptable, one can follow the argument of Keraani just described above;
that is, one can use the asymptotic orthogonality of parameters to derive an
estimate like (4.10), which in turn gives (4.11).

In the energy-critical setting, i.e. $s_{c}=1$, one can instead use the
pointwise estimate

$\bigg{|}\nabla\bigg{(}F\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}-\sum_{j=1}^{J}F(v_{n}^{j})\bigg{)}\bigg{|}\lesssim_{J}\sum_{j\neq
k}|\nabla v_{n}^{j}|\,|v_{n}^{k}|^{p}.$

A similar argument can then be used to prove (4.11); the key in both cases is
to exhibit terms that all contain some $v_{n}^{j}$ paired against some
$v_{n}^{k}$ for $j\neq k$.

In the energy-supercritical case, the authors of [33] were able to establish
analogous pointwise estimates (in terms of the Hardy–Littlewood maximal
function) for a square function of Strichartz that shares estimates with
fractional differentiation operators (see [46]). With the appropriate
pointwise estimates in place, the usual arguments can then be applied; in this
way, a potentially complicated analysis is handled quite efficiently. The
approach of [33], however, does not work in our setting, as it relies
fundamentally on the fact that $s_{c}>1$.

See also [26], which deals with the case $d=3$ and $s_{c}=\tfrac{1}{2}$ (in
which case $p=2$). In that setting, one also has to face the nonlocal nature
of $|\nabla|^{\frac{1}{2}}$; however, by using the polynomial nature of the
nonlinearity, along with the well-developed theory of paraproducts (see [12,
48]), the authors are able to place themselves back into a situation where the
usual arguments apply. In this way, they are able to overcome the difficulty
of fractional derivatives while still providing a very clean analysis.

In our case, we must deal simultaneously with a non-polynomial nonlinearity
and a fractional number of derivatives; as we will see, this necessitates a
fairly delicate and technical analysis. The main difficulty of our task stems
from the fact the nonlocal operator $|\nabla|^{s_{c}}$ does not respect
pointwise estimates in the spirit of (4.13). We will deal with this problem by
opening up the proof of the fractional chain rule (Lemma 2.4) as given in [48,
$\S 2.4$]; in particular, we will employ the Littlewood–Paley square function
(specifically, Lemma 2.2), which allows us to work at the level of individual
frequencies. By making use of maximal function and vector maximal function
estimates, we can then find a way to adapt the standard arguments.

###### Proof of (4.11).

By induction, it will suffice to treat the case of two summands; to simplify
notation, we write $f=v_{n}^{j}$ and $g=v_{n}^{k}$ for some $j\neq k$, and we
are left to show

$\displaystyle\big{\|}|\nabla|^{s_{c}}\big{(}|f+g|^{p}(f+g)-|f|^{p}f-|g|^{p}g\big{)}\big{\|}_{N^{0}([0,\infty))}\to
0$ (4.14)

as $n\to\infty$.

As alluded to above, the key will be to perform a decomposition in such a way
that all of the resulting terms we need to estimate have $f$ paired against
$g$ inside of a single integrand; for such terms, we will be able to use the
asymptotic orthogonality of parameters (2.4) to our advantage.

We first rewrite

$\displaystyle|f+g$ $\displaystyle|^{p}(f+g)-|f|^{p}f-|g|^{p}g$
$\displaystyle=\big{(}|f+g|^{p}-|f|^{p}\big{)}f+\big{(}|f+g|^{p}-|g|^{p}\big{)}g.$

By symmetry, it will suffice to treat the first term. We turn therefore to
estimating

$\big{\|}|\nabla|^{s_{c}}\big{[}(|f+g|^{p}-|f|^{p})f\big{]}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$

By Lemma 2.2, it will suffice to consider

$\bigg{\|}\left(\sum\big{|}N^{s_{c}}P_{N}\big{[}(|f+g|^{p}-|f|^{p})f\big{]}\big{|}^{2}\right)^{1/2}\bigg{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
(4.15)

Thus, we restrict our attention to a single frequency $N\in 2^{\mathbb{Z}}$.
We let $\delta_{y}f(x):=f(x-y)-f(x)$, and let $\check{\psi}$ denote the
convolution kernel of the Littlewood–Paley projection $P_{1}$. As $\psi(0)=0$,
we have

$\textstyle\int\check{\psi}(y)\,dy=0,$

so that exploiting cancellation, we can write

$\displaystyle
P_{N}\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}$
$\displaystyle\quad=\textstyle\int
N^{d}\check{\psi}(Ny)\delta_{y}\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}\,dy.$
(4.16)

We now rewrite

$\displaystyle\delta_{y}$
$\displaystyle\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}$
$\displaystyle=\delta_{y}f(x)\big{[}|f(x-y)+g(x-y)|^{p}-|f(x-y)|^{p}\big{]}$
(4.17) $\displaystyle\
+\\!f(x)\big{[}|f(x)+g(x-y)|^{p}-|f(x)+g(x)|^{p}\big{]}$ (4.18)
$\displaystyle\
+\\!f(x)\big{[}|f(x-y)\\!+\\!g(x-y)|^{p}\\!-\\!|f(x-y)|^{p}\\!+\\!|f(x)|^{p}\\!-\\!|f(x)\\!+\\!g(x-y)|^{p}\big{]}.$
(4.19)

We estimate each term individually. First, we have

$\displaystyle|\eqref{expansion1}|\lesssim|\delta_{y}f(x)|\,|g(x-y)|\left\\{|f(x-y)|^{p-1}+|g(x-y)|^{p-1}\right\\}.$

Next, we see

$\displaystyle|\eqref{expansion2}|\lesssim|f(x)|\,|\delta_{y}g(x)|\left\\{|f(x)|^{p-1}+|g(x)|^{p-1}+|g(x-y)|^{p-1}\right\\}.$

We now turn to (4.19). First, if $1<p\leq 2$, a simple argument using the
fundamental theorem of calculus implies

$\displaystyle|\eqref{expansion3}|\lesssim|f(x)|\,|\delta_{y}f(x)|\,|g(x-y)|^{p-1}$

(see Lemma A.1 for details). For $p>2$, one instead finds

$\displaystyle|\eqref{expansion3}|\lesssim|f(x)|\,|\delta_{y}f(x)|\,|g(x-y)|\left\\{|f(x)|^{p-2}+|f(x-y)|^{p-2}+|g(x-y)|^{p-2}\right\\}.$

######  Remark 4.3.

Let us pause here to note that if $p=1$, the approach above breaks down.
Notice that each term in the bounds for (4.17), (4.18), and (4.19) has two
essential properties: (i) it features $f$ paired against powers of $g$, and
(ii) the derivative (in the form of $\delta_{y}$) lands on either $f$ or $g$.
When $p=1$, the same approach does not yield a decomposition that is
satisfactory in this sense; it is for this reason that we have excluded the
case $(d,s_{c})=(5,\tfrac{1}{2})$ from this paper.

To ease the exposition, we will restrict our attention here and below to the
more difficult case $1<p\leq 2$; once we have dealt with this case, it should
be clear how to proceed when $p>2$.

Collecting terms, we continue from (4.16) to see

$\displaystyle\big{|}$ $\displaystyle
P_{N}\big{(}\big{[}|f(x)+g(x)|^{p}-|f(x)|^{p}\big{]}f(x)\big{)}\big{|}$
$\displaystyle\lesssim\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}f(x)|\,|g(x-y)|\left\\{|f(x-y)|^{p-1}+|g(x-y)|^{p-1}\right\\}\\!\,dy$
(4.20) $\displaystyle\ +\int
N^{d}|\check{\psi}(Ny)|\,|f(x)|\,|\delta_{y}g(x)|\left\\{|f(x)|^{p-1}\\!+\\!|g(x)|^{p-1}\\!+\\!|g(x-y)|^{p-1}\right\\}\,\\!dy$
(4.21) $\displaystyle\ +\int
N^{d}|\check{\psi}(Ny)|\,|f(x)|\,|\delta_{y}f(x)||g(x-y)|^{p-1}\,dy.$ (4.22)

One can see that we are already faced with several terms to estimate;
moreover, to estimate any single term will require further decomposition.
However, in the end, the same set of tools will suffice to handle every term
that appears. Thus, let us deal with only (4.20) in detail; once we have seen
how to handle this term, it should be clear that the same techniques apply to
handle (4.21) and (4.22).

Turning to (4.20), we first write

$\displaystyle\eqref{11}$ $\displaystyle=\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}f(x)|\,|g(x-y)|\,|f(x-y)|^{p-1}\,dy$
(4.23) $\displaystyle\ \ +\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}f(x)|\,|g(x-y)|^{p}\,dy.$ (4.24)

For both of these terms, we will need to make use of some auxiliary
inequalities in the spirit of [48, $\S$2.3], which we record in Lemma A.2.

We turn to (4.23). If we first write

$\displaystyle|\delta_{y}$ $\displaystyle
f(x)|\lesssim|f_{>N}(x)|+|f_{>N}(x-y)|+\sum_{K\leq N}|\delta_{y}f_{K}(x)|,$
(4.25)

then putting Lemma A.2 to use, we arrive at

$\displaystyle\eqref{11f}\lesssim$ $\displaystyle\
|f_{>N}(x)|\,M(g\,|f|^{p-1})(x)$ (4.26)
$\displaystyle+M(f_{>N}\,g\,|f|^{p-1})(x)$ (4.27) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(f_{K})(x)M(g\,|f|^{p-1})(x)$ (4.28) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(M(f_{K})\,g\,|f|^{p-1})(x).$ (4.29)

Similarly, we can decompose

$\displaystyle\eqref{11g}\lesssim$ $\displaystyle\ |f_{>N}(x)|\,M(|g|^{p})(x)$
(4.30) $\displaystyle+M(f_{>N}|g|^{p})(x)$ (4.31) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(f_{K})(x)M(|g|^{p})(x)$ (4.32) $\displaystyle+\sum_{K\leq
N}\tfrac{K}{N}M(M(f_{K})|g|^{p})(x).$ (4.33)

Let us now consider the contribution of (4.26) to the left-hand side of
(4.14). Comparing with (4.15), we see it will suffice to estimate

$\displaystyle\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}f_{>N}M(g\,|f|^{p-1})\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$

Using Hölder’s inequality and maximal function estimates, we can control this
term by

$\displaystyle\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}f_{>N}\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}|g|\,|f|^{p-1}\big{\|}_{L_{t,x}^{\frac{d+2}{2}}}.$

We now recall that $f=v_{n}^{j}$ and $g=v_{n}^{k}$ for some $j\neq k$. Then,
the first term is controlled by
$\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{S^{0}}$ (cf. Lemma 2.2), which in
turn is bounded (recall that by assumption, all of the $v_{n}^{j}$ have
scattering size $\lesssim E_{c}$). The second term can be handled in the
standard way; that is, this term vanishes in the limit due to the asymptotic
orthogonality of parameters (2.4) (cf. [28, Lemma 2.7]). Thus, we see that
(4.26) is under control. A similar approach (this time using the vector
maximal inequality) handles (4.27).

To estimate the contribution of (4.28) to the left-hand side of (4.14), we
need to estimate

$\displaystyle\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}\sum_{K\leq
N}\tfrac{K}{N}M(f_{K})M(g\,|f|^{p-1})\big{|}^{2}\big{)}^{1/2}\big{\|}_{N^{0}([0,\infty))}.$
(4.34)

For this term, we need to make use of the following basic inequality: for a
nonnegative sequence $\\{a_{K}\\}_{K\in 2^{\mathbb{Z}}}$ and $0<s<1$, one has

$\sum_{N\in 2^{\mathbb{Z}}}N^{2s}\big{|}\sum_{K\leq
N}\tfrac{K}{N}a_{K}\big{|}^{2}\lesssim\sum_{K\in
2^{\mathbb{Z}}}K^{2s}|a_{K}|^{2}$ (4.35)

(cf. [48, Lemma 4.2]). Using this inequality, along with Hölder, we can
estimate

$\displaystyle\eqref{113 again}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{K}\big{|}K^{s_{c}}M(f_{K})\big{|}^{2}\big{)}^{1/2}M(g\,|f|^{p-1})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{K}|K^{s_{c}}M(f_{K})|^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}|g|\,|f|^{p-1}\big{\|}_{L_{t,x}^{\frac{d+2}{2}}}\to
0$

as $n\to\infty$, exactly as before. Thus, (4.28) is under control; the same
approach handles (4.29) (after an application of the vector maximal
inequality).

Let us now turn to (4.30). As before, we sum over $N\in 2^{\mathbb{Z}}$ and
find that we need to estimate

$\bigg{\|}\left(\sum\big{|}N^{s_{c}}f_{>N}\big{|}^{2}\right)^{1/2}M(|g|^{p})\bigg{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$
(4.36)

Recalling that $f=v_{n}^{j}$ and $g=v_{n}^{k}$ for some $j\neq k$, we see that
we are once again in a position to use the argument from [28]. To begin, we
may assume without loss of generality that both

$\Phi_{1}:=\left(\sum|N^{s_{c}}P_{>N}v^{j}|^{2}\right)^{1/2}\quad\text{and}\quad\Phi_{2}:=M(|v^{k}|^{p})$

belong to $C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$; indeed,
$C_{c}^{\infty}$-functions are dense in both $L_{t,x}^{\frac{2(d+2)}{d}}$ and
$L_{t,x}^{\frac{d+2}{2}}.$ We now wish to use the asymptotic orthogonality of
parameters, that is,

$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}+\frac{\lambda_{n}^{k}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{k}|^{2}}{\lambda_{n}^{j}\lambda_{n}^{k}}+\frac{|t_{n}^{j}(\lambda_{n}^{j})^{2}-t_{n}^{k}(\lambda_{n}^{k})^{2}|}{\lambda_{n}^{j}\lambda_{n}^{k}}\to\infty\quad\text{as}\
n\to\infty,$ (4.37)

to show (4.36)$\to 0$.

Consider first the case $\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\to c>0$
(along a subsequence, say). If we unravel the definition of the nonlinear
profiles and change variables to move the symmetries onto $\Phi_{2}$, we
arrive at

$\displaystyle\eqref{115again}^{\frac{2(d+2)}{d+4}}$
$\displaystyle=\left(\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\right)^{\frac{4(d+2)}{d+4}}\iint\bigg{|}\Phi_{1}(s,y)\Phi_{2}(\big{(}t_{n}^{k}+\big{(}\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\big{)}^{2}(s-t_{n}^{j}),\big{(}\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\big{)}y+\tfrac{x_{n}^{j}-x_{n}^{k}}{\lambda_{n}^{k}}\big{)}\bigg{|}^{\frac{2(d+2)}{d+4}}\,dy\,ds.$

Then, recalling (4.37), we see that as $n\to\infty$, either the spatial or
temporal argument of $\Phi_{2}$ must escape the support of $\Phi_{1}$. Thus,
in this case, we get (4.36)$\to 0$.

If instead we have $\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\to 0$, then
continuing from above, we can estimate

$\displaystyle\eqref{115again}$
$\displaystyle\lesssim\big{(}\tfrac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\big{)}^{2}\big{\|}\Phi_{1}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}\big{\|}\Phi_{2}\big{\|}_{L_{t,x}^{\infty}}.$

As $\Phi_{1},\Phi_{2}\in C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$, we
see that (4.36)$\to 0$ in this case, as well.

Finally, we can treat the case
$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}\to\infty$ just like the previous
case; the only difference is that we change variables to move the symmetries
onto $\Phi_{1}$, instead of $\Phi_{2}$. Thus, we have that (4.36)$\to 0$ in
this third and final case.

We have now shown that (4.30) is under control. The same ideas can be used to
handle (4.31), (4.32), and (4.33).

As mentioned above, this same set of ideas suffices to deal with all the
remaining terms stemming from (4.11). ∎

###### Proof of (4.12).

For this term, we will need to make use (2.3). As we will see, the terms in
which $e^{it\Delta}w_{n}^{J}$ appears without derivatives will be relatively
easy to handle, as (2.3) will apply directly. On the other hand, the terms
that only contain $|\nabla|^{s_{c}}e^{it\Delta}w_{n}^{J}$ will require a more
careful analysis; in particular, we will need to carry out a local smoothing
argument before we can make effective use of (2.3).

Defining $g:=\sum_{j=1}^{J}v_{n}^{j}$ and $h:=e^{it\Delta}w_{n}^{J}$, we are
left to show

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\left(|g+h|^{p}(g+h)-|g|^{p}g\right)\big{\|}_{N^{0}([0,\infty))}=0.$
(4.38)

We write

$\displaystyle|g+h|^{p}(g+h)-|g|^{p}g=$ $\displaystyle\ |g+h|^{p}h$ (4.39)
$\displaystyle+(|g+h|^{p}-|g|^{p})g$ (4.40)

and first restrict our attention to (4.39). We proceed as before, working at a
single frequency and exploiting cancellation to write

$\displaystyle|P_{N}(|g+h|^{p}h)(x)|$ $\displaystyle=\big{|}\textstyle\int
N^{d}\check{\psi}(Ny)\delta_{y}\big{[}|g(x)+h(x)|^{p}h(x)\big{]}\,dy\big{|}$
$\displaystyle\leq\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|\delta_{y}h(x)|\,dy$ (4.41)
$\displaystyle\quad+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}\big{[}|g(x)+h(x)|^{p}\big{]}\,|h(x)|\,dy.$
(4.42)

We will deal only with (4.41), which is the more difficult term. Indeed, in
all of the terms that stem from (4.42), we will have a copy of
$e^{it\Delta}w_{n}^{J}$ appearing without derivatives, so that (2.3) will
suffice. (For completeness, we will later show how to handle such a term; cf.
(4.54) below.)

Proceeding as in (4.25), we write

$\displaystyle\eqref{221}$ $\displaystyle\lesssim\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|h_{>N}(x)|\,dy$ (4.43)
$\displaystyle+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|h_{>N}(x-y)|\,dy$ (4.44)
$\displaystyle+\sum_{K\leq N}\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)+h(x-y)|^{p}\,|\delta_{y}h_{K}(x)|\,dy.$
(4.45)

Let us now deal only with (4.45); in doing so, we will see all of the ideas
necessary to handle (4.43) and (4.44), as well. We first write

$\displaystyle\eqref{2213}$ $\displaystyle\lesssim\sum_{K\leq N}\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x-y)|^{p}|\delta_{y}h_{K}(x)|\,dy$ (4.46)
$\displaystyle+\sum_{K\leq N}\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|h(x-y)|^{p}|\delta_{y}h_{K}(x)|\,dy.$ (4.47)

We only consider (4.46), as the contribution of (4.47) is easier to estimate
(again, due to the presence of $e^{it\Delta}w_{n}^{J}$ without derivatives).
Employing the inequalities of Lemma A.2, we find

$\displaystyle\eqref{22131}\lesssim\sum_{K\leq
N}\tfrac{K}{N}M(|g|^{p})(x)M(h_{K})(x)+\sum_{K\leq
N}\tfrac{K}{N}M(|g|^{p}M(h_{K}))(x).$

Let us now concern ourselves only with the first term above, as the second is
similar. As before, to estimate the contribution of this term to (4.38) (and
thereby complete our treatment of (4.39)), we need to sum over $N\in
2^{\mathbb{Z}}$. Using (4.35) and recalling the definitions of $g$ and $h$, we
write

$\displaystyle\big{\|}\big{(}$
$\displaystyle\sum_{N}\big{|}N^{s_{c}}\sum_{K\leq
N}\tfrac{K}{N}M(|g|^{p})M(h_{K})\big{|}^{2}\big{)}^{1/2}\big{\|}_{N^{0}}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(h_{N})\big{|}^{2}\big{)}^{1/2}M(|g|^{p})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}$
$\displaystyle\lesssim\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})|^{2}\big{)}^{1/2}M\big{(}\big{|}\sum_{j=1}^{J}v_{n}^{j}\big{|}^{p}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$

Thus, to complete our treatment of (4.39), we are left to show

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})|^{2}\big{)}^{1/2}M\big{(}\big{|}\sum_{j=1}^{J}v_{n}^{j}\big{|}^{p}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{d+2}{2}}=0.$
(4.48)

To begin, we let $\eta>0$; then using (4.5), we see that there exists some
$J_{1}=J_{1}(\eta)$ so that

$\sum_{j\geq
J_{1}}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}<\eta.$

Using Hölder’s inequality, maximal function and vector maximal function
estimates, and Lemma 2.2, we can argue as we did to obtain (4.8) to see

$\displaystyle\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}$
$\displaystyle\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})|^{2}\big{)}^{1/2}M\big{(}\big{|}\sum_{j\geq
J_{1}}v_{n}^{j}\big{|}^{p}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{d+2}{2}}$
$\displaystyle\lesssim\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{d+2}{2}}\sum_{j\geq
J_{1}}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}$
$\displaystyle\lesssim\eta.$

As $\eta>0$ was arbitrary, we see that to establish (4.48), it will suffice to
show

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\big{)}^{1/2}M(|v_{n}^{j}|^{p})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}=0$
(4.49)

for $1\leq j<J_{1}$.

Restricting our attention to a single $j$ and recalling the definition of
$v_{n}^{j}$, we change variables and find we need to estimate

$\big{\|}\big{(}\sum_{N}\big{|}(\lambda_{n}^{j})^{\frac{2}{p}}N^{s_{c}}MP_{N}\big{[}e^{i[(\lambda_{n}^{j})^{2}(t-t_{n}^{j})]\Delta}w_{n}^{J}(\lambda_{n}^{j}x+x_{n}^{j})\big{]}\big{|}^{2}\big{)}^{1/2}M(|v^{j}|^{p})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}.$

We will now carry out some reductions, inspired by the proof of [28,
Proposition 3.4]: as $M(|v^{j}|^{p})$ shares bounds with $|v^{j}|^{p}$, and
$v^{j}$ obeys good bounds (it has scattering size $\lesssim E_{c}$), we may
replace $M(|v^{j}|^{p})$ with some function $\Phi$ in
$C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{d})$. If we then use Hölder’s
inequality, we find it suffices to estimate the first term in
$L_{t,x}^{2}(K)$, where $K$ is the (compact) support of this function $\Phi$.
The next step will be to use a local smoothing estimate on this (fixed) set
$K$. Now, the norms that will appear in these estimates will have critical
scaling; that is, they will be invariant under the change of variables that
eliminates the parameters $\lambda_{n}^{j}$, $x_{n}^{j},$ and $t_{n}^{j}$.
Thus, without loss of generality, we will ignore them from the start.

To establish (4.49) and complete our treatment of (4.39), we are therefore
left to show

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{2}(K)}=0$
(4.50)

for a fixed compact set $K\subset\mathbb{R}\times\mathbb{R}^{d}$.

To establish (4.50), we will need to rely on the fact that we are working on a
compact set, so that we can carry out a local smoothing argument. Indeed, the
term appearing above is morally like $|\nabla|^{s_{c}}e^{it\Delta}w_{n}^{J}$,
over which we do not have sufficient control (cf. (2.6)). However, we do have
good control over $e^{it\Delta}w_{n}^{J}$, in the form of (2.3). Thus, to
succeed, we need to find a way to estimate the term above using fewer than
$s_{c}$ derivatives; this is exactly the role of local smoothing.

For the proof of (4.50), we will use a standard local smoothing result for the
free propagator (Lemma 2.9), along with a few results from [45, Chapter V]. In
particular, we need the following: if we choose $\varepsilon>0$ so that
$-d<-1-\varepsilon$, then $|x|^{-1-\varepsilon}$ is an $A_{2}$ weight, so that
$M$ is bounded on $L^{2}(|x|^{-1-\varepsilon}\,dx)$.

###### Proof of (4.50).

We can write $K\subset[-T,T]\times\\{|x|\leq R\\}$ for some $T,R>0$. We fix
some $N_{0}\in 2^{\mathbb{Z}}$ and break into low and high frequencies:

$\displaystyle\iint_{K}\sum_{N}\big{|}N^{s_{c}}M(P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt\lesssim$
$\displaystyle\sum_{N\leq
N_{0}}\iint_{K}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt$
$\displaystyle+\sum_{N>N_{0}}\iint_{K}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt.$

For the low frequencies, we use Hölder and maximal function estimates to write

$\displaystyle\sum_{N\leq N_{0}}\iint_{K}\big{|}$ $\displaystyle
M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt$
$\displaystyle\lesssim\sum_{N\leq
N_{0}}T^{\frac{p(d+2)-4}{p(d+2)}}R^{\frac{d(p(d+2)-4)}{p(d+2)}}\big{\|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{2}$
$\displaystyle\lesssim_{K}\sum_{N\leq
N_{0}}N^{2s_{c}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{2}$
$\displaystyle\lesssim_{K}N_{0}^{2s_{c}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{2}.$

For the high frequencies, we choose $\varepsilon>0$ so that
$-d<-1-\varepsilon$. Then, using Lemma 2.9, Bernstein, and the fact that
$|x|^{-1-\varepsilon}\in A_{2}$, we can estimate

$\displaystyle\sum_{N>N_{0}}\iint_{K}\big{|}$ $\displaystyle
M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\,dx\,dt$
$\displaystyle\lesssim
R^{1+\varepsilon}\sum_{N>N_{0}}\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})|^{2}\langle
x\rangle^{-1-\varepsilon}\,dx\,dt$
$\displaystyle\lesssim_{K}\sum_{N>N_{0}}N^{2s_{c}}\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\big{|}P_{N}e^{it\Delta}w_{n}^{J}\big{|}^{2}\langle
x\rangle^{-1-\varepsilon}\,dx\,dt$
$\displaystyle\lesssim_{K}\sum_{N>N_{0}}N^{2s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}}P_{N}w_{n}^{J}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\lesssim_{K}\sum_{N>N_{0}}N^{-1}\big{\|}|\nabla|^{s_{c}}w_{n}^{J}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\lesssim_{K}N_{0}^{-1}\big{\|}|\nabla|^{s_{c}}w_{n}^{J}\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}^{2}.$

Optimizing in the choice of $N_{0}$ now yields

$\big{\|}\big{(}\sum_{N}\big{|}M(N^{s_{c}}P_{N}e^{it\Delta}w_{n}^{J})\big{|}^{2}\big{)}^{1/2}\big{\|}_{L_{t,x}^{2}(K)}\lesssim_{K}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{1}{2s_{c}+1}}\big{\|}w_{n}^{J}\big{\|}_{\dot{H}_{x}^{s_{c}}(\mathbb{R}^{d})}^{\frac{2s_{c}}{2s_{c}+1}},$

which, by (2.3), gives (4.50). ∎

We have now dealt with (4.39), and so we finally turn to (4.40). As usual, we
first restrict our attention to a single frequency $N$. We have dealt with a
term of this form before (cf. (4.16)); proceeding in exactly the same way, we
arrive at

$\displaystyle\big{|}$ $\displaystyle
P_{N}\big{(}[|g(x)+h(x)|^{p}-|g(x)|^{p}]g(x)\big{)}\big{|}$
$\displaystyle\lesssim\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}g(x)|\,|h(x-y)|\big{\\{}|g(x-y)|^{p-1}+|h(x-y)|^{p-1}\big{\\}}\,dy$
(4.51) $\displaystyle+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x)|\,|\delta_{y}g(x)|\,|h(x-y)|^{p-1}\,dy$ (4.52)
$\displaystyle+\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|g(x)|\,|\delta_{y}h(x)|\,\big{\\{}|g(x)|^{p-1}+|h(x)|^{p-1}+|h(x-y)|^{p-1}\big{\\}}\,dy,$
(4.53)

at least in the case $p\leq 2$ (as above, we will only consider this case).

Note that all of the terms above are similar to terms we have handled before.
Thus, we proceed in the same way, decomposing terms exactly as before.
Whenever a term includes a copy of $e^{it\Delta}w_{n}^{J}$ without
derivatives, things will be relatively straightforward, as one can rely on
(2.3) (see (4.54) below for details); for the one term stemming from (4.53) in
which $e^{it\Delta}w_{n}^{J}$ only appears with derivatives, we have to go
through the same local smoothing argument given above (cf. the proof of
(4.50)).

Thus, to conclude the proof of (4.12), we will see how to estimate the
contribution of the term

$\textstyle\int
N^{d}|\check{\psi}(Ny)|\,|\delta_{y}g(x)|\,|h(x-y)|\,|g(x-y)|^{p-1}\,dy.$
(4.54)

Estimating $|\delta_{y}g(x)|$ as before, we find we need to bound the terms

$\displaystyle M(h|g$ $\displaystyle|^{p-1})g_{>N}+M(h|g|^{p-1}g_{>N})$
$\displaystyle+\sum_{K\leq N}\tfrac{K}{N}M(h|g|^{p-1})M(g_{K})+\sum_{K\leq
N}M(h|g|^{p-1}M(g_{K})).$

Let us now see how to handle the contribution of the first term only, as the
other three are similar. We begin by summing over $N\in 2^{\mathbb{Z}}$ and
recalling the definitions of $g$ and $h$; then, using Hölder, maximal function
estimates, and Lemma 2.2, we can argue as we did to obtain (4.8) to see

$\displaystyle\lim_{J\to\infty}$
$\displaystyle\limsup_{n\to\infty}\big{\|}\big{(}\sum_{N}\big{|}N^{s_{c}}g_{>N}\big{|}^{2}\big{)}^{1/2}M(h|g|^{p-1})\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{p(d+2)}{2(p-1)}}$
$\displaystyle\lesssim\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}^{\frac{p(d+2)}{2(p-1)}}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2(p-1)}}\sum_{j=1}^{J}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}.$
(4.55)

We turn to estimating the first term above. We first write

$\displaystyle\lim_{J\to\infty}$
$\displaystyle\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}$
$\displaystyle\lesssim\lim_{J\to\infty}\limsup_{n\to\infty}\bigg{(}\sum_{j=1}^{J}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}+\sum_{j\neq
k}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}|\nabla|^{s_{c}}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{d+2}{d}}}\bigg{)}.$
(4.56)

Arguing as we did to obtain (4.10), we immediately get that

$\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j\neq
k}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}|\nabla|^{s_{c}}v_{n}^{k}\big{\|}_{L_{t,x}^{\frac{d+2}{d}}}=0.$
(4.57)

Next, we let $\eta>0$; then, using (2.5), we can find $J(\eta)>0$ so that

$\sum_{j>J(\eta)}\big{\|}|\nabla|^{s_{c}}\phi^{j}\big{\|}_{L_{x}^{2}}^{2}<\eta.$

Taking $\eta$ sufficiently small and applying a standard bootstrap argument,
we find

$\sum_{j>J(\eta)}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}\lesssim\sum_{j>J(\eta)}\big{\|}|\nabla|^{s_{c}}\phi^{j}\big{\|}_{L_{x}^{2}}^{2}\lesssim\eta.$
(4.58)

On the other hand, the fact that each $v_{n}^{j}$ has scattering size
$\lesssim E_{c}$ implies

$\sum_{j=1}^{J(\eta)}\big{\|}|\nabla|^{s_{c}}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}\lesssim_{E_{c}}1.$
(4.59)

Combining (4.57), (4.58), and (4.59), we can continue from (4.56) to see

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}_{L_{t,x}^{\frac{2(d+2)}{d}}}^{2}\lesssim_{E_{c}}1.$

Thus, continuing from (4.55) and using (4.5) and (2.3), we find

$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{s_{c}}\big{(}\sum_{j=1}^{J}v_{n}^{j}\big{)}\big{\|}^{\frac{p(d+2)}{2(p-1)}}_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{\|}e^{it\Delta}w_{n}^{J}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2(p-1)}}\sum_{j=1}^{J}\big{\|}v_{n}^{j}\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}}^{\frac{p(d+2)}{2}}=0,$

as needed. This completes the proof of (4.12). ∎

Having established (4.11) and (4.12), we are now done with the proof of Lemma
4.2, as well as the sketch of the proof of Proposition 4.1.

##  5\. Long-time Strichartz estimates

In this section, we prove a long-time Strichartz estimate. Such estimates were
first developed by Dodson [17] in the study of the mass-critical NLS, but have
since appeared in the energy-critical setting (see [36, 57]). In this paper,
we establish a long-time Strichartz estimate for the first time in the inter-
critical setting, modeling our approach after [17, 36, 57]. The long-time
Strichartz estimate will be an important technical tool in Section 6, in which
we rule out rapid frequency-cascade solutions, as well as in Section 7, in
which we establish a frequency-localized interaction Morawetz inequality.

We will prove long-time Strichartz estimates for $(d,s_{c})$ satisfying (1.5).
This guarantees $p>1$, which simplifies the proof. Actually, as we will point
out below, the same ideas can be used to handle $(d,s_{c})=(5,\tfrac{1}{2})$,
in which case $p=1$.

######  Proposition 5.1 (Long-time Strichartz estimates).

Take $(d,s_{c})$ satisfying (1.5). Let
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1) with $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}\subset[0,T_{max})$. Then on any compact time interval
$I\subset[0,T_{max})$, which is a union of contiguous characteristic
subintervals $J_{k}$, and for any $N>0$, we have

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}\lesssim_{u}1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}},$
(5.1)

where $K:=\int_{I}N(t)^{3-4s_{c}}\,dt.$ Moreover, for any $\varepsilon>0$,
there exists $N_{0}=N_{0}(\varepsilon)$ such that for all $N\leq N_{0}$,

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}\lesssim_{u}\varepsilon(1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}).$
(5.2)

We also note that the implicit constants in (5.1) and (5.2) are independent of
$I$.

###### Proof.

Fix a compact interval $I\subset[0,T_{max})$, which is a contiguous union of
characteristic subintervals $J_{k}$; throughout the proof, all spacetime norms
will be taken over $I\times\mathbb{R}^{d}$ unless stated otherwise. Let
$\eta_{0}>0$ and $\eta>0$ be small parameters to be chosen later, and note
that by Remark 1.8, we may find $c=c(\eta)$ so that

$\big{\|}|\nabla|^{s_{c}}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\leq\eta.$ (5.3)

For $N>0$, we define

$A(N):=\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}\quad\text{and}\quad
A_{k}(N):=\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(J_{k}\times\mathbb{R}^{d})}$

for an individual characteristic subinterval $J_{k}$. We first note that by
Lemma 1.11, (5.1) holds whenever $N\geq\sup_{J_{k}\subset I}N_{k}$. Indeed, in
this case, we have

$\displaystyle A(N)$
$\displaystyle\lesssim_{u}1+\left(\int_{I}N(t)^{2}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}1+\left(\int_{I}N(t)^{3-4s_{c}}N^{4s_{c}-1}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$

We will establish (5.1) for arbitrary $N>0$ by induction, beginning by
establishing a recurrence relation for $A(N)$:

######  Lemma 5.2 (Recurrence relation for $A(N)$).

$\displaystyle A(N)$ $\displaystyle\lesssim_{u}\ \inf_{t\in
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}$
$\displaystyle\quad+\eta^{\nu}A\big{(}\tfrac{N}{\eta_{0}}\big{)}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M)$
(5.4)

uniformly in $N$, for some positive constants $B(\eta,\eta_{0})$ and $\nu$.

###### Proof of Lemma 5.2.

We first apply Strichartz to see

$A(N)\lesssim\inf_{t\in I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}}+\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(|u|^{p}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}.$ (5.5)

Our next step is to decompose the nonlinearity $|u|^{p}u$ and estimate the
resulting pieces; the particular decomposition we choose depends on the
ambient dimension.

Case 1. When $d=3$, we have $2\leq p<4$, and we decompose as follows:

$\displaystyle|u|^{p}u=$ $\displaystyle\ (|u|^{p}+|u|^{p-2}\bar{u}u_{\leq
N/\eta_{0}})u_{>N/\eta_{0}}$ $\displaystyle+|u|^{p-2}\bar{u}(P_{>cN(t)}u_{\leq
N/\eta_{0}})u_{\leq N/\eta_{0}}$ (5.6) $\displaystyle+|u|^{p-2}\bar{u}(P_{\leq
cN(t)}u_{\leq N/\eta_{0}})u_{\leq N/\eta_{0}}.$

To estimate the contribution of the first term on the right-hand side of (5.6)
to (5.5), we let

$G:=|u|^{p}+|u|^{p-2}\bar{u}u_{\leq N/\eta_{0}}$

and use Bernstein, Lemma 2.6, and Hölder to estimate

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(Gu_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(Gu_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\displaystyle\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.7)

To estimate the contribution of the first term above, we first use the
fractional chain rule and Sobolev embedding to see

$\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{3p+4}}}\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}\lesssim_{u}1,$

while by the fractional product rule, the fractional chain rule, and Sobolev
embedding we get

$\displaystyle\big{\|}|\nabla|^{\frac{1}{2}s_{c}}$
$\displaystyle(|u|^{p-2}\bar{u}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}$
$\displaystyle\lesssim\
\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-12}}}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{3p+4}}}$
$\displaystyle\lesssim\
\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-2}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{3p+4}}}+\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}$
$\displaystyle\lesssim\
\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}$
$\displaystyle\lesssim_{u}\ 1.$

Thus, continuing from (5.7), we see

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle\big{(}(|u|^{p}+|u|^{p-2}\bar{u}u_{\leq
N/\eta_{0}})u_{>N/\eta_{0}}\big{)}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim_{u}\\!\\!\sum_{M>N/\eta_{0}}\\!\\!\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}\\!A(M).$
(5.8)

Next, we turn to estimating the contribution of the second term in (5.6) to
(5.5). We begin by restricting our attention to an individual
$J_{k}\times\mathbb{R}^{d}$. Note that we only need to consider the case
$cN_{k}\leq N/\eta_{0}$; in this case, we can use Bernstein, Hölder, Sobolev
embedding, Lemma 1.11, and the fact that $s_{c}\geq\tfrac{1}{2}$ to estimate

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u|^{p-2}\bar{u}(P_{>cN_{k}}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}|u|^{p-2}\bar{u}(P_{>cN_{k}}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$
$\displaystyle\lesssim_{u}N^{s_{c}}(cN_{k})^{-s_{c}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}^{2}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}$
(5.9)

for some positive constant $B(\eta,\eta_{0})$. Summing the estimates (5.9)
over the characteristic subintervals $J_{k}\subset I$ then gives

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u|^{p-2}\bar{u}(P_{>cN(t)}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.10)

Before proceeding to the next term in (5.6), we note that in obtaining
estimate (5.9), we could have held onto the term
$\big{\|}|\nabla|^{s_{c}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}},$
which (by interpolation) we can estimate by

$\displaystyle\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}.$

In this case, summing the estimates yields

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u|^{p-2}\bar{u}(P_{>cN(t)}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim_{u}\sup_{J_{k}\subset I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{2}}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.11)

This variant of (5.10) will be important when we eventually need to exhibit
smallness in (5.2).

To estimate the contribution of the final term in (5.6) to (5.5), we begin
with an application of the fractional product rule and Hölder to see

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u|^{p-2}\bar{u}(P_{\leq cN(t)}u_{\leq N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim$
$\displaystyle\
\big{\|}|\nabla|^{s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{6p}{7p-8}}}\big{\|}P_{\leq
cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$ (5.12)
$\displaystyle+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}|\nabla|^{s_{c}}P_{\leq
cN(t)}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$ (5.13)
$\displaystyle+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-1}\big{\|}P_{\leq
cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}.$ (5.14)

We first note that by the fractional chain rule and Sobolev embedding, we get

$\big{\|}|\nabla|^{s_{c}}(|u|^{p-2}\bar{u})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{6p}{7p-8}}}\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{3p}{2}}}^{p-2}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}1.$

Using Sobolev embedding, interpolation, and (5.3), we also see

$\displaystyle\big{\|}P_{\leq cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}P_{\leq cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{3}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}P_{\leq cN(t)}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}P_{\leq
cN(t)}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{6}}^{\frac{1}{2}}$
$\displaystyle\lesssim\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}^{\frac{1}{2}}.$

Estimating similarly gives

$\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{6p}{4-p}}}\lesssim_{u}A\big{(}\tfrac{N}{\eta_{0}}\big{)}^{\frac{1}{2}}.$

Plugging these last three estimates into (5.12), (5.13), and (5.14) and
employing a few more instances of Sobolev embedding and (5.3) finally gives

$\big{\|}|\nabla|^{s_{c}}P_{\leq N}(|u|^{p-2}\bar{u}(P_{\leq cN(t)}u_{\leq
N/\eta_{0}})u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}\lesssim_{u}\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
(5.15)

Collecting the estimates (5.8), (5.10), and (5.15), we see that in the case
$d=3$, the estimate (5.5) becomes

$\displaystyle A(N)\lesssim_{u}$ $\displaystyle\inf_{t\in
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}$
$\displaystyle\quad+\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.16)

Comparing (5.16) to (5.2), we see that Lemma 5.2 holds for $d=3$.

Case 2. In this case, we have $d\in\\{4,5\\}$ and $\tfrac{4}{d-1}\leq
p<\tfrac{4}{d-2}$, with both inequalities strict for $d=5$. In particular, we
have $1<p<2$.

Again, we wish to decompose the nonlinearity and continue from (5.5). This
time, we decompose as follows:

$\displaystyle|u|^{p}u$ $\displaystyle=|u|^{p}u_{>N/\eta_{0}}$
$\displaystyle\quad+|u_{>cN(t)}|^{p}P_{\leq cN(t)}u_{\leq N/\eta_{0}}$ (5.17)
$\displaystyle\quad+|u_{>cN(t)}|^{p}P_{>cN(t)}u_{\leq N/\eta_{0}}$
$\displaystyle\quad+(|u|^{p}-|u_{>cN(t)}|^{p})u_{\leq N/\eta_{0}}.$

We estimate the contribution of the first term on the right-hand side of
(5.17) to (5.5) similarly to the case $d=3$; in particular, by Bernstein,
Hölder, and Lemma 2.6, we have

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u|^{p}u_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(|u|^{p}u_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.18)

As we can use the fractional chain rule and Sobolev embedding to estimate

$\big{\|}|\nabla|^{\frac{1}{2}s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{p(d+8)-4}}}\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{4dp}{dp+4}}}\lesssim\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p}\lesssim_{u}1,$

we can continue from (5.18) to get

$\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(|u|^{p}u_{>N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim_{u}\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.19)

Next, we turn to estimating the second term in (5.17) to (5.5). Restricting
our attention to an individual characteristic subinterval $J_{k}$, we first
apply Bernstein, Hölder, and the fractional product rule to see

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u_{>cN_{k}}|^{p}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$ $\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}\big{\|}|\nabla|^{\frac{1}{4}}(|u_{>cN_{k}}|^{p}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}\big{\|}|\nabla|^{\frac{1}{4}}|u_{>cN_{k}}|^{p}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2dp}{p(d+3)-4}}}\big{\|}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2dp}{4-p}}}$ (5.20)
$\displaystyle\ \ \
+\\!N^{s_{c}-\frac{1}{4}}\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{4dp^{2}}{p(2d+5)-8}}}^{p}\big{\|}|\nabla|^{\frac{1}{4}}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{4dp}{8-p}}}.$ (5.21)

Using Hölder, the fractional chain rule, Sobolev embedding, Bernstein,
interpolation, (5.3), and Young’s inequality, we can estimate

$\displaystyle\eqref{young 1}$ $\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{\frac{1}{4}}u_{>cN_{k}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}|\nabla|^{s_{c}}P_{\leq
cN_{k}}u_{\leq N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}$
$\displaystyle\lesssim_{u}N^{s_{c}-\frac{1}{4}}(cN_{k})^{\frac{1}{4}-s_{c}}\big{\|}|\nabla|^{s_{c}}u_{>cN_{k}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}B(\eta)\left(\tfrac{N}{N_{k}}\right)^{s_{c}-\frac{1}{4}}\eta^{\frac{1}{2}}A_{k}\big{(}\tfrac{N}{\eta_{0}}\big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}B(\eta)\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}+\eta
A_{k}\big{(}\tfrac{N}{\eta_{0}}\big{)},$

for some positive constant $B(\eta)$. Using Lemma 1.11 as well, we can
estimate similarly

$\displaystyle\eqref{young 2}$ $\displaystyle\lesssim
N^{s_{c}-\frac{1}{4}}(cN_{k})^{\frac{1}{4}-s_{c}}\big{\|}|\nabla|^{(s_{c}-\frac{1}{4})/p}u_{>cN_{k}}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{4dp^{2}}{p(2d+5)-8}}}^{p}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}$ $\displaystyle\lesssim
B(\eta)\left(\tfrac{N}{N_{k}}\right)^{s_{c}-\frac{1}{4}}\big{\|}|\nabla|^{s_{c}}u_{>cN_{k}}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{2dp}{dp-1}}}^{p}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}B(\eta)\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}+\eta
A_{k}\big{(}\tfrac{N}{\eta_{0}}\big{)}.$

Collecting the estimates for (5.20) and (5.21) and summing over the intervals
$J_{k}\subset I$, we arrive at

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(|u_{>cN(t)}|^{p}P_{\leq cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim_{u}B(\eta)N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta
A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$ (5.22)

Before proceeding, we note that for both (5.20) and (5.21), we could have
instead estimated

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
cN_{k}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{4}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{4}}$
$\displaystyle\lesssim\eta^{\frac{1}{4}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{4}}.$

If we had done this, upon summing we could have ended up with the alternate
estimate

$\displaystyle\big{\|}|\nabla$ $\displaystyle|^{s_{c}}P_{\leq
N}(|u_{>cN(t)}|^{p}P_{\leq cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\sup_{J_{k}\subset I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\frac{1}{2}}B(\eta)N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta^{\frac{1}{2}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
(5.23)

This variant of (5.22) will be important when we need to exhibit smallness in
(5.2).

To estimate the contribution of the third term in (5.17) to (5.5), we first
define the following:

$\left\\{\begin{array}[]{ll}\theta:=\tfrac{dp-4-p}{4-p}\in[0,1),&\sigma:=\tfrac{p^{2}(d^{2}+2d-2)-4p(4d+1)+48}{4p(dp-8)}\in(0,s_{c}),\\\
\\\
r_{1}:=\tfrac{4dp(dp-8)}{p^{2}(d^{2}-2d-2)+p(28-8d)-16},&r_{2}:=\tfrac{4dp(dp-8)}{p^{2}(d^{2}+2d-2)-4p(2d+1)-16}.\end{array}\right.$

With this choice of parameters, we have

$\left\\{\begin{array}[]{ll}s_{c}+\theta(\tfrac{d-1}{2}-s_{c})=2s_{c}-\tfrac{1}{2},\\\
\\\
-\theta(s_{c}+\tfrac{1}{2})-2\sigma(1-\theta)=-(2s_{c}-\tfrac{1}{2})\end{array}\right.$

and (by Sobolev embedding)

$\begin{array}[]{ll}\dot{H}^{s_{c},\frac{2d}{d-2}}\hookrightarrow\dot{H}^{\sigma,r_{1}},&\dot{H}^{s_{c},2}\hookrightarrow\dot{H}^{\sigma,r_{2}}.\end{array}$

Then restricting our attention to an individual $J_{k}$, we can use Bernstein,
Hölder, the bilinear Strichartz estimate (Lemma 2.10), and Sobolev embedding
to estimate

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u_{>cN_{k}}|^{p}P_{>cN_{k}}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}u_{>cN_{k}}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t,x}^{2}}^{\theta}$
$\displaystyle\quad\times\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{r_{1}}}^{1-\theta}\big{\|}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{r_{2}}}^{1-\theta}$
$\displaystyle\lesssim_{u}N^{s_{c}}\left(\tfrac{N}{\eta_{0}}\right)^{\theta(\frac{d-1}{2}-s_{c})}(cN_{k})^{-\theta(s_{c}+\frac{1}{2})}$
$\displaystyle\quad\times\big{\|}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{r_{1}}}^{1-\theta}\big{\|}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{r_{2}}}^{1-\theta}$
$\displaystyle\lesssim_{u}B(\eta_{0})N^{2s_{c}-\frac{1}{2}}(cN_{k})^{-\theta(s_{c}+\frac{1}{2})-2\sigma(1-\theta)}$
$\displaystyle\quad\times\big{\|}|\nabla|^{\sigma}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{r_{1}}}^{1-\theta}\big{\|}|\nabla|^{\sigma}P_{>cN_{k}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{r_{2}}}^{1-\theta}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}\big{\|}|\nabla|^{s_{c}}u_{>cN_{k}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{1-\theta}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{1-\theta}$
$\displaystyle\lesssim_{u}B(\eta,\eta_{0})\left(\tfrac{N}{N_{k}}\right)^{2s_{c}-\frac{1}{2}}$
(5.24)

for some positive constant $B(\eta,\eta_{0})$. If we sum the estimates (5.24)
over the intervals $J_{k}\subset I$, we arrive at

$\big{\|}|\nabla|^{s_{c}}P_{\leq N}(|u_{>cN(t)}|^{p}P_{>cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim_{u}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.25)

Before moving on to the fourth (and final) term in (5.17), we note that if we
had held on to the term $\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{1-\theta}$ when deriving
(5.24), then upon summing we would get

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}(|u_{>cN(t)}|^{p}P_{>cN(t)}u_{\leq
N/\eta_{0}})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim_{u}\sup_{J_{k}\subset I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{1-\theta}B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$
(5.26)

This variant of (5.25) will be important when we eventually need to exhibit
smallness in (5.2).

We now turn to the final term in (5.17), beginning with an application of the
fractional product rule and Hölder:

$\displaystyle\big{\|}|\nabla|^{s_{c}}$ $\displaystyle P_{\leq
N}\left((|u|^{p}-|u_{>cN(t)}|^{p})u_{\leq
N/\eta_{0}}\right)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}(|u|^{p}-|u_{>cN(t)}|^{p})\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2dp}{p(d+4)-4}}}\big{\|}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{dp}{2-p}}}$ (5.27)
$\displaystyle\quad+\big{\|}|u|^{p}-|u_{>cN(t)}|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{d}{2}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}.$ (5.28)

By Lemma 2.5, Sobolev embedding, and (5.3), we first estimate

$\displaystyle\eqref{differences 1}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}A\big{(}\tfrac{N}{\eta_{0}}\big{)}$
$\displaystyle\quad+\big{\|}|\nabla|^{s_{c}}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}A\big{(}\tfrac{N}{\eta_{0}}\big{)}$
$\displaystyle\lesssim_{u}(\eta^{p-1}+\eta)A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$

On the other hand, by Sobolev embedding, Hölder, and (5.3), we get

$\displaystyle\eqref{differences 2}$
$\displaystyle\lesssim\big{(}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}+\big{\|}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{)}\big{\|}u_{\leq
cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}\lesssim_{u}\eta
A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$

Thus we can estimate the contribution of the final term in (5.17) by

$\big{\|}|\nabla|^{s_{c}}P_{\leq N}\left((|u|^{p}-|u_{>cN(t)}|^{p})u_{\leq
N/\eta_{0}}\right)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim_{u}\eta^{p-1}A\big{(}\tfrac{N}{\eta_{0}}\big{)}.$
(5.29)

Collecting the estimates (5.19), (5.22), (5.25), and (5.29), we see that in
Case 2, the estimate (5.5) becomes

$\displaystyle A(N)$ $\displaystyle\lesssim_{u}\inf_{t\in
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}$
$\displaystyle\quad+\eta^{\min\\{\frac{1}{2},p-1\\}}A\big{(}\tfrac{N}{\eta_{0}}\big{)}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M).$
(5.30)

Comparing (5.30) to (5.2), we see that Lemma 5.2 holds for $d\in\\{4,5\\}$. ∎

######  Remark 5.3.

We have omitted the case $(d,s_{c})=(5,\tfrac{1}{2})$, in which $p=1$; this
scenario is not handled under Case 2 due to the use of Lemma 2.5. However, by
using the alternate decomposition

$|u|u=|u|u_{>N/\eta_{0}}+(|u|-|u_{\leq cN(t)}|)u_{\leq N/\eta_{0}}+|u_{\leq
cN(t)}|u_{\leq N/\eta_{0}},$

one can use the same ideas as above to establish the recurrence relation in
this case.

With the recurrence relation (5.2) in hand, we can now use induction to
complete the proof of Proposition 5.1. First, recall that (5.1) holds for
$N\geq\sup_{J_{k}\subset I}N_{k}.$; i.e. we have

$A(N)\leq C(u)\left[1+N^{2s_{c}-1/2}K^{1/2}\right]$ (5.31)

for $N\geq\sup_{J_{k}\subset I}N_{k}.$ Of course, this inequality remains true
if we replace $C(u)$ by any larger constant.

We now suppose (5.31) holds at frequency $N$ and use the recurrence relation
(5.2) to show it holds at frequency $N/2$. Let us first rewrite (5.2) as

$\displaystyle A(N)$
$\displaystyle\leq\widetilde{C}(u)\big{[}1+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta^{\nu}A(\tfrac{N}{\eta_{0}})+\\!\\!\\!\\!\\!\\!\sum_{M>N/\eta_{0}}\\!\\!\\!\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A(M)\big{]}.$
(5.32)

To simplify notation, we will let $\alpha:=2s_{c}-\frac{1}{2}$ and write
$B(\eta,\eta_{0})=B$; then, if we take $\eta_{0}<\tfrac{1}{2}$ and use the
inductive hypothesis, (5.32) becomes

$\displaystyle A(\tfrac{N}{2})$
$\displaystyle\leq\widetilde{C}(u)\big{[}1+B(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}+\eta^{\nu}C(u)(1+\eta_{0}^{-\alpha}(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}})$
$\displaystyle\quad+C(u)\\!\\!\\!\\!\\!\\!\sum_{M>N/2\eta_{0}}\\!\\!\\!\left(\tfrac{N}{2M}\right)^{\frac{3}{2}s_{c}}(1+M^{\alpha}K^{\frac{1}{2}})\big{]}$
$\displaystyle\leq\widetilde{C}(u)\big{[}1+B(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}+\eta^{\nu}C(u)(1+\eta_{0}^{-\alpha}(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}})$
$\displaystyle\quad+C(u)\eta_{0}^{\frac{3}{2}s_{c}}+C(u)\eta_{0}^{\frac{1}{2}(1-s_{c})}(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\big{]}$
$\displaystyle=\widetilde{C}(u)\left[1+B(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\right]+C(u)\big{[}(\eta^{\nu}+\eta_{0}^{\frac{3}{2}s_{c}})\widetilde{C}(u)$
$\displaystyle\quad+\big{(}\eta_{0}^{-\alpha}\eta^{\nu}+\eta_{0}^{\frac{1}{2}(1-s_{c})}\big{)}\widetilde{C}(u)(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\big{]}.$
(5.33)

Notice that we had convergence of the sum above precisely because $s_{c}<1$.
If we choose $\eta_{0}$ possibly even smaller depending on $\widetilde{C}(u)$,
and $\eta$ sufficiently small depending on $\widetilde{C}(u)$ and $\eta_{0}$,
we can guarantee

$\displaystyle\eqref{induction
term}\leq\widetilde{C}(u)\left[1+B(\eta,\eta_{0})(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\right]+\tfrac{1}{2}C(u)\left[1+(\tfrac{N}{2})^{\alpha}K^{\frac{1}{2}}\right].$

If we now choose $C(u)$ possibly larger so that $C(u)\geq
2(1+B(\eta,\eta_{0}))\widetilde{C}(u)$, then this inequality implies that
(5.31) holds at $N/2$, as was needed to show. This completes the proof of
(5.1).

It remains to establish (5.2). To begin, fix $\varepsilon>0$. To exhibit the
smallness in (5.2), we need to revisit the proof of the recurrence relation
for $A(N)$, paying closer attention to the terms that gave rise to the
expression $N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}.$ More precisely, if we use
(5.11) instead of (5.10); (5.23) instead of (5.22); and (5.26) instead of
(5.25); then continuing from (5.5), the recurrence relation for $A(N)$ takes
the form

$\displaystyle A(N)\lesssim_{u}$ $\displaystyle\
f(N)+f(N)N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}}+\eta^{\nu}A(\tfrac{N}{\eta_{0}})+\sum_{M>N/\eta_{0}}(\tfrac{N}{M})^{\frac{3}{2}s_{c}}A(M),$
(5.34)

where $f(N)$ has the form

$\displaystyle f(N)=\ $ $\displaystyle\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}$
$\displaystyle+B(\eta,\eta_{0})\sum_{i=1}^{4}\sup_{J_{k}\subset
I}\big{\|}|\nabla|^{s_{c}}u_{\leq
N/\eta_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(J_{k}\times\mathbb{R}^{d})}^{\theta_{i}}$
(5.35)

for some $\theta_{i}\in(0,1]$. Here the particular values of the $\theta_{i}$
are not important; we will only need the fact that each $\theta_{i}>0$.
Combining the updated recurrence relation (5.34) with the newly proven
estimate (5.1) and once again simplifying notation via
$\alpha=2s_{c}-\frac{1}{2}$, we see

$\displaystyle A(N)$
$\displaystyle\lesssim_{u}f(N)+f(N)N^{\alpha}K^{\frac{1}{2}}+\eta^{\nu}(1+\eta_{0}^{-\alpha}N^{\alpha}K^{\frac{1}{2}})+\eta_{0}^{\frac{3}{2}s_{c}}(1+\eta_{0}^{-\alpha}N^{\alpha}K^{\frac{1}{2}})$
$\displaystyle\lesssim_{u}f(N)+\eta^{\nu}+\eta_{0}^{\frac{3}{2}s_{c}}+\left[f(N)+\eta^{\nu}\eta_{0}^{-\alpha}+\eta_{0}^{\frac{1}{2}(1-s_{c})}\right]N^{\alpha}K^{\frac{1}{2}}.$
(5.36)

To complete the argument, we will need the fact that for fixed
$\eta,\eta_{0}>0$, we have

$\lim_{N\to 0}f(N)=0,$ (5.37)

which is a consequence of almost periodicity and the fact that

$\inf_{t\in[0,T_{max})}N(t)\geq 1.$

Then, continuing from (5.36), we choose $\eta_{0}$ small enough that
$\eta_{0}^{\frac{3}{2}s_{c}}+\eta_{0}^{\frac{1}{2}(1-s_{c})}<\varepsilon$, and
choose $\eta$ sufficiently small depending on $\eta_{0}$ so that
$\eta^{\nu}+\eta_{0}^{-\alpha}\eta^{\nu}<\varepsilon$. Finally, using (5.37),
we choose $N_{0}=N_{0}(\varepsilon)$ so that $f(N)<\varepsilon$ for $N\leq
N_{0}$. With this choice of parameters, (5.36) becomes

$A(N)\lesssim_{u}\varepsilon(1+N^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}})$

for $N\leq N_{0},$ which completes the proof of (5.2). ∎

##  6\. The rapid frequency-cascade scenario

In this section, we preclude the existence of almost periodic solutions as in
Theorem 1.13 for which $\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty.$ We show
that their existence is inconsistent with the conservation of mass. The main
tool we will use is the long-time Strichartz estimate established in the
previous section; as such, we will prove the following result for $(d,s_{c})$
satisfying (1.5).

######  Theorem 6.1 (No rapid frequency-cascades).

Let $(d,s_{c})$ satisfy (1.5). Then there are no almost periodic solutions
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) with $N(t)\equiv
N_{k}\geq 1$ on each characteristic subinterval $J_{k}\subset[0,T_{max})$ such
that

$\big{\|}u\big{\|}_{L_{t,x}^{\frac{p(d+2)}{2}}([0,T_{max})\times\mathbb{R}^{d})}=\infty$
(6.1)

and