[E. Bryden, D. Kazaras, R. Perales, and C. Sormani]Brian Allen Scalar
Curvature Stability: Tools, Theorems, and Questions

##  1\. Introduction

Although scalar curvature is the simplest curvature invariant, our
understanding of scalar curvature has not matured to the same level as Ricci
or sectional curvature. Despite this fact, many rigidity phenomenon have been
established which give some of the strongest insights into scalar curvature.
Important examples include Geroch’s conjecture, the positive mass theorem, and
Llarull’s theorem. In order to further understand scalar curvature we ask
corresponding geometric stability questions, where the hypotheses of the
rigidity phenomenon are relaxed, and one would like to show that Riemannian
manifolds which satisfy the relaxed conditions are close to the rigid objects
in some topology. In this talk we will survey what is known for scalar
curvature stability, discuss what the questions are in this area, and
introduce important tools which have been useful so far.

##  2\. Scalar Curvature Phenomenon

When exploring sequences of Riemannian manifolds with scalar curvature lower
bounds we see that splines, bubbles, and drawstrings, depicted in Figure 1,
are persistent phenomenon which need to either be ruled out by making an
assumption or addressed by the choice of topology one makes when choosing an
appropriate notion of convergence. Examples with splines and bubbles are
constructed using Gromov-Lawson tunnels [27] with estimates. Careful
constructions of this type have been carried out by Basilio, Dodziuk, and
Sormani [14], Basilio and Sormani [16], Dodziuk [22], Basilio, Kazaras and
Sormani [15], and Sweeney [33]. If one allows a sequence of manifolds which
are non-diffeomorphic to the limit manifold then Basilio and Sormani [16] have
also shown that sewing is possible, something we will mostly avoid in this
note by fixing the topology. The existence of drawstrings in dimensions $n\geq
4$ with scalar curvature bounds was constructed by Lee, Naber, and Neumayer
[38] and in dimension $n=3$ by Kazaras and Xu [35].

Figure 1. From left to right, splines, bubbling, and a drawstring where a
circle has been pulled to almost a point. Along such a sequence the splines
would become arbitrarily thin, the neck of the bubble would pinch to a point,
and the circle would be pulled to a point by the drawstring.

Due to the presence of splines in all scalar curvature stability problems we
know that Gromov-Hausdorff stability is not appropriate for these problems
since a sequence of manifolds with increasingly many splines cannot converge
in the Gromov-Hausdorff sense by Gromov’s compactness theorem [19]. Hence we
are left to look for other notions of convergence for sequences of Riemannian
manifolds which are resilient to the phenomenon depicted in Figure 1. In this
note we will give examples of three different choices of convergence one can
use: $d_{p}$ convergence (see Lee, Naber, Neumayer [38] for the definition),
Gromov-Hausdorff convergence of the restricted length metric to a good set, or
Sormani-Wenger Intrinsic Flat (SWIF) convergence (see Sormani and Wenger [45]
for the definition). We emphasize that there is no clear hierarchy between the
three choices of convergence of Riemannian manifolds we will discuss in this
note since each has drawbacks in how they handle the scalar curvature
phenomenon depicted in Figure 1. Hence it is the philosophy of the author that
our goal should be to prove all three notions of convergence, under different
assumptions, for most scalar curvature rigidity results. For example, one
would rule out splines and bubbles if one wants to show $d_{p}$ stability and
one would rule out bubbles and drawstrings if one would like to show SWIF
stability. In concert, the three stability results would tell us the most
about sequences of Riemannian manifolds with scalar curvature bounds, as will
be discussed throughout the rest of this note.

##  3\. Geroch Conjecture Stability

######  Theorem 3.1 (Schoen and Yau [42], Gromov and Lawson [27]).

If $(\mathbb{T}^{n},g)$ is a Riemannian metric with non-negative scalar
curvature then $(\mathbb{T}^{n},g)$ is isometric to a flat torus.

Various special cases of Geroch stability have already been addressed. Single
and double warped products were addressed by Allen, Vazquez, Parise, Payne,
and Wang [3], the graph case was studied by Pacheco, Ketterer, and Perales
[41], and conformal cases by Allen [2], and Chu and Lee [20], and under
isoperimetric bounds and integral Ricci bounds by Allen, Bryden, and Kazaras
[9]. In these special cases, many different notions of convergence were
considered. In general, since splines, bubbles, and drawstrings are possible
in the case of Geroch stability, one needs to add a hypothesis to remove two
of the three scalar curvature phenomenon to show $d_{p}$ or SWIF stability or
one needs to cut these phenomenon out altogether. In the following stability
result, the authors choose to assume an entropy bound, which rules out splines
and bubbles, and show $d_{p}$ convergence to a flat torus. One can find many
examples in [38] where it is demonstrated that $d_{p}$ convergence is
resilient to drawstrings. See [8] for an explanation of the fact that $d_{p}$
convergence is not well suited in the presence of splines and bubbles.

######  Theorem 3.2 (Lee, Naber, and Neumayer [38]).

Fix $n\geq 2$ and $p\geq n+1$. There exists a $\delta=\delta(n,p)>0$ and
$V_{0}=v_{0}(n,p)>0$ such that the following holds: For any $V>V_{0}$ and
$(\mathbb{T}^{n},g_{j})$ a sequence of Riemannian tori such that

(1) $\displaystyle
R_{g_{j}}\geq-\frac{1}{j},\qquad\operatorname{Vol}(\mathbb{T}^{n},g_{j})\leq
V_{0},\qquad\nu(\mathbb{T}^{n},g_{j})\geq-\delta,$

a subsequence of $(\mathbb{T}^{n},g_{j})$ converges to a flat torus
$(\mathbb{T}^{n},g_{F})$ in the $d_{p}$ sense.

######  Question 3.3.

Given that the entropy bound of Theorem 3.2 removes the possibility of splines
and bubbles forming along a sequence, can we formulate and prove Geroch
stability where one removes the possibility of bubbles and drawstrings and
shows volume preserving SWIF convergence to a flat torus?

##  4\. Positive Mass Theorem Stability

######  Theorem 4.1 (Schoen and Yau [43], Witten [46]).

If $(M^{3},g)$ is an asympotically flat manifold with non-negative scalar
curvature then the ADM mass is non-negative. If the ADM mass is zero then
$(M^{3},g)$ is isometric to Euclidean space.

The stability of the positive mass theorem has been studied in many special
cases. Assuming the existence of a smooth IMCF by Allen [4, 5, 6, 7], in terms
of the Brown-York Mass for graphs in Euclidean space by Alaee, Cabrera
Pacheco, and McCormick [1], metrics conformal to Euclidean space by Corvino
[21], Lee [36], axially symmetric metrics by Bryden [18], using spinors by
Finster and Bray [17], Finster and Kath [25], and Finster [24], in the
geometrostatic setting by Stavrov-Allen and Sormani [44], in the rotationally
symmetric setting by Lee and Sormani [37], assuming a lower Ricci bound by
Kazaras, Lee, and Khuri [34], and under isoperimetric bounds and integral
Ricci bounds by Allen, Bryden, and Kazaras [9]. In these special cases, many
different notions of convergence were considered.

In the following result, the authors address a conjecture by Huisken and
Ilmanen [32] where the idea is to remove all of the scalar curvature
phenomenon depicted in Figure 1 and show that the remaining set converges to
Euclidean space.

######  Theorem 4.2 (Dong and Song [23]).

Let $(M_{i}^{3},g_{i})$ be a sequence of asymptotically flat Riemannian
manifolds with non-negative scalar curvature and suppose that the ADM mass
$m_{ADM}(M_{i},g_{i})\searrow 0$. Then for all $i\in\mathbb{N}$ and each end
in $M_{i}$, there is a domain $Z_{i}\subset M_{i}$ with smooth boundary so
that $|\partial Z_{i}|_{g_{i}}\searrow 0$, $M_{i}\setminus Z_{i}$ contains the
given end, and

(2) $\displaystyle(M_{i}\setminus
Z_{i},\hat{d}_{g_{i}},p_{i})\rightarrow(\mathbb{R}^{3},d_{\mathbb{E}^{3}},0),$

in the pointed measured Gromov-Hausdorff sense, where $p_{i}\in M_{i}\setminus
Z_{i}$ is any choice of base point, and $\hat{d}_{g_{i}}$ is the restricted
length metric on $M_{i}\setminus Z_{i}$ induced by $g_{i}$.

######  Question 4.3.

Can one formulate and prove a version of Geroch stability in the style of
Theorem 4.2?

######  Question 4.4.

In light of Theorem 4.2, one can ask to say more about what is happening on
the bad set $Z_{i}$. For instance, is $Z_{i}$ made up of splines, bubbles,
drawstrings, and sewing only? One way of providing some evidence in this
direction is to formulate a stability conjecture where one assumes a condition
which rules out splines and bubbles and concludes $d_{p}$ stability and
another version of a stability conjecture where one rules out bubbles,
drawstrings, and sewing which concludes volume preserving SWIF convergence. In
concert, these three stability results would be giving strong evidence that
the bad sets $Z_{i}$ are made up of only the known phenomenon, reinforcing the
belief that these are the only phenomenon which one needs to worry about when
discussing sequences with scalar curvature lower bounds.

##  5\. Llarull Stability

Here we state a slightly less general version of Llarull’s rigidity theorem.

######  Theorem 5.1 (Llarull [40]).

If $(\mathbb{S}^{n},g)$ is a Riemannian metric so that $g\geq
g_{\mathbb{S}^{n}}$, where $g_{\mathbb{S}^{n}}$ is the round sphere, whose
scalar curvature $R_{g}\geq n(n-1)$ then $(\mathbb{S}^{n},g)$ is isometric to
$(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$.

In this case, when we consider the stability of Theorem 5.1 we notice that
drawstrings are not possible due to the metric lower bound assumption which is
necessary in the rigidity theorem. Hence, volume preserving SWIF convergence
is most appropriate for stability in this case. In fact, any time that
rigidity is phrased in terms of a $1-$Lipschitz map $F:(M,g)\rightarrow(N,h)$
we claim that volume preserving Sormani-Wenger Intrinsic Flat convergence will
be the correct choice for proving the corresponding stability result.

######  Theorem 5.2 (Allen, Bryden, and Kazaras [10]).

Let $V,D,\overline{m},\Lambda>0$. If a sequence $(\mathbb{S}^{3},g_{i})$ of
Riemannian $3$-spheres satisfies

(3) $\displaystyle g_{i}$ $\displaystyle\geq
g_{\mathbb{S}^{3}},\qquad\operatorname{Vol}(\mathbb{S}^{3},g_{i})\leq V,$ (4)
$\displaystyle\operatorname{Diam}(\mathbb{S}^{3},g_{i})$ $\displaystyle\leq
D,\qquad\inf_{\Omega\subset\mathbb{S}^{3}}\tfrac{\operatorname{Area}(\partial\Omega,g_{i})}{\min(\operatorname{Vol}(\Omega,g_{i}),\operatorname{Vol}(\mathbb{S}^{3}\setminus\Omega,g_{i}))}\geq\Lambda,$

and

(5)
$\displaystyle\left\|\left(6-R_{g_{i}}\right)^{+}\right\|_{L^{2}(g_{i})}^{1/2}\rightarrow
0,$

then it converges in the volume preserving SWIF sense:

(6) $\displaystyle
d_{\mathcal{VF}}((\mathbb{S}^{3},g_{i}),(\mathbb{S}^{3},g_{\mathbb{S}^{3}}))\rightarrow
0.$

The proof uses two main tools from the literature: spacetime harmonic
functions developed by Hirsch, Kazaras, and Khuri [29] and used to give a new
proof of Llarull’s theorem by Hirsch, Kazaras, Khuri, and Zang [30], and the
Volume Above Distance Below (VADB) theorem of Allen, Perales, and Sormani
[12]. Shortly after Theorem 5.2 was established, a more general result was
established using the spinor formulas which Llarull used to establish rigidity
and the VADB theorem.

######  Theorem 5.3 (Hirsch and Zhang [31]).

Let $V,D,\overline{m},\Lambda>0$. If a sequence $(\mathbb{S}^{n},g_{i})$,
$n\geq 3$ of Riemannian $n$-spheres satisfies

(7) $\displaystyle g_{i}\geq
g_{\mathbb{S}^{n}},\qquad\operatorname{Diam}(\mathbb{S}^{n},g_{i})$
$\displaystyle\leq D,\qquad\inf_{u\in W^{1,2}(\mathbb{S}^{n})}\tfrac{\|\nabla
u\|_{L^{2}(\mathbb{S}^{n})}^{2}}{\inf_{k\in\mathbb{R}}\|u-k\|_{L^{2}(\mathbb{S}^{n})}^{2}}\geq\Lambda,$

and

(8) $\displaystyle R_{g_{i}}\geq n(n-1)-\frac{1}{i}$

then it converges in the volume preserving SWIF sense:

(9) $\displaystyle
d_{\mathcal{VF}}((\mathbb{S}^{n},g_{i}),(\mathbb{S}^{n},g_{\mathbb{S}^{n}}))\rightarrow
0.$

It should be noted that Hirsch and Zhang prove two other versions of Llarull
stability, one with an $L^{p}$ assumption on the scalar curvature below
$n(n-1)$ and another result with a good set bad set decomposition similar to
Theorem 4.2.

######  Question 5.4.

Can one use spinors in order to establish Geroch stability and positive mass
theorem stability in dimensions $n\geq 3$?

######  Question 5.5.

Can one generalize Theorem 5.3 to the case of the results of Goette and
Semmelmann [26]?

##  6\. Volume Above Distance Below Theorem

We end this note by stating and discussing the VADB theorem which has proved
useful so far in proving stability of Llarull’s theorem as well as many
special cases of Geroch stability and positive mass theorem stability.

######  Theorem 6.1 (Allen, Perales, and Sormani [12]).

Suppose we have a fixed closed, oriented, Riemannian manifold,
$(M^{n},g_{0})$, and a sequence of continuous Riemannian manifolds $(M,g_{j})$
such that

(10) $g_{j}(v,v)\geq\left(1-C_{j}\right)g_{0}(v,v),\quad\forall p\in M,v\in
T_{p}M,\quad C_{j}\searrow 0,$

and a uniform upper bound on diameter, $\operatorname{Diam}(M_{j})\leq D_{0}$,
and volume convergence

(11) $\operatorname{Vol}(M,g_{j})\to\operatorname{Vol}(M,g_{0}),$

then we find volume preserving SWIF convergence
$d_{\mathcal{VF}}((M,g_{j}),(M,g_{0}))\rightarrow 0$.

It should also be noted that Allen and Perales [11] have proved a version of
Theorem 6.1 for manifolds with boundary. Hence one can use the VADB theorem
with boundary to discuss stability of rigidity results on manifolds with
boundary.

######  Question 6.2.

Can one prove a version of Theorem 6.1 for sequences which are not
diffeomorphic to the limit manifold? Addressing this question would allow us
to extend Theorem 5.2 and Theorem 5.3 to $1-$Lipschitz maps from manifolds
which are not diffeomorphic to the sphere.

######  Question 6.3.

Theorem 6.1 motivates a notion of convergence for sequences of Riemannian
manifolds. We say that a sequence of Riemannian manifolds $(M,g_{j})$
converges to a Riemannian manifold $(M,g_{\infty})$ in the VADB sense if
$\operatorname{Diam}(M,g_{j})\leq D$,
$\operatorname{Vol}(M,g_{j})\rightarrow\operatorname{Vol}(M,g_{\infty})$ and
$g_{j}(v,v)\geq(1-C_{j})g_{\infty}(v,v)$, $\forall p\in M,v\in
T_{p}M,C_{j}\searrow 0$. If we consider a sequence of Riemannian manifolds
$(M,g_{j})$ so that $R_{g_{j}}\geq 0$, which converge in the VADB sense to
$(M,g_{\infty})$, then is it the case that $R_{g_{\infty}}\geq 0$? It was
shown by Gromov [28] and Bamler [13] that if $g_{j}\rightarrow g_{\infty}$ in
the $C^{0}$ sense then the answer to the question is yes. It was shown by Lee
and Topping [39] that if $d_{g_{j}}\rightarrow d_{g_{\infty}}$ in the uniform
sense then the answer is no. Considering this question for VADB convergence is
interesting because VADB convergence lies in between $C^{0}$ convergence of
Riemannian metrics and uniform convergence of distance functions. It is also
interesting since VADB convergence allows for the presence of splines along
the sequences which persist in the limit.

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$\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt<\infty.$ (6.2)

###### Proof.

We argue by contradiction. Suppose $u$ were such a solution; then by Corollary
1.10, we have

$\lim_{t\to T_{max}}N(t)=\infty,$

whether $T_{max}$ is finite or infinite (recall $s_{c}>\tfrac{1}{4}$). Thus by
Remark 1.8, we see

$\lim_{t\to T_{max}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}=0\indent\text{for any }N>0.$ (6.3)

Now, we let $I_{n}$ be a nested sequence of compact subintervals of
$[0,T_{max})$, each of which is a contiguous union of characteristics
intervals $J_{k}$. On each $I_{n}$, we will now apply Proposition 5.1;
specifically, for fixed $\eta,\eta_{0}>0$, we use the recurrence relation
(5.2), the estimate (5.1), and the hypothesis (6.2) to see

$\displaystyle A_{n}(N)$ $\displaystyle:=\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I_{n}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}\left(\int_{I_{n}}N(t)^{3-4s_{c}}\,dt\right)^{\frac{1}{2}}$
$\displaystyle\quad+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A_{n}(M)$
$\displaystyle\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}\bigg{(}\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt\bigg{)}^{\frac{1}{2}}$
$\displaystyle\quad+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A_{n}(M)$
$\displaystyle\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+B(\eta,\eta_{0})N^{2s_{c}-\frac{1}{2}}+\sum_{M>N/\eta_{0}}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}A_{n}(M).$

Arguing as we did to obtain (5.1), we conclude

$A_{n}(N)\lesssim_{u}\inf_{t\in I_{n}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}+N^{2s_{c}-\frac{1}{2}}.$

Letting $n\to\infty$ and using (6.3) then gives

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}([0,T_{max})\times\mathbb{R}^{d})}\lesssim_{u}N^{2s_{c}-\frac{1}{2}}\indent\text{for
all }N>0.$ (6.4)

We now claim that (6.4) implies

######  Lemma 6.2.

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim_{u}N^{2s_{c}-\frac{1}{2}}\indent\text{for
all }N>0.$ (6.5)

###### Proof of Lemma 6.2.

Fix $N>0$; we first use Proposition 1.14 and Strichartz to estimate

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim\big{\|}|\nabla|^{s_{c}}P_{\leq
N}(|u|^{p}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}([0,T_{max})\times\mathbb{R}^{d})}.$
(6.6)

To proceed, we decompose the nonlinearity and estimate the individual pieces;
as before, the particular decomposition we use depends on the ambient
dimension. In the estimates that follow, spacetime norms will be taken over
$[0,T_{max})\times\mathbb{R}^{d}$.

Case 1. When $d=3$, we decompose

$|u|^{p}u=|u|^{p-2}\bar{u}u_{\leq
N}^{2}+(|u|^{p-2}\bar{u}u_{>N}+2|u|^{p-2}\bar{u}u_{\leq N})u_{>N}.$

We can use Hölder, the fractional product rule, fractional chain rule, Sobolev
embedding, interpolation, and (6.4) to estimate the contribution of the first
piece as follows:

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

To estimate the contribution of the second piece, we denote

$G=|u|^{p-2}\bar{u}u_{>N}+2|u|^{p-2}\bar{u}u_{\leq N}$

and use Bernstein, Hölder, Lemma 2.6, and (6.4) to see

$\displaystyle\big{\|}|\nabla|^{s_{c}}P_{\leq N}$
$\displaystyle(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim
N^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\big{\|}|\nabla|^{-\frac{1}{2}s_{c}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{3}{2}s_{c}}\big{\|}|\nabla|^{s_{c}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{\frac{1}{2}s_{c}}G\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{12p}{11p-4}}}N^{2s_{c}-\frac{1}{2}}.$
(6.7)

A few applications of the fractional product rule, fractional chain rule, and
Sobolev embedding give

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

so that continuing from (6.7), we get

$\big{\|}|\nabla|^{s_{c}}P_{\leq
N}\big{(}(|u|^{p-2}\bar{u}u_{>N}+2|u|^{p-2}\bar{u}u_{\leq
N})u_{>N}\big{)}\big{\|}_{L_{t}^{2}L_{x}^{6/5}}\lesssim_{u}N^{2s_{c}-1/2}.$

Thus we see that the claim holds in this first case.

Case 2. When $d\in\\{4,5\\}$, we decompose

$|u|^{p}u=|u|^{p}u_{\leq N}+|u|^{p}u_{>N}.$

We employ Hölder, the fractional product rule, the fractional chain rule,
Sobolev embedding, and (6.4) to estimate the contribution of the first piece
as follows:

$\displaystyle\big{\|}|\nabla$ $\displaystyle|^{s_{c}}P_{\leq
N}(|u|^{p}u_{\leq N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}|u|^{p}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2dp}{p(d+4)-4}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{dp}{2-p}}}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}+N^{2s_{c}-\frac{1}{2}}$
$\displaystyle\lesssim_{u}N^{2s_{c}-\frac{1}{2}}.$

For the second piece, we use Hölder, Bernstein, Lemma 2.6, the fractional
chain rule, and Sobolev embedding to see

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

Thus we see that the claim holds in this second case, completing the proof of
Lemma 6.2. ∎

We now wish to use (6.5) to prove

######  Lemma 6.3.

$u\in
L_{t}^{\infty}\dot{H}_{x}^{-\varepsilon}([0,T_{max})\times\mathbb{R}^{d})\indent\text{for
some }\varepsilon>0.$

###### Proof of Lemma 6.3.

For $s_{c}>\tfrac{1}{2},$ this is easy; indeed, choosing $\varepsilon>0$ such
that $s_{c}-\tfrac{1}{2}-\varepsilon>0$, we can use Bernstein and (6.5) to see

$\displaystyle\big{\|}|\nabla|^{-\varepsilon}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}N^{-s_{c}-\varepsilon}\big{\|}|\nabla|^{s_{c}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}N^{-s_{c}-\varepsilon}\big{\|}|\nabla|^{s_{c}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq
1}N^{-s_{c}-\varepsilon}N^{2s_{c}-\frac{1}{2}}+1$
$\displaystyle\lesssim_{u}1.$

When $s_{c}=\tfrac{1}{2}$ (that is, $p=\tfrac{4}{d-1}$), we need to work a bit
harder. To begin, we note that by Bernstein and (6.5), we have

$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}N^{-\frac{1}{10}}\big{\|}|\nabla|^{\frac{1}{2}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}N^{-\frac{1}{10}}\big{\|}|\nabla|^{\frac{1}{2}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq 1}N^{\frac{2}{5}}+1$
$\displaystyle\lesssim_{u}1.$ (6.8)

We wish to show that in fact, we have the more quantitative statement

$\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim_{u}N^{\frac{1}{2}}\indent\text{for
all }N>0.$ (6.9)

Once we have established (6.9), we can complete the proof of Lemma 6.3 as
follows: choosing $0<\varepsilon<\tfrac{1}{10}$, we use Bernstein, (6.8), and
(6.9) to estimate

$\displaystyle\big{\|}|\nabla|^{-\varepsilon}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}N^{-\frac{2}{5}-\varepsilon}\big{\|}|\nabla|^{\frac{2}{5}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}N^{-\frac{2}{5}-\varepsilon}\big{\|}|\nabla|^{\frac{2}{5}}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq 1}N^{\frac{1}{10}-\varepsilon}+1$
$\displaystyle\lesssim_{u}1.$

Thus, to complete the proof of Lemma 6.3, it remains to establish (6.9). We
begin by fixing $N>0$. The proof of (6.9) will be a second iteration of the
arguments that gave (6.5), this time using (6.8) as additional input.

We first use (6.8) (and the uniqueness of weak limits) to see that the no-
waste Duhamel formula (Proposition 1.14) also holds in the weak
$\dot{H}_{x}^{\frac{2}{5}}$ topology; thus, using Strichartz as well, we can
estimate

$\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}\lesssim\big{\|}|\nabla|^{\frac{2}{5}}P_{\leq
N}(|u|^{\frac{4}{d-1}}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}([0,T_{max})\times\mathbb{R}^{d})}.$

Once again, we decompose the nonlinearity, and again the decomposition depends
on the ambient dimension. The estimates that follow will be very similar in
spirit to the estimates that gave (6.5); all estimates will be taken over
$[0,T_{max}).$

Case 1. When $d=3$, we decompose

$|u|^{2}u=\bar{u}u_{\leq N}^{2}+(\bar{u}u_{>N}+2\bar{u}u_{\leq N})u_{>N}.$

We estimate the first piece as follows: by Hölder, the fractional product
rule, the fractional chain rule, Sobolev embedding, interpolation, (6.4) and
(6.8),

$\displaystyle\big{\|}$ $\displaystyle|\nabla|^{\frac{2}{5}}P_{\leq
N}(\bar{u}u_{\leq N}^{2})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}{u}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{4}L_{x}^{6}}^{2}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{30/11}}\big{\|}|\nabla|^{\frac{2}{5}}(u_{\leq
N})^{2}\big{\|}_{L_{t}^{2}L_{x}^{15/7}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{4}L_{x}^{3}}^{2}+\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{3}}\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{15/2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\quad+\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{6}}$ $\displaystyle\lesssim_{u}N^{\frac{1}{2}}.$

For the second piece, we first let $G:=\bar{u}u_{>N}+2\bar{u}u_{\leq N},$ and
use Bernstein, Hölder, Lemma 2.6, Sobolev embedding, and (6.4) to see

$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}P_{\leq
N}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$ $\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{-\frac{2}{5}}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}$
$\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}\big{\|}|\nabla|^{-\frac{2}{5}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{15/2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{15/2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{1}{2}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{6}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}N^{\frac{1}{2}}.$
(6.10)

A few applications of the fractional product rule, Sobolev embedding, and
(6.8) give

$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}G\big{\|}_{L_{t}^{\infty}L_{x}^{6/5}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{3}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}1,$

so that (6.10) becomes

$\big{\|}|\nabla|^{\frac{2}{5}}P_{\leq
N}(Gu_{>N})\big{\|}_{L_{t}^{2}L_{x}^{6/5}}\lesssim_{u}N^{\frac{1}{2}}.$

We see that (6.9) holds in this first case.

Case 2. When $d\in\\{4,5\\}$, we decompose

$|u|^{\frac{4}{d-1}}u=|u|^{\frac{4}{d-1}}u_{\leq
N}+|u|^{\frac{4}{d-1}}u_{>N}.$

We first note that interpolating between $u\in
L_{t}^{\infty}\dot{H}_{x}^{\frac{2}{5}}$ and $u\in
L_{t}^{\infty}\dot{H}_{x}^{\frac{1}{2}},$ we have

$\displaystyle u\in L_{t}^{\infty}\dot{H}_{x}^{\frac{21-d}{40}}.$ (6.11)

Thus, to estimate the contribution of the first piece, we can use Hölder, the
fractional product rule, the fractional chain rule, Sobolev embedding, (6.4),
(6.5), and (6.11) to see

$\displaystyle\big{\|}$
$\displaystyle|\nabla|^{\frac{2}{5}}(|u|^{\frac{4}{d-1}}u_{\leq
N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{\frac{2}{5}}|u|^{\frac{4}{d-1}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{5}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-3}}}+\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{40d}{21(d-1)}}}^{\frac{4}{d-1}}\big{\|}|\nabla|^{\frac{2}{5}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{10d}{5d-11}}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{d-1}}}^{\frac{5-d}{d-1}}\\!\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}\\!+\\!\big{\|}|\nabla|^{\frac{21-d}{40}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{4}{d-1}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\frac{5-d}{d-1}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}N^{\frac{1}{2}}+N^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}.$

For the second piece, we use Bernstein, Hölder, Lemma 2.6, the fractional
chain rule, Sobolev embedding, (6.4), (6.5), and (6.8) to see

$\displaystyle\big{\|}|\nabla|^{\frac{2}{5}}$ $\displaystyle P_{\leq
N}(|u|^{\frac{4}{d-1}}u)\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{-\frac{2}{5}}(|u|^{\frac{4}{d-1}}u_{\geq
N})\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$ $\displaystyle\lesssim
N^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}|u|^{\frac{4}{d-1}}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{5}}}\big{\|}|\nabla|^{-\frac{2}{5}}u_{>N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{10d}{5d-11}}}$
$\displaystyle\lesssim\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{2d}{d-1}}}^{\frac{5-d}{d-1}}\big{\|}|\nabla|^{\frac{2}{5}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{2}{5}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{10d}{5d-11}}}$
$\displaystyle\lesssim_{u}\sum_{M>N}\left(\tfrac{N}{M}\right)^{\frac{4}{5}}\big{\|}|\nabla|^{\frac{1}{2}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}.$

Thus (6.9) holds in this second case. This completes the proof of Lemma 6.3.∎

With Lemma 6.3 at hand, we are ready to complete the proof of Theorem 6.1. Fix
$t\in[0,T_{max})$ and $\eta>0$. By Remark 1.8, we may find $c(\eta)>0$ so that

$\int_{|\xi|\leq
c(\eta)N(t)}|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta.$

Interpolating with $u\in L_{t}^{\infty}\dot{H}_{x}^{-\varepsilon},$ we get

$\int_{|\xi|\leq c(\eta)N(t)}|\widehat{u}(t,\xi)|^{2}\
d\xi\lesssim_{u}\eta^{\frac{\varepsilon}{s_{c}+\varepsilon}}.$

On the other hand, we have

$\int_{|\xi|\geq
c(\eta)N(t)}|\widehat{u}(\xi,t)|^{2}\,d\xi\leq(c(\eta)N(t))^{-2s_{c}}\int|\xi|^{2s_{c}}|\widehat{u}(t,\xi)|^{2}\,d\xi\lesssim_{u}(c(\eta)N(t))^{-2s_{c}}.$

Adding these last estimates and using Plancherel, we conclude that for all
$t\in[0,T_{max}),$ we have

$0\leq{M}(u(t)):=\int|u(t,x)|^{2}\,dx\lesssim_{u}\eta^{\frac{\varepsilon}{s_{c}+\varepsilon}}+(c(\eta)N(t))^{-2s_{c}}.$

Thus, recalling $\lim_{t\to T_{max}}N(t)=\infty$, we can conclude that for all
$\eta>0$, we may find $t_{0}\in[0,T_{max})$ so that for all
$t\in(t_{0},T_{max}),$ we have ${M}(u(t))\leq\eta.$ But by conservation of
mass, ${M}(u(t))\equiv{M}(u_{0}),$ and so we find that ${M}(u_{0})\leq\eta$
for all $\eta>0$. Of course, this gives $u\equiv 0$, which contradicts that
$u$ blows up (cf. (6.1)). ∎

######  Remark 6.4.

We have omitted the case $(d,s_{c})=(5,\tfrac{1}{2})$ from Theorem 6.1 only
because we omitted this case from the long-time Strichartz estimate,
Proposition 5.1. Of course, as remarked in the proof of Proposition 5.1, the
long-time Strichartz estimates continue to hold when
$(d,s_{c})=(5,\tfrac{1}{2})$; thus we see that Theorem 6.1 holds in this case
as well.

##  7\. The frequency-localized interaction Morawetz inequality

In this section, we prove spacetime bounds for the high-frequency portions of
almost periodic solutions to (1.1); these bounds can be used to preclude the
existence of quasi-solitons (Section 8). As we will see, establishing these
bounds will lead to the most non-trivial restrictions on the set of
$(d,s_{c})$ to which our main theorem (Theorem 1.3) applies; see the proof
below for a more detailed discussion. The main result of this section is the
following

######  Proposition 7.1 (Frequency-localized interaction Morawetz inquality).

Let $(d,s_{c})$ satisfy (1.4). Suppose
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ is an almost periodic
solution to (1.1) such that $N(t)\equiv N_{k}\geq 1$ on each characteristic
subinterval $J_{k}\subset[0,T_{max})$, and let $I\subset[0,T_{max})$ be a
compact time interval, which is a union of contiguous subintervals $J_{k}$.
Then for any $\eta>0$, there exists $N_{0}=N_{0}(\eta)$ such that for any
$N\leq N_{0}$, we have

$-\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|u_{\geq
N}(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u_{\geq
N}(t,x)|^{2}\,dx\,dy\,dt\lesssim_{u}\eta(N^{1-4s_{c}}+K),$ (7.1)

where $K:=\int_{I}N(t)^{3-4s_{c}}\,dt.$ Furthermore, $N_{0}$ and the implicit
constants above do not depend on $I$.

Before we begin the proof of Proposition 7.1, we recall a general form of the
interaction Morawetz inequality, introduced originally in [14] (for more
discussion, see also [31] and the references cited therein). We will
essentially follow the presentation in [55, Section 5].

For a fixed function $a:\mathbb{R}^{d}\to\mathbb{R}$ and $\varphi$ solving
$(i\partial_{t}+\Delta)\varphi=\mathcal{N}$, we define the interaction
Morawetz action by

$M(t)=2\,\text{Im}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}a_{k}(x-y)(\varphi_{k}\bar{\varphi})(t,x)\,dx\,dy,$

where subscripts denote spatial derivatives and repeated indices are summed.
If we define the mass bracket

$\\{f,g\\}_{m}:=\text{Im}(f\bar{g})$

and the momentum bracket

$\\{f,g\\}_{\mathcal{P}}:=\text{Re}(f\nabla\bar{g}-g\nabla\bar{f}),$

then one can show

$\displaystyle\partial_{t}M(t)=$
$\displaystyle-\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}a_{jjkk}(x-y)|\varphi(t,x)|^{2}\,dx\,dy$
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}4a_{jk}(x-y)\text{Re}(\bar{\varphi}_{j}\varphi_{k})(t,x)\,dx\,dy$
(7.2) $\displaystyle-\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\indent
2\,\text{Im}(\bar{\varphi}\varphi_{k})(t,y)a_{jk}(x-y)2\,\text{Im}(\bar{\varphi}\varphi_{j})(t,x)\,dx\,dy$
(7.3)
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}2\\{\mathcal{N},\varphi\\}_{m}(t,y)a_{j}(x-y)\
2\,\text{Im}(\bar{\varphi}\varphi_{j})(t,x)\,dx\,dy$
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\
2\,\nabla a(x-y)\cdot\\{N,\varphi\\}_{\mathcal{P}}(t,x)\,dx\,dy.$

To prove Proposition 7.1, we will use $a(x)=|x|$. Note that in this case, we
have

$\left\\{\begin{array}[]{ll}a_{j}(x)=\tfrac{x_{j}}{|x|},\\\ \\\
a_{jk}(x)=\tfrac{\delta_{jk}}{|x|}-\tfrac{x_{j}x_{k}}{|x|^{3}},\\\ \\\ \Delta
a(x)=\tfrac{d-1}{|x|},\\\ \\\ \Delta\Delta
a(x)=-(d-1)\Delta(\tfrac{1}{|x|}).\end{array}\right.$

For this choice of $a$, one can also show $\eqref{mor2}+\eqref{mor3}\geq 0$
(for details, see for example [55, Lemma 5.4]). Thus, integrating
$\partial_{t}M$ over $I$, we arrive at the following

######  Lemma 7.2 (Interaction Morawetz inequality).

$\displaystyle-\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|\varphi(t,x)|^{2}\,dx\,dy\,dt$
$\displaystyle+\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\tfrac{x-y}{|x-y|}\cdot\left\\{\mathcal{N},\varphi\right\\}_{\mathcal{P}}(t,x)\,dx\,dy\,dt$
$\displaystyle\ \ \ \lesssim\sup_{t\in
I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi(t,y)|^{2}\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(t,x)\bar{\varphi}(t,x)\,dx\,dy$
$\displaystyle\ \ \ \
+\bigg{|}\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\left\\{\mathcal{N},\varphi\right\\}_{m}(t,y)\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(t,x)\bar{\varphi}(t,x)\,dx\,dy\,dt\
\bigg{|}.$

To prove Proposition 7.1, we will apply this estimate with $\varphi=u_{\geq
N}$, with $N$ chosen small enough to capture ‘most’ of the solution. To make
this idea more precise, we first need to record the following corollary of
Proposition 5.1.

######  Corollary 7.3 (Low and high frequencies control).

Let $(d,s_{c})$ satisfy (1.5), and 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 continuous subintervals $J_{k}$,
and for any frequency $N>0$, we have

$\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}\lesssim_{u}N^{-s_{c}}(1+N^{4s_{c}-1}K)^{\frac{1}{q}}$
(7.4)

for all $\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$ with $q>4-\tfrac{2p}{dp-4},$
where $K:=\int_{I}N(t)^{3-4s_{c}}\,dt.$

Moreover, for any $\eta>0$, there exists $N_{0}=N_{0}(\eta)$ such that for all
$N\leq N_{0}$, we have

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}\lesssim_{u}\eta(1+N^{4s_{c}-1}K)^{\frac{1}{q}}$
(7.5)

for all $\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$ with $q\geq 2$.

Furthermore, $N_{0}$ and the implicit constants in (7.4) and (7.5) do not
depend on $I$.

###### Proof of Corollary 7.3.

We first show (7.4). For fixed $\alpha>s_{c}-\tfrac{1}{2}$, we can use
Bernstein and (5.1) to see

$\displaystyle\big{\|}|\nabla|^{-\alpha}u_{\geq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\sum_{M\geq N}M^{-\alpha-
s_{c}}\big{\|}|\nabla|^{s_{c}}u_{M}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{u}\sum_{M\geq N}M^{-\alpha-
s_{c}}(1+M^{2s_{c}-\frac{1}{2}}K^{\frac{1}{2}})$
$\displaystyle\lesssim_{u}N^{-\alpha-s_{c}}(1+N^{4s_{c}-1}K)^{\frac{1}{2}}.$
(7.6)

Now, take $(q,r)$ with $2<q\leq\infty$ and
$\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$, and define
$\alpha=\tfrac{(q-2)(dp-4)}{4p}.$ Notice that $\alpha>s_{c}-\tfrac{1}{2}$
exactly when $q>4-\tfrac{2p}{dp-4}.$ Thus, in this case, we get by
interpolation and (7.6) that

$\displaystyle\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{-\alpha}u_{\geq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}(I\times\mathbb{R}^{d})}^{\frac{2}{q}}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}^{1-\frac{2}{q}}$
$\displaystyle\lesssim_{u}\left[N^{-\frac{qs_{c}}{2}}(1+N^{4s_{c}-1}K)^{\frac{1}{2}}\right]^{\frac{2}{q}},$

which gives (7.4). As for (7.5), we first note that since $\inf_{t\in
I}N(t)\geq 1$, for any $\eta>0$ we may find $N_{0}(\eta)$ so that

$\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}(I\times\mathbb{R}^{d})}\leq\eta$

for all $N\leq N_{0}$ (cf. Remark 1.8). The estimate (7.5) then follows by
interpolating with (5.1). ∎

We are now ready for the

###### Proof of Proposition 7.1.

Take $I\subset[0,T_{max})$, a compact time interval, which is a contiguous
union of subintervals $J_{k}$, and let $K:=\int_{I}N(t)^{3-4s_{c}}\,dt$.
Throughout the proof, all spacetime norms will be taken over
$I\times\mathbb{R}^{d}$.

Fix $\eta>0$, and choose $N_{0}=N_{0}(\eta)$ small enough that (7.5) holds;
recall that (7.4) holds without any restriction on $N$. Next, we claim that
for $N_{0}$ possibly even smaller, we can guarantee that for $N\leq N_{0}$, we
have

$\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}\eta^{10}N^{-s_{c}}$ (7.7)

and

$\big{\|}|\nabla|^{1-s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}\eta^{10}N^{1-2s_{c}}.$ (7.8)

Indeed, by Remark 1.8 and the fact that $\inf_{t\in I}N(t)\geq 1$, we may find
$c(\eta)>0$ so that

$\big{\|}|\nabla|^{s_{c}}u_{\leq
c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\leq\eta^{10};$

combining this inequality with Bernstein, we get

$\displaystyle N^{s_{c}}\big{\|}u_{\geq N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim N^{s_{c}}\big{\|}u_{N\leq\cdot\leq
c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+N^{s_{c}}\big{\|}u_{>c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\tfrac{N^{s_{c}}}{c(\eta)^{s_{c}}}\big{\|}|\nabla|^{s_{c}}u_{>c(\eta)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\eta^{10}+N^{s_{c}}.$

Thus, taking $N$ sufficiently small, we recover (7.7). A similar argument
yields (7.8).

Next, we record the following inequality that will be useful below:

$\sup_{y\in\mathbb{R}^{d}}\bigg{|}\int_{\mathbb{R}^{d}}\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(x)\bar{\varphi}(x)\
dx\bigg{|}\lesssim\big{\|}|\nabla|^{s}\varphi\big{\|}_{2}\big{\|}|\nabla|^{1-s}\varphi\big{\|}_{2}$
(7.9)

for $0\leq s\leq 1$. Indeed, for fixed $y\in\mathbb{R}^{d}$, we can first
write

$\displaystyle\bigg{|}\int_{\mathbb{R}^{d}}\tfrac{x-y}{|x-y|}\cdot\nabla\varphi(x)\bar{\varphi}(x)\
dx\bigg{|}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s}\tfrac{x-y}{|x-y|}\varphi\big{\|}_{2}\big{\|}|\nabla|^{-s}\nabla\varphi\big{\|}_{2}$
$\displaystyle\sim\big{\|}|\nabla|^{s}\tfrac{x-y}{|x-y|}\varphi\big{\|}_{2}\big{\|}|\nabla|^{1-s}\varphi\big{\|}_{2}.$

Thus, to prove (7.9), we need to see that the operator
$|\nabla|^{s}\tfrac{x-y}{|x-y|}|\nabla|^{-s}$ is bounded on $L_{x}^{2}$
(uniformly in $y$). When $s=0$, this is clear. When $s=1$, this follows from
the chain rule, Hardy’s inequality, and the boundedness of Riesz transforms.
The general case then follows from complex interpolation.

We now wish to apply the interaction Morawetz inequality (Lemma 7.2) with
$\varphi=u_{\geq N}$ and $\mathcal{N}=P_{\geq N}(|u|^{p}u)$, with $N\leq
N_{0}$. Together with (7.7), (7.8), (7.9), Bernstein, and the fact that $u\in
L_{t}^{\infty}\dot{H}_{x}^{s_{c}}(I\times\mathbb{R}^{d})$, an application of
Lemma 7.2 gives

$\displaystyle-$ $\displaystyle\iiint|u_{\geq
N}(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u_{\geq N}(t,x)|^{2}\,dx\,dy\,dt$
$\displaystyle+\quad\\!\\!\\!\\!\\!\\!\iiint|u_{\geq
N}(t,y)|^{2}\tfrac{x-y}{|x-y|}\cdot\\{P_{\geq N}(|u|^{p}u),u_{\geq
N}\\}_{\mathcal{P}}(t,x)\,dx\,dy\,dt$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}|\nabla|^{1/2}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}$
$\displaystyle\quad+\big{\|}\\{P_{\geq N}(|u|^{p}u),u_{\geq
N}\\}_{m}\big{\|}_{L_{t,x}^{1}}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}|\nabla|^{1-s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\eta^{20}N^{1-4s_{c}}+\eta^{10}N^{1-2s_{c}}\big{\|}\\{P_{\geq
N}(|u|^{p}u),u_{\geq N}\\}_{m}\big{\|}_{L_{t,x}^{1}}.$ (7.10)

Thus, to prove Proposition 7.1, we need to get sufficient control over the
mass and momentum bracket terms appearing above.

To begin, we consider the contribution of the momentum bracket term. We can
write

$\displaystyle\\{P_{\geq N}$ $\displaystyle(|u|^{p}u),u_{\geq
N}\\}_{\mathcal{P}}$ $\displaystyle=\\{|u|^{p}u,u\\}_{\mathcal{P}}-\\{|u_{\leq
N}|^{p}u_{\leq N},u_{\leq N}\\}_{\mathcal{P}}$
$\displaystyle\quad\quad-\\{|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N},u_{\leq
N}\\}_{\mathcal{P}}-\\{P_{\leq N}(|u|^{p}u),u_{\geq N}\\}_{\mathcal{P}}$
$\displaystyle=-\tfrac{p}{p+2}\nabla(|u|^{p+2}-|u_{\leq
N}|^{p+2})-\\{|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N},u_{\leq N}\\}_{\mathcal{P}}$
$\displaystyle\quad\quad-\\{P_{\leq N}(|u|^{p}u),u_{\geq N}\\}_{\mathcal{P}}$
$\displaystyle=:I+II+III.$

After an integration by parts, we see that term $I$ contributes to the left-
hand side of (7.10) a multiple of

$\displaystyle\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq
N}(t,x)|^{p+2}}{|x-y|}\,dx\,dy\,dt$ $\displaystyle-\iiint\frac{|u_{\geq
N}(t,y)|^{2}(|u|^{p+2}-|u_{\geq N}|^{p+2}-|u_{\leq
N}|^{p+2})(t,x)}{|x-y|}\,dx\,dy\,dt.$

For term $II$, we use
$\\{f,g\\}_{\mathcal{P}}=\nabla\text{\O}(fg)+\text{\O}(f\nabla g);$ when the
derivative hits the product, we integrate by parts, while for the second term
we simply bring absolute values inside the integral. In this way, we find that
term $II$ contributes to the right-hand side of (7.10) a multiple of

$\displaystyle\iiint\frac{|u_{\geq N}(t,y)|^{2}|(|u|^{p}u-|u_{\leq
N}|^{p}u_{\leq N})u_{\leq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$
$\displaystyle+\iiint|u_{\geq N}(t,y)|^{2}|(|u|^{p}u-|u_{\leq N}|^{p}u_{\leq
N})(t,x)|\ |\nabla u_{\leq N}(t,x)|\,dx\,dy\,dt.$

Finally, for term $III$, we integrate by parts when the derivative falls on
$u_{\geq N}$; in this way, we see that term $III$ contributes to the right-
hand side of (7.10) a multiple of

$\displaystyle\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|P_{\leq
N}(|u|^{p}u)(t,x)|}{|x-y|}\,dx\,dy\,dt$ $\displaystyle+\iiint|u_{\geq
N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|\nabla P_{\leq
N}(|u|^{p}u)(t,x)|\,dx\,dy\,dt.$

We next consider the mass bracket term in (7.10). Exploiting the fact that

$\\{|u_{\geq N}|^{p}u_{\geq N},u_{\geq N}\\}_{m}=0,$

we can write

$\displaystyle\\{P_{\geq N}(|u|^{p}u),u_{\geq N}\\}_{m}$
$\displaystyle=\\{P_{\geq N}(|u|^{p}u)-|u_{\geq N}|^{p}u_{\geq N},u_{\geq
N}\\}_{m}$ $\displaystyle=\\{P_{\geq N}(|u|^{p}u-|u_{\geq N}|^{p}u_{\geq
N}-|u_{\leq N}|^{p}u_{\leq N}),u_{\geq N}\\}_{m}$ $\displaystyle\ +\\{P_{\geq
N}(|u_{\leq N}|^{p}u_{\leq N}),u_{\geq N}\\}_{m}-\\{P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N}),u_{\geq N}\\}_{m}.$

We will now collect the contributions of the mass and momentum bracket terms
and insert them back into (7.10). We will also make use of the pointwise
inequalities

$\displaystyle\big{|}|f+g|^{p}(f+g)-|f|^{p}f\big{|}\lesssim|g|^{p+1}+|g|\
|f|^{p},$
$\displaystyle\big{|}|f+g|^{p+2}-|f|^{p+2}-|g|^{p+2}\big{|}\lesssim|f|\
|g|^{p+1}+|f|^{p+1}|g|.$

In this way, (7.10) becomes

$\displaystyle-\iiint|u_{\geq
N}(t,y)|^{2}\Delta(\tfrac{1}{|\cdot|})(x-y)|u_{\geq N}(t,x)|^{2}\,dx\,dy\,dt$
(7.11) $\displaystyle\quad+\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq
N}(t,x)|^{p+2}}{|x-y|}\,dx\,dy\,dt$
$\displaystyle\lesssim_{u}\eta^{20}N^{1-4s_{c}}$ (7.12)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}|u_{\leq N}|^{p}u_{\geq
N}^{2}\big{\|}_{L_{t,x}^{1}}$ (7.13)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}|u_{\geq N}|^{p+1}u_{\leq
N}\big{\|}_{L_{t,x}^{1}}$ (7.14)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}P_{\geq N}(|u_{\leq N}|^{p}u_{\leq
N})u_{\geq N}\big{\|}_{L_{t,x}^{1}}$ (7.15)
$\displaystyle+\eta^{10}N^{1-2s_{c}}\big{\|}P_{\leq N}(|u_{\geq N}|^{p}u_{\geq
N})u_{\geq N}\big{\|}_{L_{t,x}^{1}}$ (7.16)
$\displaystyle+\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|u_{\leq
N}(t,x)|^{p+1}}{|x-y|}\,dx\,dy\,dt$ (7.17) $\displaystyle+\iiint\frac{|u_{\geq
N}(t,y)|^{2}|u_{\geq N}(t,x)|^{p+1}|u_{\leq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$
(7.18) $\displaystyle+\iiint\frac{|u_{\geq N}(t,y)|^{2}|P_{\leq
N}(|u|^{p}u)(t,x)|\,|u_{\geq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$ (7.19)
$\displaystyle+\iiint|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|u_{\leq
N}(t,x)|^{p}|\nabla u_{\leq N}(t,x)|\,dx\,dy\,dt$ (7.20)
$\displaystyle+\iiint|u_{\geq N}(t,y)|^{2}|u_{\geq N}(t,x)|^{p+1}|\nabla
u_{\leq N}(t,x)|\,dx\,dy\,dt$ (7.21) $\displaystyle+\iiint|u_{\geq
N}(t,y)|^{2}|u_{\geq N}(t,x)|\,|\nabla P_{\leq
N}(|u|^{p}u)(t,x)|\,dx\,dy\,dt.$ (7.22)

To complete the proof of Proposition 7.1, we need to show that the error terms
(7.12) through (7.22) are acceptable, in the sense that they can be controlled
by $\eta(N^{1-4s_{c}}+K).$ Clearly, (7.12) is acceptable.

Next, we consider (7.13). Using Hölder, Sobolev embedding, (7.4), and (7.5),
we get

$\displaystyle\big{\|}|u_{\leq N}|^{p}u_{\geq N}^{2}\big{\|}_{L_{t,x}^{1}}$
$\displaystyle\lesssim\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{dp}}^{p}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}^{2}$
$\displaystyle\lesssim\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{\frac{2dp}{dp-2}}}^{p}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}^{2}$
$\displaystyle\lesssim_{u}\eta^{p}N^{-2s_{c}}(1+N^{4s_{c}-1}K),$

which renders (7.13) acceptable.

We now turn to (7.14). For this term, we can again use Hölder, Sobolev
embedding, (7.4), and (7.5) to see

$\displaystyle\big{\|}|u_{\geq N}|^{p+1}u_{\leq N}\big{\|}_{L_{t,x}^{1}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$ $\displaystyle\lesssim_{u}\eta
N^{-2s_{c}}(1+N^{4s_{c}-1}K).$

Thus this term is acceptable as well. Before proceeding, however, we note that
it is this term that has forced us to exclude the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ from this paper; we postpone
further discussion until Remark 7.4 below.

We next turn to (7.15); using Hölder, Bernstein, the fractional chain rule,
Sobolev embedding, (7.4), and (7.5), we see

$\displaystyle\big{\|}P_{\geq N}$ $\displaystyle(|u_{\leq N}|^{p}u_{\leq
N})u_{\geq N}\big{\|}_{L_{t,x}^{1}}$ $\displaystyle\lesssim
N^{-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}|\nabla|^{s_{c}}(|u_{\leq
N}|^{p}u_{\leq N})\big{\|}_{L_{t}^{\frac{4}{3}}L_{x}^{\frac{2d}{d+1}}}$
$\displaystyle\lesssim N^{-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{2dp}{3}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$ $\displaystyle\lesssim
N^{-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{4p}L_{x}^{\frac{2dp}{dp-1}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\eta^{p+1}N^{-2s_{c}}(1+N^{4s_{c}-1}K),$

so that (7.15) is also acceptable.

For the final term originating from the mass bracket, (7.16), we use Hölder,
Bernstein, Sobolev embedding, (7.4), and (7.7) to see

$\displaystyle\big{\|}P_{\leq N}(|u_{\geq N}|^{p}u_{\geq N})u_{\geq
N}\big{\|}_{L_{t,x}^{1}}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d+4}}}$
$\displaystyle\lesssim N^{s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|u_{\geq N}|^{p}u_{\geq
N}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3dp}{3dp+2p-6}}}$
$\displaystyle\lesssim N^{s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$ $\displaystyle\lesssim
N^{s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{3}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}N^{-2s_{c}}(1+N^{4s_{c}-1}K),$

which shows that (7.16) is acceptable.

We now turn to the terms originating from the momentum bracket. First,
consider (7.17). By Hölder, Hardy–Littlewood–Sobolev, Sobolev embedding,
Bernstein, (7.4), (7.5), and (7.7), we can estimate

$\displaystyle\eqref{error 6}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{3d}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$
$\displaystyle\quad\times\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{3dp}{6-p}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-8}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim\big{\|}|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3d}{3d-2}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$
$\displaystyle\quad\times\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}N^{1-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}$
$\displaystyle\lesssim_{u}\eta^{12}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

so that (7.17) is acceptable.

For (7.18), we consider two cases. If $|u_{\leq N}|\leq 10^{-100}|u_{\geq
N}|,$ then we can absorb this term into the left-hand side of the inequality,
provided we can show

$\iiint\frac{|u_{\geq N}(t,y)|^{2}|u_{\geq
N}(t,x)|^{p+2}}{|x-y|}\,dx\,dy\,dt<\infty.$ (7.23)

On the other hand, if $|u_{\geq N}|\leq 10^{100}|u_{\leq N}|$, then we are
back in the situation of (7.17), which we have already handled. Thus, to
render (7.18) acceptable, it remains to prove (7.23). To this end, we define

$\theta=\tfrac{4dp-16-3p}{2(dp-4)}\in(0,p+2),$

and use Hölder, Hardy–Littlewood–Sobolev, Sobolev embedding, Lemma 1.11, and
interpolation to estimate

$\displaystyle\text{LHS}\eqref{finiteness of bootstrap terms}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{4}L_{x}^{4d}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\theta}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\frac{2p(2dp-5)}{3(dp-4)}}L_{x}^{\frac{dp(2dp-5)}{dp+2}}}^{p+2-\theta}$
$\displaystyle\lesssim\big{\|}|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{4}L_{x}^{\frac{4d}{4d-3}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{\theta}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{\frac{2p(2dp-5)}{3(dp-4)}}L_{x}^{\frac{2dp(2dp-5)}{2(dp)^{2}-11dp+24}}}^{p+2-\theta}$
$\displaystyle\lesssim_{u}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{4d}{2d-3}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{1+\theta}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}\big{\|}|\nabla|^{1/4}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}N^{-s_{c}(1+\theta)}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}N^{1/4-s_{c}(2+\theta)}\big{\|}|\nabla|^{s_{c}}u_{\geq
N}\big{\|}_{L_{t}^{4}L_{x}^{\frac{2d}{d-1}}}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}N^{1-4s_{c}}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right)^{\frac{1}{4}+\frac{(p+2-\theta)(3(dp-4))}{2p(2dp-5)}}$
$\displaystyle\lesssim_{u}N^{1-4s_{c}}\left(1+\textstyle\int_{I}N(t)^{2}\,dt\right),$

which gives (7.23), and thereby shows that (7.18) is acceptable.

Next, we turn to (7.19). Denoting

$G=|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N}-|u_{\geq N}|^{p}u_{\geq N},$

we begin by writing

$\displaystyle\eqref{error 8}$ $\displaystyle\quad=\iiint\frac{|u_{\geq
N}(t,y)|^{2}|P_{\leq N}(|u_{\leq N}|^{p}u_{\leq N})(t,x)|\,|u_{\geq
N}(t,x)|}{|x-y|}\,dx\,dy\,dt$ (7.24)
$\displaystyle\quad\quad+\iiint\frac{|u_{\geq N}(t,y)|^{2}|P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})(t,x)|\,|u_{\geq N}(t,x)|}{|x-y|}\,dx\,dy\,dt$ (7.25)
$\displaystyle\quad\quad+\iiint\frac{|u_{\geq N}(t,\\!y)|^{2}|P_{\leq
N}G(t,x)|\,|u_{\geq N}(t,x)|}{|x-y|}\,dx\,dy\,dt.$ (7.26)

For (7.24), we can write

$\displaystyle\eqref{8a}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{3d}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{3dp}{6-p}}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-8}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{12}N^{1-4s_{c}}(1+N^{4s_{c}-1}K)$

by the same arguments that dealt with (7.17).

For (7.25), we can use Hölder, Hardy–Littlewood–Sobolev, Bernstein, Sobolev
embedding, (7.4), (7.5), and (7.7) to estimate

$\displaystyle\eqref{8b}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}\tfrac{1}{|x|}\ast(P_{\leq
N}(|u_{\geq N}|^{p}u_{\geq N})u_{\geq
N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3d}{2}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})u_{\geq
N}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3d}{3d-1}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\big{\|}P_{\leq N}(|u_{\geq N}|^{p}u_{\geq
N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d-2}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}N^{1+s_{c}}\big{\|}P_{\leq N}(|u_{\geq
N}|^{p}u_{\geq N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3dp}{3dp+2p-6}}}$
$\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{6}L_{x}^{\frac{6d}{3d-2}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}N^{1+s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{10}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

which renders (7.25) acceptable.

For (7.26), we first note

$|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N}-|u_{\geq N}|^{p}u_{\geq
N}=\text{\O}(u_{\geq N}u_{\leq N}|u|^{p-1}),$

so that using Hölder, Hardy–Littlewood–Sobolev, Bernstein, Sobolev embedding,
(7.4), (7.5), and (7.7), we get

$\displaystyle\eqref{8c}$
$\displaystyle\lesssim\big{\|}\tfrac{1}{|x|}\ast|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{3d}}\big{\|}P_{\leq N}(\text{\O}(u_{\geq
N}u_{\leq
N}|u|^{p-1}))\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d+2}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$ $\displaystyle\lesssim
N\big{\|}|u_{\geq
N}|^{2}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3d}{3d-2}}}\big{\|}\text{\O}(u_{\geq
N}u_{\leq
N}|u|^{p-1})\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d+8}}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$ $\displaystyle\lesssim
N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{21}N^{1-4s_{c}}(1+N^{4s_{c}-1}K).$

Thus (7.26), and so (7.19), is acceptable.

We now turn to (7.20). By Hölder, Sobolev embedding, Bernstein, (7.5), and
(7.7), we estimate

$\displaystyle\eqref{error 9}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{3}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{dp}}^{p}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$ $\displaystyle\lesssim
N^{1-s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{3}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{\frac{2dp}{dp-2}}}^{p}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\eta^{p+31}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

so that (7.20) is acceptable.

For (7.21), we use Hölder, Sobolev embedding, Bernstein, (7.4), (7.5), and
(7.7) to get

$\displaystyle\eqref{error 10}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}$
$\displaystyle\lesssim_{u}N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}$
$\displaystyle\lesssim_{u}\eta^{21}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

which renders (7.21) acceptable.

Finally, we consider (7.22). We begin by writing

$\displaystyle\eqref{error 11}\lesssim$ $\displaystyle\,\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq N}\nabla P_{\leq
N}(|u_{\leq N}|^{p}u_{\leq N})\big{\|}_{L_{t,x}^{1}}$ (7.27) $\displaystyle+$
$\displaystyle\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq N}\nabla P_{\leq
N}(|u_{\geq N}|^{p}u_{\geq N})\big{\|}_{L_{t,x}^{1}}$ (7.28) $\displaystyle+$
$\displaystyle\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq N}\nabla P_{\leq
N}(|u|^{p}u-|u_{\leq N}|^{p}u_{\leq N}-|u_{\geq N}|^{p}u_{\geq
N})\big{\|}_{L_{t,x}^{1}}.$ (7.29)

To begin, we use Hölder, the chain rule, and the arguments that gave (7.20) to
see

$\displaystyle\eqref{11a}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{3}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{2p}L_{x}^{dp}}^{p}\big{\|}\nabla u_{\leq
N}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}$
$\displaystyle\lesssim_{u}\eta^{p+31}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

so that (7.27) is acceptable.

For (7.28), we argue essentially as we did for (7.16). That is, we use Hölder,
Bernstein, Sobolev embedding, (7.4), and (7.7) to estimate

$\displaystyle\eqref{11b}$ $\displaystyle\lesssim\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}\nabla P_{\leq
N}(|u_{\geq N}|^{p}u_{\geq
N})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d+4}}}$
$\displaystyle\lesssim N^{1+s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|u_{\geq N}|^{p}u_{\geq
N}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{3dp}{3dp+2p-6}}}$
$\displaystyle\lesssim N^{1+s_{c}}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{3}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{20}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

which gives that (7.28) is acceptable.

For (7.29), we argue similarly to the case of (7.26). In particular, we use
Hölder, Bernstein, Sobolev embedding, (7.4), (7.5), and (7.7) to see

$\displaystyle\eqref{11c}$ $\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}\text{\O}(u_{\leq
N}u_{\geq N}|u|^{p-1})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{6d}{3d+4}}}$
$\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{3dp}{6-2p}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{p-1}$
$\displaystyle\lesssim N\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{2}\big{\|}u_{\geq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}^{2}\big{\|}|\nabla|^{s_{c}}u_{\leq
N}\big{\|}_{L_{t}^{3}L_{x}^{\frac{6d}{3d-4}}}\big{\|}|\nabla|^{s_{c}}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}^{p-1}$
$\displaystyle\lesssim_{u}\eta^{21}N^{1-4s_{c}}(1+N^{4s_{c}-1}K),$

which gives that (7.29). Collecting the estimates for (7.27), (7.28), and
(7.29), we see that (7.22) is acceptable. This completes the proof of
Proposition 7.1. ∎

######  Remark 7.4.

Let us discuss why (7.14) has forced us to exclude the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ from this paper. As one can see in
the proof above, in the cases we consider, this term is fairly harmless.
However, once $s_{c}>\tfrac{3}{4}$ in dimension $d=3$ (which corresponds to
$p>\tfrac{8}{3}$), this term becomes a problem; put simply, we end up with too
many copies of $u_{\geq N}$ to deal with.

This problem has already been encountered in the energy-critical setting
($s_{c}=1$) in dimension $d=3$; in this case, one can overcome the hurdle by
applying a spatial truncation to the weight $a$. One can refer to [15] for the
original argument, wherein spatial truncation is applied at various levels and
subsequently averaged. The authors of [36] revisit the result of [15] in the
context of minimal counterexamples; at this point in the argument, they choose
to work with a more carefully designed spatial truncation, which removes the
need for any subsequent averaging argument.

This discussion begs the question: why doesn’t spatial truncation work in our
setting? To answer this, we need to understand how spatial truncations affect
the argument that leads to Proposition 7.1. What we find is that spatial
truncations ruin the convexity properties of $a$ that made some of the terms
in the proof of Lemma 7.2 positive; thus, to establish Proposition 7.1 with a
further spatial truncation, we have to control additional error terms. It
turns out that one of these additional error terms requires uniform control
over $\big{\|}u\big{\|}_{L_{x}^{p+2}}$, while another requires uniform control
over $\big{\|}\nabla u\big{\|}_{L_{x}^{2}}$ (see [36, Lemma 6.5 and Lemma
6.6]). In the energy-critical case, one can use the conservation of energy to
push the argument through, while in our cases, we cannot proceed without some
significant new input. We have therefore abandoned the cases
$(d,s_{c})\in\\{3\\}\times(\tfrac{3}{4},1)$ in this paper.

For a further discussion of these issues, refer to [36], especially Remark 6.9
therein.

##  8\. The quasi-soliton 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 will
show that their existence is inconsistent with the frequency-localized
interaction Morawetz inequality (Proposition 7.1).

Before we begin, we note that

$-\Delta(\tfrac{1}{|x|})=\left\\{\begin{array}[]{ll}4\pi\delta&d=3\\\
\tfrac{d-3}{|x|^{3}}&d\geq 4.\end{array}\right.$

Thus, if $d=3$, the conclusion of Proposition 7.1 reads

$\int_{I}\int_{\mathbb{R}^{3}}|u_{\geq
N}(t,x)|^{4}\,dx\,dt\lesssim_{u}\eta(N^{1-4s_{c}}+K)$ (8.1)

for $N\leq N_{0}(\eta)$, while if $d\in\\{4,5\\}$, the conclusion of
Proposition 7.1 reads

$\int_{I}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{|u_{\geq
N}(t,x)|^{2}|u_{\geq
N}(t,y)|^{2}}{|x-y|^{3}}\,dx\,dy\,dt\lesssim_{u}\eta(N^{1-4s_{c}}+K)$ (8.2)

for $N\leq N_{0}(\eta)$.

We now turn to

######  Theorem 8.1 (No quasi-solitons).

Let $(d,s_{c})$ satisfy (1.4). 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 $J_{k}\subset[0,T_{max})$ that
satisfy both

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

and

$\int_{0}^{T_{max}}N(t)^{3-4s_{c}}\,dt=\infty.$ (8.3)

###### Proof.

Suppose towards a contradiction that such a solution $u$ exists. We will need
the following

######  Lemma 8.2.

There exist $N_{0}>0$ and $C(u)>0$ so that

$\inf_{t\in[0,T_{max})}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{2}\,dx\gtrsim_{u}1\indent\text{for all }N\leq N_{0}.$ (8.4)

We provide the proof of Lemma 8.2 below; let us first take it for granted and
use it to complete the proof of Theorem 8.1. We let $I$ be a compact time
interval, which is a union of contiguous subintervals $J_{k}$, and let
$\eta>0$ be a small parameter. We take $C(u)$ and $N_{0}$ as in Lemma 8.2;
then, choosing $N_{0}$ possibly smaller, we can guarantee by Proposition 7.1
that (8.1) or (8.2) holds (depending on the dimension) for all $N\leq N_{0}$.

We now consider two cases:

Case 1. When $d=3$, we first note by Hölder’s inequality that

$\displaystyle\big{(}\textstyle\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}$
$\displaystyle|u_{\geq
N}(t,x)|^{2}\,dx\big{)}^{2}\lesssim_{u}N(t)^{-3}\textstyle\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{4}\,dx.$

Using this inequality, followed by (8.4), we find

$\displaystyle\int_{I}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}$
$\displaystyle|u_{\geq N}(t,x)|^{4}\,dx\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\left(\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{2}\,dx\right)^{2}N(t)^{3}\,dt$
$\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-4s_{c}}\,dt$

for all $N\leq N_{0}$. Thus, appealing to (8.1), we find

$\displaystyle\eta\left[N^{1-4s_{c}}+\int_{I}N(t)^{3-4s_{c}}\right]$
$\displaystyle\gtrsim_{u}\int_{I}\int_{\mathbb{R}^{3}}|u_{\geq
N}(t,x)|^{4}\,dx\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{4}\,dx\,dt$ $\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-4s_{c}}\,dt$
(8.5)

for all $N\leq N_{0}$.

Case 2. If $d\in\\{4,5\\}$, we once again use (8.4), but use (8.2) instead of
(8.1). We find

$\displaystyle\eta\bigg{[}N^{1-4s_{c}}$
$\displaystyle+\int_{I}N(t)^{3-4s_{c}}\,dt\bigg{]}$
$\displaystyle\gtrsim_{u}\int_{I}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|u_{\geq
N}(t,x)|^{2}|u_{\geq N}(t,y)|^{2}}{|x-y|^{3}}\,dx\,dy\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\iint_{|x-y|\leq\frac{2C(u)}{N(t)}}\big{[}\tfrac{N(t)}{2C(u)}\big{]}^{3}|u_{\geq
N}(t,x)|^{2}|u_{\geq N}(t,y)|^{2}\,dx\,dy\,dt$
$\displaystyle\gtrsim_{u}\int_{I}\big{[}\tfrac{N(t)}{C(u)}\big{]}^{3}\left(\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\geq
N}(t,x)|^{2}\,dx\right)^{2}\,dt$
$\displaystyle\gtrsim_{u}\int_{I}N(t)^{3-4s_{c}}\,dt$ (8.6)

for all $N\leq N_{0}$. Thus, continuing from (8.5) or (8.6), we see that in
either case, for $\eta$ sufficiently small depending on $u$, we get

$\int_{I}N(t)^{3-4s_{c}}\,dt\lesssim_{u}N^{1-4s_{c}}\indent\text{for all
}I\subset[0,T_{max})\text{ and all }N\leq N_{0}.$

Recalling (8.3), we now reach a contradiction by taking $I$ sufficiently large
inside $[0,T_{max}).$ ∎

Finally, we turn to the

###### Proof of Lemma 8.2.

Let $\eta_{0}>0$ be a small parameter to be determined later. As
$\inf_{t\in[0,T_{max})}N(t)\geq 1$, for $N_{0}=N_{0}(\eta_{0})$ sufficiently
small, we can guarantee

$\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}([0,T_{max})\times\mathbb{R}^{d})}\leq\eta_{0}\indent\text{for
}N\leq N_{0}.$

Then, given _any_ $C(u)>0$ and $N\leq N_{0}$, we can use Hölder’s inequality
and Sobolev embedding to estimate

$\displaystyle\bigg{|}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{>N}(t,x)|^{2}-|u(t,x)|^{2}\,dx\bigg{|}$
$\displaystyle\lesssim_{u}N(t)^{-2s_{c}}\big{\|}u_{\leq
N}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}$
$\displaystyle\lesssim_{u}\eta_{0}N(t)^{-2s_{c}}$

for all $t\in[0,T_{max}).$ Thus, if we can show that for $C(u)$ sufficiently
large, we have

$\inf_{t\in[0,T_{max})}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{2}\,dx\gtrsim_{u}1,$
(8.7)

we will have (8.4) by choosing $\eta_{0}=\eta_{0}(u)$ sufficiently small.

We now turn to (8.7). We first choose $c=c(\eta_{0})$ sufficiently large that

$\big{\|}|\nabla|^{s_{c}}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{2}([0,T_{max})\times\mathbb{R}^{d})}<\eta_{0}.$
(8.8)

We then notice that by Hölder, Bernstein, Sobolev embedding, and (8.8), we
have

$\displaystyle\bigg{|}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|$ $\displaystyle
u(t,x)|^{2}-|u_{\leq cN(t)}(t,x)|^{2}\,dx\bigg{|}$ $\displaystyle\lesssim
N(t)^{-s_{c}}\big{\|}u_{>cN(t)}(t)\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}\big{\|}u(t)\big{\|}_{L_{x}^{\frac{dp}{2}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{u}\eta_{0}N(t)^{-2s_{c}}$ (8.9)

for all $t\in[0,T_{max})$. Thus, if we can show that for $C(u)$ sufficiently
large, we have

$\inf_{t\in[0,T_{max})}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx\gtrsim_{u}1,$ (8.10)

then (8.7) will follow by taking $\eta_{0}=\eta_{0}(u)$ sufficiently small.

Let us therefore turn to establishing (8.10). We begin by choosing $C(u)$
sufficiently large that

$\inf_{t\in[0,T_{max})}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{dp}{2}}\,dx\gtrsim_{u}1$

(cf. Remark 1.7). Then, with $c=c(\eta_{0})$ as above, we see by Hölder’s
inequality, Sobolev embedding, and (8.8) that

$\displaystyle\bigg{|}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)$
$\displaystyle|^{\frac{dp}{2}}-|u_{\leq
cN(t)}(t,x)|^{\frac{dp}{2}}\,dx\bigg{|}$
$\displaystyle\lesssim\big{\|}u_{>cN(t)}\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}\big{\|}u\big{\|}_{L_{t}^{\infty}L_{x}^{\frac{dp}{2}}}^{\frac{dp}{2}-1}$
$\displaystyle\lesssim_{u}\eta_{0}$

for all $t\in[0,T_{max}).$ Thus for $\eta_{0}=\eta_{0}(u)$ sufficiently small,
we have

$\inf_{t\in[0,T_{max})}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{\frac{dp}{2}}\,dx\gtrsim_{u}1.$ (8.11)

Finally, by Hölder and Bernstein, we estimate

$\displaystyle\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|$ $\displaystyle u_{\leq
cN(t)}(t,x)|^{\frac{dp}{2}}\,dx$ $\displaystyle\lesssim\big{\|}u_{\leq
cN(t)}(t)\big{\|}_{L_{x}^{\infty}(\mathbb{R}^{d})}^{\frac{dp}{2}-2}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx$
$\displaystyle\lesssim_{u}N(t)^{2s_{c}}\big{\|}u(t)\big{\|}_{L_{x}^{\frac{dp}{2}}(\mathbb{R}^{d})}^{\frac{dp}{2}-2}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx.$
$\displaystyle\lesssim_{u}N(t)^{2s_{c}}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u_{\leq
cN(t)}(t,x)|^{2}\,dx$ (8.12)

for all $t\in[0,T_{max})$. Combining (8.11) and (8.12) now yields (8.10),
which completes the proof of Lemma 8.2. ∎

##  Appendix A Some basic estimates

We collect here some basic estimates that are useful in Section 4. We begin
with

######  Lemma A.1.

Let $1<p\leq 2$. Then

$\displaystyle\big{|}\,|a+c|^{p}-|a|^{p}-|b+c|^{p}+|b|^{p}\,\big{|}\lesssim|a-b|\,|c|^{p-1}$
(A.1)

for all $a,b,c\in\mathbb{C}$.

###### Proof.

Defining $G(z):=|z+c|^{p}-|z|^{p}$, the fundamental theorem of calculus gives

$\displaystyle\text{LHS}\eqref{stupid stupid
stupid}=\bigg{|}(a-b)\int_{0}^{1}G_{z}(b+\theta(a-b))\,d\theta+\overline{(a-b)}\int_{0}^{1}G_{\bar{z}}(b+\theta(a-b))\,d\theta\bigg{|}.$

Thus, to establish (A.1), it will suffice to establish

$\displaystyle|G_{z}(z)|+|G_{\bar{z}}(z)|\lesssim|c|^{p-1}\quad\text{uniformly
for }z\in\mathbb{C}.$

That is, we need to show

$\big{|}\,|z+c|^{p-2}(z+c)-|z|^{p-2}z\,\big{|}\lesssim|c|^{p-1}$

uniformly in $z$. If $c=0$, this inequality is obvious. Otherwise, setting
$z=c\,\zeta$ reduces the problem to showing

$\big{|}\,|z+1|^{p-2}(z+1)-|z|^{p-2}z\,\big{|}\lesssim 1$ (A.2)

uniformly in $z$. For $|z|\lesssim 1$, we immediately get (A.2) from the
triangle inequality. For $|z|\gg 1$, we can use the fundamental theorem of
calculus and the fact that $p\leq 2$ to see

$\big{|}\,|z+1|^{p-2}(z+1)-|z|^{p-2}z\,\big{|}\lesssim|z|^{p-2}\lesssim 1.$

Thus, we see that (A.2) holds, which completes the proof of Lemma A.1. ∎

Next, we record a few inequalities in the spirit of [48, $\S$2.3].

######  Lemma A.2.

Let $M$ denote the Hardy–Littlewood maximal function, and let $\check{\psi}$
denote the convolution kernel of the Littlewood–Paley projection $P_{1}$. For
a fixed function $f,$ $y\in\mathbb{R}^{d},$ and $N\in 2^{\mathbb{Z}}$, we have

$\displaystyle\textstyle\int_{\mathbb{R}^{d}}N^{d}|\check{\psi}(Ny)|\,|f(x-y)|\,dy$
$\displaystyle\lesssim M(f)(x),$ (A.3) $\displaystyle|\delta_{y}f_{N}(x)|$
$\displaystyle\lesssim N|y|\left\\{M(f_{N})(x)+M(f_{N})(x-y)\right\\},$ (A.4)
$\displaystyle\textstyle\int_{\mathbb{R}^{d}}N^{d}|y|\,|\check{\psi}(Ny)|\,\,dy$
$\displaystyle\lesssim\tfrac{1}{N}.$ (A.5)

###### Proof.

We begin with (A.3). Note first that

$\eta:=N^{d}|\check{\psi}(Ny)|$

is a spherically symmetric, decreasing function of radius; thus, we can write

$\eta(y)=\int_{0}^{\infty}\chi_{B(0,r)}(y)(-\eta^{\prime}(r))\,dr,$

where $\eta^{\prime}:=\tfrac{\partial\eta}{\partial r}$. We can then use the
definition of the Hardy–Littlewood maximal function and integrate by parts to
estimate

$\displaystyle\text{LHS}\eqref{aux 1}$
$\displaystyle\lesssim\int_{0}^{\infty}\left(\int_{|y|\leq
r}|f(x-y)|\,dy\right)(-\eta^{\prime}(r))\,dr$
$\displaystyle\lesssim\left(\int_{0}^{\infty}\eta(r)r^{d-1}\,dr\right)M(f)(x)$
$\displaystyle\lesssim_{\psi}M(f)(x).$

For (A.4), we begin by defining
$\psi_{0}(\xi)=\psi(2\xi)+\psi(\xi)+\psi(\xi/2)$, the ‘fattened’
Littlewood–Paley multiplier. Then we can write

$|\delta_{y}f_{N}(x)|=\big{|}\textstyle\int[N^{d}\check{\psi_{0}}(N(z-y))-N^{d}\check{\psi_{0}}(Nz)]f_{N}(x-z)\,dz\big{|}.$
(A.6)

If $N|y|\geq 1$, we can use the triangle inequality and argue as above to see
that

$|\delta_{y}f_{N}(x)|\leq M(f_{N})(x-y)+M(f_{N})(x),$

giving (A.4) in this case. If instead $N|y|\leq 1$, we can use the fact that
$\check{\psi_{0}}$ is Schwartz to estimate

$|\check{\psi_{0}}(N(z-y))-\check{\psi_{0}}(Nz)|\lesssim
N|y|(1+N|z|)^{-100d}.$

Then continuing from (A.6) and once again arguing as for (A.3), we find

$|\delta_{y}f_{N}(x)|\lesssim N|y|M(f_{N})(x),$

which gives (A.4) in this case.

Finally, we note that since $\check{\psi}$ is Schwartz, we have

$\int_{\mathbb{R}^{d}}N^{d}|Ny|\,|\check{\psi}(Ny)|\,dy\lesssim 1,$

which immediately gives (A.5). ∎

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