# Minimal non-scattering solutions for the Zakharov System

Timothy Candy  Department of Mathematics and Statistics, University of Otago,
PO Box 56, Dunedin 9054, New Zealand<EMAIL_ADDRESS>
###### Abstract.

We consider the Zakharov system in the energy critical dimension $d=4$ with
energy below the ground state. It is known that below the ground state
solutions exist globally in time, and scatter in the radial case. Scattering
below the ground state in the non-radial case is an open question. We show
that if scattering fails, then there exists a minimal energy non-scattering
solution below the ground state. Moreover the orbit of this solution is
precompact modulo translations. The proof follows by a concentration
compactness argument, together with a refined small data theory for energy
dispersed solutions.

###### Key words and phrases:

Zakharov system, global well-posedness, concentration compactness, scattering

###### 2010 Mathematics Subject Classification:

Primary: 35Q55. Secondary: 42B37, 35L70, 35P25

##  1\. Introduction

We are interested in understanding the global dynamics of solutions to the
Zakharov system

$\begin{split}i\partial_{t}u+\Delta u&=vu,\\\
\alpha^{-2}\partial_{t}^{2}v-\Delta v&=\Delta|u|^{2},\end{split}$ (1.1)

in four spatial dimensions, where
$(u,V)(t,x):\mathbb{R}\times\mathbb{R}^{4}\to\mathbb{C}\times\mathbb{C}$ and
$\alpha>0$ is a fixed real constant. The equation (1.2) was introduced by
Zakharov [34] as a model for Langmuir turbulence in plasma, where $u$ denotes
the slow oscillation of the magnetic field, $v$ is the ion density, and
$\alpha>0$ is the ion sound speed. We refer the reader to [28, 33, 34] for
further background on the derivation of the Zakharov system and its physical
origin. The Zakharov system has received significant attention in the
mathematical community, see for instance [7, 4, 3, 5, 12, 14, 16, 18, 21, 26]
and the references therein.

It is more convenient for our purposes to set $\alpha=1$ and consider the
first order formulation of the Zakharov system

$\begin{split}i\partial_{t}u+\Delta u&=\Re(V)u,\\\
i\partial_{t}V+|\nabla|V&=-|\nabla||u|^{2},\\\ (V,u)(0)&=(f,g)\in H^{s}\times
H^{\ell},\end{split}$ (1.2)

which can be obtained from (1.1) by letting $V=v-i|\nabla|^{-1}\partial_{t}v$
(and so $v=\Re(V)$). Sufficient regular solutions $(u,V)$ to (1.2) conserve
both the energy

$\mathcal{E}_{Z}\big{(}u(t),V(t)\big{)}=\int_{\mathbb{R}^{4}}\frac{1}{2}|\nabla
u(t,x)|^{2}+\frac{1}{4}|V(t,x)|^{2}+\frac{1}{2}\Re(V)(t,x)|u(t,x)|^{2}dx,$
(1.3)

and the (Schrödinger) mass

$\mathcal{M}\big{(}u(t)\big{)}=\int_{\mathbb{R}^{4}}\frac{1}{2}|u(t,x)|^{2}dx.$
(1.4)

Although the system (1.2) lacks a natural scaling invariance, it should be
thought of as energy critical in four spatial dimensions. For instance, the
energy regularity $H^{1}\times L^{2}$ lies on the boundary of the admissible
regularities for local well-posedness (in fact we have some form of ill-
posedness if the wave regularity is lowered to $H^{s}$ for $s<0$, see for
instance [15] are the references therein). Furthermore, the potential energy
in (1.3) is just barely controlled by the kinetic energy in four dimensions,
namely the sharp Sobolev embedding $\dot{H}^{1}(\mathbb{R}^{4})\subset
L^{4}(\mathbb{R}^{4})$ implies that

$\int_{\mathbb{R}^{4}}\Re(V)|u|^{2}dx\leqslant\|V\|_{L^{2}_{x}}\|u\|_{L^{4}_{x}}^{2}\lesssim\|V\|_{L^{2}_{x}}\|u\|_{\dot{H}^{1}}^{2}.$

The Zakharov system (1.2) is closely connected to the nonlinear Schrödinger
equation (NLS). For instance, in the subsonic limit as the ion sound speed
$\alpha\to\infty$, we have $v=-|u|^{2}$ and so formally the Zakharov system
(1.1) reduces to the focussing cubic NLS

$i\partial_{t}u+\Delta u=-|u|^{2}u.$ (1.5)

This convergence can be made rigorous, see for instance [27, 25] and the
references therein. As the cubic NLS is energy critical in four dimensions,
this gives another reason for considering the Zakharov system to be energy
critical in four dimensions. The connection between the Zakharov system and
cubic NLS also appears on the level of the energy, for instance we can write

$\mathcal{E}_{Z}(f,g)=\mathcal{E}_{NLS}(f)+\frac{1}{4}\big{\|}g+|f|^{2}\big{\|}_{L^{2}}^{2}$

where

$\mathcal{E}_{NLS}(f)=\int_{\mathbb{R}^{4}}\frac{1}{2}|\nabla
f|^{2}-\frac{1}{4}|f|^{4}dx$

is the conserved energy (or Hamiltonian) for the focussing cubic NLS on
$\mathbb{R}\times\mathbb{R}^{4}$. Furthermore, a key role in the global
dynamics of both the cubic NLS (1.5) and the Zakharov system (1.2) is played
by the famous Aubin-Talenti function [1, 29]

$Q(x)=\frac{8}{8+|x|^{2}}$

which serves as a ground state solution for both the NLS and Zakharov
equations. In particular the stationary solutions $u(t)=Q$ to (1.5) and
$(u,v)(t)=(Q,-Q^{2})$ to (1.1) (and their rescalings) give global solutions
which do not disperse for large times. The existence of non-dispersing
solutions is a typical feature of focussing equations, and immediately rules
out the possibility of large data global existence and scattering.

The ground state solution $Q$ in fact gives a sharp threshold for linear
scattering. More precisely, for the NLS (1.5) it is known that any solution
with energy (strictly) below the ground state $Q$ is global and scatters to a
free Schrödinger wave as $t\to\pm\infty$ [20, 13]. This is sharp as $u(t)=Q$
gives a (global, stationary) solution to (1.5) which clearly does not scatter
to linear solution to the Schrödinger equation. The corresponding result for
the Zakharov equation (1.1) in full generality is open. However recent work
has shown that _radial_ solutions to (1.1) with energy below the ground state
$(Q,-Q^{2})$ are global and scatter [16]. On the other hand, in the non-radial
case, it has recently been shown [9] that all solutions with data below the
ground state are global in time. As we require this result later, we recall
the precise statement from [9].

######  Theorem 1.1 (GWP below ground state [9]).

Let $(f,g)\in H^{1}\times L^{2}$ with

$4\mathcal{E}_{Z}(f,g)<\|Q\|_{\dot{H}^{1}}^{2},\qquad\|g\|_{L^{2}}\leqslant\|Q\|_{\dot{H}^{1}}.$

Then there exists a unique global solution $(u,V)\in C(\mathbb{R},H^{1}\times
L^{2})$ to (1.2) with $u\in L^{2}_{t,loc}W^{\frac{1}{2},4}_{x}$ and data
$(u,V)(0)=(f,g)$. Moreover the energy $\mathcal{E}_{Z}(u,V)$ is conserved and
we have the global bounds

$\max\big{\\{}\|u\|_{L^{\infty}_{t}\dot{H}^{1}_{x}},\|V\|_{L^{\infty}_{t}L^{2}_{x}}\big{\\}}\leqslant
4\mathcal{E}_{Z}(f,g).$

######  Remark 1.2.

The variational characterisation of the ground state $Q$ implies that we have

$\mathcal{E}_{Z}(Q,-Q^{2})=\mathcal{E}_{NLS}(Q)=\frac{1}{4}\|Q\|_{\dot{H}^{1}}^{2},\qquad\|Q^{2}\|_{L^{2}}=\|Q\|_{\dot{H}^{1}}$

and thus we could alternatively state the assumed bound on the data in Theorem
1.1 as

$\mathcal{E}_{Z}(f,g)<\mathcal{E}_{Z}(Q,-Q^{2}),\qquad\|g\|_{L^{2}}\leqslant\|Q^{2}\|_{L^{2}}.$

It is also possible to replace the condition
$\|g\|_{L^{2}}\leqslant\|Q^{2}\|_{L^{2}}$ with
$\min\\{\|f\|_{\dot{H}^{1}},\|g\|_{L^{2}}\\}\leqslant\|Q\|_{\dot{H}^{1}}$ (see
for instance Lemma 7.2 below). This is not surprising in view of the
corresponding assumption in the NLS case [20, 13].

The asymptotic behaviour of the global solutions given by Theorem 1.1 is
currently open, although in case of radial solutions it is known that
solutions scatter to free solutions to the Zakharov equation [16]. However, a
number of scattering criteria have been obtained. For instance all solutions
which are dispersive, in the sense that a Strichartz type norm is finite over
the time of evolution, can be shown to scatter [10].

######  Theorem 1.3 (Scattering criterion [10]).

Let $(u,V)\in C(\mathbb{R};H^{\frac{1}{2}}\times L^{2})$ be a global solution
to (1.2) with

$\|u\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}(\mathbb{R}\times\mathbb{R}^{4})}<\infty\qquad\text{and}\qquad(u,V)(0)\in
H^{1}\times L^{2}.$

Then in fact $(u,V)\in C(\mathbb{R};H^{1}\times L^{2})$ and moreover there
exists $(f_{\pm},g_{\pm})\in H^{1}\times L^{2}$ such that

$\lim_{t\to\pm\infty}\big{\|}\big{(}e^{-it\Delta}u,e^{-it|\nabla|}V\big{)}(t)-(f_{\pm},g_{\pm})\big{\|}_{H^{1}\times
L^{2}}=0.$

It is worth pointing out that the sharp threshold regularity threshold for
local well-posedness for the Zakharov equation in 4 dimensions is $(f,g)\in
H^{\frac{1}{2}}\times L^{2}$, see for instance [3, 10, 18]. In particular the
energy regularity $H^{1}\times L^{2}$ is $\frac{1}{2}$ a derivative (at least
for the Schrödinger component of the evolution) above the endpoint space
$H^{\frac{1}{2}}\times L^{2}$. This should help to explain the regularity
assumption $u\in L^{2}_{t}W^{\frac{1}{2},4}_{x}$ in Theorem 1.3. On the other
hand, the wave regularity cannot be lowered, and in fact it is the lack of
room in the wave regularity which is the key difficulty in the above theorems.
In particular, even proving local well-posedness in the energy space
$H^{1}\times L^{2}$ is slightly subtle (and in some sense more challenging
than the endpoint $H^{\frac{1}{2}}\times L^{2}$) and in [3] was obtained by an
additional indirect compactness argument. An alternative more direct argument
was later found in [10].

In view of the above results, together with the known linear scattering in the
radial case and corresponding know results in the case of energy critical NLS
[20, 13], it is generally expected that the following conjecture holds.

######  Conjecture 1.4 (Scattering below ground state).

Let $(f,g)\in H^{1}\times L^{2}$ with

$4\mathcal{E}_{Z}(f,g)<\|Q\|_{\dot{H}^{1}}^{2},\qquad\|g\|_{L^{2}}\leqslant\|Q\|_{\dot{H}^{1}}.$

Then there exists a unique global solution $(u,V)\in C(\mathbb{R},H^{1}\times
L^{2})$ to (1.2) with data $(u,V)(0)=(f,g)$ such that

$\|u\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}(\mathbb{R}^{1+4})}<\infty.$

If Conjecture 1.4 holds, then Theorem 1.3 immediately implies that any
solution with energy below the ground state scatters to a linear solution as
$t\to\pm\infty$. Moreover, for global solutions to (1.1), it is not so
difficult to prove that scattering is equivalent to boundedness of the
dispersive norm $L^{2}_{t}W^{\frac{1}{2},4}_{x}$. In particular Conjecture 1.4
could also be stated in terms of a scattering condition as $t\to\pm\infty$.
However it is technically more convenient to work with a dispersive condition
such as $\|u\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}}<\infty$.

Our goal in the current article is to make progress towards the resolution of
Conjecture 1.4. Unfortunately, we are unable to resolve this question,
essentially due to the lack of a suitable way to control the translation
symmetry. In the case of the NLS or nonlinear wave equations, translations are
typically controlled by using Galilei invariance/Lorentz invariance to reduce
to the case of zero momentum. However the Zakharov equation is not invariant
under either transformation, and thus removing translations seems to be a
difficult obstruction. On other hand, in the positive direction, we can show
that if Conjecture 1.4 fails, then there must be a _minimal_ counterexample,
namely a minimal energy non-scattering solution. Moreover this minimal
counterexample must be _almost periodic_ in the sense that it satisfies a
strong compactness property modulo spatial translations. It is a well-known
phenomenon in dispersive PDE that the failure of scattering implies the
existence of a minimal non-scattering solutions with strong compactness
properties. In fact in [20] a general road map was developed to prove results
of the form in Conjecture 1.4 via a concentration-compactness argument (to
show the existence of a minimal energy non-scattering solution) together with
a rigidity argument. Thus our results give the first step of this argument,
but we are unable to provide the ‘rigidity’ step. The existence of almost
periodic solutions (or minimal energy/mass non-scattering solutions) was first
obtained in [20, 22] in the context of the NLS. These arguments are closely
related to original induction on energy approach [6] (see for instance [11,
32, 23] and the references therein) which produced _approximate_ almost
periodic solutions.

To state out result precisely, we require additional notation. Given $E>0$, we
define the set

$\begin{split}\Omega(E)=\Big{\\{}(u,V)\in C(\mathbb{R};H^{1}\times
L^{2})\,\,\Big{|}\,\,(u,V)&\text{ solves \eqref{eqn:Zak} with }u\in
L^{2}_{t,loc}W^{\frac{1}{2},4}_{x}\\\ &\text{ and
}\mathcal{E}_{Z}(u,V)\leqslant
E,\,\,\|V\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant\|Q\|_{\dot{H}^{1}}\Big{\\}}.\end{split}$
(1.6)

Thus $\Omega(E)\subset C(\mathbb{R};H^{1}\times L^{2})$ is the set of all
$H^{1}\times L^{2}$ solutions to the Zakharov equation with energy below a
threshold $E$, and wave mass uniformly below the ground state $Q$. Note that
if $(f,g)\in H^{1}\times L^{2}$ with $\mathcal{E}_{Z}(f,g)\leqslant
E<\frac{1}{4}\|Q\|_{\dot{H}^{1}}^{2}$ and
$\|g\|_{L^{2}}\leqslant\|Q\|_{\dot{H}^{1}}$, then by Theorem 1.1 there exists
a (unique) global solution $(u,V)\in\Omega(E)$. In particular, $\Omega(E)$ is
non-empty. The condition $u\in L^{2}_{t,loc}W^{\frac{1}{2},4}$ is needed to
ensure uniqueness, and is harmless in practise as all solutions constructed in
this paper always satisfy this condition. There are a number of different ways
to characterise the set $\Omega(E)$ when
$E<\frac{1}{4}\|Q\|_{\dot{H}^{1}}^{2}$, see for instance Lemma 7.1 and Remark
7.2.

It is tempting to use the set $\Omega(E)$ as the basis for a concentration
compactness argument. However, due to the lack of a good well-posedness theory
in the homogeneous energy space $\dot{H}^{1}\times L^{2}$, we impose an
additional constraint on the mass of our solutions to ensure that the limiting
critical element remains in $H^{1}\times L^{2}$ (instead of just
$\dot{H}^{1}\times L^{2}$). More precisely, given $E,M>0$ we define

$L(M,E)=\sup\Big{\\{}\|u\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}(\mathbb{R}^{1+4})}\,\,\Big{|}\,\,(u,V)\in\Omega(E)\text{
and }\mathcal{M}(u)\leqslant M\Big{\\}}.$ (1.7)

Note that, in view of Theorem 1.3, scattering with mass/energy $(M,E)$ holds
precisely when $L(M,E)<\infty$.

######  Definition 1.5 (Mass/energy threshold).

We say a pair $(E_{c},M_{c})\in(0,\infty)\times(0,\infty)$ is an _mass/energy
threshold_ for the Zakharov equation if $L(M_{c},E_{c})=\infty$ and for any
$\epsilon>0$ we have

$L(M_{c},E_{c}-\epsilon)+L(M_{c}-\epsilon,E_{c})<\infty.$

In other words $(M,E)$ is a mass/energy threshold if: (i) the dispersive norm
$\|\cdot\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}}$ blows up along some sequence of
solutions with mass/energy at most $(M,E)$, and (ii) any solution with either
less energy or less mass scatters. The existence of a mass/energy threshold
with $4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$ is inconsistent with Conjecture 1.4. In
fact the failure of Conjecture 1.4 implies the existence of a mass/energy
threshold, see Lemma 9.1 below. More importantly, the failure of scattering
must be witnessed by the existence of a minimal or critical element in the
energy space. The following is our main result.

######  Theorem 1.6 (Reduction to almost periodic solutions).

Suppose that Conjecture 1.4 fails. Then there exists a mass/energy threshold
$(M_{c},E_{c})$ with $E_{c}<\frac{1}{4}\|Q\|_{\dot{H}^{1}}^{2}$ and a global
solution $(\psi,\phi)\in C(\mathbb{R},H^{1}\times L^{2})$ to (1.2) with
$\psi\in L^{2}_{t,loc}W^{\frac{1}{2},4}_{x}$ such that

$\mathcal{E}_{Z}(\psi,\phi)=E_{c},\qquad\mathcal{M}(\psi)=M_{c},\qquad\|\phi\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant\|Q\|_{\dot{H}^{1}},$

and

$\|\psi\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}((-\infty,0]\times\mathbb{R}^{4})}=\|\psi\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}([0,\infty)\times\mathbb{R}^{4})}=\infty.$

Moreover, there exists $x(t):\mathbb{R}\to\mathbb{R}^{4}$ such that the orbit
$\\{(\psi,\phi)(t,x+x(t))\mid t\in\mathbb{R}\\}$ is precompact in $H^{1}\times
L^{2}$.

The conclusion of Theorem 1.6 implies that the orbit
$\\{(\psi,\phi)(t)|t\in\mathbb{R}\\}$ is compact modulo translations, such
solutions are often called _almost periodic modulo translations_ , see the
discussion in [23]. These solutions clearly do not scatter to free solutions
as $t\to\pm\infty$, and should be thought as essentially ‘soliton like’ as
they retain their profile for large times. Theorem 1.6 is a stepping stone
towards a resolution of Conjecture 1.4 as it reduces the proof of scattering
to ruling out the existence of almost periodic solutions below the ground
state. It is important to observe that the compactness is only modulo
translations, in particular there is no scaling/frequency symmetries required.
This is perhaps not surprising in view of the fact that the Zakharov equation
is not invariant under rescalings. In some sense the lack of frequency scaling
is a consequence of the fact that the energy space is subcritical in the
Schrödinger regularity (but still critical in the wave regularity!), but it
can also be thought of as saying that the various asymptotic limits of (1.2)
are not an obstruction to scattering.

The proof of Theorem 1.6 proceeds via the usual concentration compactness
argument essentially as in [22, 20], see also [23] for an outline of this
approach. Roughly the idea is that if Conjecture 1.4 failed, then there exists
a mass/energy threshold with $(M_{c},E_{c})$ with
$4E_{c}<\|Q\|_{\dot{H}^{1}}^{2}$. In particular, there exists a sequence of
$H^{1}\times L^{2}$ solutions $(u_{n},V_{n})$ to (1.1) with mass and energy
converging to $(M_{c},E_{c})$ and
$\|u_{n}\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}}\to\infty$. Theorem 1.6 then
follows by extracting a converging subsequence via a concentration compactness
argument. It is here that we meet the first major difficulty. Typically
extracting some compactness from the sequence $(u_{n},V_{n})$ relies on a
profile decomposition in the spirit of [2] with an error going to zero in some
dispersive norm. The issue is that there is no profile decomposition known in
the endpoint Strichartz space $L^{2}_{t}L^{4}_{x}$. Moreover, obtaining such a
profile decomposition seems extremely difficult. Consequently the small data
theory in [9, 10] which required smallness in $L^{2}_{t}L^{4}_{x}$ is not
sufficient to conclude Theorem 1.6.

An alternative to trying to obtain a profile decomposition in the endpoint
Strichartz space is to extend the known small data theory to rely on a more
tractable dispersive norm. This is the approach we take here. In particular,
we obtain a refined small data theory with smallness measured in an “energy
dispersion” type norm. More precisely, define the norm

$\|u\|_{Y(I)}=\sup_{\lambda\in
2^{\mathbb{N}}}\lambda^{-4}\|u_{\lambda}\|_{L^{\infty}_{t,x}(I\times\mathbb{R}^{4})}$

where $u_{\lambda}$ is the standard (inhomogeneous) Littlewood-Paley
projection to frequencies $|\xi|+1\approx\lambda$. The norm $\|u\|_{Y}$
roughly measures how dispersed a solution is. In particular, if $\|u\|_{Y}$ is
small compared to the energy, then $u$ cannot be concentrated either spatially
or in frequency. Moreover, as free Schrödinger waves disperse, we have
$\|e^{it\Delta}f\|_{Y([T,\infty))}\to 0$ as $T\to\infty$. The $Y$ norm is much
weaker than $H^{\frac{1}{2}}$, and at least at large frequencies, roughly
scales like $\dot{H}^{-2}$. In fact a short computation using Bernstein’s
inequality and the endpoint Strichartz estimate gives

$\|u\|_{Y(I)}\lesssim\min\Big{\\{}\|u\|_{L^{\infty}_{t}L^{2}_{x}(I\times\mathbb{R}^{4})},\|u\|_{L^{\infty}_{t}\dot{H}^{1}_{x}(I\times\mathbb{R}^{4})}\Big{\\}}\qquad\text{and}\qquad\|e^{it\Delta}f\|_{Y(I)}\lesssim\|e^{it\Delta}f\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}(I\times\mathbb{R}^{4})}.$
(1.8)

This bound can clearly be improved (see (2.4) below) but suffices for our
purposes. In particular, smallness in either mass or $\dot{H}^{1}$ implies
that the energy dispersion norm $Y$ is also small. This ability to transfer
smallness in homogeneous Sobolev spaces is crucial, as we eventually need to
conclude that if a solution is close to a dispersive solution in energy, then
it is also close in the norm $\|\cdot\|_{Y}$. Note that this is not true for
the inhomogeneous dispersive norm $L^{2}_{t}W^{\frac{1}{2},4}_{x}$. We can now
state a precise version of the small data theory that we require.

######  Theorem 1.7 (Refined small data theory in $H^{s}\times L^{2}$).

Let $A>0$, $0<B<\|Q\|_{\dot{H}^{1}}$, and $\frac{1}{2}<s<2$. There exists
$\epsilon=\epsilon(A,B,s)>0$ and $C=C(A,B,s)>0$ such that for any interval
$0\in I\subset\mathbb{R}$, and any $(f,g)\in H^{s}\times L^{2}$ satisfying

$\|f\|_{H^{s}}\leqslant
A,\qquad\|e^{it\Delta}f\|_{Y(I)}\leqslant\epsilon,\qquad\|g\|_{L^{2}_{x}}\leqslant
B,$ (1.9)

there exists a (unique) solution $(u,V)\in C(I,H^{s}\times L^{2})$ to (1.2)
with data $(u,V)(0)=(f,g)$ and

$\|u\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}(I\times\mathbb{R}^{4})}\leqslant
C\|f\|_{H^{\frac{1}{2}}},\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}(I)}\leqslant\tfrac{1}{2}\big{(}B+\|Q\|_{\dot{H}^{1}}\big{)}.$

Strictly speaking, we also require a generalisation of Theorem 1.7 that
implies dispersive solutions are stable, see Theorem 5.2 for a precise
statement. In view of the bound (1.8), Theorem 1.7 is more general than the
small data theory developed in [10] which required smallness in the endpoint
Strichartz space $L^{2}_{t}W^{\frac{1}{2},4}_{x}$. It is also worth noting
that showing that smallness in $L^{2}$ suffices for scattering is
straightforward via the convexity type bound

$\|f\|_{H^{\frac{1}{2}}}\lesssim\|f\|_{L^{2}}^{\theta}\|f\|_{H^{s}}^{1-\theta}.$

On the other hand, proving that smallness in $\dot{H}^{1}$ suffices is
significantly more challenging, as the low frequencies can be badly behaved.
In particular there does not seem to be an easy direct argument to conclude
that smallness in energy suffices for scattering, and instead it seems
necessary to argue via the norm $\|\cdot\|_{Y}$ (or some similar alternative).

The proof of Theorem 1.7 is somewhat involved, and there are essentially three
main obstructions to overcome: (i) Large wave data, (ii) Gaining a power of
the $Y$ norm, (iii) Propagating smallness over large time intervals. We detail
each of these obstructions below.

  1. (i)

(Large wave data.) The wave data $g\in L^{2}$ in Theorem 1.7 is not small, and
consequently we need to absorb the free wave into the left hand side of the
equation. This issue was resolved in [9] by proving a uniform Strichartz
estimate for the Schrödinger equation with a free wave potential, and we
exploit this estimate here. As a simple toy model, consider the equation

$i\partial_{t}u+\Delta u-\Re(e^{it|\nabla|}g)u=F.$

Since $g$ is not small, it cannot be regarded as a perturbation over long time
intervals. There are two approaches to resolving this issue. One is to exploit
the dispersive effect of the free wave, and essentially observe that
$e^{it|\nabla|}g$ converges to zero in $L^{2}+L^{\infty}$, see for instance
[3, 10]. Although this argument works for arbitrary wave data in $L^{2}$, it
leads to estimates which are _not uniform_ in $g$. The lack of a uniform
estimate is a major issue for the large data theory, as below the ground state
we can control $\|V(t)\|_{L^{2}}$, but not its profile. In particular the key
step in the global theory [9] was the proof of a _uniform_ Strichartz estimate
with a free wave potential, see also [16] for the radial case. The uniform
Strichartz estimate from [9] also plays a key role in the proof of Theorem
1.7, see Theorem 2.3 below for a precise statement.

  2. (ii)

(Gaining power of $Y$ norm.) This step is the heart of the proof of Theorem
1.7, and shows that the nonlinear terms in (1.1) are perturbative provided
that $\|u\|_{Y(I)}\ll 1$. The small data theory for (1.1) essentially relies
on a fixed point argument in a Banach space based on the endpoint Strichartz
norm $L^{2}_{t}L^{4}_{x}$. Roughly speaking (ignoring derivatives for
simplicity), after discarding the non-resonant terms and applying the endpoint
Strichartz estimate [19], the argument reduces to a simple application of
Hölder’s inequality via the estimate

$\|Vu\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}\lesssim\|V\|_{L^{\infty}_{t}L^{2}_{x}}\|u\|_{L^{2}_{t}L^{4}_{x}}.$
(1.10)

Our goal is to then improve (1.10) by adding some power of the $Y$ norm to the
righthand side, i.e. essentially replacing $\|u\|_{L^{2}_{t}L^{4}_{x}}$ with
$\|u\|_{L^{\infty}_{t,x}}^{\theta}\|u\|_{L^{2}_{t}L^{4}_{x}}^{1-\theta}$. The
issue is that in the fully resonant case (so both inputs and outputs have
Fourier support close to their relevant characteristic surfaces) it seems very
hard to improve on the Strichartz bound (1.10) as there is no room in the
exponents. In particular, it does not seem possible to extract a factor
$\|u\|_{L^{\infty}_{t,x}}$. Note that even placing $V\in L^{q}_{t}L^{r}_{x}$
for some wave admissable pair $(q,r)$ does not help, since the _only_
admissable wave pair for which we have

$\|Vu\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}\lesssim\|V\|_{L^{q}_{t}L^{r}_{x}}\|u\|_{L^{a}_{t}L^{b}_{x}}$

with $(a,b)$ Schrödinger admissable, is $(q,r)=(\infty,2)$ and $(a,b)=(2,4)$.
In the radial case, there is additional room in the Strichartz exponents, and
thus the above issue can be resolved by placing $u\in
L^{2}_{t}L^{4-\delta}_{x}$ for some $\delta>0$. In the non-radial case, the
argument is more involved, and to create some room in the above chain of
estimates, we instead require the inhomogeneous bilinear restriction estimates
from [9]. These estimates essentially extend the bilinear restriction
estimates for the paraboloid obtained by Tao [30] to _inhomogeneous_
Schrödinger waves. More precisely, [9] proved that for any $r>\frac{5}{3}$ and
$\mu\in\mathbb{Z}$ we have the bound

$\|\dot{P}_{\mu}(\phi\psi)\|_{L^{1}_{t}L^{r}_{x}}\lesssim\mu^{2-\frac{4}{r}}\Big{(}\|\phi(0)\|_{L^{2}}+\|(i\partial_{t}+\Delta)\phi\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}\Big{)}\Big{(}\|\psi(0)\|_{L^{2}}+\|(i\partial_{t}+\Delta)\psi\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}\Big{)}$
(1.11)

where $\dot{P}_{\mu}$ denotes the homogeneous Littlewood-Paley projection to
frequencies $|\xi|\approx\mu\in 2^{\mathbb{Z}}$. Note that the case $r=2$
follows from Hölder’s inequality and the endpoint Strichartz estimate. The
point of the bilinear restriction type estimate (1.11) is that it allows
$r<2$, and adapting the argument from [9], this additional room essentially
allows us to improve (1.10) to something of the form

$\Big{\|}\int_{0}^{t}e^{i(t-s)\Delta}\big{(}\Re(V)u\big{)}(s)ds\Big{\|}_{L^{2}_{t}L^{4}_{x}}\lesssim\|V\|_{W}\|u\|_{Y}^{\theta}\|u\|_{S}^{1-\theta}$

where $S$ is our iteration norm for the Schrödinger component of the
evolution, and $W$ is the corresponding norm for the wave evolution, see
Section 4 for the details.

  3. (iii)

(Propagating smallness over large time intervals.) The bilinear estimates
obtained in (ii) essentially allow use to conclude that the nonlinearity in
(1.1) are perturbative provided we have smallness in the energy dispersed norm
$\|\cdot\|_{Y}$. In particular, after replacing the wave component of the
evolution by the corresponding Duhamel formula, (ii) allows us to
schematically prove a bound of the form

$\|u\|_{S}\lesssim\text{data}+\|u\|_{S}^{3-\theta}\|u\|_{Y}^{\theta}.$

However there is a technical issue in that the power $\theta>0$ can be very
small (and much smaller than $1$), and thus even if the data is small in
$\|\cdot\|_{Y}$, the estimates in (ii) are not sufficient to directly conclude
that the solution will remain small in $Y$. In particular, iterating this
smallness over a potentially large number of intervals becomes technically
challenging. Potentially this obstruction could be addressed by proving
sharper estimates with $\|u\|_{Y}$ on the left hand side (or working harder in
the iteration stage). However, here we take a simple and more flexible
approach to this issue, by first obtaining a small data theory based on a
tractable controlling quantity, and then second, proving that this controlling
quantity can be bounded by the energy dispersed norm $\|\cdot\|_{Y}$.

To illustrate the general idea, suppose we were trying to construct a fixed
point to the (cubic) nonlinear problem

$u=u_{0}+\mathcal{I}[u^{3}]$ (1.12)

where as usual $u_{0}$ is the free evolution of the data and $\mathcal{I}$
denotes the corresponding Duhamel term. Suppose we have a Banach space $X$
satisfying the nonlinear estimate

$\|\mathcal{I}[uvw]\|_{X}\lesssim\|u\|_{X}\|v\|_{X}\|w\|_{X}.$ (1.13)

The bound (1.13) together with a standard fixed point argument implies that if
$u_{0}\in X$ is small, then (1.12) has a solution $u\in X$. Our key
observation is that the smallness condition on $u_{0}$ can be weakened
significantly by the following simple observation. Namely, define the
_controlling quantity_

$\rho(u)=\sup_{\|\phi\|_{X}\leqslant 1}\|\mathcal{I}[\phi^{2}u]\|_{X}$

which roughly measures the nonlinear interaction of $u$ with a generic element
$\phi\in X$. Then for any $A>0$ there exists $\epsilon>0$ such that if the
(free evolution of the) data satisfies

$\|u_{0}\|_{X}\leqslant A,\qquad\rho(u_{0})\leqslant\epsilon$ (1.14)

then a fixed point $u\in X$ exists, and moreover we have
$\rho(u)\lesssim_{A}\epsilon$ (see Subsection 3.2 below). Clearly in view of
the bound (1.13) if $u_{0}$ is small in $X$, then (1.14) holds. On the other
hand, if say we could improve (1.13) to

$\|\mathcal{I}[u^{3}]\|_{X}\lesssim\|u\|_{D}^{\theta}\|u\|_{X}^{3-\theta}$
(1.15)

for some $\theta>0$ and some dispersive norm $\|\cdot\|_{D}$, then assuming
$\|u_{0}\|_{D}$ is small the definition of $\rho$ together with (1.15) would
imply that (1.14) holds. Of course, assuming the bound (1.15) holds, it is
also possible to argue directly (i.e. without proceed via $\rho$) that we have
a solution $u\in X$ to (1.12). However arguing via the controlling quantity
$\rho$ has the advantage that: (a) smallness $\rho(u)\lesssim_{A}\epsilon$ is
preserved, (b) it is flexible as it completely separates the issue of the
improved nonlinear estimate (1.15) from the existence theory, (c) we only need
to prove smallness of the free evolution $\rho(u_{0})$. To illustrate (a),
note that if $\theta<1$ in (1.15), then arguing via $Y$ it is not possible to
conclude that $\|u\|_{Y}\lesssim_{A}\epsilon$. The advantage of (b) is that if
at a later point a more convenient dispersive norm $D^{\prime}$ arises (or
alternatively some more involved condition on the data, for instance
concentration along light rays as in [31]), it is not necessary to repeat the
small data theory. It would simply suffice to prove that smallness in
$D^{\prime}$ implies the controlling quantity $\rho(u_{0})$ is small (for
instance via a version of the improved nonlinear estimate (1.15)). Finally,
(c) is a somewhat minor point, and simply makes the observation that as we
only need to prove $\rho(u_{0})$ is small, we may replace $u^{3}$ with
$u^{2}u_{0}$ on the lefthand side of (1.15) which in some settings may be
helpful. In the application to the Zakharov equation, the necessary
controlling quantity $\rho$ is slightly more involved (as we are dealing with
a system), but the underlying idea remains the same. Thus we first provide a
small data theory (relying on the estimates from [9, 10]) with smallness
measured in some controlling quantity $\rho$, and then later prove refined
bilinear estimates which allows us to conclude that smallness in $Y$ implies
smallness of the controlling quantity.

###  1.1. Outline of Paper

Section 2 fixes the notation used throughout this paper, defines the key
function spaces that are used to control solutions to the Zkharov equation,
and recalls a number of key estimates from the papers [9, 10]. In Section 3,
we turn to the small data theory for the Zakharov equation, assuming that a
certain controlling norm is small. In particular, we give a persistence of
regularity result, and prove a key stability property for the Zakharov
equation. The results in Section 3 only rely on the estimates from [9, 10].
The improved bilinear estimates giving a gain in the energy dispersed norm are
then proved in Section 4. In Section 5 we apply the improved bilinear
estimates together with the small data theory from Section 3 to give the proof
of Theorem 1.7, as well as give a stability theory based on the energy
dispersed norm $\|\cdot\|_{Y}$.

The final four sections contain the concentration compactness and variational
arguments needed to prove Theorem 1.6. More precisely, in Section 6 we state a
version of the profile decomposition of Bahouri-Gérard [2] while in Section 7
we recall the variational properties of the Zakharov equation from [16]. The
key step in the proof of existence of minimal mass/energy blow up solutions is
given in Section 8 where we prove a fundamental Palais-Smale type condition
which gives a crucial compactness property of bounded sequences of solutions
to the Zakharov equation. Finally in Section 9 we prove the existence of a
mass/energy threshold (under the assumption that Conjecture 1.4 fails), and
conclude the proof of Theorem 1.6.

##  2\. Notation

###  2.1. Fourier Multipliers

We let $\chi\in C^{\infty}_{0}(\mathbb{R})$ denote a smooth function
satisfying the standard Littlewood-Paley conditions

$\operatorname*{supp}\chi\subset\\{\tfrac{1}{2}\leqslant r\leqslant
2\\},\qquad 0\leqslant\chi\leqslant 1,\qquad\sum_{\lambda\in
2^{\mathbb{Z}}}\chi(\lambda^{-1}r)=1\,\,\text{ for all $r>0$}.$

The corresponding standard inhomogeneous Littlewood-Paley projections are then
given by

$P_{1}=\sum_{\begin{subarray}{c}\mu\in 2^{\mathbb{Z}}\\\ \mu\leqslant
1\end{subarray}}\chi(\mu^{-1}|\nabla|),\qquad
P_{\lambda}=\chi(\lambda^{-1}|\nabla|)\,\,\text{ for $\lambda\in
2^{\mathbb{N}}$, $\lambda>1$.}$

We also require the homogeneous version. To this end, we take
$\dot{P}_{\lambda}$ with $\lambda\in 2^{\mathbb{Z}}$ to be the homogeneous
Littlewood-Paley multipliers. We often use the short hand
$u_{\lambda}=P_{\lambda}u$, this is reserved exclusively for the inhomogeneous
Littlewood-Paley projection, thus $\lambda\in 2^{\mathbb{N}}$.

Given $\mu\in 2^{\mathbb{Z}}$, we let

$C_{\leqslant\mu}=\sum_{\begin{subarray}{c}\nu\in 2^{\mathbb{Z}}\\\
\nu\leqslant\mu\end{subarray}}\chi\big{(}\nu^{-1}|i\partial_{t}+\Delta|\big{)}$

be a smooth Fourier multiplier with support in the set
$\\{|\tau+|\xi|^{2}|\leqslant 2\mu\\}$. We also require a temporal version of
the Littlewood-Paley decomposition. To this end, we let

$P^{(t)}_{\leqslant\mu}=\sum_{\begin{subarray}{c}\nu\in 2^{\mathbb{Z}}\\\
\nu\leqslant\mu\end{subarray}}\chi\big{(}\nu^{-1}|\partial_{t}|\big{)}.$

Thus $P^{(t)}_{\leqslant\mu}$ restricts the temporal Fourier support to the
set $|\tau|\leqslant 2\mu$.

###  2.2. Function Spaces

We fix the spatial dimension to be four, and often use the short hand
$L^{q}_{t}L^{r}_{x}=L^{q}_{t}L^{r}_{x}(\mathbb{R}^{1+4})$. The arguments to
follow unfortunately involve a large number of norms. Similar to [9] (except
for the dispersive norm $Y$), these norms can be seen as special cases of the
norms appearing in [10]. These come in two flavours, the Schrödinger variants
$S^{s}$, $\underline{S}^{s}$, and $N^{s}$, and the wave variants $W^{\ell}$,
$\underline{W}^{\ell}$, and $R^{\ell}$. The solution $(u,V)$ is controlled in
$S^{s}\times W^{\ell}$, with the underlined versions of the spaces giving
slightly stronger control. The nonlinear terms are then estimated in
$N^{s}\times R^{\ell}$.

In more detail, for $\lambda\in 2^{\mathbb{N}}$ and $s\in\mathbb{R}$, we
control the frequency localised Schrödinger component of the evolution using
the norm

$\|u\|_{S_{\lambda}^{s}}=\lambda^{s}\|u\|_{L^{\infty}_{t}L^{2}_{x}(\mathbb{R}^{1+4})}+\lambda^{s}\|u\|_{L^{2}_{t}L^{4}_{x}(\mathbb{R}^{1+4})}+\lambda^{s-1}\|(i\partial_{t}+\Delta)u\|_{L^{2}_{t,x}(\mathbb{R}^{1+4})}$

and the stronger norm

$\|u\|_{\underline{S}^{s}_{\lambda}}=\lambda^{s}\|u\|_{L^{\infty}_{t}L^{2}_{x}(\mathbb{R}^{1+4})}+\|(i\partial_{t}+\Delta)u\|_{N^{s}_{\lambda}}$

where

$\|F\|_{N^{s}_{\lambda}}=\lambda^{s}\|C_{\leqslant
2^{-16}\lambda^{2}}F\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}(\mathbb{R}^{1+4})}+\lambda^{s-1}\|F\|_{L^{2}_{t,x}(\mathbb{R}^{1+4})}.$

Similarly, to estimate the frequency localised wave evolution, given
$\lambda\in 2^{\mathbb{N}}$ and $\ell\in\mathbb{R}$ we use the norm

$\|v\|_{W^{\ell}_{\lambda}}=\lambda^{\ell}\|v\|_{L^{\infty}_{t}L^{2}_{x}(\mathbb{R}^{1+4})}+\lambda^{\ell-1}\|(i\partial_{t}+|\nabla|)v\|_{L^{2}_{t,x}(\mathbb{R}^{1+4})}$

together with the stronger variant

$\|v\|_{\underline{W}^{\ell}_{\lambda}}=\lambda^{\ell}\|v\|_{L^{\infty}_{t}L^{2}_{x}(\mathbb{R}^{1+4})}+\|(i\partial_{t}+|\nabla|)v\|_{R^{\ell}_{\lambda}}$

where

$\|G\|_{R^{\ell}_{\lambda}}=\lambda^{\ell}\|P^{(t)}_{\leqslant
2^{-16}\lambda^{2}+2^{16}}G\|_{L^{1}_{t}L^{2}_{x}(\mathbb{R}^{1+4})}+\lambda^{\ell-1}\|G\|_{L^{2}_{t,x}(\mathbb{R}^{1+4})}$

and $\lambda\in 2^{\mathbb{Z}}$ and $\ell\in\mathbb{R}$. The iteration spaces
are then formed by summing up over frequencies $\lambda\in 2^{\mathbb{N}}$ in
$\ell^{2}(\mathbb{N})$, thus

$\|u\|_{S^{s}}=\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|u_{\lambda}\|_{S^{s}_{\lambda}}^{2}\Big{)}^{\frac{1}{2}},\qquad\|u\|_{\underline{S}^{s}}=\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|u_{\lambda}\|_{\underline{S}^{s}_{\lambda}}^{2}\Big{)}^{\frac{1}{2}}$

and

$\|v\|_{W^{\ell}}=\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|v_{\lambda}\|_{W^{\ell}_{\lambda}}^{2}\Big{)}^{\frac{1}{2}},\qquad\|v\|_{\underline{W}^{\ell}}=\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|v_{\lambda}\|_{\underline{W}^{\ell}_{\lambda}}^{2}\Big{)}^{\frac{1}{2}}$

and

$\|F\|_{N^{s}}=\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|F_{\lambda}\|_{N^{s}_{\lambda}}^{2}\Big{)}^{\frac{1}{2}},\qquad\|G\|_{R^{\ell}}=\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|G_{\lambda}\|_{R^{\ell}_{\lambda}}^{2}\Big{)}^{\frac{1}{2}}.$

The spaces $S^{s}$ and $\underline{S}^{s}$ are nested, contain all $H^{s}$
free solutions to the Schrödinger equation, and moreover also control the
Strichartz spaces $L^{q}_{t}L^{r}_{x}$. More precisely, an application of [10,
Lemma 2.1] and the endpoint Strichartz estimate [19] shows that for any
Schrödinger admissible exponents $(q,r)$ on $\mathbb{R}\times\mathbb{R}^{4}$,
thus $2\leqslant q,r\leqslant\infty$ and $\frac{1}{q}+\frac{2}{r}=1$, and any
$\lambda\in 2^{\mathbb{N}}$ we have the bounds

$\lambda^{s}\|u_{\lambda}\|_{L^{q}_{t}L^{r}_{x}(\mathbb{R}^{1+4})}\lesssim\|u_{\lambda}\|_{S^{s}_{\lambda}}\lesssim\|u_{\lambda}\|_{\underline{S}^{s}_{\lambda}},\qquad\|e^{it\Delta}f\|_{\underline{S}^{s}}\lesssim\|f\|_{H^{s}}.$
(2.1)

A similar bound applies in the wave case, although strictly speaking only the
stronger space $\underline{W}^{\ell}$ controls the (wave admissible) spaces
$L^{a}_{t}L^{b}_{x}$. In fact this was the key reason to introduce the
stronger variant $\underline{W}^{\ell}$. In more detail, if $(a,b)$ are wave
admissible on $\mathbb{R}\times\mathbb{R}^{4}$, thus $2\leqslant
a,b\leqslant\infty$ and $\frac{1}{a}+\frac{3}{2r}=\frac{3}{2}$, then we have
bounds

$\lambda^{\ell-\frac{5}{3a}}\|V_{\lambda}\|_{L^{a}_{t}L^{b}_{x}(\mathbb{R}^{1+4})}\lesssim\|V_{\lambda}\|_{\underline{W}^{\ell}_{\lambda}},\qquad\|V_{\lambda}\|_{W^{\ell}_{\lambda}}\lesssim\|V_{\lambda}\|_{\underline{W}^{\ell}_{\lambda}},\qquad\|e^{it|\nabla|}g\|_{\underline{W}^{\ell}}\lesssim\|f\|_{H^{\ell}}.$
(2.2)

Strictly speaking the bounds in (2.2) are not contained in [10], but follow
from an identical argument. For instance, after decomposing
$V_{\lambda}=P^{(t)}_{<2^{-16}\lambda+2^{16}}V_{\lambda}+P^{(t)}_{\geqslant
2^{-16}\lambda+2^{16}}V_{\lambda}$ an application of Bernstein’s inequality
and the endpoint free Strichartz estimate for the wave equation [19] gives for
any $\lambda\in 2^{\mathbb{N}}$

$\displaystyle\|V_{\lambda}\|_{L^{2}_{t}L^{6}_{x}}$
$\displaystyle\leqslant\|P^{(t)}_{<2^{-16}\lambda+2^{16}}V_{\lambda}\|_{L^{2}_{t}L^{6}_{x}}+\lambda^{\frac{4}{3}}\|P^{(t)}_{\geqslant
2^{-16}\lambda+2^{16}}V_{\lambda}\|_{L^{2}_{t,x}}$
$\displaystyle\lesssim\lambda^{\frac{5}{6}}\Big{(}\|P^{(t)}_{<2^{-16}\lambda+2^{16}}V_{\lambda}(0)\|_{L^{2}_{x}}+\|P^{(t)}_{<2^{-16}\lambda+2^{16}}V_{\lambda}\|_{L^{1}_{t}L^{2}_{x}}\Big{)}+\lambda^{\frac{1}{3}-1}\|(i\partial_{t}+|\nabla|)V_{\lambda}\|_{L^{2}_{t,x}}$
$\displaystyle\lesssim\lambda^{\frac{5}{6}-\ell}\|V_{\lambda}\|_{\underline{W}^{\ell}}.$

The convexity of $L^{p}$ norms, together with the trivial bound
$\|V_{\lambda}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\|V_{\lambda}\|_{\underline{W}^{\ell}}$,
then gives the first inequality in (2.2). The remaining bounds in (2.2) are
immediate from the definition of the norms.

We also require some way to measure how dispersed a solution is, this is
accomplished via the “dispersive” norms

$\|u\|_{D}=\|u\|_{L^{2}_{t}W^{\frac{1}{2},4}_{x}(\mathbb{R}^{1+4})},\qquad\|u\|_{Y}=\sup_{\lambda\in
2^{\mathbb{N}}}\lambda^{-4}\|u_{\lambda}\|_{L^{\infty}_{t,x}(\mathbb{R}^{1+4})}.$

These norms form a crucial component of the analysis, and are used to extract
some smallness out of the error terms in the profile decomposition. Note that
the Strichartz control (2.1) implies that

$\|u\|_{D}\lesssim\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\lambda\|u_{\lambda}\|_{L^{2}_{t}L^{4}_{x}}^{2}\Big{)}^{\frac{1}{2}}\lesssim\|u\|_{S^{\frac{1}{2}}}$
(2.3)

and thus the dispersive norm $\|\cdot\|_{D}$ is controlled by the iteration
norm $\|\cdot\|_{S^{\frac{1}{2}}}$. On the other hand, the norm
$\|\cdot\|_{Y}$ can be controlled by the _homogeneous_ Sobolev spaces. More
precisely, a short computation using Bernstein’s inequality gives for any
$-2\leqslant s<2$

$\displaystyle\|u\|_{Y}$ $\displaystyle\leqslant\sum_{\mu\in
2^{\mathbb{Z}},\mu\leqslant
1}\|\dot{P}_{\mu}u\|_{L^{\infty}_{t,x}}+\sup_{\lambda\in
2^{\mathbb{N}},\lambda>1}\lambda^{-4}\|u_{\lambda}\|_{L^{\infty}_{t,x}}$
$\displaystyle\lesssim\sum_{\mu\in 2^{\mathbb{Z}},\mu\leqslant
1}\mu^{2}\|\dot{P}_{\mu}u\|_{L^{\infty}_{t}L^{2}_{x}}+\sup_{\lambda\in
2^{\mathbb{N}},\lambda>1}\lambda^{-2}\|u_{\lambda}\|_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\sup_{\mu\in
2^{\mathbb{Z}}}\mu^{s}\|\dot{P}_{\mu}u\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\|u\|_{L^{\infty}_{t}\dot{H}^{s}_{x}}.$
(2.4)

Moreover, at least for free Schrödinger waves, the two dispersive norms can be
compared by noting that another application of Bernstein’s inequality (in both
$t$ and $x$) together with the endpoint Strichartz estimate gives

$\|e^{it\Delta}f\|_{Y}\lesssim\sup_{\lambda\in
2^{\mathbb{N}}}\lambda^{-2}\|e^{it\Delta}f_{\lambda}\|_{L^{2}_{t}L^{4}_{x}}\lesssim\|e^{it\Delta}f\|_{D}.$

Note that there is a derivative loss in this bound. Roughly the $Y$ norm
scales like $\dot{H}^{-2}_{x}$, while $D$ scales like $\dot{H}^{\frac{1}{2}}$
(at least at high frequencies). The choice of scaling for the $Y$ norm is
slightly arbitrary however. In fact, due to the fact that the energy is
subcritical (at least for the Schrödinger component of the Zakharov
evolution), we could equally well have worked with a norm that scales at high
frequencies like $\dot{H}^{-N}$ for any $N$.

Finally, a key step in proof of the main bilinear estimate requires the
bilinear restriction theory for the paraboloid. As in [9], the precise
estimates we need are phrased in terms of the inhomogeneous Strichartz type
norm

$\|u\|_{Z}=\|u\|_{L^{\infty}_{t}L^{2}_{x}}+\|(i\partial_{t}+\Delta)u\|_{L^{1}_{t}L^{2}_{x}+L^{2}_{t}L^{4}_{x}}.$

Given a Banach space of functions $X$ defined on
$\mathbb{R}\times\mathbb{R}^{d}$ together with an interval
$I\subset\mathbb{R}$, we define the temporally restricted spaces $X(I)$ via
the usual construction

$\|u\|_{X(I)}=\inf_{\tilde{u}\in X\text{ and }\tilde{u}|I=u}\|\tilde{u}\|_{X}$

with the convention that $\|u\|_{X(I)}=\infty$ if no such extension exists. As
we are using the standard convention
$L^{q}_{t}L^{r}_{x}=L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{4})$, the
above definition together with a simple argument gives the identity

$\|u\|_{L^{q}_{t}L^{r}_{x}(I)}=\|u\|_{L^{q}_{t}L^{r}_{x}(I\times\mathbb{R}^{4})}.$

This short hand is used frequently in the following.

###  2.3. Solution Operators

Define the inhomogeneous solution operators for the Schrödinger equation and
wave equation with zero data at $t=t_{0}$ as

$\mathcal{I}_{0,t_{0}}[F]=-i\int_{t_{0}}^{t}e^{i(t-s)\Delta}F(s)ds,\qquad\mathcal{J}_{0,t_{0}}[G]=-i\int_{t_{0}}^{t}e^{i(t-s)|\nabla|}G(s)ds.$

In the special case $t_{0}=0$ we simply write
$\mathcal{I}_{0,t_{0}}=\mathcal{I}_{0}$ and
$\mathcal{J}_{0,t_{0}}=\mathcal{J}_{0}$. The arguments in [10, Lemma 2.4 and
Lemma 2.6] give for any $s,\ell\in\mathbb{R}$ the energy type bounds

$\|\mathcal{I}_{0,t_{0}}[F]\|_{\underline{S}^{s}}\lesssim\|F\|_{N^{s}},\qquad\|\mathcal{J}_{0,t_{0}}[G]\|_{\underline{W}^{\ell}}\lesssim\|G\|_{G^{\ell}}.$
(2.5)

In the following, we frequently have to work with the Duhamel expressions
$\mathcal{I}_{0,t_{0}}[F]$ in the local in time spaces $S^{s}(I)$. To make
this slightly easier, we have the following useful relationship on the local
spaces $S^{s}(I)$ which roughly gives a distinguished extension of
$\mathcal{I}_{0,t_{0}}[F]$ from the interval $I$ to $\mathbb{R}$.

######  Lemma 2.1 (Local in time characterisation).

Let $s,\ell\in\mathbb{R}$. There exists a constant $C>0$ such that for any
interval $I\subset\mathbb{R}$ and any initial time $t_{0}\in I$ we have

$C^{-1}\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{s}(I)}\leqslant\|\mathcal{I}_{0}[\mathbbold{1}_{I}F]\|_{S^{s}}\leqslant
C\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{s}(I)}$

and

$C^{-1}\|\mathcal{J}_{0,t_{0}}[G]\|_{W^{\ell}(I)}\leqslant\|\mathcal{J}_{0}[\mathbbold{1}_{I}G]\|_{W^{\ell}}\leqslant
C\|\mathcal{J}_{0,t_{0}}[G]\|_{W^{\ell}(I)}.$

###### Proof.

We only prove the Schrödinger case as the wave case is similar. We begin by
observing that for any $\tau,\tau^{\prime}\in\mathbb{R}$ we have the identity

$\mathcal{I}_{0,\tau}[F](t)=\mathcal{I}_{0,\tau^{\prime}}[F](t)-e^{i(t-\tau)\Delta}\mathcal{I}_{0,\tau^{\prime}}[F](\tau).$
(2.6)

Hence, after noting that

$\|e^{i(t-\tau)\Delta}\mathcal{I}_{0,\tau^{\prime}}[F](\tau)\|_{S^{s}}\lesssim\|\mathcal{I}_{0,\tau^{\prime}}[F](\tau)\|_{H^{s}}\lesssim\|\mathcal{I}_{0,\tau^{\prime}}[F]\|_{L^{\infty}_{t}H^{s}_{x}}\lesssim\|\mathcal{I}_{0,\tau^{\prime}}[F]\|_{S^{s}},$

we conclude that for any $\tau\in\mathbb{R}$ we have the useful bound

$\|\mathcal{I}_{0}[F]\|_{S^{s}}\approx\|\mathcal{I}_{0,\tau}[F]\|_{S^{s}}.$
(2.7)

Let $t_{0}\in I\subset\mathbb{R}$ and suppose that $\psi\in S^{s}$ is an
extension of $\mathcal{I}_{0,t_{0}}[F]$ from $I$ to $\mathbb{R}$. Then letting
$t_{-}<t_{+}$ denote the endpoints of the interval $I$ we have

$\mathcal{I}_{0,t_{0}}[\mathbbold{1}_{I}F](t)=\begin{cases}e^{i(t-t_{+})\Delta}\psi(t_{+}),&t_{+}<t,\\\
\psi(t),&t_{-}\leqslant t\leqslant t_{+},\\\
e^{i(t-t_{-})\Delta}\psi(t_{-}),&t<t_{-}.\end{cases}$

and therefore the definition of the norm $\|\cdot\|_{S^{s}_{\lambda}}$ gives
for any $\lambda\in 2^{\mathbb{N}}$

$\displaystyle\|\mathcal{I}_{0,t_{0}}[\mathbbold{1}_{I}F_{\lambda}]\|_{S^{s}_{\lambda}}$
$\displaystyle\lesssim\lambda^{s}\Big{(}\|\psi_{\lambda}\|_{L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{4}_{x}(I)}+\|e^{i(t-t_{+})\Delta}\psi_{\lambda}(t_{+})\|_{L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{4}_{x}(t>t_{+})}$
$\displaystyle\qquad\qquad+\|e^{i(t-t_{-})\Delta}\psi_{\lambda}(t_{-})\|_{L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{4}_{x}(t<t_{-})}+\lambda^{-1}\|(i\partial_{t}+\Delta)\psi_{\lambda}\|_{L^{2}_{t,x}(I)}\Big{)}$
$\displaystyle\lesssim\lambda^{s}\Big{(}\|\psi_{\lambda}\|_{L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{4}_{x}}+\lambda^{-1}\|(i\partial_{t}+\Delta)\psi_{\lambda}\|_{L^{2}_{t,x}}\Big{)}=\|\psi_{\lambda}\|_{S^{s}_{\lambda}}.$

Consequently, after summing up over frequencies and applying (2.7), we obtain

$\|\mathcal{I}_{0}[\mathbbold{1}_{I}F]\|_{S^{s}}\approx\|\mathcal{I}_{0,t_{0}}[\mathbbold{1}_{I}F]\|_{S^{s}}\lesssim\|\psi\|_{S^{s}}.$

Taking the infimum over all possible extensions, the definition of the time
restricted spaces $S^{s}(I)$ gives

$\|\mathcal{I}_{0}[\mathbbold{1}_{I}F]\|_{S^{s}}\lesssim\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{s}(I)}.$

To prove the converse inequality, we simply note that since $t_{0}\in I$ we
have the identity

$\mathbbold{1}_{I}\mathcal{I}_{0,t_{0}}[F]=\mathbbold{1}_{I}\mathcal{I}_{0,t_{0}}[\mathbbold{1}_{I}F]$

and hence again applying (2.7) we see that

$\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{s}(I)}\leqslant\|\mathcal{I}_{0,t_{0}}[\mathbbold{1}_{I}F]\|_{S^{s}}\lesssim\|\mathcal{I}_{0}[\mathbbold{1}_{I}F]\|_{S^{s}}.$

∎

Due to the presence of the derivative in time, bounding $\|u\|_{S^{s}(I\cup
J)}$ by the sum $\|u\|_{S^{s}(I)}+\|u\|_{S^{s}(J)}$ involves losing a factor
which depends on the size of the intervals [10, Lemma 2.8]. This is a slight
technical inconvenience. However the Duhamel operator $\mathcal{I}_{0,t_{0}}$
_does_ satisfy a useful decomposability bound which is independent of the
intervals.

######  Lemma 2.2 (Duhamel operator is decomposable).

Let $s\in\mathbb{R}$ and $I,I_{j}\subset\mathbb{R}$, $1\leqslant j\leqslant
N$, be intervals with $I=\cup_{j=1}^{N}I_{j}$. Then for any $t_{0}\in I$ and
$t_{j}\in I_{j}$ we have

$\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{s}(I)}\lesssim\sum_{j=1}^{N}\|\mathcal{I}_{0,t_{j}}[F]\|_{S^{s}(I_{j})}$

###### Proof.

The proof is a direct consequence of Lemma 2.1. Choose disjoint intervals
$I_{j}^{*}\subset I_{j}$ such that $t_{j}\in I_{j}^{*}$ and $I=\cup
I_{j}^{*}$. Then as $t_{0}\in I$, two applications of Lemma 2.2 give

$\displaystyle\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{s}(I)}\approx\|\mathcal{I}_{0}[\mathbbold{1}_{I}F]\|_{S^{s}}\leqslant\sum_{j=1}^{N}\|\mathcal{I}_{0}[\mathbbold{1}_{I_{j}^{*}}F]\|_{S^{s}}\approx\sum_{j=1}^{N}\|\mathcal{I}_{0,t_{j}}[F]\|_{S^{s}(I_{j}^{*})}\lesssim\sum_{j=1}^{N}\|\mathcal{I}_{0,t_{j}}[F]\|_{S^{s}(I_{j})}.$

∎

###  2.4. Solution Operators with Potentials

Our eventual goal is to understand solutions to the Zakharov equation with
data close to the ground state. In particular, this means we need to consider
the Schrödinger flow with a free wave potential which is not small. To this
end, following [9] we introduce notation for the solution operator for the
linear flow perturbed by a potential. More precisely, given a time dependent
potential $v(t,x):\mathbb{R}^{1+4}\to\mathbb{R}$, we let
$\mathcal{U}_{v,t_{0}}(t)f$ denote the solution to the homogeneous problem

$i\partial_{t}u+\Delta u-vu=0,\qquad u(t_{0})=f.$

Similarly, we let $\mathcal{I}_{v,t_{0}}[F]$ denote the solution to the
inhomogeneous problem

$(i\partial_{t}+\Delta-v)u=F,\qquad u(t_{0})=0.$

As previously, in the special case where the initial time $t_{0}=0$, we write
$\mathcal{U}_{v,t_{0}}=\mathcal{U}_{v}$,
$\mathcal{I}_{v,t_{0}}=\mathcal{I}_{v}$. Under reasonable assumptions on the
potential $v$, data $f$, and forcing term $F$, the operators
$\mathcal{U}_{v,t_{0}}f,\mathcal{I}_{v,t_{0}}[F]\in C(I;H^{s})$ are always
well-defined [9]. For instance if the potential $v=\Re(e^{it|\nabla|}g)\in
L^{\infty}_{t}L^{2}_{x}$ is a free wave, then for any $0\leqslant s<1$ we have
the energy estimate

$\|\mathcal{I}_{v,t_{0}}[F]\|_{\underline{S}^{s}(I)}\lesssim_{g}\|F\|_{N^{s}(I)},$

see [10]. This can be improved to a _uniform_ in $g$ estimate provided that
$g$ has $L^{2}$ norm below the ground state. In fact provided
$0<B<\|Q\|_{\dot{H}^{1}}$ we have for any $\|g\|_{L^{2}}\leqslant B$ the
uniform bound

$\|\mathcal{I}_{v,t_{0}}[F]\|_{\underline{S}^{s}(I)}\lesssim_{B}\|F\|_{N^{s}(I)}$

where the implied constant depends only on $B$ (and $0\leqslant s<1$) [9].
More formally, we have the following improvement to (2.5).

######  Theorem 2.3 (Strichartz with free wave potential [9, Theorem 6.1]).

Let $0\leqslant s<1$ and $0<B<\|Q\|_{\dot{H}^{1}}$. There is a constant
$C=C(B)>0$ such that if $I\subset\mathbb{R}$ is an interval, then for any
$g\in L^{2}$ with $\|g\|_{L^{2}}\leqslant B$ we have

$\|u\|_{\underline{S}^{s}(I)}\leqslant C\Big{[}\inf_{t\in
I}\|u(t)\|_{H^{s}}+\big{\|}\big{(}i\partial_{t}+\Delta-\Re(e^{it|\nabla|}g)\big{)}u\big{\|}_{N^{s}(I)}\Big{]}.$

###  2.5. Standard Bilinear Estimates

We frequently rely on the bilinear estimates contained in [10]. Although the
estimates in [10] are valid in the full region of admissible regularities for
the Zakharov equation (including the energy space $H^{1}\times L^{2}$), the
general statement requires more complicated iteration spaces than those we
require here. For our purposes, it suffices to work in the energy subcritical
setting $H^{s}\times L^{2}$ with $\frac{1}{2}\leqslant s<1$. In this regime,
the estimates in [10] can be stated as follows.

######  Theorem 2.4 (Standard bilinear estimates [10]).

Let $0\leqslant s<1$ and $0\leqslant\ell<\frac{1}{2}$. Then for any interval
$0\in I\subset\mathbb{R}$ we have

$\displaystyle\|vu\|_{N^{s}(I)}$
$\displaystyle\lesssim\|v\|_{W^{0}(I)}\|u\|_{S^{s}(I)},$
$\displaystyle\big{\|}\mathcal{J}_{0}\big{[}|\nabla|(\overline{u}w)\big{]}\big{\|}_{\underline{W}^{\ell}(I)}$
$\displaystyle\lesssim\|u\|_{S^{\ell+\frac{1}{2}}(I)}\|w\|_{S^{\ell+\frac{1}{2}}(I)},$

and the corresponding endpoint counterparts

$\displaystyle\|vu\|_{N^{\frac{1}{2}}(I)}$
$\displaystyle\lesssim\|v\|_{W^{0}(I)}\Big{(}\|u\|_{D(I)}\|u\|_{S^{\frac{1}{2}}(I)}\Big{)}^{\frac{1}{2}},$
$\displaystyle\big{\|}\mathcal{J}_{0}\big{[}|\nabla|(\overline{u}w)\big{]}\big{\|}_{\underline{W}^{0}(I)}$
$\displaystyle\lesssim\Big{(}\|u\|_{D(I)}\|w\|_{D(I)}\|u\|_{S^{\frac{1}{2}}(I)}\|w\|_{S^{\frac{1}{2}}(I)}\Big{)}^{\frac{1}{2}}.$

The bilinear estimates in Theorem 2.4 suffice for the small data global
theory. However the proof of Theorem 1.6 requires a substantial improvement
over these estimates. In particular, partially due to the fact that we don’t
have a good profile decomposition in the endpoint Strichartz spaces
$L^{2}_{t}L^{4}_{x}$, we require a version of Theorem 2.4 with a factor of the
norm $\|\cdot\|_{Y}$ on the righthand side, see Theorem 4.1 for a precise
statement.

##  3\. The Small Data Theory

In this section we collect together a number of consequences of the bilinear
estimates contained in Theorem 2.4. In particular all results proved in this
section can be seen as consequences of the estimates contained in [9, 10].

###  3.1. Persistence of Regularity

We start with a persistence of regularity type result which follows from the
estimates in [9, 10]. It closely related to a more general persistence of
regularity result [10, Theorem 8.1] (which is valid for general regularities).
The advantage here is that, via the uniform Strichartz estimate in Theorem 2.3
we obtain a uniform bound in the iteration spaces
$\underline{S}^{s}\times\underline{W}^{0}$ which is independent of the profile
of the solution $(u,V)$. This uniformness is particularly useful later, as we
frequently have good control over the size of $(u,V)$ in various norms, but
not its profile.

Note that the energy space $H^{1}\times L^{2}$ is excluded from the result
below. However this is not an issue, as for our purposes it suffices to work
in $H^{s}\times L^{2}$ with $\frac{1}{2}\leqslant s<1$. As explained in the
introduction, once we have a global solution with $\|u\|_{D}<\infty$,
scattering in the energy space $H^{1}\times L^{2}$ follows from Theorem 1.3.

######  Theorem 3.1 (Persistence of regularity).

Let $\frac{1}{2}\leqslant s<1$, $A>0$, and $0<B<\|Q\|_{\dot{H}^{1}}$. There
exists a constant $C_{A,B}>0$ such that for any interval $I\subset\mathbb{R}$,
if $(u,V)\in C(I;H^{\frac{1}{2}}\times L^{2})$ solves (1.2) on
$I\times\mathbb{R}^{4}$ with

$\|u\|_{L^{\infty}_{t}H^{\frac{1}{2}}_{x}(I\times\mathbb{R}^{4})}+\|u\|_{D(I)}\leqslant
A,\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}(I\times\mathbb{R}^{4})}\leqslant B,$
(3.1)

then

$\|u\|_{\underline{S}^{s}(I)}\leqslant C_{A,B}\inf_{t\in
I}\|u(t)\|_{H^{s}},\qquad\|V\|_{\underline{W}^{0}(I)}\leqslant\inf_{t\in
I}\|V(t)\|_{L^{2}}+C_{A,B}\inf_{t\in I}\|u(t)\|_{H^{\frac{1}{2}}}^{2}.$

###### Proof.

We start by noting that an application of Theorem 2.4 gives for any
$0\leqslant s<1$ and $0\in I^{\prime}\subset\mathbb{R}$ the bounds

$\|Vu\|_{N^{s}(I^{\prime})}\lesssim\|V\|_{W^{0}(I^{\prime})}\|u\|_{S^{s}(I^{\prime})}$
(3.2)

and

$\big{\|}\mathcal{J}_{0}\big{[}|\nabla||u|^{2}\big{]}\big{\|}_{\underline{W}^{0}(I^{\prime})}\lesssim\|u\|_{D(I^{\prime})}\|u\|_{S^{\frac{1}{2}}(I^{\prime})}.$
(3.3)

Together with the uniform Strichartz estimate, Theorem 2.3, we conclude that
provided

$\|V(0)\|_{L^{2}}\leqslant B<\|Q\|_{\dot{H}^{1}}$

we have for any $\frac{1}{2}\leqslant s<1$

$\displaystyle\|u\|_{S^{s}(I^{\prime})}$
$\displaystyle\lesssim_{B}\|u(0)\|_{H^{s}}+\big{\|}[i\partial_{t}+\Delta-\Re(e^{it|\nabla|}V(0))]u\big{\|}_{N^{s}(I)}$
$\displaystyle\lesssim_{B}\|u(0)\|_{H^{s}}+\big{\|}\mathcal{J}_{0}\big{[}|\nabla||u|^{2}\big{]}\big{\|}_{\underline{W}^{0}(I^{\prime})}\|u\|_{S^{s}(I^{\prime})}\lesssim_{B}\|u(0)\|_{H^{s}}+\|u\|_{D(I^{\prime})}\|u\|_{S^{\frac{1}{2}}(I^{\prime})}\|u\|_{S^{s}(I^{\prime})}.$

Time translation invariance and the assumption (3.1), together with the local
well-posedness in $S^{s}\times W^{0}$ from [10] and a continuity argument
(starting with the case $s=\frac{1}{2}$ and then $s>\frac{1}{2}$) implies that
there exists $\delta\ll_{A,B}1$ such that for any interval $I^{\prime}\subset
I$ we have the implication

$\|u\|_{D(I^{\prime})}\leqslant\delta\qquad\Longrightarrow\qquad\|u\|_{S^{s}(I^{\prime})}\lesssim_{B}\inf_{t\in
I^{\prime}}\|u(t)\|_{H^{s}}.$ (3.4)

To extend this bound to the whole interval $I\subset\mathbb{R}$, we again
exploit the assumption (3.1). More precisely, the upper bound on
$\|u\|_{D(I)}$ implies that we can find a collection of $J\in\mathbb{N}$
intervals $I_{j}\subset I$ and times $t_{j}\in I_{j}\cap I_{j+1}$ such that

$I=\cup_{j=1}^{J}I_{j},\qquad\|u\|_{D(I_{j})}\leqslant\delta,\qquad
J\lesssim\delta^{-2}A^{2}\approx_{A,B}1,\qquad t_{j}\in I_{j}\cap I_{j+1}.$

A short argument using the implication (3.4) then implies that

$\sum_{j=1}^{J}\|u\|_{S^{s}(I_{j})}\lesssim_{A,B}\inf_{t\in
I}\|u(t)\|_{H^{s}}.$ (3.5)

Hence applying this bound with $s=\frac{1}{2}$ together with the energy
inequality (2.5) and the bilinear estimate (3.3) gives

$\sup_{j}\|V\|_{W^{0}(I_{j})}\lesssim
B+\sup_{j}\big{\|}\mathcal{J}_{0}\big{[}|\nabla||u|^{2}\big{]}\big{\|}_{\underline{W}^{0}(I_{j})}\lesssim
B+\sup_{j}\|u_{j}\|_{S^{\frac{1}{2}}(I_{j})}^{2}\lesssim_{A,B}1.$

Therefore, applying Lemma 2.2, the energy inequality (2.5), and (3.2) we
conclude that

$\displaystyle\|u\|_{S^{s}(I)}\lesssim\inf_{t\in
I}\|u(t)\|_{H^{s}}+\sum_{j=1}^{J}\|\Re(V)u\|_{N^{s}(I_{j})}\lesssim_{A,B}\Big{(}1+\sup_{j}\|V\|_{W^{0}(I_{j})}\Big{)}\inf_{t\in
I}\|u(t)\|_{H^{s}}\lesssim_{A,B}\inf_{t\in I}\|u(t)\|_{H^{s}}.$

To upgrade this bound to the space $\underline{S}^{s}(I)$, we simply plug in
the equation once more and observe that

$\|V\|_{\underline{W}^{0}(I)}\leqslant\inf_{t\in
I}\|V(t)\|_{L^{2}}+C\|u\|_{S^{\frac{1}{2}}(I)}^{2}\leqslant\inf_{t\in
I}\|V(t)\|_{L^{2}}+C_{A,B}\inf_{t\in
I}\|u(t)\|_{H^{\frac{1}{2}}}^{2}\lesssim_{A,B}1$

and

$\|u\|_{\underline{S}^{s}(I)}\lesssim\inf_{t\in
I}\|u(t)\|_{H^{s}}+\|\Re(V)u\|_{N^{s}(I)}\leqslant C_{A,B}\inf_{t\in
I}\|u(t)\|_{H^{s}}.$

∎

######  Remark 3.2.

By arguing as in Section 7 of [10], at a cost of allowing the constant
$C_{A,B}$ to depend on the profile of $V$ and not just its size, the
assumption $\|V\|_{L^{\infty}_{t}L^{2}_{x}}\leqslant B<\|Q\|_{\dot{H}^{1}}$ in
Theorem 3.1 can be removed. This is a consequence of the dispersive properties
of the free wave evolution. Moreover, although the energy scaling $s=1$ is
excluded, a version of the above persistence of regularity result is true for
all (admissible) regularities $H^{s}\times H^{\ell}$, see [10] for details.

Theorem 3.1 implies that for the purpose of proving global well-posedness and
scattering, the condition $\|u\|_{D}<\infty$ is equivalent to
$\|u\|_{L^{2}_{t}B^{s}_{4,2}}<\infty$. In particular we could equally well
have taken our dispersive norm to be the non-endpoint Besov variant
$\|\cdot\|_{L^{2}_{t}B^{s}_{4,2}}$.

###  3.2. Motivation for Controlling Quantity

Suppose (for instance) we were trying to construct a fixed point to the
(cubic) nonlinear problem

$u=u_{0}+\mathcal{I}[u^{3}]$

where as usual $u_{0}$ is the free evolution of the data, and $\mathcal{I}$
denotes the corresponding Duhamel term. Assume we have a Banach space $X$
together with the nonlinear bound

$\|\mathcal{I}[uvw]\|_{X}\lesssim\|u\|_{X}\|v\|_{X}\|w\|_{X}.$

One approach to construct the fixed point $u$ would be to consider the
sequence $(u_{k})_{k\in\mathbb{N}}\subset X$ defined iteratively as usual via

$u_{k}=u_{0}+\mathcal{I}[u_{k-1}^{3}].$

If we are working in a small data setting, we expect that the fixed point $u$
should be close to the free evolution $u_{0}$. This suggests that we should
expand the nonlinearity into factors $u-u_{0}$. Together with our assumed
nonlinear bound, we would then obtain

$\displaystyle\|\mathcal{I}[u^{3}]\|_{X}$
$\displaystyle=\|\mathcal{I}[(u-u_{0})^{3}+3(u-u_{0})^{2}u_{0}+3(u-u_{0})u_{0}^{2}+u_{0}^{3}]\|_{X}$
$\displaystyle\lesssim\|u-u_{0}\|_{X}^{3}+\|u-u_{0}\|_{X}^{2}\|u_{0}\|_{X}+\|\mathcal{I}[(u-u_{0})u_{0}^{2}]\|_{X}+\|\mathcal{I}[u_{0}^{3}]\|_{X}.$

and

$\displaystyle\|\mathcal{I}[u^{3}-v^{3}]\|_{X}$
$\displaystyle=\|\mathcal{I}[(u-v)^{3}+3(u-v)^{2}v+3(u-v)v^{2}]\|_{X}$
$\displaystyle\lesssim\|u-v\|_{X}\Big{(}\|u-v\|_{X}^{2}+\|u-v\|_{X}\|v\|_{X}+\|v-u_{0}\|_{X}\|v\|_{X}+\|v-u_{0}\|_{X}^{2}\Big{)}+\|\mathcal{I}[(u-v)u_{0}^{2}]\|_{X}.$

In other words, if we define the controlling quantity

$\rho(v)=\sup_{\|\phi\|_{X}\leqslant 1}\|\mathcal{I}[\phi v^{2}]\|_{X},$

then provided $\|u_{0}\|_{X}\leqslant A$ we conclude that

$\|u_{k+1}-u_{0}\|_{X}\lesssim\|u_{k}-u_{0}\|_{X}^{3}+A\|u_{k}-u_{0}\|_{X}^{2}+\big{(}\|u_{k}-u_{0}\|_{X}+A\big{)}\rho(u_{0})$

and

$\|u_{k+1}-u_{k}\|_{X}\lesssim\|u_{k}-u_{k-1}\|_{X}\Big{(}\|u_{k}-u_{k-1}\|_{X}^{2}+(\|u_{k}-u_{k-1}\|_{X}+\|u_{k-1}-u_{0}\|_{X})(\|u_{k-1}-u_{0}\|_{X}+A)+\rho(u_{0})\Big{)}.$

Therefore, assuming that $\epsilon=\epsilon(A)>0$ is sufficiently small, under
the smallness assumption

$\|u_{0}\|_{X}\leqslant A,\qquad\rho(u_{0})\leqslant\epsilon$

we conclude the existence of a fixed point $u\in X$ to the problem
$u=u_{0}+\mathcal{I}[u^{3}]$. In other words, we do not require the data to be
small in $X$, but only the nonlinear interaction of the free solution with a
generic element of our iteration space $X$ must be small. Note that the
nonlinear estimate immediately implies

$\rho(u_{0})\lesssim\|u_{0}\|_{X}^{2}$

and so clearly it suffices to have smallness in $X$. On the other hand,
typically in applications we try to prove an improved estimate of the form

$\rho(u_{0})\leqslant\|u_{0}\|_{Y}^{\theta}\|u_{0}\|_{X}^{2-\theta}$

where $Y$ is some intermediate norm, thus improving the assumption of
smallness in $X$ to requiring just smallness in $Y$. The advantage of
considering the quantity $\rho$ is that we only have to prove this estimate
for the free solution $u_{0}$, as opposed to all elements of $X$ (which is
what we would be forced to do if we added it into the iteration argument).
Moreover the quantity $\rho$ is very flexible, as we can potentially formulate
the improved estimate with a variety of choices of norms $Y$.

###  3.3. A Stability Theorem

Motivated by the discussion in the previous subsection, given an interval
$I\subset\mathbb{R}$ and $u\in C(I;H^{\frac{1}{2}})$, we define the
controlling quantity

$\rho_{I}(u)=\sup_{\|v\|_{\underline{W}^{0}}\leqslant
1}\big{\|}\mathcal{I}_{0}\big{[}\mathbbold{1}_{I}\Re(v)u\big{]}\big{\|}_{S^{\frac{1}{2}}}+\sup_{\|w\|_{\underline{S}^{\frac{1}{2}}}\leqslant
1}\big{\|}\mathcal{J}_{0}\big{[}\mathbbold{1}_{I}|\nabla|\Re(\overline{w}u)\big{]}\big{\|}_{W^{0}}.$
(3.6)

In view of (2.7), the initial time $t=0$ in the definition of $\rho_{I}$ could
be replaced with any $t_{0}\in\mathbb{R}$ at a cost of multiplying by some
absolute constant. A quick application of Lemma 2.1 gives for any $t_{0}\in I$
the bounds

$\|\mathcal{I}_{0,t_{0}}[\Re(v)u]\|_{S^{\frac{1}{2}}(I)}\lesssim\rho_{I}(u)\|v\|_{\underline{W}^{0}(I)},\qquad\|\mathcal{J}_{0,t_{0}}[|\nabla|\Re(\overline{w}u)]\|_{W^{0}(I)}\lesssim\rho_{I}(u)\|w\|_{\underline{S}^{\frac{1}{2}}(I)}.$

Moreover, a short computation via Theorem 2.4 and (2.5) gives

$\rho_{I}(u)\lesssim\|u\|_{S^{\frac{1}{2}}(I)},\qquad\rho_{I}(u+u^{\prime})\leqslant\rho_{I}(u)+\rho_{I}(u^{\prime}).$

Finally, we make the straightforward observation that if $I\subset J$, then by
Lemma 2.1 we always have

$\rho_{I}(u)\lesssim\rho_{J}(u).$

The small data scattering theory in [9, 10] relied on knowing that the data
was small in either $H^{\frac{1}{2}}\times L^{2}$ or the dispersive norm
$\|\cdot\|_{D}$. Our goal is to extend this to only requiring that the free
evolution of the data is small with respect to the controlling quantity
$\rho_{I}$. Note that smallness of the controlling quantity is much more
general than smallness in norm, for instance an application of Theorem 2.4
gives

$\rho_{I}(e^{it\Delta}f)\lesssim\Big{(}\|f\|_{H^{\frac{1}{2}}}\|e^{it\Delta}f\|_{D(I)}\Big{)}^{\frac{1}{2}}\lesssim\|f\|_{H^{\frac{1}{2}}}.$
(3.7)

On the other hand, as we shall see later, it is also possible to bound
$\rho_{I}$ by the a factor of the “energy dispersion” or $L^{\infty}_{t,x}$
type norm $\|\cdot\|_{Y}$. This additional flexibility provided by $\rho_{I}$
is extremely useful when we eventually come to studying compactness properties
of a hypothetical counterexample to Conjecture 1.4. In particular, we can
avoid the technically challenging issue to trying to obtain a profile
decomposition in the endpoint Strichartz space $L^{2}_{t}L^{4}_{x}$.

The construction of a minimal counter example relies on a crucial stability
property of solutions to the Zakharov equation. Roughly speaking, we need to
that if we have a solution $(\psi,\phi)\in C(I;H^{\frac{1}{2}}\times L^{2})$
is a solution to the perturbed equation

$\begin{split}(i\partial_{t}+\Delta)\psi&=\Re(\phi)\psi+F,\\\
(i\partial_{t}+|\nabla|)\phi&=|\nabla||\psi|^{2}+G,\end{split}$ (3.8)

which is dispersive, thus $\psi\in S^{\frac{1}{2}}(I)$, then under a suitable
smallness assumption on $F$ and $G$, there exists a genuine solution $(u,V)$
to the Zakharov equation (1.2) which is also dispersive and ‘close’ to
$(\psi,\phi)$. A precisely formulation of this stability property is as
follows.

######  Theorem 3.3 (Stability).

Let $A>0$, $0<B<\|Q\|_{\dot{H}^{1}}$, and $\frac{1}{2}\leqslant s<1$. There
exists $\epsilon_{0}(A,B)>0$ such that if
$0<\epsilon\leqslant\epsilon_{0}(A,B)$, $t_{0}\in I\subset\mathbb{R}$ is an
interval, and

$(f,g)\in H^{s}\times L^{2},\qquad(F,G)\in N^{\frac{1}{2}}(I)\times
M^{0}(I),\qquad(\psi,\phi)\in C(I,H^{\frac{1}{2}}\times L^{2})$

satisfies the boundedness condition

$\|f\|_{H^{\frac{1}{2}}}+\|\psi\|_{S^{\frac{1}{2}}(I)}\leqslant
A,\qquad\|\phi\|_{L^{\infty}_{t}L^{2}_{x}(I\times\mathbb{R}^{4})}\leqslant B,$
(3.9)

the smallness condition

$\rho_{I}\big{(}e^{i(t-t_{0})\Delta}[f-\psi(t_{0})]\big{)}+\|g-\phi(t_{0})\|_{L^{2}}+\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{\frac{1}{2}}(I)}+\|\mathcal{J}_{0,t_{0}}[G]\|_{W^{0}(I)}<\epsilon,$
(3.10)

and $(\psi,\phi)$ solve the perturbed Zakharov equation (3.8) on
$I\times\mathbb{R}^{4}$, then there exists a (unique) solution $(u,V)\in
C(I,H^{s}\times L^{2})$ to (1.2) with data $(u,V)(t_{0})=(f,g)$. Moreover, we
have the bounds

$\rho_{I}(u-\psi)+\big{\|}u-\psi-e^{i(t-t_{0})\Delta}[f-\psi(t_{0})]\big{\|}_{S^{\frac{1}{2}}(I)}+\|V-\phi\|_{W^{0}(I)}\lesssim_{A,B}\epsilon$

and

$\|u\|_{\underline{S}^{s}(I)}\lesssim_{A,B,s}\|f\|_{H^{s}},\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}(I)}\leqslant\tfrac{1}{2}\big{(}B+\|Q\|_{\dot{H}^{1}}\big{)}$

where the implied constants depend only on $A$, $B$, and $s$.

###### Proof.

We begin by considering the case $s=\frac{1}{2}$ under the additional
assumption that the dispersive norm $\|\psi\|_{D(I)}$ is sufficiently small.
This additional smallness assumption is then removed by arguing as in the
proof of Theorem 3.1. Upgrading the regularity from $s=\frac{1}{2}$ to
$\frac{1}{2}<s<1$ follows from a direct application of Theorem 3.1. We now
turn to the details. Suppose for the moment that the following claim holds:

> Claim: Let $A>0$ and $0<B<\|Q\|_{\dot{H}^{1}}$. There exists
> $\epsilon^{*}(A,B),\delta^{*}(A,B)>0$ such that if
>
> $0<\epsilon\leqslant\epsilon^{*}(A,B)\qquad\text{ and }\qquad
> 0<\delta\leqslant\delta^{*}(A,B),$
>
> and $(\psi,\phi)\in C(I;H^{\frac{1}{2}}\times L^{2})$ is a solution to (3.8)
> satisfying (3.9), (3.10), and
>
> $\|\psi\|_{D(J)}<\delta,$ (3.11)
>
> for some $t_{0}\in J\subset I$, then there exists a solution $(u,V)\in
> C(J;H^{\frac{1}{2}}\times L^{2})$ to the Zakharov system (1.2) such that
> $(u,V)(t_{0})=(f,g)$ and we have
>
>
> $\big{\|}u-\psi-e^{i(t-t_{0})\Delta}[f-\psi(t_{0})]\big{\|}_{S^{\frac{1}{2}}(J)}\lesssim_{B}(1+A)^{2}\epsilon$
>
> and
>
>
> $\big{\|}V-\phi\big{\|}_{W^{0}(J\times\mathbb{R}^{4})}\lesssim_{B}(1+A)^{3}\epsilon$
>
> where the implied constant depends only on $B$.

To prove that the claim implies the required result, we argue as in the proof
of Theorem 3.1 and decompose the initial interval $I$ into subintervals on
which we have the required smallness (3.11). Fix $A>0$,
$0<B<\|Q\|_{\dot{H}^{1}}$, and assume that we have $(\psi,\phi)\in
C(I;H^{\frac{1}{2}}\times L^{2})$ such that the conditions (3.9), (3.10), and
(3.8) hold where $\epsilon>0$ is to be determined later. Since
$\|\psi\|_{D(I)}\leqslant\|\psi\|_{S^{\frac{1}{2}}(I)}\leqslant A$, we can
find a collection of intervals $I_{j}\subset I$ such that

$I=\cup_{j=-N}^{N}I_{j},\qquad\|\psi\|_{D(I_{j})}\leqslant\delta,\qquad
N\approx\delta^{-2}A^{2}\approx_{A,B}1,\qquad I_{j}\cap
I_{j+1}\not=\varnothing,\qquad t_{0}\in I_{0}$

where $\delta=\delta^{*}(2A,B)>0$ is as in the statement of the above claim.
To construct the solution $(u,V)$ forward in time, i.e. on the interval
$\cup_{j\geqslant 0}I_{j}$, we first choose times $t_{j+1}\in I_{j}\cap
I_{j+1}$. Assuming we have a solution on the interval $I_{j}$, we note that

$\displaystyle\rho_{I}\big{(}e^{i(t-t_{j+1})\Delta}[u-\psi](t_{j+1})\big{)}$
$\displaystyle-\rho_{I}\big{(}e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{)}$
$\displaystyle\lesssim\rho_{I}\Big{(}e^{i(t-t_{j+1})\Delta}\big{(}[u-\psi](t_{j+1})-e^{i(t_{j+1}-t_{j})\Delta}[u-\psi](t_{j})\big{)}\Big{)}$
$\displaystyle\lesssim\big{\|}[u-\psi](t_{j+1})-e^{i(t_{j+1}-t_{j})\Delta}[u-\psi](t_{j})\big{\|}_{H^{\frac{1}{2}}}$
$\displaystyle\lesssim\big{\|}u-\psi-e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{\|}_{S^{\frac{1}{2}}(I_{j})}$

and similarly

$\displaystyle\|[u-\psi](t_{j+1})\|_{H^{\frac{1}{2}}}-\|[u-\psi](t_{j})\|_{H^{\frac{1}{2}}}$
$\displaystyle\leqslant\big{\|}[u-\psi](t_{j+1})-e^{i(t_{j+1}-t_{j})\Delta}[u-\psi](t_{j})\big{\|}_{H^{\frac{1}{2}}}$
$\displaystyle\lesssim\big{\|}u-\psi-e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{\|}_{S^{\frac{1}{2}}(I_{j})}.$

In view of (2.7) (and a similar version for the wave Duhamel term), for any
$j$,

$\|\mathcal{I}_{0,t_{j}}[F]\|_{S^{\frac{1}{2}}(I)}+\|\mathcal{J}_{0,t_{j}}[G]\|_{W^{0}(I)}\lesssim\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{\frac{1}{2}}(I)}+\|\mathcal{J}_{0,t_{0}}[G]\|_{W^{0}(I)}\lesssim\epsilon.$

Therefore, provided we have say

$\epsilon\leqslant\frac{1}{N+1}\big{[}C_{B}(1+2A)^{3}\big{]}^{-(N+1)}\epsilon^{*}(2A,B)$

where $C_{B}$ is a suitably chosen constant depending only on $B$ (via the
above implied constants), repeatedly applying the above claim, together with
the uniqueness of solutions in $D(I)$, gives a solution $(u,V)\in
C(I;H^{\frac{1}{2}}\times L^{2})$ such that for any $|j|\leqslant N$ we have

$\rho_{I}\big{(}e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{)}+\big{\|}u-\psi-e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{\|}_{S^{\frac{1}{2}}(I_{j})}+\|V-\phi\|_{W^{0}(I_{j})}\lesssim_{A,B}\epsilon,$

and

$\|u\|_{S^{\frac{1}{2}}(I_{j})}+\|V\|_{W^{0}(I_{j})}\lesssim_{A,B}1.$

To extend these bounds to the whole interval $I$, we observe that an
application of Lemma 2.2 gives

$\displaystyle\big{\|}\mathcal{I}_{0,t_{0}}\big{[}\Re(V)u-\Re(\phi)\psi\big{]}\big{\|}_{S^{\frac{1}{2}}(I)}$
$\displaystyle\lesssim\sum_{j=-N}^{N}\Big{(}\big{\|}\mathcal{I}_{0,t_{0}}\big{[}\Re(V-\phi)\psi\big{]}\big{\|}_{S^{\frac{1}{2}}(I_{j})}$
$\displaystyle\qquad+\big{\|}\mathcal{I}_{0,t_{0}}\big{[}\Re(V)\big{(}u-\psi-e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{)}\big{]}\big{\|}_{S^{\frac{1}{2}}(I_{j})}+\big{\|}\mathcal{I}_{0,t_{0}}\big{[}\Re(V)e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{]}\big{\|}_{S^{\frac{1}{2}}(I_{j})}\Big{)}$
$\displaystyle\lesssim\sum_{j=-N}^{N}\Big{[}\|V-\phi\|_{W^{0}(I_{j})}\|\psi\|_{S^{\frac{1}{2}}(I)}$
$\displaystyle\qquad+\|V\|_{\underline{W}^{0}(I_{j})}\Big{(}\|u-\psi-e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\|_{S^{\frac{1}{2}}(I_{j})}+\rho_{I}\big{(}e^{i(t-t_{j})\Delta}[u-\psi](t_{j})\big{)}\Big{)}\Big{]}$
$\displaystyle\lesssim_{A,B}\epsilon$

where we used the bound $N\lesssim_{A,B}1$ together with

$\|V\|_{\underline{W}^{0}(I_{j})}\lesssim\|V(t_{j})\|_{L^{2}_{x}}+\|u\|_{S^{\frac{1}{2}}(I_{j})}^{2}\lesssim_{A,B}1.$

Therefore we obtain the global smallness bound

$\displaystyle\big{\|}u-\psi-e^{i(t-t_{0})}[f-\psi(t_{0})]\big{\|}_{S^{\frac{1}{2}}(I)}$
$\displaystyle=\big{\|}\mathcal{I}_{0,t_{0}}\big{[}\Re(V)u-\Re(\phi)\psi-F\big{]}\big{\|}_{S^{\frac{1}{2}}(I)}\lesssim_{A,B}\epsilon.$

After observing that

$\displaystyle\rho_{I}(u-\psi)$
$\displaystyle\leqslant\rho_{I}\big{(}u-\psi-e^{i(t-t_{0})\Delta}[u-\psi](t_{0})\big{)}+\rho_{I}\big{(}e^{i(t-t_{0})\Delta}[u-\psi](t_{0})\big{)}$
$\displaystyle\lesssim\|u-\psi-e^{i(t-t_{0})\Delta}[u-\psi](t_{0})\big{\|}_{S^{\frac{1}{2}}(I)}+\epsilon\lesssim_{A,B}\epsilon$

we have the required smallness bounds on the Schrödinger evolution. Moreover,
an application of Theorem 3.1 gives for any $\frac{1}{2}\leqslant s<1$ the
higher regularity bounds

$\|u\|_{\underline{S}^{s}(I)}\lesssim_{A,B,s}\|f\|_{H^{s}}.$

To bound the wave evolution we simply observe that since

$|u|^{2}-|\psi|^{2}=\Re\big{[}(\overline{u}+\overline{\psi})(u-\psi)\big{]}=\Re\big{[}(\overline{u}+\overline{\psi}-\mathcal{I}_{0,t_{0}}[F])(u-\psi)+\mathcal{I}_{0,t_{0}}[F](u-\psi)\big{]}$

and

$\|\psi-\mathcal{I}_{0,t_{0}}[F]\|_{\underline{S}^{\frac{1}{2}}(I)}\lesssim\|\phi\|_{W^{0}(I)}\|\psi\|_{S^{\frac{1}{2}}(I)}\lesssim\big{(}\|\phi(t_{0})\|_{L^{2}}+\|\psi\|_{S^{\frac{1}{2}}(I)}^{2}\big{)}\|\psi\|_{S^{\frac{1}{2}}(I)}\lesssim_{A,B}1$

we conclude that

$\displaystyle\|V-\phi\|_{W^{0}(I)}$
$\displaystyle\leqslant\|g-\phi(t_{0})\|_{L^{2}}+\big{\|}\mathcal{J}_{0,t_{0}}\big{[}|u|^{2}-|\psi|^{2}-G\big{]}\big{\|}_{W^{0}(I)}$
$\displaystyle\lesssim\epsilon+\big{\|}\mathcal{J}_{0,t_{0}}\big{[}\Re((u+\psi-\mathcal{I}_{0,t_{0}}[F])(u-\psi)\big{)}\big{]}\big{\|}_{W^{0}(I)}+\big{\|}\mathcal{J}_{0,t_{0}}\big{[}\Re\big{(}\mathcal{I}_{0,t_{0}}[F](u-\psi)\big{)}\big{]}\big{\|}_{W^{0}(I)}$
$\displaystyle\lesssim\epsilon+\big{(}\|u\|_{\underline{S}^{\frac{1}{2}}(I)}+\|\psi-\mathcal{I}_{0,t_{0}}[F]\|_{S^{\frac{1}{2}}(I)}\big{)}\rho_{I}(u-\psi)+\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{\frac{1}{2}}(I)}\big{(}\|\psi\|_{S^{\frac{1}{2}}(I)}+\|u\|_{S^{\frac{1}{2}}(I)}\big{)}$
$\displaystyle\lesssim_{A,B}\epsilon.$

Therefore theorem follows from the claimed local version under the additional
constraint (3.11).

It remains to verify the claim. By translation invariance, we may fix
$t_{0}=0$. Let $\phi_{0}=e^{it|\nabla|}\phi(0)$. Our goal is to construct a
solution to the problem

$\begin{split}\big{(}i\partial_{t}+\Delta-\Re(\phi_{0})\big{)}u^{\prime}&=\Re(V^{\prime}+\phi)(u^{\prime}+\psi)-\Re(\phi)\psi-\Re(\phi_{0})u^{\prime}-F,\\\
(i\partial_{t}+|\nabla|)V^{\prime}&=|\nabla|\big{(}|u^{\prime}+\psi|^{2}-|\psi|^{2}\big{)}-G,\end{split}$
(3.12)

with data

$(u^{\prime},V^{\prime})(0)=(f^{\prime},g^{\prime}):=\big{(}f-\psi(0),g-\phi(0)\big{)}.$

Provided we obtain appropriate bounds on $(u^{\prime},V^{\prime})$, the claim
then follows by taking $(u,V)=(\psi+u^{\prime},\phi+V^{\prime})$. We begin by
observing that an application of Theorem 2.4 together with (3.11) and (3.9)
gives

$\|\phi-\phi_{0}\|_{W^{0}(J)}=\big{\|}\mathcal{J}_{0}[|\nabla||\psi|^{2}+G]\big{\|}_{W^{0}(J)}\lesssim\|\psi\|_{D(J)}\|\psi\|_{S^{\frac{1}{2}}(J)}+\|\mathcal{J}_{0}[G]\|_{W^{0}(J)}\lesssim\delta
A+\epsilon$ (3.13)

and (assuming $\epsilon\ll 1$ and $\delta A\ll 1$)

$\displaystyle\|\psi-\mathcal{I}_{0}[F]\|_{\underline{S}^{\frac{1}{2}}(J)}\lesssim\|\psi(0)\|_{H^{\frac{1}{2}}}+\|\mathcal{I}_{0}[\Re(\phi)\psi]\|_{\underline{S}^{\frac{1}{2}}(J)}\lesssim
A+\|\phi\|_{W^{0}(J)}\|\psi\|_{S^{\frac{1}{2}}(J)}\lesssim A$ (3.14)

Define $u_{0}^{\prime}=e^{it\Delta}f^{\prime}$ and for any $k\in\mathbb{N}$
consider the sequence

$\displaystyle u_{k+1}^{\prime}$
$\displaystyle=\mathcal{U}_{\Re(\phi_{0})}(t)f^{\prime}+\mathcal{I}_{\Re(\phi_{0})}\big{[}\Re(V^{\prime}_{k+1}+\phi)(u_{k}^{\prime}+\psi)-\Re(\phi)\psi-\Re(\phi_{0})u_{k}^{\prime}-F\big{]}$
$\displaystyle V_{k+1}^{\prime}$
$\displaystyle=e^{it|\nabla|}g^{\prime}+\mathcal{J}_{0}\big{[}|\nabla|\big{(}|u_{k}^{\prime}+\psi|^{2}-|\psi|^{2}\big{)}-G\big{]}.$

We make the inductive assumption that

$\|u^{\prime}_{k}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\leqslant A.$ (3.15)

This bound clearly holds when $k=0$, and the estimates below substantially
improve on this weak upper bound, however it simplifies the notation somewhat.
Write

$\displaystyle|u^{\prime}_{k}+\psi|^{2}-|\psi|^{2}=|u_{k}^{\prime}-u_{0}^{\prime}|^{2}+2\Re[\overline{u_{0}^{\prime}}(u^{\prime}_{k}-u_{0}^{\prime}+\mathcal{I}_{0}[F])]+2\Re[\overline{u_{0}^{\prime}}(u_{0}^{\prime}+\psi-\mathcal{I}_{0}[F])]+2\Re[\overline{\psi}(u_{k}^{\prime}-u_{0}^{\prime})].$

Then the definition of $\rho_{I}(u_{0}^{\prime})$, Theorem 2.3, and the bounds
(3.9), (3.10), (3.11), and (3.14) give

$\displaystyle\|V_{k+1}^{\prime}\|_{W^{0}(J)}$
$\displaystyle\lesssim\|g^{\prime}\|_{L^{2}}+\big{\|}|\nabla|\mathcal{J}_{0}\big{[}|u_{k}^{\prime}-u_{0}^{\prime}|^{2}\big{]}\big{\|}_{W^{0}(J)}+\big{\|}|\nabla|\mathcal{J}_{0}\big{[}\overline{u_{0}^{\prime}}(u_{0}^{\prime}+\psi-\mathcal{I}_{0}[F])\big{]}\big{\|}_{W^{0}(J)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\big{\|}|\nabla|\mathcal{J}_{0}\big{[}\Re\big{(}\overline{u_{0}^{\prime}}(u_{0}^{\prime}+\psi-\mathcal{I}_{0}[F])\big{)}\big{]}\big{\|}_{W^{0}(J)}+\big{\|}|\nabla|\mathcal{J}_{0}[\overline{\psi}(u_{k}^{\prime}-u_{0}^{\prime})]\big{\|}_{W^{0}(J)}$
$\displaystyle\lesssim\epsilon+\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}^{2}+\|u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{(}\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\|\mathcal{I}_{0}[F]\|_{S^{\frac{1}{2}}(J)}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\rho_{I}(u_{0}^{\prime})\big{(}\|u_{0}^{\prime}\|_{\underline{S}^{\frac{1}{2}}(J)}+\|\psi-\mathcal{I}_{0}[F]\|_{\underline{S}^{\frac{1}{2}}(J)}\big{)}+\|\psi\|_{S^{\frac{1}{2}}(J)}\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim(1+A)\epsilon+A\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}$

where we also applied the inductive assumption (3.15). Similarly, for the
differences $V_{k+1}^{\prime}-V_{j+1}^{\prime}$, after writing

$V_{k+1}^{\prime}-V_{j+1}^{\prime}=|\nabla|\mathcal{J}_{0}\big{[}|u_{k}^{\prime}+\psi|^{2}-|u_{j}^{\prime}+\psi|^{2}\big{]}=|\nabla|\mathcal{J}_{0}\big{[}|u_{k}^{\prime}-u_{j}^{\prime}|^{2}+2\Re\big{(}(\overline{u}_{k}^{\prime}-\overline{u}_{j}^{\prime})(u_{j}^{\prime}+\psi)\big{)}\big{]}$

we can bound the difference in the stronger space $\underline{W}^{0}(J)$ via

$\displaystyle\|V_{k+1}^{\prime}-V_{j+1}^{\prime}\|_{\underline{W}^{0}(J)}$
$\displaystyle\lesssim\|u_{j}^{\prime}-u_{k}^{\prime}\|_{S^{\frac{1}{2}}(J)}^{2}+\|u_{j}^{\prime}-u_{k}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{(}\|u_{j}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\|u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\|\psi\|_{S^{\frac{1}{2}}(J)}\big{)}$
$\displaystyle\lesssim
A\|u_{j}^{\prime}-u_{k}^{\prime}\|_{S^{\frac{1}{2}}(J)}.$

The fact that we don’t assume a good bound for $\mathcal{J}_{0}[G]$ in the
stronger space $\underline{W}^{0}(J)$ prevents us from improving the above
bound for $V_{k}^{\prime}$ from $W^{0}(J)$ to $\underline{W}^{0}(J)$. However
this is possible for components of the wave evolution. More precisely, we have
(again using the inductive assumption (3.15))

$\displaystyle\|V_{k+1}^{\prime}+\phi-\phi_{0}\|_{\underline{W}^{0}(J)}$
$\displaystyle=\big{\|}e^{it|\nabla|}g^{\prime}+\mathcal{J}_{0}\big{[}|\nabla||u_{k}^{\prime}+\psi|^{2}\big{]}\big{\|}_{\underline{W}^{0}(J)}\lesssim\epsilon+\|u^{\prime}_{k}+\psi\|_{S^{\frac{1}{2}}(J)}^{2}\lesssim
A^{2}.$

We now turn to the bounds for the Schrödinger component of the evolution. The
first step is to observe that the identities

$\mathcal{I}_{\Re(\phi_{0})}[\Phi]=\mathcal{I}_{0}[\Phi]+\mathcal{I}_{\Re(\phi_{0})}[\Re(\phi_{0})\mathcal{I}_{0}[\Phi]]$

and

$\mathcal{U}_{\Re(\phi_{0})}f=e^{it\Delta}f+\mathcal{I}_{\Re(\phi_{0})}[\Re(\phi_{0})e^{it\Delta}f]=e^{it\Delta}f+\mathcal{I}_{0}[\Re(\phi_{0})e^{it\Delta}f]+\mathcal{I}_{\Re(\phi_{0})}\big{[}\Re(\phi_{0})\mathcal{I}_{0}[\Re(\phi_{0})e^{it\Delta}f]\big{]}$

together with the uniform energy estimate in Theorem 2.3 and the bilinear
estimates in Theorem 2.4 gives (since $\|\phi(0)\|_{L^{2}}\leqslant
B<\|Q\|_{\dot{H}^{1}}$)

$\|\mathcal{I}_{\Re(\phi_{0})}[\Phi]\|_{S^{\frac{1}{2}}(J)}\lesssim_{B}\|\mathcal{I}_{0}[\Phi]\|_{S^{\frac{1}{2}}(J)}+\|\phi_{0}\mathcal{I}_{0}[\Phi]\|_{N^{\frac{1}{2}}(J)}\lesssim_{B}\|\mathcal{I}_{0}[\Phi]\|_{S^{\frac{1}{2}}(J)}$

and

$\displaystyle\|\mathcal{U}_{\Re(\phi_{0})}f^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}\|\mathcal{I}_{0}[\Re(\phi_{0})u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\|\Re(\phi_{0})\mathcal{I}_{0}[\Re(\phi_{0})u_{0}^{\prime}]\|_{N^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}\|\mathcal{I}_{0}[\Re(\phi_{0})u_{0}^{\prime}]\|_{S^{\frac{1}{2}}(J)}\lesssim_{B}\|\phi_{0}\|_{\underline{W}^{0}(J)}\rho_{I}(u_{0}^{\prime})\lesssim_{B}\epsilon.$

Hence after decomposing

$\Re(V^{\prime}_{k+1}+\phi)(u_{k}^{\prime}+\psi)-\Re(\phi)\psi-\phi_{0}u_{k}^{\prime}=\Re(V_{k+1}^{\prime}+\phi-\phi_{0})(u_{k}^{\prime}-u_{0}^{\prime})+\Re(V_{k+1}^{\prime}+\phi-\phi_{0})u_{0}^{\prime}+\Re(V^{\prime}_{k+1})\psi$

we see that

$\displaystyle\|u^{\prime}_{k+1}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\leqslant\|\mathcal{U}_{\Re(\phi_{0})}f^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\big{\|}\mathcal{I}_{\Re(\Phi_{0})}\big{[}\Re(V^{\prime}_{k+1}+\phi)(u_{k}^{\prime}+\psi)-\Re(\phi)\psi-\phi_{0}u_{k}^{\prime}\big{]}\big{\|}_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}\epsilon+\big{\|}\mathcal{I}_{0}\big{[}\Re(V_{k+1}^{\prime}+\phi-\phi_{0})(u_{k}^{\prime}-u_{0}^{\prime})+\Re(V_{k+1}^{\prime}+\phi-\phi_{0})u_{0}^{\prime}+\Re(V^{\prime}_{k+1})\psi\big{]}\big{\|}_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}\epsilon+\|V_{k+1}^{\prime}+\phi-\phi_{0}\|_{W^{0}(J)}\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\qquad\qquad\qquad+\|V_{k+1}^{\prime}+\phi-\phi_{0}\|_{\underline{W}^{0}(J)}\rho_{I}(u_{0}^{\prime})+\|V^{\prime}_{k+1}\|_{W^{0}(J)}\big{(}\|\psi\|_{D(J)}\|\psi\|_{S^{\frac{1}{2}}(J)}\big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim_{B}\epsilon+\big{[}(1+A)\epsilon+\delta
A+A\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{]}\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\qquad\qquad\qquad+A^{2}\epsilon+\big{[}(1+A)\epsilon+A\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{]}(\delta
A)^{\frac{1}{2}}$
$\displaystyle\lesssim_{B}(1+A^{2})\epsilon+\big{[}(1+A)\epsilon+A^{\frac{3}{2}}\delta^{\frac{1}{2}}+A\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{]}\|u_{k}^{\prime}-u_{0}\|_{S^{\frac{1}{2}}(J)}.$

Similarly, for differences, we write

$\Re(V_{k+1}^{\prime}+\phi)(u_{k}^{\prime}+\psi)-\Re(\phi_{0})u_{k}^{\prime}-\Re(V_{j+1}^{\prime}+\phi)(u_{j}^{\prime}+\psi)+\Re(\phi_{0})u_{j}^{\prime}=\Re(V_{k+1}^{\prime}-V_{j+1}^{\prime})(u_{k}^{\prime}+\psi)+\Re(V_{j+1}^{\prime}+\phi-\phi_{0})(u_{k}^{\prime}-u_{j}^{\prime})$

and observe that the above bounds give

$\displaystyle\|u_{k+1}^{\prime}-u_{j+1}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle=\big{\|}\mathcal{I}_{\Re(\phi_{0})}\big{[}\Re(V_{k+1}^{\prime}-V_{j+1}^{\prime})(u_{k}^{\prime}+\psi)+\Re(V_{j+1}^{\prime}+\phi-\phi_{0})(u_{k}^{\prime}-u_{j}^{\prime})\big{]}\big{\|}_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}\big{\|}\mathcal{I}_{0}\big{[}\Re(V_{k+1}^{\prime}-V_{j+1}^{\prime})(u_{k}^{\prime}+\psi)\big{]}\big{\|}_{S^{\frac{1}{2}}(J)}+\big{\|}\mathcal{I}_{0}\big{[}(V^{\prime}_{j+1}+\phi-\phi_{0})(u_{k}^{\prime}-u_{j}^{\prime})\big{]}\big{\|}_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}\|V^{\prime}_{k+1}-V_{j+1}^{\prime}\|_{\underline{W}^{0}(J)}\Big{(}\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\rho_{I}(u_{0}^{\prime})+\big{(}\|\psi\|_{S^{\frac{1}{2}}(J)}\|\psi\|_{D(J)}\big{)}^{\frac{1}{2}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+\big{(}\|V_{j+1}^{\prime}\|_{W^{0}(J)}+\|\phi-\phi_{0}\|_{W^{0}(J)}\big{)}\|u_{k}^{\prime}-u_{j}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}A\|u_{k}^{\prime}-u_{j}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{(}\|u^{\prime}_{k}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\epsilon+(A\delta)^{\frac{1}{2}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+\big{(}(1+A)\epsilon+\delta
A+A\|u_{j}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{)}\|u_{k}^{\prime}-u_{j}^{\prime}\|_{S^{\frac{1}{2}}(J)}$
$\displaystyle\lesssim_{B}(1+A)\big{(}\epsilon+(A\delta)^{\frac{1}{2}}+\|u_{j}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}+\|u_{k}^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\big{)}\|u_{j}^{\prime}-u_{k}^{\prime}\|_{S^{\frac{1}{2}}(J)}.$

Therefore, a standard argument shows that there exists
$\epsilon^{*}(A,B),\delta^{*}(A,B)>0$ such that if
$0<\epsilon<\epsilon^{*}(A,B)$ and $0<\delta<\delta^{*}(A,B)$ then there
exists a solution $(u^{\prime},V^{\prime})$ to (3.12) such that

$\|u^{\prime}-u_{0}^{\prime}\|_{S^{\frac{1}{2}}(J)}\lesssim_{B}(1+A^{2})\epsilon,\qquad\|V^{\prime}-e^{it\Delta}f^{\prime}\|_{W^{0}(J)}\lesssim_{B}(1+A^{3})\epsilon.$

Letting $(u,V)=(\psi+u^{\prime},\phi+V^{\prime})$, and noting that for any
$s\in J$ we have

$\displaystyle\rho_{I}\big{(}e^{i(t-s)\Delta}[u(s)-\psi(s)]\big{)}$
$\displaystyle\leqslant\rho_{I}\big{(}e^{it\Delta}[f-\psi(0)]\big{)}+\rho_{I}\Big{(}e^{i(t-s)\Delta}[u(s)-\psi(s)-e^{is\Delta}[f-\psi(0)]\big{]}\Big{)}$
$\displaystyle\lesssim\epsilon+\big{\|}u(s)-\psi(s)-e^{is\Delta}[f-\psi(0)]\big{\|}_{H^{\frac{1}{2}}}\lesssim_{B}(1+A^{2})\epsilon$

the claim now follows. ∎

A quick corollary of the previous theorem is an existence theorem for the
Zakharov equation where the time of existence depends only on the size of the
controlling quantity $\rho_{I}$.

######  Corollary 3.4 (Refined small data theory).

Let $A>0$, $0<B<\|Q\|_{\dot{H}^{1}}$, and $\frac{1}{2}\leqslant s<1$. There
exists $\epsilon_{0}(A,B)>0$ such that if
$0<\epsilon\leqslant\epsilon_{0}(A,B)$, $I\subset\mathbb{R}$ is an interval,
and $(f,g)\in H^{s}\times L^{2}$ such that for some $t_{0}\in I$ we have the
boundedness condition

$\|f\|_{H^{\frac{1}{2}}}\leqslant A,\qquad\|g\|_{L^{2}_{x}}\leqslant B,$
(3.16)

and the smallness condition

$\rho_{I}\big{(}e^{i(t-t_{0})\Delta}f\big{)}<\epsilon,$ (3.17)

then there exists a (unique) solution $(u,V)\in C(I,H^{s}\times L^{2})$ to
(1.2) with data $(u,V)(0)=(f,g)$. Moreover, we have the bounds

$\rho_{I}(u)+\big{\|}u-e^{i(t-t_{0})\Delta}f\big{\|}_{S^{\frac{1}{2}}(I)}+\|V-e^{i(t-t_{0})|\nabla|}g\|_{W^{0}(I)}\lesssim_{A,B}\epsilon$

and

$\|u\|_{\underline{S}^{s}(I)}\lesssim_{A,B,s}\|f\|_{H^{s}},\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}(I)}\leqslant\tfrac{1}{2}\big{(}B+\|Q\|_{\dot{H}^{1}}\big{)}$

where the implied constants depend only on $A$, $B$, and $s$.

###### Proof.

We simply apply Theorem 3.3 with $\psi=F=G=0$ and $\phi=e^{it|\nabla|}g$. ∎

###  3.4. Existence of Wave Operators

To extract some compactness from a bounded sequence of solutions to the
Zakharov equation, we require solutions with prescribed data at infinity (i.e
the existence of wave operators). This is a simple consequence of the results
in [9, 10].

######  Theorem 3.5 (Existence of wave operators below ground state).

Let $(f,g)\in H^{1}\times L^{2}$ with

$2\|f\|_{\dot{H}^{1}}^{2}+\|g\|_{L^{2}}^{2}<\|Q\|_{\dot{H}^{1}}^{2}.$

Then there exists a unique global solution $(u,V)\in C(\mathbb{R},H^{1}\times
L^{2})$ to (1.2) such that $u\in L^{2}_{t,loc}W^{\frac{1}{2},4}_{x}$ and

$\lim_{t\to-\infty}\big{\|}(u,V)(t)-\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}\big{\|}_{H^{1}\times
L^{2}}=0.$

Moreover, the energy $\mathcal{E}_{Z}(u,V)$ is conserved, and

$4\mathcal{E}_{Z}(u,V)=\lim_{t\to-\infty}4\mathcal{E}_{Z}\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}=2\|f\|_{\dot{H}^{1}}^{2}+\|g\|_{L^{2}}^{2},\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}}^{2}\leqslant
2\|f\|_{\dot{H}^{1}}^{2}+\|g\|_{L^{2}}^{2}.$

###### Proof.

The estimates in [10] (see Theorem 2.4 and the energy inequality (2.5))
together with a standard fixed point argument give a time $T\in\mathbb{R}$ and
a unique solution $(u,V)\in C((-\infty,T],H^{1}\times L^{2})$ to (1.2) with
$u\in L^{2}_{t}W^{\frac{1}{2},4}_{x}((-\infty,T]\times\mathbb{R}^{4})$ and

$\lim_{t\to-\infty}\big{\|}(u,V)(t)-\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}\big{\|}_{H^{1}\times
L^{2}}=0.$

Moreover this solution conserves energy, and hence as

$\displaystyle\big{|}\mathcal{E}_{Z}(u,v)$
$\displaystyle-\mathcal{E}_{Z}\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}\big{|}$
$\displaystyle\lesssim\big{(}\|u(t)\|_{H^{1}}+\|V(t)\|_{L^{2}}+\|f\|_{H^{1}}+\|g\|_{L^{2}}\big{)}\big{(}\|u(t)-e^{it\Delta}f\|_{H^{1}}+\|V(t)-e^{it|\nabla|}g\|_{L^{2}}\big{)}$

we have

$\mathcal{E}_{Z}(u,V)=\lim_{t\to-\infty}\mathcal{E}_{Z}\big{(}e^{it\Delta}f,e^{it|\nabla|}g\big{)}=\frac{1}{2}\|f\|_{H^{1}}^{2}+\frac{1}{4}\|g\|_{L^{2}}^{2}$

where the last identity follows from the fact that, due to the dispersive
properties of the Schrödinger evolution, we have
$\lim_{t\to-\infty}\|e^{it\Delta}f\|_{L^{4}_{x}}=0$. Therefore in view of the
assumptions on the asymptotic data $(f,g)$ we have

$4\mathcal{E}_{Z}(u,V)<\|Q\|_{\dot{H}^{1}}^{2},\qquad\lim_{t\to-\infty}\|V(t)\|_{L^{2}}=\|g\|_{L^{2}}<\|Q\|_{\dot{H}^{1}}.$

Consequently we can find some $T^{*}\in(-\infty,T]$ such that

$4\mathcal{E}_{Z}\big{(}u(T^{*}),V(T^{*})\big{)}<\|Q\|_{\dot{H}^{1}}^{2},\qquad\|V(T^{*})\|_{L^{2}}<\|Q\|_{\dot{H}^{1}}$

and hence taking $(u,V)(T^{*})$ as data, Theorem 1.1 extends the solution
$(u,V)$ from $(-\infty,T]$ to $\mathbb{R}$ with $(u,V)\in
C(\mathbb{R},H^{1}\times L^{2})$. ∎

##  4\. Bilinear Estimates

This section contains the key bilinear estimates that form the foundation of
the proof of our main results. Our main goal is to prove the nonlinear terms
on the righthand side of (1.2) are perturbative provided that the $Y$ norm is
small. As mentioned in the introduction, the fundamental issue is that in the
fully resonant case (so both inputs and outputs have Fourier support close to
their relevant characteristic surfaces) it seems very hard to improve on the
Strichartz bounds

$\|\mathcal{I}_{0}[\Re(V)u]\|_{L^{2}_{t}L^{4}_{x}}\lesssim\|\Re(V)u\|_{L^{2}_{t}L^{\frac{4}{3}}_{x}}\lesssim\|V\|_{L^{\infty}_{t}L^{2}_{x}}\|u\|_{L^{2}_{t}L^{4}_{x}}.$
(4.1)

In particular, it does not seem possible to extract a factor
$\|u\|_{L^{\infty}_{t,x}}$. Instead, following the approach in [9], we require
a version of the bilinear restriction estimates for the paraboloid. More
precisely, we recall that provided $r>\frac{5}{3}$ and $\mu\in 2^{\mathbb{Z}}$
we have the (homogeneous) bilinear restriction estimate for the paraboloid

$\big{\|}\dot{P}_{\mu}(\overline{e^{it\Delta}f}e^{it\Delta}g)\big{\|}_{L^{1}_{t}L^{r}_{x}}\lesssim\mu^{2-\frac{4}{r}}\|f\|_{L^{2}}\|g\|_{L^{2}}.$
(4.2)

Note that the Stricharz estimates only give the case $r=2$. The key advantage
of (4.2) is that we can take $r<2$ and thus gain some additional room over
simply applying Hölder’s inequality followed by the linear Strichartz
estimate. The range $r>\frac{5}{3}$ is sharp, in the sense that (4.2) fails if
$r<\frac{5}{3}$. The endpoint $r=\frac{5}{3}$ is open. The estimate (4.2) was
obtained in [8]. Bilinear restriction estimates for the paraboloid in
$L^{p}_{t,x}$ up to the sharp region were obtain by Tao [30], the mixed norm
case $L^{q}_{t}L^{r}_{x}$ with $q>1$ was then considered in [24].

As it stands, the estimate (4.2) is not particularly useful in the nonlinear
problem as the restriction to homogeneous solutions is too strong. However
recently stronger estimates have been developed that improve on (4.2) by
allowing general functions belonging to suitable adapted function spaces [8].
Of particular importance for our purposes are the _inhomogeneous_ bilinear
restriction estimates proved in [9], see Theorem 4.3 below. In fact, by
adapting the arguments from [9] we obtain the following crucial bilinear
estimate.

######  Theorem 4.1 (Improved Bilinear Estimates).

Let $s>\frac{1}{2}$. There exists $C>0$ and $0<\theta<1$ such that for any
interval $0\in I\subset\mathbb{R}$ we have

$\big{\|}\mathcal{I}_{0}\big{[}\Re(V)u\big{]}\big{\|}_{S^{\frac{1}{2}}(I)}\leqslant
C\Big{(}\|V\|_{Y(I)}\|u\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|V\|_{\underline{W}^{0}(I)}\|u\|_{\underline{S}^{s}(I)}\Big{)}^{1-\theta}$

and

$\big{\|}\mathcal{J}_{0}\big{[}|\nabla|\Re(\overline{w}u)\big{]}\big{\|}_{W^{0}(I)}\leqslant
C\Big{(}\|w\|_{Y(I)}\|u\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|w\|_{\underline{S}^{\frac{1}{2}}(I)}\|u\|_{\underline{S}^{s}(I)}\Big{)}^{1-\theta}.$

We give the proof of Theorem 4.1 at the end of this section, after first
recalling the inhomogeneous bilinear restriction estimates from [9], and
proving frequency localised versions of the bilinear estimates in Theorem 4.1.

It is worth noting that there is a loss of (Schrödinger) regularity in Theorem
4.1. In particular, we require $u\in\underline{S}^{s}$ with $s>\frac{1}{2}$,
but can only place the Duhamel term $\mathcal{I}_{0}[\Re(V)u]$ into
$S^{\frac{1}{2}}$. This reflects the fact that obtaining a factor of
$\|u\|_{L^{\infty}_{t,x}}$ is surprisingly challenging, and even after using
the bilinear restriction estimates above, we are essentially forced to place
$u\in L^{a}_{t}L^{b}_{x}$ for some $a>2$ (or at least some space which has
temporal component of the form $L^{a}_{t}$). In turn, this means we have to
control $V$ in $L^{q}_{t}$ for some $q<\infty$. As we see below, this is
possible, but there is a derivative loss. It is unclear if this can be
avoided. In any case, as our eventual goal is to study solutions with
regularity $H^{1}\times L^{1}$, allowing some loss in the Schrödinger
regularity is not a major obstruction. However, it does force some technical
complications, which are mostly alleviated by considering smallness in terms
of the controlling quantity $\rho$ (as in Section 3), as opposed to trying to
propagate smallness in $Y(I)$ via Theorem 4.1 directly in an iteration
argument.

###  4.1. Bilinear Restriction Estimates

The proof of global existence for the Zakharov equation relied crucially on
the following bilinear restriction type estimates for the paraboloid and cone.
The first estimate gives a useful bilinear estimate in $L^{2}_{t,x}$ in the
case where $u$ can be an _inhomogeneous_ solution to the Schrödinger equation.
Recall that we have defined

$\|u\|_{Z}=\|u\|_{L^{\infty}_{t}L^{2}_{x}}+\|(i\partial_{t}+\Delta)u\|_{L^{1}_{t}L^{2}_{x}+L^{2}_{x}L^{4}_{x}}.$

######  Theorem 4.2 (Bilinear $L^{2}_{t,x}$ for inhomogeneous wave-Schrödinger
interactions [9, Theorem 4.1]).

Let $0\leqslant\alpha<\frac{1}{2}$. For all $\lambda\in 2^{\mathbb{N}}$ and
$\mu\in 2^{\mathbb{Z}}$ with $\mu\lesssim\lambda$, we have

$\|u_{\lambda}e^{it|\nabla|}\dot{P}_{\mu}g\|_{L^{2}_{t,x}(\mathbb{R}^{1+4})}\lesssim\Big{(}\frac{\mu}{\lambda}\Big{)}^{\alpha}\mu\|u_{\lambda}\|_{Z}\|\dot{P}_{\mu}g\|_{L^{2}_{x}}.$
(4.3)

The second bilinear estimate we require is an inhomogeneous version of a
bilinear restriction estimate for the paraboloid.

######  Theorem 4.3 (Bilinear restriction for inhomogeneous Schrödinger [9,
Theorem 4.2]).

Let $r>\frac{5}{3}$. For any $\mu\in 2^{\mathbb{Z}}$ we have

$\|\dot{P}_{\mu}(\overline{w}u)\|_{L^{1}_{t}L^{r}_{x}(\mathbb{R}^{1+4})}\lesssim\mu^{2-\frac{4}{r}}\|w\|_{Z}\|u\|_{Z}.$
(4.4)

For technical reasons, and to slightly simplify the arguments below, we
require the following extension of the above bilinear restriction estimates.

######  Corollary 4.4 (Bilinear restriction in iteration norms).

Let $r>\frac{5}{3}$. If $\tilde{\mu}\in 2^{\mathbb{Z}}$ and $\mu,\lambda\in
2^{\mathbb{N}}$ with $\mu\lesssim\lambda$ then

$\|u_{\lambda}\dot{P}_{\tilde{\mu}}v_{\mu}\|_{L^{2}_{t,x}}\lesssim\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{1}{4}}\tilde{\mu}\|u_{\lambda}\|_{\underline{S}^{0}_{\lambda}}\|v_{\mu}\|_{\underline{W}^{0}_{\mu}}.$

Moreover, for any $\tilde{\mu}\in 2^{\mathbb{Z}}$ and
$\lambda_{1},\lambda_{2}\in 2^{\mathbb{N}}$ we have

$\|\dot{P}_{\tilde{\mu}}(\overline{w}u)\|_{L^{1}_{t}L^{r}_{x}}\lesssim\tilde{\mu}^{2-\frac{4}{r}}\min\big{\\{}\|w\|_{Z},\|w\|_{\underline{S}^{0}_{\lambda_{1}}}\big{\\}}\|u\|_{\underline{S}^{0}_{\lambda_{2}}}.$

###### Proof.

The proof follows from Theorem 4.2 and Theorem 4.3 by adapting the standard
$X^{s,b}$ transference argument. We start by replacing the norm $\|u\|_{Z}$
with $\|u\|_{\underline{S}^{0}_{\lambda}}$. To this end, we write

$(C_{d}u)(t,x)=\int_{\mathbb{R}}e^{it\tau}(e^{it\Delta}f^{(\tau)}_{d})(x)d\tau$

with
$\widehat{f}^{(\tau)}_{d}(\xi)=\mathcal{F}_{t,x}[C_{d}u](\tau+|\xi|^{2},\xi)$
and note that

$\displaystyle\int_{\mathbb{R}}\|f^{(\tau)}_{d}\|_{L^{2}_{x}}d\tau\approx\int_{|\tau|\approx
d}\|f^{(\tau)}_{d}\|_{L^{2}_{x}}d\tau\lesssim
d^{\frac{1}{2}}\|C_{d}u\|_{L^{2}_{t,x}}\approx
d^{-\frac{1}{2}}\|(i\partial_{t}+\Delta)u\|_{L^{2}_{t,x}}$

In particular, by definition of the norm
$\|\cdot\|_{\underline{S}^{0}_{\lambda}}$, we have the bound

$\|C_{\ll\lambda^{2}}u\|_{Z}+\sum_{d\gtrsim\lambda^{2}}d^{-\frac{1}{2}}\int_{\mathbb{R}}\|f^{(\tau)}_{d}\|_{L^{2}_{x}}d\tau\lesssim\|u\|_{\underline{S}^{0}_{\lambda}}$

and hence, applying Theorem 4.2, we see that

$\displaystyle\|u_{\lambda}e^{it|\nabla|}\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{t,x}}$
$\displaystyle\lesssim\|C_{\ll\lambda^{2}}u_{\lambda}e^{it|\nabla|}\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{t,x}}+\sum_{d\gtrsim\lambda^{2}}\|C_{d}u_{\lambda}e^{it|\nabla|}\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{t,x}}$
$\displaystyle\lesssim\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{1}{4}}\tilde{\mu}\|C_{\ll\lambda^{2}}u_{\lambda}\|_{Z}\|\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{x}}+\sum_{d\gtrsim\lambda^{2}}\int_{\mathbb{R}}\|e^{it\Delta}f^{(\tau)}_{\lambda,d}e^{it|\nabla|}\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{t,x}}d\tau$
$\displaystyle\lesssim\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{1}{4}}\tilde{\mu}\Big{(}\|C_{\ll\lambda^{2}}u_{\lambda}\|_{Z}+\sum_{d\gtrsim\lambda^{2}}\int_{\mathbb{R}}\|f^{(\tau)}_{\lambda,d}\|_{L^{2}_{x}}d\tau\Big{)}\|\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{x}}$
$\displaystyle\lesssim\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{1}{4}}\tilde{\mu}\|u_{\lambda}\|_{\underline{S}^{0}_{\lambda}}\|\dot{P}_{\tilde{\mu}}g\|_{L^{2}_{x}}.$
(4.5)

Similarly, applying Theorem 4.3, we have

$\displaystyle\|\dot{P}_{\tilde{\mu}}(\overline{w}u)\|_{L^{1}_{t}L^{r}_{x}}$
$\displaystyle\lesssim\|\dot{P}_{\tilde{\mu}}(\overline{w}C_{\ll\lambda_{2}^{2}}u\|_{L^{1}_{t}L^{r}_{x}}+\sum_{d\gtrsim\lambda_{2}^{2}}\|\dot{P}_{\tilde{\mu}}(\overline{w}C_{d}u)\|_{L^{1}_{t}L^{r}_{x}}$
$\displaystyle\lesssim\tilde{\mu}^{2-\frac{4}{r}}\|w\|_{Z}\|C_{\ll\lambda_{2}^{2}}u\|_{Z}+\sum_{d\gtrsim\lambda_{2}^{2}}\int_{\mathbb{R}}\|\dot{P}_{\tilde{\mu}}(\overline{w}e^{it\Delta}f^{(\tau)}_{d})\|_{L^{1}_{t}L^{r}_{x}}d\tau$
$\displaystyle\lesssim\tilde{\mu}^{2-\frac{4}{r}}\|w\|_{Z}\Big{(}\|C_{\ll\lambda^{2}_{2}}u\|_{Z}+\sum_{d\gtrsim\lambda_{2}^{2}}\int_{\mathbb{R}}\|f^{(\tau)}_{d}\|_{L^{2}_{x}}d\tau\Big{)}$
$\displaystyle\lesssim\tilde{\mu}^{2-\frac{4}{r}}\|w\|_{Z}\|u\|_{\underline{S}^{0}_{\lambda_{2}}}.$

Repeating this argument for $w$ shows that we can replace $\|w\|_{Z}$ with
$\|w\|_{\underline{S}^{0}_{\lambda_{1}}}$. Thus it only remains to replace the
free wave $e^{it|\nabla|}g$ in (4.5) with a general function
$v\in\underline{W}^{0}_{\mu}$, but this follows from a similar argument.
Namely we write

$\displaystyle v_{\mu}$
$\displaystyle=P^{(t)}_{\ll\mu^{2}}v_{\mu}+\sum_{d\gtrsim\mu^{2}}P^{(t)}_{d}v_{\mu}$

and apply the Duhamel formula together with (4.5) for the first term. On the
other hand, for the second term we argue via the standard $X^{s,b}$
transference argument, and observe that we can write

$P^{(t)}_{d}v_{\mu}=\int_{|\tau|\approx
d}e^{it\tau}e^{it|\nabla|}g^{(\tau)}_{\mu,d}d\tau$

with
$\widehat{g}^{(\tau)}_{\mu,d}(\xi)=\mathcal{F}_{t,x}[P^{(t)}_{d}v_{\mu}](\tau+|\xi|,\xi)$.
Hence the required bound follows by arguing as in the proof of (4.5). ∎

###  4.2. Schrödinger Nonlinearity

The next step is to combine the above bilinear restriction type estimates to
give a gain when bounding the Schrödinger nonlinearity. The key observation is
that we have additional regularity to exploit, namely in the applications to
follow, the Schrödinger evolution lies in the energy space $H^{1}$. In
particular, if we only wish to obtain an improved bound at lower regularities,
we have some room to exploit. On way to do this is via the following.

######  Theorem 4.5 (Frequency localised bound for Schrödinger nonlinearity).

Let $0<s\leqslant s^{\prime}<1$ and $\ell\geqslant 0$. There exists
$0<\theta<1$ such that for any interval $I\subset\mathbb{R}$ and any
$\mu,\lambda_{0},\lambda_{1}\in 2^{\mathbb{N}}$ we have

$\big{\|}\mathcal{I}_{0}\big{[}\mathbbold{1}_{I}P_{\lambda_{0}}(v_{\mu}u_{\lambda_{1}})\big{]}\big{\|}_{S^{s}_{\lambda_{0}}}\lesssim\Big{(}\frac{\lambda_{min}}{\lambda_{max}}\Big{)}^{\theta}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\theta\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda_{1}}\|_{\underline{S}^{s^{\prime}}_{\lambda_{1}}}\Big{)}^{1-\theta\sigma}$

where $9\sigma=\ell+s^{\prime}-s$,
$\lambda_{min}=\min\\{\mu,\lambda_{0},\lambda_{1}\\}$, and
$\lambda_{max}=\max\\{\mu,\lambda_{0},\lambda_{1}\\}$.

###### Proof.

Fix $0<s<1$. It suffices to consider the case where
$9\sigma=\ell+s^{\prime}-s$ is sufficiently small, and
$0\leqslant\ell<\frac{1}{4}$. We begin by observing that if
$\frac{1}{q}=\frac{\sigma}{2(1-\sigma)}$ and
$\frac{1}{r}=\frac{1}{2}-\frac{2}{3q}$, then provided $\sigma\geqslant 0$ is
sufficiently small, the pair $(q,r)$ is wave Strichartz admissible. Hence the
Strichartz estimates (2.1) and (2.2) together with an application of
Bernstein’s inequality gives for any $\mu,\lambda_{0},\lambda_{1}\in
2^{\mathbb{N}}$, and any $\tilde{\mu}\in 2^{\mathbb{Z}}$

$\displaystyle\|$ $\displaystyle
P_{\lambda_{0}}(\dot{P}_{\tilde{\mu}}v_{\mu}u_{\lambda_{1}})\|_{L^{2}_{t,x}(I)}$
$\displaystyle\lesssim(\min\\{\lambda_{0},\lambda_{1},\tilde{\mu}\\})^{1-\frac{13\sigma}{3}}\Big{(}\|v_{\mu}\|_{L^{\infty}_{t,x}(I)}\|u_{\lambda_{1}}\|_{L^{\infty}_{t,x}(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{L^{q}_{t}L^{r}_{x}}\|u_{\lambda_{1}}\|_{L^{2}_{t}L^{4}_{x}}\Big{)}^{1-\sigma}$
$\displaystyle\lesssim(\min\\{\lambda_{0},\lambda_{1},\tilde{\mu}\\})^{1-\frac{13\sigma}{3}}\mu^{\frac{29\sigma}{6}-\ell(1-\sigma)}\lambda_{1}^{4\sigma-s^{\prime}(1-\sigma)}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda_{1}}\|_{S^{s^{\prime}}_{\lambda_{1}}}\Big{)}^{1-\sigma}.$

Note that this bound is only nontrivial when
$\langle\tilde{\mu}\rangle\approx\mu$, and two largest frequencies are
comparable. Thus the additional homogeneous $\dot{P}_{\tilde{\mu}}$ multiplier
is only interesting when $\mu=1$, as then $\dot{P}_{\tilde{\mu}}$ forces an
additional localisation to the frequency region
$|\xi|\approx\tilde{\mu}\lesssim 1$ (compared to the inhomogeneous
localisation $|\xi|\leqslant 2$ provided by $v_{\mu}$ when $\mu=1$). In
particular, a short computation shows that provided $\sigma\geqslant 0$ is
sufficiently small, if $\mu,\lambda\in 2^{\mathbb{N}}$ with
$\mu\lesssim\lambda$, and $\tilde{\mu}\in 2^{\mathbb{Z}}$, then

$\lambda^{s-1}\|\dot{P}_{\tilde{\mu}}v_{\mu}u_{\lambda}\|_{L^{2}_{t,x}(I)}\lesssim\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{3}{4}}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda}\|_{S^{s^{\prime}}_{\lambda}}\Big{)}^{1-\sigma}$
(4.6)

while if
$\mu\approx\max\\{\lambda_{0},\lambda_{1}\\}\gg\min\\{\lambda_{0},\lambda_{1}\\}$
(and $\lambda_{0},\lambda_{1}\in 2^{\mathbb{N}}$) we instead have

$\begin{split}\lambda^{s-1}_{0}\|P_{\lambda_{0}}(v_{\mu}u_{\lambda_{1}})\|_{L^{2}_{t,x}(I)}&\lesssim\Big{(}\frac{\min\\{\lambda_{0},\lambda_{1}\\}}{\mu}\Big{)}^{\min\\{\frac{s}{2},\frac{1-s}{2}\\}}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda_{1}}\|_{S^{s^{\prime}}_{\lambda_{1}}}\Big{)}^{1-\sigma}.\end{split}$
(4.7)

This clearly suffices to bound the $L^{2}_{t,x}$ component of the
$S^{s}_{\lambda_{0}}$ norm after potentially decomposing into small
frequencies via $v_{\mu}=\sum_{\tilde{\mu}\in
2^{\mathbb{Z}},\langle\tilde{\mu}\rangle\approx\mu}\dot{P}_{\tilde{\mu}}v_{\mu}$.

To bound the Stricharz component of the $S^{s}_{\lambda}$ norm, we consider
separately the cases $\mu\lesssim\lambda_{0}\approx\lambda_{1}$ and
$\mu\gg\min\\{\lambda_{0},\lambda_{1}\\}$, and adapt the argument used to
prove [9, Theorem 5.1] together with Corollary 4.4. More precisely, in view of
the duality (see [9, Lemma 2.2]), to deal with the case
$\mu\lesssim\lambda_{0}\approx\lambda_{1}$ it suffices to prove that if
$1\lesssim\mu\lesssim\lambda$ then we have $\theta>0$ such that

$\Big{|}\int_{I\times\mathbb{R}^{4}}v_{\mu}\overline{w}u_{\lambda}\,dx\,dt\Big{|}\lesssim\lambda^{-s}\Big{(}\frac{\mu}{\lambda}\Big{)}^{\theta}\|w\|_{Z}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\theta\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda}\|_{S^{s^{\prime}}_{\lambda}}\Big{)}^{1-\theta\sigma}.$
(4.8)

Let $K_{\tilde{\mu}}\in L^{1}$ denote the convolution kernel of the Fourier
multiplier $\dot{P}_{\approx{\tilde{\mu}}}$, and take
$\frac{1}{r}=\frac{1}{2}+\frac{\theta}{4(1-\theta)}$ where $0<\theta<1$. Then
provided $\theta>0$ is sufficiently small, for any $w\in Z$ we have via
Hölder’s inequality and Corollary 4.4

$\displaystyle\Big{|}\int_{I\times\mathbb{R}^{4}}v_{\mu}\overline{w}u_{\lambda}\,dx\,dt\Big{|}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}\tilde{\mu}\in
2^{\mathbb{Z}}\\\
\langle\tilde{\mu}\rangle\approx\mu\end{subarray}}\Big{|}\int_{I\times\mathbb{R}^{4}}\dot{P}_{\tilde{\mu}}v_{\mu}\overline{w}u_{\lambda}\,dx\,dt\Big{|}$
$\displaystyle\leqslant\sum_{\begin{subarray}{c}\tilde{\mu}\in
2^{\mathbb{Z}}\\\
\langle\tilde{\mu}\rangle\approx\mu\end{subarray}}\|\dot{P}_{\tilde{\mu}}v_{\mu}\dot{P}_{\approx\tilde{\mu}}(\overline{w}u_{\lambda})\|_{L^{1}_{t}L^{\frac{4}{3}}_{x}(I)}^{\theta}\|\dot{P}_{\tilde{\mu}}v_{\mu}\dot{P}_{\approx\tilde{\mu}}(\overline{w}u_{\lambda})\|_{L^{1}_{t}L^{\frac{2r}{2+r}}_{x}}^{1-\theta}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}\tilde{\mu}\in
2^{\mathbb{Z}}\\\
\langle\tilde{\mu}\rangle\approx\mu\end{subarray}}\Big{(}\int_{\mathbb{R}^{4}}|K_{\tilde{\mu}}(y)|\|(\dot{P}_{\tilde{\mu}}v_{\mu})(t,x)\overline{w}(t,x-y)u_{\lambda}(t,x-y)\|_{L^{1}_{t}L^{\frac{4}{3}}_{x}(I)}dy\Big{)}^{\theta}$
$\displaystyle\qquad\qquad\qquad\qquad\times\Big{(}\|v_{\mu}\|_{L^{\infty}_{t}L^{2}_{x}}\|\dot{P}_{\mu}(\overline{w}u_{\lambda})\|_{L^{1}_{t}L^{r}_{x}}\Big{)}^{1-\theta}$
$\displaystyle\lesssim\lambda^{-s}\|w\|_{Z}\sum_{\begin{subarray}{c}\tilde{\mu}\in
2^{\mathbb{Z}}\\\
\langle\tilde{\mu}\rangle\approx\mu\end{subarray}}\Big{(}{\tilde{\mu}^{-1}}\lambda^{s}\sup_{y\in\mathbb{R}^{4}}\|(\dot{P}_{\tilde{\mu}}v_{\mu})(t,x)u_{\lambda}(t,x-y)\|_{L^{2}_{t,x}(I)}\Big{)}^{\theta}\Big{(}\|v_{\mu}\|_{L^{\infty}_{t}L^{2}_{x}}\|u_{\lambda}\|_{\underline{S}^{s}_{\lambda}}\Big{)}^{1-\theta}.$

To bound the $L^{2}_{t,x}$ norm in the above inequality, we combine Corollary
4.4 and (4.6) which gives

$\displaystyle\tilde{\mu}^{-1}\lambda^{s}\|\dot{P}_{\tilde{\mu}}v_{\mu}u_{\lambda}\|_{L^{2}_{t,x}(I)}$
$\displaystyle\lesssim\Big{[}\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{-\frac{1}{4}}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda}\|_{S^{s^{\prime}}_{\lambda}}\Big{)}^{1-\sigma}\Big{]}^{\frac{1}{4}}\Big{[}\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{1}{4}}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda}\|_{\underline{S}^{s^{\prime}}_{\lambda}}\Big{]}^{\frac{3}{4}}$
$\displaystyle\approx\Big{(}\frac{\tilde{\mu}}{\lambda}\Big{)}^{\frac{1}{4}}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\frac{\sigma}{4}}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\mu}}\|u_{\lambda}\|_{\underline{S}^{s^{\prime}}_{\lambda}}\Big{)}^{1-\frac{\sigma}{4}}.$

Therefore the bound (4.8) follows from the translation invariance of the norm
$\|\cdot\|_{\underline{S}^{s}_{\lambda}}$, and summing up over low frequencies
$\tilde{\mu}\lesssim\mu$.

It only remains to consider the case
$\mu\approx\max\\{\lambda_{0},\lambda_{1}\\}\gg\min\\{\lambda_{0},\lambda_{1}\\}$.
Let $\frac{1}{q}=\frac{\sigma}{2(1-\sigma)}$,
$\frac{1}{r}=\frac{1}{2}-\frac{2}{3q}$, and
$\frac{1}{a}=\frac{1}{2}+\frac{7\sigma}{12(1-\sigma)}$. Then provided
$\sigma\geqslant 0$ is sufficiently small, the pair $(q,r)$ is wave Strichartz
admissible and hence an application of (2.2) together with Corollary 4.4
(noting that
$\overline{w}_{\lambda_{0}}u_{\lambda_{1}}=P_{\approx\mu}(\overline{w}_{\lambda_{0}}u_{\lambda_{1}})$
) and the endpoint Strichartz estimate for the Schrödinger equation gives

$\displaystyle\Big{|}\int_{I\times\mathbb{R}^{4}}v_{\mu}\overline{w}_{\lambda_{0}}u_{\lambda_{1}}\,dx\,dt\Big{|}$
$\displaystyle\lesssim\Big{(}\|v_{\mu}\|_{L^{\infty}_{t,x}(I)}\|w_{\lambda_{0}}\|_{L^{2}_{t}L^{4}_{x}}\|u_{\lambda_{1}}\|_{L^{\infty}_{t,x}(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{L^{q}_{t}L^{r}_{x}}\|\overline{w}_{\lambda_{0}}u_{\lambda_{1}}\|_{L^{1}_{t}L^{a}_{x}}\Big{)}^{1-\sigma}$
$\displaystyle\lesssim\|w\|_{Z}\Big{(}\|v_{\mu}\|_{L^{\infty}_{t,x}(I)}\|u_{\lambda_{1}}\|_{L^{\infty}_{t,x}(I)}\Big{)}^{\sigma}\Big{(}\mu^{\frac{5}{3q}-\ell}\lambda_{1}^{2-\frac{4}{a}-s^{\prime}}\|v_{\mu}\|_{\underline{W}^{\ell}_{\lambda}}\|u_{\lambda_{1}}\|_{\underline{S}^{s^{\prime}}_{\lambda}}\Big{)}^{1-\sigma}$
$\displaystyle\lesssim\mu^{6\sigma-\ell}\lambda_{1}^{3\sigma-s^{\prime}}\|w\|_{Z}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\lambda}}\|u_{\lambda_{1}}\|_{\underline{S}^{s^{\prime}}_{\lambda}}\Big{)}^{1-\sigma}$

where the last line again used the assumption $0\leqslant\ell<\frac{1}{4}$ and
$0<s^{\prime}<1$ to slightly simplify the frequency exponents. Therefore
another application of duality (see [9, Lemma 2.2]) gives for any
$\mu\approx\max\\{\lambda_{0},\lambda_{1}\\}\gg\min\\{\lambda_{0},\lambda_{1}\\}$
the bound

$\lambda_{0}^{s}\|\mathcal{I}_{0}[\mathbbold{1}_{I}P_{\lambda_{0}}(v_{\mu}u_{\lambda_{1}})]\|_{L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{4}_{x}}\lesssim\mu^{9\sigma-\ell+s}\lambda_{1}^{-s^{\prime}}\Big{(}\|v_{\mu}\|_{Y(I)}\|u_{\lambda_{1}}\|_{Y(I)}\Big{)}^{\sigma}\Big{(}\|v_{\mu}\|_{\underline{W}^{\ell}_{\lambda}}\|u_{\lambda_{1}}\|_{\underline{S}^{s^{\prime}}_{\lambda}}\Big{)}^{1-\sigma}.$

Since $9\sigma=\ell+s^{\prime}-s$ and $s>0$, this suffices when
$\mu\approx\lambda_{1}\gg\lambda_{0}$. On the other hand, when
$\mu\approx\lambda_{0}\gg\lambda_{1}$, we instead interpolate with the high-
low frequency gain

$\lambda_{0}^{s}\|\mathcal{I}_{0}[\mathbbold{1}_{I}P_{\lambda_{0}}(v_{\mu}u_{\lambda_{1}})]\|_{L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}L^{4}_{x}}\lesssim\Big{(}\frac{\lambda_{1}}{\mu}\Big{)}^{1-s}\|v_{\mu}\|_{\underline{W}^{0}_{\lambda}}\|u_{\lambda_{1}}\|_{\underline{S}^{s}_{\lambda}}$

which is a consequence of (the proof of) [9, Theorem 3.1] (see (3.5) and
equation below (3.3) in [9]) together with the energy inequality (2.5). Note
that the temporal cutoff $\mathbbold{1}_{I}$ can be removed by applying (2.7)
and Lemma 2.1. ∎

Note that in the previous theorem, we have two options. We can either take
$\ell=0$ and $s^{\prime}>s$ (so spend additional Schrödinger regularity), or
$\ell>0$ and $s^{\prime}=s$ (so spend additional wave regularity).

###  4.3. Wave Nonlinearity

We have a similar frequency localised improved bilinear estimate for the wave
nonlinearity.

######  Theorem 4.6 (Frequency localised control of wave nonlinearity).

Let $s>\frac{1}{2}$. There exists $\theta>0$ such that for any interval
$I\subset\mathbb{R}$ and any $\mu,\lambda_{1},\lambda_{2}\in 2^{\mathbb{N}}$
we have

$\big{\|}\mathcal{J}_{0}\big{[}\mathbbold{1}_{I}|\nabla|\Re\big{(}P_{\mu}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\big{)}\big{]}\big{\|}_{W^{0}_{\mu}}\lesssim\Big{(}\frac{\lambda_{min}}{\lambda_{max}}\Big{)}^{\theta}\Big{(}\|w_{\lambda_{1}}\|_{Y(I)}\|u_{\lambda_{2}}\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}\Big{)}^{1-\theta}$

where $\lambda_{min}=\min\\{\mu,\lambda_{1},\lambda_{2}\\}$ and
$\lambda_{max}=\max\\{\mu,\lambda_{1},\lambda_{2}\\}$.

###### Proof.

It suffices to consider the case $\frac{1}{2}<s\leqslant 1$. Let $\theta>0$
(sufficiently small), and take $\frac{1}{q}=\frac{1}{4(1-\theta)}$ and
$\frac{1}{r}=\frac{1}{2}-\frac{1}{2q}$. An application of (2.1) together with
Bernstein’s inequality gives

$\displaystyle\mu^{-1}\big{\|}|\nabla|\Re\big{(}P_{\mu}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\big{)}\big{\|}_{L^{2}_{t,x}(I)}$
$\displaystyle\lesssim\Big{(}\|w_{\lambda_{1}}\|_{L^{\infty}_{t,x}(I)}\|u_{\lambda_{2}}\|_{L^{\infty}_{t,x}(I)}\Big{)}^{\theta}\big{\|}P_{\mu}\big{(}\overline{w}_{\lambda_{1}}u_{\lambda_{2}}\big{)}\big{\|}_{L^{\frac{q}{2}}_{t,x}}^{1-\theta}$
$\displaystyle\lesssim\Big{(}\|w_{\lambda_{1}}\|_{L^{\infty}_{t,x}(I)}\|u_{\lambda_{2}}\|_{L^{\infty}_{t,x}(I)}\Big{)}^{\theta}\Big{(}\lambda_{min}^{4(\frac{2}{r}-\frac{2}{q})}\big{\|}P_{\mu}\big{(}\overline{w}_{\lambda_{1}}u_{\lambda_{2}}\big{)}\big{\|}_{L^{\frac{q}{2}}_{t}L^{\frac{r}{2}}_{x}}\Big{)}^{1-\theta}$
$\displaystyle\lesssim\lambda_{min}^{1-4\theta}\lambda_{max}^{8\theta-(1-\theta)(\frac{1}{2}+s)}\Big{(}\|w_{\lambda_{1}}\|_{Y(I)}\|u_{\lambda_{2}}\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}\Big{)}^{1-\theta}.$

After observing that
$\lambda_{min}^{1-4\theta}\lambda_{max}^{8\theta-(1-\theta)(\frac{1}{2}+s)}\lesssim(\frac{\lambda_{min}}{\lambda_{max}})^{1-4\theta}\lambda_{max}^{\frac{1}{2}-s+6\theta}$,
the previous bound suffices to control the $L^{2}_{t,x}$ component of the norm
$W^{0}_{\mu}$ provided that $s\geqslant\frac{1}{2}+6\theta$. To bound the
$L^{\infty}_{t}L^{2}_{x}$ component, we take the wave admissible exponents
$\frac{1}{q}=\theta$ and $\frac{1}{r}=\frac{1}{2}-\frac{2\theta}{3}$, and let
$\frac{1}{a}=\frac{1}{1-\theta}(\frac{1}{2}+\frac{2\theta}{3})$. The bilinear
restriction type estimate in Corollary 4.4 gives for any $\tilde{\mu}\in
2^{\mathbb{Z}}$

$\displaystyle\|\dot{P}_{\tilde{\mu}}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}_{x}(I)}$
$\displaystyle\lesssim\Big{(}\|w_{\lambda_{1}}\|_{L^{\infty}_{t,x}(I)}\|u_{\lambda_{2}}\|_{L^{\infty}_{t,x}(I)}\Big{)}^{\theta}\|\dot{P}_{\tilde{\mu}}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\|_{L^{1}_{t}L^{a}_{x}(I)}^{1-\theta}$
$\displaystyle\lesssim(\lambda_{1}\lambda_{2})^{4\theta}\big{(}\tilde{\mu}^{2-\frac{4}{a}}\lambda_{1}^{-\frac{1}{2}}\lambda_{2}^{-s}\big{)}^{1-\theta}\Big{(}\|w_{\lambda_{1}}\|_{Y(I)}\|u_{\lambda_{2}}\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}\Big{)}^{1-\theta}$
$\displaystyle\lesssim\tilde{\mu}^{-(\frac{5}{3}\theta+1)}\Big{(}\frac{\tilde{\mu}}{\lambda_{1}}\Big{)}^{\frac{1}{2}-3\theta}\Big{(}\frac{\tilde{\mu}}{\lambda_{2}}\Big{)}^{\frac{1}{2}}\lambda_{2}^{\frac{1}{2}-s+7\theta}\Big{(}\|w_{\lambda_{1}}\|_{Y(I)}\|u_{\lambda_{2}}\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}\Big{)}^{1-\theta}.$

Consequently, as the pair $(q,r)$ is wave admissable, provided
$s>\frac{1}{2}+7\theta$ the inhomogeneous Strichartz estimate for the wave
equation gives

$\displaystyle\big{\|}\mathcal{J}_{0}\big{[}\mathbbold{1}_{I}|\nabla|\Re\big{(}P_{\mu}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\big{)}\big{]}\big{\|}_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}\tilde{\mu}\in
2^{\mathbb{N}}\\\
\langle\tilde{\mu}\rangle\approx\mu\end{subarray}}\tilde{\mu}\big{\|}\mathcal{J}_{0}\big{[}\mathbbold{1}_{I}\Re\big{(}\dot{P}_{\tilde{\mu}}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\big{)}\big{]}\big{\|}_{L^{\infty}_{t}L^{2}_{x}}$
(4.9) $\displaystyle\lesssim\sum_{\begin{subarray}{c}\tilde{\mu}\in
2^{\mathbb{N}}\\\
\langle\tilde{\mu}\rangle\approx\mu\end{subarray}}\tilde{\mu}^{1+\frac{5}{3}\theta}\|\dot{P}_{\tilde{\mu}}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\|_{L^{q^{\prime}}_{t}L^{r^{\prime}}_{x}(I)}$
$\displaystyle\lesssim\Big{(}\frac{\mu}{\lambda_{1}}\Big{)}^{\frac{1}{2}-3\theta}\Big{(}\frac{\mu}{\lambda_{2}}\Big{)}^{\frac{1}{2}}\Big{(}\|w_{\lambda_{1}}\|_{Y(I)}\|u_{\lambda_{2}}\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}\Big{)}^{1-\theta}.$

This is enough to bound the $L^{\infty}_{t}L^{2}_{x}$ contribution in the case
$\mu\approx\lambda_{\min}$, thus it only remains to consider the case
$\mu\approx\lambda_{\max}\gg\lambda_{min}$. But this follows by combining
(4.9) with the derivative gain

$\displaystyle\big{\|}\mathcal{J}_{0}\big{[}\mathbbold{1}_{I}|\nabla|\Re\big{(}P_{\mu}(\overline{w}_{\lambda_{1}}u_{\lambda_{2}})\big{)}\big{]}\big{\|}_{L^{\infty}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\mu^{1-2s}\|w_{\lambda_{1}}\|_{\underline{S}^{s}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}$
$\displaystyle\approx\mu^{1-2s}\lambda_{1}^{s-\frac{1}{2}}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}\lesssim\mu^{\frac{1}{2}-s}\|w_{\lambda_{1}}\|_{\underline{S}^{\frac{1}{2}}_{\lambda_{1}}}\|u_{\lambda_{2}}\|_{\underline{S}^{s}_{\lambda_{2}}}$

which is a consequence of [10, Proof of (4.1) case 1] and [10, Lemma 2.6] in
the case $\mu\approx\lambda_{max}\gg\lambda_{min}$ (note that
$W^{\ell}=W^{\ell,0,0}$ and $S^{s}=S^{s,0,0}$, and so $a=b=0$ in the notation
of [10]). ∎

###  4.4. Proof of Theorem 4.1

The proof of Theorem 4.1 is now a direct application of the corresponding
frequency localised bounds.

###### Proof of Theorem 4.1.

We only give the proof of the Schrödinger bound, as the wave bound is similar.
Let $I\subset\mathbb{R}$ be an interval. In view of (2.4), have
$\|u_{\lambda}\|_{Y}\lesssim\|u_{\lambda}\|_{S^{s}_{\lambda}}$ and
$\|V_{\mu}\|_{Y}\lesssim\|V_{\mu}\|_{W^{0}_{\mu}}$, and hence we may take
$\theta>0$ as small as we like. An application of Theorem 4.5 together with a
standard summation argument gives for any $s>\frac{1}{2}$

$\big{\|}\mathcal{I}_{0}\big{[}\mathbbold{1}_{I}\Re(V)u\big{]}\big{\|}_{S^{\frac{1}{2}}}\lesssim\Big{(}\|u\|_{Y(I)}\|V\|_{Y(I)}\Big{)}^{\theta}\Big{(}\sum_{\mu\in
2^{\mathbb{N}}}\|V_{\mu}\|_{W^{0}_{\mu}}^{4(1-\theta)}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\lambda\in
2^{\mathbb{N}}}\|u_{\lambda}\|_{S^{s}_{\lambda}}^{4(1-\theta)}\Big{)}^{\frac{1}{2}}.$

In particular, choosing $\theta>0$ slightly smaller if necessary, we can
ensure that $4(1-\theta)\geqslant 2$ and hence conclude that

$\big{\|}\mathcal{I}_{0}\big{[}\mathbbold{1}_{I}\Re(V)u\big{]}\big{\|}_{S^{\frac{1}{2}}}\lesssim\Big{(}\|u\|_{Y(I)}\|V\|_{Y(I)}\Big{)}^{\theta}\Big{(}\|V\|_{W^{0}}\|u\|_{S^{s}}\Big{)}^{1-\theta}.$

By definition of the temporally restricted spaces $S^{s}(I)$ and $W^{0}(I)$,
we can replace the norms on the right hand side by their corresponding
localised counterparts. Therefore, provided $0\in I$, the bound for the
Schrödinger nonlinearity now follows from Lemma 2.1. ∎

##  5\. Small Data Theory II

In this section give a small data theory based on the dispersive norm
$\|\cdot\|_{Y}$. In particular, we give the proof of Theorem 1.7 as well as a
version of the stability result from Theorem 3.3 adapted to the concentration
compactness arguments to follow. The idea is simply to combine the refined
bilinear estimates from the previous section together with the small data
theory developed in Section 3. The first step is to observe that the improved
bilinear estimate in Theorem 4.1 implies that the controlling quantity
$\rho_{I}$ defined in (3.6) can be bounded in terms of the dispersive norm
$\|\cdot\|_{Y(I)}$.

######  Corollary 5.1 (Controlling norm bounded by $Y(I)$).

Let $s>\frac{1}{2}$. There exists $C>0$ and $0<\theta<1$ such that for any
interval $0\in I\subset\mathbb{R}$ we have

$\rho_{I}(u)\leqslant
C\|u\|_{Y(I)}^{\theta}\|u\|_{\underline{S}^{s}}^{1-\theta}.$

###### Proof.

This follows directly from the definition of the controlling quantity
$\rho_{I}$, (3.6), together with Lemma 2.1 and Theorem 4.1. ∎

We can now state the version of the stability result in Theorem 3.3 that we
exploit in the following.

######  Theorem 5.2 (Stability II).

Let $A>0$, $0<B<\|Q\|_{\dot{H}^{1}}$, and $\frac{1}{2}<s<2$. There exists
$\epsilon_{0}(A,B,s)>0$, $\theta=\theta(A,B,s)>0$, and $C=C(A,B,s)>0$ such
that if $0<\epsilon<\epsilon_{0}(A,B,s)$, $t_{0}\in I\subset\mathbb{R}$ is an
interval, and

$(f,g)\in H^{s}\times L^{2},\qquad(F,G)\in N^{\frac{1}{2}}(I)\times
M^{0}(I),\qquad(\psi,\phi)\in C(I,H^{\frac{1}{2}}\times L^{2})$

satisfies for some $t_{0}\in I$ the boundedness condition

$\|f\|_{H^{s}}+\|\psi\|_{S^{\frac{1}{2}}(I)}\leqslant
A,\qquad\|\phi\|_{L^{\infty}_{t}L^{2}_{x}(I\times\mathbb{R}^{4})}\leqslant B,$
(5.1)

the smallness condition

$\big{\|}e^{i(t-t_{0})\Delta}\big{(}f-\psi(t_{0})\big{)}\big{\|}_{Y(I)}+\|g-\phi(t_{0})\|_{L^{2}}+\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{\frac{1}{2}}(I)}+\|\mathcal{J}_{0,t_{0}}[G]\|_{W^{0}(I)}<\epsilon,$
(5.2)

and $(\psi,\phi)$ solve the perturbed Zakharov equation (3.8) on
$I\times\mathbb{R}^{4}$, then there exists a (unique) solution $(u,V)\in
C(I,H^{s}\times L^{2})$ to the Zakharov equation (1.2) with data
$(u,V)(t_{0})=(f,g)$ and the bounds

$\big{\|}u-\psi-e^{i(t-t_{0})\Delta}\big{(}f-\psi(t_{0})\big{)}\big{\|}_{S^{\frac{1}{2}}(I)}\leqslant
C\epsilon^{\theta},\qquad\|u\|_{D(I)}\leqslant
C\|f\|_{H^{\frac{1}{2}}},\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}(I)}\leqslant\tfrac{1}{2}\big{(}B+\|Q\|_{\dot{H}^{1}}\big{)}.$

###### Proof.

In view of the persistence of regularity from [10, Theorem 8.1], it suffices
to consider the case $\frac{1}{2}<s<1$. An application of Corollary 5.1
together with (2.1) implies that

$\rho_{I}(e^{it\Delta}f)\lesssim\|e^{it\Delta}f\|_{Y(I)}^{\theta}\|e^{it\Delta}f\|_{\underline{S}^{s}(I)}^{1-\theta}\lesssim\epsilon^{\theta}\|f\|_{H^{s}}^{1-\theta}\lesssim\epsilon^{\theta}A^{1-\theta}$

where $\theta$ and the implied constant depend only on $s>\frac{1}{2}$. In
particular, choosing $\epsilon>0$ sufficiently small to ensure that
$\epsilon^{\theta}A^{1-\theta}\ll\epsilon_{0}(A,B)$ where $\epsilon_{0}$ is as
in Theorem 3.3, the conditions (5.1) and (5.2) imply the corresponding
conditions (3.9) and (5.2) in Theorem 3.3. Consequently an application of
Theorem 3.3 implies that there exists a unique solution $(u,V)\in
C(I;H^{s}\times L^{2})$ with data $(u,V)(t_{0})=(f,g)$ satisfying

$\big{\|}u-\psi-e^{i(t-t_{0})\Delta}\big{(}f-\psi(t_{0})\big{)}\big{\|}_{S^{\frac{1}{2}}(I)}\lesssim_{A,B,s}\epsilon^{\theta},\qquad\|V\|_{L^{\infty}_{t}L^{2}_{x}(I)}\leqslant\tfrac{1}{2}\big{(}B+\|Q\|_{\dot{H}^{1}}\big{)}$

where (as in the remainder of the proof) the implied constant depends only on
$A$, $B$, and $S$. To obtain the claimed bound in the Strichartz norm, we note
that via (2.3) and the homogeneous Strichartz estimate, we have

$\displaystyle\|u\|_{D(I)}$
$\displaystyle\lesssim\|u-\psi-e^{i(t-t_{0})}(f-\psi(t_{0})\big{)}\big{\|}_{D(I)}+\|e^{i(t-t_{0})\Delta}\big{(}f-\psi(t_{0})\big{)}\big{\|}_{D(I)}$
$\displaystyle\lesssim\|u-\psi-e^{i(t-t_{0})}(f-\psi(t_{0})\big{)}\big{\|}_{S^{\frac{1}{2}}(I)}+\|f-\psi(t_{0})\|_{H^{\frac{1}{2}}}\lesssim\epsilon^{\theta}+A.$

In particular, $\|u\|_{D(I)}\lesssim_{A,B,S}1$, and hence the claimed
Strichartz bound follows from Theorem 3.1. ∎

Although it is not needed in the following, strictly speaking the bounds
implied by Theorem 3.3 are slightly stronger than those stated in the
conclusion in of Theorem 5.2. For later use, we observe that in view of the
bound (2.4), we note that the smallness condition (5.2) can be replaced with

$\min\big{\\{}\|f-\psi(t_{0})\|_{L^{2}},\|f-\psi(t_{0})\|_{\dot{H}^{1}}\big{\\}}+\|g-\phi(t_{0})\|_{L^{2}}+\|\mathcal{I}_{0,t_{0}}[F]\|_{S^{\frac{1}{2}}(I)}+\|\mathcal{J}_{0,t_{0}}[G]\|_{W^{0}(I)}<\epsilon.$
(5.3)

Thus we only need the Schrödinger data $f$ and $\psi(t_{0})$ to be close in
the _homogeneous_ spaces $L^{2}$ or $\dot{H}^{1}$. This smallness in the
homogeneous spaces is crucial, as we want to conclude that solutions which are
close to a dispersive solution in either mass $\mathcal{M}$ (so $L^{2}$) or
energy $\mathcal{E}_{Z}$ (so roughly $\dot{H}^{1}$), are also dispersive. It
is worth noting that, since we are working in the subcritical setting
$s>\frac{1}{2}$, proving that smallness in $L^{2}$ suffices is straightforward
via the convexity type bound

$\|f\|_{H^{\frac{1}{2}}}\lesssim\|f\|_{L^{2}}^{\theta}\|f\|_{H^{s}}^{1-\theta}.$

On the other hand, proving that smallness in $\dot{H}^{1}$ suffices is
significantly more challenging, as the low frequencies can be badly behaved.
In particular there does not seem to be an easy direct argument to conclude
that smallness in energy suffices for stability.

We now come to the proof of Theorem 1.7.

###### Proof of Theorem 1.7.

Similar to the proof of Corollary 3.4, this follows by letting $F=G=0$ and
$\phi=e^{it|\nabla|}g$ in Theorem 5.2. ∎

##  6\. Profile Decomposition

The proof of Theorem 1.6 requires a precise understanding of how a bounded
sequence in $H^{1}\times L^{2}$ can fail to be compact. As usual, this is
accomplished via the profile decomposition of Bahouri-Gérard [2]. To describe
the profile decomposition we require, we first introduce our notation for the
group elements responsible for the lack of compactness. Given
$(s,y)\in\mathbb{R}\times\mathbb{R}^{4}$ we define the group elements
$\mathfrak{g}[s,y]:H^{1}\times L^{2}\to H^{1}\times L^{2}$ as

$\mathfrak{g}[s,y](f,g)=\big{(}e^{-is\Delta}\tau_{y}f,e^{-is|\nabla|}\tau_{y}g\big{)}$

where $(\tau_{y}\phi)(x)=\phi(x-y)$ is the translation operator on
$\mathbb{R}^{4}$. Note that there is no frequency parameter in the group
elements. This is due to the fact that sequences in $H^{1}\times L^{2}$ with
frequency parameters going to either $0$ or $\infty$, are small in the $Y$
norm. For instance, if we consider the sequence

$(f_{n},g_{n})(x)=\big{(}\lambda_{n}^{2}\langle\lambda_{n}\rangle^{-1}f(\lambda_{n}x),\lambda_{n}^{2}g(\lambda_{n}x)\big{)}$

with say $\operatorname*{supp}(\widehat{f},\widehat{g})\subset\\{|\xi|\approx
1\\}$. Then $\|(f_{n},g_{n})\|_{H^{1}\times
L^{2}}\approx\|(f,g)\|_{H^{1}\times L^{2}}$ is a bounded sequence with no
limit in $H^{1}\times L^{2}$ if $\lambda_{n}+\lambda_{n}^{-1}\to\infty$. On
the other hand, a short computation gives
$\|(e^{it\Delta}f_{n},e^{it|\nabla|}g_{n})\|_{Y}\to 0$ if
$\lambda_{n}+\lambda_{n}^{-1}\to\infty$, and hence we can absorb sequences