2.1 Setup: dFT and the quadratic spectral distribution

We refer to Section 3 for a more detailed presentation of the distorted Fourier transform (dFT) and admit for the moment the existence of generalised eigenfunctions $\psi = \psi (x,\xi )$ such that

(2.1) $$ \begin{align} \forall \quad \xi \in {\mathbb R}, \qquad (- \partial_x^2 + V ) \psi(x,\xi) = \xi^2 \psi(x,\xi), \end{align} $$

and the familiar formulas relating the Fourier transform and its inverse in dimension $d=1$ hold if one replaces (up to a constant) $e^{i\xi x}$ by $\psi (x,\xi )$ :

(2.2) $$ \begin{align} \widetilde{f}(\xi) = \int_{\mathbb R} \overline{\psi(x,\xi)} f(x)\,dx \qquad \mbox{and} \qquad f(x) = \int_{\mathbb R} {\psi(x,\xi)} \widetilde{f}(\xi) \,d\xi. \end{align} $$

Let us consider a solution of the equation

$$ \begin{align*} \partial_t^2 u + (- \partial_x^2 + V(x) + 1) u = a(x)u^2, \qquad (u,u_t)(t=0) = (u_0,u_1). \end{align*} $$

Defining the profile g by

(2.3) $$ \begin{align} \begin{aligned} & g(t,x):=e^{it\sqrt{H+ 1}}\big(\partial_t -i \sqrt{ H + 1 } \big)u, \qquad \widetilde{g}(t,\xi) = e^{it\langle \xi \rangle}\big(\partial_t - i\langle \xi \rangle \big) \widetilde{u}, \end{aligned} \end{align} $$

and denoting $\widetilde {g}_+=\widetilde {g}$ , $\widetilde {g}_-= \overline {\widetilde {g}}$ , one sees that $\widetilde {g}$ satisfies an equation of the form

(2.4) $$ \begin{align} \partial_t \widetilde{g}(t,\xi) & = - \sum_{\iota_1,\iota_2\in\{+,-\}} \iota_1\iota_2 \iint e^{it \Phi_{\iota_1\iota_2}(\xi,\eta,\sigma)} \widetilde{g}_{\iota_1}(t,\eta) \widetilde{g}_{\iota_2}(t,\sigma) \, \frac{\mu_{\iota_1\iota_2}(\xi,\eta,\sigma)}{4\langle \eta \rangle \langle \sigma \rangle} \, d\eta \, d\sigma, \end{align} $$

where the oscillatory phase is given by

(2.5) $$ \begin{align} \Phi_{\iota_1 \iota_2}(\xi,\eta,\sigma) = \langle \xi \rangle - \iota_1 \langle \eta \rangle - \iota_2 \langle \sigma \rangle, \end{align} $$

and

(2.6) $$ \begin{align} \mu_{\iota_1\iota_2}(\xi,\eta,\sigma) := \int a(x) \overline{\psi(x,\xi)} \psi_{\iota_2}(x,\eta) \psi_{\iota_1}(x,\sigma) \, dx \end{align} $$

is what we refer to as the (quadratic) ‘nonlinear spectral distribution’ (NSD).

For the sake of exposition we will drop the signs $(\iota _1,\iota _2)$ from $\widetilde {g}$ and $\mu $ since they do not play any major role. We will instead keep the relevant signs in equation (2.5) and the analogous expressions for cubic interactions. We also drop the factor $\langle \eta \rangle \langle \sigma \rangle $ in equation (2.4). With this simplifications, integrating equation (2.4) over time gives

(2.7) $$ \begin{align} \widetilde{g}(t,\xi) = \widetilde{g_0}(\xi) -i \sum_{\iota_1,\iota_2 \in \{+,-\}} \int_0^t \iint e^{is \Phi_{\iota_1 \iota_2}(\xi,\eta,\sigma)} \widetilde{g}(s,\eta) \widetilde{g}(s,\sigma) \mu(\xi,\eta,\sigma) \,d\eta\, d\sigma\,ds. \end{align} $$

The first task is to analyse $\mu $ in equation (2.6), and we immediately see an essential difference from the flat case $V=0$ : in the absence of a potential, the generalised eigenfunctions $\psi (x,\xi )$ should be replaced by $e^{i\xi x}$ , in which case $\mu (\xi ,\eta ,{\sigma }) = \delta (\xi -\eta -{\sigma })$ – in particular, the sum of the frequencies of the two inputs: that is, $\eta $ and ${\sigma }$ , gives the output frequency $\xi $ . This can be thought of as a ‘conservation of momentum’ or ‘correlation’ between the frequencies. But if $V\neq 0$ , the structure of $\mu $ becomes more involved, and there is no a priori relation between the frequencies. This can be seen as a ‘decorrelation’ or ‘uncertainty’ due to the presence of the potential.

For the sake of this presentation, we can essentially think that

(2.8) $$ \begin{align} \begin{aligned} \mu(\xi,\eta,\sigma) & = \sum_{\mu,\nu \in\{+,-\}} \Big[ A_{\mu,\nu}(\xi,\eta,\sigma) \delta(\xi + \mu \eta + \nu {\sigma}) \\ &\quad +\, B_{\mu,\nu}(\xi,\eta,\sigma) \, \mathrm{p.v.} \frac{1}{\xi + \mu \eta + \nu {\sigma}} \Big] + C(\xi,\eta,\sigma), \end{aligned} \end{align} $$

where $A_{\mu ,\nu }$ , $B_{\mu ,\nu }$ and C are smooth functions and ‘ $\mathrm {p.v.}$ ’ stands for principal value.

The $\delta $ component of $\mu $ gives a contribution to equation (2.7) that is essentially the same as in the flat case, only algebraically more complicated due to the different signs combinations and the coefficients (which are related to the transmission and reflection coefficients of the potential). One could expect to treat these terms as in the classical flat case, that is, using a normal form transformation to eliminate the quadratic term in favour of cubic ones [Reference Shatah66, Reference Delort11, Reference Hayashi and Naumkin30].

The $\mathrm {p.v.}$ term in equation (2.8) seriously impacts the nature of the problem at hand. When the variable $\xi + \mu \eta + \nu {\sigma }$ that determines the singularity is very small, one could think that the corresponding interactions are not so different from those allowed by the $\delta $ distribution, possibly only logarithmically worse. When instead $\xi + \mu \eta + \nu {\sigma }$ is not too small, we have in essence a smooth kernel. While this might seem like a favourable situation, it is in fact a major complication. The decorrelation between the input and output frequencies prevents the application of a normal form transformation (quadratic terms cannot be eliminated); even more, it creates a genuinely nonlinear phenomenon of loss of regularity (in Fourier space) at specific bad frequencies. We explain this in more detail in the following paragraphs.

2.2 Oscillations and resonances: Singular vs. regular terms

Let us consider the quadratic interactions in equation (2.7) and, according to equation (2.8), write them as

(2.9) $$ \begin{align} \int_0^t \iint e^{is \Phi_{\iota_1 \iota_2}(\xi,\eta,\sigma)} \widetilde{g}(s,\eta) \widetilde{g}(s,\sigma) \, \mathfrak{m}(\xi,\eta,\sigma) \,d\eta\, d\sigma \, ds, \end{align} $$

where $\mathfrak {m}(\xi ,\eta ,\sigma )$ can be a distribution (i.e., a $\delta $ or a $\mathrm {p.v.}$ ) or a smooth function. The properties of equation (2.9) are dictated by the oscillations of the exponential factor and the structure of the singularities of $\mathfrak {m}$ . More precisely,

• • If $\mathfrak {m}=\delta (\xi - \mu \eta - \nu \sigma )$ or $\mathfrak {m} = \mathrm {p.v.} \frac {1}{\xi - \mu \eta - \nu \sigma }$ , resonant oscillations can be characterised as the stationary points of the phase $s \Phi _{\iota _1 \iota _2}$ , restricted to the singular hypersurface $\{ \xi - \mu \eta - \nu \sigma = 0 \}$ . Up to changing coordinates, we can reduce to the phase

  (2.10) $$ \begin{align} \Phi^S_{\iota_1 \iota_2}(\xi,\eta) = \langle \xi \rangle - \iota_1 \langle \eta \rangle - \iota_2 \langle \xi-\eta \rangle \end{align} $$
  (where we added the superscript S to emphasise that we consider a singular $\mathfrak {m}$ ), for which stationary points satisfy
  (2.11) $$ \begin{align} \Phi^S_{\iota_1 \iota_2}(\xi,\eta) = \partial_{\eta} \Phi^S_{\iota_1 \iota_2}(\xi,\eta) = 0. \end{align} $$
  These are the classical resonances.

• • If $\mathfrak {m}$ is smooth, we need to look at the unrestricted stationary points of the phase $s \Phi ^R_{\iota _1 \iota _2} = s(\langle \xi \rangle -\iota _1\langle \eta \rangle -\iota _2\langle \sigma \rangle )$ (where we added the superscript R to emphasise that we consider a regular $\mathfrak {m}$ ): that is,

  (2.12) $$ \begin{align} \Phi^R_{\iota_1 \iota_2}(\xi,\eta,\sigma) = \partial_{\eta} \Phi^R_{\iota_1 \iota_2}(\xi,\eta,\sigma) = \partial_{\sigma} \Phi_{\iota_1 \iota_2}^R(\xi,\eta,\sigma) = 0. \end{align} $$

This simple and natural distinction has important implications on the behaviour of equation (2.9), hence on the solution of the nonlinear equation, which we now discuss.

2.3 Regular quadratic terms and bad frequencies

Let us first look at the case when $\mathfrak {m}$ is smooth. The regular quadratic phase $\Phi ^R_{\iota _1 \iota _2}(\xi ,\eta ,\sigma ) = \langle \xi \rangle - \iota _1 \langle \eta \rangle - \iota _2 \langle \sigma \rangle $ leads to rather harmless interactions if $(\iota _1\iota _2) \neq (++)$ since in this case, there are no solutions to equation (2.12). For the $(\iota _1\iota _2)=(++)$ interaction, we have that

(2.13) $$ \begin{align} \Phi^R_{++}= \partial_{\eta} \Phi^R_{++} = \partial_{\sigma} \Phi^R_{++} = 0 \quad \Longleftrightarrow \quad (\xi,\eta,{\sigma})=(\pm\sqrt{3},0,0). \end{align} $$

This is a full resonance or coherent interaction and it is the source of many of the difficulties. Notice that this sort of interaction is generic in dimension $1$ in the presence of a potential, since in equation (2.12) there are $3$ variables and as many equations to solve. Obviously, a similar phenomenon would occur already in the case $V=0$ and a nonlinear term of the form $a(x)u^2$ .

Recall that the classical theory of quadratic/cubic one-dimensional dispersive problems revolves around trying to control weighted-type norms of the form $\|x g\|_{L^2}$ . The natural candidate in our context is then $\| \partial _{\xi } \widetilde {g}\|_{L^2_{\xi }}$ . In some cases, such as equation (KG), or the more standard examples of flat cubic NLS and cubic KG equations, one knows that a uniform-in-time bound cannot be achieved due to long-range effects already present in the corresponding flat problem. As the next best thing, one can try to establish

(2.14) $$ \begin{align} \| \partial_{\xi} \widetilde{g}\|_{L^2_{\xi}} \lesssim \langle t \rangle^{\alpha} \end{align} $$

for some small $\alpha>0$ .

Let us now explain how equation (2.14) is incompatible with the nonlinear resonance equation (2.13). Since our assumptions will always guarantee $\widetilde {g}(0) = 0$ , equation (2.14) implies

(2.15) $$ \begin{align} |\widetilde{g}(\xi)| \lesssim \langle t \rangle^{\alpha} |\xi|^{1/2}. \end{align} $$

Consider then the main $(++)$ contribution to the right-hand side of equation (2.9), namely

(2.16) $$ \begin{align} \mathcal{Q}^R_{++}(t,\xi) := \int_0^t \iint e^{is \Phi_{++}(\xi,\eta,\sigma)} \widetilde{g}(s,\eta) \widetilde{g}(s,\sigma) \mathfrak{q}(\xi,\eta,\sigma) \,d\eta\, d\sigma\,ds, \end{align} $$

where $\mathfrak {q}$ is a smooth symbol. Up to lower-order terms,

(2.17) $$ \begin{align} \partial_{\xi} \mathcal{Q}^R_{++}(t,\xi) \approx \int_0^t \iint s \frac{\xi}{\langle \xi \rangle} e^{is \Phi_{++}(\xi,\eta,\sigma)} \mathfrak{q}(\xi,\eta,\sigma) \widetilde{g}(s,\eta) \widetilde{g}(s,\sigma) \, d\eta\,d\sigma. \end{align} $$

Observe that $|s \Phi _{++} | \ll 1$ if $|\xi - \sqrt {3}| + |\eta |^2 + |\sigma |^2 \ll \langle s \rangle ^{-1}$ and that in this region there are no oscillations that can help. Thus, when $\mathfrak {q}(\pm \sqrt {3},0,0) \neq 0$ , we are led to the following heuristic lower bound: for $||\xi | - \sqrt {3}| \approx r$ ,

(2.18) $$ \begin{align} \big| \partial_{\xi} \mathcal{Q}^R_{++}(t,\xi) \big| \gtrsim \int_1^{\mathrm{min}(\frac{1}{r},t)} s \cdot \langle s \rangle^{2\alpha} \int_{|\eta|^2 + |\sigma|^2 \leq s^{-1}} |\eta|^{1/2} |\sigma|^{1/2} \,d\eta\,d\sigma \,ds \approx \mathrm{min}\Big(\frac{1}{r},t\Big)^{\frac{1}{2} + 2\alpha}. \end{align} $$

This implies that, if $\langle t \rangle ^{-1} \leq r \ll \langle t \rangle ^{-1/2}$ ,

(2.19) $$ \begin{align} {\big\| \partial_{\xi} \mathcal{Q}^R_{++} \big\|}_{L^2(|\xi-\sqrt{3}| \approx r)} \gtrsim r^{-2\alpha} \gg \langle t \rangle^{\alpha}, \end{align} $$

which is inconsistent with the bootstrap hypothesis in equation (2.14). We then need to modify the bootstrap norm to a version of $\| \partial _{\xi } \widetilde {f} \|_{L^2}$ that is localised dyadically around $\pm \sqrt {3}$ and degenerates as $|\xi | \rightarrow \sqrt {3}$ . The analysis needed to propagate such a degenerate norm turns out to be quite delicate. A phenomenon similar to the one described above was previously observed in [Reference Deng, Ionescu and Pausader14, Reference Deng, Ionescu, Pausader and Pusateri15] in the two-dimensional (unperturbed) setting.

Note that in the heuristics in equation (2.19), one would get better bounds, consistent with equation (2.15), when $\mathfrak {q}(\pm \sqrt {3},0,0) = 0$ . For the model in equation (KG2), one has $\mathfrak {q}(\pm \sqrt {3},0,0) \neq 0$ when V is nongeneric (case (E)) and $\widetilde {a}(\pm \sqrt {3}) \neq 0$ . A true degeneracy in frequency space will then occur for these models. For equation (KG), under the assumptions (A) or (B) or (C), it is instead possible to show that $\mathfrak {q}(\xi ,0,0) = 0$ ; this is connected to the discussion at the end of Remark (3) and the possibility of simplifying the functional framework in this case.

Remark 2.1. The argument above also shows that, if $\widetilde {g}(0) \neq 0$ , then

$$ \begin{align*} \mathcal{Q}^R_{++}(t,\xi) \approx \int_0^t e^{is (\langle \xi \rangle-2)} \mathfrak{q}(\xi,0,0) \big( \widetilde{g}(s,0) \big)^2 \frac{ds}{s+1} + \cdots \end{align*} $$

so that $\mathcal {Q}^R_{++}(t,\pm \sqrt {3})$ is logarithmically diverging if $\mathfrak {q}(\pm \sqrt {3},0,0) \neq 0$ . This suggests that $\widetilde {g}$ is not uniformly bounded, which in turn implies that the solution cannot decay pointwise at the linear rate; see equation (1.12). In the case of equation (1.14) with localised $a(x)$ such that $\widehat {a}(\pm \sqrt {3}) \neq 0$ (and no cubic terms), this has been rigorously proved in [Reference Lindblad, Lührmann and Soffer55], where the authors construct global solutions that decay in $L^{\infty }_x$ at the optimal rate of $\log t /t$ . This result was then extended in [Reference Lindblad, Lührmann, Schlag and Soffer56] to the case of any nongeneric potential with the corresponding condition $\widetilde {a}(\pm \sqrt {3}) = 0 $ .

Also note that $\widetilde {g}(0)\neq 0$ will give an asymptotic of the form $\partial _{\xi } \mathcal {Q}^R_{++}(t,\xi ) \approx |\langle \xi \rangle -2|^{-1}$ . When localised at the scale $||\xi |-\sqrt {3}| \approx 2^{\ell }$ , this gives an $L^2$ norm of size $2^{-\ell /2}$ . The functional framework that we will adopt does not quite allow for such a singularity, as this would correspond to choosing the parameter $\beta =1/2$ in the definition of the norm in equation (2.30) (this is the norm in which we will measure the derivative of our [renormalised] profile in frequency space). However, we can allow essentially any slightly less singular behaviour; this seems to suggest that a zero-energy resonance may be treated by our methods at least for long times.

2.4 Singular quadratic and cubic terms

Let us now consider the quadratic interactions in equation (2.7) that correspond to the first two terms in equation (2.8). Disregarding the irrelevant signs $\mu ,\nu $ and the coefficients $A,B$ , let us denote them by

(2.20) $$ \begin{align} \mathcal{Q}^M_{\iota_1\iota_2}(t,\xi) := \int_0^t \iint e^{is \Phi^S_{\iota_1\iota_2}(\xi,\eta,\sigma)} \widetilde{g}(s,\eta) \widetilde{g}(s,\sigma) M(\xi-\eta-\sigma) \,d\eta\, d\sigma\,ds, \quad M \in \{ \delta,\mathrm{p.v.}\}. \end{align} $$

The $\delta $ case

The case of the $\delta $ distribution corresponds to the Euclidean ( $V=0$ ) quadratic Klein-Gordon, which is not resonant (in any dimension), in the sense that for any $\xi ,\eta \in {\mathbb R}$ and $\iota _1,\iota _2\in \{+,-\}$ , equation (2.10) never vanishes, and more precisely

(2.21) $$ \begin{align} \big| \langle \xi \rangle -\iota_1 \langle \eta \rangle -\iota_2 \langle \xi -\eta \rangle \big| \gtrsim \mathrm{min}(\langle \xi \rangle, \langle \eta \rangle,\langle\xi-\eta\rangle)^{-1}.\end{align} $$

This implies that the quadratic interactions $\mathcal {Q}^{\delta }_{\iota _1\iota _2}(t,\xi )$ can be eliminated by a normal form transformation. This was first shown in the seminal work of Shatah [Reference Shatah66] in $3$ d and crucially used in the $1$ d case in [Reference Delort11] and [Reference Hayashi and Naumkin30].

Applying a normal form transformation to equation (2.20) gives quadratic boundary terms that we disregard for simplicity and cubic terms when $\partial _t$ hits the profile $\widetilde {g}$ . From equations (2.7)–(2.8), we see that these cubic terms can be of several types depending on the various combinations of convolutions between $\delta ,\mathrm {p.v.}$ and smooth functions. Without going into the details of these (we refer the reader to Section 5), we concentrate on the simplest interaction: that is, the ‘flat’ one

(2.22) $$ \begin{align} \begin{aligned} \mathcal{C}^{S}_{\iota_1 \iota_2 \iota_3} (t,\xi) & = \iint e^{it \Phi^S_{\iota_1 \iota_2 \iota_3}(\xi,\eta,\zeta)} \mathfrak{c}^S_{\iota_1 \iota_2 \iota_3}(\xi,\eta,\zeta) \, \widetilde{g}_{\iota_1}(t,\xi-\eta) \widetilde{g}_{\iota_2}(t,\xi-\eta-\zeta) \widetilde{g}_{\iota_3}(t,\xi-\zeta) \, d\eta \, d\zeta \end{aligned} \end{align} $$

with a smooth symbol $\mathfrak {c}^S_{\iota _1 \iota _2 \iota _3}$ and phase functions

$$ \begin{align*}\Phi^S_{\iota_1 \iota_2 \iota_3} (\xi,\eta,\zeta) = \langle \xi \rangle - \iota_1 \langle \xi - \eta \rangle - \iota_2 \langle \xi - \eta -\zeta \rangle -\iota_3 \langle \xi - \zeta \rangle. \end{align*} $$

We observe that if $\{\iota _1,\iota _2,\iota _3\} \neq \{+,+,-\}$ , the equations $\partial _{\eta } \Phi ^S_{\iota _1 \iota _2 \iota _3} = \partial _{\zeta } \Phi ^S_{\iota _1 \iota _2 \iota _3} = \Phi ^S_{\iota _1 \iota _2 \iota _3} = 0$ have no solutions, and therefore the case $\{\iota _1,\iota _2,\iota _3\} = \{+,+,-\}$ is the main one. If we look at the $(+-+)$ phase for simplicity, we see that, for every fixed $\xi $ ,

$$ \begin{align*}\Phi^S_{+-+} = \partial_{\eta} \Phi^S_{+-+} = \partial_{\zeta} \Phi^S_{+-+} = 0 \quad \Longleftrightarrow \quad \eta = \zeta = 0. \end{align*} $$

This resonance is responsible for the logarithmic phase correction appearing in equation (1.13). We refer the reader to [Reference Kato and Pusateri39, Reference Ionescu and Pusateri32, Reference Germain, Pusateri and Rousset24] where a similar phenomenon has been dealt with. We should point out, however, that in our case, the asymptotic behaviour in equation (1.13) is slightly harder to capture because of the degenerate weighted norm and the algebraic complications due to the treatment of potentials with general transmission and reflection coefficients.

2.5 The functional framework

To measure the evolution of our solutions, we need to take into account various aspects including pointwise decay, spatial localisation (which we measure through regularity on the distorted Fourier side), the coherent space-time resonance phenomenon in equation (2.13) (which dictates the choice of our $L^2$ -based norm) and long-range asymptotics. We describe our functional setting below after introducing the necessary notation.

2.5.1 Notation

To introduce our functional framework, we first define the Littlewood-Paley frequency decomposition.

Frequency decomposition. We fix a smooth even cutoff function $\varphi : {\mathbb R} \to [0,1]$ supported in $[-8/5,8/5]$ and equal to $1$ on $[-5/4,5/4]$ . Note that the choice of the number $8/5$ for the support of $\varphi $ is fairly arbitrary, and other choices are possible; however, this number is chosen to be less than $\sqrt {3}$ so that when we define the cutoffs $\chi _{\ell }$ centred around $\pm \sqrt {3}$ in equation (2.27), we can start the indexing at $0$ .

For $k \in {\mathbb Z}$ , we define $\varphi _k(x) := \varphi (2^{-k}x) - \varphi (2^{-k+1}x)$ , so that the family $(\varphi _k)_{k \in {\mathbb Z}}$ forms a partition of unity,

$$ \begin{align*} \sum_{k\in{\mathbb Z}}\varphi_k(\xi)=1, \quad \xi \neq 0. \end{align*} $$

We let

(2.23) $$ \begin{align} \varphi_{I}(x) := \sum_{k \in I \cap {\mathbb Z}}\varphi_k, \quad \text{for any} \quad I \subset {\mathbb R}, \quad \varphi_{\leq a}(x) := \varphi_{(-\infty,a]}(x), \quad \varphi_{> a}(x) = \varphi_{(a,\infty)}(x), \end{align} $$

with similar definitions for $\varphi _{< a},\varphi _{\geq a}$ . We will also denote $\varphi _{\sim k}$ a generic smooth cutoff function that is supported around $|\xi | \approx 2^k$ , for example $\varphi _{[k-2,k+2]}$ or $\varphi ^{\prime }_k$ .

We denote by $P_k$ , $k\in {\mathbb Z}$ , the Littlewood-Paley projections adapted to the regular Fourier transform:

$$ \begin{align*} \widehat{P_k f}(\xi) = \varphi_k(\xi) \widehat{f}(\xi), \quad \widehat{P_{\leq k} f}(\xi) = \varphi_{\leq k}(\xi) \widehat{f}(\xi), \quad \textrm{ and so on.} \end{align*} $$

We will avoid using, as a recurrent notation, the distorted analogue of these projections.

We also define the cutoff functions

(2.24) $$ \begin{align} \varphi_k^{(k_0)}(\xi) = \left\{ \begin{array}{ll} \varphi_k(\xi) \quad & \mbox{if} \quad k> \lfloor k_0 \rfloor, \\ \varphi_{\leq \lfloor k_0 \rfloor }(\xi) \quad & \mbox{if} \quad k= \lfloor k_0 \rfloor, \end{array} \right. \end{align} $$

and

(2.25) $$ \begin{align} \varphi_k^{[k_0,k_1]}(\xi) = \left\{ \begin{array}{ll} \varphi_k(\xi) \quad & \mbox{if} \quad k \in (\lfloor k_0 \rfloor, \lfloor k_1 \rfloor) \cap {\mathbb Z}, \\ \varphi_{\leq \lfloor k_0 \rfloor}(\xi) \quad & \mbox{if} \quad k= \lfloor k_0 \rfloor, \\ \varphi_{\geq \lfloor k_1 \rfloor}(\xi) \quad & \mbox{if} \quad k= \lfloor k_1 \rfloor. \end{array} \right. \end{align} $$

We are adopting the standard notation $\lfloor x \rfloor $ to denote the largest integer smaller than x. Note that the indexes $k_0$ and $k_1$ in equations (2.24)–(2.25) do not need to be integers. We also adopt the convention that if $k_0=k_1$ , then $\varphi _k^{[k_0,k_1]}=1$ .

We will denote by T a positive time, and always work on an interval $[0,T]$ for our bootstrap estimates; see, for example, Proposition 7.1. To decompose the time integrals such as equation (2.7) for any $t \in [0,T]$ (this is first done in equation (8.12) and then systematically throughout Sections 8–11), we will use a suitable decomposition of the indicator function $\mathbf {1}_{[0,t]}$ by fixing functions $\tau _0,\tau _1,\cdots , \tau _{L+1}: {\mathbb R} \to [0,1]$ , for an integer L with $|L-\log _2 (t+2)| < 2$ , with the properties that

(2.26) $$ \begin{align} \begin{aligned} & \sum_{n=0}^{L+1}\tau_n(s) = \mathbf{1}_{[0,t]}(s), \qquad \,\mbox{supp}\, (\tau_0) \subset [0,2], \quad \,\mbox{supp}\, (\tau_{L+1}) \subset \big[{\tfrac{1}{4}}t,t\big], \\ & \mbox{and} \quad \,\mbox{supp}\,(\tau_n) \subseteq [2^{n-1},2^{n+1}], \quad |\tau_n'({s})|\lesssim 2^{-n}, \quad \mbox{for} \quad n= 1,\dots, L. \end{aligned} \end{align} $$

In all our arguments, we also will often restrict to $n\geq 1$ , as the contribution for $n=0$ is always trivial to handle.

In light of the coherent phenomenon explained in Section 2.3, we also need cutoff functions

(2.27) $$ \begin{align} \chi_{\ell,\sqrt 3}(z) = \varphi_{\ell}(|z|-\sqrt{3}), \quad \ell \in {\mathbb Z} \cap (-\infty,0], \end{align} $$

which localise around $\pm \sqrt {3}$ at a scale $\approx 2^{\ell }$ . In analogy with equations (2.23) and (2.24), we also define

(2.28) $$ \begin{align} \chi_{\ast,\sqrt 3}(z) = \varphi_\ast(|z|-\sqrt{3}), \qquad \chi_{\ell,\sqrt 3}^{\ast}(z) = \varphi_{\ell}^{\ast}(|z|-\sqrt{3}).\end{align} $$

More notation.

For any $k \in {\mathbb Z}$ , let $k^+ := \mathrm {max}(k,0)$ and $k^- := \mathrm {min}(k,0)$ .

We denote as $\mathbf {1}_A$ the characteristic function of a set $A\subset {\mathbb R}$ and let $\mathbf {1}_\pm $ be the characteristic function of $\{\pm x> 0\}$ .

We use $a\lesssim b$ when $a \leq Cb$ for some absolute constant $C>0$ independent on a and b. $a \approx b$ means that $a\lesssim b$ and $b\lesssim a$ . When a and b are expressions depending on variables or parameters, the inequalities are assumed to hold uniformly over these.

Given $c\in {\mathbb R}$ , we will use the notation $c+$ to denote a number d larger than c but that can be chosen arbitrarily close to it. Similarly, we will use $c-$ for a number smaller than c that can be chosen arbitrarily close to it; see, for example, equation (6.9). We will sometimes use this convention also with $c=\infty $ to denote an arbitrarily large number (see, for example, equation (6.27)).

We will denote by $\mathrm {min}(x_1,x_2,\dots )$ , respectively $\mathrm {max}(x_1,x_2,\dots )$ , the minimum, respectively maximum, over the set $\{x_1,x_2,\dots \}$ . We will also denote by $\mathrm {min}_2(x_1,x_2,\dots )$ , respectively $\mathrm {max}_2(x_1,x_2,\dots )$ , the second smallest, respectively second largest, element in the set $\{x_1,x_2,\dots \}$ . We are also using $\mathrm {med}(x_1,x_2,x_3)$ for $\mathrm {max}_2(x_1,x_2,x_3)$ ; see, for example, equation (5.60) or equation (8.76).

We denote by

(2.29) $$ \begin{align} \widehat{f}=\widehat{{\mathcal{F}}}(f):= \frac{1}{\sqrt{2\pi}} \int_{\mathbb R} e^{-ix\xi} f(x) \, dx \end{align} $$

the standard Fourier transform of f.

We use the standard notation for Lebesgue $L^p$ spaces and for Sobolev spaces $W^{k,p}$ and $H^k=W^{k,2}$ .

2.5.2 Norms

For $T>0$ , we let $W_T$ be the space given by the norm

(2.30) $$ \begin{align} {\| h \|}_{W_T} := \sup_{n \geq 0} \sup_{\ell \in \mathbb{Z} \cap [\lfloor -\gamma n\rfloor,0]} {\big\| \chi_{\ell,\sqrt{3}}^{[-\gamma n,0]}(\,\cdot\,) \, \tau_n(t) \, h(t,\cdot) \mathbf{1}_{0 \leq t \leq T} \big\|}_{L^{\infty}_t L^2_{\xi}} 2^{\beta \ell} 2^{-\alpha n}, \end{align} $$

whereFootnote ⁹ $\tau _n$ here denotes a partition of unity as in equation (2.26) with T in place of t, and where the parameters $0 < \alpha , \beta , \gamma < \frac {1}{2}$ satisfy

(2.31) $$ \begin{align} \gamma \beta' < \alpha < \frac{\beta'}{2}, \quad \beta' \ll 1, \qquad \beta' := \frac{1}{2} - \beta, \quad \gamma' := \frac 1 2 - \gamma. \end{align} $$

$\beta '$ is a fixed constant that needs to be chosen small enough to satisfy various inequalities that we will impose in the course of the proof. Note that we automatically have $\gamma <1/2$ and that one possible way to impose all of the conditions in equation (2.31) is to choose $\alpha $ sufficiently small and

$$ \begin{align*}\beta' = 2\alpha + 2\alpha^2, \quad \gamma' = 2\alpha + \alpha^2.\end{align*} $$

Let us briefly explain the choice of the norm and parameters:

• • The norm in equation (2.30) will be used to measure our solution on the Fourier side. More precisely, we will show that ${\|\langle \xi \rangle \partial _{\xi } \widetilde {f}\|}_{W_T} \lesssim {\varepsilon }_0$ , where $\widetilde {f}$ is a renormalised version of the profile $\widetilde {g}$ in equation (1.6). As already pointed out, measuring $\partial _{\xi }$ on the Fourier side is akin to measuring a weighted norm in real space.

• • The quantity $2^{\ell }$ measures the distance from $\pm \sqrt {3}$ starting at smallest scale $2^{-{\gamma } n}$ , where $2^n \approx |t|$ and the norm is penalised by the factor $2^{\beta \ell }$ . The additional penalization of $2^{-\alpha n}$ is added globally to take into account long-range effects that are present at every frequency.

• • To make sure that localisation and derivation in the $W_T$ norm commute (under the hypothesis that $\widetilde {f}$ is uniformly bounded), one needs $\beta ' \gamma \leq \alpha $ .

• • In order to deduce from a bound on the $W_T$ norm (together with a bound on the $\widetilde {{\mathcal {F}}}^{-1}\langle \xi \rangle ^{-3/2}L^{\infty }$ ) the necessary linear decay estimate at the optimal rate of $\langle t \rangle ^{-1/2}$ , we need $\alpha + \beta \gamma < 1/4$ ; see Proposition 3.11. Since

  $$ \begin{align*}\alpha + \beta \gamma = \alpha + 1/4 - \beta{\gamma}' - \beta'/2,\end{align*} $$
  it suffices to impose $\beta ' \geq 2\alpha $ .

2.6 The main bootstrap and proof of Theorem 1.1

For $T>0$ , consider a local solution $u\in C([0,T],H^5({\mathbb R})) \cap C^1([0,T],H^4({\mathbb R}))$ of equation (KG) constructed by standard methods. Our proof is based on showing an a priori estimate for the following norm:

(2.32) $$ \begin{align} {\| u \|}_{X_T} = \sup_{t\in[0,T]} \Big[ \langle t \rangle^{-p_0} {\big\| (\sqrt{H+1}\,u,u_t)(t) \big\|}_{H^4} + \langle t \rangle^{1/2} {\| (\partial_t,\partial_x)u(t) \|}_{L^{\infty}} \Big], \qquad 0<p_0<\alpha. \end{align} $$

Under the initial smallness condition in equation (1.3), we will assume the a priori bound

(2.33) $$ \begin{align} {\|u \|}_{X_T} \leq {\varepsilon}_1, \end{align} $$

and show that this implies

(2.34) $$ \begin{align} {\| u \|}_{X_T} \leq C{\varepsilon}_{0} + C{\varepsilon}_{1}^2, \end{align} $$

for some absolute constant $C>0$ . Picking ${\varepsilon }_0$ sufficiently small and using a standard bootstrap argument with ${\varepsilon }_1 = 2 C {\varepsilon }_0$ , equation (2.34) gives global existence of solutions that are small in the space $X_\infty $ . Also, using time reversibility, we obtain solutions for all times.

The structure of the paper and the proof of Theorem 1.1, with details on how the main bootstrap equation (2.34) will be proved, are described below.

2.7 Structure of the paper and the proof of Theorem 1.1

In this subsection, we discuss the organization of the paper, describe the overall structure of the proof, and give more details about the various estimates needed to show equation (2.34), under the a priori assumption in equation (2.33).

• • Section 3 contains an exposition of the elements of the scattering theory for Schrödinger operators $H:=-\partial _x^2 + V$ on ${\mathbb R}$ , which we will need.

  After introducing the Jost functions $f_\pm $ (see equation (3.1)) and the transmission and reflection coefficients T and $R_\pm $ (see equations (3.7) and (3.13)), we define the distorted Fourier transform (dFT) as in equation (2.2) (see equation (3.21)), with the ‘distorted’ (or generalised) exponentials (or eigenfunctions) $\psi (x,\xi )$ given by equation (3.19).

  Some basic properties of the dFT are discussed in Section 3.2.1. Then the $\psi (x,\xi )$ are analysed in detail in Section 3.3 and decomposed into a singular and a regular part. The singular part behaves at spatial infinity like linear combinations of (standard) complex exponentials, while the regular part is fast decaying. This decomposition is also at the heart of the decomposition of the nonlinear spectral distribution $\mu $ defined in equation (2.6).

  In Section 3.4, we prove the first nontrivial result involving the dFT: that is, the estimate for the linear flow $e^{it\sqrt {H+1}} = e^{it\langle \widetilde {D}\rangle }$ (see the notation for Fourier multipliers in Section 3.2.2) given in equation (3.32), which involves the degenerate norm $W_T$ . This estimate shows that sharp $L^{\infty }_x$ decay (i.e., at the rate of $t^{-1/2}$ ) for the evolution $e^{it\sqrt {H+1}} h(t)$ of a (time-dependent) profile h on an interval $[0,T]$ , follows from controlling the norms

  (2.35) $$ \begin{align} \sup_{t \in [0,T]} {\big\| \langle \xi \rangle^{3/2} \widetilde{h}(t) \big\|}_{L^{\infty}_{\xi}} \leq C, \qquad {\| \langle \xi \rangle \partial_{\xi} \widetilde{h} \|}_{W_T} \leq C, \qquad \sup_{t \in[0,T]} \big( \langle t \rangle^{-p_0} {\| \langle \xi \rangle^4 \widetilde{h} \|}_{L^2} \big) \leq C, \end{align} $$
  where $W_T$ is the norm defined in equation (2.30), with the restriction on the parameters in equation (2.31), $p_0$ sufficiently small, and C an absolute constant independent of T. As it turns out, we cannot control the norms in equation (2.35) for the profile associated to the solution u and infer the bound for the $L^{\infty }_x$ norm in equation (2.32) from this. Instead, we need to take a longer route and estimate norms as in equation (2.35) for a renormalised profile, which is defined in Section 5; see equation (5.53).

• • Before moving on to the analysis of the nonlinear time evolution, we study more precisely the nonlinear spectral measure in Section 4.

  The main Proposition 4.1 describes the precise structure of the NSD $\mu $ (see equation (2.6)); for lighter notation, we omit the indexes $\iota _1,\iota _2$ here. The main goal is to decompose $\mu $ into a ‘singular’ and a ‘regular’ part.

  The singular part, denoted $\mu ^S$ , is a linear combination of $\delta $ and $\mathrm {p.v.}$ distributions, as anticipated in equation (2.8); the precise definition is given by equations (4.2)–(4.3), with formulas for the coefficients given in equations (4.4)–(4.5). Notice that these coefficients may not be smooth at $\xi =0$ (e.g., in the generic case). As is apparent, handling formulas involving these coefficients requires quite a lot of somewhat tedious bookkeeping; however, this is necessary for two main reasons: first, we need the exact expressions to calculate the final asymptotics for the solution of equation (KG); and second, we will need to check some smoothness properties for the multipliers of the trilinear terms that will appear after a normal form transformation and involve these coefficients.

  The regular part of the NSD, denoted $\mu ^R$ , is defined in equation (4.6) with equation (4.7), and it is essentially a smooth function of the three frequencies $(\xi ,\eta ,{\sigma })$ up to possible jump singularities on the axes. The mapping properties of the associated bilinear operator are established in Section 4.2, with equation (4.29) showing that it essentially behaves like multiplication by a localised function.

• • In Section 5, we begin the analysis of the time evolution by defining the profile associated to u as

  (2.36) $$ \begin{align} g := e^{it\sqrt{H+1}} (\partial_t - i\sqrt{H+1})u; \end{align} $$
  see equations (5.5) and (5.2). From the main equation (KG), we write the nonlinear evolution for $\widetilde {g}$ as in equations (5.7)–(5.8) (which is the same as the formula in equation (2.4)).

  Using the decomposition of $\mu = \mu ^S + \mu ^R$ , we would like to decompose accordingly the quadratic terms in the formula for $\partial _t\widetilde {g}$ into singular terms and regular terms. However, as briefly mentioned in equation (2.4), because of the presence of the $\mathrm {p.v.}$ term coming from $\mu ^S$ , we cannot do this decomposition directly. We instead need a further distinction within the terms containing the $\mathrm {p.v.}$ into ‘truly’ singular terms, where the $\mathrm {p.v.}$ is restricted close to its singularity, and more regular ones that are supported away from the singularity. This is the role of the cutoff $\varphi ^{\ast }$ defined in equation (5.12) and appearing in equation (5.11). The singular terms are then defined according to equations (5.10)–(5.11). The precise choice of $\varphi ^{\ast }$ is made so that, on its support, we can derive lower bounds for the oscillating phases $\Phi $ in equation (5.8).

  The main motivation for the splitting

  $$ \begin{align*}\partial_t\widetilde{g} = \mathcal{Q}^S + \mathcal{Q}^R,\end{align*} $$
  as done in Section 5.2, is that the singular quadratic terms resemble the quadratic terms that one would get for a flat ( $V=0$ ) quadratic KG equation. In particular, the oscillating phases are lower bounded on the support of $\mathcal {Q}^S$ , as established in Lemma 5.2; then we can apply a normal form transformation to recast these terms into cubic ones.

  The algebra for the normal form step is carried out in Section 5.4. Starting from the simple identity in equation (5.27), we naturally define the bilinear normal form transformation Footnote ¹⁰ T in equation (5.29), which arises from the boundary terms in the time-integration by parts. More precisely (but still omitting the various sums over the signs such as $\iota _1$ and $\iota _2$ ) we have

  (2.37) $$ \begin{align} \int_0^t\mathcal{Q}^S \,ds = \widetilde{{\mathcal{F}}}{T}(g,g)(t) - \widetilde{{\mathcal{F}}}{T}(g,g)(0) + \int_0^t B_1(s) + B_2(s) \, ds \end{align} $$
  where the bulk terms $B_1$ and $B_2$ are the expressions defined in equations (5.34) and (5.35) and are cubic in g. The only quadratic terms left are then the $\mathcal {Q}^R$ terms that include the contribution from $\mu ^R$ , and the $\mathrm {p.v.}$ part restricted outside the support of $\varphi ^{\ast }$ .

  In Section 5.5, respectively, Section 5.6, we analyse the leading order symbol $\mathfrak {b}^1$ , respectively, the lower-order symbol $\mathfrak {b}^2$ , of the bulk term $B_1$ , respectively, $B_2$ above. These are somewhat complicated expressions since they involve the symbol equation (5.11) and its variant without the cutoff $\varphi ^{\ast }$ and therefore combinations of the (nonsmooth) coefficients in equation (4.4). The leading order symbol is made by the convolution of $\delta $ and $\mathrm {p.v.}$ -type distributions; see equations (5.36)–(5.38). For the later nonlinear analysis, we need to make sure that this symbol is nice enough so that the associated trilinear operators satisfy Hölder type bounds. An important technical point then is the verification of the smoothness with respect to variable in which the convolution is performed; this is done in Section 5.5.1. In Section 5.5.2, we then calculate precisely the top order (singular $\delta $ and $\mathrm {p.v.}$ -type) contribution from $\mathfrak {b}^1$ : that is, the symbol $\mathfrak {c}^S$ in equation (5.46); the associated trilinear operator will be denoted by $\mathcal {C}^S$ . Other contributions from $\mathfrak {b}^1$ and the symbol $\mathfrak {b}^2$ are analysed in Section 5.6; the associated trilinear operator will be denoted by $\mathcal {C}^R$ . The mapping properties of these trilinear operators are analysed in Section 6.

  In the last Section 5.7, we finally arrive at the definition in equation (5.53) of the renormalised profile

  (2.38) $$ \begin{align} f = g - T(g,g). \end{align} $$
  We see that f satisfies an equation where the only quadratic terms are regular ones, and the cubic terms are those analysed in the previous subsections. Equation (5.55) for $\widetilde {f}$ is the starting point for the nonlinear analysis, and we record it here for ease of reference in a slightly simplified form (omitting the easier regular cubic terms; see equations (5.59)–(5.60))
  (2.39) $$ \begin{align} \partial_t \widetilde{f} = \mathcal{Q}^R(g,g) + \mathcal{C}^{S}(g,g,g); \end{align} $$
  see the definitions in equations (5.56)–(5.57).

  The heart of the proof of the bootstrap equation (2.34) is another bootstrap argument for the renormalised profile f involving the norms in equation (2.35) and is based on (a renormalisation of) equation (2.39). See the description of the contents of Section 7 below.

• • Section 6 contains bilinear and trilinear estimates for the various operators appearing in our problem. Here we need to analyse different types of pseudo-product operators, from the standard bilinear ones (equation (6.1)) to trilinear ones involving a $\mathrm {p.v.}$ (equation (6.5)). Bounds for general bilinear and trilinear operators of the types that appear in our proof are established in Lemmas 6.5 and 6.7, and basic criteria to check the assumptions in these lemmas are also given.

  In Section 6.3, we analyse in detail the normal form operator T and establish, in Lemma 6.10, that it satisfies Hölder-type bounds with a gain of regularity on the inputs.

  The other main results in this Section are Lemma 6.11, which gives improved Hölder-type inequalities for the smooth bilinear operator $\mathcal {Q}^R$ , and Lemma 6.13, which gives sharp Hölder-type bounds with some gain of regularity for the singular cubic terms $\mathcal {C}^S$ .

• • In Section 7, we set up the proof of the main bootstrap bound in equation (2.34). As mentioned above, these estimates will mostly involve the renormalised profile f, but we first need to relate the desired bounds for g (and u, as stated in equations (1.4)–(1.5)) to the necessary bounds on f. Here is how we proceed.

  With ${\varepsilon }_0$ as in equation (1.3), we let ${\varepsilon }_2 \gg {\varepsilon }_1 \gg {\varepsilon }_0$ . Proposition 7.1 gives an a priori bootstrap on g for the norms

  (2.40) $$ \begin{align} \sup_{t\in[0,T]} \Big[ \langle t \rangle^{-p_0} {\big\| \langle \xi \rangle^{4} \widetilde{g}(t) \big\|}_{L^{2}} + \langle t \rangle^{1/2} {\| e^{-it\langle \partial_x \rangle} \mathbf{1}_{\pm}(D)\mathcal{W}^* g(t) \|}_{L^{\infty}} \Big], \end{align} $$
  where $\mathcal {W}^{\ast }$ is the adjoint of the wave operator defined in equation (3.24); we assume that equation (2.40) is bounded by $2{\varepsilon }_2$ and claim that it can bounded by ${\varepsilon }_2$ . Proposition 7.2 instead gives the main bootstrap for the following norms of f
  (2.41) $$ \begin{align} \sup_{t\in[0,T]} \Big[ \langle t \rangle^{-p_0} {\big\| \langle \xi \rangle^{4} \widetilde{f}(t) \big\|}_{L^{2}} + {\| \langle \xi \rangle \partial_{\xi} \widetilde{f} \|}_{W_t} + {\| \langle \xi \rangle^{3/2} \widetilde{f}(t) \|}_{L^{\infty}} \Big]; \end{align} $$
  assuming that equation (2.41) is bounded by $2{\varepsilon }_1$ , we claim that it can be bounded by ${\varepsilon }_1$ . Note that we are assuming much stronger information on f than on g.

  Section 7.1 is dedicated to showing how the a priori bound on equation (2.41) by $2{\varepsilon }_1$ can be used to close the claimed bootstrap for the norms in equation (2.40). This is not too hard to do using the relation $g = f + T(g,g)$ (see equation (2.38)), the bilinear bounds on the T operator established in Section 6.3 and the linear estimate in equation (3.32).

  Note that once we have proven a bound for equation (2.40) by ${\varepsilon }_2 = C{\varepsilon }_0$ , we can immediately deduce the Sobolev bound in equation (1.5) from equation (2.36) and the boundedness of wave operators (Theorem 3.10). The decay bound in equation (1.4) does not follow directly from the $L^{\infty }$ bound in equation (2.40) (because wave operators may be unbounded on $L^{\infty }$ ), and it is proved separately in equation (7.9). The weighted bound in equation (1.7) is proved in Lemma 7.6. Since equations (1.4)–(1.7) follow from the bound on equation (2.40), the proof of the main theorem has been reduced to proving the bootstrap Proposition 7.2. As part of the arguments needed to prove these bootstrap estimates on f, we will also establish its asymptotic behaviour; see equation (1.13) (and Section 10).

  The rest of Section 7 prepares for later analysis and the proof of Proposition 7.2. Section 7.2 contains some preliminary bounds on f that follow from the a priori bound on the norms in equation (2.41) (Lemma 7.5). Then, in Section 7.3, using equation (2.38), we rewrite the equation for f (see equation (2.39)) as

  (2.42) $$ \begin{align} \widetilde{f}(t) - \widetilde{f}(0) = \int_0^t \mathcal{Q}^R(f,f) \,ds + \int_0^t \mathcal{C}^{S}(f,f,f) \,ds + \cdots + \mathcal{R}(t), \end{align} $$
  where the ‘ $\dots $ ’ denotes other cubic and quartic terms in f and $\mathcal {R}$ s denote terms that have a higher degree of homogeneity in f and g and can be treated as remainders. We actually use expansions at different orders depending on which norm we are trying to estimate.

  To close the bootstrap for f, we then need to estimate the terms on the right-hand side of equation (2.42). Lemmas 7.8 and 7.9 give, among other things, suitable bounds on the remainders $\mathcal {R}$ , in all the norms in equation (2.41).

  In Section 7.4, for the convenience of the reader, we summarise the bounds obtained thus far and list all the bounds that are left to prove.

• • Sections 8 and 9 constitute the heart of the paper and the more technical part of the analysis. The goal of these two sections is to carry out the main parts of the estimates for the weighted $L^2$ norm in equation (2.30) of the regular quadratic terms, $\mathcal {Q}^R(f,f)$ , and of the singular cubic terms, $\mathcal {C}^S(f,f,f)$ .The desired weighted bound for $\mathcal {Q}^R(f,f)$ is equation (8.1) in Proposition 8.1. Section 8 is then entirely dedicated to proving this key bound when the interactions are restricted to the main resonant ones: that is, $(\eta ,\sigma ) = (0,0) \longrightarrow \xi = \pm \sqrt {3}$ (see the notation used in equations (8.8) and (8.2)).

  Sections 8.1–8.3 give some preliminary bounds and reductions. We first take care of frequencies $\xi $ that are very close to $\pm \sqrt {3}$ and reduce the desired bound to showing equation (8.35) with equation (8.36) for the localised operatorFootnote ¹¹ $I(t,\xi )$ defined in equation (8.32); these reductions are summarised in Lemma 8.4.

  Note that the estimate in equation (8.32) involves localisation in the size of the input variables $|\eta |$ and $|{\sigma }|$ , in the distance $||\xi |-\sqrt {3}|$ , in the size of the oscillating phase $|\Phi |$ , and in the integrated time s, at dyadic scales with respective parameters $k_1,k_2,\ell ,p$ and m. These localisations allow us to distinguish various cases and to exploit the oscillations efficiently in either frequency space or time depending on the relative size of the quantities involved. The estimates are split into four main regions, as described in equation (8.48), and treating each of these regions occupies one of the Sections 8.4–8.7.

  We remark that a useful quantity is the one defined in equation (8.17), which incorporates some improved decay properties of the solution (for small frequencies).

• • In Section 9, we estimate the weighted $L^2$ norm for the singular cubic terms $\mathcal {C}^S(f,f,f)$ of the form in equation (2.22) (see equation (5.57) with equation (5.46) for the precise definition), focusing on the case of the main resonant interactions $\pm (\sqrt {3},\sqrt {3},\sqrt {3}) \rightarrow \pm \sqrt {3}$ . In particular, we achieve the main step in the proof of Proposition 9.1, which deals with the interactions of the type $(+-+)$ , where the signs correspond to the signs of the oscillating factors in the cubic phases in equation (5.57).

  Section 9.2 treats the terms that involve a $\delta $ factor, while Section 9.3 treats those with a $\mathrm {p.v.}$ (recall the form of the cubic symbols in equation (5.46)). Once again, we need to distinguish various cases depending on the distance of the input and output variables from the bad frequency $\sqrt {3}$ , relative to time and the size of their differences (see, for example, the dyadic localisations in equation (9.16)).

• • With Sections 8 and 9, we have taken care of estimating the weighted norm for the leading-order terms on the right-hand side of equation (2.42) in the case of the main resonant interactions. All of the other nonresonant interactions are estimated in Section 11.

  Section 10 contains the main part of the proof for the control of the Fourier- $L^{\infty }$ norm in equation (2.41): that is, the proof of Proposition 10.1, which gives asymptotics for the singular cubic terms $\mathcal {C}^S$ (see equation (10.2), where the Hamiltonian function is given in equation (10.27)). From this, we can then derive an asymptotic ODE for $\widetilde {f}$ and thus the asymptotic behaviour of the solution as in equation (10.4) (see equation (1.13)).

  Section 10.1 provides first a formal computation for the asymptotics, based on the stationary phase lemma. Section 10.2 utilises these computations to give the exact structure of the long-range asymptotics and the form of the Hamiltonian H appearing in the statement of Proposition 10.1). Rigorous bounds are then proved in equation (10.3).

• • Section 11 contains the estimates needed to control all the contributions from the nonlinear terms on the right-hand side of equation (2.42) that have not been dealt with in Sections 7–10, since they are lower-order compared to the main ones. We refer the reader to the first paragraph of Section 11 for a list of the estimates that are carried out there and the details on how they complete the proofs of the main propositions stated in the previous sections.

  The estimates of Section 11 complete the bootstrap on the norm in equation (2.41).

• • Finally, Appendix A contains a verification of the spectral assumptions needed to apply our results to the double sine-Gordon model in equation (1.22) and obtain Corollary 1.6.

3.1 Linear scattering theory

3.1.1 Jost solutions

Define $f_{+}(x,\xi )$ and $f_-(x,\xi )$ by the requirements that

(3.1) $$ \begin{align} (- \partial_x^2 + V) f_{\pm} = \xi^2 f_{\pm}, \quad \mbox{for all}\ x,\xi\in{\mathbb R}, \quad \mbox{and} \quad \left\{ \begin{array}{ll} \lim_{x\rightarrow \infty} |f_{+}(x,\xi) - e^{ix\xi}| = 0, \\ \lim_{x\rightarrow -\infty} |f_{-}(x,\xi) - e^{- ix\xi}| = 0. \end{array} \right. \end{align} $$

Define further

(3.2) $$ \begin{align} m_{+}(x,\xi) = e^{-i\xi x} f_{+}(x,\xi) \quad \mbox{and} \quad m_{-}(x,\xi) = e^{i\xi x} f_{-}(x,\xi), \end{align} $$

so that $m_{\pm }$ is a solution of

(3.3) $$ \begin{align} \partial_x^2 m_{\pm} \pm 2i\xi \partial_x m_{\pm} = Vm_{\pm}, \qquad m_\pm(x,\xi) \to 1 \;\mbox{as}\ x \to \pm \infty. \end{align} $$

The functions $m_\pm $ satisfy symbol type bounds for $\pm x>0$ , as stated in the following lemma.

Lemma 3.1. For all nonnegative integers $\alpha , \beta , N$ ,

(3.4) $$ \begin{align} & \big| \partial_x^{\alpha} \partial_{\xi}^{\beta} (m_{\pm}(x,\xi) - 1) \big| \lesssim \langle x \rangle^{-N} \langle \xi \rangle^{-1-\beta}, \quad \pm x \geq -1, \end{align} $$

(3.5) $$ \begin{align} & \big| \partial_x^{\alpha} \partial_{\xi}^{\beta} (m_{\pm}(x,\xi) - 1) \big| \lesssim \langle x \rangle^{1+\beta}\langle \xi \rangle^{-1-\beta}, \quad \pm x \leq 1.\quad \end{align} $$

The estimates in equations (3.4)–(3.5) can be obtained from the integral form of equation (3.3)

(3.6) $$ \begin{align} & m_+(x,\xi) = 1 + \int_x^{\infty} D_{\xi}(y-x) V(y) m_+(y,\xi) \,dy, \nonumber\\ & m_-(x,\xi) = 1 + \int_{-\infty}^x D_{\xi}(x-y) V(y) m_-(y,\xi) \,dy, \end{align} $$

where

$$ \begin{align*}D_{\xi}(z) = \frac{e^{2i\xi z} - 1}{2i\xi}. \end{align*} $$

Since the proof is fairly standard, we skip the details and refer the reader to [Reference Delort12, Appendix A].

3.1.2 Transmission and reflection coefficients

A classical reference for the formulas that we recall here is [Reference Deift and Trubowitz10] (see also [Reference Weder76], [Reference Yafaev73], for example). Denote as $T(\xi )$ and $R_{\pm }(\xi )$ , respectively, the transmission and reflection coefficients associated to the potential V. These coefficients are such that

(3.7) $$ \begin{align} &f_+ (x,\xi) = \frac{1}{T_+(\xi)} f_-(x,-\xi) + \frac{R_-(\xi)}{T_+(\xi)} f_-(x,\xi), \nonumber\\ &f_- (x,\xi) = \frac{1}{T_-(\xi)} f_+(x,-\xi) + \frac{R_+(\xi)}{T_-(\xi)} f_+(x,\xi), \end{align} $$

or, equivalently,

$$ \begin{align*} &f_+ (x,\xi) \sim \frac{1}{T_+(\xi)} e^{i\xi x} + \frac{R_-(\xi)}{T_+(\xi)} e^{-i\xi x} \quad \mbox{as}\ x \to - \infty, \\ &f_- (x,\xi) \sim \frac{1}{T_-(\xi)} e^{-i\xi x} + \frac{R_+(\xi)}{T_-(\xi)} e^{i\xi x} \quad \mbox{as}\ x \to \infty. \end{align*} $$

In the equalities above, $T_+$ and $T_-$ do a priori differ; however, since the Wronskian

(3.8) $$ \begin{align} W(\xi) := W(f_+(\xi),f_-(\xi)), \qquad W(f,g) = f'g - fg' \end{align} $$

is independent of the point x, where it is computed for solutions of equation (3.1), one sees (taking $x\rightarrow \pm \infty )$ that $T_+ = T_- = T$ and

(3.9) $$ \begin{align} W(\xi) = \frac{2i\xi}{T(\xi)}. \end{align} $$

Since $\overline {f_{\pm }(x,\xi )} = f_{\pm }(x,-\xi )$ , we obtain furthermore that

(3.10) $$ \begin{align} \overline{T(\xi)} = T(-\xi) \qquad \mbox{and} \qquad \overline{R_{\pm}(\xi)} = R_{\pm}(-\xi). \end{align} $$

Finally, computing $W(f_{+}(\xi ),f_-(\xi ))$ , $W(f_{+}(\xi ),f_+(-\xi ))$ , $W(f_{-}(\xi ),f_-(-\xi ))$ at $x = \pm \infty $ gives

(3.11) $$ \begin{align} & |R_{\pm}(\xi)|^2 + |T(\xi)|^2 = 1, \qquad \mbox{and} \qquad T(\xi)\overline{R_-(\xi)} + R_+(\xi)\overline{T(\xi)} = 0. \end{align} $$

As a consequence, the scattering matrix associated to the potential V is unitary:

(3.12) $$ \begin{align} S(\xi) := \left( \begin{array}{cc} T(\xi) & R_+(\xi) \\ R_-(\xi) & T(\xi) \end{array} \right), \qquad S^{-1}(\xi) := \left( \begin{array}{cc} \overline{T(\xi)} & \overline{R_-(\xi)} \\ \overline{R_+(\xi)} & \overline{T(\xi)} \end{array} \right). \end{align} $$

Starting from the integral formula in equation (3.6) giving $m_{\pm }$ , letting $x \to \mp \infty $ and relating it to the definition of T and $R_{\pm }$ gives

(3.13) $$ \begin{align} \begin{aligned} &T(\xi) = \frac{2i\xi}{2i\xi - \int V(x) m_{\pm} (x,\xi)\,dx},\\[6pt] & R_{\pm}(\xi) = \frac{\int e^{\mp 2i\xi x} V(x) m_{\mp}(x,\xi)\,dx}{2i\xi - \int V(x) m_{\pm} (x,\xi)\,dx}. \end{aligned} \end{align} $$

These formulas are only valid for $\xi \neq 0$ a priori. But a moment of reflection shows that T and $R_{\pm }$ can be extended to be smooth functions on the whole real line. Combining these formulas with Lemma 3.1 gives the following lemma.

Lemma 3.2. Let T and $R_\pm $ be defined as in equation (3.13). Then under our assumptions on V, for any $\beta $ and N, we have

(3.14) $$ \begin{align} |\partial_{\xi}^{\beta} [T(\xi) - 1]| \lesssim \langle \xi \rangle^{-1-\beta}, \qquad |\partial_{\xi}^{\beta} R_{\pm}(\xi) | \lesssim \langle \xi \rangle^{-N}.\end{align} $$

3.1.3 Generic and exceptional potentials

We call the potential V

• • generic if $\displaystyle \int V(x) m_{\pm } (x,0)\,dx \neq 0$

• • exceptional if $\displaystyle \int V(x) m_{\pm } (x,0)\,dx = 0$

• • very exceptional if $\displaystyle \int V(x) m_{\pm } (x,0)\,dx = \int x V(x) m_{\pm } (x,0)\,dx = 0$

Lemma 3.3. The following four assertions are equivalent:

1. (i) V is generic.

2. (ii) $\displaystyle T(0) = 0, R_{\pm }(0) = -1$ .

3. (iii) $W(0) \neq 0$ .

4. (iv) The potential V does not have a resonance at $\xi =0$ ; in other words, there does not exist a bounded nontrivial solution in the kernel of $-\partial _x^2 + V$ .

Checking the equivalence of these assertions is easy based on the formulas in equation (3.13).

Proposition 3.4 (Low-energy scattering)

If V is generic, there exists $\alpha \in i \mathbb {R}$ such that

(3.15) $$ \begin{align} T(\xi) = \alpha \xi + O(\xi^2). \end{align} $$

If V is exceptional, let

$$ \begin{align*}a := f_+(-\infty,0) \in \mathbb{R} \setminus \{ 0 \}. \end{align*} $$

Then

(3.16) $$ \begin{align} T(0) = \frac{2a}{1+a^2}, \qquad R_+(0) = \frac{1 - a^2}{1+ a^2}, \qquad \mbox{and} \qquad R_-(0) = \frac{a^2-1}{1+ a^2}. \end{align} $$

Proof. In the generic case, observe that

$$ \begin{align*}T(\xi) = \frac{2i}{-\int V(x) m_{\pm}(x,0)\,dx} \xi + O(\xi^2), \end{align*} $$

hence the desired result since $m_{\pm }(\cdot ,0)$ is real-valued.

We now turn to the exceptional case. Denoting

$$ \begin{align*}b = \int V(x) \partial_{\xi} m_{\pm}(x,0) \,dx, \quad \mbox{and} \quad c_{\pm} = \int V(x) x m_{\pm}(x,0)\,dx, \end{align*} $$

$T(0)$ and $R_{\pm }(0)$ can, thanks to equation (3.13), be expressed as

(3.17) $$ \begin{align} T(0) = \frac{2i}{2i-b}, \qquad R_{\pm}(0) = \frac{b \mp 2ic_{\mp}}{2i-b}. \end{align} $$

It remains to determine the values of b and $c_{\pm }$ . In order to determine $c_{+}$ , recall the integral equation (3.6) satisfied by $m_+$ , and let $\xi \to 0$ and $x \to -\infty $ in that formula. Taking advantage of the condition $\int V(y) m_+(y,0)\,dy=0$ , we observe that

(3.18) $$ \begin{align} a = m_+(-\infty,0) = 1 + \int_{-\infty}^{\infty} y V(y) m_+(y,0)\,dy = 1 + c_+. \end{align} $$

Similarly, we find $\frac {1}{a} = 1 - c_-$ .

Turning to b, we first claim that it is purely imaginary. Indeed, differentiating equation (3.3), setting $\xi = 0$ and taking the real part, we obtain that

$$ \begin{align*}\mathfrak{Re} \left[ (\partial_x^2 - V) \partial_{\xi} m_+(x,0) \right] = 0. \end{align*} $$

Since $\partial _{\xi } m_+(x,0) \to 0$ as $x \to \infty $ , we deduce that $\mathfrak {Re} \partial _{\xi } m_+ = 0$ . Using this fact and plugging the formulas in equation (3.17) into the identity $|T(0)|^2 + |R_{-}(0)|^2 = 1$ , we find

$$ \begin{align*}b = \left(2 - a - \frac{1}{a}\right)i. \end{align*} $$

The formulas giving b and c in terms of a now lead to the desired formulas for $T(0)$ and $R(0)$ .

Remark 3.5. From equation (3.18), we see that in the very exceptional case (see the definition before Lemma 3.3), we have $a=1$ , and therefore $T(0) = 1$ and $R_\pm (0) = 0$ . Also, notice that $a=1$ in the exceptional case when the zero-energy resonance is even. When instead it is odd, we have $a=-1$ , and therefore $T(0)=-1$ and $R_\pm (0)=0$ .

3.1.4 Resolvent and spectral projection

If $\mathfrak {Im} \lambda \neq 0$ , the resolvent of H is defined by $R_V(\lambda ) = (H - \lambda ^{-1})^{-1}$ .

Assuming first that $\mathfrak {Re}(\lambda )>0$ and $\mathfrak {Im} \lambda> 0$ , we let $\xi + i \eta = \sqrt {\lambda }$ , with $\xi ,\eta>0$ . Then $f_{\pm }(\xi + i \eta )$ can be defined through natural extensions of the above definition, and the resolvent $R_V(\lambda )$ is given by the kernel

$$ \begin{align*}R_V(\lambda)(x,y) = -\frac{1}{W(\xi + i \eta)} [f_+(\mathrm{max}(x,y),\xi+i\eta) f_-(\mathrm{min}(x,y),\xi+i\eta)]. \end{align*} $$

Letting $\mathfrak {Im} \lambda \to 0$ (and still with the convention that $\xi>0$ ),

$$ \begin{align*}R_V(\xi^2 + i0) = - \frac{1}{W(\xi)} f_+(\mathrm{max}(x,y),\xi) f_-(\mathrm{min}(x,y),\xi). \end{align*} $$

Similarly,

$$ \begin{align*}R_V(\xi^2 - i0) = - \frac{1}{W(-\xi)} f_+(\mathrm{max}(x,y),-\xi) f_-(\mathrm{min}(x,y),-\xi). \end{align*} $$

By Stone’s formula, the spectral measure associated to H is, for $\lambda>0$ ,

$$ \begin{align*}E(d\lambda) = \frac{1}{2\pi i} [R_V(\lambda + i 0) - R_V(\lambda - i0)] d\lambda. \end{align*} $$

The formulas above for $R_V(\lambda \pm i 0)$ lead to

$$ \begin{align*}E(d\lambda)(x,y) = \frac{1}{4\pi} \frac{|T(\sqrt \lambda)|^2}{\sqrt \lambda} [f_-(x,\sqrt \lambda) \overline{f_-(y,\sqrt \lambda)} + f_+(x,\sqrt \lambda) \overline{f_+(y,\sqrt \lambda)}] d\lambda. \end{align*} $$

3.2 Distorted Fourier transform

3.2.1 Definition and first properties

We adopt the following normalisation for the (flat) Fourier transform on the line

$$ \begin{align*}\widehat{\mathcal{F}} \phi (\xi) = \widehat{\phi}(\xi) = \frac{1}{\sqrt{2\pi}} \int e^{-i\xi x} \phi(x) \, dx. \end{align*} $$

As is well-known,

$$ \begin{align*}\widehat{\mathcal{F}}^{-1} \phi = \frac{1}{\sqrt{2\pi}} \int e^{i\xi x} \phi(\xi) \, d\xi= \widehat{\mathcal{F}}^* \phi, \end{align*} $$

and $\mathcal {F}$ is an isometry on $L^2(\mathbb {R})$ .

We now define the wave functions associated to H:

(3.19) $$ \begin{align} \psi(x,\xi) := \frac{1}{\sqrt{2\pi}} \left\{ \begin{array}{ll} T(\xi) f_+(x,\xi) & \mbox{for}\ \xi \geq 0 \\ \\ T(-\xi) f_-(x,-\xi) & \mbox{for}\ \xi < 0. \end{array} \right. \end{align} $$

Once again, this definition a priori only makes sense for $\xi \neq 0$ , but it can be extended by continuity to $\xi =0$ . It follows from the estimates on T and $f_{\pm }$ that for any $\alpha $ , $\beta $ and $\xi \neq 0$ ,

(3.20) $$ \begin{align} |\partial_x^{\alpha} \partial_{\xi}^{\beta} \psi (x,\xi)| \lesssim \langle x \rangle^{\beta} \langle \xi \rangle^{\alpha}. \end{align} $$

The distorted Fourier transform is then defined by

(3.21) $$ \begin{align} \widetilde{\mathcal{F}} \phi(\xi) = \widetilde {\phi}(\xi) = \int_{\mathbb{R}} \overline{\psi(x,\xi)} \phi(x)\,dx. \end{align} $$

Proposition 3.6 (Mapping properties of the distorted Fourier transform)

With $\widetilde {{\mathcal {F}}}$ defined in equation (3.21),

1. (i) $\widetilde {\mathcal {F}}$ is a unitary operator from $L^2$ onto $L^2$ . In particular, its inverse is

   $$ \begin{align*}\widetilde{\mathcal{F}}^{-1} \phi(x) = {\widetilde{\mathcal{F}}}^{\ast} \phi(x) = \int_{\mathbb R} \psi(x,\xi) \phi(\xi)\,d\xi. \end{align*} $$

2. (ii) $\widetilde {\mathcal {F}}$ maps $L^1(\mathbb {R})$ to functions in $L^{\infty }(\mathbb {R})$ that are continuous at every point except $0$ and converge to $0$ at $\pm \infty $ .

3. (iii) $\widetilde {\mathcal {F}}$ maps the Sobolev space $H^s(\mathbb {R})$ onto the weighted space $L^2(\langle \xi \rangle ^{2s} \,d\xi )$ .

4. (iv) If $\widetilde {f}$ is continuous at zero, then for any integer $s \geq 0$ ,

   $$ \begin{align*} \|\langle \xi \rangle^s \partial_{\xi} \widetilde{f} \|_{L^2} \lesssim \| f \|_{H^s} + \| \langle x \rangle f \|_{H^s}. \end{align*} $$

Proof. As in other parts of this section, we follow Yafaev [Reference Yafaev73, Chap. 5].

$(i)$ To see that $\mathcal {F}$ is an isometry, we use the Stone formula derived in the previous subsection to write, for any functions $g,h \in L^2$ (recall that E is the spectral measure associated to H),

$$ \begin{align*} \langle g, h \rangle & = \int E (d\lambda) g \, \overline{h} \\ & = \frac{1}{4\pi} \iiint \frac{|T(\sqrt{\lambda})|^2}{\sqrt \lambda} \left[ f_-(x, \sqrt{\lambda}) \overline{f_-(y,\sqrt \lambda)} + f_+(x,\sqrt \lambda) \overline{f_+(y,\sqrt \lambda)} \right] g(y) \overline{h(x)} \,dy \,dx \,d\lambda. \end{align*} $$

Changing the integration variable to $\xi = \sqrt {\lambda }$ , this is

$$ \begin{align*} \frac{1}{2\pi} \int_{\xi \in {\mathbb R}_+} \int_{{\mathbb R}_x} \int_{{\mathbb R}_y} |T(\xi)|^2 \left[ f_-(x,\xi) \overline{f_-(y,\xi)} + f_+(x,\xi) \overline{f_+(y,\xi)} \right] g(y) \overline{h(x)} \,dy \,dx \, d\xi = \langle \widetilde{g} \,,\, \widetilde{h} \rangle. \end{align*} $$

To see that the range of $\widetilde {\mathcal {F}}$ is $L^2$ , we argue by contradiction. If this was not the case, there would exist $g \in L^2$ not zero such that, for any $f \in \mathcal {C}^{\infty }_0$ and any $0<R_0<R_1$ ,

$$ \begin{align*}\langle g \,,\,\widetilde{\mathcal{F}} E([R_0^2,R_1^2]) f \rangle = 0.\end{align*} $$

Using the spectral theorem representation and the intertwining identity $\widetilde {{\mathcal {F}}} H = k^2 \widetilde {{\mathcal {F}}}$ , we deduce that

$$ \begin{align*}\widetilde{\mathcal{F}} \left[ E([R_0^2,R_1^2]) f \right] (\xi) = \big( \mathbf{1}_{[-R_1,-R_0]}(\xi) + \mathbf{1}_{[R_0,R_1]}(\xi) \big) \widetilde{f}(\xi). \end{align*} $$

Therefore, for any $f \in \mathcal {C}^{\infty }_0$ ,

$$ \begin{align*} 0 = \langle g \,,\,\widetilde{\mathcal{F}} E([R_0^2,R_1^2]) f \rangle = \int_{R_0}^{R_1} \int \big[ \psi(x,\xi) g(\xi) + \psi(x,-\xi) g(-\xi) \big] \overline{f(x)} \,dx \,d\xi. \end{align*} $$

This implies that

$$ \begin{align*}\int_{R_0}^{R_1} \big[ \psi(x,\xi)g(\xi) + \psi(x,-\xi) g(-\xi) \big] \,d\xi = 0. \end{align*} $$

Since $R_0,R_1$ are arbitrary, we deduce $ \psi (x,\xi )g(\xi ) + \psi (x,-\xi ) g(-\xi ) = 0, $ a.e. $\xi $ . Since $x \mapsto \psi (x,\xi )$ and $x \mapsto \psi (x,-\xi )$ are independent functions (nonvanishing Wronskian), this implies $g = 0$ .

$(ii)$ is a consequence of equation (3.20) and the Riemann-Lebesgue lemma.

$(iii)$ is a consequence of Theorem 3.10 below.

$(iv)$ Focusing on $x>0$ (through a smooth cutoff function $\chi _+$ ) and $\xi>0$ , the distorted Fourier transform can be written as a pseudodifferential operator

$$ \begin{align*}\widetilde{\mathcal{F}} \left[ \chi_+ f \right] (\xi) = \int a(x,\xi) e^{-ix\xi} f(x)\,dx \end{align*} $$

with symbol

$$ \begin{align*}a(x,\xi) = \frac{1}{\sqrt{2\pi}} \overline{T(\xi) m_+(x,\xi)}. \end{align*} $$

Taking a derivative in $\xi $ ,

$$ \begin{align*}\partial_{\xi} \widetilde{\mathcal{F}} \left[ \chi_+ f \right] (\xi) = \int \partial_{\xi} a(x,\xi) e^{-ix\xi} f(x)\,dx + \int a(x,\xi) e^{-ix\xi} (-ix) f(x)\,dx. \end{align*} $$

From the bounds in equations (3.4) and (3.14), along with a classical theorem on the boundedness of pseudo-differential operators, the statement $(iv)$ follows for $s = 0$ . If $s\in \mathbb {N}$ , it suffices to multiply the above by $\langle \xi \rangle ^s$ and integrate by parts in x in the integrals. We only discussed the case of positive frequencies, but the case of negative frequencies is identical. It remains to check that no singularity arises at $\xi =0$ when applying $\xi>0$ , which is ensured by the assumption that $\widetilde {f}$ is continuous.

Lemma 3.7. If the potential V is even, then the distorted Fourier transform preserves evenness and oddness.

Proof. Observe that when V is even, we have the relation $f_+(x,\xi ) = f_-(-x,\xi )$ between the generalised eigenfunctions in equation (3.1), by uniqueness of solutions for the ODE. From this and the definition in equation (3.19), we see that $\psi (x,\xi )=\psi (-x,-\xi )$ . The preservation of parity for the distorted Fourier transform then follows directly from the definition in equation (3.21).

As appears in Proposition 3.6, one of the main differences between the mapping properties of $\widehat {\mathcal {F}}$ and $\widetilde {\mathcal {F}}$ has to do with zero frequency. Since the zero frequency furthermore plays a key role in the nonlinear analysis developed in the present paper, we investigate this question a bit more.

• • If V is generic, then $\psi (x,0) = 0$ and $\widetilde {f}(0) = 0$ if $f \in L^1$ . Furthermore, assuming better integrability properties at $\infty $ ,

  (3.22) $$ \begin{align} \begin{aligned} & \mbox{if}\ \xi> 0, \qquad \widetilde{f}(\xi) = - \frac{\alpha\xi}{\sqrt{2\pi}} \int f(x) f_+(x,0)\,dx +O(\xi^2) \\ & \mbox{if}\ \xi < 0, \qquad \widetilde{f}(\xi) = \frac{\alpha\xi}{\sqrt{2\pi}} \int f(x) f_-(x,0)\,dx+O(\xi^2), \end{aligned} \end{align} $$
  where $\alpha $ was defined in equation (3.15). Thus, $\widetilde {f}$ is typically continuous but not continuously differentiable at zero.

• • If V is exceptional, then

  $$\begin{align*}\sqrt{2\pi} \, \psi(x,0+) = \frac{2a}{1+a^2} f_+(x,0), \qquad \mbox{and} \qquad \sqrt{2\pi} \, \psi(x,0-) = \frac{1}{a} \psi(x,0+), \end{align*}$$
  where a was defined in Proposition 3.4. Therefore, if $f \in L^1$ ,
  (3.23) $$ \begin{align} \begin{aligned} \widetilde{f}(0+) = \frac{2a}{1+a^2} \frac{1}{\sqrt{2\pi}} \int f(x) f_+(x,0) \,dx \qquad \mbox{and} \qquad \widetilde{f}(0-) = \frac{1}{a} \widetilde{f}(0+). \end{aligned} \end{align} $$
  As a consequence, $\widetilde {f}$ is continuous if $a=1$ but might not be otherwise.

3.2.2 Fourier multipliers

Given m a function on the real line, the flat and distorted Fourier multipliers are defined by

$$ \begin{align*} & m(D) = \widehat{\mathcal F}^{-1} m(\xi) \widehat{\mathcal F} \\ & m(\widetilde D) = \widetilde{\mathcal F}^{-1} m(\xi) \widetilde{\mathcal F}. \end{align*} $$

Denoting $H_0$ and H for the flat and perturbed Schrödinger operators

$$ \begin{align*}H_0 = - \partial_x^2, \qquad H = - \partial_x^2 + V, \end{align*} $$

these operators are diagonalised by $\widehat {\mathcal {F}}$ and $\widetilde {\mathcal {F}}$ , giving the functional calculus

$$ \begin{align*} & f(H_0) = \widehat{\mathcal{F}}^{-1} f(\xi^2)\widehat{\mathcal{F}}\\ & f(H) = \widetilde{\mathcal{F}}^{-1} f(\xi^2) \widetilde{\mathcal{F}}. \end{align*} $$

In particular,

$$ \begin{align*}e^{it \sqrt{1 + H_0}} = e^{it \langle D \rangle} \qquad \mbox{and} \qquad e^{it \sqrt{1 + H}} = e^{it \langle \widetilde D \rangle}. \end{align*} $$

Lemma 3.8. Assume that f is real-valued and that m is even and real-valued. Then $m(\widetilde {D}) f$ is real-valued.

Proof. This follows from the simple observation that f is real-valued if and only if

$$ \begin{align*}\left\{ \begin{array}{ll} T(\xi) \widetilde{f}(\xi) = - T(-\xi) R_+(\xi) \overline{\widetilde{f}(\xi)} + \overline{\widetilde{f}(-\xi)} &\mbox{for}\ \xi>0 \\[4pt] T(-\xi) \widetilde{f}(\xi) = - T(\xi) R_-(-\xi) \overline{\widetilde{f}(\xi)} + \overline{\widetilde{f}(-\xi)} & \mbox{for}\ \xi<0. \end{array} \right. \end{align*} $$

3.2.3 The wave operator

The wave operator $\mathcal {W}$ is given by

$$ \begin{align*}\mathcal{W} = \operatorname{s-lim}_{t\to \infty} e^{itH} e^{-itH_0}. \end{align*} $$

Proposition 3.9. The wave operator is unitary on $L^2$ and given by

$$ \begin{align*}\mathcal{W} = \widetilde{\mathcal{F}}^{-1} \widehat{\mathcal{F}}. \end{align*} $$

As a consequence,

(3.24) $$ \begin{align} \mathcal{W}^{-1} = \mathcal{W}^* = \widehat{\mathcal{F}}^{-1} \widetilde{\mathcal{F}}, \end{align} $$

and the wave operator intertwines H and $H_0$ :

$$ \begin{align*}f(H) = \mathcal{W} f(H_0) \mathcal{W}^*. \end{align*} $$

Proof. In order to prove the desired formula for the wave operator, it suffices to check that, for any $f \in L^2$ ,

$$ \begin{align*}\left\| e^{itH} e^{-itH_0} f - \widetilde{\mathcal{F}}^{-1} \widehat{\mathcal{F}} f \right\|_2 \rightarrow 0. \end{align*} $$

By the functional calculus, this is equivalent to

$$ \begin{align*}\left\| \widehat{\mathcal{F}}^{-1} e^{it \xi^2} f - \widetilde{\mathcal{F}}^{-1} e^{it \xi^2} f \right\|_2 \rightarrow 0. \end{align*} $$

By unitarity, it suffices to check the above for a dense subset of f, and thus we might assume $f \in \mathcal {C}^{\infty }_0$ . By symmetry between positive and negative frequencies, we can furthermore assume that $\operatorname {Supp} f \subset (0,\infty )$ . Therefore, matters reduce to proving that

$$ \begin{align*}\left\| \int_0^{\infty} e^{i(x\xi + t\xi^2)} (1- T(\xi) m_+(x,\xi)) f(\xi)\,d\xi \right\|_{L^2_x} \rightarrow 0. \end{align*} $$

To see that the above is true, we split the function whose $L^2$ norm we want to estimate into

$$ \begin{align*} & \int_0^{\infty} e^{i(x\xi + t\xi^2)} \mathbf{1}_+(x) (1-T(\xi) m_+(x,\xi)) f(\xi)\,d\xi - \int_0^{\infty} e^{i(-x\xi + t\xi^2)} \mathbf{1}_-(x) R_-(\xi) m_-(x,\xi) f(\xi)\,d\xi \\ & \qquad + \int_0^{\infty} e^{i(x\xi + t\xi^2)} \mathbf{1}_-(x) (1- m_-(x,\xi)) f(\xi)\,d\xi \\ & = I + II + III. \end{align*} $$

The terms I and $II$ have nonstationary phases, from which it follows that they converge to zero as $t \to \infty $ . As for $III$ , it goes to zero pointwise by the stationary phase lemma and is uniformly (in t) bounded by a decaying function of x, as follows from the estimates on $m_-$ ; therefore, it goes to zero in $L^2$ by the dominated convergence theorem.

Finally, the following theorem gives the boundedness of the wave operators on Sobolev spaces.

Theorem 3.10 (Weder [Reference Weder75])

$\mathcal {W}$ and $\mathcal {W}^*$ extend to bounded operators on $W^{k,p}({\mathbb R})$ for any k and $1 < p < \infty $ . Furthermore, in the exceptional case, if $f_+(-\infty ,0) = 1$ , this remains true if $p=1$ or $\infty $ .

3.2.4 What if discrete spectrum is present?

The above discussion relied on the assumption that

$$ \begin{align*}L^2_{ac} = P_{ac} L^2 = L^2, \end{align*} $$

where we denoted $P_{ac}$ the projector on the absolutely continuous spectrum of H. Since we are assuming $V \in \mathcal {S}$ , we can exclude singularly continuous spectrum as well as embedded discrete spectrum, but there might be a finite number of negative eigenvalues $\lambda _N < \dots < \lambda _1 < 0$ with corresponding eigenfunctions $\phi _1,\dots ,\phi _N$ ; see [Reference Deift and Trubowitz10]. Then all the statements made above require small adaptations. Indeed, $\widetilde {\mathcal {F}}$ is zero on $\phi _j$ for all j and unitary from $L^2_{ac}$ to $L^2$ . Thus,

$$ \begin{align*} \widetilde{{\mathcal{F}}} \widetilde{{\mathcal{F}}}^{-1} = \operatorname{Id}_{L^2} \qquad \mbox{and} \qquad \widetilde{{\mathcal{F}}}^{-1} \widetilde{{\mathcal{F}}} = P_{ac}. \end{align*} $$

3.3 Decomposition of $\psi (x,\xi )$

Let $\rho $ be an even, smooth, nonnegative function equal to $0$ outside of $B(0,2)$ and such that $\int \rho = 1$ . Define $\chi _{\pm }$ by

(3.25) $$ \begin{align} \chi_+(x) = H * \rho = \int_{-\infty}^{x} \rho(y)\,dy , \quad \mbox{and} \quad \chi_+(x) + \chi_-(x) = 1, \end{align} $$

where H is the Heaviside function, $H=\mathbf {1}_{+}$ . Notice that

$$ \begin{align*}\chi_+(x) = \chi_-(-x). \end{align*} $$

With $\chi _\pm $ as above, and using the definition of $\psi $ in equation (3.19) and $f_\pm $ and $m_\pm $ in equations (3.1)–(3.2), as well as the identity in equation (3.7), we can write

(3.26) $$ \begin{align} \begin{aligned} \mbox{for} \quad \xi>0 \qquad \sqrt{2\pi}\psi(x,\xi) & = \chi_+(x)T(\xi)m_+(x,\xi)e^{ix\xi} \\ &\quad + \chi_-(x) \big[ m_-(x,-\xi)e^{i\xi x} + R_-(\xi) m_-(x,\xi)e^{-i\xi x} \big], \end{aligned} \end{align} $$

and

(3.27) $$ \begin{align} \begin{aligned} \mbox{for} \quad \xi<0 \qquad \sqrt{2\pi}\psi(x,\xi) & = \chi_-(x)T(-\xi)m_-(x,-\xi)e^{ix\xi} \\ &\quad + \chi_+(x) \big[m_+(x,\xi) e^{i\xi x} + R_+(-\xi)m_+(x,-\xi) e^{-i\xi x} \big]. \end{aligned} \end{align} $$

We then decompose

(3.28) $$ \begin{align} \sqrt{2\pi}\psi(x,\xi) = \psi^S(x,\xi) + \psi^R(x,\xi) , \end{align} $$

where, on the one hand, the singular part (nondecaying in x) is

(3.29) $$ \begin{align} \begin{aligned} \mbox{for} \quad \xi>0 \qquad \psi^S(x,\xi) & := \chi_+(x) T(\xi) e^{i\xi x } + \chi_-(x) (e^{i\xi x} + R_-(\xi) e^{-i\xi x}), \\ \mbox{for} \quad \xi<0 \qquad \psi^S(x,\xi) & := \chi_-(x) T(-\xi) e^{i\xi x} + \chi_+(x) (e^{i\xi x} + R_+(-\xi) e^{-i\xi x}), \end{aligned} \end{align} $$

and the regular part is

(3.30) $$ \begin{align} \begin{aligned} \mbox{for} \quad \xi>0 \qquad \psi^R(x,\xi) & := \chi_+(x) T(\xi) (m_+(x,\xi)-1) e^{i\xi x} \\ &\quad + \chi_-(x) \big[ (m_-(x,-\xi) - 1)e^{i\xi x} + R_-(\xi)(m_-(x,\xi) - 1) e^{-ix\xi} \big], \\ \mbox{for} \quad \xi<0 \qquad \psi^R(x,\xi) & := \chi_-(x) T(-\xi) (m_-(x,-\xi)-1) e^{i\xi x} \\ &\quad + \chi_+(x) \big[ (m_+(x,\xi) - 1)e^{i\xi x} + R_+(-\xi)(m_+(x,-\xi) - 1) e^{-ix\xi} \big]. \end{aligned} \end{align} $$

3.4 Linear estimates

Recall that $\langle D \rangle = \sqrt { -\partial _x^2 + 1 }$ and $\langle \widetilde {D} \rangle = \sqrt { -\partial _x^2 + V + 1} = \widetilde {{\mathcal {F}}}^{-1} \langle \xi \rangle \widetilde {{\mathcal {F}}}$ .

A more precise asymptotic formula with an explicit leading order term can be read off the proof of Proposition 3.11; in particular, up to a faster-decaying remainder of the same form of those appearing in equation (3.32), we have

(3.33) $$ \begin{align} e^{it\langle \widetilde D \rangle} f \approx \frac{e^{i \frac{\pi}{4}}}{\sqrt{2t}} \langle \xi_0 \rangle^{3/2} e^{i t \langle \xi_0 \rangle + i x \xi_0} \widetilde{f} \left( \xi_0 \right) \qquad \mbox{as}\ t \to \infty, \quad \frac{\xi_0}{\langle \xi_0 \rangle} = - \frac{x}{t}. \end{align} $$

Remark 3.12. Note that, in view of equation (2.31), we have $\alpha + \beta \gamma < 1/4$ . Therefore, uniform-in-time control of the profile in $\widetilde {{\mathcal {F}}}^{-1}\langle \xi \rangle ^{-3/2} L^{\infty }$ and $W_t$ , and in $H^4$ with small time growth gives the sharp $|t|^{-1/2}$ decay for linear solutions through equation (3.32).

Furthermore, let $\mathbf {a}$ be any of the coefficients defined in equation (4.5). In view of equation (3.31) and the regularity of $T(\xi )$ and $R(\xi )$ in equation (3.14), we have

(3.34) $$ \begin{align} \begin{aligned} {\big\| e^{\pm it \langle D \rangle} \widehat{{\mathcal{F}}}^{-1} \big(\mathbf{a}(\xi) \widetilde{f} \, \big) \big\|}_{L^{\infty}_x} & \lesssim \frac{1}{\langle t \rangle^{1/2}} {\| \langle \xi \rangle^{3/2} \widetilde{f} \|}_{L^{\infty}_{\xi}} + \frac{1}{\langle t \rangle^{3/4-\alpha -\beta\gamma}} \big( {\| \langle \xi \rangle \partial_{\xi} \widetilde{f} \|}_{W_t} + {\| \widetilde{f} \|}_{L^2} \big) \\ &\quad + \frac{1}{\langle t \rangle^{7/12}} {\| \langle \xi \rangle^4 \widetilde{f} \|}_{L^2}. \end{aligned} \end{align} $$

Remark 3.13. Besides the pointwise decay estimates of Proposition 3.11 above, we will also use the following variant: for $k \geq 5$ ,

(3.35) $$ \begin{align} {\| e^{\pm it\langle D \rangle} \varphi_k(D) f \|}_{L^{\infty}} \lesssim \frac{1}{t^{1/2}} 2^{3k/2} {\big\| \varphi_k\widehat{f} \big\|}_{L^2}^{1/2} \big( {\big\| \varphi_k\partial_{\xi}\widehat{f} \big\|}_{L^2} + {\big\| \varphi_k\widehat{f} \big\|}_{L^2} \big)^{1/2}, \end{align} $$

which follows from the standard $L^1\rightarrow L^{\infty }$ decay and the interpolation inequality

$$ \begin{align*} {\big\| \varphi_k(D) f \big\|}_{L^1} \lesssim {\big\| \varphi_k(D) f \big\|}_{L^2}^{1/2} {\big\| x \, \varphi_k(D) f \big\|}_{L^2}^{1/2}. \end{align*} $$

Notice that equation (3.35) also implies (see equation (3.24))

(3.36) $$ \begin{align} {\| e^{\pm it\langle D \rangle} \mathcal{W}^* \varphi_k(\widetilde{D}) f \|}_{L^{\infty}} \lesssim \frac{1}{|t|^{1/2}} 2^{3k/2} {\big\| \varphi_k\widetilde{f} \big\|}_{L^2}^{1/2} \big( {\big\| \varphi_k\partial_{\xi}\widetilde{f} \big\|}_{L^2} + {\big\| \varphi_k\widetilde{f} \big\|}_{L^2} \big)^{1/2}. \end{align} $$

To prove Proposition 3.11, we use the following stationary phase lemma:

Lemma 3.14. Consider for $X \in \mathbb {R}$ , $t \geq 0$ , $x\in {\mathbb R}$ the integrals

$$ \begin{align*} I_{\mu, \nu}(t,X,x) = \int_{{\mathbb R}_{\mu}} e^{it ( \nu \langle \xi \rangle - \xi X)} a(x,\xi) g(\xi)\, d\xi, \qquad \mu,\nu \in \{+,-\}, \end{align*} $$

and assume that

(3.37) $$ \begin{align} \sup_{x \in {\mathbb R}, \, \xi \in {\mathbb R}_\mu}\left(|a(x,\xi)| + \langle \xi \rangle |\partial_{\xi}a(x,\xi)| \right)\lesssim 1. \end{align} $$

Then we have the estimate

(3.38) $$ \begin{align} \begin{aligned} \big| I_{\mu, \nu}(t,X,x) \big| \lesssim \frac{1}{\langle t \rangle^{1/2}} {\| \langle \xi \rangle^{3/2} g(\xi) \|}_{L^{\infty}} + \frac{1}{\langle t \rangle^{3/4 - \alpha - \beta\gamma}} {\| \langle \xi \rangle \partial_{\xi} g \|}_{W_T} + \frac{1}{\langle t \rangle^{7/12}} {\| \langle \xi \rangle^4 g \|}_{L^2}. \end{aligned} \end{align} $$

We postpone the proof of the lemma and give first the proof of Proposition 3.11.

Proof of Proposition 3.11

In order to prove equation (3.31), we write

$$ \begin{align*} e^{\pm i t \langle D \rangle} \mathbf{1}_I (D) f = \frac{1}{\sqrt{2\pi}} \int_{I} e^{i t(\pm \langle \xi \rangle - \xi X)} \widehat{f}(\xi)\, d\xi, \quad X := - x/t \end{align*} $$

and use Lemma 3.14 on $I={\mathbb R}_{+}$ or ${\mathbb R}_{-}$ and $a\equiv 1$ .

To prove equation (3.32), we use the distorted Fourier inversion (see equation (3.6)) to write

$$ \begin{align*} e^{\pm i t \langle \widetilde{D} \rangle} f = \int_{{\mathbb R}_+} e^{\pm i t \langle \xi \rangle} \psi(x,\xi) \widetilde{f}(\xi)\, d\xi + \int_{{\mathbb R}_-} e^{\pm i t \langle \xi \rangle} \psi (x,\xi) \widetilde{f}(\xi)\, d\xi. \end{align*} $$

Let us estimate the first integral, the other one being similar. Using equation (3.26), we can write

$$ \begin{align*} \sqrt{2\pi} \int_{{\mathbb R}_{+}} e^{\pm i t \langle \xi \rangle} \psi(x,\xi) f(\xi)\, d\xi & = \chi_+(x) \int_{{\mathbb R}_+} e^{ i t(\pm \langle \xi \rangle - \xi X)} T(\xi) m_{+}(x,\xi) f(\xi)\, d\xi \\ &\quad + \chi_-(x) \int_{{\mathbb R}_+} e^{i t(\pm \langle \xi \rangle - \xi X)} m_{-}(x,-\xi) f(\xi)\, d\xi \\ &\quad + \chi_-(x) \int_{{\mathbb R}_+} e^{ i t(\pm \langle \xi \rangle + \xi X)} R_{-}(\xi) m_{-}(x,\xi) f(\xi)\, d\xi. \end{align*} $$

Then the desired estimate follows by using Lemma 3.14 with $a(x,\xi )= T(\xi ) m_{+}(x,\xi )$ , $m_{-}(x,-\xi )$ and $R_-(\xi )m_{-}(x,\xi )$ , where the assumption in equation (3.37) holds thanks to Lemmas 3.1 and 3.2.

Proof of Lemma 3.14

It suffices to consider only the case $\mu =+$ , $\nu =+$ and $t \geq 1$ , $X \geq 0$ ; all other cases are similar or easier. We let

(3.39) $$ \begin{align} I_{++} = \sum_{k\in{\mathbb Z}} I_k , \qquad I_k(t,X) := \int_{{\mathbb R}_+} e^{it ( \langle \xi \rangle - \xi X)} a(x,\xi) g(\xi) \varphi_k(\xi) \, d\xi. \end{align} $$

First, notice that since

(3.40) $$ \begin{align} \big| I_k(t,x) \big| \lesssim \int_{{\mathbb R}_+} | g(\xi) | \varphi_k(\xi) \, d\xi \lesssim \mathrm{min} \big( 2^k {\| g \|}_{L^{\infty}}, 2^{-7k/2} {\| \langle \xi \rangle^4 g \|}_{L^2} \big), \end{align} $$

we see that $|I_{++}|$ enjoys the desired bound if $2^k \gtrsim t^{1/6}$ , or $2^k \lesssim t^{-1/2}$ . From now on, we assume

(3.41) $$ \begin{align} C t^{-1/2} \leq 2^k \leq (1/C) t^{1/6} \end{align} $$

for a suitably large absolute constant $C>0$ .

Let us denote

(3.42) $$ \begin{align} \begin{aligned} & \phi_{X}(\xi) := \langle \xi \rangle - \xi X, \quad \\ & \phi_X'(\xi) = \frac{\xi}{\langle \xi \rangle} - X, \quad \phi_{X}''(\xi) = \frac{1}{\langle \xi \rangle^3}, \qquad \xi_{0} := X/\sqrt{1-X^2}, \end{aligned} \end{align} $$

and note that the phase $\phi _X$ has no stationary points if $X \geq 1$ , and a unique, nondegenerate, stationary point at $\xi _0$ for any $X \in [0, 1)$ . Consider $n \in {\mathbb Z}_+$ such that $t \in [2^{n-1},2^n]$ , and let $q_0 \in {\mathbb Z}$ be the smallest integer such that $2^{q_0} \geq 2^{(3/2)k^+} 2^{-n/2} \approx \langle \xi \rangle ^{3/2} t^{-1/2}$ . Note that $2^{q_0} \ll \mathrm {min}(2^k,1)$ if C in equation (3.41) is large enough.

In what follows, we may assume that $|\xi _0|\approx 2^k$ , for otherwise there is no stationary point on the support of equation (3.39), $|\phi _X'(\xi )| \gtrsim 2^k 2^{-3k^+}$ , and the proof of the statement is easier. In other words, for fixed $\xi _0$ , we may assume that there is a finite number of indexes k for which $I_k$ does not vanish.

Using the notation in equation (2.25), we decompose

(3.43) $$ \begin{align} I_k = \sum_{q \in [q_0,\infty) \cap {\mathbb Z}} I_{k,q}, \qquad I_{k,q}(t,X) := \int_{{\mathbb R}_+} e^{it ( \langle \xi \rangle - \xi X)} a(x,\xi) \, \varphi_q^{(q_0)}(\xi-\xi_0) \, \varphi_k(\xi) \, g(\xi)\, d\xi. \end{align} $$

Bounding the contribution to the sum over k of the term with $q = q_0$ is immediate. Let us then consider $q> q_0$ and note that on the support of the integral in equation (3.43), we have

(3.44) $$ \begin{align} |\phi_X'(\xi)| \approx |\xi-\xi_0| |\phi_X''| \approx 2^{q} 2^{-3k^+} \gtrsim 2^{-n/2} 2^{-(3/2)k^+}. \end{align} $$

Integrating by parts using $(it \phi _{X}')^{-1} \partial _{\xi } e^{it \phi _X} = e^{it \phi _X}$ , we obtain

(3.45) $$ \begin{align} \begin{aligned} & I_{k,q} = \frac{1}{it} \big[ J_{k,q}^{(1)} + J_{k,q}^{(2)} + J_{k,q}^{(3)} + J_{k,q}^{(4)} \big] , \\ & J_{k,q}^{(1)}(t,X) = \int_{{\mathbb R}_+} e^{it ( \langle \xi \rangle - \xi X)} \frac{\phi_X''}{(\phi_X')^2} a(x,\xi) \, \varphi_q(\xi-\xi_0) \, \varphi_k(\xi) \, g(\xi) \, d\xi, \\ & J_{k,q}^{(2)}(t,X) = -\int_{{\mathbb R}_+} e^{it ( \langle \xi \rangle - \xi X)} \frac{1}{\phi_X'} \, \partial_{\xi} a(x,\xi) \, \varphi_q(\xi-\xi_0) \, \varphi_k(\xi) \, g(\xi) \, d\xi, \\ & J_{k,q}^{(3)}(t,X) = - \int_{{\mathbb R}_+} e^{it ( \langle \xi \rangle - \xi X)} \frac{1}{\phi_X'} a(x,\xi) \, \partial_{\xi} \big[ \varphi_q(\xi-\xi_0) \, \varphi_k(\xi) \big] \, g(\xi) \, d\xi, \\ & J_{k,q}^{(4)}(t,X) = - \int_{{\mathbb R}_+} e^{it ( \langle \xi \rangle - \xi X)} \frac{1}{\phi_X'} a(x,\xi) \, \varphi_q(\xi-\xi_0) \, \varphi_k(\xi) \, \partial_{\xi} g(\xi) \, d\xi. \end{aligned} \end{align} $$

Using equations (3.42) and (3.44) and changing the index of summation $q \mapsto p+(3/2)k^+$ , we have

(3.46) $$ \begin{align} \begin{aligned} \Big| \sum_{q\geq q_0, k} J_{k,q}^{(1)}(t,X) \Big|& \lesssim \sum_{p \geq -n/2, k} \int_{{\mathbb R}_+} \frac{\langle \xi \rangle^3}{(\xi-\xi_0)^2} \, \varphi_{\sim p}\big( (\xi-\xi_0)2^{-(3/2)k^+} \big) \, \varphi_k(\xi) \, |g(\xi)| \, d\xi \\ & \lesssim t^{1/2} {\| \langle \xi \rangle^{3/2} g \|}_{L^{\infty}}. \end{aligned} \end{align} $$

For the second term, using $|\partial _{\xi } a|\leq 1$ , we can estimate

(3.47) $$ \begin{align} \begin{aligned} \Big| \sum_{q\geq q_0, k} J_{k,q}^{(2)}(t,X) \Big| &\lesssim \sum_{p \geq -n/2, k} \int_{{\mathbb R}_+} \frac{\langle \xi \rangle^3}{|\xi-\xi_0|} \, \varphi_{\sim p}\big( (\xi-\xi_0) 2^{-(3/2)k^+} \big) \, \varphi_k(\xi) \, |g(\xi)| \, d\xi \\ & \lesssim t^{1/2} {\| g \|}_{L^{\infty}}. \end{aligned} \end{align} $$

The third term in equation (3.45) is similar to the first one:

(3.48) $$ \begin{align} \begin{aligned} \Big| \sum_{q\geq q_0, k} J_{k,q}^{(3)}(t,X) \Big| \lesssim t^{1/2} {\| \langle \xi \rangle^{3/2} g \|}_{L^{\infty}}. \end{aligned} \end{align} $$

The upper bounds in equations (3.46)–(3.48), after being multiplied by $t^{-1}$ , are bounded by the first term on the right-hand side of equation (3.38).

For the last integral in equation (3.45), we want to distinguish cases depending on the location of $\xi $ relative to the frequency $\sqrt {3}$ . We insert cutoffs $\varphi _{\ell }^{(\ell _0)}(\xi -\sqrt {3})$ , for $\ell _0 := -\gamma n$ , and bound

(3.49) $$ \begin{align} \begin{aligned} & t^{-1} \big| J_{k,q}^{(4)} \big| \lesssim t^{-1} \big[ K_{\leq \ell_0} + \sum_{\ell_0 < \ell \leq 0} K_{\ell} + K_{>0} \big] \\ & K_{\ast}(t,X) = \int_{{\mathbb R}_+} \frac{1}{|\phi_X'|} |a(x,\xi)| \, \varphi_q(\xi-\xi_0) \, \varphi_k(\xi) \, \varphi_{\ast}\big(\xi-\sqrt{3}\big) \, |\partial_{\xi} g(\xi)| \, d\xi. \end{aligned} \end{align} $$

The first term can be estimated as follows:

$$ \begin{align*} t^{-1} \sum_{q> q_0} \big| K_{\leq \ell_0}(t,X) \big| & \lesssim t^{-1} \sum_{q>q_0} 2^{-q} \cdot 2^{\mathrm{min}(q,-{\gamma} n)/2} \cdot {\big\| \varphi_{\leq -{\gamma} n} \big(\xi-\sqrt{3}\big)\partial_{\xi} g \big\|}_{L^2} \\ & \lesssim t^{-1} \sum_{q>q_0} 2^{-q/2} 2^{({\beta} {\gamma} +\alpha) n} {\| \chi_{\leq 0,\sqrt{3}} \, \partial_{\xi} g \|}_{W_T} \\ & \lesssim t^{-1} 2^{-q_0/2} t^{{\beta}{\gamma} +\alpha} {\| \chi_{\leq 0,\sqrt{3}} \, \partial_{\xi} g \|}_{W_T}, \end{align*} $$

consistently with equation (3.38), since we must have $|k|\leq 5$ and $2^{q_0} \approx t^{-1/2}$ . The last term in equation (3.49) can be estimated similarly:

(3.50) $$ \begin{align} \begin{aligned} t^{-1} \sum_{q>q_0} \big| K_{>0}(t,X) \big| & \lesssim t^{-1} \sum_{q>q_0} 2^{3k^+-q} \cdot 2^{\mathrm{min}(q,k)/2} \cdot {\big\| \varphi_k \varphi_{>0} \big(\xi-\sqrt{3}\big) \partial_{\xi} g \big\|}_{L^2} \\ & \lesssim t^{-3/4} 2^{(5/4)k^+} \cdot t^{\alpha} {\| \chi_{> 0,\sqrt{3}} \langle \xi \rangle \partial_{\xi} g \|}_{W_T}. \end{aligned} \end{align} $$

Upon summing over $2^k \leq t^{1/6}$ the right-hand side of equation (3.50), we obtain a contribution bounded by the second term on the right-hand side of equation (3.38), since $-3/4 + (5/4) (1/6) + \alpha < -3/4 + \alpha + \beta \gamma $ with our choice of parameters in equation (2.31) (provided, for example, that $\beta ',\gamma ' < 1/24$ ).

Finally, we estimate

$$ \begin{align*} t^{-1} \big| K_{\ell}(t,X) \big| & \lesssim t^{-1} 2^{-q} \cdot 2^{\mathrm{min}(q,\ell)/2} \cdot {\big\| \varphi_{\ell}\big(\xi-\sqrt{3}\big)\partial_{\xi} g \big\|}_{L^2} \\ & \lesssim t^{-1} 2^{-q/2} \cdot 2^{-{\beta} \ell} 2^{\alpha n} {\| \chi_{\leq 0,\sqrt{3}} \, \partial_{\xi} g \|}_{W_T}, \end{align*} $$

and, using again that $|k|\leq 5$ for $\ell \leq 0$ , we see that this contributions can be summed over $q>q_0$ with $2^{q_0} \gtrsim t^{-1/2}$ , and $\ell> \ell _0$ with $2^{\ell _0} \gtrsim t^{-{\gamma }}$ , and be bounded by the second term on the right-hand side of equation (3.38).

4.1 The structure of the quadratic spectral distribution

Recall that we denote, for a function f,

$$ \begin{align*}f_+(x) = f(x) \quad \mbox{and} \quad f_-(x) = \overline{f(x)}. \end{align*} $$

Proposition 4.1. Under the assumptions on $V=V(x)$ and $a=a(x)$ in Theorem 1.1, there exists a tempered distribution $\mu _{\iota _1 \iota _2} \in \mathcal {S}'({\mathbb R}^3)$ for $\iota _1,\iota _2 \in \{+,-\}$ such that, if $f,g \in \mathcal {S}$ ,

$$ \begin{align*}\widetilde{\mathcal{F}} \left[ a(x) f_{\iota_1} g_{\iota_2} \right] (\xi) = \iint \widetilde{f}_{\iota_1} (\eta) \widetilde{g}_{\iota_2}(\sigma) \mu_{\iota_1 \iota_2} (\xi,\eta,\sigma) \,d\eta \,d\sigma. \end{align*} $$

The distribution $\mu _{\iota _1 \iota _2}$ can be decomposed into

(4.1) $$ \begin{align} \begin{aligned} 2\pi \mu_{\iota_1 \iota_2} (\xi,\eta,\sigma) = \mu_{\iota_1 \iota_2}^S(\xi,\eta,\sigma) + \mu_{\iota_1 \iota_2}^R(\xi,\eta,\sigma), \end{aligned} \end{align} $$

where the following hold:

• • The ‘ $\mathrm {singular}$ ’ part of the distribution can be written as

  (4.2) $$ \begin{align} \mu_{\iota_1 \iota_2}^S(\xi,\eta,\sigma) = \mu_{\iota_1 \iota_2} ^{S,-}(\xi,\eta,\sigma) + \mu_{\iota_1 \iota_2} ^{S,+}(\xi,\eta,\sigma), \end{align} $$
  with $\epsilon \in \{+,-\}$ ,
  (4.3) $$ \begin{align} \begin{aligned} \mu_{\iota_1 \iota_2}^{S,\epsilon} (\xi,\eta,\sigma) & := \ell_{\epsilon\infty} \sum_{\lambda,\mu,\nu \in \{+,-\}} a_{\substack{- \iota_1 \iota_2 \\ \lambda \mu \nu}}^{\epsilon} (\xi,\eta, \sigma) \, \Big[\sqrt{\frac \pi 2} \delta(p) + \epsilon \mathrm{p.v.} \frac{\widehat \phi(p)}{ip} \Big], \\ p &:= \lambda \xi - \iota_1 \mu \eta - \iota_2 \nu \sigma, \end{aligned} \end{align} $$
  where $\phi $ is a smooth, even, real-valued, compactly supported function with integral one; the coefficients are given by
  (4.4) $$ \begin{align} a_{\substack{\iota_0 \iota_1 \iota_2 \\ \lambda \mu \nu}}^{\epsilon}(\xi,\eta,\sigma) = \mathbf{a}_{\lambda,\iota_0}^{\epsilon}(\xi) \mathbf{a}_{\mu,\iota_1}^{\epsilon}(\eta) \mathbf{a}_{\nu,\iota_2}^{\epsilon}(\sigma) \qquad \mbox{with} \qquad \mathbf{a}_{\mu,\iota}^{\epsilon} = \big( \mathbf{a}_{\mu}^{\epsilon} \big)_{\iota} \end{align} $$
  and
  (4.5) $$ \begin{align} \left\{ \begin{array}{l} \mathbf{a}^-_+(\xi) = \mathbf{1}_+(\xi) + \mathbf{1}_-(\xi) T(-\xi) \\ \mathbf{a}^-_-(\xi) = \mathbf{1}_+(\xi) R_-(\xi) \end{array} \right. \qquad \left\{ \begin{array}{l} \mathbf{a}^+_+(\xi) = T(\xi) \mathbf{1}_+(\xi) + \mathbf{1}_-(\xi) \\ \mathbf{a}^+_-(\xi) = \mathbf{1}_-(\xi) R_+(-\xi). \end{array} \right. \end{align} $$

• • The ‘ $\mathrm {regular}$ ’ part of the distribution $\mu ^R_{\iota _1\iota _2}$ can be written as a linear combination of the form

  (4.6) $$ \begin{align} \mu_{\iota_1 \iota_2}^R(\xi,\eta,\sigma) = \sum_{\epsilon_1,\epsilon_2,\epsilon_3 \in \{+,-\}} \mathbf{1}_{\epsilon_1}(\xi)\mathbf{1}_{\epsilon_2}(\eta)\mathbf{1}_{\epsilon_3}({\sigma}) \mathfrak{r}_{\epsilon_1\epsilon_2\epsilon_3}(\xi,\eta,\sigma), \end{align} $$
  where the symbols $\mathfrak {r}_{\epsilon _1\epsilon _2\epsilon _3}: {\mathbb R}^3 \rightarrow {\mathbb C}$ are smooth and satisfy, for any nonnegative integer N and $a,b,c$ ,
  (4.7) $$ \begin{align} \begin{aligned} \big|\partial_{\xi}^a \partial_{\eta}^b \partial_{\sigma}^c \mathfrak{r}_{\epsilon_1\epsilon_2\epsilon_3}(\xi,\eta,{\sigma}) \big| \lesssim \langle \inf_{\mu,\nu} |\xi - \mu \eta - \nu \sigma| \rangle^{-N}. \end{aligned} \end{align} $$

Proof. We proceed in a few steps.

The Fourier transform of $(\chi _{\pm })^3$ . By the choice in equation (3.25) of $\chi _{-}$ , $ \partial _{x} (\chi _-)^3$ is a $\mathcal {C}^{\infty }_{c}$ function, which we can write as $ \partial _{x} (\chi _-)^3= \phi ^o - \phi ^e$ , where $\phi ^o$ and $\phi ^e$ are, respectively, odd and even and $\mathcal {C}^{\infty }_{c}$ . Furthermore, since $\phi ^o$ is odd, we can write $\phi ^o = \partial _{x} \psi $ , where $\psi \in \mathcal {C}^{\infty }_{c}$ and $\psi $ is even. We have thus obtained that

$$ \begin{align*} (\chi_-)^3= \psi + \int_{x}^{+ \infty} \phi^e (y)\, dy = \psi + \phi^e \ast \mathbf{1}_-, \qquad \int_{\mathbb R}\phi^e (y)\,dy =1, \end{align*} $$

where we denoted by $\mathbf {1}_\pm $ the characteristic function of $\{\pm x> 0\}$ . Taking the Fourier transform and using the classical formulas

(4.8) $$ \begin{align} \widehat{f \ast g} = \sqrt{2\pi} \widehat{f} \cdot \widehat{g}, \qquad \widehat{1} = \sqrt{2\pi} \delta_0, \qquad \widehat{ \mbox{sign} x} = \mathrm{p.v.} \sqrt{\frac{2}{\pi}}\frac{1}{i\xi}, \end{align} $$

we see that $\widehat {\mathbf {1}_-} = \sqrt {\dfrac {\pi }{2}} \delta - \dfrac {1}{\sqrt {2\pi }} \mathrm {p.v.}\dfrac {1}{i\xi }$ , and therefore, since $\widehat {\phi ^e}(0) = \dfrac {1}{\sqrt {2\pi }}$ ,

$$ \begin{align*} \widehat{(\chi_-)^3} - \widehat{\psi} = \widehat{\mathcal{F}} \big(\phi^e \ast \mathbf{1}_- \big) = \sqrt{2\pi} \widehat{\mathbf{1}_-}(\xi) \widehat{\phi^e}(\xi) = \sqrt{\frac{\pi}{2}} \delta(\xi) - \mathrm{p.v.}\frac{\widehat{\phi^e}(\xi)}{i\xi}. \end{align*} $$

Since $\chi _+(-x) = \chi _-(x)$ , this implies a corresponding formula for $\chi _+$ . To summarise, setting $\phi = \phi ^e$ ,

(4.9) $$ \begin{align} \widehat{(\chi_{\pm})^3}(\xi) = \sqrt{\frac{\pi}{2}} \delta(\xi) \pm \mathrm{p.v.} {\widehat{\phi}(\xi)\over i\xi} + \widehat{\psi}(\xi). \end{align} $$

The regularization step. Consider for simplicity the case $\iota _1 = \iota _2 = +$ . If $f,g,h \in \mathcal {S}$ , denoting w a cutoff function,

(4.10) $$ \begin{align} \nonumber \langle \widetilde{a(x)fg}(\xi) ,\widetilde{h}(\xi) \rangle & = \iint \overline{\psi(x,\xi)} a(x) \int \widetilde{f}(\eta) \psi(x,\eta)\,d\eta \int \widetilde{g}(\sigma) \psi(x,\sigma)\,d\sigma \,dx \, \overline{\widetilde{h}(\xi)} \,d\xi \\ \nonumber & = \lim_{R \to \infty} \iint a(x) w(x/R) \overline{\psi(x,\xi)} \int \widetilde{f}(\eta) \psi(x,\eta)\,d\eta \int \widetilde{g}(\sigma) \psi(x,\sigma)\,d\sigma \,dx \, \overline{\widetilde{h}(\xi)} \,d\xi \\ \nonumber & = \lim_{R \to \infty} \iiint \widetilde{f}(\eta) \widetilde{g}(\sigma) \overline{\widetilde{h}(\xi)} \Big( \int a(x) w(x/R) \overline{\psi(x,\xi)} \psi(x,\eta) \psi(x,\sigma) \,dx \Big) \,d\eta \,d\sigma \,d\xi \\ & = \iiint \widetilde{f}(\eta) \widetilde{g}(\sigma) \overline{\widetilde{h}(\xi)} \mu_{++} (\xi,\eta,\sigma) \,d\eta \,d\sigma \,d\xi, \end{align} $$

where $\mu _{++}$ is defined as the limit in the sense of (tempered) distributions

(4.11) $$ \begin{align} \mu_{++} (\xi,\eta,\sigma) = \lim_{R \to \infty} \int a(x) w(x/R) \overline{\psi(x,\xi)} \psi(x,\eta) \psi(x,\sigma) \,dx. \end{align} $$

Note that while the limit in equation (4.11) can be easily seen to exist in the sense of (tempered) distribution, the limit leading to equation (4.10) needs to be understood in a different topology. In fact, although $\mu _{++}$ is a tempered distribution, $\widetilde {f}$ is not a Schwartz function, even if f is a Schwartz function: $\widetilde {f}$ might not be smooth, or may even be discontinuous, at zero; see equations (3.22) and (3.23). Nevertheless, one can still make sense rigorously of the limit and the pairing in equation (4.10) for $f,g,h \in \mathcal {S}$ . It actually suffices to consider $\widetilde {f},\widetilde {g},\widetilde {h} \in L^1\cap L^{\infty }$ , for example.

First, let us see that $\mu _{++}$ can be integrated against $\widetilde {f}(\eta ) \widetilde {g}(\sigma ) \overline {\widetilde {h}(\xi )}$ provided that $\widetilde {f}$ , $\widetilde {g}$ and $\widetilde {h}$ are in $L^1 \cap L^2$ (which is the case if $f,g,h \in \mathcal {S}$ ). Indeed, we will see that, up to more regular terms, $\mu _{++}$ is a linear combination of $\delta (p)$ and $\mathrm {p.v.} \frac {1}{p}$ distributions, where p is as in equation (4.3), with piecewise smooth coefficients in the variables $\xi ,\eta $ and ${\sigma }$ . The coefficients do not matter, so it suffices to look at the cases $\mu _{++} = \delta (\xi - \eta - \sigma )$ or $\mathrm {p.v.} \frac {1}{\xi - \eta - \sigma }$ (since the signs $\lambda , \mu , \nu $ in the definition of p also are not relevant). Then in the case of the $\delta $ distribution, we have

$$ \begin{align*}\iiint \delta(\xi - \eta - \sigma) \widetilde{f}(\eta) \widetilde{g}(\sigma) \overline{\widetilde{h}(\xi)} d\eta \, d\sigma \,d\xi = \iint \widetilde{f}(\eta) \widetilde{g}(\sigma) \overline{\widetilde{h}(\eta + \sigma)} d\eta \, d\sigma, \end{align*} $$

which is well defined by the Cauchy-Schwarz inequality if $\widetilde {f}, \widetilde {g}, \widetilde {h} \in L^1 \cap L^2$ . In the case of the $\mathrm {p.v.} \frac {1}{p}$ distribution, denoting by H the (standard) Hilbert transform, we have

$$ \begin{align*}\iiint \mathrm{p.v.} \frac{1}{\xi - \eta - \sigma} \widetilde{f}(\eta) \widetilde{g}(\sigma) \overline{\widetilde{h}(\xi)} d\eta \, d\sigma \,d\xi = \iint \widetilde{f}(\eta) \widetilde{g}(\sigma) \overline{[H \widetilde{h}](\eta + \sigma)} d\eta \, d\sigma, \end{align*} $$

which is well defined by the boundedness of the Hilbert transform on $L^2$ , and the Cauchy-Schwarz inequality.

Second, to justify the limit in equation (4.10), let us split

$$ \begin{align*}\widetilde{f} = \widetilde{f} (1 - \chi(\cdot / \epsilon) ) + \widetilde{f} \chi(\cdot / \epsilon) = \widetilde{f_{1,\epsilon}} + \widetilde{f_{2,\epsilon}}, \end{align*} $$

where $\chi $ is a smooth cutoff function equal to $1$ in a neighbourhood of $0$ , and similarly for g and h. We can then write

(4.12) $$ \begin{align} \langle \widetilde{{\mathcal{F}}}\big( a(x)w(x/R) fg\big)(\xi), \widetilde{h}(\xi) \rangle & = \langle \widetilde{{\mathcal{F}}}\big( a(x)w(x/R) f_{1,\epsilon} g_{1,\epsilon}\big)(\xi),\widetilde{h}_{1,\epsilon}(\xi) \rangle \qquad\qquad\quad \end{align} $$

(4.13) $$ \begin{align} &\phantom{\qquad\langle \widetilde{{\mathcal{F}}}\big( a(x)w(x/R) fg\big)(\xi), \widetilde{h}(\xi) \rangle} + \langle \widetilde{{\mathcal{F}}}\big(a(x)w(x/R)f_{2,\epsilon} g_{1,\epsilon}\big)(\xi),\widetilde{h}_{1,\epsilon}(\xi) \rangle + \{ \mbox{similar terms}\}. \end{align} $$

Here, the ‘similar terms’ contain at least one factor with an index $2$ , namely $g_{2,\epsilon }$ or $h_{2,\epsilon }$ . We claim that the terms in equation (4.13) are $O(\epsilon )$ remainder terms uniformly in R. If this is the case, then

$$ \begin{align*}\langle \widetilde{{\mathcal{F}}}\big( a(x)w(x/R)fg\big)(\xi), \widetilde{h}(\xi) \rangle = \langle \widetilde{{\mathcal{F}}} \big(a(x) w(x/R) f_{1,\epsilon} g_{1,\epsilon} \big)(\xi), \widetilde{h_{1,\epsilon}}(\xi) \rangle + O(\epsilon). \end{align*} $$

For the main term on the right-hand side above, the limit as in equation (4.10) is justified, since the functions involved are Schwartz. Therefore, one can let $R \to \infty $ first, then let $\epsilon \to 0$ , and obtain the desired formula.

To show that the remainder terms in equation (4.13) are $O(\epsilon )$ , we use the properties of the distorted Fourier transform:

$$ \begin{align*} \Big| \langle \widetilde{{\mathcal{F}}} \big( a(x)w(x/R)f_{2,\epsilon} g_{1,\epsilon} \big) (\xi), \widetilde{h_{1,\epsilon}}(\xi) \rangle \Big| & = \Big| \langle a(x) w(x/R) f_{2,\epsilon} g_{1,\epsilon}, h_{1,\epsilon} \rangle \Big| \\ &\lesssim {\| f_{2,\epsilon} \|}_{L^{\infty}} {\| g_{1,\epsilon} \|}_{L^2} \| h_{1,\epsilon} \|_{L^2} \lesssim {\| \widetilde{f_{2,\epsilon}} \|}_{L^1} {\| g_{1,\epsilon} \|}_{L^2} {\| h_{1,\epsilon} \|}_{L^2} \\ &\lesssim \epsilon {\| \widetilde{f} \|}_{L^{\infty}} {\| g \|}_{L^2} {\| h \|}_{L^2} \lesssim \epsilon. \end{align*} $$

In the following, we simply denote

$$ \begin{align*}\mu_{\iota_1 \iota_2} (\xi,\eta,\sigma) = \int a(x) \overline{\psi(x,\xi)} \psi_{\iota_1}(x,\eta) \psi_{\iota_2}(x,\sigma) \, dx. \end{align*} $$

Decomposition of the quadratic spectral distribution. We can write $\mu _{\iota _1\iota _2}$ as a sum of terms of the form

(4.14) $$ \begin{align} \frac{1}{(2\pi)^{3/2}} \int a(x) \overline{\psi^A(x,\xi)}\psi_{\iota_1}^B(x,\eta) \psi_{\iota_2}^C(x,\sigma) \, dx, \qquad A,B,C \in \{S,R\}, \end{align} $$

where we are using our main decomposition of $\psi $ in equation (3.28).

The singular part $\mu _S$ . The main singular component comes from part of the contribution to equation (4.14) with $A,B,C = S$ . The decomposition equation (3.29) can be written under the form

(4.15) $$ \begin{align} \begin{aligned} \psi^S(x,\xi)& = \chi_+(x) \sum_{\lambda \in \{+,-\} } \mathbf{a}_{\lambda}^+(\xi) e^{i \lambda \xi x} + \chi_-(x) \sum_{\lambda \in \{+,-\} } \mathbf{a}_{\lambda}^-(\xi) e^{i \lambda \xi x} \\ & =: \chi_+(x) \psi^{S,+}(x,\xi) + \chi_-(x) \psi^{S,-}(x,\xi), \end{aligned} \end{align} $$

where

(4.16) $$ \begin{align} \mathbf{a}_{\lambda}^-(\xi) = \left\{ \begin{array}{llll} 1 & \mbox{if} \qquad \lambda = + & & \mbox{and} \qquad \xi> 0, \\ R_-(\xi) & \mbox{if} \qquad \lambda = - & & \mbox{and} \qquad \xi > 0, \\ T(-\xi) & \mbox{if} \qquad \lambda= + & & \mbox{and} \qquad \xi < 0, \\ 0 & \mbox{if} \qquad \lambda = - & & \mbox{and} \qquad \xi < 0, \end{array} \right. \end{align} $$

and

(4.17) $$ \begin{align} \mathbf{a}_{\lambda}^+(\xi) = \left\{ \begin{array}{llll} T(\xi) & \mbox{if} \qquad \lambda = + & & \mbox{and} \qquad \xi> 0, \\ 0 & \mbox{if} \qquad \lambda = - & & \mbox{and} \qquad \xi > 0, \\ 1 & \mbox{if} \qquad \lambda= + & & \mbox{and} \qquad \xi < 0, \\ R_+(-\xi) & \mbox{if} \qquad \lambda = - & & \mbox{and} \qquad \xi < 0, \end{array} \right. \end{align} $$

or, equivalently,

(4.18) $$ \begin{align} \begin{aligned} & \mathbf{a}^-_+(\xi) = \mathbf{1}_+(\xi) + \mathbf{1}_-(\xi) T(-\xi) \\ & \mathbf{a}^-_-(\xi) = \mathbf{1}_+(\xi) R_-(\xi) \\ & \mathbf{a}^+_+(\xi) = T(\xi) \mathbf{1}_+(\xi) + \mathbf{1}_-(\xi) \\ & \mathbf{a}^+_-(\xi) = \mathbf{1}_-(\xi) R_+(-\xi). \end{aligned} \end{align} $$

Consider the terms in equation (4.14) with $A,B,C = S$ and such that in each decomposition of $\psi ^S$ there are only contributions containing $\chi _+$ (that is, $\psi ^{S,+}$ ) or $\chi _-$ (that is, $\psi ^{S,-}$ ). We can write this as

(4.19) $$ \begin{align} \begin{aligned} & \frac{1}{\sqrt{2\pi}} \int_{\mathbb R} a(x) \chi_\pm^3(x) \overline{\psi^{S,\pm}(x,\xi)} \psi^{S,\pm}_{\iota_1}(x,\eta) \psi^{S,\pm}_{\iota_2}(x,{\sigma}) \,dx \\ &\quad = \frac{1}{\sqrt{2\pi}} \sum_{\lambda,\mu,\nu \in \{\pm \}} \int_{\mathbb R} a(x) \chi_\pm^3(x) \, {a}^{\pm}_{\substack{- \iota_1 \iota_2 \\ \lambda \mu \nu}}(\xi,\eta,\sigma) \, \overline{e^{\lambda i\xi x}} e^{\iota_1 \mu i\eta x}e^{\iota_2 \nu i \sigma x} \, dx, \\ &\quad = \sum_{\lambda,\mu,\nu \in \{\pm\}} \widehat{{\mathcal{F}}}\big(a(x)(\chi_{\pm})^3\big)(\lambda \xi - \iota_1 \mu \eta - \iota_2 \nu \sigma) \, a^{\pm}_{\substack{- \iota_1 \iota_2 \\ \lambda \mu \nu}}(\xi,\eta,\sigma). \end{aligned} \end{align} $$

We then write $a(x)(\chi _{\pm })^3 = \ell _{\pm \infty }(\chi _{\pm })^3 + (a(x)-\ell _{\pm \infty })(\chi _{\pm })^3$ , where this last function is Schwartz. Using the formula in equation (4.9) for $\widehat {(\chi _{\pm })^3}$ , we see that the first terms in the right-hand side of equation (4.9), namely $\frac {\sqrt {\pi }}{2} \delta \pm \mathrm {p.v.} \frac {\widehat {\phi }}{i\xi }$ , make up the singular part of the distribution $\mu ^{S,\pm }$ in equation (4.3). The contribution corresponding to the last term, $\widehat {\psi }(\xi )$ , together with the one from $\widehat {{\mathcal {F}}} \big (a(x)-\ell _{\pm \infty }\big )$ , can be absorbed into the regular part of the distribution $\mu ^R$ ; see equation (4.27).

The regular part $\mu _R$ . The regular part $\mu _R$ contains all other contributions. These are of two main types: terms of the form in equation (4.14) when one of the indexes $A,B,C$ is R, or contributions where both $\chi _+$ and $\chi _-$ appear; see equation (4.15). More precisely, we can write

(4.20) $$ \begin{align} \mu_{\iota_1 \iota_2}^R(\xi,\eta,\sigma) = \mu_{\iota_1 \iota_2}^{R1}(\xi,\eta,\sigma) + \mu_{\iota_1 \iota_2}^{R2}(\xi,\eta,\sigma), \end{align} $$

where, if we let $X_R = \{ (A_1,A_2,A_3) \, : \, \exists \, j=1,2,3 \,\, \mbox { s.t.} \, A_j = R\}$ ,

(4.21) $$ \begin{align} \mu_{\iota_1 \iota_2}^{R1}(\xi,\eta,\sigma) & := \sum_{(A,B,C) \in X_R} \int a(x) \, \overline{\psi^A(x,\xi)}\psi^B_{\iota_1}(x,\eta) \psi^C_{\iota_2}(x,\sigma) \, dx \end{align} $$

and

(4.22) $$ \begin{align} \begin{aligned} \mu_{\iota_1 \iota_2}^{R2}(\xi,\eta,\sigma) := \sum_{A,B,C = S} \int a(x) \, \overline{\psi^A(x,\xi)} \psi^B_{\iota_1}(x,\eta) \psi^C_{\iota_2}(x,\sigma) \, dx - \mu^S_{\iota_1\iota_2} (\xi,\eta,\sigma). \end{aligned} \end{align} $$

In the remainder of the proof, we verify the properties in equations (4.6)–(4.7) for equations (4.21)–(4.22).

To understand equation (4.21), we start by looking at the case $A=R$ and $B,C=S$ . We restrict our analysis to $\xi>0$ (see equation (3.30)); $\xi <0$ can be treated in the same way. According to equation (3.30), this gives the terms

(4.23) $$ \begin{align} \begin{aligned} & \int a(x) \overline{\chi_+(x) T(\xi) (m_+(x,\xi)-1) e^{i\xi x} } \psi^S(x,\eta) \psi^S(x,\sigma) \, dx \\ &\quad + \int a(x) \overline{\chi_-(x) \big[ (m_-(x,-\xi) - 1)e^{i\xi x} + R_-(\xi)(m_-(x,\xi) - 1) e^{-ix\xi} \big]} \psi^S(x,\eta) \psi^S(x,\sigma) \, dx. \end{aligned} \end{align} $$

Let us look at the first term above and only at the contributions to $\psi ^S$ coming from $\psi ^{S,+}$ (see equation (4.15)): that is,

(4.24) $$ \begin{align} R_{\mu\nu}(\xi,\eta,{\sigma}) & := \overline{T(\xi)} \mathbf{a}_\mu^+(\eta) \mathbf{a}_\nu^+(\sigma) \int a(x) \chi_+^3(x) (\overline{m_+(x,\xi)}-1) \, e^{-i\xi x} e^{i\mu \eta x} e^{i\nu{\sigma} x} \, dx. \end{align} $$

Notice that the coefficients in front of the integral are products of indicator functions and smooth functions, consistent with equations (4.6)–(4.7). Dropping the irrelevant signs $\mu ,\nu $ , it then suffices to treat

(4.25) $$ \begin{align} R(\xi,\eta,{\sigma}) & := \int a(x) \chi^3_+(x) (\overline{m_+(x,\xi)}-1) \, e^{-i x\xi} \, e^{ix(\eta+{\sigma})} \, dx. \end{align} $$

We use the fast decay and smoothness of $m_+-1$ from Lemma 3.1 to integrate by parts. More precisely, for any M, we write

$$ \begin{align*} \big| R(\xi,\eta,{\sigma}) \big| & = \Big| \frac{1}{[i(\xi-\eta-{\sigma})]^M} \int e^{-i x\xi} \, e^{ix(\eta+{\sigma})} \partial_x^M \big[ a(x) \chi_+^3(x) \overline{(m_+(x,\xi)-1)} \big] \, dx \Big| \\ & \lesssim \frac{1}{|\xi-\eta-{\sigma}|^M} \sum_{0 \leq \alpha \leq M} \int_{x\geq -1} \big| \partial_x^{\alpha} (m_+(x,\xi)-1) \big|\, dx \lesssim \frac{1}{|\xi-\eta-{\sigma}|^M}, \end{align*} $$

having used equation (3.4) for the last inequality.

We have therefore bounded the expression in equation (4.24) by the right-hand side of equation (4.7) for $a=b=c=0$ .

To estimate the derivatives, notice that applying multiple $\eta $ - and ${\sigma }$ -derivatives is harmless since these result in additional powers of x, but $m_+-1$ decays as fast as desired. Similarly, again from equation (3.4), we see that $\partial _{\xi }$ derivatives can also be handled easily since $\partial _{\xi }^{\alpha } m$ decays fast as well. Notice that the second line in equation (4.23) can be treated exactly like the first one, using the properties of $m_-$ from equation (3.4). All the other terms in equation (4.21) can be treated the same way.

Let us look at the remaining piece in equation (4.22). We can write, according to the notation in equation (4.15) and the definition in equation (4.19),

(4.26) $$ \begin{align} \begin{aligned} \mu_{\iota_1 \iota_2}^{R2}(\xi,\eta,\sigma) = \sum \int a(x) \, \chi_{\epsilon_1}(x)\chi_{\epsilon_2}(x)\chi_{\epsilon_3}(x) \overline{\psi^{S,\epsilon_1}(x,\xi)} \psi^{S,\epsilon_2}(x,\eta) \psi^{S,\epsilon_3}(x,\sigma) \, dx, \qquad \end{aligned} \end{align} $$

where the sum is over $(\epsilon _1,\epsilon _2,\epsilon _3) \neq (+,+,+), (-,-,-)$ . In particular, this means that $a \, \chi _{\epsilon _1}\chi _{\epsilon _2}\chi _{\epsilon _3}$ is a smooth compactly supported function, which we denote by $\chi $ (omitting the dependence on the signs, which is not relevant here), and equation (4.26) is a linear combination of terms of the form

(4.27) $$ \begin{align} \begin{aligned} \int \chi(x) \overline{\mathbf{a}_\lambda^{\epsilon_1}(\xi) e^{i\lambda \xi x}} \mathbf{a}_\mu^{\epsilon_2}(\eta) e^{i\mu \eta x} \mathbf{a}_\nu^{\epsilon_3}({\sigma}) e^{i\nu {\sigma} x} \, dx = \widehat{\chi}(\lambda\xi -\mu\eta-\nu{\sigma}) \overline{\mathbf{a}_\lambda^{\epsilon_1}(\xi)}\mathbf{a}_\mu^{\epsilon_2}(\eta)\mathbf{a}_\nu^{\epsilon_3}({\sigma}). \end{aligned} \end{align} $$

The desired conclusion in equations (4.6)–(4.7) follows from the properties of the coefficients $\mathbf {a}_{\epsilon }^{\lambda }$ and the fact that $\widehat \chi $ is Schwartz.

4.2 Mapping properties for the regular part of the quadratic spectral distribution

The product operation $(f,g) \mapsto fg$ obviously satisfies Hölder’s inequality; but it is natural to ask about the mapping properties of the bilinear operators associated to the distributions $\mu ^S$ and $\mu ^R$

$$ \begin{align*}(f,g) \mapsto \widetilde{\mathcal{F}}^{-1} \int \mu^{S,R}(\xi,\eta,\sigma) \widetilde{f}(\eta) \widetilde{f}(\sigma) \,d\eta\,d\sigma. \end{align*} $$

The singular part $\mu ^S$ can be thought of as the leading order term; and indeed, it does satisfy Hölder’s inequality, and this is optimal. The regular part is lower-order in that it gains integrability ‘at $\infty $ ’, but it does not gain regularity. Thus, it can essentially be thought of as an operator of the type $(f,g) \mapsto F f g$ , where F is bounded and rapidly decaying. The following lemma gives a rigorous statement along these lines.

Lemma 4.2 Bilinear estimate for $\mu ^R$

Under the same assumptions and with the same notations as in Proposition 4.1, consider the measure $\mu ^R = \mu ^R_{\iota _1\iota _2}$ and the corresponding bilinear operator

(4.28) $$ \begin{align} M_R[a,b] := \widetilde{\mathcal{F}}^{-1} \iint \mu^R(\xi,\eta,{\sigma}) \widetilde{a}(\eta) \widetilde{b}({\sigma}) d\eta\,d{\sigma}. \end{align} $$

Then for all

$$ \begin{align*} p_1,p_2\in[2,\infty), \qquad \frac{1}{p_1}+\frac{1}{p_2} \leq \frac{1}{2} \end{align*} $$

it holds that

(4.29) $$ \begin{align} \begin{aligned} {\big\| M_R[a,b] \big\|}_{L^2} \lesssim {\| a \|}_{L^{p_1}} {\| b \|}_{L^{p_2}}. \end{aligned} \end{align} $$

Moreover, for $p_1,p_2$ as above and $p_3,p_4$ another pair satisfying the same assumptions, we have, for any integer $l\geq 0$ ,

(4.30) $$ \begin{align} \begin{aligned} {\big\| \langle \partial_x \rangle^l M_R[a,b] \big\|}_{L^2} \lesssim {\| \langle \partial_x \rangle^l a \|}_{L^{p_1}} {\| b \|}_{L^{p_2}} + {\| a \|}_{L^{p_3}} {\| \langle \partial_x \rangle^l b \|}_{L^{p_4}}. \end{aligned} \end{align} $$

Proof of Lemma 4.2

The starting point is the splitting of $\mu ^R$ in equations (4.20)–(4.22). We will omit the irrelevant signs $\iota _1\iota _2$ in what follows and just denote $\mu ^{R1,2}=\mu _{\iota _1 \iota _2}^{R1,2}$ . Also notice that in the definition of $M_R$ , we can replace all the distorted Fourier transforms by flat Fourier transforms, in view of the boundedness of the (adjoint) wave operator $\mathcal {W}^* := \widehat {{\mathcal {F}}}^{-1} \widetilde {{\mathcal {F}}}$ on $L^p$ , $p\in [2,\infty )$ ; see Proposition 3.9 and Theorem 3.10.

Proof of equation (4.29). From equation (4.21), we see that $\mu ^{R1}$ is a linear combination of terms of the form

(4.31) $$ \begin{align} \int \overline{\psi^A(x,\xi)}\psi^B(x,\eta) \psi^C(x,\sigma) \, dx, \end{align} $$

where at least one of the apexes $A,B$ or C is equal to R; recall the definition of $\psi ^S$ and $\psi ^R$ in equations (3.29) and (3.30). It suffices to look at the two cases $A=R$ or $B=R$ .

Let us first look at the case $A=R$ and further restrict our attention to $\xi>0$ and the contribution from $\chi _+$ ; all the other contributions can be handled the same way. We are then looking at the distribution

(4.32) $$ \begin{align} \mu_1(\xi,\eta,{\sigma}) := \int \overline{\chi_+(x) T(\xi) (m_+(x,\xi)-1) e^{i\xi x}} \psi^B(x,\eta) \psi^C(x,\sigma) \, dx, \qquad B,C = S \mbox{ or } R. \end{align} $$

The bilinear operator associated to it is

(4.33) $$ \begin{align} \begin{aligned} M_1[a,b] &= \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x} \iint \mu_1(\xi,\eta,{\sigma}) \widehat{a}(\eta) \widehat{b}({\sigma}) d\eta\,d{\sigma} \\ & = \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x}\! \int \! \chi_+(y) \overline{T(\xi) (m_+(y,\xi)-1)} e^{-i\xi y} \Big( \int_{\mathbb R} \widehat{a}(\eta) \psi^B(y,\eta) d\eta \Big) \Big( \int_{\mathbb R} \widehat{b}({\sigma}) \psi^C(y,\sigma) \, d{\sigma} \Big) \, dy. \end{aligned} \end{align} $$

If we define

(4.34) $$ \begin{align} u^A(x) := \int_{\mathbb R} \widehat{u}(\xi) \psi^A(x,\xi)\,d\xi, \qquad A=S,R, \end{align} $$

and the symbol $m(y,\xi ):= \langle y \rangle \chi _+(y) \overline {T(\xi ) (m_+(y,\xi )-1)}$ , we see that

$$ \begin{align*} {\| M_1[a,b] \|}_{L^2} & \lesssim {\Big\|\int_{\mathbb R} m(y,\xi) e^{-i\xi y} \cdot \langle y \rangle^{-1} \cdot a^B(y) \cdot b^C(y) \, dy \Big\|}_{L^2_{\xi}}. \end{align*} $$

In view of Lemmas 3.1 and 3.2, we see that $m=m(y,\xi )$ satisfies standard pseudo-differential symbol estimates and deduce that the associated operator is bounded $L^2\mapsto L^2$ . It follows that

(4.35) $$ \begin{align} \begin{aligned} {\| M_1[a,b] \|}_{L^2} & \lesssim {\| \langle y \rangle^{-1} \cdot a^B \cdot b^C \|}_{L^2_y} \lesssim {\| a^B \|}_{L^{p_1}} {\| b^C \|}_{L^{p_2}}. \end{aligned} \end{align} $$

The estimate in equation (4.35) gives us the right-hand side of equation (4.29), provided we show that $u \mapsto u^S , u^R$ as defined in equation (4.34) are bounded on $L^p$ , $p\in [2,\infty )$ . Since $u^S + u^R = u$ , it suffices to show ${\| u^S \|}_{L^p} \lesssim {\| u \|}_{L^p}$ . From the definition of $\psi ^S$ in equations (3.29) and (4.15)–(4.18), we see that this reduces to proving

(4.36) $$ \begin{align} {\Big\| \int_{\mathbb R} e^{i\lambda x\xi} \mathbf{a}_{\lambda}^{\pm}(\xi) \widehat{u}(\xi) \,d\xi \Big\|}_{L^p} \lesssim {\|u\|}_{L^p}. \end{align} $$

In view of the boundedness of the Hilbert transform, it is enough to obtain the same bound where the coefficients $\mathbf {a}_{\lambda }^{\pm }$ are replaced just by $T(\pm \xi )$ or $R_{\pm }(\mp \xi )$ . The desired bound then follows since $T(\pm \xi )-1$ and $R_{\pm }(\mp \xi )$ are $H^1$ functions (see equation (3.14)) so that their Fourier transforms are in $L^1$ .

Consider next the case $B=R$ . Again, without loss of generality, we may restrict our attention to $\eta>0$ and the contribution from $\chi _+$ : that is, we look at the measure

(4.37) $$ \begin{align} \mu_1'(\xi,\eta,{\sigma}) := \int \chi_+(x) \psi^S(x,\xi) \overline{T(\eta) (m_+(x,\eta)-1) e^{i\eta x}} \psi^C(x,\sigma) \, dx, \qquad C = S \mbox{ or } R. \end{align} $$

Letting the associated operator be

(4.38) $$ \begin{align} \begin{aligned} M_1'[a,b] &:= \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x} \iint \mu_1'(\xi,\eta,{\sigma}) \widehat{a}(\eta) \widehat{b}({\sigma}) \,d\eta\,d{\sigma} \\ & = \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x} \int \chi_+(y) \psi^S(y,\xi) \Big( \int_{\mathbb R} \widehat{a}(\eta) \overline{T(\eta) (m_+(y,\eta)-1) e^{i\eta y}} \, d\eta \Big) \Big( \int_{\mathbb R} \widehat{b}({\sigma}) \psi^C(y,\sigma) \, d{\sigma} \Big) \, dy, \end{aligned} \end{align} $$

we see that

$$ \begin{align*} {\| M_1'[a,b] \|}_{L^2} & \lesssim {\Big\| \langle y \rangle \int_{\mathbb R} \overline{T(\eta) (m_+(y,\eta)-1) e^{i\eta y}} \, \widehat{a}(\eta) \, d\eta \Big\|}_{L^{p_1}} {\| b^C \|}_{L^{p_2}} \end{align*} $$

having used that $\psi ^S$ defines a bounded PDO on $L^2$ , as we showed above. The desired conclusion in equation (4.29) then follows since $\langle x \rangle T(\eta ) (m_+(x,\eta )-1)$ is the symbol of a bounded PDO on $L^p$ , for $p\in (2,\infty )$ in view of Lemmas 3.1, 3.2 and standard results on PDOs; see, for example, [Reference Caldéron and Vaillancourt3].

We now analyse the $\mu ^{R,2}$ component from equation (4.22) by looking at the more explicit expression for it in equation (4.27). From this, we see that it suffices to look at bilinear operators of the form

(4.39) $$ \begin{align} M_2[a,b] := \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x} \iint_{{\mathbb R}\times{\mathbb R}} \widehat{\chi}(\lambda\xi -\mu\eta-\nu{\sigma}) \overline{\mathbf{a}_\lambda^{\epsilon_1}(\xi)}\mathbf{a}_\mu^{\epsilon_2}(\eta)\mathbf{a}_\nu^{\epsilon_3}({\sigma}) \widehat{a}(\eta) \widehat{b}({\sigma}) \,d\eta\,d{\sigma}, \end{align} $$

where $\chi $ is Schwartz. By boundedness of the Fourier multipliers $\mathbf {a}^{\epsilon }_{\lambda }$ ,

$$ \begin{align*} {\| M_2[a,b] \|}_{L^2} & \lesssim {\big\| \chi \cdot \widehat{{\mathcal{F}}}^{-1}\big( \mathbf{a}_\mu^{\epsilon_2} \widehat{a}\big) \widehat{{\mathcal{F}}}^{-1}\big( \mathbf{a}_\nu^{\epsilon_3}\widehat{b} \big) \big\|}_{L^2} \\ & \lesssim {\big\| \widehat{{\mathcal{F}}}^{-1}\big( \mathbf{a}_\mu^{\epsilon_2} \widehat{a}\big) \big\|}_{L^{p_1}} {\big\| \widehat{{\mathcal{F}}}^{-1}\big( \mathbf{a}_\nu^{\epsilon_3}\widehat{b} \big) \big\|}_{L^{p_2}} \lesssim {\| a \|}_{L^{p_1}} {\| b \|}_{L^{p_2}}. \end{align*} $$

Proof of equation (4.30). We proceed similarly to the proof of equation (4.29) and reduce to estimating derivatives of the bilinear operators $M_1, M_1^{\prime }$ and $M_2$ , respectively, defined in equation (4.33), equation (4.38) and (4.39);

Applying derivatives to $M_1$ gives

(4.40) $$ \begin{align} \begin{aligned} \partial_x^l M_1[a,b] & = \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x} \int_{\mathbb R} \chi_+(y) \overline{T(\xi) (m_+(y,\xi)-1)} \, (i\xi)^l e^{-i\xi y} a^A(y) \, b^B(y) \, dy. \end{aligned} \end{align} $$

Integrating by parts in y and distributing derivatives on $a^A$ , $b^B$ and $m_+-1$ gives a linear combination of terms of the form

(4.41) $$ \begin{align} \begin{aligned} M_{l_1,l_2}[a,b] := \widehat{{\mathcal{F}}}^{-1}_{\xi\mapsto x} \int_{\mathbb R} \overline{T(\xi)} \, \partial_y^{l_1} \big( \chi_+(y) (\overline{m_+(y,\xi)-1} ) \big) \, e^{-i\xi y} \cdot \partial_y^{l_2} \big(a^A(y) \, b^B(y) \big) \, dy \end{aligned} \end{align} $$

with $l_1+l_2=l$ . From Lemmas 3.1, 3.2, we see that $m_{l_1}(x,\xi ) := \overline {T(\xi )} \, \partial _x^{l_1} \big ( \chi _+(x) (\overline {m_+(x,\xi )-1} ) \big )$ gives rise to a standard PDO bounded on $L^2$ . Therefore, to bound in equation (4.41) by the right-hand side of equation (4.30), it suffices to use product Sobolev inequalities and ${\| \partial _x^l u^S \|}_{L^p} \lesssim {\| \langle \partial _x \rangle ^l u \|}_{L^p}$ , $p\in [2,\infty )$ , which follows from the inequality in equation (4.36) with $\partial _x^lu$ instead of u.

A similar argument can be used for $M_1^{\prime }$ : from equation (4.38), we see that x-derivatives become powers of $\xi $ , which in turn can be transformed to y-derivatives since $\psi ^S(y,\xi )$ is a linear combination of exponentials $e^{\pm i y\xi }$ by harmless $\xi $ -dependent coefficients; integrating by parts in y and using the boundedness on $L^p$ of the PDO with symbol $\langle x \rangle m_{l_1}(x,\xi ) $ gives the desired bound.

The argument for $M_2$ is straightforward, using that $T-1$ is a bounded multiplier and $\chi $ is Schwartz:

$$ \begin{align*} {\| \langle \partial_x \rangle^l M_2[a,b] \|}_{L^2} \lesssim {\Big\| \langle \partial_x \rangle^l \Big[ \chi \cdot \widehat{{\mathcal{F}}}^{-1}\big( a_\mu^{\epsilon_2} \widehat{a}\big) \widehat{{\mathcal{F}}}^{-1}\big( a_\nu^{\epsilon_3}\widehat{b} \big) \Big] \Big\|}_{L^2} \lesssim {\Big\| \langle \partial_x \rangle^l \Big[ \widehat{{\mathcal{F}}}^{-1}\big( a_\mu^{\epsilon_2} \widehat{a}\big) \widehat{{\mathcal{F}}}^{-1}\big( a_\nu^{\epsilon_3}\widehat{b} \big) \Big] \Big\|}_{L^q} \end{align*} $$

with $1/q=1/p_1+1/p_2$ , and we can use standard Gagliardo-Nirenberg-Sobolev inequalities and equation (4.36) to obtain equation (4.30).