frequency information carried by the input functions $f,g$ and $h$.616161This
is transparent in the “size-energy” formulation stated below in Proposition
2.1, Section 2.8.

The appeasement of this conflict is obtained through a “half-way” compromise:

  * •

The first action, from the $T_{m}$ side, is to substantially refine (96), in
particular increasing its symmetry by also producing the time-frequency
localization for $h\cdot w_{k,n,v}^{e}(\lambda)$.

  * •

The second action comes from the direction of the BHT analysis and relies on
modifying/adapting the original concepts of size and energy to the presence of
the weight $w_{k,n,v}^{e}(\lambda)$ and, consequently, to the distinct roles
played by $(f,g)$ and $h$ in our problem.

Now the first action is part of the transitional step to be discussed in the
next section, while the second action is part of the multi-scale time-
frequency approach that will be the subject of Section 2.8.

We end this section by providing another angle on the above mentioned conflict
that follows from inspecting the structure of $BC^{a}$: on the multiplier side
– see _e.g._ (LABEL:multm) – there is a merging between the bilinear Hilbert
transform behavior focusing on the Whitney decomposition relative to the
$\xi-\eta=0$ “singularity” and the Carleson-type behavior encapsulated in the
expression $e^{ic(\xi-\eta)^{\frac{a}{a-1}}\lambda(x)^{-\frac{1}{a-1}}}$.
These two operators end up on the multiplier side by acting on the same
variable $\xi-\eta$. Equivalently, on the physical space side, there is an
intertwinement between the structure of the input functions – $f(x-t)g(x+t)$ –
and that corresponding to the kernel $\frac{e^{i\lambda(x)t^{a}}}{t}$,
respectively. This phenomenon makes more difficult the process of identifying
a way to decouple/untwist the behaviors of $B$ and $C^{a}$ within $BC^{a}$,
which plays a key role in our approach.626262In contrast with this – see the
discussion within Section 1.2.3, II.4 – the decoupling process within the
analysis of the Bi-Carleson operator (33) is easier since in this latter
situation the BHT singularity and the Carleson-type singularity do not act on
the same variable.

###  2.7. (II) A transitional step: refinement of the single-scale estimate

The main task here is to obtain a trilinear dualized version of (96) with a
sharp time-frequency localization of the input functions $f,g$ and $h$ that
simultaneously quantifies the presence of the linearizing function $\lambda$
and separates the information carried by the input functions $f,g$ from the
information carried by $h$ and $w_{k,n,v}^{e}(\lambda)$.

The first key insight here is to notice that at a heuristic level one can
pretend that

(97)
$\underline{S}_{k,\ell,r}^{n,v,p^{\prime}}(f,g)(x):=S_{k,\ell,r}^{n,v,p^{\prime}}(f,g)(x)e^{-i2\ell
2^{am\over 2}(2^{{am\over 2}-k}x-p_{r})}\>\>\textrm{is constant
on}\>\underline{I}_{k,r}^{p^{\prime}}\>,$

where here we used the notation
$\displaystyle\underline{I}_{k,r}^{p^{\prime}}:=\Big{[}2^{k}(r-1)+\frac{p^{\prime}}{2^{am-k}},2^{k}(r-1)+\frac{p^{\prime}+1}{2^{am-k}}\Big{]}$.

As a consequence, for any positive $w_{k,n,v}\in L^{1}_{loc}(\mathbb{R})$, one
may think that

(98)
$\displaystyle\int_{\mathbb{R}}\underline{S}_{k,\ell,r}^{n,v,p^{\prime}}(f_{\ell+1},$
$\displaystyle g_{\ell-1})(x)e^{i2\ell 2^{am\over 2}(2^{{am\over
2}-k}x-p_{r})}h(x)\rho_{am-ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x)dx$
$\displaystyle\approx 2^{-\frac{am-k}{2}}\big{\langle}h\rho_{am-
ak}(\lambda)w_{k,n,v}^{e}(\lambda),\check{\varphi}_{am-k,2\ell-1}^{0,2^{am}(r-1)+p^{\prime}}\big{\rangle}$
$\displaystyle\hskip
50.00008pt\cdot\frac{\int_{\underline{I}_{k,r}^{p^{\prime}}}\underline{S}_{k,\ell,r}^{n,v,p^{\prime}}(f_{\ell+1},g_{\ell-1})(x)w_{k,n,v}(x)dx}{\int_{\underline{I}_{k,r}^{p^{\prime}}}w_{k,n,v}(x)dx}.$

After a Cauchy-Schwarz argument, one then concludes that

(99) $\displaystyle\big{|}\underline{\mathcal{L}}_{m,k,\ell,r}(f,g,h)\big{|}$
$\displaystyle\hskip 10.00002pt\lesssim\sum_{n,v\sim 2^{am\over
2}}\sum_{p^{\prime}\sim 2^{am}}\frac{\big{|}\langle h\rho_{am-
ak}(\lambda(\cdot))w_{k,n,v}^{e}(\lambda),\check{\varphi}_{am-k,2\ell-1}^{0,2^{am}(r-1)+p^{\prime}}\rangle\big{|}}{\big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\underline{I}_{k,r}^{p^{\prime}}}w_{k,n,v}(x)dx\big{)}^{\frac{1}{2}}}$
$\displaystyle\hskip
20.00003pt\cdot\left(\int_{\underline{I}_{k,r}^{p^{\prime}}}\big{|}\underline{S}_{k,\ell,r}^{n,v,p^{\prime}}(f_{\ell+1},g_{\ell-1})(x)\big{|}^{2}w_{k,n,v}(x)dx\right)^{\frac{1}{2}}.$

More precisely, (98) and (LABEL:eq:est:w:aver1) have to be understood on the
right hand side as a superposition of similar terms with fast decaying
amplitudes – that will be made more rigorous in Section 6.

As a quick remark, the liberty of choosing any positive $w_{k,n,v}\in
L^{1}_{loc}(\mathbb{R})$ in (LABEL:eq:est:w:aver1) will offer us great
flexibility later when analyzing the behavior of each of the terms involved in
decomposition (100) below.

Once at this point we will need to consider three key parameters636363Recall
relation (82).:

  * •

the “mass” of the weight $w_{k,n,v}(\lambda)$, roughly defined as
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\underline{I}_{k,r}^{p^{\prime}}}w_{k,n,v}(\lambda)(x)dx$
and encoding the behavior of the linearizing function $\lambda$;

  * •

the size distribution of the input function $f$; and

  * •

the size distribution of the input function $g$.

The behavior of these parameters will determine the partitioning of the set of
indices

$\mathcal{I}:=\Big{\\{}(p^{\prime},n,v)|p^{\prime}\sim 2^{am},n,v\sim
2^{am\over 2}\Big{\\}}$

into three components

$\mathcal{I}=\mathfrak{l}\cup\mathfrak{u}\cup\mathfrak{c}$

with $\mathfrak{l}$ the light mass set, $\mathfrak{u}$ the heavy mass-uniform
input distribution set and $\mathfrak{c}$ the heavy mass-clustered input
distribution set. This will reflect into the corresponding splitting

$\displaystyle\underline{\mathcal{L}}_{m,k,\ell,r}(f,g,h)$
$\displaystyle=\underline{\mathcal{L}}_{m,P}(f,g,h)$ (100)
$\displaystyle=\underline{\mathcal{L}}_{m,P}^{\mathfrak{l}}(f,g,h)+\underline{\mathcal{L}}_{m,P}^{\mathfrak{u}}(f,g,h)+\underline{\mathcal{L}}_{m,P}^{\mathfrak{c}}(f,g,h)\>,$

with each of the above forms analyzed accordingly:

  * •

the light mass component $\underline{\mathcal{L}}_{m,P}^{\mathfrak{l}}(f,g,h)$
is composed of the terms that correspond to the indices where the associated
mass of $w_{k,n,v}(\lambda)$ is small;

  * •

the heavy mass-uniform input component
$\underline{\mathcal{L}}_{m,P}^{\mathfrak{u}}(f,g,h)$ addresses precisely the
situation that captures the cancellation offered by the non-zero curvature and
relies in a key fashion on the estimate (96); and

  * •

the heavy mass-clustered input
$\underline{\mathcal{L}}_{m,P}^{\mathfrak{c}}(f,g,h)$ captures the situation
when the size of the input functions $f,g$ is larger than the expected average
value.

The conclusion of this analysis will be provided by Proposition 7.1, Section
7, which states that

(101)
$\underline{\mathcal{L}}_{m,P}^{\mathfrak{l}}(f,g,h)\lesssim|I_{P}|^{-\frac{1}{2}}f(P_{1})g(P_{2})h_{\mathfrak{l}}(P_{3}),$
(102) $\underline{\mathcal{L}}_{m,P}^{\mathfrak{u}}(f,g,h)\lesssim
2^{-\frac{\delta_{1}am}{2}}|I_{P}|^{-\frac{1}{2}}f(P_{1})g(P_{2})h_{\mathfrak{u}}(P_{3}),$

for some $\delta_{1}>0$, and

(103)
$\underline{\mathcal{L}}_{m,P}^{\mathfrak{c}}(f,g,h)\lesssim|I_{P}|^{-\frac{1}{2}}f(P_{1})g(P_{2})h_{\mathfrak{c}}(P_{3}),$

where here $f(P_{1})$ and $g(P_{2})$ represent local $\ell^{2}$-sums involving
the Fourier coefficients of the corresponding function adapted to the family
$\\{s\\}_{s\in\mathbb{S}(\mathbb{P})}$ while $h_{*}(P_{3})$ quantifies a
suitable weighted version of similar local $\ell^{2}$-sums associated to $h$
and restricted to $*\in\\{\mathfrak{l},\mathfrak{u},\mathfrak{c}\\}$ and
involving both the function $h$ itself and the weight
$w_{\cdot,\cdot,\cdot}(\lambda)$. The decay obtained in (102) is a direct
manifestation of the non-zero curvature analysis performed at stage (I), while
for (101) and (103) the $m$-exponential decay is hidden in the structure of
$h_{*}(P_{3})$.

This puts an end to the role played by the non-zero curvature in our problem
and concurrently to the first part of our proof.

###  2.8. (III) The low resolution, multi-scale analysis

In this final part of the proof’s exposition we address the underlying
modulation invariance structure of our problem encoded in (P2). As already
explained in Section 2.3, our plan is to appeal to the time-frequency
techniques originating in the work of Lacey and Thiele as modeled by the
following general abstract result646464For the definitions of mass and energy
(and other related concepts) please see Section 7.656565This general abstract
result stated above was necessary when treating objects having a structure
more complex than the standard BHT as revealed by the following examples: 1)
the Bi-est operator regarded as BHT composed with another BHT, or, 2) the Bi-
Carleson operator $Bi\text{-}C$ already mentioned in Section 1.2.3 (see (33))
that can be seen as a composition between BHT and the Carleson operator.:

######  Proposition 2.1 (Proposition 6.5 of [82]).

Let $\mathbb{P}\subset{{\mathbb{P}}_{\textrm{BHT}}}$ be a finite collection of
tri-tiles and $(a_{P_{j}})_{P\in\mathbb{P}}$ a sequence of complex numbers.
Then for any $\theta_{1},\theta_{2},\theta_{3}\in[0,1)$ with
$\theta_{1}+\theta_{2}+\theta_{3}=1$

$\displaystyle\big{|}\sum_{P\in\mathbb{P}}\frac{1}{|I_{P}|^{\frac{1}{2}}}a_{P_{1}}a_{P_{2}}a_{P_{3}}\big{|}\lesssim\prod_{j=1}^{3}\big{(}\mathrm{Size\,}^{\mathbb{P}}_{j}((a_{P_{j}})_{P\in\mathbb{P}})\big{)}^{\theta_{j}}\big{(}\mathrm{Energy\,}^{\mathbb{P}}_{j}((a_{P_{j}})_{P\in\mathbb{P}})\big{)}^{1-\theta_{j}}.$

Defining now
$\underline{\mathcal{L}}^{*}_{m}(f,g,h):=\sum_{P\in{{\mathbb{P}}_{\textrm{BHT}}}}\underline{\mathcal{L}}_{m,P}^{*}(f,g,h)$
we insert (101)-(103) into the above proposition – with the latter used for
the most part as a black-box – where for
$*\in\\{\mathfrak{l},\mathfrak{u},\mathfrak{c}\\}$ and
$P\in{{\mathbb{P}}_{\textrm{BHT}}}$ we set $a_{P_{1}}:=f(P_{1})$,
$a_{P_{2}}:=g(P_{2})$ and $a_{P_{3}}:=h_{*}(P_{3})$.

In order to obtain though (69), we have to further address the symmetry
conflict exposed in Section 2.6 and, in particular, the different scaling
manifested in the “two-resolution analysis”: the above proposition involves
the $2^{am}$-area tiles part of the definition of tri-tiles $P$ while the
structure of the coefficients $f(P_{1})$, $g(P_{2})$ and $h_{*}(P_{3})$
depends on the interaction among area one tiles part of the tri-tile family
$\mathbb{S}(P)$.

In answering the above we will have to re-work the classical energy and size
estimates and adapt them to our context. In particular, due to the high-
resolution discretization – see Sections 2.4 and 2.7 – the sizes for $f$ and
$g$ will be local $L^{2}$ averages, with the associated energy still remaining
an $L^{2}$-based quantity; for $h$, we will need to consider different sizes
that also take into account the information carried by $w_{k,n,v}(\lambda)$,
each developing from the light, uniform, or clustered components.

The key step for obtaining our desired $m$-decay is revealed in Proposition
7.2 of Section 7 whose proof goes through estimating size and energy
quantities and can be summarized as follows: For any subcollection
$\mathbb{P}\subset{{\mathbb{P}}_{\textrm{BHT}}}$ we have

  * •

$L^{2}$ control over the size of the uniform component
$\\{h_{\mathfrak{u}}(P_{3})\\}_{P\in\mathbb{P}}$; i.e.,

(104)
$\mathrm{Size\,}^{\mathbb{P}}_{3}(\\{h_{\mathfrak{u}}(P_{3})\\}_{P\in\mathbb{P}})\lesssim\sup_{P\in\mathbb{P}}\frac{1}{|I_{P}|^{\frac{1}{2}}}\|h\cdot\tilde{\chi}_{I_{P}}\|_{L^{2}}.$

The simple but relevant observation here is that due to the presence of
$\lambda$, the orthogonality among the involved terms is a consequence of the
disjointness of the scales.

  * •

$m$-decay within the $L^{2}$ control over the size of the light component
$\\{h_{\mathfrak{l}}(P_{3})\\}_{P\in\mathbb{P}}$; i.e.,

(105)
$\mathrm{Size\,}^{\mathbb{P}}_{3}(\\{h_{\mathfrak{l}}(P_{3})\\}_{P\in\mathbb{P}})\lesssim
2^{-\frac{\delta_{1}am}{4}}\sup_{P\in\mathbb{P}}\frac{1}{|I_{P}|^{\frac{1}{2}}}\|h\cdot\tilde{\chi}_{I_{P}}\|_{L^{2}}.$

This is a direct consequence of the “lightness” of the weight
$w_{\cdot,\cdot,\cdot}$.

  * •

$m$-decay within the $L^{2}$ control over the size of the clustered component
$\\{h_{\mathfrak{c}}(P_{3})\\}_{P\in\mathbb{P}}$, i.e. for some $\mu>0$

(106)
$\mathrm{Size\,}^{\mathbb{P}}_{3}(\\{h_{\mathfrak{c}}(P_{3})\\}_{P\in\mathbb{P}})\lesssim
2^{-\frac{\mu
am}{4}}\sup_{P\in\mathbb{P}}\frac{1}{|I_{P}|^{\frac{1}{2}}}\|h\cdot\tilde{\chi}_{I_{P}}\|_{L^{2}}.$

Here one relies on the fact that there are at most
$O(2^{(1-\mu)\frac{am}{2}})$ values of $n$ for which there exists $v\sim
2^{am\over 2}$ such that $(p^{\prime},n,v)\in\mathfrak{c}$, where $\mu$ is a
numerical constant chosen small enough.

Putting (101)-(106) all together, we obtain – via the proposition above – the
desired estimate (69) within the local-$L^{2}$ range. A standard interpolation
argument, having as end-points our earlier obtained local-$L^{2}$ estimate and
the result in [60] proves the result claimed by our Theorem 1.2 in the
interior of the full range – see Section 7. We point out that in order to get
the estimates corresponding to one of the exponents $p$ or $q$ in Theorem 1.2
being $\infty$, we need to go through an extra interpolation step at the level
of the spatially localized “sizes”, see Subsection 7.5.2.

##  3\. Initial reductions

In this section we will operate via various decompositions several
simplifications on our bilinear Hilbert-Carleson operator $T:=BC^{a}$. Indeed,
departing from the strategy outlined in Section 2.1 – recall here the key
relations (65)–(68) – we will address the following items:

  * •

the rigorous decomposition of $T$ into the sum of two components: the low
oscillatory component $T_{0}$ and the high oscillatory component
$T_{\mathrm{osc}}$;

  * •

the complete treatment of the low oscillatory component $T_{0}$;

  * •

the rigorous decomposition of the high oscillatory component into a sum of
operators according to the level sets of the phase of the multiplier thus
clarifying the decomposition $T_{\mathrm{osc}}=\sum_{m\in\mathbb{N}}T_{m}$
evoked in (68);

  * •

the reduction to the local-$L^{2}$ case for the boundedness range of $T_{m}$;

  * •

the scale decomposition $T_{m}=\sum_{k\in\mathbb{Z}}T_{m,k}$ and the
identification of the main component of the multiplier associated with each
$T_{m,k}$.

###  3.1. The decomposition $T=T_{0}+T_{\mathrm{osc}}$

By a standard argument, we start by constructing a smooth non-negative even
bump function $\rho$ supported in $[-2,-1/2]\cup[1/2,2]$ such that for every
$t\neq 0$

(107) $\sum_{k\in\mathbb{Z}}\rho(2^{-k}t)=1.$

Similarly, we can identify a smooth non-negative even bump function
$\tilde{\rho}=\tilde{\rho}_{a}$ supported in
$[-2^{a},-2^{-a}]\cup[2^{-a},2^{a}]$ such that for every $\lambda\neq 0$

$\sum_{j\in\mathbb{Z}}\tilde{\rho}(2^{-aj}\lambda)=1.$

Next we set

$\chi(\lambda,t):=\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\ j+k\leq
0\end{subarray}}\tilde{\rho}(2^{-aj}\lambda)\rho(2^{-k}t)$

and notice that this function has the following properties: 1)
$\chi(\lambda,t)\lesssim 1$ for all $\lambda,t\neq 0$, and 2) if
$\chi(\lambda,t)\neq 0$ then $|\lambda t^{a}|\lesssim 1$.
Selecting now in the rôle of $\lambda$ the function $\lambda(x)$ arising from
the linearization of the supremum in the definition of $BC^{a}$, we have that

$BC^{a}(f,g)(x)=:T(f,g)(x)=\int f(x-t)g(x+t)e^{i\lambda(x)t^{a}}\frac{dt}{t}$

which decomposes as

(108) $\displaystyle\int
f(x-t)g(x+t)e^{i\lambda(x)t^{a}}\chi(\lambda(x),t)\frac{dt}{t}$
$\displaystyle\hskip 10.00002pt+\int
f(x-t)g(x+t)e^{i\lambda(x)t^{a}}(1-\chi(\lambda(x),t))\frac{dt}{t}$
$\displaystyle=:T_{0}(f,g)(x)+T_{\mathrm{osc}}(f,g)(x),$

thus completing the decomposition of our operator $T$ into a sum of low and
high oscillatory components as stated in the introduction.

###  3.2. The boundedness of $T_{0}$

In this section we show that $T_{0}$ is bounded in the same range as the one
corresponding to the bilinear Hilbert transform. This will be achieved by
showing that

(109) $|T_{0}(f,g)|\lesssim H_{\ast}(f,g)+M(f,g),$

where here

  * •

$H_{\ast}(f,g)$ stands for the maximal (smooth) truncation of the bilinear
Hilbert transform

(110) $H_{\ast}(f,g)(x):=\sup_{r>0}\Big{|}\int
f(x-t)g(x+t)\theta(t/r)\frac{dt}{t}\Big{|},$

with $\theta$ any smooth bump function supported near the origin, and

  * •

$M(f,g)$ stands for the bilinear maximal function

(111) $M(f,g)(x):=\sup_{r>0}\frac{1}{2r}\int_{-r}^{r}|f(x-t)||g(x+t)|dt.$

Notice that, indeed, if (109) holds, then the claim about the boundedness
range for $T_{0}$ follows from Lacey’s work in [60] (see also [24] for further
generalisations).

Turning now towards the proof of (109), we split $T_{0}$ as

$\displaystyle T_{0}(f,g)(x)$ $\displaystyle=\int
f(x-t)g(x+t)\chi(\lambda(x),t)\frac{dt}{t}$ $\displaystyle\hskip
10.00002pt+\int
f(x-t)g(x+t)(e^{i\lambda(x)t^{a}}-1)\chi(\lambda(x),t)\frac{dt}{t}$
$\displaystyle=:I+II.$

We now see that the first term on the right-hand side is bounded by
$H_{\ast}(f,g)(x)$:

$\displaystyle|I|$ $\displaystyle=\Big{|}\int
f(x-t)g(x+t)\chi(\lambda(x),t)\frac{dt}{t}\Big{|}$
$\displaystyle=\Big{|}\sum_{j\in\mathbb{Z}}\tilde{\rho}(2^{-aj}\lambda(x))\;\int
f(x-t)g(x+t)\sum_{k\leq-j}\rho(2^{-k}t)\frac{dt}{t}\Big{|}$
$\displaystyle=\Big{|}\sum_{j\in\mathbb{Z}}\tilde{\rho}(2^{-aj}\lambda(x))\;\int
f(x-t)g(x+t)\theta(2^{j}t)\frac{dt}{t}\Big{|}$
$\displaystyle\leq\Big{(}\sum_{j\in\mathbb{Z}}\tilde{\rho}(2^{-aj}\lambda(x))\Big{)}H_{\ast}(f,g)(x)=H_{\ast}(f,g)(x),$

where here we have let $\theta(t):=\sum_{k\leq 0}\rho(2^{-k}t)$ (a smooth bump
function supported near the origin).

For the second term, we use the relations $|e^{is}-1|\leq|s|$ and
$\chi(\lambda,t)\lesssim\mathbf{1}_{[-C,C]}(\lambda t^{a})$ in order to deduce
that

$\displaystyle|II|$ $\displaystyle\lesssim\int_{\\{t:|\lambda(x)t^{a}|\lesssim
1\\}}|f(x-t)||g(x+t)||\lambda(x)t^{a}|\frac{dt}{|t|}$
$\displaystyle\lesssim\sum_{\ell\in\mathbb{N}}2^{-\ell}\int_{\\{t:|\lambda(x)t^{a}|\sim
2^{-\ell}\\}}|f(x-t)||g(x+t)|\frac{dt}{|t|}$
$\displaystyle\lesssim\sum_{\ell\in\mathbb{N}}2^{-\ell}\frac{1}{2^{-\ell/a}|\lambda(x)|^{-1/a}}\int_{\\{t:|t|\lesssim
2^{-\ell/a}|\lambda(x)|^{-1/a}\\}}|f(x-t)||g(x+t)|dt$ $\displaystyle\lesssim
M(f,g)(x).$

This ends the proof of (109).

###  3.3. Level set partition of the multiplier’s phase: the decomposition
$T_{\mathrm{osc}}=\sum_{m\in\mathbb{N}}T_{m}$

We start by splitting the operator into components in which both the spatial
scale and the parameter666666Since any bound we will obtain below will be
independent of the measurable function $\lambda(\cdot)$, from now on we
consider it fixed once and for all and omit it when convenient from the
notation. $\lambda$ are localized. Then we will reassemble the pieces
according to the magnitude of the oscillatory phase – it is in this latter
parameter that we will obtain decay in our norm bounds, allowing us to sum all
the contributions together.

Observe that by construction we have676767Notice that $1-\chi(\lambda,t)$
nicely factors into terms in which $\lambda$ is localized and terms in which
$t$ is localized, this being the very reason for which we defined
$\chi(\lambda,t)$ the way we did, instead of something simpler such as
$\theta(\lambda t^{a})$ with $\theta$ a bump function supported near the
origin (which would have sufficed for the purposes of the previous
subsection).

$1-\chi(\lambda,t)=\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j+k>0\end{subarray}}\tilde{\rho}(2^{-aj}\lambda)\rho(2^{-k}t),$

and therefore we can decompose $T_{\mathrm{osc}}$ accordingly as

$T_{\mathrm{osc}}(f,g)(x)=\sum_{\begin{subarray}{c}j,k\in\mathbb{Z}\\\
j+k>0\end{subarray}}\tilde{\rho}(2^{-aj}\lambda(x))\int
f(x-t)g(x+t)e^{i\lambda(x)t^{a}}\rho(2^{-k}t)\frac{dt}{t}.$

Letting $m=k+j$ we see that we can collect the terms according to $m$,
obtaining

$\displaystyle T_{\mathrm{osc}}(f,g)(x)$
$\displaystyle=\sum_{m\in\mathbb{N}}\sum_{k\in\mathbb{Z}}\tilde{\rho}(2^{-a(m-k)}\lambda(x))\int
f(x-t)g(x+t)e^{i\lambda(x)t^{a}}\rho(2^{-k}t)\frac{dt}{t}$ (112)
$\displaystyle=:\sum_{m\in\mathbb{N}}\sum_{k\in\mathbb{Z}}T_{m,k}(f,g)(x)=:\sum_{m\in\mathbb{N}}T_{m}(f,g)(x).$

Observe that in each of the integrals above for a given $m$ we have
$|\lambda(x)t^{a}|\sim 2^{am}$; in other words, we have localized the
magnitude of the phase thus achieving the decomposition we desired.

###  3.4. The boundedness range of $T_{m}$ – reduction to the local-$L^{2}$
case

As explained in the introduction, the rest of the paper will be devoted to the
study of

$T_{m}(f,g):=\sum_{k\in\mathbb{Z}}T_{m,k}(f,g).$

Given $m\in\mathbb{N}$, our final aim is to show that $T_{m}$ obeys
$L^{p}\times L^{q}\to L^{r}$ bounds that decay exponentially in the parameter
$m$.

More precisely, our main result will be the following:

######  Theorem 3.1.

For any exponents $p,q,r$ such that $1<p,q\leq\infty$, $2/3<r<\infty$ and
$1/p+1/q=1/r$ there exists $\delta=\delta(p,q)>0$ such that for all $f\in
L^{p}(\mathbb{R}),g\in L^{q}(\mathbb{R})$ and for all $m\in\mathbb{N}$

(113) $\|T_{m}(f,g)\|_{L^{r}}\lesssim_{a,p,q}2^{-\delta
am}\|f\|_{L^{p}}\|g\|_{L^{q}}.$

Moreover, the implicit constant is independent of the fixed measurable
function $\lambda$.

To begin with, we notice that each $T_{m}$ is controlled pointwise by the
bilinear maximal function $M$ defined in (111):

(114) $\displaystyle|T_{m}(f,g)(x)|$
$\displaystyle=\Big{|}\sum_{k\in\mathbb{Z}}\tilde{\rho}(2^{-a(m-k)}\lambda(x))\int
f(x-t)g(x+t)e^{i\lambda(x)t^{a}}\rho(2^{-k}t)\frac{dt}{t}\Big{|}$
$\displaystyle\leq\sum_{k\in\mathbb{Z}}\tilde{\rho}(2^{-a(m-k)}\lambda(x))\int|f(x-t)||g(x+t)||\rho(2^{-k}t)|\frac{dt}{|t|}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\tilde{\rho}(2^{-a(m-k)}\lambda(x))M(f,g)(x)\lesssim
M(f,g)(x).$

Hence the boundedness of $T_{m}$ follows from that of $M$; in particular, from
[60], $T_{m}$ is $L^{p}\times L^{q}\to L^{r}$ bounded for any
$1<p,q\leq\infty$, $2/3<r<\infty$ with $1/p+1/q=1/r$. Therefore we also have
uniformly in $m$

$\|T_{m}(f,g)\|_{L^{r}}\lesssim_{a,p,q}\|f\|_{L^{p}}\|g\|_{L^{q}}.$

Because of this and by means of interpolation,686868See for example [44] for a
precise statement about multilinear interpolation which keeps track of
constants identity (113) for a certain tuple $(p_{0},q_{0},r_{0})$ – with
$1<p_{0},q_{0}<\infty,2/3<r_{0}<\infty$ – easily extends across the full
_open_ range of Theorem 3.1. This leaves out the boundary tuples
$(p,\infty,p)$ and $(\infty,q,q)$, but these cases will also follow via
(linear) interpolation, once (113) is known to hold for some tuples
$(p_{0},\infty,p_{0})$ or $(\infty,q_{0},q_{0})$.

In order to obtain the full range in Theorem 3.1, we proceed in the following
manner:

  * •

we show that (113) holds for all tuples $(p,q,r)$ belonging to the _strict_
local-$L^{2}$ range, i.e. such that $2<p,q,r^{\prime}<\infty$;

  * •

interpolating this with the estimates we have for $M$, we obtain (113) for all
tuples $(p,q,r)$, except for endpoints corresponding to $p=\infty$ or
$q=\infty$;

  * •

using a different type of interpolation – this time at the level of localized
maximal averages (see Section 7.5.2) – we prove that (113) extends to all
tuples $(p,q,r)$ with $2<p,q\leq\infty,1<r<\infty$, $1/p+1/q=1/r$;

  * •

finally, we use interpolation on the boundary696969We interpolate between
$\|T_{m}\|_{L^{p_{1}}\times L^{\infty}\to
L^{p_{1}}}\lesssim_{a,p_{1}}2^{-\delta am}$ for $p_{1}>2$ and
$\|T_{m}\|_{L^{p_{2}}\times L^{\infty}\to L^{p_{2}}}\lesssim 1$ for
$1<p_{2}<2$. This reduces to linear interpolation for the operator $f\mapsto
T_{m}(f,g)$, where $g\in L^{\infty}$ is fixed. of the range to deduce the full
result of Theorem 3.1.

As a consequence of the above discussion, Theorem 3.1 follows from a similar
result for input functions in the local-$L^{2}$ range:

######  Theorem 3.2 (local-$L^{2}$ input functions).

For any exponents $p,q,r$ such that $2<p,q\leq\infty,1<r<\infty$ and
$1/p+1/q=1/r$, there exists $\delta=\delta(p,q)>0$ such that for all $f\in
L^{p}(\mathbb{R}),g\in L^{q}(\mathbb{R})$ and for all $m\in\mathbb{N}$

(115) $\|T_{m}(f,g)\|_{L^{r}}\lesssim_{a,p,q}2^{-\delta
am}\|f\|_{L^{p}}\|g\|_{L^{q}},$

where the constant is independent of the fixed measurable function $\lambda$.

The central point of our paper will be the proof of the $m$-decaying estimates
within the local-$L^{2}$ range.

###  3.5. The main component of $T_{m,k}$: stationary phase analysis.

In this section we will focus on the analysis of the multiplier symbol
associated with $T_{m,k}$ as represented by the formula

$T_{m,k}(f,g)(x)=\tilde{\rho}\Big{(}\frac{\lambda(x)}{2^{a(m-k)}}\Big{)}\iint\widehat{f}(\xi)\widehat{g}(\eta)e^{i(\xi+\eta)x}\mathfrak{M}_{m,k}(\xi-\eta;\lambda(x))d\xi
d\eta,$

where

(116) $\mathfrak{M}_{m,k}(\zeta;\lambda):=\int e^{i\lambda t^{a}}e^{-i\zeta
t}\frac{\rho(2^{-k}t)}{t}dt.$

Since $\mathfrak{M}_{m,k}$ is an oscillatory integral, we will make use of the
stationary phase principle in order to decompose our operator $T_{m,k}$ into a
principal component that will become the main object of interest for the
remainder of the paper and an error term that will be controlled by the
bilinear maximal function defined in (111). In the present section we will
only provide a heuristic argument for identifying the dominant term of
$T_{m,k}$ (respectively $\mathfrak{M}_{m,k}$), while the detailed but largely
standard analysis involving more laborious computations and reductions can be
find in Appendix B.

These being said, we turn out attention towards (116) and notice that the
phase of $\mathfrak{M}_{m,k}$, that is $\mu(t):=\lambda t^{a}-\zeta t$, has –
up to a sign – a single critical real point at
$|t_{c}|:=\left|\zeta/a\lambda\right|^{1/(a-1)}$. A heuristic application of
the stationary phase principle (see for example Ch. VIII of [95]) suggests
that the main contribution to the oscillatory integral $\mathfrak{M}_{m,k}$ is
given, up to an absolute constant, by

$e^{i\mu(t_{c})}\frac{\rho(2^{-k}t_{c})}{t_{c}(\mu^{\prime\prime}(t_{c}))^{1/2}},$

provided that $\zeta$ is such that $|t_{c}|\sim 2^{k}$. Expanding the
definitions of $t_{c}$ and $\mu$ in the above formula and taking into account
that $|t_{c}|\sim 2^{k},\lambda\sim 2^{a(m-k)}$ and $|\zeta|\sim 2^{am-k}$, we
necessarily have that the second factor above is of magnitude $\sim
2^{-am/2}$, and thus the main contribution to $\tilde{\rho}(\lambda
2^{-a(m-k)})\mathfrak{M}_{m,k}(\zeta;\lambda)$ is morally of the
form707070Here one can compute exactly the value of $c_{a}$ as given by the
formula $c_{a}:=a^{-a/(a-1)}-a^{-1/(a-1)}$.

(117)
$\mathfrak{m}_{m,k}(\zeta;\lambda):=\frac{1}{2^{am/2}}e^{ic_{a}\zeta^{a/(a-1)}\lambda^{-1/(a-1)}}\rho\Big{(}\frac{\lambda}{2^{a(m-k)}}\Big{)}\psi\Big{(}\frac{\zeta}{2^{am-k}}\Big{)}$

for some bump functions $\rho(\lambda),\psi(\zeta)$ supported in
$|\lambda|\sim 1$, $|\zeta|\sim 1$ respectively.717171Notice that we are
abusing notation once again: the latter $\rho$ is not necessarily the same one
as that introduced in (107). For later convenience, we will denote from now on
$\rho_{am-ak}(\lambda):=\rho(\lambda 2^{-a(m-k)})$.

Based on the above heuristic (once again made precise in Appendix B) we have
that Theorem 3.2 will follow from the following:

######  Theorem 3.3.

With $\mathfrak{m}_{m,k}$ defined by (117) we let (with a further abuse of
notation)

(118)
$T_{m,k}(f,g)(x):=\iint\widehat{f}(\xi)\widehat{g}(\eta)e^{i(\xi+\eta)x}\mathfrak{m}_{m,k}(\xi-\eta;\lambda(x))d\xi
d\eta$

and $T_{m}(f,g):=\sum_{k\in\mathbb{Z}}T_{m,k}(f,g)$.
For any $p,q,r$ such that $2<p,q\leq\infty,1<r<\infty$ and $1/p+1/q=1/r$,
there exists $\delta=\delta(p,q)>0$ such that for all
$m\in\mathbb{N}$727272Here the implicit constant depends at most polynomially
on the $C^{1}$-norms of $\rho,\psi$ appearing in (117).

(119) $\Big{\|}T_{m}(f,g)\Big{\|}_{L^{r}}\lesssim_{a,p,q}2^{-\delta
am}\|f\|_{L^{p}}\|g\|_{L^{q}}.$

This result will be derived as a consequence of Proposition 4.1, Corollary
7.3, and the extension of the range of boundedness performed in Section 7.5.2.

######  Remark 3.4.

We end this section by mentioning that in the non-resonant regime, our
operator $BC^{a}$ may be thought of as a bilinear modulation invariant737373In
the sense of (P2). version of Stein-Wainger’s (polynomial) Carleson operator
discussed in [99]. This parallelism is preserved when performing the
stationary phase analysis of the multiplier as one can notice by comparing the
discussion in our Appendix B with the corresponding analysis in [99] or [96].

##  4\. Two-resolution analysis: Discretization

###  4.1. Preliminaries

In this section we will introduce the model operators that are critical for
the study of $BC^{a}$. Generically speaking, the role of such a model is to
express the original operator, say $T$, as a superposition of objects of lower
complexity (thus easier to be analyzed) while properly capturing the key
features of $T$ (e.g. dilation, translation, modulation invariance with all
these regarded as part of the larger interaction between the structure of the
input functions and that of the kernel). Of course, the geometry of the time-
frequency regions associated with these lower complexity objects plays the
determinative role in the partition, organization of the terms and finally in
regrouping the information carried by them in order to obtain the global
control over $T$.

Loosely speaking, through our discretization procedure(s) we aim to handle the
two competing features of $BC^{a}$: the non-zero curvature (phase) and the
modulation invariance (input).

Indeed, the current section will be divided into two major components:

  * •

_the high resolution modelling_ : designed with the purpose of extracting the
cancellation encoded in the curvature of the phase of our kernel/multiplier
within every single scale, its analysis is further split into three stages:

    * –

construction of the _discrete phase-linearized wave-packet model_ : this is
performed in Section 4.2 and follows a methodology – detailed in Section 4.2.2
– that involves the phase linearization of the multiplier and the Gabor
discretization of the input functions. While this model will be essential in
the later stages of our proof when we need to perform a multiscale analysis of
BHT-flavor it comes with a major constraint: it is _not_ an $m$-decaying
absolutely summable model thus making difficult the task of capturing the
cancellation at the single scale level. This is the reason for which we need
to appeal to a second, auxiliary model, that will be described immediately
below.

    * –

construction of the _continuous phase-linearized spatial model_ : discussed in
Section 4.3 – see also the helpful heuristic in Section 4.3.3 – this relies on
a linearization process that is applied directly on the kernel (spatial) side
of our operator but _without_ performing any Fourier analysis (wave-packet
discretization) on the input functions. It is this coarser model that will be
employed in Section 5 in order to obtain the desired single scale estimate
with exponential decay in the parameter $m$.

    * –

transition between the two models above: this is explained in Section 4.3.2
and will become an important ingredient in Section 6 when refining the single
scale estimate in order to capture the time-frequency localization properties
of the dualizing function $h$ and thus prepare the implementation of the low
resolution multi-scale analysis.

Finally, a very concise antithetical view on these two high resolution models
that summarizes the need for both of them is discussed in Section 4.4.

  * •

_the low resolution modelling_ : presented in Section 4.5, this model will be
quintessential in approaching the zero-curvature feature of our operator
$BC^{a}$. Its design relies on properly regrouping the information contained
in the discrete phase-linearized wave-packet model in order to be able to
extract the almost orthogonality across multiple scales thus involving the
multi-scale nature of our problem. It is through this model that we will be
able to exploit the bilinear Hilbert type features of our operator in order to
propagate the exponential decay in $m$ from the local level (single scale) to
the global level (multi-scale model).

The success of our approach depends on the compatibility between the high and
low resolution analysis as it is this joint contribution that will deliver the
desired result.

###  4.2. The high resolution model (I): a phase-linearized wave-packet
discretization

In the present section we construct a first model operator relying on a wave-
packet discretization of the input functions.

####  4.2.1. Preparatives

Recalling the decompositions (108) and (112) and relation (118) in Section 3,
throughout this section we fix $m\in\mathbb{N}$, $k\in\mathbb{Z}$ and focus
our attention on the operator $T_{m,k}$ given by

(120) $\displaystyle T_{m,k}(f,g)(x):=$
$\displaystyle\iint\widehat{f}(\xi)\widehat{g}(\eta)e^{i(\xi+\eta)x}\frac{\rho_{am-
ak}(\lambda(x))}{2^{am\over
2}}e^{ic_{a}(\xi-\eta)^{a^{\prime}}\lambda(x)^{-\frac{1}{a-1}}}\psi\Big{(}\frac{\xi-\eta}{2^{am-k}}\Big{)}d\xi
d\eta,$

where from now on, for notational simplicity, we set $a^{\prime}=a/(a-1)$.

Notice that in the formula above the frequency information is essentially
supported in the strip $\\{\xi-\eta\sim 2^{am-k}\\}$. As a first step we
partition this strip in square regions, thus decoupling the non-oscillatory
information in $\xi$ and $\eta$. For this, we consider a suitable partition of
unity

$1=\sum_{\ell\in\mathbb{Z}}\phi\Big{(}\zeta-\ell\Big{)},$

where $\phi$ is a smooth function which is compactly supported say in
$[\frac{1}{2},2]$. Inserting this into the formula of our operator, in the
first instance, we get

$\displaystyle
T_{m,k}(f,g)(x)=\sum_{\ell,\ell^{\prime}\in\mathbb{Z}}\iint\widehat{f}(\xi)\widehat{g}(\eta)\phi\Big{(}{\xi\over
2^{am-k}}-\ell\Big{)}\phi\Big{(}{\eta\over 2^{am-k}}-\ell^{\prime}\Big{)}$
$\displaystyle e^{i(\xi+\eta)x}\frac{\rho_{am-ak}(\lambda(x))}{2^{am\over
2}}e^{ic_{a}(\xi-\eta)^{a^{\prime}}\lambda(x)^{-\frac{1}{a-1}}}\psi\Big{(}\frac{\xi-\eta}{2^{am-k}}\Big{)}d\xi
d\eta.$

Since the function $\psi$ introduced in the stationary phase analysis of the
multiplier in (116) is supported around $\zeta\sim 1$, simple
considerations747474By writing each of the functions $\psi$ and $\phi$ as a
superposition of, say, $20$ functions, we can further shrink the support of
our functions and assume without loss of generality that $\psi$ is supported
in $[2-\frac{1}{10},2+\frac{1}{10}]$ and $\phi$ in
$[\frac{3}{4},\frac{5}{4}]$. imply that without loss of generality we may
assume that

(121) $\displaystyle
T_{m,k}(f,g)(x)=\sum_{\ell\in\mathbb{Z}}\iint\widehat{f}(\xi)\widehat{g}(\eta)\phi\Big{(}{\xi\over
2^{am-k}}-(\ell+1)\Big{)}\phi\Big{(}{\eta\over 2^{am-k}}-(\ell-1)\Big{)}$
$\displaystyle e^{i(\xi+\eta)x}\frac{\rho_{am-ak}(\lambda(x))}{2^{am\over
2}}e^{ic_{a}(\xi-\eta)^{a^{\prime}}\lambda(x)^{-\frac{1}{a-1}}}\psi\Big{(}\frac{\xi-\eta}{2^{am-k}}\Big{)}d\xi
d\eta.$

Having reached this point, we turn our attention towards the crucial part of
the multiplier – its oscillatory component – the one that will dictate the
further discretization of our operator. Before entering into details, we make
a short detour in order to explain the general methodology that we will
follow.

####  4.2.2. The LGC-method outlined

In this section we briefly discuss the key characteristics of the LGC-
methodology employed in [71]; these are as follows: phase linearization, Gabor
(local Fourier) analysis and almost orthogonality via time-frequency
correlation757575Throughout the paper the notion of time-frequency correlation
designates the existence of a conditionality/relationship among the time and
frequency parameters appearing in the Gabor discretizations of the input
functions. It should be noted that in general the “correlation” aspect of the
LGC method can also incorporate the standard frequency correlation of a given
function relative to a Fourier mode; however, this latter notion of
correlation is not used in the present paper. and phase level set analysis.
Before elaborating more on the above elements we emphasise that each of them
has been commonly used in the math literature: for example phase linearization
is extensively exploited in PDE, Gabor decompositions are standard tools in
signal processing and more recently in time-frequency analysis while the study
of the frequency correlations proves its utility in several areas with a
special highlight on additive combinatorics. Thus, the novelty and power of
the method767676This was introduced in order to provide a unifying perspective
on several themes belonging to the non-zero curvature realm. For more on this,
the reader is invited to consult Section 1.3 in the Introduction. does not
rely on the nature of each of its constitutive elements but rather on their
combined effect in analyzing problems that belong to the general class of
singular and maximal integral operators.

With these being said, we now present a very concise overview of these key
characteristics:

  1. (L)

_linearization_ : this represents the design of the spatial/frequency
discretization procedure through which the phase of the operator’s kernel (if
regarded from the spatial side) or the phase of its associated multiplier (if
regarded from the frequency side) is confined to oscillate at a linear level
(when restricted to the regions arising from this discretization);

  2. (G)

_adapted Gabor frame analysis_ : within each of the regions obtained at the
first item, one performs an adapted Gabor frame decomposition of the functions
on which the operator acts; this produces a new expression which now involves
a sum of local Fourier (Gabor) coefficients multiplied by variable weights
resulted from the kernel/multiplier discretization discussed at the previous
step;

  3. (C)

_cancellation via $TT^{*}$-method and (time-frequency) correlation_: the
resulting discrete operator is now analyzed at the purely $L^{2}$ level via a
$TT^{*}$ argument. This analysis relies on the cancellation hidden in the
oscillation of the discretized kernel/multiplier, and has as an output an
expression involving $\ell^{2}$ spatial/frequency coefficients associated to
the input functions, multiplied by elementary expressions that depend on the
time and frequency parameters involved in the discretization. Eventually, one
exploits the created time-frequency correlations by employing various elements
such as777777This list is not complete, and depending on the nature of the
problem, not all these elements need to appear in the proof. the analysis of
the size distribution of the (input dependent) coefficients, phase level set
estimates, elements of combinatorics and number theory (see _e.g._ discrete
Van der Corput or more general bounds for exponential sums) thus achieving the
desired generic decay estimate shaped by (9).

With these explained, we are now ready to follow the above steps in order to
achieve our first model operator.

####  4.2.3. Phase linearization

In this section, for a fixed $m\in\mathbb{N}$ and $k,\ell\in\mathbb{Z}$ we are
going to subdivide each of the frequency squares
$[2^{am-k}(\ell+1),2^{am-k}(\ell+2)]\times[2^{am-k}(\ell-1),2^{am-k}\ell]$
obtained in Section 4.2.1 into a collection of smaller dyadic squares of a
prescribed length $2^{\frac{am}{2}-k}$, designed such that the phase of the
multiplier oscillates at a linear level within each such smaller square.

In what follows we assume that $(x,\xi,\eta)$ are such that
$\mathfrak{m}_{m,k}(\xi-\eta;\lambda(x))\not=0$, see (117). Then, if we let
$\mu(\xi,\eta)$ stand for the phase of the multiplier
$\mathfrak{m}_{m,k}(\xi-\eta;\lambda(x))$ as given by (117), we notice that

$\mu(\xi,\eta)=\mu(\xi-\eta)=c_{a}(\xi-\eta)^{a/(a-1)}\lambda(x)^{-1/(a-1)},$

with $|\lambda(x)|\sim 2^{a(m-k)}$ and $|\xi-\eta|\sim 2^{am-k}$.

From the above we deduce that

(122)
$\displaystyle|\partial^{2}_{\xi}\mu(\xi,\eta)|=|\partial^{2}_{\eta}\mu(\xi,\eta)|=|\partial^{2}_{\xi\eta}\mu(\xi,\eta)|$
$\displaystyle=|c_{a}||\xi-\eta|^{\frac{2-a}{a-1}}|\lambda(x)|^{-\frac{1}{a-1}}$
$\displaystyle\sim 2^{2k-am}=(2^{\frac{am}{2}-k})^{-2}.$

Consequently, if we want to impose that the second order term in the Taylor
expansion of $\mu(\xi,\eta)$ around a given point $(\xi_{0},\eta_{0})$ is at
most $O(1)$, we need to impose the condition
$|\xi-\xi_{0}|,|\eta-\eta_{0}|\lesssim 2^{\frac{am}{2}-k}$. Now, within such a
smaller square region we have that the phase can be decomposed as a linear
component plus an – essentially non-oscillatory – nonlinear component.

In light of the above it becomes natural to subdivide the frequency support of
$\mathfrak{m}_{m,k}(\xi-\eta;\lambda(x))$ into (essentially) squares of size-
length $2^{\frac{am}{2}-k}$. This process will be made precise in the next
section.

####  4.2.4. Gabor frame discretization

Following the above description we will perform a Gabor frame (windowed
Fourier series) decomposition of the input functions that visualize the
frequency universe in “unit” lengths of size $2^{\frac{am}{2}-k}$. Concretely,
we first introduce the $L^{2}$-normalized Gabor frame (wave-packet)

(123)
$\varphi^{u,n}_{\frac{am}{2}-k,\ell}(\xi):=\frac{1}{2^{\frac{am-2k}{4}}}\varphi\Big{(}\frac{\xi}{2^{\frac{am}{2}-k}}-u-2^{\frac{am}{2}}(\ell-1)\Big{)}e^{-in\xi
2^{-(\frac{am}{2}-k)}}$

with $\varphi$ a suitable smooth function compactly supported and then
decompose

(124)
$\displaystyle\qquad\qquad\qquad\qquad\widehat{f}_{\ell+1}(\xi):=\widehat{f}(\xi)\phi({\xi\over
2^{am-k}}-(\ell+1))$ $\displaystyle=$
$\displaystyle\sum_{u,n\in\mathbb{Z}}\langle\widehat{f}_{\ell+1},\varphi^{u,n}_{\frac{am}{2}-k,\ell}\rangle\varphi^{u,n}_{\frac{am}{2}-k,\ell}(\xi)=\sum_{2\cdot
2^{\frac{am}{2}}<u\leq 3\cdot
2^{\frac{am}{2}}}\sum_{n\in\mathbb{Z}}\langle\widehat{f}_{\ell+1},\varphi^{u,n}_{\frac{am}{2}-k,\ell}\rangle\varphi^{u,n}_{\frac{am}{2}-k,\ell}(\xi),$

and

(125)
$\displaystyle\qquad\qquad\quad\quad\widehat{g}_{\ell-1}(\eta):=\widehat{g}(\eta)\phi({\eta\over
2^{am-k}}-(\ell-1))$ $\displaystyle=$
$\displaystyle\sum_{v,p\in\mathbb{Z}}\langle\widehat{g}_{\ell-1},\varphi^{v,p}_{\frac{am}{2}-k,\ell}\rangle\varphi^{v,p}_{\frac{am}{2}-k,\ell}(\eta)=\sum_{1\leq
v\leq
2^{\frac{am}{2}}}\sum_{p\in\mathbb{Z}}\langle\widehat{g}_{\ell-1},\varphi^{v,p}_{\frac{am}{2}-k,\ell}\rangle\varphi^{v,p}_{\frac{am}{2}-k,\ell}(\eta),$

where in the above we made use of the support properties for
$\widehat{f}_{\ell+1}$, $\widehat{g}_{\ell-1}$ and $\varphi$.

If we now insert (124) and (125) back to (121) we deduce that

(126) $\displaystyle
T_{m,k}(f,g)(x)=\sum_{\ell\in\mathbb{Z}}\sum_{n,p\in\mathbb{Z}}$
$\displaystyle\sum_{u,v\in\mathbb{Z}}\langle\widehat{f}_{\ell+1},\varphi^{u,n}_{\frac{am}{2}-k,\ell}\rangle\langle\widehat{g}_{\ell-1},\varphi^{v,p}_{\frac{am}{2}-k,\ell}\rangle\mathcal{K}^{u,v,n,p}_{m,k,\ell}(x),$

with

(127) $\displaystyle\mathcal{K}^{u,v,n,p}_{m,k,\ell}(x)$
$\displaystyle:=\frac{1}{2^{\frac{am}{2}}}\rho_{am-ak}(\lambda(x))\iint
e^{ic_{a}(\xi-\eta)^{a^{\prime}}\lambda(x)^{-\frac{1}{a-1}}}e^{i\xi x}e^{i\eta
x}$ $\displaystyle\hskip
70.0001pt\cdot\varphi^{u,n}_{\frac{am}{2}-k,\ell}(\xi)\varphi^{v,p}_{\frac{am}{2}-k,\ell}(\eta)\psi\Big{(}\frac{\xi-\eta}{2^{am-k}}\Big{)}d\xi
d\eta.$

This ends the linearized wave-packet decomposition of our operator, of course,
up to a better understanding of the kernel $\mathcal{K}^{u,v,n,p}_{m,k,\ell}$.

####  4.2.5. Analyzing the kernel

In this section we focus on obtaining an explicit form for the kernel
$\mathcal{K}^{u,v,n,p}_{m,k,\ell}$ by exploiting the time-frequency
localization of the wave-packets in relation to the oscillatory factor
$\displaystyle e^{ic_{a}(\xi-\eta)^{a^{\prime}}\lambda(x)^{-\frac{1}{a-1}}}$
in (127).

Notice that the change of variables $\xi\rightarrow 2^{\frac{am}{2}-k}\xi$ and
$\eta\rightarrow 2^{\frac{am}{2}-k}\eta$ in (127) together with the definition
of the wave-packets (123) allows us to rewrite the above as

$\displaystyle\mathcal{K}^{u,v,n,p}_{m,k,\ell}(x)=$
$\displaystyle\frac{\rho_{am-ak}(\lambda(x))}{2^{k}}\iint
e^{ic_{a}\frac{\big{(}2^{\frac{am}{2}-k}(\xi-\eta)\big{)}^{a^{\prime}}}{\lambda(x)^{1/(a-1)}}}\cdot
e^{i(2^{\frac{am}{2}-k}\xi x+2^{\frac{am}{2}-k}\eta x-\xi n-\eta p)}$
$\displaystyle\cdot\varphi(\xi-u-2^{\frac{am}{2}}(\ell-1))\varphi(\eta-v-2^{\frac{am}{2}}(\ell-1))\psi\Big{(}\frac{\xi-\eta}{2^{\frac{am}{2}}}\Big{)}d\xi
d\eta.$

Applying now a second change of variables $\xi-\eta=\xi_{1}$ and
$\xi+\eta=\eta_{1}$ we rewrite the kernel
$\mathcal{K}^{u,v,n,p}_{m,k,\ell}(x)$ as

(128) $\displaystyle\frac{\rho_{am-ak}(\lambda(x))}{2^{k}}\iint
e^{ic_{a}(2^{\frac{am}{2}-k}\xi_{1})^{a^{\prime}}\lambda(x)^{-\frac{1}{a-1}}}e^{i2^{\frac{am}{2}-k}\eta_{1}x}e^{-i\frac{\xi_{1}+\eta_{1}}{2}n}e^{-i\frac{\eta_{1}-\xi_{1}}{2}p}$
$\displaystyle\cdot\varphi\Big{(}\frac{\xi_{1}+\eta_{1}}{2}-u-2^{\frac{am}{2}}(\ell-1)\Big{)}\varphi\Big{(}\frac{\eta_{1}-\xi_{1}}{2}-v-2^{\frac{am}{2}}(\ell-1)\Big{)}\psi\Big{(}\frac{\xi_{1}}{2^{\frac{am}{2}}}\Big{)}d\xi_{1}d\eta_{1}.$

We claim that the expression under the integral sign behaves essentially as a
tensor product of the form $a(\xi_{1})b(\eta_{1})$ for some suitable functions
$a$ and $b$. Indeed, given the support localization for the function $\varphi$
our kernel can be safely approximated as

(129) $\mathcal{K}^{u,v,n,p}_{m,k,\ell}(x)\approx\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}w_{k,\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)(x)\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}}(x),$

where here787878For simplicity of the exposition, we abuse the notation and in
the definitions of $w_{k,\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)(x)$ and
$\frac{1}{2^{\frac{am-2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}}(x)$
we are using the same symbol $\varphi$ as the one introduced earlier though of
course, strictly speaking, the functions involved in these definitions while
having similar properties do not need to agree with $\varphi$.

(130) $\displaystyle
w_{k,\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)(x):=\int\varphi(\xi_{1}-(u-v))e^{i\frac{p-n}{2}\xi_{1}}e^{ic_{a}2^{({am\over
2}-k)a^{\prime}}\xi_{1}^{a^{\prime}}\lambda(x)^{-{1\over{a-1}}}}\psi\Big{(}\frac{\xi_{1}}{2^{\frac{am}{2}}}\Big{)}d\xi_{1},$

and the wave-packet
$\frac{1}{2^{\frac{am-2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}}(x)$
stands for the expression

(131) $\int
e^{i2^{\frac{am}{2}-k}\eta_{1}x}e^{-i\frac{p+n}{2}\eta_{1}}\varphi(\eta_{1}-(u+v)-2^{\frac{am}{2}}(2\ell-2))d\eta_{1}.$

The rigorous, formal approach to the informal statement (129) follows some
standard reasonings in Fourier analysis and thus, in what follows, we will
only provide an outline of them:

Since $\varphi$ is smooth and compactly supported – for the sake of the
argument one may assume without loss of generality that
$\textrm{supp}\varphi\subseteq[0,1]$ – we can write pointwise within $[0,1]$
the equality between $\varphi$ and its Fourier series, that is

$\varphi(\zeta)=\sum_{s\in\mathbb{Z}}c_{s}e^{is\zeta},\>\>\>\>\forall\>\zeta\in[0,1],$

where the coefficients $(c_{s})_{s}$ are fast-decaying (in a way that agrees
with the regularity of $\varphi$).

Deduce from the above that on the support of the function797979Here we abuse
the notation as we allow the function $\varphi$ to slightly change from line
to line.
$(\xi_{1},\eta_{1})\mapsto\varphi(\xi_{1}-(u-v))\varphi(\eta_{1}-(u+v)-2^{am\over
2}(2\ell-2))$ one has that

$\varphi(\frac{\xi_{1}+\eta_{1}}{2}-u-2^{\frac{am}{2}}(\ell-1))\varphi(\frac{\eta_{1}-\xi_{1}}{2}-v-2^{\frac{am}{2}}(\ell-1))$

equals

$\displaystyle\sum_{s,t\in\mathbb{Z}}c_{s}c_{t}e^{is(\frac{\xi_{1}+\eta_{1}}{2}-u-2^{\frac{am}{2}}(\ell-1))}e^{it(\frac{\eta_{1}-\xi_{1}}{2}-v-2^{\frac{am}{2}}(\ell-1))}.$

Thus, (129) becomes

$\displaystyle\mathcal{K}^{u,v,n,p}_{m,k,\ell}(x)$
$\displaystyle=\sum_{s,t\in\mathbb{Z}}c_{s}c_{t}\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}w_{k,\frac{s-t}{2}+\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)(x)\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}-\frac{s+t}{2}}(x)$
$\displaystyle\qquad\qquad\quad
e^{-is(u+2^{\frac{am}{2}}(\ell-1))}e^{-it(v+2^{\frac{am}{2}}(\ell-1))}$
$\displaystyle=:\sum_{s,t\in\mathbb{Z}}c_{s}c_{t}\tilde{\mathcal{K}}^{u,v,n+t,p+s}_{m,k,\ell}(x).$

Due to the fast decay of the coefficients $\\{c_{s}\\}_{s}$ we can focus on
the case $s=t=0$, justifying our claim (129).

We conclude this section with the following remark: by a simple inspection of
(129) we notice that the behavior of the rough linearizing function is now
separated and isolated within the term
$w_{k,\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)$ which can be regarded as a
weight attached to the standard wave-packet
$\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}}$.

####  4.2.6. Achieving the final form of our first model

Based on the previous section’s reasonings, we were able to reduce the initial
operator $T_{m,k}(f,g)$ defined in (120) to the study of an expression
realized as a superposition of simpler operators that reads as808080In this
precise context, for notational simplicity, we are abusing our convention on
the notation $v\sim 2^{am\over 2}$ (which as defined originally is equivalent
with $|v|\sim 2^{am\over 2}$). Indeed, in formula (132) the domain of the
summation $\\{u,v\sim 2^{\frac{am}{2}}\\}$ is in fact restricted to
$2^{\frac{am}{2}}<u\leq 2\cdot 2^{\frac{am}{2}}$ and $-\frac{3}{2}2^{am\over
2}<v\leq-\frac{1}{2}2^{\frac{am}{2}}$.

(132) $\displaystyle\sum_{\ell\in\mathbb{Z}}\ \sum_{n,p\in\mathbb{Z}}\
\sum_{u,v\in\mathbb{Z}}$
$\displaystyle\langle\hat{f}_{\ell+1},\varphi^{u,n}_{\frac{am}{2}-k,\ell}\rangle\langle\hat{g}_{\ell-1},\varphi^{v,p}_{\frac{am}{2}-k,\ell}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}}(x)w_{k,\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)(x)$
$\displaystyle=\sum_{\ell\in\mathbb{Z}}\ \sum_{n,p\in\mathbb{Z}}\
\sum_{u,v\sim 2^{\frac{am}{2}}}$
$\displaystyle\langle\hat{f}_{\ell+1},\varphi^{u,n}_{\frac{am}{2}-k,\ell}\rangle\langle\hat{g}_{\ell-1},\varphi^{v,p}_{\frac{am}{2}-k,\ell}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{u+v,\frac{p+n}{2}}(x)w_{k,\frac{p-n}{2},\frac{v-u}{2}}^{e}(\lambda)(x).$

By applying Parseval’s identity and the change of variables
$(u,v,n,p)\to(u-v,u+v,p-n,p+n)$ we have that the expression in (132) may be
rewritten as

(133)
$\displaystyle\sum_{\ell\in\mathbb{Z}}\sum_{n,p\in\mathbb{Z}}\sum_{u,v\in\mathbb{Z}}$
$\displaystyle\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p+n}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p}(x)w_{k,n,v}^{e}(\lambda)(x).$

The first task for this section is to show that the study of (133) can be
further reduced to a simpler expression – which, for notational simplicity,
will be redefined as $T_{m,k}$ – and which involves the summation only over
$n\sim_{a}2^{\frac{am}{2}}$.818181We remind here that the definition of
$\sim_{a}$ was provided in footnote 53.828282This will be an immediate
consequence of the structure of the decaying term in the right-hand side of
(140). Indeed, our claim is that it is enough to consider838383From now on,
for notational simplicity, we will drop the $a$ dependence in $\sim_{a}$.
However, it should be noted that, strictly speaking, whenever we read $n,p\sim
2^{\frac{am}{2}}$ we have in fact that $n,p\sim_{a}2^{\frac{am}{2}}$.

(134) $\displaystyle
T_{m,k}(f,g)(x)=\sum_{\ell\in\mathbb{Z}}\sum_{\begin{subarray}{c}n\sim
2^{\frac{am}{2}}\\\
p\in\mathbb{Z}\end{subarray}}\sum_{u,v\in\mathbb{Z}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p+n}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p}(x)w_{k,n,v}^{e}(\lambda)(x)$
.

If is to believe for a moment (134), then the restriction $n\sim
2^{\frac{am}{2}}$ naturally848484Notice the structure $p-n$ and $p+n$ in the
Gabor coefficients $\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p-n}\rangle$ and
$\langle g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p+n}\rangle$,
respectively. The deeper, geometric motivation for the regrouping of the sum
in $p$ will become transparent in Section 4.5. invites to a regrouping of the
sum in $p$ in subcomponents spanning precisely the reference length
$2^{\frac{am}{2}}$. More precisely, we will write the set of integers as a
disjoint union of intervals containing $2^{\frac{am}{2}}$ consecutive
integers, and using the notation

(135) $p_{r}:=2^{\frac{am}{2}}(r-1)+p,\quad\text{ with }\quad
r\in\mathbb{Z}\quad\textrm{and}\quad 2^{am\over 2}\leq p<2\cdot 2^{am\over 2}$

we will rewrite (134) as

(136) $\displaystyle T_{m,k}(f,g)(x)=\sum_{\ell,r\in\mathbb{Z}}\sum_{n,p\sim
2^{\frac{am}{2}}}\sum_{u,v\in\mathbb{Z}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}(x)w_{k,n,v}^{e}(\lambda)(x)$
.

Returning now to (134), the key for proving our claim resides in the behavior
of the oscillatory term

(137) $\displaystyle\rho_{am-ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x)$
$\displaystyle=\rho_{am-
ak}(\lambda(x))\int\varphi(\xi+2v)e^{in\xi}e^{ic_{a}2^{({am\over
2}-k)a^{\prime}}\xi^{a^{\prime}}\lambda(x)^{-{1\over{a-1}}}}\psi\Big{(}\frac{\xi}{2^{\frac{am}{2}}}\Big{)}d\xi.$

Denote now the phase in (137) by

(138) $\nu(\xi):=n\xi+c_{a}2^{({am\over
2}-k)a^{\prime}}\xi^{a^{\prime}}\lambda(x)^{-{1\over{a-1}}},$

and integrate by parts twice in the right-hand side of (137) in order to
obtain under the integral sign a decaying factor of the form
$\frac{1}{1+|\nu^{\prime}(\xi)|^{2}}$. Once here, denoting by $\xi_{c}$
the858585Up to a sign, this real number is uniquely determined. critical point
of $\nu$ (_i.e._ , $\nu^{\prime}(\xi_{c})=0$) we have two situations:

  * •

either $|\xi_{c}|\sim 2^{\frac{am}{2}}$ case in which, using the phase
linearization performed in Section 4.2.3 we have by construction that
$|\nu^{\prime\prime}(\xi)|\sim 1$ for $|\xi|\sim 2^{\frac{am}{2}}$ and hence,
via a Taylor series argument, we deduce that
$|\nu^{\prime}(\xi)|\sim|\xi-\xi_{c}|\sim|\xi_{c}+2v|$;

  * •

or else we must have $|\nu^{\prime}(\xi)|\gtrsim 2^{\frac{am}{2}}$.

In either of the above situations, observing that

(139) $\xi_{c}=\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{a(\frac{am}{2}-k)}},$

we obtain a first time-frequency correlation condition as given by

(140) $\displaystyle\big{|}\rho_{am-
ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x)\big{|}$
$\displaystyle\lesssim\rho_{am-
ak}(\lambda(x))\frac{1}{1+\Big{|}\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{a(\frac{am}{2}-k)}}+2v\Big{|}^{2}}$
$\displaystyle\hskip 10.00002pt=:w_{k,n,v}(\lambda)(x)$

Now the decay in (140) offers us the intuition that the main contribution in
(133) is expected to come from parameters $n$ and $v$ satisfying

$-2v\approx\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{a(\frac{am}{2}-k)}}.$

However, since $v\sim 2^{\frac{am}{2}}$, that effectively restricts $n$ to
$|n|\sim 2^{\frac{am}{2}}$ which by symmetry may be reduced to $n\sim
2^{\frac{am}{2}}$ thus providing an informal argument supporting our claim in
(134).

The formal, rigorous argument is presented now. First, recall from Section 3.4
that we only need to prove operator bounds for $\sum_{k}T_{m,k}$ in the
local-$L^{2}$ range (_i.e._ $2<p,q\leq\infty$) with a $2^{-\delta am}$ decay.
Thus, our claim is a consequence of the following dualized statement:

######  Proposition 4.1.

Let $\mathfrak{N}=\mathfrak{N}_{m}$ be the subset of indices868686Here we use
$A\ll_{a}B$ to denote the inequality $A\leq c_{a}B$ for a constant $c_{a}>0$
depending on $a$ and sufficiently small that any subsequent arguments
appealing to this inequality are enabled (with the obvious modifications for
the situation $A\gg_{a}B$).

$\mathfrak{N}:=\\{n\in\mathbb{Z}:|n|\ll_{a}2^{\frac{am}{2}}\textrm{ or
}|n|\gg_{a}2^{\frac{am}{2}}\\}.$

For each $k\in\mathbb{Z}$ and $m\in\mathbb{N}$ define now the trilinear
form878787This is obtained from (133) by restricting $n$ to $\mathfrak{N}$ and
then dualizing against the function $h$.

$\displaystyle\underline{\mathcal{L}}_{m,k}^{\mathfrak{N}}(f,g,h):=\sum_{\ell\in\mathbb{Z}}$
$\displaystyle\sum_{\begin{subarray}{c}n\in\mathfrak{N}\\\
p\in\mathbb{Z}\end{subarray}}\sum_{u,v\sim 2^{am\over
2}}\frac{1}{2^{\frac{am+2k}{4}}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p+n}\rangle$
$\displaystyle\cdot\int\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p}(x)h(x)\rho_{am-
ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x)dx.$

Then for every $N>0$ and any exponents $p,q,r$ such that $2<p,q\leq\infty$ and
$1/p+1/q=1/r$ we have

$\big{\|}\sum_{k\in\mathbb{Z}}\underline{\mathcal{L}}_{m,k}^{\mathfrak{N}}\big{\|}_{L^{p}\times
L^{q}\times L^{r^{\prime}}\to{\mathbb{C}}}\lesssim_{a,p,q,N}2^{-Nam}$

uniformly in $m\in\mathbb{N}$.

###### Proof.

Recalling the definition of $\nu(\xi)$ in (138) we have

$\nu^{\prime}(\xi)=n+c_{a}a^{\prime}2^{({am\over
2}-k)a^{\prime}}\Big{(}\frac{\xi}{\lambda(x)}\Big{)}^{{1\over{a-1}}}$

with $|\xi|\sim 2^{\frac{am}{2}}$ (because of the support of $\psi$); for
$n\in\mathfrak{N}$ the phase function has no critical point, since
$|\nu^{\prime}(\xi)|\sim 2^{\frac{am}{2}}$ if $|n|\ll_{a}2^{\frac{am}{2}}$ and
$|\nu^{\prime}(\xi)|\sim|n|$ if $|n|\gg_{a}2^{\frac{am}{2}}$. We can thus
summarize this as

$|\nu^{\prime}(\xi)|\sim(2^{\frac{am}{2}}+|n|).$

Therefore, repeated integration by parts in (137) yields for any888888The
value of $N$ is allowed to vary from line to line. $N>0$

$\rho_{am-
ak}(\lambda(x))|w_{k,n,v}^{e}(\lambda)(x)|\lesssim_{N}(2^{\frac{am}{2}}+|n|)^{-N}\rho_{am-
ak}(\lambda(x)).$

For any interval $I$, we introduce now the $L^{\infty}$-normalised weight
adapted to $I$ as

(141)
$\tilde{\chi}_{I}(x):=\Big{(}1+\frac{\mathrm{dist}(x,I)}{|I|}\Big{)}^{-M}$

for some large $M>0$ – also allowed to change from line to line. Recalling the
definition of our wave-packet in (123) we have

$|\check{\varphi}^{2u,p}_{\frac{am}{2}-k,2\ell-1}(x)|\lesssim
2^{\frac{am-2k}{4}}\tilde{\chi}_{I^{p}_{k}}(x),$

where $I^{p}_{k}$ denotes the interval

(142)
$I^{p}_{k}:=\\{x\in\mathbb{R}:2^{\frac{am}{2}-k}x-p\in[0,1]\\}=\Big{[}\frac{p}{2^{\frac{am}{2}-k}},\frac{p+1}{2^{\frac{am}{2}-k}}\Big{]}.$

Taking absolute values and applying the pointwise bounds above we have then
for the trilinear form

$\displaystyle|\underline{\mathcal{L}}_{m,k}^{\mathfrak{N}}(f,g,h)|\lesssim
2^{-k}\sum_{\ell\in\mathbb{Z}}$
$\displaystyle\sum_{\begin{subarray}{c}n\in\mathfrak{N}\\\
p\in\mathbb{Z}\end{subarray}}\sum_{u,v\sim 2^{am\over 2}}|\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p-n}\rangle||\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p+n}\rangle|$
$\displaystyle\cdot(2^{\frac{am}{2}}+|n|)^{-N}\int\tilde{\chi}_{I^{p}_{k}}(x)|h(x)|\rho_{am-
ak}(\lambda(x))dx,$

where the last factor does not depend anymore on $u,v,\ell$. Using Cauchy-
Schwarz in $u,v$ and $\ell$ yields without difficulty

$\displaystyle|\underline{\mathcal{L}}_{m,k}^{\mathfrak{N}}(f,g,h)|$
$\displaystyle\lesssim
2^{\frac{am}{2}-k}\sum_{\begin{subarray}{c}n\in\mathfrak{N}\\\
p\in\mathbb{Z}\end{subarray}}\Big{(}\int|f(y)|^{2}\tilde{\chi}_{I_{k}^{p-n}}(y)dy\Big{)}^{1/2}\Big{(}\int|g(z)|^{2}\tilde{\chi}_{I_{k}^{p+n}}(z)dz\Big{)}^{1/2}$
$\displaystyle\hskip
40.00006pt\cdot(2^{\frac{am}{2}}+|n|)^{-N}\int\tilde{\chi}_{I_{k}^{p}}(x)|h(x)|\rho_{am-
ak}(\lambda(x))dx.$

Now for every $x\in\mathbb{R}$, standard reasonings imply that

$\displaystyle\Big{(}\int|f(y)|^{2}\tilde{\chi}_{I_{k}^{p-n}}(y)dy\Big{)}^{1/2}\leq
2^{\frac{1}{2}(k-{am\over
2})}\Big{(}|n|+\frac{\text{dist}(x,I_{k}^{p})}{2^{{am\over
2}-k}}\Big{)}^{1\over 2}M_{2}(f)(x),$

with a similar inequality for the average on $g$ (here
$M_{2}(f)=M(|f|^{2})^{1/2}$). Deduce from here that

$\displaystyle|\underline{\mathcal{L}}_{m,k}^{\mathfrak{N}}(f,g,h)|$
$\displaystyle\lesssim\sum_{p\in\mathbb{Z}}\sum_{n\in\mathfrak{N}}(2^{\frac{am}{2}}+|n|)^{-N}\int\Big{(}|n|+\frac{\text{dist}(x,I_{k}^{p})}{2^{{am\over
2}-k}}\Big{)}M_{2}(f)(x)$ $\displaystyle\qquad\qquad\cdot
M_{2}(g)(x)\tilde{\chi}_{I_{k}^{p}}(x)|h(x)|\rho_{am-ak}(\lambda(x))dx$
$\displaystyle\lesssim 2^{-Nam}\sum_{p\in\mathbb{Z}}\int
M_{2}(f)(x)M_{2}(g)(x)\tilde{\chi}_{I_{k}^{p}}(x)|h(x)|\rho_{am-
ak}(\lambda(x))dx.$

Given the definition of $\tilde{\chi}_{I_{k}^{p}}$ the sum in $p$ is $\lesssim
1$. The factor of $\rho_{am-ak}(\lambda(x))$ enables us to further sum over
$k$ and thus

$\displaystyle\sum_{k}|\underline{\mathcal{L}}_{m,k}^{\mathfrak{N}}(f,g,h)|$
$\displaystyle\lesssim 2^{-Nam}\sum_{k\in\mathbb{Z}}\int
M_{2}(f)(x)M_{2}(g)(x)|h(x)|\rho_{am-ak}(\lambda(x))dx$ $\displaystyle\lesssim
2^{-Nam}\int M_{2}(f)(x)M_{2}(g)(x)|h(x)|dx.$

Via Hölder, the latter expression is now trivially bounded in the stated range
$2<p,q\leq\infty$ and $1/p+1/q=1/r$ thus ending our proof. ∎

We now record the dualized version of what we achieved via our reductions:

(143)
$\displaystyle\quad\underline{\mathcal{L}}(f,g,h):=\sum_{m\in\mathbb{N}}\underline{\mathcal{L}}_{m}(f,g,h):=\sum_{m\in\mathbb{N}}\sum_{k\in\mathbb{Z}}\underline{\mathcal{L}}_{m,k}(f,g,h):=\sum_{m\in\mathbb{N}}\sum_{k\in\mathbb{Z}}\langle
T_{m,k}(f,g),h\rangle=$
$\displaystyle\sum_{{m\in\mathbb{N}}\atop{k,\ell,r\in\mathbb{Z}}}\sum_{n,p,u,v\sim
2^{\frac{am}{2}}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle\langle
h,\frac{\rho_{am-
ak}(\lambda)}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}w_{k,n,v}^{e}(\lambda)\rangle.$

As already mentioned in Section 4.1, the major problem that we are confronted
with when dealing with the above model is the failure of the $m$-decay
absolute summability property – see the Appendix A for a proof. As a
consequence, we have to design an alternative route that will help us in
Section 5 to extract the cancellation encoded in the curvature of the phase in
our operator’s kernel. This route is the subject of our next section.

###  4.3. The high resolution model (II): a continuous phase-linearized
spatial model

####  4.3.1. Motivation and intuition

The aim of this subsection is to highlight a simple though deep manifestation
of the fact the trigonometric system is not an unconditional basis for
$L^{p}(\mathbb{T})$ whenever $1\leq p\not=2\leq\infty$. The example that we
will provide below reveals the subtle cancellation hidden in the Fourier
coefficients of a given function and serves as a memento that many of
otherwise simple objects when expressed as sums involving Fourier coefficients
give rise to only conditionally summable models. Then the challenge in
operating with such a model is how to properly regroup the terms (see _e.g._
the case of Littlewood-Paley theory) such that enough cancellation can be
captured within the newly formed terms.

However, as we will soon see, sometimes reshaping the entire problem from the
frequency to the physical side completely removes this difficult issue of
conditional summability. It is this last remark that we would particularly
like to emphasise via the example below. Its key underlying philosophy was
inspirational for our problem, and, in our search for proving the single scale
estimate in Section 5 motivated the idea of substituting the discrete phase-
linearized wave-packet model by the continuous phase-linearized spatial
model.898989For this substitution/transition between the two models one is
also invited to consult Section 4.3.2.

Let $F,G\in L^{2}(\mathbb{T})$ with their Fourier representation given by

$F(t)\simeq\sum_{n\in\mathbb{Z}}a_{n}e^{int}\>\>\>\textrm{and}\>\>\>G(t)\simeq\sum_{k\in\mathbb{Z}}b_{k}e^{ikt}.$

Then, a simple application of Hölder implies

(144) $FG\in L^{1}(\mathbb{T}),$

while from the basic properties of the Fourier series we have

(145)
$(FG)(t)\simeq\sum_{n\in\mathbb{Z}}(\sum_{k\in\mathbb{Z}}a_{n-k}b_{k})e^{int}.$

Taking now $H\in L^{\infty}(\mathbb{T})$ with
$H(t)\simeq\sum_{n\in\mathbb{Z}}c_{n}e^{int}$ we let

(146) $\Lambda(F,G,H):=((FG)*H)(0)=\int_{\mathbb{R}}F(t)G(t)H(-t)dt.$

Now, we conclude that

  * •

on the one hand, from (144), (145) and (146), we have

$\displaystyle\Lambda(F,G,H)\simeq\sum_{n,k\in\mathbb{Z}}a_{n-k}b_{k}c_{n},$
$\displaystyle|\Lambda(F,G,H)|\leq\|F\|_{L^{2}(\mathbb{T})}\|G\|_{L^{2}(\mathbb{T})}\|H\|_{L^{\infty}(\mathbb{T})}.$

  * •

on the other hand, for generic
$\|F\|_{L^{2}(\mathbb{T})}=\|G\|_{L^{2}(\mathbb{T})}=\|H\|_{L^{\infty}(\mathbb{T})}=1$

$\sum_{n,k\in\mathbb{Z}}|a_{n-k}||b_{k}||c_{n}|=\infty.$

As already hinted earlier, the valuable lesson that we are learning from the
above example is that the sums involving (product) of Fourier coefficients (of
some functions) are encoding hidden cancellations that can be often exploited
by reframing the problem from the Fourier (frequency) setting into a spatial
setting.

With this in mind, we return to the high resolution wave-packet discretized
model as reflected in (136) and notice that we are in a very similar
situation: our model may be visualised as a superposition of _product of local
Fourier coefficients_ attached to $f$ and $g$ (and with a modified weight for
$h$ if regarded in the dualized version).

Thus, judging by analogy with the expression in (145), we are invited to
consider909090In this motivational argument we are fixing all the parameters
but $u$. the following elementary block within (136):

(147) $\displaystyle
S_{k,\ell,r}^{n,v,p}(f,g)(x):=\sum_{u\in\mathbb{Z}}\frac{1}{2^{\frac{am+2k}{4}}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}(x),$

whose joint frequency support regarded from the input side, i.e. the support
of $(\hat{f},\hat{g})$, is depicted in Figure 6.

Recalling now (123) and (131) we further deduce that

(148) $\displaystyle S_{k,\ell,r}^{n,v,p}(f,g)(x)$
$\displaystyle=\frac{1}{2^{k}}\Big{(}\sum_{u\in\mathbb{Z}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle
e^{i(2^{\frac{am}{2}-k}x-p_{r})2u}\Big{)}$ $\displaystyle\hskip
20.00003pt\cdot\check{\varphi}(2^{\frac{am}{2}-k}x-p_{r})e^{i2^{\frac{am}{2}}(2^{\frac{am}{2}-k}x-p_{r})(2\ell-2)}.$

$v$ fixed in the square ${\footnotesize\begin{aligned}
&[2^{am-k}(\ell+1),2^{am-k}(\ell+2)]\\\
&\times[2^{am-k}(\ell-1),2^{am-k}\ell]\end{aligned}}$ Figure 6. Frequency
portrait of the information associated with $S_{k,\ell,r}^{n,v,p}$. It is at
this level that we manage to exploit the curvature of the phase (see Section
5) and also obtain the frequency localization of $h\cdot w_{k,n,v}^{e}$ (see
Section 6).

Analyzing the right-hand side of (148) we deduce the following time-frequency
characteristics:

  * •

in space, thanks to the presence of $\check{\varphi}$, the information is
concentrated within $I_{k}^{p_{r}}$;

  * •

in frequency – see Figure 6 – the joint support in $(\xi,\eta)$ encoded within
the wave-packets contributing to the sum stays within the strip

$\xi-\eta=-2^{\frac{am}{2}-k}2v+O(2^{\frac{am}{2}-k}).$

Thus, inspecting (148) in parallel with (145), intuitively we expect

(149)
$S_{k,\ell,r}^{n,v,p}(f,g)(x)\approx\frac{1}{2^{k}}e^{2inv}\Big{(}\int_{I_{k}^{n}}f_{\ell+1}(x-t)g_{\ell-1}(x+t)e^{-i2^{\frac{am}{2}-k}2vt}dt\Big{)}\mathbf{1}_{I_{k}^{p_{r}}}(x).$

The rigorous realization of (149) is the subject of our next section.

####  4.3.2. Transition from the discrete phase-linearized wave-packet model
to the continuous phase-linearized spatial model

In this section we focus on making (149) precise. For reasons that will become
transparent momentarily, we will reshape the information in (147) by isolating
the Gabor coefficient of $f$, which we write as:

$\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle=\int_{\mathbb{R}}f_{\ell+1}(s)\overline{\check{\varphi}_{\frac{am}{2}-k,\ell}^{0,p_{r}-n}}(s)e^{-i(2^{\frac{am}{2}-k}s-(p_{r}-n))(u-v)}\,ds.$

After the change of variable $2^{\frac{am}{2}-k}s-(p_{r}-n)=y$ and a
periodization argument (similar with the one used in order to prove the
Poisson summation formula) this becomes

$\frac{1}{2^{\frac{am-2k}{4}}}\int_{\mathbb{T}}\Big{(}\sum_{j_{1}\in\mathbb{Z}}f_{\ell+1}\big{(}\frac{y+j_{1}+(p_{r}-n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(y+j_{1})\Big{)}e^{-iy(u-v)}dy,$

where here
$\check{\varphi}_{0,\ell}^{0,0}(y)=\check{\varphi}(y)e^{iy2^{\frac{am}{2}}(\ell-1)}$.

Applying a similar reasoning for the coefficient $\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle$ we can
now rewrite (147) as

(150) $\displaystyle
S_{k,\ell,r}^{n,v,p}(f,g)(x)=\frac{1}{2^{\frac{am+2k}{4}}}\frac{1}{2^{\frac{am}{2}-k}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{0,p_{r}}(x)$
$\displaystyle\cdot\sum_{u\in\mathbb{Z}}\Big{(}\int_{\mathbb{T}}\Big{(}\sum_{j_{1}\in\mathbb{Z}}f_{\ell+1}\big{(}\frac{y+j_{1}+(p_{r}-n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(y+j_{1})\Big{)}e^{-iy(u-v)}dy\Big{)}$
$\displaystyle\cdot\Big{(}\int_{\mathbb{T}}\Big{(}\sum_{j_{2}\in\mathbb{Z}}g_{\ell-1}\big{(}\frac{y+j_{2}+(p_{r}+n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(y+j_{2})\Big{)}e^{-iy(u+v)}dy\Big{)}e^{i2u(2^{\frac{am}{2}-k}x-p_{r})}.$

Once at this point, observe that we have the following identity: if $a,b\in
L^{2}(\mathbb{T})$, then

(151)
$\displaystyle\sum_{u\in\mathbb{Z}}\Big{(}\int_{\mathbb{T}}a(y)e^{-iy(u-v)}dy\Big{)}\cdot\Big{(}\int_{\mathbb{T}}b(y)e^{-iy(u+v)}dy\Big{)}e^{i2ux}$
$\displaystyle=\int_{\mathbb{T}}a(x-t)b(x+t)e^{-2ivt}dt.$

Applying now (151) for
$a(y):=\sum_{j_{1}\in\mathbb{Z}}f_{\ell+1}\big{(}\frac{y+j_{1}+(p_{r}-n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(y+j_{1})$
and
$b(y):=\sum_{j_{2}\in\mathbb{Z}}g_{\ell-1}\big{(}\frac{y+j_{2}+(p_{r}+n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(y+j_{2})$
and inserting the result in (150) we deduce

(152) $\displaystyle
S_{k,\ell,r}^{n,v,p}(f,g)(x)=\sum_{j_{1},j_{2}\in\mathbb{Z}}\frac{1}{2^{\frac{am+2k}{4}}}\frac{1}{2^{\frac{am}{2}-k}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{0,p_{r}}(x)$
$\displaystyle\cdot\int_{\mathbb{T}}\Big{(}f_{\ell+1}\big{(}\frac{2^{\frac{am}{2}-k}x-p_{r}-t+j_{1}+(p_{r}-n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(2^{\frac{am}{2}-k}x-p_{r}-t+j_{1})\Big{)}$
$\displaystyle\cdot\Big{(}g_{\ell-1}\big{(}\frac{2^{\frac{am}{2}-k}x-p_{r}+t+j_{2}+(p_{r}+n)}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(2^{\frac{am}{2}-k}x-p_{r}+t+j_{2})\Big{)}e^{-2ivt}dt.$

Finally, after applying the change of variable $t\mapsto
2^{\frac{am}{2}-k}t-n$, we conclude

(153) $\displaystyle
S_{k,\ell,r}^{n,v,p}(f,g)(x)=\sum_{j_{1},j_{2}\in\mathbb{Z}}\frac{1}{2^{k}}\check{\varphi}(2^{\frac{am}{2}-k}x-p_{r})e^{i(2^{\frac{am}{2}-k}x-p_{r})2^{\frac{am}{2}}(2\ell-2)}$
$\displaystyle\cdot\int_{I^{n}_{k}}f_{\ell+1}\big{(}x-t+\frac{j_{1}}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(2^{\frac{am}{2}-k}(x-t)-p_{r}+n+j_{1})$
$\displaystyle\cdot
g_{\ell-1}\big{(}x+t+\frac{j_{2}}{2^{\frac{am}{2}-k}}\big{)}\overline{\check{\varphi}_{0,\ell}^{0,0}}(2^{\frac{am}{2}-k}(x+t)-p_{r}-n+j_{2})e^{-2iv(2^{\frac{am}{2}-k}t-n)}\,dt$
$\displaystyle=:\sum_{j_{1},j_{2}\in\mathbb{Z}}\tilde{S}_{k,p_{r}}^{n,v}[j_{1},j_{2}](f_{\ell+1},g_{\ell-1})(x).$

######  Observation 4.2.

Formulation (153) ends the rigorous transition from the discrete wave-packet
model to the continuous spatial model. As already mentioned, we will make use
of it in Section 5 when proving the single estimate (183) and then in Section
6 when working at a refined version of (183). In all these instances one could
work directly with the formulation (153) proved above. However, for notational
simplicity and fluency of the exposition we will reduce (153) to a simpler,
approximative form – represented by (154) below – and this last formulation
will be embraced from now on in our paper. While we will present an outline of
why indeed it is enough to only work with (154) we will leave the technical
details to the reader.

In light of the above observation, we now provide a brief argumentation for
the following claim:

If $x\in I^{p_{r}}_{k}$ then without loss of generality one can assume from
now on that

(154)
$S_{k,\ell,r}^{n,v,p}(f,g)(x)=\tilde{S}_{k}^{n,v}(f_{\ell+1,r},g_{\ell-1,r})(x),$

with

(155)
$\tilde{S}_{k}^{n,v}(F,G)(x):=\frac{1}{2^{k}}\int_{I^{n}_{k}}F\big{(}x-t\big{)}G\big{(}x+t\big{)}e^{-2iv(2^{\frac{am}{2}-k}t-n)}dt.$

and

(156) $\displaystyle f_{\ell+1,r}:=\sum_{u\in\mathbb{Z}}\sum_{p\sim
2^{am/2}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}\rangle\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}=\sum_{u,p\sim
2^{am/2}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}\rangle\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}},$
$\displaystyle g_{\ell-1,r}:=\sum_{u\in\mathbb{Z}}\sum_{p\sim 2^{am/2}}\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}\rangle\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}=\sum_{u,p\sim
2^{am/2}}\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}\rangle\check{\varphi}_{\frac{am}{2}-k,\ell}^{u,p_{r}}.$

Indeed, we first start by noticing that since
$\check{\varphi}_{0,1}^{0,0}(y)=\check{\varphi}(y)$, we have

$\displaystyle\overline{\check{\varphi}_{0,\ell}^{0,0}}(2^{\frac{am}{2}-k}(x-t)-p_{r}+n+j_{1})\overline{\check{\varphi}_{0,\ell}^{0,0}}(2^{\frac{am}{2}-k}(x+t)-p_{r}-n+j_{2})$
$\displaystyle=$
$\displaystyle\overline{\check{\varphi}}(2^{\frac{am}{2}-k}(x-t)-p_{r}+n+j_{1})\overline{\check{\varphi}}(2^{\frac{am}{2}-k}(x+t)-p_{r}-n+j_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\cdot
e^{-i(2(2^{\frac{am}{2}-k}x-p_{r})+j_{1}+j_{2})2^{\frac{am}{2}}(\ell-1)}.$

Isolating now the first term in the last line above and ignoring the
conjugation we use the information that $\check{\varphi}$ is compactly
supported in frequency for applying a standard Taylor series argument – here
$x\in I^{p_{r}}_{k}$ is assumed to be fixed – that has as an effect the
tensorization (separation of variables in $x$ and $t$):

$\displaystyle\check{\varphi}(2^{\frac{am}{2}-k}(x-t)-p_{r}+n+j_{1})=\int_{\mathbb{R}}e^{i\xi(2^{\frac{am}{2}-k}(x-t)-p_{r}+n+j_{1})}\varphi(\xi)d\xi$
$\displaystyle=\sum_{k_{1},k_{2}\in\mathbb{Z}}\int_{\mathbb{R}}\frac{\left(i\xi(2^{\frac{am}{2}-k}x-p_{r}+1)\right)^{k_{1}}}{k_{1}!}\frac{\left(-i\xi(2^{\frac{am}{2}-k}t-n+1)\right)^{k_{2}}}{k_{2}!}e^{i\xi
j_{1}}\varphi(\xi)d\xi$
$\displaystyle=\sum_{k_{1},k_{2}\in\mathbb{Z}}\frac{\left(2^{\frac{am}{2}-k}x-p_{r}+1\right)^{k_{1}}}{k_{1}!}\frac{\left(2^{\frac{am}{2}-k}t-n+1\right)^{k_{2}}}{k_{2}!}\check{\varphi}_{k_{1},k_{2}}(j_{1}),$

where in the last line $\check{\varphi}_{k_{1},k_{2}}$ is a Schwartz function
depending on $k_{1},k_{2}$ such that
$|\check{\varphi}_{k_{1},k_{2}}(j_{1})|\lesssim\frac{C^{k_{1}+k_{2}}}{|j_{1}|^{10}+1}$
with $C>0$ a suitable absolute constant.

Deduce thus from here that for any $j_{1},j_{2}\in\mathbb{Z}$ and $x\in
I^{p_{r}}_{k}$ we have

(157)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\tilde{S}_{k,p_{r}}^{n,v}[j_{1},j_{2}](f_{\ell+1},g_{\ell-1})(x)=$
$\displaystyle\frac{1}{2^{k}}e^{-i(j_{1}+j_{2})2^{\frac{am}{2}}(\ell-1)}\sum_{{k_{1},l_{1}\in\mathbb{N}}\atop{k_{2},l_{2}\in\mathbb{N}}}\frac{\left(2^{\frac{am}{2}-k}x-p_{r}+1\right)^{k_{1}+l_{1}}}{k_{1}!l_{1}!}\check{\varphi}(2^{\frac{am}{2}-k}x-p_{r})\check{\varphi}_{k_{1},k_{2}}(j_{1})\check{\varphi}_{l_{1},l_{2}}(j_{2})$
$\displaystyle\cdot\int_{I^{n}_{k}}f_{\ell+1}\big{(}x-t+\frac{j_{1}}{2^{\frac{am}{2}-k}}\big{)}g_{\ell-1}\big{(}x+t+\frac{j_{2}}{2^{\frac{am}{2}-k}}\big{)}e^{-2iv(2^{\frac{am}{2}-k}t-n)}\frac{\left(2^{\frac{am}{2}-k}t-n+1\right)^{k_{2}+l_{2}}}{k_{2}!l_{2}!}dt$

A simple inspection of (157) reveals the following:

  * •

the presence of the term
$\check{\varphi}_{k_{1},k_{2}}(j_{1})\check{\varphi}_{l_{1},l_{2}}(j_{2})$
provides a polynomial decay in terms of the size of the parameters
$j_{1},j_{2}$ thus allowing us to sum in $j_{1}$ and $j_{2}$ in (153);

  * •

for $x\in I^{p_{r}}_{k}$ the expressions $2^{\frac{am}{2}-k}x-p_{r}+1$ and
$\check{\varphi}(2^{\frac{am}{2}-k}x-p_{r})$ are of size $\approx 1$;

  * •

for $t\in I^{n}_{k}$ the expression $2^{\frac{am}{2}-k}t-n+1$ is of size
$\approx 1$.919191It might be instructive to verify – for example – the proof
of (191) in Section 5 to see why factors such as
$\frac{\left(2^{\frac{am}{2}-k}t-n+1\right)^{k_{2}+l_{2}}}{k_{2}!l_{2}!}$ and
$\frac{\left(2^{\frac{am}{2}-k}x-p_{r}+1\right)^{k_{1}+l_{1}}}{k_{1}!l_{1}!}$
are indeed harmless.

The above explains why, for $x\in I^{p_{r}}_{k}$, the right-hand side in (153)
can be reduced to the (dominant) term
$\tilde{S}_{k,p_{r}}^{n,v}[0,0](f_{\ell+1},g_{\ell-1})$ and moreover, why the
latter can be further reduced to the expression
$\tilde{S}_{k}^{n,v}(f_{\ell+1},g_{\ell-1})$ which, given the imposed range
for $x$, is well approximated929292Strictly speaking
$\mathbf{1}_{[r2^{k},(r+1)2^{k})}(x)\tilde{S}_{k}^{n,v}(f_{\ell+1},g_{\ell-1})(x)$
is a superposition of terms of the form
$c_{r-r^{\prime}}\tilde{S}_{k}^{n,v}(f_{\ell+1,r^{\prime}},g_{\ell-1,r^{\prime}})(x)$,
$r^{\prime}\in\mathbb{Z}$ with
$\\{c_{r^{\prime}}\\}_{r^{\prime}\in\mathbb{Z}}\in\ell^{\infty}(\mathbb{Z})$ a
fast-decaying sequence obeying $c_{r^{\prime}}=O(|r^{\prime}|^{-2})$ as
$|r^{\prime}|\rightarrow\infty$. by
$\tilde{S}_{k}^{n,v}(f_{\ell+1,r},g_{\ell-1,r})(x)$, thus certifying (154).

####  4.3.3. The continuous phase-linearized spatial model revisited – bird’s
eye view. The heuristic equivalence of the two high resolution models

This section is meant solely for presenting an alternate, more direct way of
visualizing our continuous model that in particular reveals the important role
played by the linearization process. Indeed, returning to the original form of
our operator $T_{m,k}$ we show the very natural way in which one can achieve a
decomposition of $T_{m,k}$ in elementary blocks resembling the expression in
the right-hand side of (149). Since the previous section was covered in great
detail, the description provided here will be expository, and hence stripped
of technicalities.

These being said, let us recall the original formulation

(158) $\displaystyle T_{m,k}(f,g)(x)$ $\displaystyle=\rho_{am-
ak}(\lambda(x))\int f(x-t)g(x+t)e^{i\lambda(x)t^{a}}\rho(2^{-k}t)\frac{dt}{t}$
$\displaystyle\approx\frac{1}{2^{k}}\rho_{am-
ak}(\lambda(x))\int_{[2^{k},2^{k+1}]}f(x-t)g(x+t)e^{i\lambda(x)t^{a}}dt.$

Following the same phase linearization approach as the one in Section 4.2.3
with the only distinction that we are now working in space as opposed to
frequency there, we first isolate the phase of our kernel

$\tau_{x}(t):=\lambda(x)t^{a},$

and notice that in the non-trivial regime deduced from (158) we must have
$|t|\approx 2^{k}$ and $|\lambda(x)|\approx 2^{a(m-k)}$. As a consequence, we
further have that

$\tau_{x}^{\prime\prime}(t)=a(a-1)\lambda(x)t^{a-2}\approx_{a}2^{am-2k},$

and hence, in order to focus only on the linear growth around a given point
$t_{0}\in[2^{k},2^{k+1}]$ one must have the size of the spatial region
restricted to $|t-t_{0}|\lesssim 2^{k}2^{-\frac{am}{2}}$.

Thus, recalling that $I^{n}_{k}$ stands for
$\Big{[}\frac{n}{2^{\frac{am}{2}-k}},\frac{n+1}{2^{\frac{am}{2}-k}}\Big{]}$,
we have that

(159) $T_{m,k}(f,g)(x)\approx\frac{1}{2^{k}}\rho_{am-
ak}(\lambda(x))\sum_{n\sim
2^{\frac{am}{2}}}\int_{I^{n}_{k}}f(x-t)g(x+t)e^{i\lambda(x)t^{a}}dt.$

Given the very construction of $I^{n}_{k}$, taking
$t^{n}_{k}:=n2^{k-\frac{am}{2}}$ and applying a Taylor series argument around
the chosen point $t^{n}_{k}$, we have that

(160)
$\tau_{x}(t)=\lambda(x)t^{a}=\lambda(x)(t^{n}_{k})^{a}+(t-t^{n}_{k})a\lambda(x)(t^{n}_{k})^{a-1}+O(1),$

as long as $|\lambda(x)|\approx 2^{a(m-k)}$ and $t\in I^{n}_{k}$.

Thus, ignoring939393This can be done rigorously via a Taylor series argument.
the $O(1)$ term appearing in the phase, we have reached the final form of our
(continuous) spatial linearized model which is given by

(161) $\displaystyle T_{m,k}(f,g)(x)\approx\frac{1}{2^{k}}\rho_{am-
ak}(\lambda(x))$ $\displaystyle\sum_{p\in\mathbb{Z}}\sum_{n\sim
2^{\frac{am}{2}}}e^{i(1-a)\lambda(x)(t^{n}_{k})^{a}}\chi_{I^{p}_{k}}(x)$
$\displaystyle\cdot\Big{(}\int_{I^{n}_{k}}f(x-t)g(x+t)e^{ia\lambda(x)(t^{n}_{k})^{a-1}t}dt\Big{)}.$

We end this section with an informative discussion on how can one visualize
the equivalence between the continuous model expressed above in (161) and the
discretized wave-packet model expressed in (136). Starting from the latter and
employing definition (147) we have

(162) $\displaystyle T_{m,k}(f,g)(x)=$
$\displaystyle\sum_{\ell,r\in\mathbb{Z}}\sum_{n,p\sim
2^{\frac{am}{2}}}\sum_{u,v\sim 2^{\frac{am}{2}}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}(x)w_{k,n,v}^{e}(\lambda)(x)$
$\displaystyle=$ $\displaystyle\sum_{\ell,r\in\mathbb{Z}}\sum_{n,p\sim
2^{\frac{am}{2}}}\sum_{v\sim
2^{\frac{am}{2}}}S_{k,\ell,r}^{n,v,p}(f,g)(x)\rho_{am-
ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x).$

Now based on the heuristic in (149) we further have

(163) $\displaystyle T_{m,k}(f,g)(x)\approx$
$\displaystyle\sum_{\ell,r\in\mathbb{Z}}\sum_{n,p\sim
2^{\frac{am}{2}}}\sum_{v\sim
2^{\frac{am}{2}}}\mathbf{1}_{I_{k}^{p_{r}}}(x)\rho_{am-
ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x)$
$\displaystyle\cdot\frac{1}{2^{k}}e^{2inv}\left(\int_{I_{k}^{n}}f_{\ell+1}(x-t)g_{\ell-1}(x+t)e^{-i2^{\frac{am}{2}-k}2vt}dt\right).$

Once at this point, relying on the time-frequency correlation as captured by
(140), we notice that the main contribution is attained when

(164) $-2v\approx\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{a(\frac{am}{2}-k)}},$

while from relations (137)–(140) via a stationary phase argument we deduce

(165) $\rho_{am-ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x)\approx\rho_{am-
ak}(\lambda(x))e^{i(1-\frac{1}{a^{\prime}})\bar{c}_{a}\frac{n^{a}\lambda(x)}{2^{a(\frac{am}{2}-k)}}}\varphi\big{(}\xi+\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{a(\frac{am}{2}-k)}}\big{)}.$

Putting together (163), (164) and (165), at the informal level, we conclude

(166) $\displaystyle T_{m,k}(f,g)(x)$
$\displaystyle\approx\sum_{\ell,r\in\mathbb{Z}}\sum_{n,p\sim
2^{\frac{am}{2}}}\mathbf{1}_{I_{k}^{p_{r}}}(x)\rho_{am-
ak}(\lambda(x))\frac{1}{2^{k}}e^{-i\frac{1}{a^{\prime}}\bar{c}_{a}\frac{n^{a}\lambda(x)}{2^{a(\frac{am}{2}-k)}}}$
$\displaystyle\hskip
30.00005pt\cdot\Big{(}\int_{I_{k}^{n}}f_{\ell+1}(x-t)g_{\ell-1}(x+t)e^{i2^{\frac{am}{2}-k}\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{a(\frac{am}{2}-k)}}t}dt\Big{)},$

thus certifying for $\bar{c}_{a}=a$ the equivalence with the formulation given
in (161) as claimed.

###  4.4. A brief contrasting conclusion regarding the two high resolution
models

Our intention here is to summarize the need for utilizing both phase-
linearized high-resolution models constructed earlier:

  * •

On the one hand – see Section 5, the integral formulation in (161) captures
enough cancellation to yield the desired single scale version of (69), that is

(167) $\|T_{m,k}(f,g)\|_{L^{r}}\lesssim_{a,p,q}2^{-\delta
am}\|f\|_{L^{p}}\|g\|_{L^{q}},$

even if one takes absolute values in the summand of (161).

As already explained, (167) or its dual correspondent cannot be though
attained if one ignores the signs of the Gabor coefficients in either (134) or
in its dual version (143), respectively.

  * •

On the other hand – see Sections 6 and 7, for upgrading the single scale
estimate (167) to the global estimate (69) one has to involve the interactions
among the wave-packet associated with the input functions $f,g$ and $h$, thus
necessitating the use of the first model operator discussed in Section 4.2
that will now need to be reshaped in order to exploit the BHT-like structure
of our operator – this latter key step is part of the design of the low
resolution model introduced in the next section.

###  4.5. The low resolution model: identifying the underlying BHT structures

In this section we highlight the built-in BHT frame as suggested by the zero-
curvature features (P1) and (P2) and set the scene for the multi-scale time-
frequency analysis that will be performed later on in Section 7. Based on the
previous reasonings, we are naturally led to consider two kinds of time-
frequency tiling:

  * •

the first one captures the local, single-scale behavior949494As the name
suggests, this discretization that is part of the high resolution analysis
concerns the local properties of our operator when restricted to a single “big
frequency square” – see Figures 2 and 3. This process has as a result the
subdivision of such a big frequency square into smaller, same-scale frequency
squares. Of course, each such big frequency square comes with its own family
of smaller, uniform-in-scale frequency squares and as a result at the global
level we will deal with a multiscale problem whose aim is to quantify the
interactions among these families of smaller frequency squares. of our
operator and involves tiles of area $1$; this model is simply the geometric,
tile rephrasing of the high resolution wave-packet model discussed in Section
4.2.

  * •

the second one accounts for the global, multi-scale behavior of our operator
and consists of tiles having area $2^{am}$. This model, referred to as the low
resolution wave-packet model, is the expression of the BHT features of
$BC^{a}$, and amounts to regrouping the terms within the first model above in
order to later exploit the almost orthogonality along multiple scales via tree
structures.

####  4.5.1. The bilinear Hilbert transform frequency portrait: identifying
the structure of the time-frequency regions

The initial decomposition ${1\over t}=\sum_{k\in\mathbb{Z}}{\rho(2^{-k}t)\over
t}$ of the singular kernel, followed by the level set decomposition of the
phase of the multiplier for the high oscillatory component, and the stationary
phase analysis led to the study of the operator
$T_{m}=\sum_{k\in\mathbb{Z}}T_{m,k}$, with each individual term $T_{m,k}$ (as
defined in (120)) having frequency information supported in the region
$\\{\xi-\eta\sim 2^{am-k}\\}$. The subsequent step, designed for independently
extracting the frequency information for $f$ and $g$, led to considering “big
square” frequency regions

(168)
$\big{[}2^{am-k}(\ell+1),2^{am-k}(\ell+2)\big{]}\times\big{[}2^{am-k}(\ell-1),2^{am-k}\ell\big{]}.$

As $k$ and $\ell$ vary over the set of integers, this collection of squares
identifies with (a part of) a Whitney decomposition of the region
$\\{\xi>\eta\\}$ that may be interpreted as a $2^{am}$-dilation of the
classical frequency portrait of the bilinear Hilbert transform.959595The
multiplier associated to BHT is $sgn(\xi-\eta)$, and $\\{\xi=\eta\\}$
represents its singularity set relative to which one performs the above
mentioned Whitney decomposition. Thus, inspired by and in parallel with the
standard BHT analysis, we expect the collection induced by the squares in
(168) to play an analogue role in the analysis of our operator $BC^{a}$.

However, unlike the BHT case, when analyzing the information contained in the
above frequency regions (168) we will not involve wave-packets adapted to the
scale of these big squares. Instead, in order to be able to exploit the
curvature component of our $BC^{a}$ which is responsible for the strategy
presented in Section 2.1 and particularly (69), our time-frequency analysis
relies on wave-packets adapted to the smaller scale $2^{{am\over 2}-k}$ as
indicated by the LGC methodology. This produces

(169) $\displaystyle T_{m,k,\ell}(f,g)(x)=\sum_{\begin{subarray}{c}n\sim
2^{\frac{am}{2}}\\\
\bar{p}\in\mathbb{Z}\end{subarray}}\sum_{u,v\in\mathbb{Z}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,\bar{p}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,\bar{p}+n}\rangle$
$\displaystyle\cdot\frac{\rho_{am-
ak}(\lambda(x))}{2^{\frac{am+2k}{4}}}\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,\bar{p}}(x)w_{k,n,v}^{e}(\lambda)(x)$
,

which has the frequency information supported in (168). In order to control
the information contained in the Whitney squares, a certain spatial
localization that indicates an (almost) orthogonality relation between objects
having the same frequency support is necessary.

Notice that for every $\bar{p}\in\mathbb{Z}$, the spatial information for $f$
is adapted to $\Big{[}\frac{\bar{p}-n}{2^{{am\over
2}-k}},\frac{\bar{p}-n+1}{2^{{am\over 2}-k}}\Big{]}$, and that of $g$ to
$\Big{[}\frac{\bar{p}+n}{2^{{am\over 2}-k}},\frac{\bar{p}+n+1}{2^{{am\over
2}-k}}\Big{]}$. Since $n\sim 2^{am\over 2}$, a $2^{am\over 2}$-separation
between the values of $\bar{p}$ warrants separation for not only the spatial
information for $f$, but also for that of $g$ and of the output
$T_{m,k,\ell}$. This leads to the regrouping mentioned in (136), the
definition (135) and the representation of $T_{m}$ in the form of
$T_{m}(f,g)=\sum_{k,\ell,r\in\mathbb{Z}}T_{m,k,\ell,r}(f,g)$, where

(170) $\displaystyle T_{m,k,\ell,r}(f,g)(x)$
$\displaystyle:=\sum_{\begin{subarray}{c}p\sim 2^{\frac{am}{2}}\\\ n\sim
2^{\frac{am}{2}}\end{subarray}}\sum_{\begin{subarray}{c}u\sim
2^{\frac{am}{2}}\\\ v\sim
2^{\frac{am}{2}}\end{subarray}}\frac{1}{2^{\frac{am+2k}{4}}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle$
$\displaystyle\hskip
70.0001pt\cdot\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}(x)\rho_{am-
ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x).$

The spatial information for $f$ relevant to $T_{m,k,\ell,r}$ is thus
concentrated on

$\bigcup_{p,n\sim 2^{am\over 2}}\Big{[}\frac{p_{r}-n}{2^{{am\over
2}-k}},\frac{p_{r}-n+1}{2^{{am\over 2}-k}}\Big{]},$

which can be approximated969696Up to considering $O(1)$ similar terms. by
$[r2^{k},(r+1)2^{k}]$. Similarly, the spatial information for $g$ is, for the
most part, contained in

$\bigcup_{p,n\sim 2^{am\over 2}}\Big{[}\frac{p_{r}+n}{2^{{am\over
2}-k}},\frac{p_{r}+n+1}{2^{{am\over 2}-k}}\Big{]},$

and this is also comparable to $[r2^{k},(r+1)2^{k}]$.

In conclusion, for different values of $r$, the spatial information for $f$
encoded in $T_{m,k,\ell,r}(f,g)$ is located on disjoint intervals; the same
holds for $g$ and for the output $T_{m,k,\ell,r}(f,g)$. On the other hand, for
a fixed $r$, the spatial information for $f$, $g$ and the output
$T_{m,k,\ell,r}(f,g)$ is adapted to the same interval $[r2^{k},(r+1)2^{k}]$.

The regrouping in (136), which consisted in gathering $2^{am\over 2}$
consecutive integer values of $p$, had two important consequences: (i) the
time-frequency information for any of $f,g$ or $T_{m,k,\ell,r}(f,g)$ featured
in $T_{m,k,\ell,r}$, uniquely determines the time-frequency information for
the remaining input/output data; (ii) as $\ell$ and $r$ vary, the time-
frequency information is associated to disjoint regions.

So the analogue of the tri-tiles appearing in the BHT analysis are precisely
the time-frequency regions of area $2^{am}$

(171) $\displaystyle P_{1}$
$\displaystyle:=I_{P}\times\boldsymbol{\omega}_{P_{1}}:=[r2^{k},(r+1)2^{k}]\times\big{[}2^{am-k}(\ell+1),2^{am-k}(\ell+2)\big{]}$
$\displaystyle P_{2}$
$\displaystyle:=I_{P}\times\boldsymbol{\omega}_{P_{2}}:=[r2^{k},(r+1)2^{k}]\times\big{[}2^{am-k}(\ell-1),2^{am-k}\ell\big{]},$
$\displaystyle P_{3}$
$\displaystyle:=I_{P}\times\boldsymbol{\omega}_{P_{3}}:=[r2^{k},(r+1)2^{k}]\times\big{[}2^{am-k}(2\ell),2^{am-k}(2\ell+1)\big{]},$

where $k,\ell,r$ vary in $\mathbb{Z}$. We denote
${{\mathbb{P}}_{\textrm{BHT}}}$ the collection of such tri-tiles
$P:=(P_{1},P_{2},P_{3})$, in clear analogy to the bilinear Hilbert transform
operator:

(172) ${{\mathbb{P}}_{\textrm{BHT}}}:=\\{P=(P_{1},P_{2},P_{3})|\text{ with
}P_{1},P_{2},P_{3}\text{ given by }\eqref{eq:tri:tiles:def:S4},\text{ for
}k,\ell,r\in\mathbb{Z}\\}.$

Since each of the $P_{j}$ has area $2^{am}$, we refer to them as “expanded
(tri-)tiles”.

In fact, ${{\mathbb{P}}_{\textrm{BHT}}}$ is characterized (such a collection
is sometimes called of _rank 1_) by the following properties:

  * (i)

the collection of frequency squares
$(\boldsymbol{\omega}_{P_{1}}\times\boldsymbol{\omega}_{P_{2}})_{P\in{{\mathbb{P}}_{\textrm{BHT}}}}$
is a Whitney covering of the half-plane $\\{\xi>\eta\\}$, which means

(173)
$\text{dist}(\boldsymbol{\omega}_{P_{1}}\times\boldsymbol{\omega}_{P_{2}},\Gamma)\sim|\boldsymbol{\omega}_{P_{1}}|=|\boldsymbol{\omega}_{P_{2}}|,$

where $\Gamma:=\\{\xi=\eta\\}$ is the boundary of the half-plane;

  * (ii)

For $P\in{{\mathbb{P}}_{\textrm{BHT}}}$, the three tiles $P_{1},P_{2},P_{3}$
have the same spatial support $I_{P}$ and are all of area $2^{am}$;

  * (iii)

For $P\in{{\mathbb{P}}_{\textrm{BHT}}}$,
$\boldsymbol{\omega}_{P_{3}}\sim\boldsymbol{\omega}_{P_{1}}+\boldsymbol{\omega}_{P_{2}}$,
i.e. $\boldsymbol{\omega}_{P_{3}}$ is an interval of the same length as
$\boldsymbol{\omega}_{P_{1}}$ and $\boldsymbol{\omega}_{P_{2}}$, from the same
dyadic lattice, and contained in
$\boldsymbol{\omega}_{P_{1}}+\boldsymbol{\omega}_{P_{2}}$.

####  4.5.2. Tile-rephrasing of the wave-packet information

In this section we reinterpret the discretization (170) of our operator from a
geometric point of view by phrasing the action of the wave-packets in terms of
tiles.

We start by isolating the wave-packet
$\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}$ part of the term
$\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle$ in
(170). We remark now that its time-frequency portrait is captured within the
area one tile

$\displaystyle s_{1}$
$\displaystyle:=I^{p_{r}-n}_{k}\times\omega^{u_{\ell}-v}_{k}$
$\displaystyle:=\Big{[}(r-1)2^{k}+\frac{p-n}{2^{\frac{am}{2}-k}},(r-1)2^{k}+\frac{p-n+1}{2^{\frac{am}{2}-k}}\Big{]}$
$\displaystyle\hskip
30.00005pt\times\Big{[}\frac{\ell}{2^{-(am-k)}}+\frac{u-v}{2^{-(\frac{am}{2}-k)}},\frac{\ell}{2^{-(am-k)}}+\frac{u-v+1}{2^{-(\frac{am}{2}-k)}}\Big{]}.$

Notice now that for $r,k$ fixed, as $v,n,p$ range over their respective $\sim
2^{\frac{am}{2}}$ possible values, $s_{1}$ will “span” a time-frequency region
which is roughly a $2^{\frac{am}{2}}$-dilate of any other chosen $s_{1}$ with
similar characteristics, region that can be essentially identified with
$P_{1}$ from (171).

Deduce that, up to a decomposition of the family of parameters into $O(1)$
subfamilies, we have979797For a better geometric intuition, we refer the
reader to Figure 5 in Section 2.

$s_{1}\subset P_{1}$

for all $n,v,p\sim 2^{\frac{am}{2}}$ .
Similar considerations hold for $g$ as well, and in particular we can
introduce

$\displaystyle s_{2}$
$\displaystyle:=I^{p_{r}+n}_{k}\times\omega^{u_{\ell-1}+v}_{k}$
$\displaystyle=\Big{[}(r-1)2^{k}+\frac{p+n}{2^{\frac{am}{2}-k}},(r-1)2^{k}+\frac{p+n+1}{2^{\frac{am}{2}-k}}\Big{]}$
$\displaystyle\hskip
30.00005pt\times\Big{[}\frac{\ell-1}{2^{-(am-k)}}+\frac{u+v}{2^{-(\frac{am}{2}-k)}},\frac{\ell-1}{2^{-(am-k)}}+\frac{u+v+1}{2^{-(\frac{am}{2}-k)}}\Big{]}$

so that for $p,n,v\sim 2^{\frac{am}{2}}$ we always have $s_{2}\subset P_{2}$.

Turning towards the wave-packet
$\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}$ we are also led to
introduce

$\displaystyle s_{3}$
$\displaystyle:=I^{p_{r}}_{k}\times\omega^{2u_{2\ell-1}}_{k}$
$\displaystyle=\Big{[}(r-1)2^{k}+\frac{p}{2^{\frac{am}{2}-k}},(r-1)2^{k}+\frac{p+1}{2^{\frac{am}{2}-k}}\Big{]}$
$\displaystyle\hskip
30.00005pt\times\Big{[}\frac{2\ell-1}{2^{-(am-k)}}+\frac{2u}{2^{-(\frac{am}{2}-k)}},\frac{2\ell-1}{2^{-(am-k)}}+\frac{2u+1}{2^{-(\frac{am}{2}-k)}}\Big{]}$

with (again for all $p,n,v\sim 2^{\frac{am}{2}}$) $s_{3}\subset P_{3}.$

We summarize the roles of the parameters appearing in the definition of the
tritiles $s=(s_{1},s_{2},s_{3})$, $P=(P_{1},P_{2},P_{3})$, and in the model
operator (170):

  * •

$k$ defines the main scale of the decomposition and determines the strip in
the frequency plane;

  * •

the frequency parameter $\ell$ indicates the joint frequency localization of
$f$ and $g$ at the scale $2^{am-k}$;

  * •

the spatial parameter $r$ indicates the spatial moral support of the
input/output function(s) arising from the action of $T_{m,k,\ell,r}$;

  * •

the frequency parameters $u$ and $v$ provide a more accurate localization for
$\hat{f}$ and $\hat{g}$ at the smaller scale $2^{\frac{am}{2}-k}$ (this arises
from the linearization of the oscillatory phase of the multiplier symbol);

  * •

the spatial parameter $p$ indicates a more accurate spatial information of the
output corresponding to the wave-packet
$\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}$.

  * •

the spatial parameter $n$ accounts for the spatial localization in the
continuous $t$ variable (at the same smaller scale $2^{\frac{am}{2}-k}$)
exploiting the bilinear Hilbert structure of the input functions that reflects
in the $\frac{2n}{2^{\frac{am}{2}-k}}$ difference989898This is immediate from
the spatial structure of the arguments of the input functions: $x+t$ for $f$
and $x-t$ for $g$. between the moral spatial supports of $f$ and $g$,
respectively.

We notice that $k,\ell,r\in\mathbb{Z}$ are associated with the “expanded
tiles” $P$, while the regular $s$ tiles of area $1$ require also the
parameters $p,n,u,v\sim 2^{am\over 2}$ in order to be located as we are
zooming into a given $P$. This explains why we refer to a model involving the
tri-tiles $s$ as a _high resolution model_ and to the model involving the
“expanded tiles” $P$ as a _low resolution model_.

The expanded tri-tile $P=(P_{1},P_{2},P_{3})$ and their subordinated tri-
tile(s) $s=(s_{1},s_{2},s_{3})$ are represented in Figure 4 while their inter-
relationship is illustrated in Figure 5. For future reference, we collect all
the above definitions :

(174) $\displaystyle s_{1}$
$\displaystyle:=I^{p_{r}-n}_{k}\times\omega^{u_{\ell}-v}_{k},$ $\displaystyle
s_{2}$ $\displaystyle:=I^{p_{r}+n}_{k}\times\omega^{u_{\ell-1}+v}_{k},$
$\displaystyle s_{3}$
$\displaystyle:=I^{p_{r}}_{k}\times\omega^{2u_{2\ell-1}}_{k},$ $\displaystyle
P_{1}$
$\displaystyle:=[r2^{k},(r+1)2^{k}]\times\big{[}2^{am-k}(\ell+1),2^{am-k}(\ell+2)\big{]},$
$\displaystyle P_{2}$
$\displaystyle:=[r2^{k},(r+1)2^{k}]\times\big{[}2^{am-k}(\ell-1),2^{am-k}\ell\big{]},$
$\displaystyle P_{3}$
$\displaystyle:=[r2^{k},(r+1)2^{k}]\times\big{[}2^{am-k}(2\ell),2^{am-k}(2\ell+1)\big{]}.$

Since later on we will prefer to drop the explicit form of both collections of
tri-tiles, we also write for $1\leq j\leq 3$,
$s_{j}=I_{s_{j}}\times\omega_{s_{j}}$, where $I_{s_{j}}$ is one of the
intervals obtained by dividing $I_{P}$ into $2^{am\over 2}$ pieces of equal
length, and $\omega_{s_{j}}$ is one of the intervals obtained by dividing
$\boldsymbol{\omega}_{P_{j}}$ into $2^{am\over 2}$ identical pieces. In such a
situation we write

(175) $\hat{s}_{j}=P_{j},\quad\text{for }1\leq j\leq 3,\quad\text{or simply
}\hat{s}:=P.$

More generally we use the notation

(176) $\hat{J}=I$

if $J$ is obtained by dividing $I$ into $2^{am\over 2}$ congruent
subintervals.

Notice that each of the intervals $I_{P}$ is dyadic while each of its high-
resolution associate $I_{s_{j}}$ obeys $\hat{I}_{s_{j}}=I_{P}$.
Correspondingly, each of the intervals $\boldsymbol{\omega}_{P_{j}}$ are part
of a $2^{am}$ dilation of the dyadic grid with
$\hat{\omega}_{s_{j}}=\boldsymbol{\omega}_{P_{j}}$.

######  Remark 4.3.

As mentioned previously, ${{\mathbb{P}}_{\textrm{BHT}}}$ is a rank-1 family:
once $P_{i}$ determined, so are the other two tiles forming the expanded tri-
tiles $P=(P_{1},P_{2},P_{3})$ and thus only the parameters $k,r,\ell$ will be
of relevance. In contrast, the collection of tri-tiles $s=(s_{1},s_{2},s_{3})$
from (174) involves the extra parameters $n,p,u,v\sim 2^{am\over 2}$, and once
$s_{1}$ is set, there are still $2^{am\over 2}$ choices for the spatial
interval $I_{s_{2}}$ and $2^{am\over 2}$ choices for the frequency interval
$\omega_{s_{2}}$ in order to fully determine $s_{2}$. The rank-1 property
makes ${{\mathbb{P}}_{\textrm{BHT}}}$ especially suitable for the later multi-
scale time-frequency analysis, but, as we will see, we still need to rely on
the tri-tiles of area 1 and on their associated wave-packets when estimating
the almost orthogonality among the expanded tiles that is quintessential in
obtaining the control over $T_{m}$ and in extenso over our original operator
$BC^{a}$.

Finally, for any $s=(s_{1},s_{2},s_{3})$ as in (174), we let

(177) $\displaystyle\phi_{s_{1}}$
$\displaystyle:=\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n},\quad$
$\displaystyle\rho_{I_{P}}(\lambda)$ $\displaystyle:=\rho_{am-
ak}(\lambda(x)),$ $\displaystyle\phi_{s_{2}}$
$\displaystyle:=\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n},\quad$
$\displaystyle w^{e}_{s}(\lambda)$ $\displaystyle:=w_{k,n,v}^{e}(\lambda),$
$\displaystyle\phi_{s_{3}}$
$\displaystyle:=\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}},\quad$
$\displaystyle|I_{s}|$ $\displaystyle:=2^{-(\frac{am}{2}-k)}.$

####  4.5.3. The low-resolution model

With the notations and conventions from above, we are now ready to formally
introduce the low-resolution model. This will be delivered in two equivalent
forms utilized depending on the context and our convenience.

  * •

the low resolution model - the parametric format

In this setting, our low resolution model is given by

$T(f,g)(x)=\sum_{m\in\mathbb{N}}T_{m}(f,g)(x)=\sum_{m\in\mathbb{N}}\sum_{k\in\mathbb{Z}}T_{m,k}(f,g)(x),$

with

(178) $\displaystyle T_{m,k}(f,g)(x)$
$\displaystyle=\sum_{\ell,r\in\mathbb{Z}}T_{m,k,\ell,r}(f,g)(x),\qquad\text{
where}$ $\displaystyle T_{m,k,\ell,r}(f,g)(x)$
$\displaystyle:=\sum_{\begin{subarray}{c}p\sim 2^{\frac{am}{2}},\\\ n\sim
2^{\frac{am}{2}}\end{subarray}}\sum_{\begin{subarray}{c}u\sim
2^{\frac{am}{2}},\\\ v\sim
2^{\frac{am}{2}}\end{subarray}}\frac{1}{2^{\frac{am+2k}{4}}}\langle
f_{\ell+1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u-v,p_{r}-n}\rangle\langle
g_{\ell-1},\check{\varphi}_{\frac{am}{2}-k,\ell}^{u+v,p_{r}+n}\rangle$
$\displaystyle\hskip
70.0001pt\cdot\check{\varphi}_{\frac{am}{2}-k,2\ell-1}^{2u,p_{r}}(x)\rho_{am-
ak}(\lambda(x))w_{k,n,v}^{e}(\lambda)(x).$

Notice here, that for fixed $m\in\mathbb{N}$ and $k,\ell,r\in\mathbb{Z}$ the
discretization of $T_{m,k,\ell,r}$ in (178) represents precisely the high-
resolution linearized wave-packet discretization obtained in (136).

Since we will work extensively with the dualization of these operators, we
define the associated trilinear forms by

(179) $\displaystyle\underline{\mathcal{L}}_{m}(f,g,h)$
$\displaystyle:=\sum_{k\in\mathbb{Z}}\underline{\mathcal{L}}_{m,k}(f,g,h),$
$\displaystyle\underline{\mathcal{L}}_{m,k}(f,g,h)$ $\displaystyle:=\langle
T_{m,k}(f,g),h\rangle=\sum_{\ell,r\in\mathbb{Z}}\underline{\mathcal{L}}_{m,k,\ell,r}(f,g,h),$
$\displaystyle\underline{\mathcal{L}}_{m,k,\ell,r}(f,g,h)$
$\displaystyle:=\langle T_{m,k,\ell,r}(f,g),h\rangle.$

  * •

the low resolution model - the tile format

Alternatively, our operator can be written in terms of elementary building
blocks associated with (expanded) tri-tiles. This format, anticipated in (76)
of Section 2, will be used in Section 7 where the time-frequency analysis for
the low resolution model will be performed.

In this situation we have

(180) $\displaystyle
T(f,g)(x)=\sum_{m\in\mathbb{N}}T_{m}(f,g)(x)=\sum_{m\in\mathbb{N}}\sum_{P\in{{\mathbb{P}}_{\textrm{BHT}}}}T_{m,P}(f,g)(x),\qquad\text{
where }$ $\displaystyle
T_{m,P}(f,g)(x):=2^{-\frac{am}{2}}\sum_{s:\hat{s}=P}\frac{1}{|I_{s}|^{\frac{1}{2}}}\langle
f,\phi_{s_{1}}\rangle\langle
g,\phi_{s_{2}}\rangle\phi_{s_{3}}(x)\rho_{I_{P}}(\lambda(x))w^{e}_{s}(\lambda)(x).$

Here the operators $T_{m,P}$ correspond to the operators $T_{m,k,\ell,r}$ from
the parametrized setting where we assumed that $P=(P_{1},P_{2},P_{3})$ and
$s=(s_{1},s_{2},s_{3})$ are given by (174).

Maintaining the parallelism with the parametric format we also define the
associated trilinear forms

(181) $\displaystyle\underline{\mathcal{L}}_{m}(f,g,h)$
$\displaystyle:=\sum_{P\in{{\mathbb{P}}_{\textrm{BHT}}}}\underline{\mathcal{L}}_{m,P}(f,g,h),$
$\displaystyle\underline{\mathcal{L}}_{m,P}(f,g,h)$ $\displaystyle:=\langle
T_{m,P}(f,g),h\rangle.$

We end this section with the following

######  Remark 4.4.

The low resolution analysis will be critical in Section 7 where we will need
to employ time-frequency methods similar in spirit with those used for
approaching the bilinear Hilbert transform. While both (178) and (180) offer a
good intuition on the essence of the low resolution format in reality we will
need a slight refinement of these formulations that is described in Section 6.

In what follows we will stick to the parametric format relying on the indices
$k,\ell,r,p,n,u,v$ as it will be more convenient for performing a high-
resolution study of the operator. Once we have completed the finer analysis,
we will pass to the tile format. This latter, more geometric description, will
be employed along the entire Section 7.

##  5\. The high resolution, single-scale analysis (I): Extracting the
cancellation encoded in the phase curvature

###  5.1. Preparatives

In this section we provide the first key ingredient for the strategy discussed
in Section 2 – see in particular (69) – that is expressed in the following
single scale $m$-exponential decay estimate:999999While not needed later in a
larger range, it is not difficult to see that this $L^{2}$-based estimate may
be extended via interpolation arguments to the full Banach triangle, i.e.
$\|\underline{\mathcal{L}}^{a}_{m,k}\|_{L^{p}\times L^{q}\times
L^{r^{\prime}}\to\mathbb{C}}\lesssim_{a,p,q,r}2^{-\delta_{0}^{\prime}am}$ with
$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $1<p,q<\infty$ and some exponent
$\delta_{0}^{\prime}$ depending on $p,q$.

######  Proposition 5.1.

Fix $k,\ell,r\in\mathbb{Z}$ and $m\in\mathbb{N}$ and assume
$a\in(0,\infty)\setminus\\{1,2\\}$ is a given parameter. With the notations
from the previous section we let100100100While the same as the expression in
Section 4.5.3 we restate for convenience the formula of our form here, and
also – for only this section – insert the superscript $a$ in our formulations
in order to stress the dependence of our form as well as of our estimates on
this parameter.

(182) $\displaystyle\underline{\mathcal{L}}^{a}_{m,k,\ell,r}(f,g,h):=$
$\displaystyle\sum_{n,v,p\sim
2^{\frac{am}{2}}}\int_{\mathbb{R}}S_{k,\ell,r}^{n,v,p}(f,g)(x)\,\rho_{am-
ak}(\lambda(x))\,w_{k,n,v}^{e}(\lambda)(x)\,h(x)\,dx.$

Then, there exists $\delta_{0}\in(0,1)$ absolute constant such that

(183)
$|\underline{\mathcal{L}}^{a}_{m,k,\ell,r}(f,g,h)|\lesssim_{a}2^{-\delta_{0}am}\,2^{-\frac{k}{2}}\|f\|_{L^{2}}\|g\|_{L^{2}}\|h\|_{L^{2}}.$

We start by elaborating on several subtleties pertaining to the above
statement and its proof.

We first notice that the conclusion of Proposition 5.1 may be reshaped up to
tails101101101 _E.g._ a superposition of similar flavor terms with fast
decaying coefficients.

(184)
$\displaystyle|\underline{\mathcal{L}}_{m,k,\ell,r}^{a}(f,g,h)|\lesssim_{a}2^{-\delta_{0}am}|I_{k,r}|$
$\displaystyle\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{k,r}}|\pi_{\omega_{k,\ell+1}}f|^{2}\Big{)}^{1/2}$
$\displaystyle\cdot\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{k,r}}|\pi_{\omega_{k,\ell-1}}g|^{2}\Big{)}^{1/2}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{k,r}}|h|^{2}\Big{)}^{1/2},$

where here $I_{k,r}:=[r2^{k},(r+1)2^{k}]$,
$\omega_{k,\ell+1}:=[2^{am-k}\ell,2^{am-k}(\ell+1)]$, $r,\ell\in\mathbb{Z}$,
$\pi_{\omega}$ stands for the frequency projection operator on the interval
$\omega$ and
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I}f:=\frac{1}{|I|}\int_{I}f$.

Recalling now that $\underline{\mathcal{L}}_{m,k,\ell,r}^{a}(f,g,h)$ may be
rewritten as $\underline{\mathcal{L}}_{m,P}^{a}(f,g,h)$ with
$P=(P_{1},P_{2},P_{3})\in{{\mathbb{P}}_{\textrm{BHT}}}$ the tri-tile defined
by (174), we notice that (184) would correspond to a _single tile_ estimate if
the function $h$ would be well-localized in time-frequency.

A more thorough inspection reveals now two key features of (184):

  1. (a)

it is _asymmetric_ in the time-frequency localization of the inputs $(f,g,h)$:
indeed, the functions $f$ and $g$ are well adapted to the expected time-
frequency regions represented by the expanded tiles $P_{1}$ and $P_{2}$
respectively; however, for the function $h$ our estimate only provides the
spatial localization corresponding to $P_{3}$;

  2. (b)

it is a _purely $L^{2}$_-based estimate that focuses on the norm
$\|\underline{\mathcal{L}}_{m,P}^{a}\|_{L^{2}\times L^{2}\times
L^{2}\to\mathbb{C}}$ (and thus, in particular, goes beyond the usual Hölder
duality range).

As expected, these two features strongly relate with both our proof
methodology and the structure of our paper:

  1. (I)

as per item (a) above, the shape of the single scale estimate in (184)
indicates that its proof favors the _bilinear_ behavior $(f,g)$ within the
structure of $T_{m,k,\ell,r}^{a}$ treating the third function $h$ as an
auxiliary object. This is further on reinforced by the item (b) which hints
that (184) involves a $TT^{*}$ argument in disguise, since
$\|\underline{\mathcal{L}}_{m,k,\ell,r}^{a}\|_{L^{2}\times L^{2}\times
L^{2}\to\mathbb{C}}=\|T_{m,k,\ell,r}^{a}\|_{L^{2}\times L^{2}\to L^{2}}$.

  2. (II)

the lack of a good time-frequency localization for the function $h$ observed
in (a) hints towards the fact that one needs a further refinement of the
single scale estimate (184) in order to develop the multiscale analysis
required for providing the full boundedness range for our original operator
$BC^{a}$. Indeed, as mentioned in the introduction, the treatment of $BC^{a}$
has to include the approach for the classical BHT with the latter relying
fundamentally on the geometry/combinatorics of the (expanded) tri-tiles
$P=(P_{1},P_{2},P_{3})\in{{\mathbb{P}}_{\textrm{BHT}}}$ and hence on the good
localization of the dualizing function $h$. Also the observation in (b) will
offer the right setting for constructing and adapting suitable notions of
sizes for each of the functions $f,g,\text{ and }h$ consistent with the
multilinear time-frequency analysis for BHT.

Based on the information provided in Sections 4.1, 4.2.2 and 4.4 in what
follows we will briefly recap and provide more context for

  1. (i)

the features discussed at (a) and (b) and juxtaposed to (I) and (II),

  2. (ii)

the necessity of going back and forth between the discrete linearized wave-
packet (no $m$-decay absolute summability) model and the continuous linearized
spatial ($m$-decay absolute summability) model.102102102It might be worth
noticing that this is a direct consequence of the rough behavior of the
linearizing function $\lambda(x)$; indeed, if $\lambda(x)$ is smooth (say e.g.
polynomial in $x$) then one can work directly with the discrete phase-
linearized wave-packet model which obeys the $m$-decay absolute summability
property.

Indeed, as already revealed in Section 4, our initial approach is guided by
the frequency discretization of our operator consistent with the phase
linearization of the multiplier (see Section 4.2.3) followed by an adapted
Gabor frame discretization of the input functions $f$ and $g$. This achieves
the discretization of our dualized form
$\underline{\mathcal{L}}^{a}_{m,k}(f,g,h)$ as given by (84) and hence
completes the first two steps (L) and (G) in our LGC-methodology described in
Section 4.2.2. However, the attempt of completing our program with the final
step (C) faces a key difficulty: the wave-packet coefficients appearing in the
discretized form
$\underline{\mathcal{L}}^{a}_{m}(f,g,h)=\sum_{k\in\mathbb{Z}}\underline{\mathcal{L}}^{a}_{m,k}(f,g,h)$
do not produce an $m$-decaying absolute summable model. As already explained,
we thus have to make a detour and reverse the Gabor discretization reshaping
the linearization at the first step onto the spatial variable (kernel side) –
see Sections 4.3.2 and 4.3.3 and compare this with Section 5.2. The spatial
linearization offers a critical advantage in that it circumvents the problem
of the conditional $m$-decaying summability for
$\underline{\mathcal{L}}_{m,k,\ell,r}^{a}(f,g,h)$ (see Appendix A) since now,
via (182), the signum of the local Fourier (Gabor) coefficients within (147)
are encrypted through Parseval, into a bilinear oscillatory integral of the
form (154)–(155). Once at this point we pass to the (C) item exposed in
Section 4.2.2 and employ what one might refer to as a _bilinear
$TT^{*}$-argument_ that relies on exponential sums103103103The fact that our
treatment of a continuous nature problem is based on discrete analysis
elements such as exponential sums is quite surprising; in more standard
situations one usually sees the reverse applied - with continuous realm
serving as a model and offering assistance to discrete analysis. In our
setting, the specific shape of these exponential sums is dictated by the
spatial phase linearization discretization. and on level set estimates
exploiting the time-frequency correlation and the non-zero curvature of the
phase. This provides the insight to our proof of (184), which gives us the
desired exponential decay in $m$ but, as mentioned at item (a), without also
providing a good time-frequency localization for the function $h$. The latter
aspect however is necessary when passing from the single scale (fixed
$k\in\mathbb{Z}$) to the multiscale analysis (summation over
$k\in\mathbb{Z}$). This is why we need a refinement of the single scale
estimate that is performed in Section 6 and that – via the transition
explained in Section 4.3.2 – combines the features of both the present
approach via spatial linearization and that of the initial linearized wave-
packet discretized model ending up by providing a single scale estimate with
both exponential decay in $m$ and well localized information for the triple
$(f,g,h)$.

###  5.2. Transition from the discrete phase-linearized wave-packet model to
the continuous phase-linearized spatial model

This section can be thought as an adaptation of the reasonings presented in
Section 4.3.2 to the specific context provided by Proposition 5.1.

We start our reduction with the following simple observation: since our
estimate (183) is invariant under translation, modulation and dilation
symmetries, it is enough to only treat the case $k=0$, $\ell=1$ and $r=1$ (and
hence $p_{r}=p$). Next, for notational simplicity, throughout this section we
will refer to the wave-packets $\check{\varphi}^{u,n}_{\frac{am}{2},1}$ as
simply $\check{\varphi}^{u,n}$.

Recalling now (182), an immediate application of Cauchy-Schwarz gives

(185)
$|\underline{\mathcal{L}}^{a}_{m,0,1,1}(f,g,h)|\lesssim\Lambda_{m}^{a}(f,g)\,\|h\|_{L^{2}},$

where here we used the notation104104104Throughout the paper $\bar{c}_{a}$ is
an absolute nonzero constant that is allowed to change from line to line.
Also, throughout this section we set $\langle x\rangle:=(1+x^{2})^{1/2}$.

(186) $\displaystyle[\Lambda_{m}^{a}(f,g)]^{2}:=$
$\displaystyle\int_{\mathbb{R}}\Bigg{(}\frac{\rho_{am}(\lambda(x))}{2^{\frac{am}{4}}}\sum_{\begin{subarray}{c}n,v\\\
\sim 2^{\frac{am}{2}}\end{subarray}}\frac{\Big{|}\sum\limits_{u,p\sim
2^{\frac{am}{2}}}\langle f,\check{\varphi}^{u-v,p-n}\rangle\langle
g,\check{\varphi}^{u+v,p+n}\rangle\check{\varphi}^{2u,p}(x)\Big{|}}{\big{\langle}\bar{c}_{a}\frac{n^{a-1}\lambda(x)}{2^{\frac{a^{2}m}{2}}}+2v\big{\rangle}^{2}}\Bigg{)}^{2}\,dx.$

From (185), (186) and the symmetry invariance discussed earlier, we deduce
that (183) is an immediate consequence of

(187)
$|\Lambda_{m}^{a}(f,g)|\lesssim_{a}2^{-\delta_{0}am}\|f\|_{L^{2}}\|g\|_{L^{2}}.$

Notice now that since (186) is a positive expression, from now on we can
assume without loss of generality that

(188) $\rho_{am}(\lambda(x))=1\qquad\textrm{for any $x\in\mathbb{R}$}.$

Once at this point, we remark – via (149) (or the more precise (154)) – that
our desired estimate (187) follows if we are able to prove a similar bound for
the continuous version of our bilinear form $\Lambda_{m}^{a}(f,g)$, as given
by105105105Here, as before,
$I^{n}=[n\,2^{-\frac{am}{2}},\,(n+1)\,2^{-\frac{am}{2}}]$.:

(189) $\displaystyle[\underline{\Lambda}_{m}^{a}(f,g)]^{2}:=$
$\displaystyle\sum_{p\sim 2^{\frac{am}{2}}}\int_{I^{p}}\Bigg{(}\sum_{n,v\sim
2^{\frac{am}{2}}}\,\frac{\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{-i2^{\frac{am}{2}}\,2vt}\,dt\Big{|}}{\big{\langle}\bar{c}_{a}\frac{n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)+2v\big{\rangle}^{2}}\Bigg{)}^{2}\,dx.$

######  Observation 5.2.

In formulation (189) above, for clarity purposes, we used a standard almost-
orthogonality argument in $p$ in order to only focus on the main diagonal term
and discard the error term produced by the “tails” arising from the wave-
packet localization. We will not provide here an argumentation for this
reduction since there are several similar arguments carried over in detail in
Sections 4 and 6.

We now claim that the behavior of $\underline{\Lambda}_{m}^{a}(f,g)$ may be
reduced to that of106106106Here is where we use the time-frequency correlation
provided by the expression appearing at the denominator in the right-hand side
of (189).

(190) $\displaystyle[\bar{\Lambda}_{m}^{a}(f,g)]^{2}:=$
$\displaystyle\sum_{p\sim 2^{\frac{am}{2}}}\int_{I^{p}}\bigg{(}\sum_{n\sim
2^{\frac{am}{2}}}\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{i\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a(a-1)m}{2}}}\lambda(x)\,t}\,dt\Big{|}\bigg{)}^{2}\,dx.$

Indeed, recalling the notation $M_{b}g(x):=e^{-ibx}g(x)$, $b\in\mathbb{R}$, we
further claim that

(191) $\underline{\Lambda}_{m}^{a}(f,g)\lesssim\sum_{s\in\mathbb{Z}:|s|\leq
2^{\frac{am}{2}}}\frac{\bar{\Lambda}_{m}^{a}(f,M_{2^{\frac{am}{2}}s}g)}{s^{2}+1}.$

This last statement is based on a rather standard technical reasoning
involving a Taylor series argument. However, for reader’s convenience, we
provide below a sketch of its proof: we first notice that

$\displaystyle[\underline{\Lambda}_{m}^{a}(f,g)]^{2}\approx\sum_{p\sim
2^{\frac{am}{2}}}\int_{I^{p}}\bigg{(}\sum_{n,v\sim
2^{\frac{am}{2}}}\,\sum_{|s|\leq 2^{\frac{am}{2}}}$
$\displaystyle\mathbf{1}\Big{(}\Big{|}\frac{\bar{c}_{a}\,n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)+2v-s\Big{|}\lesssim
1\Big{)}$
$\displaystyle\cdot\frac{\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{-i2^{\frac{am}{2}}\,2vt}\,dt\Big{|}}{s^{2}+1}\bigg{)}^{2}\,dx.$

Fix now $x$, and assume
$\Big{|}\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)+2v-s\Big{|}\lesssim
1$. Isolating the $t$-integrand, we have

$\displaystyle\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{-i2^{\frac{am}{2}}\,2vt}\,dt\Big{|}$
$\displaystyle=\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{-i2^{\frac{am}{2}}st}e^{i2^{\frac{am}{2}}\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)t}$
$\displaystyle\hskip 130.0002pt\cdot
e^{-i2^{\frac{am}{2}}(t-(n-1)2^{-\frac{am}{2}})(\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)+2v-s)}\,dt\Big{|}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\frac{\Big{(}\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)+2v-s\Big{)}^{k}}{k!}\Big{|}\int_{I^{n}}f(x-t)(M_{2^{\frac{am}{2}}s}g)(x+t)$
$\displaystyle\hskip 130.0002pt\cdot
e^{i2^{\frac{am}{2}}\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a^{2}m}{2}}}\lambda(x)t}(2^{\frac{am}{2}}(t-(n-1)2^{-\frac{am}{2}}))^{k}\,dt\Big{|}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\Big{|}\int_{I^{n}}f(x-t)(M_{2^{\frac{am}{2}}s}g)(x+t)e^{i\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a(a-1)m}{2}}}\lambda(x)t}$
$\displaystyle\hskip
130.0002pt\cdot\frac{(2^{\frac{am}{2}+10}(t-(n-1)2^{-\frac{am}{2}}))^{k}}{k!}\,dt\Big{|}.$

Now, since $2^{\frac{am}{2}+10}\,(t-(n-1)\,2^{-\frac{am}{2}})$ is a function
that is morally constant one for $t\in I^{n}$ we conclude that (191) holds.

With these done, based on the above considerations, the proof of (183) is an
immediate consequence of Proposition 5.3 below.

###  5.3. The continuous phase-linearized spatial model formulation of the
single scale estimate

In this section we formulate the single scale estimate in the desired
continuous phase-linearized spatial form and perform some simple reductions
involving the distribution of the spatial information carried by the input
functions $f$ and $g$.

######  Proposition 5.3.

Let $a\in(0,\infty)\setminus\\{1,2\\}$ be a fixed parameter and
$\lambda:\mathbb{R}\to[2^{am-10},2^{am+10}]$ be an arbitrary Lebesgue
measurable function.

Then there exists107107107At the end of this proof we provide an explicit
value for $\delta_{0}$ though this is far from optimal. One can indeed
increase the value of the exponent $\delta_{0}$ but we prefer to trade off
this improvement over the clarity of the exposition since for our purposes any
non-trivial decay (i.e $\delta_{0}>0$) is good enough. an absolute constant
$\delta_{0}>0$ such that for any $m\in\mathbb{N}$, one has

(192) $\displaystyle\bigg{(}\sum_{p\sim
2^{\frac{am}{2}}}\int_{I^{p}}\bigg{(}\sum_{n\sim
2^{\frac{am}{2}}}\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{i\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a(a-1)m}{2}}}\lambda(x)t}\,dt\Big{|}\bigg{)}^{2}\,dx\bigg{)}^{\frac{1}{2}}$
$\displaystyle=\bar{\Lambda}_{m}^{a}(f,g)\lesssim_{a}2^{-\delta_{0}am}\|f\|_{L^{2}}\|g\|_{L^{2}}.$

###### Proof.

The proof of this statement will be split in two cases depending on the
distribution of the spatial information carried by the functions $f$ and $g$:

Fix $\mu\in(0,1)$ a small parameter to be chosen later and define

  * •

the _$f$ -clustered_ (heavy) component as

$C_{\mu}(f):=\Big{\\{}n\sim 2^{\frac{am}{2}}:\int_{3I^{n}}|f|^{2}\geq
2^{(\mu-1)\frac{am}{2}}\|f\|_{L^{2}}^{2}\Big{\\}}.$

Notice here that

(193) $\\#C_{\mu}(f)\lesssim 2^{(1-\mu)\frac{am}{2}}.$

  * •

the _$f$ -uniform_ (light) component as

(194) $U_{\mu}(f):=\Big{\\{}n\sim
2^{\frac{am}{2}}:\int_{3I^{n}}|f|^{2}<2^{(\mu-1)\frac{am}{2}}\|f\|_{L^{2}}^{2}\Big{\\}}.$

A similar decomposition applies to $g$.

In direct correspondence with the above we analyze the behavior of our form
$\bar{\Lambda}_{m}^{a}$ according to:

  * •

the _clustered_ component defined as

(195) $\displaystyle\bar{\Lambda}_{m}^{a,C_{\mu}}(f,g):=$
$\displaystyle\bigg{(}\sum_{p\sim
2^{\frac{am}{2}}}\int_{I^{p}}\bigg{(}\sum_{\begin{subarray}{c}n\sim
2^{\frac{am}{2}}:\\\ p-n\in C_{\mu}(f)\\\ \text{ or }\\\ p+n\in
C_{\mu}(g)\end{subarray}}\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{i\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a(a-1)m}{2}}}\lambda(x)t}\,dt\Big{|}\bigg{)}^{2}\,dx\bigg{)}^{\frac{1}{2}},$

  * •

the _uniform_ component defined as

(196) $\displaystyle\bar{\Lambda}_{m}^{a,U_{\mu}}(f,g):=$
$\displaystyle\bigg{(}\sum_{p\sim
2^{\frac{am}{2}}}\int_{I^{p}}\bigg{(}\sum_{\begin{subarray}{c}n\sim
2^{\frac{am}{2}}:\\\ p-n\in U_{\mu}(f)\\\ \text{ and }\\\ p+n\in
U_{\mu}(g)\end{subarray}}\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{i\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a(a-1)m}{2}}}\lambda(x)t}\,dt\Big{|}\bigg{)}^{2}\,dx\bigg{)}^{\frac{1}{2}},$

and notice that

(197)
$\bar{\Lambda}_{m}^{a}(f,g)\leq\bar{\Lambda}_{m}^{a,C_{\mu}}(f,g)\,+\,\bar{\Lambda}_{m}^{a,U_{\mu}}(f,g).$

With these done, we are now ready to perform our analysis.

Case 1: The clustered component.

The clustered component has a straightforward treatment. Indeed, relying on
the estimate (193) and applying (twice) a Cauchy-Schwarz argument we get

(198) $\displaystyle[\bar{\Lambda}_{m}^{a,C_{\mu}}(f,g)]^{2}$
$\displaystyle\lesssim 2^{(1-\mu)\frac{am}{2}}\sum_{p,n\sim
2^{\frac{am}{2}}}\int_{I^{p}}\Big{(}\int_{I^{n}}|f(x-t)|^{2}\,dt\Big{)}\Big{(}\int_{I^{n}}|g(x+t)|^{2}\,dt\Big{)}\,dx$
$\displaystyle\lesssim 2^{-\mu\frac{am}{2}}\sum_{p,n\sim
2^{\frac{am}{2}}}\int_{3I^{p-n}}|f|^{2}\int_{3I^{p+n}}|g|^{2}\lesssim
2^{-\mu\frac{am}{2}}\|f\|^{2}_{L^{2}}\|g\|^{2}_{L^{2}}.$

Case 2: The uniform component.

The treatment of the uniform component is the main step for our present single
scale estimate and in particular for (192). It is within this chapter of our
journey that the non-zero curvature plays a critical rôle. Because of its
later relevance the proof of the key estimate for the uniform component will
be delivered as part of an independent statement - see Proposition 5.5 below.
In what follows we show that if the latter holds then relation (192) also
holds.

For this, it is enough to notice that a simple Cauchy–Schwarz argument gives
us

(199) $\displaystyle[\bar{\Lambda}_{m}^{a,U_{\mu}}(f,g)]^{2}$
$\displaystyle\lesssim 2^{\frac{am}{2}}\sum_{p\sim
2^{\frac{am}{2}}}\sum_{\begin{subarray}{c}n\sim 2^{\frac{am}{2}}\\\ p-n\in
U_{\mu}(f)\\\ p+n\in
U_{\mu}(g)\end{subarray}}\int_{I^{p}}\Big{|}\int_{I^{n}}f(x-t)g(x+t)e^{i\frac{\bar{c}_{a}n^{a-1}}{2^{\frac{a(a-1)m}{2}}}\lambda(x)t}\,dt\Big{|}^{2}\,dx.$

However, from Proposition 5.5, the right-hand side of the above inequality is
bounded from above by $2^{-2\delta_{1}am}$ for some absolute constant
$\delta_{1}>0$ and hence

(200)
$[\bar{\Lambda}_{m}^{a,U_{\mu}}(f,g)]^{2}\lesssim_{a}2^{-2\delta_{1}am}\|f\|^{2}_{L^{2}}\|g\|^{2}_{L^{2}}.$

Putting together (197), (198) and (200) we conclude that

(201)
$|\bar{\Lambda}_{m}^{a}(f,g)|\lesssim_{a}2^{-\delta_{0}am}\|f\|_{L^{2}}\|g\|_{L^{2}},$

where here108108108In effect – see (265) and (316) – the proof of Proposition
5.5 will give us that, up to a polynomial term in $m$, (200) holds for
$\delta_{1}=\frac{1}{68}\,(1-16\mu)$ for $a\in(0,\infty)\setminus\\{1,2,3\\}$
and $\delta_{1}=\frac{1}{6}\,(\frac{1}{114}-\frac{4}{3}\mu)$ for $a=3$. Thus,
ignoring polynomial growth in $m$ factors, (201) follows with
$\delta_{0}=\frac{1}{1292}$ (for $\mu=\frac{1}{323}$).
$\delta_{0}:=\min\\{\delta_{1},\frac{\mu}{4}\\}$. ∎