# On a planar Pierce–Yung operator

David Beltran  David Beltran: Departament d’Anàlisi Matemàtica, Universitat de
València, Dr. Moliner 50, 46100 Burjassot, Spain<EMAIL_ADDRESS>,
Shaoming Guo  Shaoming Guo: Chern Institute of Mathematics, Nankai University,
Tianjin, China
and
Department of Mathematics, University of Wisconsin-Madison, Madison, WI-53706,
USA<EMAIL_ADDRESS>and  Jonathan Hickman  Jonathan Hickman: School
of Mathematics, James Clerk Maxwell Building, The King’s Buildings, Peter
Guthrie Tait Road, Edinburgh, EH9 3FD, UK<EMAIL_ADDRESS>
###### Abstract.

We show that the operator

$\mathcal{C}f(x,y):=\sup_{v\in\mathbb{R}}\Big{|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{ivt^{3}}\frac{\,\mathrm{d}t}{t}\Big{|}$

is bounded on $L^{p}(\mathbb{R}^{2})$ for every $1<p<\infty$. This gives an
affirmative answer to a question of Pierce and Yung.

###### 2020 Mathematics Subject Classification:

42B25, 42B20

##  1\. Introduction

###  1.1. Statement of the main theorem

Let $d\geq 2$ be an integer and $K\colon\mathbb{R}^{d-1}\to\mathbb{R}$ be a
Calderón–Zygmund kernel (see [21, (1.9)] for the precise requirements). Pierce
and Yung [21] considered the following operator111Indeed, they considered a
much wider class of operators, with $v(t_{1}^{3}+\cdots+t_{n-1}^{3})$ replaced
by more general polynomials. We shall come back to this point later.

$\mathcal{C}_{d}f(x,y):=\sup_{v\in\mathbb{R}}\Big{|}\int_{\mathbb{R}^{d-1}}f(x-t,y-|t|^{2})e^{iv(t_{1}^{3}+\cdots+t_{n-1}^{3})}K(t)\,\mathrm{d}t\Big{|},$
(1.1)

where $x\in\mathbb{R}^{d-1}$, $y\in\mathbb{R}$, $t\in\mathbb{R}^{d-1}$.
Moreover, in the same paper, whenever $d\geq 3$ they proved that222Given a
list of objects $L$ and real numbers $A$, $B\geq 0$, here and throughout we
write $A\lesssim_{L}B$ or $B\gtrsim_{L}A$ to indicate $A\leq C_{L}B$ for some
constant $C_{L}$ which depends only items in the list $L$. We write
$A\sim_{L}B$ to indicate $A\lesssim_{L}B$ and $B\lesssim_{L}A$.

$\|\mathcal{C}_{d}f\|_{L^{p}(\mathbb{R}^{d})}\lesssim_{p,d,K}\|f\|_{L^{p}(\mathbb{R}^{d})}$
(1.2)

holds for all $1<p<\infty$. Roughly speaking, their proof relies crucially on
a $TT^{*}$ argument and sophisticated oscillatory integral estimates, both of
which require $d\geq 3$. In particular, they asked whether (1.2) holds for
$d=2$. In this paper, we give an affirmative answer to this question.

######  Theorem 1.1.

For $x,y\in\mathbb{R}$, consider the operator

$\mathcal{C}f(x,y):=\sup_{v\in\mathbb{R}}\Big{|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{ivt^{3}}\frac{\,\mathrm{d}t}{t}\Big{|},$
(1.3)

initially defined for Schwartz functions $f\in\mathcal{S}(\mathbb{R}^{2})$. We
have

$\|\mathcal{C}f\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{2})}$

for all $1<p<\infty$.

We remark that our methods apply to more general Calderón–Zygmund kernels than
the Hilbert kernel, but we focus on this case for simplicity.

###  1.2. Earlier attempts.

Several earlier attempts were made to solve the Pierce–Yung problem. Guo,
Pierce, Roos and Yung [14] studied a weaker maximal operator than (1.3). More
precisely, they proved that

$\Big{\|}\sup_{v\in\mathbb{R}}\Big{\|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{ivt^{3}}\frac{\,\mathrm{d}t}{t}\Big{\|}_{L^{p}_{y}(\mathbb{R})}\Big{\|}_{L^{p}_{x}(\mathbb{R})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{2})}$
(1.4)

holds for every $p\in(1,\infty)$. Here $L^{p}_{y}$ (respectively, $L^{p}_{x}$)
means we take the $L^{p}$ norm in the $y$ (respectively, $x$) variable. To
better compare (1.4) with (1.3), we linearise the relevant maximal operators.

Our result in Theorem 1.1 says that

$\Big{\|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{iv(x,y)t^{3}}\frac{\,\mathrm{d}t}{t}\Big{\|}_{L^{p}(\mathbb{R}^{2})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{2})}$

holds for every $p\in(1,\infty)$ and every measurable function $v(x,y)$.
However, (1.4) is equivalent to saying that

$\Big{\|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{iv(x)t^{3}}\frac{\,\mathrm{d}t}{t}\Big{\|}_{L^{p}(\mathbb{R}^{2})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{2})}$
(1.5)

holds for every $p\in(1,\infty)$ and every measurable function $v(x)$. We
emphasise that the measurable function $v(x)$ in (1.5) does not depend on the
$y$ variable. This allows the authors in [14] to take a Fourier transform in
the $y$ variable and turn the analysis into a much simpler problem, which is
essentially one-dimensional.

More recently, Roos [24], Ramos [23] and Becker [2, 1] tried to bound planar
Pierce–Yung operators333They all consider a more general form of the
Pierce–Yung operator defined above. by viewing them as modulation-invariant
operators, in the spirit of Carleson’s operator (see Carleson [9]) and the
bilinear Hilbert transform (see Lacey and Thiele [17, 18]). However, the
Fourier multiplier for the Hilbert transform along the parabola is very rough,
in the sense that the Fourier multiplier for a Calderón–Zygmund kernel has
singularities only at the origin and at infinity, while that for the Hilbert
transform along the parabola is singular along an entire line in the frequency
space $\mathbb{R}^{2}$. Consequently, the aforementioned results [24, 23, 2,
1] all have certain logarithmic losses.

###  1.3. Why dimension matters.

Here we describe two key differences between the $d\geq 3$ and $d=2$ cases.

As mentioned in §1.1, to prove their result (1.2) for $d\geq 3$, Pierce and
Yung [21] used a $TT^{*}$ method. In particular, they wrote the kernel of the
resulting $TT^{*}$ operator as an oscillatory integral and then carried out
sophisticated oscillatory integral estimates.

To be more precise, let $\varphi_{0}\colon\mathbb{R}\to\mathbb{R}$ be a smooth
function supported on the interval $[1,2]$. Pierce and Yung [21] first
linearized the maximal operator (1.1) and aimed to show that there exists
$\varepsilon>0$, depending only on the dimension $d$ and the kernel $K$, such
that

$\big{\|}\mathcal{C}_{d,j}f\big{\|}_{L^{2}(\mathbb{R}^{d})}\lesssim_{d,K}2^{-\varepsilon
j}\|f\|_{L^{2}(\mathbb{R}^{d})}$ (1.6)

holds for all $j\in\mathbb{N}$ and $d\geq 3$, where

$\mathcal{C}_{d,j}f(x,y):=\int_{\mathbb{R}^{d-1}}f(x-t,y-|t|^{2})e^{iv(x,y)|t|^{3}}\varphi_{0}(2^{-j}|t|)K(t)\,\mathrm{d}t$

for $v(x,y)$ a measurable function taking values in the interval $[1,2]$. Once
the $L^{2}$ estimate (1.6) is proved, one may interpolate with trivial $L^{1}$
and $L^{\infty}$ estimates (without any decaying factor $2^{-\varepsilon j}$)
to obtain the full range $p\in(1,\infty)$.

The decaying factor $2^{-\varepsilon j}$ is natural to expect: by Minkowski’s
inequality, we trivially have

$\Big{\|}\int_{\mathbb{R}^{d-1}}f(x-t,y-|t|^{2})\varphi_{0}(2^{-j}|t|)K(t)\,\mathrm{d}t\Big{\|}_{L^{2}(\mathbb{R}^{d})}\lesssim_{d,K}\|f\|_{L^{2}(\mathbb{R}^{d})},$

uniformly in $j\in\mathbb{N}$. Adding the extra oscillatory term
$e^{iv(x,y)|t|^{3}}$, it is very natural to expect a decay estimate of the
form (1.6). However, proving the decay estimate (1.6) is highly non-trivial as
$v$ does not have any regularity and can be an arbitrary measurable function.
Moreover, our estimates must also be uniform over all such choices of $v$.

The strategy of Pierce and Yung [21] is to regularise the operator
$\mathcal{C}_{d,j}$ by considering $\mathcal{C}_{d,j}\mathcal{C}_{d,j}^{*}$,
the composition of the adjoint operator $\mathcal{C}_{d,j}^{*}$ with
$\mathcal{C}_{d,j}$. The operator $\mathcal{C}_{d,j}\mathcal{C}_{d,j}^{*}$ may
be better behaved than the operator $\mathcal{C}_{d,j}$ as it involves an
integral in $\mathbb{R}^{d-1}\times\mathbb{R}^{d-1}$, rather than
$\mathbb{R}^{d-1}$ itself. As the function $f$ depends on $d$ variables, when
expressing $\mathcal{C}_{d,j}\mathcal{C}_{d,j}^{*}$ as a convolution operator,
the kernel can be written as an integral in

$(d-1)+(d-1)-d=d-2$ (1.7)

variables. From (1.7), one immediately sees how the constraint $d\geq 3$ in
Pierce and Yung [21] appears: when $d=2$ the convolution kernel is not given
by an integral formula, and is just a very rough measure.

There is a second place where $d\geq 3$ plays a key role in [21]. This can be
explained by the (vague) principle that integrating along the paraboloid in
$\mathbb{R}^{d}$ for $d\geq 3$ is less ‘singular’ than integrating along the
parabola in $\mathbb{R}^{2}$. Indeed, the larger $d$ gets, the less ‘singular’
it is to integrate along the paraboloid in $\mathbb{R}^{d}$.

Following the above principle, Stein [29] proved that the spherical maximal
operator in $\mathbb{R}^{d}$ is bounded whenever $d\geq 3$. Some time later,
the case $d=2$ was resolved in Bourgain’s breakthrough work [5]. Sogge [26]
and Mockenhaupt, Seeger and Sogge [20, 19] further developed and significantly
expanded upon Bourgain’s ideas, leading to the discovery of the local
smoothing phenomenon for linear wave equations.

The difference between Stein’s approach to the spherical maximal function [29]
and Mockenhaupt, Seeger and Sogge’s approach to the circular maximal function
[19] is very similar to the difference between the approach of Pierce and Yung
[21] and the approach in the current paper. Neither Stein [29] nor Pierce and
Yung [21] required any local smoothing estimates in their work. By contrast,
local smoothing estimates are crucial in Mockenhaupt, Seeger and Sogge’s work
[19] and in our proof of Theorem 1.1. It is perhaps helpful to remark here
that Mockenhaupt, Seeger and Sogge [19] used local smoothing estimates for
linear wave operators; in the current paper, what we encounter is a local
smoothing estimate for a variable-coefficient Schrödinger operator (see
Theorem 1.4 below). These local smoothing estimates follow from the recent
work [10], which is reviewed in §1.5. At their heart lies an observation of
Wisewell [32] regarding differences between the geometry of Kakeya and Nikodym
sets of curves.

###  1.4. An $L^{2}$ bound and its surprising proof.

In this subsection we discuss the proof strategy for Theorem 1.1 and further
remark on the variable-coefficient local smoothing estimates for Schödinger
operators that feature in the argument.

To prove Theorem 1.1, we still prove a decay estimate of the form (1.6). More
precisely, we prove that there exists a universal constant $\varepsilon>0$
such that

$\big{\|}\mathcal{C}_{j}f\big{\|}_{L^{2}(\mathbb{R}^{2})}\lesssim
2^{-\varepsilon j}\|f\|_{L^{2}(\mathbb{R}^{2})}$ (1.8)

holds for all $j\in\mathbb{N}$, where

$\mathcal{C}_{j}f(x,y):=\int_{\mathbb{R}}f(x-t,y-t^{2})e^{iv(x,y)t^{3}}\varphi_{0}(2^{-j}|t|)\frac{\,\mathrm{d}t}{t}$

for $v(x,y)$ a measurable function taking values in the interval $[1,2]$. What
is surprising to us is that we do not know how to prove (1.8) directly, and
instead we show (see Theorem 2.3 below) that there exist $p_{\circ}>2$ and
$\varepsilon_{\circ}>0$ such that

$\big{\|}\mathcal{C}_{j}f\big{\|}_{L^{p_{\circ}}(\mathbb{R}^{2})}\lesssim
2^{-\varepsilon_{\circ}j}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$ (1.9)

for all $j\in\mathbb{N}$. This, together with the trivial bound

$\big{\|}\mathcal{C}_{j}f\big{\|}_{L^{1}(\mathbb{R}^{2})}\lesssim\|f\|_{L^{1}(\mathbb{R}^{2})},$

implies (1.8). In the literature related to (singular) Radon transforms in
harmonic analysis, we are not aware of another instance where an $L^{2}$ decay
estimate is proven via interpolation; usually it is natural to work directly
in $L^{2}$ because of, for instance, self-adjoint properties and Plancherel’s
theorem.

The estimate (1.9) (or, more generally, Theorem 2.3 below) follows in part
from a theorem for variable coefficient Schrödinger-type propagators recently
proved in [10]. The general result in [10] can be thought of as a local
smoothing variant of the celebrated work of Bourgain [6] on oscillatory
integral operators, and is partly inspired by important observations of
Wisewell [32] concerning Kakeya and Nikodym sets of curves. To apply the
theory from [10], one must check that the relevant phase functions satisfy
certain conditions in addition to those of Hörmander [16], called the Nikodym
non-compression hypothesis (see Definition 1.3 below). This hypothesis ensures
certain families of curves associated to the operator $\mathcal{C}_{j}$ cannot
compress into small sets (what Bourgain [6] called the Kakeya compression
phenomenon): we refer the reader to [10] for further details.

We emphasise that (1.9) is not an off-the-shelf application of the
aforementioned local smoothing results from [10]. An array of classical
oscillatory integral and $L^{2}$-based techniques are applied to reduce the
problem to analysing the most singular part of the operator where the local
smoothing result applies. These arguments are partly inspired by methods
developed by Pramanik–Seeger [22] to prove local smoothing-type results in the
presence of fold singularities. Furthermore, it transpires that, even after
the above reductions, the Nikodym non-compression hypothesis is not globally
valid, and an exceptional piece of the operator must be analysed via
alternative means. However, this piece is amenable to (variable coefficient)
decoupling theory, which is a standard tool in the study of local smoothing
estimates [33, 7, 3, 4].

###  1.5. Hörmander-type phase conditions and the Nikodym non-compression
hypothesis

In this subsection we review a key result from [10] that we shall use later.
We begin by reviewing some of the standard definitions in the oscillatory
integral operator literature (see, for example, [10, 15]), in the special case
of translation-invariant phases. For $n\geq 2$ and $\rho>0$, let

$\mathbb{D}_{\rho}^{n}:=\mathbb{R}^{n-1}\times\mathbb{B}_{\rho}^{1}\times\mathbb{B}_{\rho}^{n-1},$

where $\mathbb{B}_{\rho}^{d}$ denotes the open ball in $\mathbb{R}^{d}$ of
radius $\rho$ centred at the origin.

######  Definition 1.2.

We say $\Phi\colon\mathbb{D}_{\rho}^{n}\to\mathbb{R}$ is a translation-
invariant phase if there exists some smooth function
$\phi\colon\mathbb{B}_{\rho}^{1}\times\mathbb{B}_{\rho}^{n-1}\to\mathbb{R}$
such that

$\Phi(\mathbf{x},w;\boldsymbol{\xi}):=\mathbf{x}\cdot\boldsymbol{\xi}+\phi(w;\boldsymbol{\xi})\qquad\textrm{for
all $(\mathbf{x},w;\boldsymbol{\xi})\in\mathbb{D}_{\rho}^{n}$.}$ (1.10)

Furthermore, if $\Phi$ is of the form (1.10), then we say that $\Phi$ is a
translation-invariant, Hörmander-type phase if

$\det\partial_{w}\partial^{2}_{\boldsymbol{\xi}\boldsymbol{\xi}}\phi(w;\boldsymbol{\xi})\neq
0\quad\textrm{for all
$(w;\boldsymbol{\xi})\in\mathbb{B}_{\rho}^{1}\times\mathbb{B}_{\rho}^{2}$}.$
(1.11)

Often we pair the phase with a smooth cutoff function $a\in
C_{c}^{\infty}(\mathbb{B}_{\rho}^{1}\times\mathbb{B}_{\rho}^{n-1})$, in which
case we refer to the pairing $[\Phi;a]$ as a phase-amplitude pair.

Given $[\Phi;a]$ a phase-amplitude pair and $\lambda\geq 1$, we consider the
rescaled data

$\Phi^{\lambda}(\mathbf{x},w;\boldsymbol{\xi}):=\lambda\Phi(\mathbf{x}/\lambda,w/\lambda;\boldsymbol{\xi})\quad\textrm{and}\quad
a^{\lambda}(w;\boldsymbol{\xi}):=a(w/\lambda;\boldsymbol{\xi}).$

The variable Schrödinger propagator $U^{\lambda}=U^{\lambda}[\Phi;a]$
associated to $[\Phi;a]$ is the operator

$U^{\lambda}f(\mathbf{x},w):=\int_{\widehat{\mathbb{R}}^{2}}e^{i\Phi^{\lambda}(\mathbf{x},w;\boldsymbol{\xi})}a^{\lambda}(w;\boldsymbol{\xi})\hat{f}(\boldsymbol{\xi})\,\mathrm{d}\boldsymbol{\xi},$
(1.12)

initially defined as acting on $f\in\mathcal{S}(\mathbb{R}^{2})$.

Any variable Schrödinger propagator associated to a Hörmander-type phase
satisfies a range of universal estimates, given by Stein’s oscillatory
integral theorem [28]. Our approach requires estimates for specific operators
(related to the $\mathcal{C}_{j}$ above) which go beyond this universal range.
Examples due to Bourgain [6] and Wisewell [32] demonstrate Stein’s theorem
[28] is sharp over the class of general translation-invariant, Hörmander-type
phases. Thus, to proceed we must impose additional conditions on our class of
phase functions.

######  Definition 1.3.

For $0<\rho<1$, let $\Phi\colon\mathbb{D}^{2}_{\rho}\to\mathbb{R}$ be a real
analytic translation-invariant phase of the form (1.10). For $\gamma\in(0,1)$,
we say that $\Phi$ satisfies the Nikodym non-compression hypothesis with
parameter $\gamma$ if :

  1. a)

For all
$(w;\boldsymbol{\xi})\in\mathbb{B}^{1}_{\rho}\times\mathbb{B}^{2}_{\rho}$, we
have
$\det\partial^{2}_{\boldsymbol{\xi}\boldsymbol{\xi}}\phi(w;\boldsymbol{\xi})=0$;

  2. b)

For all
$(w;\boldsymbol{\xi})\in\mathbb{B}^{1}_{\rho}\times\mathbb{B}^{2}_{\rho}$, we
have

$\left|\det\begin{bmatrix}\partial_{w}\partial^{2}_{\xi_{1}\xi_{1}}\phi(w;\boldsymbol{\xi}),&\partial_{w}\partial^{2}_{\xi_{1}\xi_{2}}\phi(w;\boldsymbol{\xi}),&\partial_{w}\partial^{2}_{\xi_{2}\xi_{2}}\phi(w;\boldsymbol{\xi})\\\
\partial^{2}_{w}\partial^{2}_{\xi_{1}\xi_{1}}\phi(w;\boldsymbol{\xi}),&\partial^{2}_{w}\partial^{2}_{\xi_{1}\xi_{2}}\phi(w;\boldsymbol{\xi}),&\partial^{2}_{w}\partial^{2}_{\xi_{2}\xi_{2}}\phi(w;\boldsymbol{\xi})\\\
\partial^{3}_{w}\partial^{2}_{\xi_{1}\xi_{1}}\phi(w;\boldsymbol{\xi}),&\partial^{3}_{w}\partial^{2}_{\xi_{1}\xi_{2}}\phi(w;\boldsymbol{\xi}),&\partial^{3}_{w}\partial^{2}_{\xi_{2}\xi_{2}}\phi(w;\boldsymbol{\xi})\end{bmatrix}\right|\geq\gamma.$

Definition 1.3 is a quantified variant of a special case of the Nikodym non-
compression hypothesis from [10]. Under this hypothesis, we have the following
(weakly) quantified local smoothing estimate.

######  Theorem 1.4 ([10, Theorem 2.8 II)]).

There exist some $2<p_{\circ}<4$, $\beta(p_{\circ})<1/2$ and large
$M_{\circ}\geq 1$, $N_{\circ}\geq 1$ such that the following holds. Let
$\rho>0$, $0<\gamma\leq 1$ and $W\geq 1$ and $[\Phi;a]$ be a phase-amplitude
pair where $\Phi\colon\mathbb{D}^{2}_{\rho}\to\mathbb{R}$ is a real analytic
translation-invariant phase which satisfies the Nikodym non-compression
hypothesis with parameter $\gamma$. Further suppose that

$|\det\partial_{w}\partial^{2}_{\boldsymbol{\xi}\boldsymbol{\xi}}\phi(w;\boldsymbol{\xi})|\geq\gamma\qquad\textrm{for
all
$(w;\boldsymbol{\xi})\in\mathbb{B}^{1}_{\rho}\times\mathbb{B}^{2}_{\rho}$};$
(1.13)

the amplitude satisfies
$\|a\|_{L^{\infty}(\mathbb{B}_{\rho}^{1}\times\mathbb{B}_{\rho}^{2})}=1$, and

$|\partial_{w}^{\alpha}\partial^{\boldsymbol{\beta}}_{\boldsymbol{\xi}}\phi(w;\boldsymbol{\xi})|\leq
W\quad\textrm{and}\quad|\partial_{w}^{\alpha}\partial^{\boldsymbol{\beta}}_{\boldsymbol{\xi}}a(w;\boldsymbol{\xi})|\leq
W\text{ for all
}(w;\boldsymbol{\xi})\in\mathbb{B}^{1}_{\rho}\times\mathbb{B}_{\rho}^{2}$
(1.14)

and all $\alpha\in\mathbb{N}_{0}$,
$\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}^{2}$ satisfying
$1\leq\alpha+\beta_{1}+\beta_{2}\leq N_{\circ}$. Then

$\|U^{\lambda}[\Phi;a]f\|_{L^{p_{\circ}}(\mathbb{R}^{3})}\lesssim\lambda^{\beta(p_{\circ})}\big{(}W\gamma^{-1}\big{)}^{M_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
(1.15)

for all $\lambda\geq 1$.

This result follows from [10, Theorem 2.8 II)] by tracking the dependencies on
the parameters $\gamma$ and $W$ in the proof: see Appendix A for further
details.444In [10, Theorem 2.8 II)] the operators $U^{\lambda}[\Phi;a]$ are
smoothly localised in the $\mathbf{x}$ variable to a $\lambda$-ball. However,
the global variant in Theorem 1.4 easily follows by the pseudo-local
properties of the operator (see, for instance, the proof of Proposition 6.6 at
the end of §6.7 below).

Note that, by Plancherel’s theorem (since our phases are translation-
invariant), (1.15) holds with

$p_{\circ}=2,\qquad\beta(p_{\circ})=1/2.$ (1.16)

Moreover, by Stein’s oscillatory integral theorem [28] (which generalises the
classical Stein–Tomas restriction theorem), we see that (1.15) holds with

$p_{\circ}=4,\qquad\beta(p_{\circ})=1/2.$ (1.17)

Therefore, if we do not aim for $\beta(p_{\circ})<1/2$ in Theorem 1.4, then we
can just interpolate (1.16) and (1.17). Unfortunately, the resulting estimate
(1.15) with $2\leq p_{\circ}\leq 4$ and $\beta(p_{\circ})=1/2$ falls slightly
short of our needs: it is crucial that we have $\beta(p_{\circ})<1/2$ in
Theorem 1.4 in order to sum up high frequencies (see Theorem 2.3 below).

After proving Theorem 1.4, we can interpolate it with (1.16) and (1.17) to see
that Theorem 1.4 holds for all $2<p_{\circ}<4$. This key estimate is used
later in the proof of Propositions 3.4 and 3.5.

###  1.6. The full planar Pierce–Yung operator.

It may be possible to generalise the argument in the current paper to prove
that

$\sup_{v\in\mathbb{R}}\Big{|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{ivt^{m}}\frac{\,\mathrm{d}t}{t}\Big{|}$

is bounded on $L^{p}(\mathbb{R}^{2})$ for every $p\in(1,\infty)$ and every
integer $m\geq 4$. However, it remains an interesting question whether the
full Pierce–Yung operator

$\sup_{v_{3},\dots,v_{m}\in\mathbb{R}}\Big{|}\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{i(v_{3}t^{3}+\dots+v_{m}t^{m})}\frac{\,\mathrm{d}t}{t}\Big{|}$

is bounded on $L^{p}(\mathbb{R}^{2})$ for every integer $m\geq 4$ and
$p\in(1,\infty)$.

### Notational conventions

We let $\widehat{\mathbb{R}}$ denote the frequency domain, which is the
Pontryagin dual group of $\mathbb{R}$ understood here as simply a copy of
$\mathbb{R}$. Given $f\in L^{1}(\mathbb{R}^{d})$ and $g\in
L^{1}(\widehat{\mathbb{R}}^{d})$ we define the Fourier transform and inverse
Fourier transform by

$\hat{f}(\boldsymbol{\xi}):=\int_{\mathbb{R}^{d}}e^{-i\langle\mathbf{x},\boldsymbol{\xi}\rangle}f(\mathbf{x})\,\,\mathrm{d}\mathbf{x}\quad\textrm{and}\quad\check{g}(x):=\frac{1}{(2\pi)^{d}}\int_{\widehat{\mathbb{R}}^{d}}e^{i\langle\mathbf{x},\boldsymbol{\xi}\rangle}g(\boldsymbol{\xi})\,\,\mathrm{d}\boldsymbol{\xi},$

respectively. Given $a,b\in\mathbb{R}$ we write $a\wedge b:=\min\\{a,b\\}$ and
$a\vee b:=\max\\{a,b\\}$.

### Acknowledgements

D.B. is supported by the AEI grants RYC2020-029151-I and PID2022-140977NA-I00.
S.G. is partly supported by the Nankai Zhide Foundation, and partly supported
by NSF-2044828.

##  2\. Decomposition of the kernel

We begin with some elementary preliminaries for the proof of Theorem 1.1.

###  2.1. Decomposition relative to the oscillatory factor

Let $\beta_{0}\in C^{\infty}_{c}(\mathbb{R})$ be real, even and radially non-
increasing, supported in $[-2,2]$ and satisfy $\beta_{0}(s)=1$ for
$s\in[-1,1]$. Define $\beta(s):=\beta_{0}(s)-\beta_{0}(2s)$ so that

$\sum_{j\in\mathbb{Z}}\beta(2^{-j}s)=1\quad\textrm{and}\quad\beta_{0}(s)=\sum_{j\leq
0}\beta(2^{-j}s)\quad\textrm{for all $s\in\mathbb{R}\setminus\\{0\\}$.}$

For any $m\in\mathbb{Z}$, define

$\displaystyle H^{v}_{\leq m}f(x,y)$
$\displaystyle:=\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})e^{ivt^{3}}\beta_{0}(2^{-m}t)\frac{\,\mathrm{d}t}{t},$
$\displaystyle H^{v}_{m}f(x,y)$
$\displaystyle:=\int_{\mathbb{R}}f(x-t,y-t^{2})e^{ivt^{3}}\beta(2^{-m}t)\frac{\,\mathrm{d}t}{t}.$

We then decompose

$\mathcal{C}f(x,y)\leq\mathcal{C}_{\leq
0}f(x,y)+\sum_{\ell>0}\mathcal{C}_{\ell}f(x,y)$ (2.1)

where

$\mathcal{C}_{\leq 0}f(x,y):=\sup_{j\in\mathbb{Z}}\sup_{v\in
V_{j}}|H^{v}_{\leq-j}f(x,y)|,\qquad\mathcal{C}_{\ell}f(x,y):=\sup_{j\in\mathbb{Z}}\sup_{v\in
V_{j}}|H^{v}_{-j+\ell}f(x,y)|$

for $V_{j}:=[2^{3j},2^{3(j+1)}]$.

###  2.2. Non-oscillatory analysis

The term $\mathcal{C}_{\leq 0}$ has no oscillation and can be controlled by
the maximal function and the Hilbert transform along the parabola. Recall that
these are the operators

$\displaystyle M_{\mathrm{par}}f(x,y)$
$\displaystyle:=\sup_{r>0}\frac{1}{2r}\int_{-r}^{r}|f(x-t,y-t^{2})|\,\mathrm{d}t,$
$\displaystyle H_{\mathrm{par}}f(x,y)$
$\displaystyle:=\mathrm{p.v.}\int_{\mathbb{R}}f(x-t,y-t^{2})\frac{\,\mathrm{d}t}{t}$

which are both well-known to be bounded on $L^{p}(\mathbb{R}^{2})$ for
$1<p<\infty$ (see, for instance, [11, Chapter 8, §7]). Moreover, it was shown
in [21, Theorem 5.1] that $L^{p}$-boundedness also holds for the maximal
truncations

$\sup_{\varepsilon>0}|H_{\mathrm{par},\varepsilon}f(x,y)|,\qquad\text{ where
}\quad
H_{\mathrm{par},\varepsilon}f(x,y):=\int_{|y|\geq\varepsilon}f(x-t,y-t^{2})\frac{\,\mathrm{d}t}{t}.$
(2.2)

In view of these known results, we quickly deduce the following lemma.

######  Lemma 2.1.

For every $1<p<\infty$, we have

$\|\mathcal{C}_{\leq
0}f\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

###### Proof.

Write

$H_{\leq-j}^{v}=\big{(}H_{\leq-j}^{v}-H_{\leq-j}^{0}\big{)}+H_{\leq-j}^{0}=:M_{-j}^{v}+H_{\leq-j}^{0}.$

Then

$\displaystyle\sup_{v\in V_{j}}|M_{-j}^{v}f(x,y)|$ $\displaystyle=\sup_{v\in
V_{j}}\Big{|}\int_{\mathbb{R}}f(x-t,y-t^{2})\big{(}e^{ivt^{3}}-1\big{)}\beta_{0}(2^{j}t)\frac{\,\mathrm{d}t}{t}\Big{|}$
$\displaystyle\lesssim
2^{j}\int_{\mathbb{R}}|f(x-t,y-t^{2})|\beta_{0}(2^{j}t)\,\mathrm{d}t$
$\displaystyle\lesssim\frac{1}{2^{-j+2}}\int_{|t|\leq
2^{-j+1}}|f(x-t,y-t^{2})|\,\mathrm{d}t.$

Thus,

$\sup_{j\in\mathbb{Z}}\sup_{v\in V_{j}}|M_{-j}^{v}f(x,y)|\lesssim
M_{\mathrm{par}}f(x,y)$

and the $L^{p}$-boundedness for this term follows.

For the second term, since $|t|\leq 2^{-j+1}$ for all
$t\in\mathrm{supp}\,\beta_{0}(2^{j}\,\cdot\,)$, we note that

$\begin{split}H^{0}_{\leq-j}f(x,y)&=H_{\mathrm{par}}f(x,y)-H_{\mathrm{par},2^{-j+1}}f(x,y)\\\
&+\int_{|t|\leq
2^{-j+1}}f(x-t,y-t^{2})\big{(}\beta_{0}(2^{j}t)-1\big{)}\frac{\,\mathrm{d}t}{t},\end{split}$

where $H_{\mathrm{par},2^{-j+1}}f$ is a truncation as defined in (2.2). Since
$\beta_{0}(s)=1$ for $s\in[-1,1]$, we have

$\displaystyle\Big{|}\int_{|t|\leq
2^{-j+1}}f(x-t,y-t^{2})\big{(}\beta_{0}(2^{j}t)-1\big{)}\frac{\,\mathrm{d}t}{t}\Big{|}$
$\displaystyle\lesssim\frac{1}{2^{-j}}\int_{2^{-j}\leq|t|\leq
2^{-j+1}}|f(x-t,y-t^{2})|\,\mathrm{d}t\lesssim M_{\mathrm{par}}f(x,y).$

Therefore,

$\displaystyle\sup_{j\in\mathbb{Z}}\sup_{v\in
V_{j}}|H_{\leq-j}^{0}f(x,y)|\leq|H_{\mathrm{par}}f(x,y)|+\sup_{\varepsilon>0}|H_{\varepsilon}f(x,y)|+M_{\mathrm{par}}f(x,y)$

and the $L^{p}$-boundedness follows. ∎

Regarding the terms $\mathcal{C}_{\ell}$, we have the following uniform bound.

######  Lemma 2.2.

For all $1<p\leq\infty$, we have

$\|\mathcal{C}_{\ell}f\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{p}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

###### Proof.

We trivially bound

$\displaystyle|H_{-j+\ell}^{v}f(x,y)|$
$\displaystyle\leq\Big{|}\int_{\mathbb{R}}|f(x-t,y-t^{2})|\beta(2^{j-\ell}t)\frac{\,\mathrm{d}t}{|t|}\Big{|}$
$\displaystyle\lesssim\frac{1}{2^{-j+\ell+2}}\int_{|t|\leq
2^{-j+\ell+1}}|f(x-t,y-t^{2})|\,\mathrm{d}t\lesssim M_{\mathrm{par}}f(x,y)$

uniformly in $v$ and $j$, so the result follows from the
$L^{p}(\mathbb{R}^{2})$-boundedness of $M_{\mathrm{par}}$. ∎

###  2.3. Our goal

Lemma 2.2 falls short of proving Theorem 1.1 since the estimates are not
summable in $\ell$. The key ingredient in the proof of Theorem 1.1 is the
following improvement.

######  Theorem 2.3.

There exists some $p_{\circ}>2$ and $\varepsilon_{\circ}>0$ such that

$\|\mathcal{C}_{\ell}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}\lesssim
2^{-\ell\varepsilon_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$ (2.3)

holds for all $\ell\geq 1$.

Once this is established, interpolation with Lemma 2.2 immediately upgrades
(2.3) to all $1<p<\infty$. Theorem 1.1 is then an immediate consequence of
this upgrade and Lemma 2.1, by virtue of the decomposition (2.1) and the
triangle inequality.

##  3\. Reduction to key estimates

###  3.1. Frequency decomposition

Let $\beta\in C_{c}^{\infty}(\mathbb{R})$ be as in §2. For any
$m\in\mathbb{Z}$ and $i=1,2$, let $P_{m}^{(i)}$ denote the $i$th coordinate
Littlewood–Paley projection associated with $\beta(2^{-m}\,\cdot\,)$: that is,
for $f\in\mathcal{S}(\mathbb{R}^{2})$, we define

$(P_{m}^{(1)}f)\;\widehat{}\;(\xi,\eta):=\beta(2^{-m}\xi)\widehat{f}(\xi,\eta),\qquad(P_{m}^{(2)}f)\;\widehat{}\;(\xi,\eta):=\beta(2^{-m}\eta)\widehat{f}(\xi,\eta).$

Note that

$f=\sum_{m_{1},m_{2}\in\mathbb{Z}}P_{m_{1}}^{(1)}P_{m_{2}}^{(2)}f.$

It is also convenient to define the operators

$P_{\leq m}^{(i)}:=\sum_{n\leq m}P_{n}^{(i)},$

which, for $i=1$, $2$, are associated to the multipliers
$\beta_{0}(2^{-m}\xi)$ and $\beta_{0}(2^{-m}\eta)$, respectively. We shall
also make use of Littlewood–Paley projections $\widetilde{P}_{m}^{(i)}$
satisfying

$P_{m}^{(i)}=P_{m}^{(i)}\widetilde{P}_{m}^{(i)}$

and associated to a cut-off function $\widetilde{\beta}\in
C_{c}^{\infty}(\mathbb{R})$ supported in $[-4,-1/4]\cup[1/4,4]$ and that
equals $1$ on $\mathrm{supp}\,\beta\subseteq[-2,-1/2]\cup[1/2,2]$.

For $\ell\in\mathbb{N}$, $j\in\mathbb{Z}$ and $v\in V_{j}$, we decompose

$H_{-j+\ell}^{v}f=\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}\in\mathbb{Z}}H_{-j+\ell}^{v}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f.$

Moreover, we group the low frequencies and write, for $C_{1},C_{2}\geq 10^{6}$
satisfying $C_{1}=2C_{2}+12$ fixed (absolute) constants,

$\displaystyle H_{-j+\ell}^{v}f$ $\displaystyle=H_{-j+\ell}^{v}P_{\leq-
C_{1}+2\ell+j}^{(1)}P_{\leq-C_{2}+\ell+2j}^{(2)}f$
$\displaystyle\quad+\sum_{k_{1}>-C_{1}}H_{-j+\ell}^{v}P_{k_{1}+2\ell+j}^{(1)}P_{\leq-
C_{2}+\ell+2j}^{(2)}f$
$\displaystyle\quad+\sum_{k_{2}>-C_{2}}H_{-j+\ell}^{v}P_{\leq-
C_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f$
$\displaystyle\quad+\sum_{k_{1}>-C_{1}}\sum_{k_{2}>-C_{2}}H_{-j+\ell}^{v}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f.$
(3.1)

Note that the Fourier multiplier associated to
$H_{-j+\ell}^{v}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+j}^{(2)}$ is given by

$\beta(2^{-k_{1}-2\ell-j}\xi)\beta(2^{-k_{2}-\ell-2j}\eta)\int_{\mathbb{R}}e^{-i(t\xi+t^{2}\eta-t^{3}v)}\beta(2^{-\ell+j}t)\frac{\,\mathrm{d}t}{t},$
(3.2)

and similar formulas hold for the other frequency projections.

The low frequency part of the operator is easily dealt with.

######  Lemma 3.1 (Low frequency part).

For $2\leq p\leq\infty$, the inequality

$\|\sup_{j\in\mathbb{Z}}\sup_{v\in V_{j}}|H^{v}_{-j+\ell}P_{\leq-
C_{1}+2\ell+j}^{(1)}P_{\leq-
C_{2}+\ell+2j}^{(2)}f|\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{p,N,C_{1},C_{2}}2^{-\ell
N}\|f\|_{L^{p}(\mathbb{R}^{2})}$

holds for all $N\in\mathbb{N}_{0}$ and all $\ell\in\mathbb{N}$.

The proof of Lemma 3.1 is based on pointwise comparison with the strong
maximal function, which is achieved by estimating the kernel using elementary
integration-by-parts arguments. We postpone the details until §4.3.

For the high frequencies, we introduce a rescaled variant
$\mathcal{H}_{n}^{w}$ of $H_{-j+\ell}^{v}$ for which we prove the key
estimates and then relate back to $H_{-j+\ell}^{v}$. Given $n\in\mathbb{N}$,
$w>0$, define

$\mathcal{H}_{n}^{w}f(x,y):=\frac{1}{(2\pi)^{2}}\int_{\widehat{\mathbb{R}}^{2}}e^{i(x\xi+y\eta)}\int_{\mathbb{R}}e^{-i2^{n}(t\xi+t^{2}\eta-t^{3}w)}\beta(t)\frac{\,\mathrm{d}t}{t}\widehat{f}(\xi,\eta)\,\mathrm{d}\xi\,\mathrm{d}\eta$

for all $f\in\mathcal{S}(\mathbb{R}^{2})$. Furthermore, for
$(k_{1},k_{2})\in\mathbb{Z}^{2}$ we define the frequency localised operators

$\displaystyle\mathcal{H}_{n,k_{1},k_{2}}^{w}:=\mathcal{H}_{n}^{w}P_{k_{1}}^{(1)}P_{k_{2}}^{(2)},\quad\mathcal{H}_{n,\leq
k_{1},k_{2}}^{w}:=\mathcal{H}_{n}^{w}P_{\leq
k_{1}}^{(1)}P_{k_{2}}^{(2)},\quad\mathcal{H}_{n,k_{1},\leq
k_{2}}^{w}:=\mathcal{H}_{n}^{w}P_{k_{1}}^{(1)}P_{\leq k_{2}}^{(2)}.$

With this notation, our goal is to prove the following.

######  Theorem 3.2.

There exist $p_{\circ}>2$ and $\varepsilon_{\circ}>0$, $\delta_{\circ}>0$ such
that the inequalities

$\displaystyle\Big{(}\int_{1}^{8}\|\mathcal{H}^{w}_{n,k_{1},\leq-
C_{2}}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}^{p_{\circ}}\,\mathrm{d}w\Big{)}^{1/p_{\circ}}$
$\displaystyle\lesssim
2^{-n/p_{\circ}-n\varepsilon_{\circ}-k_{1}\delta_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$
(3.3) $\displaystyle\Big{(}\int_{1}^{8}\|\mathcal{H}^{w}_{n,\leq-
C_{1},k_{2}}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}^{p_{\circ}}\,\mathrm{d}w\Big{)}^{1/p_{\circ}}$
$\displaystyle\lesssim
2^{-n/p_{\circ}-n\varepsilon_{\circ}-k_{2}\delta_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$
(3.4)
$\displaystyle\Big{(}\int_{1}^{8}\|\mathcal{H}_{n,k_{1},k_{2}}^{w}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}^{p_{\circ}}\,\mathrm{d}w\Big{)}^{1/p_{\circ}}$
$\displaystyle\lesssim
2^{-n/p_{\circ}-n\varepsilon_{\circ}-(k_{1}+k_{2})\delta_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
(3.5)

hold for all $n\in\mathbb{N}$, $k_{1}>-C_{1}$ and $k_{2}>-C_{2}$.

Once the above theorem is established, Theorem 2.3 follows from standard
Littlewood–Paley, Sobolev embedding and scaling arguments.

###### Proof of Theorem 2.3 (assuming Lemma 3.1 and Theorem 3.2).

Let $\ell\in\mathbb{N}$, $j\in\mathbb{Z}$, $k_{1},k_{2}\in\mathbb{Z}$.
Performing the change of variables

$(t;\xi,\eta)\mapsto(2^{\ell-j}t;2^{2\ell+j}\xi,2^{\ell+2j}\eta)$

in (3.2), we obtain

$H_{-j+\ell}^{v}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f(x,y)=\mathcal{H}_{3\ell}^{2^{-3j}v}P_{k_{1}}^{(1)}P_{k_{2}}^{(2)}f_{\ell,j}(2^{2\ell+j}x,2^{\ell+2j}y)$

where

$f_{\ell,j}(x,y):=f(2^{-2\ell-j}x,2^{-\ell-2j}y).$

Recall that $V_{j}:=[2^{3j},2^{3(j+1)}]$. By the above scaling, for any fixed
$j\in\mathbb{Z}$, $p\geq 1$ and $M>0$, the norm inequality

$\|\sup_{v\in
V_{j}}|H_{-j+\ell}^{v}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f|\|_{L^{p}(\mathbb{R}^{2})}\leq
M\|f\|_{L^{p}(\mathbb{R}^{2})}$

holds for all $f\in\mathcal{S}(\mathbb{R}^{2})$ if and only if the norm
inequality

$\|\sup_{1\leq w\leq
8}|\mathcal{H}^{w}_{3\ell,k_{1},k_{2}}f|\|_{L^{p}(\mathbb{R}^{2})}\leq
M\|f\|_{L^{p}(\mathbb{R}^{2})}$

holds for all $f\in\mathcal{S}(\mathbb{R}^{2})$, with the same constant $M$. A
similar statement holds if either $P_{k_{1}}^{(1)}$ or $P_{k_{2}}^{(2)}$ is
replaced with $P_{\leq k_{1}}^{(1)}$ or $P_{\leq k_{2}}^{(2)}$.

For $g\in\mathcal{S}(\mathbb{R}^{2})$, by the Fundamental Theorem of Calculus
and Hölder’s inequality,

$\sup_{1\leq w\leq
8}|\mathcal{H}_{3\ell}^{w}g|^{p}\leq|\mathcal{H}^{1}_{3\ell}g|^{p}+p\Big{(}\int_{1}^{8}|\mathcal{H}^{w}_{3\ell}g|^{p}\,\mathrm{d}w\Big{)}^{1-1/p}\Big{(}\int_{1}^{8}|\partial_{w}\mathcal{H}_{3\ell}^{w}g|^{p}\,\mathrm{d}w\Big{)}^{1/p}.$

Furthermore,

$\partial_{w}\mathcal{H}^{w}_{3\ell}g=3i2^{3\ell}\widetilde{\mathcal{H}}^{w}_{3\ell}g,$

where the operator $\widetilde{\mathcal{H}}^{w}_{3\ell}$ is defined by the
Fourier multiplier

$\int_{\mathbb{R}}e^{i2^{3\ell}(-t\xi-t^{2}\eta+t^{3}w)}\beta(t)t^{2}\,\mathrm{d}t,$

which is essentially identical to the multiplier of $\mathcal{H}^{w}_{3\ell}$.
Combining the above observations and letting $g$ be suitable Littlewood–Paley
projections of $f$, provided $C_{1}$, $C_{2}\geq 1$ are chosen sufficiently
large, Theorem 3.2 implies

$\displaystyle\|\sup_{v\in
V_{j}}|H^{v}_{-j+\ell}P_{k_{1}+2\ell+j}^{(1)}P_{\leq-
C_{2}+\ell+2j}^{(2)}f|\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{-3\ell\varepsilon_{\circ}-k_{1}\delta_{\circ}}\|\widetilde{P}_{k_{1}+2\ell+j}^{(1)}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$
$\displaystyle\|\sup_{v\in V_{j}}|H^{v}_{-j+\ell}P_{\leq-
C_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f|\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{-3\ell\varepsilon_{\circ}-k_{2}\delta_{\circ}}\|\widetilde{P}^{(2)}_{k_{2}+\ell+2j}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$
$\displaystyle\|\sup_{v\in
V_{j}}|H^{v}_{-j+\ell}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{-3\ell\varepsilon_{\circ}-(k_{1}+k_{2})\delta_{\circ}}\|\widetilde{P}^{(1)}_{k_{1}+2\ell+j}f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$

where we have used that $P_{m}^{(i)}=P_{m}^{(i)}\widetilde{P}_{m}^{(i)}$ for
$m\in\mathbb{Z}$ and $i=1$, $2$.

By (one-dimensional) Littlewood–Paley theory and Fubini’s theorem, we have
that

$\Big{(}\sum_{j\in\mathbb{Z}}\|\widetilde{P}^{(i)}_{j}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\Big{)}^{1/p}\lesssim\|f\|_{L^{p}(\mathbb{R}^{2})}\qquad\textrm{for
all $2\leq p<\infty$.}$

Using that
$\sup_{j\in\mathbb{Z}}|a_{j}|^{p}\leq\sum_{j\in\mathbb{Z}}|a_{j}|^{p}$, the
above estimates therefore upgrade to

$\displaystyle\sum_{k_{1}>-C_{1}}\|\sup_{j\in\mathbb{Z}}\sup_{v\in
V_{j}}|H^{v}_{-j+\ell}P_{k_{1}+2\ell+j}^{(1)}P_{\leq-
C_{2}+\ell+2j}^{(2)}f|\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{-3\ell\varepsilon_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$
$\displaystyle\sum_{k_{2}>-C_{2}}\|\sup_{j\in\mathbb{Z}}\sup_{v\in
V_{j}}|H^{v}_{-j+\ell}P_{\leq-
C_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f|\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{-3\ell\varepsilon_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})},$
$\displaystyle\sum_{k_{1}>-C_{1}}\sum_{k_{2}>-C_{2}}\|\sup_{j\in\mathbb{Z}}\sup_{v\in
V_{j}}|H^{v}_{-j+\ell}P_{k_{1}+2\ell+j}^{(1)}P_{k_{2}+\ell+2j}^{(2)}f|\|_{L^{p_{\circ}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{-3\ell\varepsilon_{\circ}}\|f\|_{L^{p_{\circ}}(\mathbb{R}^{2})}.$

In view of the decomposition (3.1), theses estimates, together with Lemma 3.1
and an application of the triangle inequality, imply (2.3). ∎

###  3.2. Isolating the critical points

Let $m_{n}^{w}$ denote the multiplier associated to the operator
$\mathcal{H}_{n}^{w}$, so that

$m_{n}^{w}(\xi,\eta):=\int_{\mathbb{R}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}a(t)\,\mathrm{d}t$
(3.6)

where

$\phi_{\xi,\eta}^{w}(t):=-t\xi-t^{2}\eta+t^{3}w\quad\textrm{and}\quad
a(t):=\frac{\beta(t)}{t}.$ (3.7)

For $(k_{1},k_{2})\in\mathbb{Z}^{2}$, the multipliers of the frequency
localised operators $\mathcal{H}_{n,k_{1},k_{2}}^{w}$, $\mathcal{H}_{n,\leq
k_{1},k_{2}}^{w}$, $\mathcal{H}_{n,k_{1},\leq k_{2}}^{w}$ are then given by

$\displaystyle m_{n,k_{1},k_{2}}^{w}(\xi,\eta)$
$\displaystyle:=\beta(2^{-k_{1}}\xi)\beta(2^{-k_{2}}\eta)m_{n}^{w}(\xi,\eta),$
$\displaystyle m_{n,\leq k_{1},k_{2}}^{w}(\xi,\eta)$
$\displaystyle:=\beta_{0}(2^{-k_{1}}\xi)\beta(2^{-k_{2}}\eta)m_{n}^{w}(\xi,\eta),$
$\displaystyle m_{n,k_{1},\leq k_{2}}^{w}(\xi,\eta)$
$\displaystyle:=\beta(2^{-k_{1}}\xi)\beta_{0}(2^{-k_{2}}\eta)m_{n}^{w}(\xi,\eta),$

respectively.

We analyse the above multipliers using the method of stationary phase. This
relies on understanding the critical points of $\phi_{\xi,\eta}^{w}$, which
are the zeros of the function

$(\phi_{\xi,\eta}^{w})^{\prime}(t)=-\xi-2t\eta+3t^{2}w.$

These are given by

$t_{\pm}(w;\xi,\eta):=\frac{\eta\pm\sqrt{\Delta(w;\xi,\eta)}}{3w}\quad\textrm{where}\quad\Delta(w;\xi,\eta):=\eta^{2}+3w\xi,$
(3.8)

provided $\Delta(w;\xi,\eta)\geq 0$. We partition the range of
$(k_{1},k_{2})\in\mathbb{Z}^{2}$ according to whether the critical points
$t_{\pm}(w;\xi,\eta)$ exist for either
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w}$,
$(\xi,\eta)\in\mathrm{supp}\,m_{n,\leq k_{1},k_{2}}^{w}$ or
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},\leq k_{2}}^{w}$, for some choice of
$1\leq w\leq 8$. To this end, define the sets of indices

$\displaystyle\mathcal{G}_{1}$
$\displaystyle:=\\{(k_{1},k_{2})\in\mathbb{Z}^{2}\cap(-C_{1},\infty)\times(-C_{2},\infty):k_{1}>(k_{2}\vee
0)+8\\}$ $\displaystyle=\\{(k_{1},k_{2})\in\mathbb{Z}^{2}:k_{1}>8\textrm{ and
}k_{1}-8>k_{2}>-C_{2}\\}$

and

$\displaystyle\mathcal{G}_{2}$
$\displaystyle:=\\{(k_{1},k_{2})\in\mathbb{Z}^{2}\cap(-C_{1},\infty)\times(-C_{2},\infty):k_{2}>(k_{1}\vee
0)+8\\}$ $\displaystyle=\\{(k_{1},k_{2})\in\mathbb{Z}^{2}:k_{2}>8\textrm{ and
}k_{2}-8>k_{1}>-C_{1}\\}.$

We shall show below, in the proof of Lemma 5.2, that for
$(k_{1},k_{2})\in\mathcal{G}_{1}$ or $(k_{1},k_{2})\in\mathcal{G}_{2}$ there
are no critical points when $(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},\leq
k_{2}}^{w}$ or $(\xi,\eta)\in\mathrm{supp}\,m_{n,\leq k_{1},k_{2}}^{w}$,
respectively. This leads to very favourable bounds for the corresponding
multipliers: see Proposition 3.3.

It is easy to check that
$\big{(}\mathbb{Z}^{2}\cap(-C_{1},\infty)\times(-C_{2},\infty)\big{)}\setminus(\mathcal{G}_{1}\cup\mathcal{G}_{2})\subseteq\mathcal{B}_{\mathrm{diag}}$
where

$\mathcal{B}_{\mathrm{diag}}:=\\{(k_{1},k_{2})\in\mathbb{Z}:|k_{1}-k_{2}|\leq(C_{1}\vee
C_{2})+16\text{ and }k_{1}>-C_{1},k_{2}>-C_{2}-8\\}.$

These ‘diagonal’ exponents index pieces of the multiplier admitting critical
points.

###  3.3. Non-stationary case

The following proposition guarantees very strong estimates when the the
exponents $(k_{1},k_{2})$ lie in $\mathcal{G}_{1}$ or $\mathcal{G}_{2}$.

######  Proposition 3.3 (Non-stationary estimates).

Let $2\leq p<\infty$ and $n\in\mathbb{N}$.

  1. i)

If $(k_{1},k_{2})\in\mathcal{G}_{1}$, then

$\|\mathcal{H}^{w}_{n,k_{1},k_{2}}\|_{L^{p}(\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})}+\|\mathcal{H}^{w}_{n,k_{1},\leq
k_{2}}\|_{L^{p}(\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})}\lesssim_{p,N}2^{-k_{1}N-nN}$

holds for all $N\in\mathbb{N}_{0}$, uniformly in $1\leq w\leq 8$.

  2. ii)

If $(k_{1},k_{2})\in\mathcal{G}_{2}$, then

$\|\mathcal{H}^{w}_{n,k_{1},k_{2}}\|_{L^{p}(\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})}+\|\mathcal{H}^{w}_{n,\leq
k_{1},k_{2}}\|_{L^{p}(\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})}\lesssim_{p,N}2^{-k_{2}N-nN}$

holds for all $N\in\mathbb{N}_{0}$, uniformly in $1\leq w\leq 8$.

Proposition 3.3 is a consequence of elementary $L^{2}$-based arguments. We
postpone the proof until §5.

###  3.4. Stationary case

The analysis of the remaining pieces of the operator requires much deeper
tools. The key estimates are as follows.

######  Proposition 3.4 ($L^{p}$ local smoothing).

For all $2<p<5/2$ there exists $\varepsilon(p)>0$ such that for all
$n\in\mathbb{N}$ the following hold.

  1. i)

If $-C_{1}<k_{1}\leq 8$, then

$\Big{(}\int_{1}^{8}\|\mathcal{H}^{w}_{n,k_{1},\leq-
C_{2}}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{-n/p-n\varepsilon(p)}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

  2. ii)

If $-C_{2}<k_{2}\leq 8$, then

$\Big{(}\int_{1}^{8}\|\mathcal{H}^{w}_{n,\leq-
C_{1},k_{2}}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{-n/p-n\varepsilon(p)}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

We also have the following estimate for comparable frequencies.

######  Proposition 3.5 ($L^{p}$ local smoothing).

For all $2<p<5/2$ there exists $\varepsilon(p)>0$ such that for all
$n\in\mathbb{N}$ and $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ we have

$\Big{(}\int_{1}^{8}\|\mathcal{H}_{n,k_{1},k_{2}}^{w}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{-n/p-n\varepsilon(p)-(k_{1}+k_{2})\varepsilon(p)}\|f\|_{L^{p}(\mathbb{R}^{2})}.$
(3.9)

Both Proposition 3.4 and Proposition 3.5 rely on the local smoothing estimates
from Theorem 1.4. We postpone the proofs until §6.

###  3.5. Proof of Theorem 3.2

With this in hand, we can now prove the key estimate for the proof of Theorem
1.1.

###### Proof of Theorem 3.2 (assuming Proposition 3.3, 3.4 and 3.5).

We treat each of the three estimates separately, and further subdivide into
cases depending on the values of $k_{1}$, $k_{2}$. In each case we highlight
whether the non-stationary estimates from Proposition 3.3 or the local
smoothing estimates from Proposition 3.4 or Proposition 3.5 are used.

Proof of (3.3). We consider two separate regimes:

Case 1: $[k_{1}>8]$. Non-stationary. We apply Proposition 3.3 i) with
$k_{2}:=-C_{2}$, noting $(k_{1},k_{2})\in\mathcal{G}_{1}$. Integrating the
resulting bounds over $1\leq w\leq 8$, we obtain a strong form of (3.3), valid
for all $2\leq p<\infty$ with rapid decay in $2^{-n}$ and $2^{-k_{1}}$.

Case 2: $[-C_{1}<k_{1}\leq 8]$. Local smoothing. We apply Proposition 3.4 i).
Since $k_{1}$ is bounded, we do not require any decay in $2^{-k_{1}}$.

Proof of (3.4). We consider two separate regimes:

Case 1: $[k_{2}>8]$. Non-stationary. We apply Proposition 3.3 ii) with
$k_{1}:=-C_{1}$, noting $(k_{1},k_{2})\in\mathcal{G}_{2}$. Integrating the
resulting bounds over $1\leq w\leq 8$, we obtain a strong form of (3.4), valid
for all $2\leq p<\infty$ with rapid decay in $2^{-n}$ and $2^{-k_{2}}$.

Case 2: $[-C_{2}<k_{2}\leq 8]$. Local smoothing. We apply Proposition 3.4 ii).
Since $k_{2}$ is bounded, we do not require any decay in $2^{-k_{2}}$.

Proof of (3.5). We consider three separate regimes:

Case 1a: $[(k_{1},k_{2})\in\mathcal{G}_{1}]$. Non-stationary. We apply
Proposition 3.3 i), as in Case 1 of the proof of (3.3). Since $k_{1}>k_{2}+8$,
we can trade $2^{-k_{1}}$ decay for $2^{-k_{2}}$ decay in this regime.

Case 1b: $[(k_{1},k_{2})\in\mathcal{G}_{2}]$. Non-stationary. We apply
Proposition 3.4 ii), as in Case 1 of the proof of (3.4). Since
$k_{2}>k_{1}+8$, we can trade $2^{-k_{2}}$ for $2^{-k_{1}}$ decay in this
regime.

Case 2: $[(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}]$. Local smoothing. We
apply Proposition 3.5.

This completes the proof of (3.5). ∎

It remains to prove Proposition 3.3, 3.4 and 3.5.

###  3.6. Further decomposition

Before turning to the main arguments, we carry out further reductions
regarding Proposition 3.5. Here we consider two separate cases depending on
the size of $n$.

Let $\kappa>0$ be a fixed parameter to be determined.

Case 1: $[k_{1}>\kappa n]$. In this case, Proposition 3.5 is a consequence of
the following two elementary lemmas.

######  Lemma 3.6.

Let $2\leq p<\infty$ and $n\in\mathbb{N}$. If $(k_{1},k_{2})\in\mathbb{Z}^{2}$
and $k_{2}\geq 8$, then

$\sup_{1\leq w\leq
8}\|\mathcal{H}^{w}_{n,k_{1},k_{2}}\|_{L^{p}(\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})}\lesssim_{p}2^{-n/p-k_{2}/p}.$

In the case $k_{2}<8$ we have the following surrogate.

######  Lemma 3.7.

Let $2\leq p\leq\infty$, $n\in\mathbb{N}$ and
$(k_{1},k_{2})\in\mathbb{Z}^{2}$. Then

$\sup_{1\leq w\leq
8}\|\mathcal{H}^{w}_{n,k_{1},k_{2}}\|_{L^{p}(\mathbb{R}^{2})\to
L^{p}(\mathbb{R}^{2})}\lesssim_{p}1.$

Lemma 3.6 is a consequence of elementary $L^{2}$ arguments, based on van der
Corput’s lemma, whilst Lemma 3.7 follows from crude kernel estimates. We
postpone the proofs until §§4-5.

Case 2: $[k_{1}\leq\kappa n]$. Here we perform a further decomposition and
follow a similar strategy to that of Pramanik–Seeger [22] in the context of
maximal estimates for averaging operators over curves in $\mathbb{R}^{3}$:
$L^{2}$-estimates when the Fourier decay of the multiplier is slowest (since
this only happens in a localised region, one can get a gain in the
$L^{\infty}$ estimates), and $L^{p}$-local smoothing estimates for $p>2$ when
the decay is essentially $|\xi|^{-1/2}$ (this happens for a large portion of
the frequency space, so no $L^{\infty}$ gain is available and $L^{2}$-methods
fall short for summability).

We decompose

$m_{n,k_{1},k_{2}}^{w}=m_{n,k_{1},k_{2}}^{w,\leq\kappa}+m_{n,k_{1},k_{2}}^{w,>\kappa}$

where

$m_{n,k_{1},k_{2}}^{w,\leq\kappa}(\xi,\eta):=m_{n,k_{1},k_{2}}^{w}(\xi,\eta)\beta_{0}(2^{-2k_{2}+2\lfloor
n\kappa\rfloor}\Delta(w;\xi,\eta));$

here $\Delta(w;\xi,\eta)$ is as in (3.8). Let
$\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}$ and
$\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}$ the associated multiplier operators.

Case 2a: $[k_{1}\leq\kappa n;\,\,\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}]$.
For the operator $\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}$ we use
$L^{2}$-methods to obtain the following smoothing estimate.

######  Proposition 3.8.

Let $2\leq p<5/2$ and $n\in\mathbb{N}$. If
$(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ satisfies $k_{1}\leq\kappa n$,
then

$\Big{(}\int_{1}^{8}\|\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}\lesssim_{p}2^{-n/p-\kappa
n(5/p-2)}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

To prove Proposition 3.8, we argue slightly differently depending on whether
$k_{2}>8$ or $k_{2}\leq 8$. In particular, we reduce matters to proving Lemma
3.9 below.

Suppose $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ with $k_{2}\leq 8$. In
this case, we further decompose

$m_{n,k_{1},k_{2}}^{w,\leq\kappa}=\sum_{\ell=\lfloor\kappa n\rfloor}^{\lfloor
n/3\rfloor}m_{n,k_{1},k_{2}}^{w,\ell}$

where

$m_{n,k_{1},k_{2}}^{w,\ell}(\xi,\eta):=\begin{cases}m_{n,k_{1},k_{2}}^{w}(\xi,\eta)\beta(2^{-2k_{2}+2\ell}\Delta(w;\xi,\eta))\qquad&\text{
if $\lfloor\kappa n\rfloor\leq\ell<\lfloor n/3\rfloor$}\\\
m_{n,k_{1},k_{2}}^{w}(\xi,\eta)\beta_{0}(2^{-2k_{2}+2\ell}\Delta(w;\xi,\eta))\qquad&\text{
if $\ell=\lfloor n/3\rfloor$}\end{cases}.$

The rationale of this decomposition is that $\Delta(w;\xi,\eta)$ quantifies
the simultaneous vanishing of the first and second-order derivatives of the
phase function $\phi^{w}_{\xi,\eta}$.

Denoting by $\mathcal{H}^{w,\ell}_{n,k_{1},k_{2}}$ the associated multiplier
operators, the main estimate for Case 2a reads as follows.

######  Lemma 3.9.

Let $2\leq p<\infty$, $n\in\mathbb{N}$ and
$(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$.

  1. i)

If $k_{2}>8$ and $k_{1}\leq\kappa n$, then

$\Big{(}\int_{1}^{8}\|\mathcal{H}^{w,\leq\kappa}_{n,k_{1},k_{2}}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{-n/p-\kappa
n(5/p-2)-k_{2}/p}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

  2. ii)

If $k_{2}\leq 8$ and $\lfloor\kappa n\rfloor\leq\ell\leq\lfloor n/3\rfloor$,
then

$\Big{(}\int_{1}^{8}\|\mathcal{H}_{n,k_{1},k_{2}}^{w,\ell}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{-n/2-\ell(5/p-2)}\|f\|_{L^{p}(\mathbb{R}^{2})}.$
(3.10)

This is essentially a refined version of Proposition 3.8.

###### Proof of Proposition 3.8 (assuming Lemma 3.9).

This is immediate, summing the estimate (3.10) in $\ell$ for the $k_{2}\leq 8$
case. ∎

Lemma 3.9 is a consequence of elementary $L^{2}$ arguments, based on van der
Corput’s lemma. Here we exploit the fact that the additional cutoff localises
to a small $w$-interval. We postpone the proof until §5.

Case 2b: $[k_{1}\leq\kappa n;\,\,\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}]$.
The analysis of the operator $\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}$ is more
subtle, since $L^{2}$-methods alone do not suffice to prove a satisfactory
bound. The main estimate reads as follows.

######  Proposition 3.10.

There exists some $M\in\mathbb{N}$ such that the following holds. For all
$2<p<4$ there exists $\varepsilon(p)>0$ such that if $n\in\mathbb{N}$,
$0<\kappa<1$ and $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ satisfy
$k_{1}\leq\kappa n$, then

$\Big{(}\int_{1}^{8}\|\mathcal{H}^{w,>\kappa}_{n,k_{1},k_{2}}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{M\kappa
n}2^{-n/p-n\varepsilon(p)}\|f\|_{L^{p}(\mathbb{R}^{2})}.$ (3.11)

Proposition 3.10 is a consequence of the local smoothing estimate from Theorem
1.4. We postpone the proof until §6.

The preceding results combine to give Proposition 3.5.

###### Proof of Proposition 3.5 (assuming Lemma 3.6 and Proposition 3.8 and
3.10).

Let $n\in\mathbb{N}$ and $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$.
Following the above scheme, we subdivide the argument into cases depending on
the values of $k_{1},k_{2}$ and $n$. In each case we highlight whether the van
der Corput estimates from Lemma 3.6 or Proposition 3.8 or the local smoothing
estimates from Proposition 3.10 are used.

Fix any $2<p<5/2$ and let $M$ and $\varepsilon(p)>0$ be the parameters given
by Proposition 3.10. Set $\kappa\equiv\kappa(p):=\varepsilon(p)/2M>0$.

Case 1a: $[k_{1}>\kappa n]$. van der Corput. If $k_{2}\geq 8$, then we apply
Lemma 3.6 and integrate the resulting uniform bounds over $1\leq w\leq 8$.
Since $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$, we can trade $2^{-k_{2}}$
decay for $2^{-k_{1}}$ decay. Furthermore, since $k_{1}>\kappa n$, we can
trade $2^{-k_{1}}$ decay for $2^{-n}$ decay. Thus, provided this trade is
suitably executed, we can obtain an inequality of the form (3.9) for any
choice of $2\leq p<\infty$.

On the other hand, if $k_{2}<8$, then the conditions
$(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ and $\kappa n<k_{1}$ mean that
all three parameters $k_{1}$, $k_{2}$ and $n$ are bounded. Thus, we do not
require any decay in $2^{-k_{1}}$, $2^{-k_{2}}$ or $2^{-n}$, so the estimate
from Lemma 3.7 suffices.

Case 2: $[k_{1}\leq\kappa n]$. It suffices to prove bounds of the form (3.9)
with $\mathcal{H}_{n,k_{1},k_{2}}^{w}$ replaced by the each of the localised
operators $\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}$ and
$\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}$. Furthermore, since $k_{1}\leq\kappa
n$ and $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$, we can trade $2^{-n}$
decay for either $2^{-k_{1}}$ or $2^{-k_{2}}$ decay. In particular, it
suffices to prove estimates which exhibit only appropriate $2^{-n}$ decay.

  * •

van der Corput. For the operator $\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}$,
the desired bound (3.9) follows from Proposition 3.8.

  * •

Local smoothing. For the operator $\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}$,
the desired bound (3.9) follows from Proposition 3.10, since our choice of
$\kappa$ guaranteed that (3.11) holds with constant
$2^{-n/p-n\varepsilon(p)/2}$.

This concludes the proof of (3.9) for $\mathcal{H}^{w}_{n,k_{1},k_{2}}$ for
the specific choice of $p$ and therefore for all $2<p<5/2$. ∎

###  3.7. Recap

The proof of Theorem 1.1 is now reduced to showing the following:

  * •

Lemma 3.1. This follows by pointwise domination by the strong maximal function
bounds. The details are presented in §4.3.

  * •

Lemma 3.7. This follows from an elementary kernel estimate in §4.

  * •

Proposition 3.3, Lemma 3.6 and Lemma 3.9. These results are all based on
elementary $L^{2}$-based theory and are treated in §5.

  * •

Proposition 3.4 and Proposition 3.10. These results are deeper, relying on the
local smoothing inequality Theorem 1.4, and are treated in §6.

##  4\. Kernel estimates

In this section we apply elementary integration-by-parts arguments to bound
the kernels of our localised operators. This will imply $L^{\infty}$ estimate
of the corresponding operators which will be interpolated with the
$L^{2}$-theory of §5.

Furthermore, we also establish a kernel estimate for the low frequency piece
of our operator, which implies Lemma 3.1.

###  4.1. The operators $\mathcal{H}_{n,k_{1},k_{2}}$

We begin with the operators $\mathcal{H}_{n,k_{1},k_{2}}^{w}$ introduced in
§3.1 (together with their low frequency variants), whose kernels are given by
the inverse Fourier transforms of the multipliers $m_{n,k_{1},k_{2}}^{w}$ from
§3.2.

######  Lemma 4.1.

Let $n\in\mathbb{N}$, $1\leq w\leq 8$ and $k_{1},k_{2}\in\mathbb{Z}$. Then

$\|\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w}]\|_{L^{1}(\mathbb{R}^{2})}\lesssim
1,$

and the same holds if $m_{n,k_{1},k_{2}}^{w}$ is replaced by $m_{n,\leq
k_{1},k_{2}}^{w}$, $m_{n,k_{1},\leq k_{2}}^{w}$.

###### Proof.

We have

$\displaystyle|\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w}](x,y)|$
$\displaystyle\sim\Big{|}\int_{\mathbb{R}}\int_{\widehat{\mathbb{R}}^{2}}e^{i(x\xi+y\eta+2^{n}\phi_{\xi,\eta}^{w}(t))}\beta(2^{-k_{1}}\xi)\beta(2^{-k_{2}}\eta)\,\mathrm{d}\xi\,\mathrm{d}\eta\,a(t)\,\mathrm{d}t\Big{|}$
$\displaystyle\lesssim_{N}\int_{\mathbb{R}}\frac{2^{k_{1}}2^{k_{2}}}{(1+2^{k_{1}}|x-2^{n}t|+2^{k_{2}}|y-2^{n}t^{2}|)^{N}}\,a(t)\,\mathrm{d}t$

by repeated integration-by-parts. Integrating in $(x,y)$ and using Fubini’s
theorem, we obtain the result. ∎

As a consequence of Lemma 4.1, we have the basic $L^{p}$ estimate from Lemma
3.7.

###### Proof of Lemma 3.7.

Since, for all $f\in\mathcal{S}(\mathbb{R}^{2})$, we have

$\mathcal{H}^{w}_{n,k_{1},k_{2}}f=K_{n,k_{1},k_{2}}^{w}\ast
f\qquad\textrm{where}\qquad
K_{n,k_{1},k_{2}}^{w}:=\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w}],$

the desired bound follows from Lemma 4.1 and Young’s inequality. ∎

###  4.2. The operators $\mathcal{H}_{n,k_{1},k_{2}}^{2,\leq\kappa}$ and
$\mathcal{H}_{n,k_{1},k_{2}}^{w,\ell}$

We now turn to the kernels of the operators
$\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}$ and
$\mathcal{H}_{n,k_{1},k_{2}}^{w,\ell}$ introduced in §3.6.

######  Lemma 4.2.

Let $n\in\mathbb{N}$, $1\leq w\leq 8$ and
$(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$.

  1. i)

If $k_{2}>8$ and $k_{1}\leq\kappa n$, then

$\|\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w,\leq\kappa}]\|_{L^{1}(\mathbb{R}^{2})}\lesssim
2^{2\lfloor\kappa n\rfloor}.$

  2. ii)

If $k_{2}\leq 8$, then for all $\lfloor\kappa n\rfloor\leq\ell\leq\lfloor
n/3\rfloor$, we have

$\|\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w,\ell}]\|_{L^{1}(\mathbb{R}^{2})}\lesssim
2^{2\ell}.$

###### Proof.

The proof of i) follows from a minor adaptation of that of ii), and so we only
provide the latter.

It suffices to show the pointwise bound

$|\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w,\ell}](x,y)|\lesssim_{N}\int_{1/2}^{2}\frac{2^{-2\ell}}{(1+2^{-2\ell}|x-2^{n}t|+2^{-2\ell}|y-2^{n}t^{2}|)^{N}}\,\mathrm{d}t$

holds for all $N\in\mathbb{N}_{0}$, uniformly in $1\leq w\leq 8$. Write
$m_{n,k_{1},k_{2}}^{w,\ell}=a_{k_{1},k_{2}}^{w,\ell}\cdot m_{n}^{w}$ where

$a_{k_{1},k_{2}}^{w,\ell}(\xi,\eta):=\beta(2^{-2k_{2}+2\ell}\Delta(w;\xi,\eta))\beta(2^{-k_{1}}\xi)\beta(2^{-k_{2}}\eta)$

and $m_{n}^{w}$ is as defined in (3.6). Then

$\displaystyle\mathcal{F}^{-1}[m_{n,k_{1},k_{2}}^{w,\ell}](x,y)=\int_{\mathbb{R}}e^{i2^{n}wt^{3}}\mathcal{F}^{-1}[a_{k_{1},k_{2}}^{w,\ell}](x-2^{n}t,y-2^{n}t^{2})a(t)\,\mathrm{d}t.$

In light of the above, it suffices to show that

$|\mathcal{F}^{-1}[a_{k_{1},k_{2}}^{w,\ell}](x,y)|\lesssim_{N}\frac{2^{-2\ell}}{(1+2^{-2\ell}|x|+2^{-2\ell}|y|)^{N}}$

for all $N\in\mathbb{N}_{0}$. However, this follows from routine integration-
by-parts and the fact that, for fixed $\eta$, the $\xi$-support of
$a_{k_{1},k_{2}}^{w,\ell}$ lies in a fixed interval of length $O(2^{-2\ell})$,
in view of the support condition $|\Delta(w;\xi,\eta)|\lesssim 2^{-2\ell}$ and
that $\Delta(w;\xi,\eta)=\eta^{2}+3w\xi$. ∎

###  4.3. Proof of Lemma 3.1

Here we provide the proof of Lemma 3.1. The key ingredient is the following
kernel estimate, which shares similarities with the above kernel estimates.

######  Lemma 4.3.

For $\ell\in\mathbb{N}$ and $j\in\mathbb{Z}$ and $v\in V_{j}$, let
$K_{\ell,j}^{v}$ denote the kernel of the operator $H_{-j+\ell}^{v}P_{\leq-
C_{1}+2\ell+j}^{(1)}P_{\leq-C_{2}+\ell+2j}^{(2)}$. Then

$|K_{\ell,j}^{v}(x,y)|\lesssim_{N}2^{-\ell
N}2^{3(j+\ell)}(1+2^{2\ell+j}|x|+2^{\ell+2j}|y|)^{-N}\qquad\textrm{for all
$N\in\mathbb{N}_{0}$.}$

###### Proof.

By the rescaling argument from §3.1, it follows that

$K_{\ell,j}^{v}(x,y)=2^{3(j+\ell)}\frac{1}{(2\pi)^{2}}\int_{\widehat{\mathbb{R}}^{2}}e^{i(2^{2\ell+j}x\xi+2^{\ell+2j}y\eta)}\mathfrak{m}_{3\ell}^{2^{-3j}v}(\xi,\eta)\,\mathrm{d}\xi\,\mathrm{d}\eta$

where, using the definitions from (3.7), we have

$\mathfrak{m}_{n}^{w}(\xi,\eta):=\beta_{0}(2^{C_{1}}\xi)\beta_{0}(2^{C_{2}}\eta)\int_{\mathbb{R}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}a(t)\,\mathrm{d}t.$

Since $v\in V_{j}=[2^{3j},2^{3(j+1)}]$, repeated integration-by-parts shows
that

$|K_{\ell,j}^{v}(x,y)|\lesssim_{N}\sup_{1\leq w\leq
8}\|\mathfrak{m}_{3\ell}^{w}\|_{C^{N}(\widehat{\mathbb{R}}^{2})}2^{3(j+\ell)}\big{(}1+2^{2\ell+j}|x|+2^{\ell+2j}|y|\big{)}^{-N}$
(4.1)

for all $N\in\mathbb{N}$. Differentiating $\mathfrak{m}_{n}^{w}$, the problem
reduces to showing

$\sup_{1\leq w\leq 8}\sup_{\begin{subarray}{c}|\xi|\leq 2^{-C_{1}+1}\\\
|\eta|\leq
2^{-C_{2}+1}\end{subarray}}\Big{|}\int_{\mathbb{R}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}t^{m}a(t)\,\mathrm{d}t\Big{|}\lesssim_{m,M}2^{-nM}\qquad\textrm{for
all $m$, $M\in\mathbb{N}_{0}$.}$ (4.2)

Indeed, the desired bounds then follow by taking $M:=2N$, so the gain in
$2^{-2Nn}$ from (4.2) compensates for the loss of $2^{Nn}$ in (4.1).

Let $1\leq w\leq 8$, $(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$ satisfy
$|\xi|\leq 2^{-C_{1}+1}$ and $|\eta|\leq 2^{-C_{2}+1}$ and
$t\in\mathrm{supp}\,a$, so that $1/2\leq|t|\leq 2$. Since
$(\phi_{\xi,\eta}^{w})^{\prime}(t)=-\xi-2\eta t+3wt^{2}$, we have

$|(\phi_{\xi,\eta}^{w})^{\prime}(t)|\geq 3/4-|\xi|-4|\eta|\geq 1/8,$

provided $C_{1}$, $C_{2}\geq 4$. Moreover,
$|(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)|\leq 2|\eta|+6|t|w\leq 2^{6}$ and
$|(\phi_{\xi,\eta}^{w})^{\prime\prime\prime}(t)|\leq 6w\leq 2^{6}$. The
desired estimate (4.2) now follows from an application of the non-stationary
phase Lemma 5.1 below. ∎

Lemma 3.1 easily follows from the above kernel estimate.

###### Proof of Lemma 3.1.

Using a standard argument (see, for instance, [30, Chapter 2, §2.1]), Lemma
4.3 implies the pointwise bound

$\sup_{j\in\mathbb{Z}}\sup_{2^{3j}\leq v\leq
2^{3(j+1)}}|H_{-j+\ell}^{v}P_{\leq-C_{1}+2\ell+j}^{(1)}P_{\leq-
C_{2}+\ell+2j}^{(2)}f(x,y)|\lesssim_{N}2^{-N\ell}M_{\textrm{st}}f(x,y),$

where $M_{\textrm{st}}$ denotes the strong Hardy–Littlewood maximal function.
Thus, the desired bound follows from the $L^{p}$ boundedness of
$M_{\mathrm{st}}$. ∎

##  5\. $L^{2}$ Theory

Here we prove Proposition 3.3, Lemma 3.6 and Lemma 3.9, using elementary
$L^{2}$-based theory. Throughout this section,
$\phi_{\xi,\eta}^{w}(t):=-t\xi-t^{2}\eta+t^{3}w$ and $a$ are as defined in
(3.7); it is useful to note the formulæ

$(\phi_{\xi,\eta}^{w})^{\prime}(t)=-\xi-2t\eta+3t^{2}w,\quad(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)=-2\eta+6tw,\quad(\phi_{\xi,\eta}^{w})^{\prime\prime\prime}(t)=6w$
(5.1)

and that $(\phi_{\xi,\eta}^{w})^{(k)}(t)=0$ for $k\geq 4$.

###  5.1. The non-stationary case

The proof of Proposition 3.3 makes repeated use of the following elementary
oscillatory integral estimate.

######  Lemma 5.1 (Non-stationary phase).

Let $N\in\mathbb{N}$ and $\phi\in C^{N}(-1,1)$ be real valued and $a\in
C^{N}_{c}(-1,1)$ satisfy $\|a\|_{C^{N}}\leq 1$. Further suppose

  1. i)

$|\phi^{\prime}(t)|\geq 1$ for all $t\in(-1,1)$;

  2. ii)

$|\phi^{(k)}(t)|\leq M$ for all $1\leq k\leq N$ and $t\in(-1,1)$.

For all $\lambda\geq 1$, we have

$\Big{|}\int_{\mathbb{R}}e^{i\lambda\phi(t)}a(t)\,\mathrm{d}t\Big{|}\lesssim_{N,M}\lambda^{-N}.$

Here the implied constant depends on $N$ and $M$, but is otherwise independent
of the choice of $\phi$ and $a$.

Lemma 5.1 is a simple and well-known consequence of repeated integration-by-
parts. We omit the details.

######  Lemma 5.2.

Let $n\in\mathbb{N}$, $k_{1}$, $k_{2}\in\mathbb{Z}$ and
$(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$, $1\leq w\leq 8$.

  1. i)

If $2^{(k_{2}\vee 0)+8}\leq|\xi|$ and $|\eta|\leq 2^{k_{2}+1}$, then

$|m_{n}^{w}(\xi,\eta)|\lesssim_{N}|\xi|^{-N}2^{-nN}.$ (5.2)

  2. ii)

If $|\xi|\leq 2^{k_{1}+1}$ and $|\eta|\geq 2^{(k_{1}\vee 0)+8}$, then

$|m_{n}^{w}(\xi,\eta)|\lesssim_{N}|\eta|^{-N}2^{-nN}.$ (5.3)

###### Proof.

Each estimate is proved using repeated integration-by-parts (Lemma 5.1). We
make repeated use of the fact that $1/2\leq|t|\leq 2$ on the support of $a$.

i) If $1/2\leq t\leq 2$, then $|-2t\eta+3t^{2}w|\leq 2^{k_{2}+3}+3\cdot
2^{5}\leq 2^{(k_{2}\vee 0)+7}\leq|\xi|/2$. Thus, by (5.1) and the triangle
inequality,

$|(\phi_{\xi,\eta}^{w})^{\prime}(t)|\geq|\xi|/2\qquad\textrm{for all $1/2\leq
t\leq 2$.}$

A similar calculation shows that
$|(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)|\leq|\xi|$ and
$|(\phi_{\xi,\eta}^{w})^{\prime\prime\prime}(t)|\leq|\xi|$. Since all higher
order derivatives vanish, the desired bound (5.2) follows from an application
of Lemma 5.1, at least after applying suitable changes of variables and
scalings to massage the setup to match the hypotheses of the lemma.

ii) If $1/2\leq t\leq 2$, then $|-\xi+3t^{2}w|\leq 2^{k_{1}+1}+3\cdot
2^{5}\leq 2^{(k_{1}\vee 0)+7}\leq|\eta|$. Thus, by (5.1) and the triangle
inequality,

$|(\phi_{\xi,\eta}^{w})^{\prime}(t)|\geq|\eta|\qquad\textrm{for all $1/2\leq
t\leq 2$.}$

A similar calculation shows that
$|(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)|\leq 3|\eta|$ and
$|(\phi_{\xi,\eta}^{w})^{\prime\prime\prime}(t)|\leq|\eta|$. Thus, as before,
the desired bound (5.3) follows from an application of Lemma 5.1. ∎

###### Proof of Proposition 3.3.

In each case it suffices to prove the $L^{2}$ bound only. The full range of
$L^{p}$ then follows via interpolation with the $L^{\infty}$ bound from Lemma
4.1.

i) If $(k_{1},k_{2})\in\mathcal{G}_{1}$, then $k_{1}>(k_{2}\vee 0)+8$ and so
$2^{(k_{2}\vee 0)+8}\leq 2^{k_{1}-1}\leq|\xi|$ and $|\eta|\leq 2^{k_{2}+1}$
for $(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w}$ or
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},\leq k_{2}}^{w}$. The desired $L^{2}$
bound now follows from Lemma 5.2 i) and Plancherel’s theorem.

ii) If $(k_{1},k_{2})\in\mathcal{G}_{2}$, then $k_{2}>(k_{1}\vee 0)+8$ and so
$2^{(k_{1}\vee 0)+8}\leq 2^{k_{2}-1}\leq|\eta|$ and $|\xi|\leq 2^{k_{1}+1}$
for $(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w}$ or
$(\xi,\eta)\in\mathrm{supp}\,m_{n,\leq k_{1},k_{2}}^{w}$. The desired $L^{2}$
bound now follows from Lemma 5.2 ii) and Plancherel’s theorem. ∎

###  5.2. van der Corput estimates

We now turn to the proofs of Lemma 3.6 and Lemma 3.9. We first observe a basic
pointwise bound for the underlying multiplier, based on the van der Corput
oscillatory integral estimate.

######  Lemma 5.3.

Let $n\in\mathbb{N}$, $(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$ and $1\leq w\leq
8$. If $|\eta|\geq 2^{7}$, then

$|m_{n}^{w}(\xi,\eta)|\lesssim 2^{-n/2}|\eta|^{-1/2}.$

###### Proof.

By (5.1), we have $|(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)|\geq
2|\eta|-6|t|w\geq|\eta|$ for $|\eta|\geq 3\cdot 2^{5}\geq 6|t|w$. Thus, the
desired bound follows from van der Corput’s lemma. ∎

Lemma 5.3 and Lemma 4.1 easily combine to give Lemma 3.6.

###### Proof of Lemma 3.6.

For $k_{2}>8$ we have $|\eta|\geq 2^{k_{2}-1}>2^{7}$ on the support of
$m_{n,k_{1},k_{2}}^{w}$. Thus, we can apply Lemma 5.3 and Plancherel’s theorem
to deduce the desired bound for $p=2$. The $p=\infty$ case again follows from
Lemma 4.1, and we conclude the proof by interpolating between these two
endpoints. ∎

It remains to prove Lemma 3.9. The key ingredient is the following multiplier
estimate, which is again based on van der Corput’s lemma.

######  Lemma 5.4.

Let $n\in\mathbb{N}$, $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ and
$(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$ and $1\leq w\leq 8$.

  1. i)

If $k_{2}>8$ and $k_{1}\leq\kappa n$, then

$\Big{(}\int_{1}^{8}|m_{n,k_{1},k_{2}}^{w,\leq\kappa}(\xi,\eta)|^{2}\,\mathrm{d}w\Big{)}^{1/2}\lesssim
2^{-n/2-\kappa n/2-k_{2}/2}.$

  2. ii)

If $k_{2}\leq 8$ and $\lfloor\kappa n\rfloor\leq\ell\leq\lfloor n/3\rfloor$,
then

$\Big{(}\int_{1}^{8}|m_{n,k_{1},k_{2}}^{w,\ell}(\xi,\eta)|^{2}\,\mathrm{d}w\Big{)}^{1/2}\lesssim
2^{-(n+\ell)/2}.$

###### Proof.

For both i) and ii) we shall exploit the fact that the $\Delta(w;\xi,\eta)$
localisation corresponds to localising the $w$ variable to a small interval.
In particular,

$\Big{|}w+\frac{\eta^{2}}{3\xi}\Big{|}=\frac{1}{3|\xi|}|\Delta(w;\xi,\eta)|\leq
2^{-k_{1}}|\Delta(w;\xi,\eta)|$ (5.4)

for $2^{k_{1}-1}\leq|\xi|\leq 2^{k_{1}+1}$.

i) On the support of $m_{n,k_{1},k_{2}}^{w,\leq\kappa}$ we have
$|\Delta(w;\xi,\eta)|\leq 2^{2k_{2}-2\kappa n+1}$. This combines with (5.4) to
give

$\Big{|}w+\frac{\eta^{2}}{3\xi}\Big{|}\leq 2^{2k_{2}-k_{1}-2\kappa n+1}.$

Since $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ and $k_{1}\leq\kappa n$,
it follows that $k_{2}\leq k_{1}+C\leq\kappa n+C$ for $C:=(C_{1}\vee
C_{2})+16$. Thus, for fixed $(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$, we
conclude that $m_{n,k_{1},k_{2}}^{w,\leq\kappa}(\xi,\eta)$ is non-zero only if
$w$ belongs to a fixed interval of length $O(2^{-\kappa n})$ centred at
$-\eta^{2}/(3\xi)$.

On the other hand, since $k_{2}>8$ we have $|\eta|\geq 2^{k_{2}-1}>2^{7}$ and
Lemma 5.3 implies that

$|m_{n,k_{1},k_{2}}^{w,\leq\kappa}(\xi,\eta)|\lesssim 2^{-n/2-k_{2}/2}$

holds uniformly in $1\leq w\leq 8$. Combining the above observations, the
desired result immediately follows.

ii) On the support of $m_{n,k_{1},k_{2}}^{w,\ell}$ we have
$|\Delta(w;\xi,\eta)|\leq 2^{2k_{2}-2\ell+1}$. This combines with (5.4) to
give

$\Big{|}w+\frac{\eta^{2}}{3\xi}\Big{|}\leq 2^{2k_{2}-k_{1}-2\ell+1}.$

Since $k_{2}\leq 8$, it follows that $2k_{2}-k_{1}\leq 32+C_{1}$. Thus, for
fixed $(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$, we conclude that
$m_{n,k_{1},k_{2}}^{w,\ell}(\xi,\eta)$ is non-zero only if $w$ belongs to a
fixed interval of length $O(2^{-2\ell})$ centred at $-\eta^{2}/(3\xi)$.

On the other hand, we claim that

$|m_{n,k_{1},k_{2}}^{w,\ell}(\xi,\eta)|\lesssim 2^{-(n-\ell)/2}$ (5.5)

holds uniformly in $1\leq w\leq 8$. Once, this is established and combined
with our earlier observation, the desired result immediately follows.

The case $\ell=\lfloor n/3\rfloor$ of (5.5) follows from van der Corput’s
lemma with third order derivatives, since
$|(\phi_{\xi,\eta}^{w})^{\prime\prime\prime}(t)|=6w\geq 1$. For the case
$\lfloor\kappa n\rfloor\leq\ell<\lfloor n/3\rfloor$ of (5.5), we first write
$\beta=\beta^{+}+\beta^{-}$ where $\beta^{+}\in C^{\infty}_{c}(\mathbb{R})$ is
supported on $[1/2,2]$ and $\beta^{-}\in C^{\infty}_{c}(\mathbb{R})$ is
supported on $[-2,-1/2]$. We decompose

$m_{n,k_{1},k_{2}}^{w,\ell}=m_{n,k_{1},k_{2}}^{w,\ell,+}+m_{n,k_{1},k_{2}}^{w,\ell,-}$

where

$m_{n,k_{1},k_{2}}^{w,\ell,\pm}(\xi,\eta):=\beta^{\pm}(2^{-2k_{2}+2\ell}\Delta(w;\xi,\eta))m_{n,k_{1},k_{2}}^{w}(\xi,\eta).$

Since $\Delta(w;\xi,\eta)<0$ on the support of $m_{n,k_{1},k_{2}}^{w,\ell,-}$,
the phase function $\phi_{\xi,\eta}^{w}$ has no (real) critical points (see
(3.8)). Furthermore,

$(\phi_{\xi,\eta}^{w})^{\prime}(t)\geq(\phi_{\xi,\eta}^{w})^{\prime}\Big{(}\frac{\eta}{3w}\Big{)}=-\xi-\frac{\eta^{2}}{3w}\geq
2^{2k_{2}-2\ell-6}$

for all $t\in\mathrm{supp}\,a$ and
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w,\ell,-}$, since
$t\mapsto(\phi_{\xi,\eta}^{w})^{\prime}(t)$ has a minimum at $t=\eta/3w$.
Consequently, by van der Corput’s lemma with first order derivatives,

$|m_{n,k_{1},k_{2}}^{w,\ell,-}(\xi,\eta)|\lesssim
2^{-(n-2\ell)}\qquad\text{for
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w,\ell,-}$}.$ (5.6)

On the support of $m_{n,k_{1},k_{2}}^{w,\ell,+}$, we have
$\Delta(w;\xi,\eta)>0$ and the phase function $\phi_{\xi,\eta}^{w}$ admits the
critical points $t_{\pm}(w;\xi,\eta)$, which are the roots of the quadratic
polynomial $(\phi_{\xi,\eta}^{w})^{\prime}(t)$ with leading coefficient $3w$.
Thus,

$\displaystyle(\phi_{\xi,\eta}^{w})^{\prime}(t)$
$\displaystyle=3w(t-t_{+}(w;\xi,\eta))(t-t_{-}(w;\xi,\eta)),$ (5.7)
$\displaystyle(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)$
$\displaystyle=6w\Big{(}t-\frac{t_{+}(w;\xi,\eta)+t_{-}(w;\xi,\eta)}{2}\Big{)}.$
(5.8)

From the quadratic formula (3.8), we have

$\frac{t_{+}(w;\xi,\eta)-t_{-}(w;\xi,\eta)}{2}=\frac{\sqrt{\Delta(w;\xi,\eta)}}{3w}.$

Since $\Delta(w;\xi,\eta)\geq 2^{2k_{2}-2\ell-1}$ on the support of
$m_{n,k_{1},k_{2}}^{w,\ell,+}$, we conclude that there exists an absolute
constant $c>0$ such that $|t_{+}(w;\xi,\eta)-t_{-}(w;\xi,\eta)|\geq
4c2^{-\ell}$ on the support of $m_{n,k_{1},k_{2}}^{w,\ell,+}$.

We now write $m_{n}^{w}=\mathfrak{m}_{n,1}^{w}+\mathfrak{m}_{n,2}^{w}$ where

$\mathfrak{m}_{n,i}^{w}(\xi,\eta):=\int_{T_{i}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}a(t)\,\mathrm{d}t,\qquad
i=1,\,2,$

for

$\displaystyle T_{1}$ $\displaystyle:=\big{\\{}t\in\mathbb{R}:1/2\leq|t|\leq
2\textrm{ and }\min_{\pm}|t-t_{\pm}(w;\xi,\eta)|\geq c2^{-\ell}\big{\\}},$
$\displaystyle T_{2}$ $\displaystyle:=\big{\\{}t\in\mathbb{R}:1/2\leq|t|\leq
2\big{\\}}\setminus T_{1}.$

Note that each of the sets $T_{i}$ consists of a union of a bounded number of
intervals. It follows from (5.7) that $|(\phi_{\xi,\eta}^{w})^{\prime}(t)|\geq
3c^{2}2^{-2\ell}$ for all $t\in T_{1}$ whenver
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w,\ell,+}$. Thus, van der
Corput’s lemma with first order derivatives gives

$|\mathfrak{m}_{n,1}^{w}(\xi,\eta)|\lesssim 2^{-(n-2\ell)}\qquad\textrm{for
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w,\ell,+}$.}$ (5.9)

On the other hand, it follows from (5.8) that
$|(\phi_{\xi,\eta}^{w})^{\prime\prime}(t)|\geq 6c2^{-\ell}$ for all $t\in
T_{2}$ whenever $(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w,\ell,+}$.
Thus, van der Corput’s lemma with second order derivatives gives

$|\mathfrak{m}_{n,2}^{w}(\xi,\eta)|\lesssim 2^{-(n-\ell)/2}\qquad\textrm{for
$(\xi,\eta)\in\mathrm{supp}\,m_{n,k_{1},k_{2}}^{w,\ell,+}$.}$ (5.10)

Since $\ell\leq n/3$, the estimates (5.6) and (5.9) and (5.10) combine to give
(5.5). ∎

Using the preceding oscillatory integral estimates, we conclude this section
with the proof of Lemma 3.9.

###### Proof of Lemma 3.9.

We first observe that the $p=2$ case of Lemma 3.9 is a direct consequence of
Lemma 5.4 and Plancherel’s theorem.

i) Let $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ satisfy $k_{2}>8$ and
$k_{1}\leq\kappa n$. The kernel estimate from Lemma 4.2 i) implies

$\sup_{1\leq w\leq
8}\|\mathcal{H}_{n,k_{1},k_{2}}^{w,\leq\kappa}f\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim
2^{2\kappa n}\|f\|_{L^{\infty}(\mathbb{R}^{2})}.$

Interpolating this with the $p=2$ case gives the desired estimate.

ii) Let $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ satisfy $k_{2}\leq 8$.
For all $\lfloor\kappa n\rfloor\leq\ell\leq\lfloor n/3\rfloor$, the kernel
estimate from Lemma 4.2 ii) implies

$\sup_{1\leq w\leq
8}\|\mathcal{H}_{n,k_{1},k_{2}}^{w,\ell}f\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim
2^{2\ell}\|f\|_{L^{\infty}(\mathbb{R}^{2})}.$

Interpolating this with the $p=2$ case again gives the desired estimate. ∎

##  6\. $L^{p}$ local smoothing

###  6.1. Setup

Recall the multiplier $m_{n}^{w}$ defined in (3.6) is an oscillatory integral
with phase $\phi_{\xi,\eta}^{w}$. Furthermore, in (3.8) we introduced the
functions

$t_{\pm}(w;\xi,\eta):=\frac{\eta\pm\sqrt{\Delta(w;\xi,\eta)}}{3w}\qquad\textrm{for}\qquad\Delta(w;\xi,\eta):=\eta^{2}+3w\xi,$

which correspond to the two (potentially non-real) roots of
$(\phi_{\xi,\eta}^{w})^{\prime}$. By standard stationary phase computations
(see the proof of Proposition 6.8 below), provided the multiplier is suitably
localised, we expect $m_{n}^{w}$ to oscillate with a fundamental frequency
given by one of the phases

$\phi_{\pm}(w;\xi,\eta):=\phi^{w}_{\xi,\eta}\circ t_{\pm}(w;\xi,\eta).$

This leads us to consider the pair of translation-invariant phases

$\Phi_{\pm}(x,y,w;\xi,\eta):=x\xi+y\eta+\phi_{\pm}(w;\xi,\eta).$ (6.1)

In this section we prove local smoothing estimates for operators of the form
$U^{\lambda}[\Phi_{\pm};\mathfrak{a}]$ introduced in (1.12) for $\Phi_{\pm}$
as in (6.1) and $\mathfrak{a}$ suitably localised. These estimates are then
used to prove Proposition 3.4 and Proposition 3.10, thereby concluding the
proof of Theorem 1.1.

###  6.2. Localised amplitudes

Here and throughout the remainder of the section, we let $C_{\star}\geq 1$
denote a fixed absolute constant, which chosen sufficiently large for the
purposes of the forthcoming arguments.

To ensure our operators satisfy the Hörmander condition (1.11), we work with
the following class of amplitude functions.

######  Definition 6.1.

For $0<\tau\leq 1$, let $\mathfrak{A}_{\pm}(\tau)$ denote the class of
$\mathfrak{a}_{\pm}\in
C^{\infty}_{c}(\mathbb{R}\times\widehat{\mathbb{R}}^{2})$ such that for all
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{a}_{\pm}$ we have $|\xi|$,
$|\eta|\leq C_{\star}$, $\tau\leq w\leq C_{\star}$ and

$\tau\leq\Delta(w;\xi,\eta)\quad\textrm{and}\quad 1/4\leq
t_{\pm}(w;\xi,\eta)\leq 4.$ (6.2)

######  Lemma 6.2.

If $\mathfrak{a}_{\pm}\in\mathfrak{A}_{\pm}(\tau)$, then the following hold.

  1. i)

The phase $\Phi_{\pm}$ is real-analytic and satisfies (1.11) on a
neighbourhood of $\mathrm{supp}\,\mathfrak{a}_{\pm}$. In particular, for
$\boldsymbol{\xi}=(\xi,\eta)$, the matrix

$\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\partial_{w}\phi_{\pm}(w;\xi,\eta):=\begin{bmatrix}\partial_{w}\partial^{2}_{\xi\xi}\phi_{\pm}(w;\xi,\eta)&\partial_{w}\partial^{2}_{\xi\eta}\phi_{\pm}(w;\xi,\eta)\\\
\partial_{w}\partial^{2}_{\xi\eta}\phi_{\pm}(w;\xi,\eta)&\partial_{w}\partial^{2}_{\eta\eta}\phi_{\pm}(w;\xi,\eta)\end{bmatrix}$
(6.3)

satisfies

$|\det\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\partial_{w}\phi_{\pm}(w;\xi,\eta)|\gtrsim
1\qquad\textrm{for all $(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{a}_{\pm}$},$
(6.4)

where the implied constants can be taken to be absolute.

  2. ii)

We also have the derivative bounds

$|\partial_{w}^{\alpha}\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}\phi_{\pm}(w;\xi,\eta)|\lesssim_{\alpha,\beta_{1},\beta_{2}}\tau^{-\alpha-\beta_{1}-\beta_{2}+1}\qquad\textrm{for
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{a}_{\pm}$}$ (6.5)

for all $\alpha$, $\beta_{1}$, $\beta_{2}\in\mathbb{N}_{0}$ with
$\alpha+\beta_{1}+\beta_{2}\geq 1$.

###### Proof.

First note, by the discriminant hypothesis in (6.2), the function $\Phi_{\pm}$
is real analytic on a neighbourhood of $\mathrm{supp}\,\mathfrak{a}_{\pm}$.
Furthermore, the derivative bounds in ii) follow by a straightforward direct
computation: see §B.1.555For derivatives of second and higher order, there is
actually a $\tau^{1/2}$-gain over (6.5), but this is irrelevant for our
purposes.

It remains to prove (6.4). A computation (c.f. §B.2) shows that

$\det\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\partial_{w}\phi_{\pm}(w;\xi,\eta)=-\frac{9}{4}\cdot\frac{t_{\pm}(w;\xi,\eta)^{4}}{\Delta(w;\xi,\eta)^{2}}.$
(6.6)

The desired bound immediately follows from (6.2). ∎

The above observation allows us to apply the classical theory of Hörmander-
type operators to the phase/amplitude pair $[\Phi_{\pm};\mathfrak{a}_{\pm}]$
whenever $\mathfrak{a}_{\pm}\in\mathfrak{A}_{\pm}(\tau)$. However, this is
insufficient for our purposes and we wish to apply Theorem 1.4 to establish
improved estimates. For this, we consider a more restrictive class of
amplitudes.

######  Definition 6.3.

For $0<\sigma,\tau\leq 1$ and $W\geq 1$ we define the following.

  1. i)

Let $\mathfrak{N}_{\pm}(\sigma,\tau)$ denote the class of
$\mathfrak{h}_{\pm}\in\mathfrak{A}_{\pm}(\tau)$ which, in addition, satisfy

$|\xi|\geq\sigma\qquad\textrm{for all
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{h}_{\pm}$}$

and

$|\partial_{w}^{\alpha}\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}\mathfrak{h}_{\pm}(w;\xi,\eta)|\lesssim_{\alpha,\beta_{1},\beta_{2}}\tau^{-\alpha-\beta_{2}}(\tau\wedge\sigma)^{-\beta_{1}}\qquad\textrm{for
all $\alpha$, $\beta_{1}$, $\beta_{2}\in\mathbb{N}_{0}$.}$ (6.7)

  2. ii)

Let $\mathfrak{E}_{\pm}(\sigma)$ denote the class of
$\mathfrak{e}_{\pm}\in\mathfrak{A}_{\pm}(C_{\star}^{-1})$ which, in addition,
satisfy

$|\xi|\leq\sigma\qquad\textrm{for all
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{e}_{\pm}$}$ (6.8)

and

$|\partial_{w}^{\alpha}\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}\mathfrak{e}_{\pm}(w;\xi,\eta)|\lesssim_{\alpha,\beta_{1},\beta_{2}}\sigma^{-\beta_{1}-\beta_{2}}\qquad\textrm{for
all $\alpha$, $\beta_{1}$, $\beta_{2}\in\mathbb{N}_{0}$.}$ (6.9)

Here the implied constants in (6.7) and (6.9) can be thought of as a fixed
sequence, the terms of which are defined sufficiently large so as to satisfy
the forthcoming requirements of the proof.

Definition 6.3 i) is specifically tailored for application of Theorem 1.4. On
the other hand, Definition 6.3 ii) allows application of decoupling theory:
see §6.7.

######  Lemma 6.4.

If $\mathfrak{h}_{\pm}\in\mathfrak{N}_{\pm}(\sigma,\tau)$ for some
$0<\sigma,\tau<1$, then $\Phi_{\pm}$ satisfies the Nikodym non-compression
hypothesis on $\mathrm{supp}\,\mathfrak{h}_{\pm}$ with parameter $\gamma$
satisfying $\gamma\sim\sigma$. In particular, the matrix

$N_{\pm}(w;\xi,\eta):=\begin{bmatrix}\partial_{w}\partial_{\xi\xi}^{2}\phi_{\pm}(w;\xi,\eta)&\partial_{w}\partial_{\xi\eta}^{2}\phi_{\pm}(w;\xi,\eta)&\partial_{w}\partial_{\eta\eta}^{2}\phi_{\pm}(w;\xi,\eta)\\\
\partial_{w}^{2}\partial_{\xi\xi}^{2}\phi_{\pm}(w;\xi,\eta)&\partial_{w}^{2}\partial_{\xi\eta}^{2}\phi_{\pm}(w;\xi,\eta)&\partial_{w}^{2}\partial_{\eta\eta}^{2}\phi_{\pm}(w;\xi,\eta)\\\
\partial_{w}^{3}\partial_{\xi\xi}^{2}\phi_{\pm}(w;\xi,\eta)&\partial_{w}^{3}\partial_{\xi\eta}^{2}\phi_{\pm}(w;\xi,\eta)&\partial_{w}^{3}\partial_{\eta\eta}^{2}\phi_{\pm}(w;\xi,\eta)\\\
\end{bmatrix}$

satisfies

$|\det N_{\pm}(w;\xi,\eta)|\gtrsim\sigma\qquad\textrm{for all
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{h}_{\pm}$},$ (6.10)

where the implied constants can be taken to be absolute.

###### Proof.

Direct computation (c.f. §B.3) reveals that
$\det\partial^{2}_{\boldsymbol{\xi}\boldsymbol{\xi}}\phi_{\pm}(w;\xi,\eta)\equiv
0$ as a function of $(w;\xi,\eta)$ and so we only need to verify (6.10). A
further, and somewhat involved(!), computation (c.f. §B.3) shows that

$\det N_{\pm}(w;\xi,\eta)=-\frac{3^{7}}{2^{4}}\cdot\frac{\xi\cdot
t_{\pm}(w;\xi,\eta)^{8}}{\Delta(w;\xi,\eta)^{5}}.$ (6.11)

The desired bound therefore follows immediately from the support conditions in
Definition 6.1 and Definition 6.3 i). ∎

###  6.3. Local smoothing via Nikodym non-compression

We combine the observations of the preceding section with Theorem 1.4 to
deduce the following.

######  Proposition 6.5.

There exists some $M\in\mathbb{N}$ such that the following holds. For all
$2<p<4$ there exists $\varepsilon_{\mathrm{n}}(p)>0$ such that

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{h}_{\pm}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim\lambda^{1/2-\varepsilon_{\mathrm{n}}(p)}(\sigma\tau)^{-M}\|f\|_{L^{p}(\mathbb{R}^{2})}$

for all $\lambda\geq 1$, whenever
$\mathfrak{h}_{\pm}\in\mathfrak{N}_{\pm}(\sigma,\tau)$ for some
$0<\sigma,\tau<1$.

###### Proof.

By Lemma 6.2 i), we know that $\Phi_{\pm}$ is translation-invariant, real
analytic and satisfies H1) and H2) on an open neighbourhood of
$\mathrm{supp}\,\mathfrak{h}_{\pm}$. Furthermore, by Lemma 6.4, there exists
some $\gamma\sim\sigma$ such that $\Phi_{\pm}$ satisfies the Nikodym non-
compression hypothesis with parameter $\gamma$ on an open neighbourhood of
$\mathrm{supp}\,\mathfrak{h}_{\pm}$. The hypothesis (6.7) and Lemma 6.2 ii)
guarantee that the derivative bounds (1.14) hold for
$W=O((\sigma\tau)^{-M_{0}})$ for some absolute exponent $M_{0}\in\mathbb{N}$.
We may therefore directly apply Theorem 1.4 to obtain the desired result. ∎

###  6.4. Local smoothing for the exceptional piece via decoupling

To complement Proposition 6.5, we prove a local smoothing result for operators
of the form $U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]$ with
$\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$.

######  Proposition 6.6.

For all $2\leq p\leq 4$ and $\delta>0$, given $\lambda\geq 1$ and
$\lambda^{-1/2}\leq\sigma\leq 1$, we have

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim_{\delta}\lambda^{1/2+\delta}\sigma^{1/2-1/p}\|f\|_{L^{p}(\mathbb{R}^{2})}.$
(6.12)

whenever $\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$.

The mechanism behind Proposition 6.6 is very different from that of
Proposition 6.5 and, in particular, relies on $\ell^{p}$ decoupling theory.
This is a standard tool for proving $L^{p}$-local smoothing estimates for
wave-type propagators [33]; it is typically not as relevant in the Schrödinger
context. However, here decoupling allows for the (crucial) gain of
$\sigma^{1/2-1/p}$, which arises from the frequency localisation near the line
$\mathcal{L}:=\\{(0,\eta):\eta\in\mathbb{R}\\}$ in the amplitude class
$\mathfrak{E}_{\pm}(\sigma)$. The proof is complicated by the fact that our
operator is hyperbolic: the eigenvalues of the matrix (6.3) have opposite
signs. This rules out application of the more powerful $\ell^{2}$ decoupling
theory, and instead we rely on a two-step decoupling process which combines a
lower dimensional ‘small cap’ decoupling inequality, exploiting curvature
properties of $\Phi_{\pm}$ restricted to $\mathcal{L}$, with a variable
coefficient extension of the $\ell^{p}$ decoupling theory from [8]. We
postpone the details until §6.7 below.

###  6.5. Applying the local smoothing estimates

We now combine Proposition 6.5 and Proposition 6.6 with stationary phase
arguments to prove a general $L^{p}$ smoothing estimate from which Proposition
3.4 and Proposition 3.10 follows.

######  Definition 6.7.

For $0<\tau\leq 1$, let $\mathfrak{B}(\tau)$ denote the class of $b\in
C^{\infty}_{c}(\mathbb{R}\times\widehat{\mathbb{R}}^{2})$ such that:

  1. i)

$|\xi|$, $|\eta|\leq C_{\star}$, $\tau\leq w\leq C_{\star}$ and
$|\Delta(w;\xi,\eta)|\geq\tau$ for all $(w;\xi,\eta)\in\mathrm{supp}\,b$.

  2. ii)

If $\tau<C_{\star}^{-1}$, then $|\xi|>C_{\star}^{-1}$ for all
$\xi\in\mathrm{supp}_{\xi}\,b$.

  3. iii)

$|\partial_{w}^{\alpha}\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}b(w;\xi,\eta)|\lesssim_{\alpha,\beta_{1},\beta_{2}}\tau^{-\alpha-\beta_{1}-\beta_{2}}$
for all $\alpha$, $\beta_{1}$, $\beta_{2}\in\mathbb{N}_{0}$.

Given $b\in\mathfrak{B}(\tau)$, for each $w\in\mathbb{R}$ and $n\in\mathbb{N}$
let $\mathcal{H}_{n}^{w}[b]$ denote the operator with Fourier multiplier

$m_{n}^{w}[b](\xi,\eta):=b(w;\xi,\eta)m_{n}^{w}(\xi,\eta)=b(w;\xi,\eta)\int_{\mathbb{R}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}a(t)\,\,\mathrm{d}t.$

With these definitions, our general smoothing estimate reads thus.

######  Proposition 6.8.

There exists some $M\in\mathbb{N}$ such that the following holds. For all
$2<p<4$, there exists some $\varepsilon(p)>0$ such that

$\Big{(}\int_{0}^{\infty}\big{\|}\mathcal{H}_{n}^{w}[b]f\big{\|}_{L^{p}(\mathbb{R}^{2})}^{p}\,\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{-n/p-\varepsilon(p)n}\tau^{-M}\|f\|_{L^{p}(\mathbb{R}^{2})}$
(6.13)

holds whenever $b\in\mathfrak{B}(\tau)$ for some $2^{-n/4}\leq\tau\leq 1$.

###### Proof.

We break the proof into steps.

Step 1: Initial decomposition. Write
$m_{n}^{w}[b]=\sum_{\pm}m_{n,\pm}^{w}[b]+m_{n,\mathrm{error}}^{w}[b]$ where

$m_{n,\pm}^{w}[b](\xi,\eta):=b_{\mathrm{main}}(w;\xi,\eta)\int_{\mathbb{R}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}a_{\pm}(w;\xi,\eta;t)\,\,\mathrm{d}t$

and, for $C_{\star}\geq 1$ as above,

$\displaystyle b_{\mathrm{main}}(w;\xi,\eta)$
$\displaystyle:=b(w;\xi,\eta)\chi_{(0,\infty)}\big{(}\Delta(w;\xi,\eta)\big{)},$
$\displaystyle a_{\pm}(w;\xi,\eta;t)$
$\displaystyle:=a(t)\beta_{0}\big{(}C_{\star}^{2}\tau^{-1/2}(t-t_{\pm}(w;\xi,\eta))\big{)}.$

The hypotheses on the support of $b$ ensures the function $b_{\mathrm{main}}$
is smooth. Furthermore, for $(w;\xi,\eta)\in\mathrm{supp}\,b_{\mathrm{main}}$,
the critical points $t_{\pm}(w;\xi,\eta)$ are real valued and so the $a_{\pm}$
are well-defined. Recall that the $t_{\pm}(w;\xi,\eta)$ satisfy

$\frac{t_{+}(w;\xi,\eta)-t_{-}(w;\xi,\eta)}{2}=\frac{\sqrt{\Delta(w;\xi,\eta)}}{3w}\geq(3C_{\star})^{-1}\tau^{1/2}.$

It therefore follows that for each fixed
$(w;\xi,\eta)\in\mathrm{supp}\,b_{\mathrm{main}}$, the $t$-supports of the
$a_{\pm}$ are disjoint.

Step 2: Non-stationary term. Here we argue as in the proof of Lemma 5.4 ii).

Let $b_{\mathrm{error}}:=b-b_{\mathrm{main}}$. For
$(w;\xi,\eta)\in\mathrm{supp}\,b_{\mathrm{error}}$ we have
$\Delta(w;\xi,\eta)<0$ and

$(\phi_{\xi,\eta}^{w})^{\prime}(t)\geq(\phi_{\xi,\eta}^{w})^{\prime}\Big{(}\frac{\eta}{3w}\Big{)}=-\frac{\Delta(w;\xi,\eta)}{3w}\geq(3C_{\star})^{-1}\tau>0\quad\text{for
all $t\in\mathrm{supp}\,a$,}$

where in the first inequality we have used that
$t\mapsto(\phi_{\xi,\eta}^{w})^{\prime}(t)$ has a minimum at $t=\eta/3w$.
Thus, by non-stationary phase (Lemma 5.1), we see that

$|\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}m_{n}^{w}[b_{\mathrm{error}}](\xi,\eta)|\lesssim_{\beta_{1},\beta_{2},N}2^{n(\beta_{1}+\beta_{2})}2^{-Nn}\tau^{-N},\quad\textrm{$\beta_{1}$,
$\beta_{2}$, $N\in\mathbb{N}_{0}$.}$ (6.14)

On the other hand, let $a_{\mathrm{error}}:=a-a_{+}-a_{-}$. It follows from
(5.7) that, if $(w;\xi,\eta)\in\mathrm{supp}\,b_{\mathrm{main}}$, then
$|(\phi_{\xi,\eta}^{w})^{\prime}(t)|\geq 3C_{\star}^{-4}\tau^{2}$ for all
$t\in\mathrm{supp}\,a_{\mathrm{error}}(w;\xi,\eta;\,\cdot\,)$. Since

$(m_{n}^{w}-m_{n,\pm}^{w})[b_{\mathrm{main}}](\xi,\eta)=b_{\mathrm{main}}(w;\xi,\eta)\int_{\mathbb{R}}e^{i2^{n}\phi_{\xi,\eta}^{w}(t)}a_{\mathrm{error}}(w;\xi,\eta;t)\,\mathrm{d}t,$

another application of non-stationary phase yields

$|\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}(m_{n}^{w}-m_{n,\pm}^{w})[b_{\mathrm{main}}](\xi,\eta)|\lesssim_{\beta_{1},\beta_{2},N}2^{n(\beta_{1}+\beta_{2})}2^{-Nn}\tau^{-2N}$
(6.15)

for all $\beta_{1}$, $\beta_{2}$, $N\in\mathbb{N}_{0}$.

Combining (6.14) and (6.15) we have, by repeated integration-by-parts,

$|\mathcal{F}^{-1}[m_{n,\mathrm{error}}^{w}[b]](x,y)|\lesssim_{N}\frac{2^{-Nn}\tau^{-2N}}{(1+2^{-n}|x|+2^{-n}|y|)^{10}}$

for all $N\in\mathbb{N}_{0}$ and $w\in\mathbb{R}$. Thus, by Young’s
convolution inequality, (6.13) holds for the piece of the operator associated
to $m_{n,\mathrm{error}}^{w}[b]$.

Step 3: Passing to variable propagators. In light of the above observations,
it suffices to prove that (6.13) holds with the operator
$\mathcal{H}_{n}^{w}[b]$ replaced with either of the operators
$\mathcal{H}_{n,\pm}^{w}[b]$ associated to the Fourier multipliers
$m_{n,\pm}^{w}[b]$. By (5.8), we have

$(\phi_{\xi,\eta}^{w})^{\prime\prime}(t_{\pm}(w;\xi,\eta))=\pm
6w\Big{(}\frac{t_{+}(w;\xi,\eta)-t_{-}(w;\xi,\eta)}{2}\Big{)}=\pm
2\sqrt{\Delta(w;\xi,\eta)}$

and so the critical points of the phase are non-degenerate, in the sense that

$|(\phi_{\xi,\eta}^{w})^{\prime\prime}(t_{\pm}(w;\xi,\eta))|\geq
2\tau^{1/2}>0.$

Provided $C_{\star}\geq 1$ is chosen sufficiently large, we may therefore
apply standard asymptotic expansions for oscillatory integrals (see, for
instance, [27, Chapter 1]) to deduce that

$m_{n,\pm}^{w}[b](\xi,\eta)=\tau^{-1/4}2^{-n/2}e^{i2^{n}\phi_{\pm}(w;\xi,\eta)}\mathfrak{a}_{\pm}(w;\xi,\eta)$
(6.16)

where each $\mathfrak{a}_{\pm}$ is a smooth function satisfying
$\mathrm{supp}\,\mathfrak{a}_{\pm}\subseteq b_{\mathrm{main}}$ and

$|\partial^{\alpha}_{w}\partial^{\beta_{1}}_{\xi}\partial^{\beta_{2}}_{\eta}\mathfrak{a}_{\pm}(w;\xi,\eta)|\lesssim_{\alpha,\beta_{1},\beta_{2}}\tau^{-\alpha-\beta_{1}-\beta_{2}}\qquad\textrm{for
all $\alpha$, $\beta_{1}$, $\beta_{2}\in\mathbb{N}_{0}$.}$ (6.17)

In view of (6.16), for $\Phi_{\pm}$ as in (6.1), we have
$\mathcal{H}_{n,\pm}^{2^{-n}w}=\tau^{-1/4}2^{-n/2}U^{2^{n}}[\Phi_{\pm};\mathfrak{a}_{\pm}]$.
In particular, applying the rescaling $w\mapsto 2^{-n}w$, in order to prove
(6.13) it suffices to show that for all $2<p<4$ there exists some
$\varepsilon(p)>0$ such that

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{a}_{\pm}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim\lambda^{1/2-\varepsilon(p)}\tau^{-M}\|f\|_{L^{p}(\mathbb{R}^{2})}$
(6.18)

holds uniformly over all $\lambda\geq 1$, for some universal exponent
$M\in\mathbb{N}$.

Step 4: Applying $L^{p}$ local smoothing. We shall prove (6.18) by combining
Proposition 6.5 and Proposition 6.6. First of all, we verify that the
amplitudes $\mathfrak{a}_{\pm}$ lie in the class $\mathfrak{A}_{\pm}(\tau)$
from Definition 6.1. The conditions $|\xi|$, $|\eta|\leq C_{\star}$, $\tau\leq
w\leq C_{\star}$ and $|\Delta(w;\xi,\eta)|\geq\tau$ for
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{a}_{\pm}$ are inherited from the
support of $b$. It remains to show $1/4\leq t_{\pm}(w;\xi,\eta)\leq 4$ for
$(w;\xi,\eta)\in\mathrm{supp}\,\mathfrak{a}_{\pm}$. Indeed, if the above
condition on $t_{\pm}(w;\xi,\eta)$ fails and $C_{\star}\geq 4$, then it
follows that

$|t-t_{\pm}(w;\xi,\eta)|>2C_{\star}^{-2}\geq
2C_{\star}^{-2}\tau^{1/2}\qquad\textrm{for all $t\in[1/2,2]$.}$

Thus, we see from the definitions that $a_{\pm}(w;\xi,\eta;\,\cdot\,)\equiv
0$, which forces $\mathfrak{a}_{\pm}(w;\xi,\eta)=0$. These conclusions hold in
a neighbourhood of $(w;\xi,\eta)$ and so
$(w;\xi,\eta)\notin\mathrm{supp}\,\mathfrak{a}_{\pm}$.

Now fix $2<p<4$ and let $M\in\mathbb{N}$ and $\varepsilon_{\mathrm{n}}(p)>0$
be the quantities guaranteed by Proposition 6.5. Defining
$\delta_{\mathrm{n}}(p):=\varepsilon_{\mathrm{n}}(p)/(2M)\leq 1/4$, we write

$\mathfrak{a}_{\pm}=\mathfrak{h}_{\pm}+\mathfrak{e}_{\pm}\quad\textrm{where}\quad\mathfrak{e}_{\pm}(w;\xi,\eta):=\mathfrak{a}_{\pm}(w;\xi,\eta)\beta_{0}\big{(}\lambda^{\delta_{\mathrm{n}}(p)}\xi\big{)}.$

We may assume $\lambda^{\delta_{\mathrm{n}}(p)}\geq 4C_{\star}$, since
otherwise the desired bound is trivial. Thus, by property ii) from Definition
6.7, if $\tau<C_{\star}^{-1}$, then $|\xi|>C_{\star}^{-1}$ for
$\xi\in\mathrm{supp}_{\xi}\,\mathfrak{a}_{\pm}$, so $\mathfrak{e}_{\pm}\equiv
0$. Using this observation and (6.17), we see that

$\mathfrak{h}_{\pm}\in\mathfrak{N}_{\pm}(\lambda^{-\delta_{\mathrm{n}}(p)},\tau)\qquad\textrm{and}\qquad\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\lambda^{-\delta_{\mathrm{n}}(p)}).$

Thus, Proposition 6.5 implies that

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{h}_{\pm}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim_{p}\lambda^{1/2-\varepsilon_{\mathrm{n}}(p)/2}\tau^{-M}\|f\|_{L^{p}(\mathbb{R}^{2})}.$

On the other hand, Proposition 6.6 (with, say, $\delta:=1/4-1/(2p)>0$) implies
that

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}\lesssim_{p}\lambda^{1/2-\varepsilon(p)}\|f\|_{L^{p}(\mathbb{R}^{2})}$

where $\varepsilon(p):=(1/4-1/(2p))\delta_{\mathrm{n}}(p)$ satisfies
$0<\varepsilon(p)<\varepsilon_{\mathrm{n}}(p)/2$. Combining these two
estimates, we deduce the desired bound (6.18). ∎

###  6.6. Proof of Proposition 3.4 and Proposition 3.10

Both Proposition 3.4 and Proposition 3.10 follow easily from Proposition 6.8.
It is convenient to begin with the proof of the latter. Throughout what
follows, we let $\rho\in C^{\infty}_{c}(\mathbb{R})$ be a fixed bump function
satisfying $\rho(w)=1$ for all $1\leq w\leq 8$ and
$\mathrm{supp}\,\rho\subseteq[1/2,16]$.

###### Proof of Proposition 3.10.

Fix $n\in\mathbb{N}$, $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$ satisfying
$k_{1}\leq\kappa n$. Note that

$\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}f(x,y)=\mathcal{H}_{n,k_{1},k_{2}}^{w,>\kappa}f(x,y)\rho(w)=\mathcal{H}_{n}^{w}[b_{0}]f(x,y)$

for $1\leq w\leq 8$ and $(x,y)\in\mathbb{R}^{2}$, where

$b_{0}(w;\xi,\eta):=\rho(w)\beta(2^{-k_{1}}\xi)\beta(2^{-k_{2}}\eta)\big{(}1-\beta_{0}\big{(}2^{-2k_{2}+2\lfloor
n\kappa\rfloor}\Delta(w;\xi,\eta)\big{)}\big{)}.$

We apply the rescaling $(\xi,\eta)\mapsto(2^{k_{2}}\xi,2^{k_{2}}\eta)$ and
$w\mapsto 2^{k_{2}}w$ to arrive at $\mathcal{H}_{n+k_{2}}^{w}[b]f$ where

$b(w;\xi,\eta):=\rho(2^{k_{2}}w)\beta(2^{k_{2}-k_{1}}\xi)\beta(\eta)\big{(}1-\beta_{0}\big{(}2^{2\lfloor
n\kappa\rfloor}\Delta(w;\xi,\eta)\big{)}\big{)}.$

Thus, given $2<p<4$, matters are reduced to showing

$\Big{(}\int_{0}^{\infty}\big{\|}\mathcal{H}_{n}^{w}[b]f\big{\|}_{L^{p}(\mathbb{R}^{2})}^{p}\,\,\mathrm{d}w\Big{)}^{1/p}\lesssim_{p}2^{2M\kappa
n}2^{-n/p-\varepsilon(p)n}\|f\|_{L^{p}(\mathbb{R}^{2})},$ (6.19)

where $\varepsilon(p)>0$ and $M\in\mathbb{N}$ are as in the statement of
Proposition 6.8. However, using the condition
$(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$, if $C_{\star}\geq 1$ is chosen
sufficiently large and we define $\tau:=2^{-C_{2}-16}2^{-2n\kappa}$, then

$C_{\star}^{-1}\leq|\xi|\leq C_{\star},\quad|\eta|\leq C_{\star},\quad\tau\leq
w\leq C_{\star}\quad\textrm{and}\quad|\Delta(w;\xi,\eta)|\geq\tau$

for all $(w;\xi,\eta)\in\mathrm{supp}\,b$ and it is clear that
$b\in\mathfrak{B}(\tau)$. Thus, (6.19) is a direct consequence of Proposition
6.8. ∎

Combining Proposition 3.10 with the $L^{2}$-based theory from §5, we obtain
Proposition 3.5. All that remains is to verify Proposition 3.4.

###### Proof of Proposition 3.4.

Both i) and ii) easily follow by verifying that the relevant multipliers
satisfy the hypotheses of Proposition 6.8.

i) Suppose $-C_{1}<k_{1}\leq 8$ and recall that $C_{1}=2C_{2}+12$. We begin
with a technical reduction, which relies on Proposition 3.5. Defining
$\widetilde{C}_{2}:=C_{2}+8$, we may write

$\mathcal{H}^{w}_{n,k_{1},\leq-
C_{2}}=\mathcal{H}^{w}_{n,k_{1},\leq-\widetilde{C}_{2}}+\sum_{k_{2}=-\widetilde{C}_{2}+1}^{-C_{2}}\mathcal{H}^{w}_{n,k_{1},k_{2}}.$

For each $-\tilde{C}_{2}<k_{2}\leq-C_{2}$, we have $|k_{1}-k_{2}|\leq
C_{2}+16$ and so $(k_{1},k_{2})\in\mathcal{B}_{\mathrm{diag}}$. Hence, we can
apply Proposition 3.5 to favourably bound each of the operators
$\mathcal{H}^{w}_{n,k_{1},k_{2}}$ in the above sum.

It remains to estimate $\mathcal{H}^{w}_{n,k_{1},\leq-\widetilde{C}_{2}}$. As
in the proof of Proposition 3.10, matters are reduced to bounding
$\mathcal{H}_{n}^{w}[b]$ where

$b(w;\xi,\eta):=\rho(w)\beta(2^{-k_{1}}\xi)\beta_{0}(2^{\tilde{C}_{2}}\eta).$

On $\mathrm{supp}\,b$ we have $|\xi|\geq 2^{-k_{1}-1}\geq 2^{-C_{1}-2}$ and
$|\eta|\leq 2^{-\widetilde{C}_{2}+1}$. Consequently,

$|\Delta(w;\xi,\eta)|=|\eta^{2}+3\xi w|\geq 3\cdot
2^{-C_{1}-3}-2^{-2\tilde{C}_{2}+2}\geq 2^{-C_{1}-3}$

provided $\tilde{C}_{2}$ satisfies $2\tilde{C}_{2}\geq C_{1}+4$. However,
since $\tilde{C}_{2}=C_{2}+8$ and $C_{1}=2C_{2}+12$, this is indeed the case.
Recalling Definition 6.7, provided $C_{\star}\geq 1$ is chosen sufficiently
large, it is therefore clear that $b\in\mathfrak{B}(\tau)$ for $\tau\sim 1$
and Proposition 6.8 applies to give the desired result.

ii) Suppose $-C_{2}<k_{2}\leq 8$ and consider the operator
$\mathcal{H}^{w}_{n,\leq-C_{1},k_{2}}$. Arguing as before, matters are reduced
to bounding $\mathcal{H}_{n}^{w}[b]$ where666Note that it is only here we
consider $|\xi|\ll 1$ so that the decoupling argument is needed.

$b(w;\xi,\eta):=\rho(w)\beta_{0}(2^{C_{1}}\xi)\beta(2^{-k_{2}}\eta).$

On the support of $b$ we have $|\xi|\leq 2^{-C_{1}+1}$ and $|\eta|\geq
2^{-k_{2}-1}\geq 2^{-C_{2}-2}$. Consequently,

$|\Delta(w;\xi,\eta)|=|\eta^{2}+3\xi w|\geq 2^{-2C_{2}-4}-3\cdot
2^{-C_{1}+5}\geq 2^{-2C_{2}-5}>0,$

since $C_{1}=2C_{2}+12$. Thus, provided $C_{\star}\geq 1$ is chosen
sufficiently large, it is clear that $b\in\mathfrak{B}(\tau)$ for $\tau\sim 1$
and Proposition 6.8 applies to give the desired result. ∎

###  6.7. The decoupling argument

It remains to prove Proposition 6.6. We begin with some preliminary
definitions and lemmas.

For $0<\sigma\leq 1$, let $\mathcal{I}(\sigma)$ be a finitely-overlapping
cover of $[-2C_{\star},2C_{\star}]$ intervals of length $2\sigma$ and
$(\zeta_{I})_{I\in\mathcal{I}(\sigma)}$ be a smooth partition of unity
subordinate to $\mathcal{I}(\sigma)$, satisfying
$|\partial_{s}^{N}\zeta_{I}(s)|\lesssim_{N}\sigma^{-N}$ for all
$N\in\mathbb{N}_{0}$. Let $P_{I}$ be the Fourier projection operator acting on
functions on $\mathbb{R}^{2}$ with multiplier
$(\xi,\eta)\mapsto\zeta_{I}(\eta)$. Given any
$\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$, we further write
$\mathfrak{e}_{\pm,I}(w;\xi,\eta):=\mathfrak{e}_{\pm}(w;\xi,\eta)\zeta_{I}(\eta)$;
note that $\mathfrak{e}_{\pm,I}\in\mathfrak{E}_{\pm}(\sigma)$.

######  Lemma 6.9 (Coarse scale decoupling).

For $2\leq p\leq 4$ and $\delta>0$, given $\lambda\geq 1$ and
$\lambda^{-1/2}\leq\sigma\leq 1$, we have

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]f\big{\|}_{L^{p}(B_{\lambda})}\lesssim_{\delta}\sigma^{-(1/2-1/p)-\delta}\Big{(}\sum_{I\in\mathcal{I}(\sigma)}\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,I}]f\big{\|}_{L^{p}(w_{B_{\lambda}})}^{p}\Big{)}^{1/p},$
(6.20)

up to a rapidly decreasing $L^{p}$ error, whenever
$B_{\lambda}\subseteq\mathbb{R}^{3}$ is a $\lambda$-ball and
$\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$.

Here and below, we say that an inequality
$A(f;\lambda)\lesssim_{L}B(f;\lambda)$ holds up to a rapidly decreasing
$L^{p}$ error if
$A(f;\lambda)\lesssim_{L,N}B(f;\lambda)+\lambda^{-N}\|f\|_{L^{p}(\mathbb{R}^{2})}$
holds for all $N\in\mathbb{N}_{0}$. Furthermore, given a ball
$B\subset\mathbb{R}^{d}$ for $d=2$, $3$, we let
$w_{B}\colon\mathbb{R}^{d}\to(0,1]$ denote the weight

$w_{B}(\mathbf{z}):=\big{(}1+r_{B}^{-1}|\mathbf{z}-\mathbf{z}_{B}|\big{)}^{-100}\qquad\textrm{for
all $\mathbf{z}\in\mathbb{R}^{d}$,}$

where $\mathbf{z}_{B}$ and $r_{B}$ denote the centre and radius of $B$,
respectively.

Lemma 6.9 is a consequence of a ‘small cap’ decoupling result, which is
essentially a restatement of the sharp Fourier restriction theorem in the
plane.

######  Definition 6.10.

Let $\mathfrak{G}$ denote the class of $C^{2}$ functions $\gamma\colon
J\to\mathbb{R}$, for $J\subseteq\mathbb{R}$ an open interval of length at most
$2C_{\star}$, satisfying

$|\gamma^{(j)}(s)|\leq C_{\star}^{5}\quad\textrm{for $j=1$,
$2$}\quad\textrm{and}\quad|\gamma^{(2)}(s)|\geq
C_{\star}^{-5}\qquad\textrm{for all $s\in J$.}$

Given $\gamma\in\mathfrak{G}$ and $\rho>0$, let $\mathcal{Z}(\gamma;\rho)$
denote the space of $F\in\mathcal{S}(\mathbb{R}^{2})$ satisfying

$\mathrm{supp}\,\widehat{F}\subseteq\\{(\xi,\eta)\in
J\times\widehat{\mathbb{R}}:|\eta-\gamma(\xi)|<\rho\\}.$

With this definition, the small cap decoupling lemma reads thus.

######  Lemma 6.11 (Small cap decoupling in the plane (c.f. [13, Lemma 4.2])).

Let $\gamma\in\mathfrak{G}$. For all $2\leq p\leq 4$, $\delta>0$ and $R\geq
1$, we have

$\|F\|_{L^{p}(B_{R})}\lesssim_{p,\delta}R^{1/2-1/p+\delta}\Big{(}\sum_{I\in\mathcal{I}(R^{-1})}\|P_{I}F\|_{L^{p}(w_{B_{R}})}^{p}\Big{)}^{1/p}\qquad\textrm{}$

whenever $F\in\mathcal{Z}(\gamma;R^{-1})$ and $B_{R}\subseteq\mathbb{R}^{2}$
is an $R$-ball.

By standard interpolation arguments, the lemma follows from the case $p=4$
which is a consequence of the classical $L^{4}$ Fourier restriction estimate,
dating back to [12, 34]. The latter implies that, for all $\delta>0$, we have

$\displaystyle\|F\|_{L^{4}(\mathbb{R}^{2})}$
$\displaystyle\lesssim_{p,\delta}R^{-3/4+\delta}\|\widehat{F}\|_{L^{4}(\mathbb{R}^{2})}$
$\displaystyle\lesssim
R^{-3/4+\delta}\Big{(}\sum_{I\in\mathcal{I}(R^{-1})}\|(P_{I}F)\;\widehat{}\;\|_{L^{4}(\mathbb{R}^{2})}^{4}\Big{)}^{1/4}$
(6.21)

whenever $F\in\mathcal{S}(\mathbb{R}^{2})$ satisfies the above Fourier support
condition. Indeed, the first inequality in (6.21) follows from the uncertainty
principle reformulation of the restriction estimate as in [31], whilst the
second inequality is a trivial consequence of the essentially disjoint
frequency support. To pass from (6.21) to the $p=4$ case of Lemma 6.11, one
applies the Hausdorff–Young inequality to the right-hand norms in (6.21) to
pass to $L^{4/3}$ norms of the $P_{I}F$. The resulting inequality implies a
local version of itself where the integrals are essentially localised to a
spatial $R$-ball. One may then apply Hölder’s inequality to pass from the
localised $L^{4/3}$ norms to $L^{4}$ norms, at the cost of a factor of $R$.

###### Proof (of Lemma 6.9).

Fix $\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$ and $0<\delta<1/2$ and,
for notational convenience, in this proof we let
$U^{\lambda}_{\pm}:=U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]$ and
$U^{\lambda}_{\pm,I}:=U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,I}]$. Note
that, by Hölder’s inequality, (6.20) holds with the factor $\sigma^{-\delta}$
placed with $\sigma^{-1/2}$. Consequently, we assume without loss of
generality that

$\sigma\leq\min\big{\\{}6^{-1}C_{\star}^{-2},\lambda^{-\delta}\big{\\}}<1/100.$
(6.22)

Fix $B_{\lambda}\subseteq\mathbb{R}^{2}$ a $\lambda$-ball and let
$\mathcal{B}(\sigma^{-1})$ be a finitely-overlapping covering of $B_{\lambda}$
by balls of radius $\sigma^{-1}$. Fixing $2\leq p\leq 4$ and $\delta>0$, we
aim to show

$\big{\|}U^{\lambda}_{\pm}f(x,\,\cdot\,,\,\cdot\,)\big{\|}_{L^{p}(B)}\lesssim_{\delta,N}\sigma^{-(1/2-1/p)-\delta}\Big{(}\sum_{I\in\mathcal{I}(\sigma)}\big{\|}U^{\lambda}_{\pm,I}f(x,\,\cdot\,,\,\cdot\,)\big{\|}_{L^{p}(w_{B})}^{p}\Big{)}^{1/p}$
(6.23)

holds up to inclusion of a rapidly decreasing $L^{p}$ error for each
$x\in\mathbb{R}$ and every ball $B\in\mathcal{B}(\sigma^{-1})$. Once (6.23) is
established, the desired result immediately follows by first taking an
$\ell^{p}$ norm of both sides of the inequality over all
$B\in\mathcal{B}(\sigma^{-1})$ and then taking an $L^{p}$-norm in $x$.

Let $\rho\in\mathcal{S}(\mathbb{R}^{2})$ satisfy
$\mathrm{supp}\,\hat{\rho}\subseteq B(0,1)$ and $|\rho(\mathbf{z})|\gtrsim 1$
for all $\mathbf{z}\in B(0,1)$. Given $B\in\mathcal{B}(\sigma^{-1})$ with
centre $\mathbf{z}_{B}=(y_{B},w_{B})$, define
$\rho_{B}(\mathbf{z}):=\rho(\sigma(\mathbf{z}-\mathbf{z}_{B}))$. Furthermore,
for each $x\in\mathbb{R}$ let

$G_{\pm,x}^{\lambda,B}(\mathbf{z}):=U^{\lambda}_{\pm}f(x,\mathbf{z}_{B}+\mathbf{z})\rho_{B}(\mathbf{z}_{B}+\mathbf{z}),$

for $\mathbf{z}=(y,w)\in\mathbb{R}^{2}$. It then follows that

$\big{\|}U^{\lambda}_{\pm}f(x,\,\cdot\,,\,\cdot\,)\big{\|}_{L^{p}(B)}\lesssim\|G_{\pm,x}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}.$
(6.24)

Our goal is to show that the Fourier transform
$(G_{\pm,x}^{\lambda,B})\;\widehat{}\;$ is essentially supported in the
$\sigma$-neighbourhood of some curve, allowing us to apply Lemma 6.11 above.
This Fourier support property is a consequence of the spatio-temporal
localisation to the $\sigma^{-1}$-ball: at this scale the variable coefficient
phase $\Phi_{\pm}^{\lambda}$ can be approximated by a constant coefficient
phase, and the operator $U^{\lambda}_{\pm}f$ is effectively a Fourier
transform.

For $\mathbf{z}=(y,w)\in\mathbb{R}^{2}$, we may write

$G_{\pm,x}^{\lambda,B}(\mathbf{z})=\int_{\mathbb{R}^{2}}K_{\pm,x}^{\lambda,B}(\mathbf{z};u,v)f(u,v)\,\,\mathrm{d}u\,\mathrm{d}v$

where

$K_{\pm,x}^{\lambda,B}(\mathbf{z};u,v)=\rho_{B}(\mathbf{z}_{B}+\mathbf{z})\int_{\widehat{\mathbb{R}}^{2}}e^{-i(u\xi+v\eta)+i\Phi^{\lambda}_{\pm}(x,\mathbf{z}_{B}+\mathbf{z};\boldsymbol{\xi})}\mathfrak{e}_{\pm}^{\lambda}(w_{B}+w;\boldsymbol{\xi})\,\mathrm{d}\boldsymbol{\xi}$
(6.25)

for $\boldsymbol{\xi}=(\xi,\eta)$. Observe that

$\Phi_{\pm}^{\lambda}(x,\mathbf{z}_{B}+\mathbf{z};\xi,\eta)=\Phi_{\pm}^{\lambda}(x,y_{B},w_{B};\xi,\eta)+y\eta+wh_{\pm}^{\lambda,B}(\xi,\eta)+\mathcal{E}_{\pm}^{\lambda,B}(w;\xi,\eta)$
(6.26)

where
$h_{\pm}^{\lambda,B}(\xi,\eta):=\partial_{w}\phi_{\pm}(w_{B}/\lambda;\xi,\eta)$
and

$\mathcal{E}_{\pm}^{\lambda,B}(w;\xi,\eta):=\frac{w^{2}}{\lambda}\int_{0}^{1}\partial_{w}^{2}\phi_{\pm}\big{(}(w_{B}+sw)/\lambda;\xi,\eta\big{)}\,(1-s)\,\mathrm{d}s;$

in particular $\mathcal{E}_{\pm}^{\lambda,B}(w;\xi,\eta)$ satisfies

$|\partial_{w}^{\alpha}\partial_{\xi}^{\beta_{1}}\partial_{\eta}^{\beta_{2}}\mathcal{E}_{\pm}^{\lambda,B}(w;\xi,\eta)|\lesssim_{\alpha,\beta_{1},\beta_{2}}\sigma^{\alpha}\qquad\textrm{for
all $\alpha$, $\beta_{1}$, $\beta_{2}\in\mathbb{N}_{0}$.}$ (6.27)

Note that (6.27) relies on the hypothesis $\sigma^{2}\geq\lambda^{-1}$, the
support hypothesis $|w|\leq\sigma^{-1}$ and Lemma 6.2 ii).

Taking the Fourier transform of the kernel representation of
$G_{\pm,x}^{\lambda,B}$ in (6.25), applying (6.26) and Fubini’s theorem, we
obtain

$(G_{\pm,x}^{\lambda,B})\;\widehat{}\;(\tilde{\eta},\tilde{\omega})=\int_{\mathbb{R}^{2}}\int_{\widehat{\mathbb{R}}^{2}}e^{i\Phi_{\pm}^{\lambda}(x-u,y_{B}-v,w_{B};\xi,\eta)}A_{\pm}^{\lambda,B}(\tilde{\eta},\tilde{\omega};\xi,\eta)\,\mathrm{d}\xi\,\mathrm{d}\eta\,f(u,v)\,\mathrm{d}u\,\mathrm{d}v$
(6.28)

where

$A_{\pm}^{\lambda,B}(\tilde{\eta},\tilde{\omega};\xi,\eta):=\int_{\mathbb{R}^{2}}e^{-i(y(\tilde{\eta}-\eta)+w(\tilde{\omega}-h_{\pm}^{\lambda,B}(\xi,\eta)))}\mathfrak{e}_{\pm}^{\lambda,B}(y,w;\xi,\eta)\,\mathrm{d}y\,\mathrm{d}w$

for

$\mathfrak{e}_{\pm}^{\lambda,B}(y,w;\xi,\eta):=e^{i\mathcal{E}_{\pm}^{\lambda,B}(w;\xi,\eta)}\mathfrak{e}_{\pm}^{\lambda}(w_{B}+w;\xi,\eta)\rho_{B}(y_{B}+y,w_{B}+w)$

and where we have used that

$-u\xi-v\eta+\Phi^{\lambda}_{\pm}(x,y_{B},w_{B};\xi,\eta)=\Phi_{\pm}^{\lambda}(x-u,y_{B}-v,w_{B};\xi,\eta).$

In view of (6.27), (6.9) and (6.22), and setting
$\boldsymbol{\xi}=(\xi,\eta)$, we have

$|\partial_{\mathbf{z}}^{\boldsymbol{\alpha}}\partial_{\boldsymbol{\xi}}^{\boldsymbol{\beta}}\mathfrak{e}_{\pm}^{\lambda,B}(\mathbf{z};\boldsymbol{\xi})|\lesssim_{\boldsymbol{\alpha},\boldsymbol{\beta}}\sigma^{-\beta_{1}-\beta_{2}}\sigma^{\alpha_{1}+\alpha_{2}}$

for all $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2})$,
$\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}^{2}$. Thus,
repeated integration-by-parts yields

$\displaystyle|\partial_{\boldsymbol{\xi}}^{\boldsymbol{\beta}}\partial_{(\tilde{\eta},\tilde{\omega})}^{\boldsymbol{\gamma}}$
$\displaystyle
A_{\pm}^{\lambda,B}(\tilde{\eta},\tilde{\omega};\boldsymbol{\xi})|$ (6.29)
$\displaystyle\lesssim_{\boldsymbol{\beta},\boldsymbol{\gamma},N}\sigma^{-\beta_{1}-\beta_{2}-\gamma_{1}-\gamma_{2}-2}\big{(}1+\sigma^{-1}|\tilde{\eta}-\eta|+\sigma^{-1}|\tilde{\omega}-h_{\pm}^{\lambda,B}(\boldsymbol{\xi})|\big{)}^{-N}$

for all $\boldsymbol{\beta}=(\beta_{1},\beta_{2})$,
$\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2})\in\mathbb{N}_{0}^{2}$ and
$N\in\mathbb{N}_{0}$; here we have used that $\rho_{B}$ decays rapidly outside
of a ball of radius $\sigma^{-1}$.

By (6.8) and Lemma 6.2 ii), there exists an absolute constant $C\geq 1$ such
that

$|\tilde{\omega}-h_{\pm}^{\lambda,B}(\xi,\eta)|\geq|\tilde{\omega}-h_{\pm}^{\lambda,B}(0,\tilde{\eta})|-C(|\tilde{\eta}-\eta|+\sigma)$
(6.30)

whenever $(\xi,\eta)\in\mathrm{supp}_{\xi,\eta}\,A_{\pm}^{\lambda,B}$. On the
other hand, given $(\xi,\eta)\in\widehat{\mathbb{R}}^{2}$, we may write
$\eta^{2}=\Delta(w;\xi,\eta)-3w\xi$ for any $w\in\mathbb{R}$. In particular,
the hypotheses $\sigma<6^{-1}C_{\star}^{-2}$ from (6.22) imply that

$2C_{\star}^{-1}\leq(2C_{\star})^{-1/2}\leq|\eta|\leq
C_{\star}\qquad\textrm{whenever
$\eta\in\mathrm{supp}_{\eta}\,\mathfrak{e}_{\pm}$.}$ (6.31)

Let $\delta_{\circ}:=\delta/100$ and define
$\mathcal{E}:=\widehat{\mathbb{R}}^{2}\setminus\mathcal{N}$ where

$\mathcal{N}:=\big{\\{}(\tilde{\eta},\tilde{\omega})\in\widehat{\mathbb{R}}^{2}:C_{\star}^{-1}\leq|\tilde{\eta}|\leq
2C_{\star},\,|\tilde{\omega}-h_{\pm}^{\lambda,B}(0,\tilde{\eta})|<\lambda^{\delta_{\circ}}\sigma\big{\\}}.$
(6.32)

We may assume $4C\lambda^{-\delta}\leq\lambda^{-\delta_{\circ}}$, otherwise
the desired result is trivial. Combining (6.29), (6.30) and (6.31) and using
the hypothesis $\sigma<\lambda^{-\delta}$ from (6.22), we see that

$|\partial_{\boldsymbol{\xi}}^{\boldsymbol{\beta}}\partial_{(\tilde{\eta},\tilde{\omega})}^{\boldsymbol{\gamma}}A_{\pm}^{\lambda,B}(\tilde{\eta},\tilde{\omega};\xi,\eta)|\lesssim_{\delta,\boldsymbol{\beta},\boldsymbol{\gamma},N}\lambda^{-N}\qquad\textrm{whenever
$(\tilde{\eta},\tilde{\omega})\in\mathcal{E}$}$ (6.33)

for all $\boldsymbol{\beta}$, $\boldsymbol{\gamma}\in\mathbb{N}_{0}^{2}$,
$N\in\mathbb{N}_{0}$. Furthermore, by repeatedly applying integration-by-parts
to (6.28) in the $(\xi,\eta)$ variables and combining the resulting expression
with the estimate (6.33), Lemma 6.2 ii) and Hölder’s inequality,

$|\partial_{(\tilde{\eta},\tilde{\omega})}^{\boldsymbol{\gamma}}(G_{\pm,x}^{\lambda,B})\;\widehat{}\;(\tilde{\eta},\tilde{\omega})|\lesssim_{\delta,\boldsymbol{\gamma},N}\lambda^{-N}\|f\|_{L^{p}(\mathbb{R}^{2})}\qquad\textrm{whenever
$(\tilde{\eta},\tilde{\omega})\in\mathcal{E}$.}$ (6.34)

Now let $P_{\pm}^{\lambda,B}$ be the Fourier projection operator with Fourier
multiplier

$\beta_{0}\big{(}\lambda^{-\delta_{\circ}}\sigma^{-1}(\tilde{\omega}-h_{\pm}^{\lambda,B}(0,\tilde{\eta}))\big{)}\big{(}\beta_{0}\big{(}(2C_{\star})^{-1}\tilde{\eta}\big{)}-\beta_{0}(C_{\star}\tilde{\eta})\big{)}$

Combining the Fourier inversion formula with a further integration-by-parts
argument and (6.34), we have

$\big{|}\big{(}I-P_{\pm}^{\lambda,B}\big{)}G_{\pm,x}^{\lambda,B}(y,w)\big{|}\lesssim_{N}\lambda^{-N}\|f\|_{L^{p}(\mathbb{R}^{2})}\qquad\textrm{for
all $N\in\mathbb{N}_{0}$.}$ (6.35)

A minor adjustment of this argument also shows that

$\big{|}P_{\tilde{I}}\big{(}I-P_{\pm}^{\lambda,B}\big{)}G_{\pm,x}^{\lambda,B}(y,w)\big{|}\lesssim_{N}\lambda^{-N}\|f\|_{L^{p}(\mathbb{R}^{2})}\qquad\textrm{for
all $N\in\mathbb{N}_{0}$}$ (6.36)

for all $\tilde{I}\in\mathcal{I}(\lambda^{\delta_{\circ}}\sigma)$.

On the other hand, the projection $P_{\pm}^{\lambda,B}G_{\pm,x}^{\lambda,B}$
has Fourier support within the set $\mathcal{N}$ defined in (6.32). This is
the vertical $2\lambda^{\delta_{\circ}}\sigma$-neighbourhood of the curve

$\big{\\{}(s,\gamma_{\pm}^{\lambda,B}(s)):C_{\star}^{-1}\leq|s|\leq
2C_{\star}\big{\\}}\qquad\textrm{where}\qquad\gamma_{\pm}^{\lambda,B}(s):=(\partial_{w}\phi_{\pm})(w_{B}/\lambda;0,s).$

A direct calculation (c.f. §B.4) shows that

$\big{(}\gamma_{\pm}^{\lambda,B}\big{)}^{\prime\prime}(s)=\frac{2^{4}}{3^{2}}\cdot\frac{s}{(\lambda^{-1}w_{B})^{3}}\quad\textrm{and
so}\quad|\big{(}\gamma_{\pm}^{\lambda,B}\big{)}^{\prime\prime}(s)|\geq
C_{\star}^{-4}\quad\textrm{for $C_{\star}^{-1}\leq|s|\leq 2C_{\star}$.}$
(6.37)

Moreover, it is not difficult to see that
$\gamma_{\pm}^{\lambda,B}\in\mathfrak{G}$, using the notation from Definition
6.10. Furthermore,
$P_{\pm}^{\lambda,B}G_{\pm,x}^{\lambda,B}\in\mathcal{Z}(\gamma_{\pm}^{\lambda,B};2\lambda^{\delta_{\circ}}\sigma)$
and so we may apply Lemma 6.11 to deduce that

$\|P_{\pm}^{\lambda,B}G_{\pm,x}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{p,\delta}\sigma^{-(1/2-1/p)}\lambda^{2\delta_{\circ}}\Big{(}\sum_{\tilde{I}\in\mathcal{I}(\lambda^{\delta_{\circ}}\sigma)}\|P_{\tilde{I}}P_{\pm}^{\lambda,B}G_{\pm,x}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}^{p}\Big{)}^{1/p}.$

Combining this with (6.35) and (6.36), we therefore have

$\|G_{\pm,x}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{p,\delta,N}\sigma^{-(1/2-1/p)}\lambda^{2\delta_{\circ}}\Big{(}\sum_{\tilde{I}\in\mathcal{I}(\lambda^{\delta_{\circ}}\sigma)}\|P_{\tilde{I}}G_{\pm,x}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}^{p}\Big{)}^{1/p},$
(6.38)

up to a rapidly decreasing $L^{p}$ error.

We may write

$G_{\pm,x}^{\lambda,B}=\sum_{I\in\mathcal{I}(\sigma)}G_{\pm,x,I}^{\lambda,B}\quad\textrm{where}\quad
G_{\pm,x,I}^{\lambda,B}(\mathbf{z}):=U^{\lambda}_{\pm,I}f(x,\mathbf{z}_{B}+\mathbf{z})\rho_{B}(\mathbf{z}_{B}+\mathbf{z})$
(6.39)

for all $I\in\mathcal{I}(\sigma)$. Each $G_{\pm,x,I}^{\lambda,B}$ is of the
same form as the function $G_{\pm,x}^{\lambda,B}$: the only change is that the
underlying amplitude $\mathfrak{e}_{\pm}$ is replaced with
$\mathfrak{e}_{\pm,I}$. Given any bounded interval $J\subseteq\mathbb{R}$, let
$\eta_{J}$ denote its centre. Since
$\mathfrak{e}_{\pm,I}\in\mathfrak{E}_{\pm}^{\lambda}(\sigma)$, we can repeat
the analysis in (6.28) and (6.29) with $G_{\pm,x}^{\lambda,B}$ replaced with
$G_{\pm,x,I}^{\lambda,B}$ to conclude that

$|\partial_{(\tilde{\eta},\tilde{\omega})}^{\boldsymbol{\gamma}}(G_{\pm,x,I}^{\lambda,B})\;\widehat{}\;(\tilde{\eta},\tilde{\omega})|\lesssim_{\delta,\boldsymbol{\gamma},N}\lambda^{-N}\|f\|_{L^{p}(\mathbb{R}^{2})}\qquad\textrm{whenever
$|\tilde{\eta}-\eta_{I}|>2\lambda^{\delta_{\circ}}\sigma$,}$

for all $\boldsymbol{\gamma}\in\mathbb{N}_{0}^{2}$. In particular, if
$\tilde{I}\in\mathcal{I}(\lambda^{\delta_{\circ}}\sigma)$,
$I\in\mathcal{I}(\sigma)$ satisfy
$|\eta_{I}-\eta_{\tilde{I}}|>10\lambda^{\delta_{\circ}}\sigma$, then

$\|P_{\tilde{I}}G_{\pm,x,I}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}\lesssim_{N}\lambda^{-N}\|f\|_{L^{p}(\mathbb{R}^{2})}.$
(6.40)

By combining (6.40) with an application of Hölder’s inequality and the
$L^{p}$-boundedness of the projection operators $P_{\tilde{I}}$, we deduce
that

$\displaystyle\sum_{\tilde{I}\in\mathcal{I}(\lambda^{\delta_{\circ}}\sigma)}\|P_{\tilde{I}}G_{\pm,x}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}^{p}$
$\displaystyle\lesssim\lambda^{\delta_{\circ}p}\sum_{I\in\mathcal{I}(\sigma)}\|G_{\pm,x,I}^{\lambda,B}\|_{L^{p}(\mathbb{R}^{2})}^{p}$
$\displaystyle\lesssim\lambda^{\delta_{\circ}p}\sum_{I\in\mathcal{I}(\sigma)}\|U_{\pm,I}^{\lambda}f(x,\,\cdot\,,\,\cdot\,)\|_{L^{p}(w_{B})}^{p},$

where the last step follows from the definition (6.39) of the
$G_{\pm,x,I}^{\lambda,B}$ and the rapid decay of $\rho$. Finally, we combine
the above display with (6.24) and (6.38) to obtain (6.23), as required. ∎

Let $\Theta(\lambda^{-1/2})$ be a finitely-overlapping cover of
$[-2C_{\star},2C_{\star}]^{2}$ by balls of radius $\lambda^{-1/2}$ and
$(\psi_{\theta})_{\theta\in\Theta(\lambda^{-1/2})}$ be a smooth partition of
unity subordinate to $\Theta(\lambda^{-1/2})$, satisfying
$|\partial_{\boldsymbol{\xi}}^{\boldsymbol{\beta}}\psi_{\theta}(\boldsymbol{\xi})|\lesssim_{\boldsymbol{\beta}}\lambda^{(\beta_{1}+\beta_{2})/2}$
for all $\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}^{2}$. Given
$\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$ and
$\theta\in\Theta(\lambda^{-1/2})$, we define
$\mathfrak{e}_{\pm,\theta}(w;\boldsymbol{\xi}):=\mathfrak{e}_{\pm}(w;\boldsymbol{\xi})\psi_{\theta}(\boldsymbol{\xi})$.

######  Lemma 6.12 (Fine scale decoupling).

For $2\leq p\leq 4$ and $\delta>0$, given $\lambda\geq 1$,
$\lambda^{-1/2}\leq\sigma\leq 1$, we have

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,I}]f\big{\|}_{L^{p}(B_{\lambda})}\lesssim_{\delta}(\lambda\sigma^{2})^{1/2-1/p+\delta}\Big{(}\sum_{\theta\in\Theta(\lambda^{-1/2})}\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}]f\big{\|}_{L^{p}(w_{B_{\lambda}})}^{p}\Big{)}^{1/p},$

up to a rapidly decreasing $L^{p}$ error, whenever $I\in\mathcal{I}(\sigma)$,
$\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$ and
$B_{\lambda}\subseteq\mathbb{R}^{3}$ is a $\lambda$-ball.

The proof follows from variable coefficient $\ell^{p}$ decoupling, via a
scaling argument.

###### Proof (sketch).

Recall that $\Phi_{\pm}$ satisfies conditions H1) and H2) on a neighbourhood
of $\mathrm{supp}\,\mathfrak{e}_{\pm,I}$ and that
$\mathrm{supp}_{\xi,\eta}\,\mathfrak{e}_{\pm,I}$ is contained in a ball of
radius $O(\sigma)$. We apply parabolic rescaling to the operator
$U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,I}]$ on right-hand side of (6.12),
sending it to an operator with amplitude localised at unit scale. This can be
decoupled using the natural variable coefficient generalisation of the
$\ell^{p}$ decoupling inequality for surfaces of non-vanishing Gaussian
curvature from [8, Theorem 1.1]. We carry out this decoupling into caps at
scale $\sigma^{-1}\lambda^{-1/2}$. Undoing the parabolic rescaling, we obtain
the desired decoupling for our original operator into caps at scale
$\lambda^{-1/2}$.

We remark that the required variable coefficient decoupling inequality can be
derived from the constant coefficient following the argument of [3] (which
treats the case of non-homogeneous phase functions); see also [25, Theorem
7.1.2.] for a closely related result. ∎

Finally, for each $\theta\in\Theta(\lambda^{-1/2})$, let

$K^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}](\mathbf{x},w):=\frac{1}{(2\pi)^{2}}\int_{\widehat{\mathbb{R}}^{2}}e^{i\Phi_{\pm}^{\lambda}(\mathbf{x},w;\boldsymbol{\xi})}\mathfrak{e}_{\pm,\theta}^{\lambda}(w;\boldsymbol{\xi})\,\,\mathrm{d}\boldsymbol{\xi}$

denote the kernel of the localised propagator
$U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}]$, viewed as a Fourier
multiplier acting on functions on $\mathbb{R}^{2}$ for fixed $w\in\mathbb{R}$.

######  Lemma 6.13 (Localised kernel estimates).

For $\lambda^{-1/2}\leq\sigma\leq 1$, we have

$|K^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}](\mathbf{x},w)|\lesssim\lambda^{-1}\big{(}1+\lambda^{-1/2}|\mathbf{x}+\partial_{\xi}\phi_{\pm}^{\lambda}(w;\boldsymbol{\xi}_{\theta})|\big{)}^{-100},\quad\theta\in\Theta(\lambda^{-1/2}),$

whenever $\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$.

In particular, Lemma 6.13 implies that

$\sup_{w\in\mathbb{R}}\|K^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}](\,\cdot\,,w)\|_{L^{1}(\mathbb{R}^{2})}\lesssim
1\qquad\textrm{for all $\theta\in\Theta(\lambda^{-1/2})$,}$

which leads to favourable $L^{\infty}$ bounds for the localised propagators.

###### Proof (of Lemma 6.13).

By a change of variables,

$K^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}](\mathbf{x},w):=\frac{\lambda^{-1}}{(2\pi)^{2}}e^{i\Phi_{\pm}(\mathbf{x},w;\boldsymbol{\xi}_{\theta})}\int_{\widehat{\mathbb{R}}^{2}}e^{i\lambda^{-1/2}\langle\mathbf{x}+\partial_{\boldsymbol{\xi}}\phi_{\pm}^{\lambda}(w;\boldsymbol{\xi}_{\theta}),\boldsymbol{\xi}\rangle}e_{\pm,\theta}^{\lambda}(w;\boldsymbol{\xi})\,\mathrm{d}\boldsymbol{\xi}$

where

$e_{\pm,\theta}^{\lambda}(w;\boldsymbol{\xi}):=e^{i\mathcal{E}^{\lambda}_{\pm,\theta}(w;\boldsymbol{\xi})}\mathfrak{e}_{\pm,\theta}(w/\lambda;\boldsymbol{\xi}_{\theta}+\lambda^{-1/2}\boldsymbol{\xi})$

for

$\mathcal{E}^{\lambda}_{\pm,\theta}(w;\boldsymbol{\xi}):=\phi_{\pm}^{\lambda}(w;\boldsymbol{\xi}_{\theta}+\lambda^{-1/2}\xi)-\phi_{\pm}^{\lambda}(w;\boldsymbol{\xi}_{\theta})-\lambda^{-1/2}\langle\partial_{\boldsymbol{\xi}}\phi_{\pm}^{\lambda}(w;\boldsymbol{\xi}_{\theta}),\boldsymbol{\xi}\rangle.$

By Lemma 6.2 ii) and (6.9), we have
$|\partial_{\boldsymbol{\xi}}^{\boldsymbol{\beta}}e_{\pm,\theta}^{\lambda}(w;\boldsymbol{\xi})|\lesssim_{\boldsymbol{\beta}}1$
for all $\boldsymbol{\beta}\in\mathbb{N}_{0}^{2}$. The desired result now
follows by repeated integration-by-parts. ∎

###### Proof (of Proposition 6.6).

Let $2\leq p\leq 4$, $\delta>0$ and fix $\lambda\geq 1$,
$\lambda^{-1/2}<\sigma\leq 1$ and
$\mathfrak{e}_{\pm}\in\mathfrak{E}_{\pm}(\sigma)$. By Lemma 6.13 and the fact
that $|w|\lesssim\lambda$ whenever
$w\in\mathrm{supp}_{w}\,K^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]$, the
operator $U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]$ is essentially local. In
particular, it suffices to prove (6.12) with the left-hand norm restricted to
some $\lambda$-ball $B_{\lambda}\subseteq\mathbb{R}^{3}$.

By combining Lemma 6.9 and Lemma 6.12, using the rapid decay of
$w_{B_{\lambda}}$, we deduce that

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm}]f\big{\|}_{L^{p}(B_{\lambda})}\lesssim_{\delta}\lambda^{\delta}(\lambda\sigma)^{1/2-1/p}\Big{(}\sum_{\theta\in\Theta(\lambda^{-1/2})}\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}^{p}\Big{)}^{1/p}$

holds up to the inclusion of a rapidly decreasing $L^{p}$ error. It remains to
show

$\Big{(}\sum_{\theta\in\Theta(\lambda^{-1/2})}\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}]f\big{\|}_{L^{p}(\mathbb{R}^{3})}^{p}\Big{)}^{1/p}\lesssim\lambda^{1/p}\|f\|_{L^{p}(\mathbb{R}^{2})}.$
(6.41)

By Plancherel’s theorem (since our phases are translation-invariant: more
generally one may appeal to Hörmander’s oscillatory integral theorem [16]),

$\big{\|}U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}]f\big{\|}_{L^{2}(\mathbb{R}^{3})}\lesssim\lambda^{1/2}\|f_{\theta}\|_{L^{2}(\mathbb{R}^{2})},$
(6.42)

where $f_{\theta}$ is defined by
$\widehat{f}_{\theta}:=\tilde{\psi}_{\theta}\cdot\widehat{f}$ for
$\tilde{\psi}_{\theta}\in C^{\infty}_{c}(\widehat{\mathbb{R}}^{2})$ satisfying
$\tilde{\psi}_{\theta}(\boldsymbol{\xi})=1$ for $\boldsymbol{\xi}\in
2\cdot\theta$ and $\mathrm{supp}\,\tilde{\psi}_{\theta}\subseteq
4\cdot\theta$. By taking the $\ell^{2}$ sum of (6.42) over all
$\theta\in\Theta(\lambda^{-1/2})$ and summing the resulting right-hand side
using Plancherel’s theorem, we obtain the $p=2$ case of (6.41).

On the other hand, the kernel estimate from Lemma 6.13 readily implies that

$\|U^{\lambda}[\Phi_{\pm};\mathfrak{e}_{\pm,\theta}]f\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim\|f\|_{L^{\infty}(\mathbb{R}^{2})}$

for all $\theta\in\Theta(\lambda^{-1/2})$. This is precisely the $p=\infty$
case of (6.41). Interpolating this with the above $L^{2}$ yields (6.41) for
all $2\leq p\leq\infty$. ∎

##  Appendix A Quantified oscillatory integral estimates

Here we discuss the polynomial dependence on $\gamma$ and $W$ in Theorem 1.4.

We first note that, by rescaling in the $\mathbf{x}$, $w$ and
$\boldsymbol{\xi}$ variables, it suffices to consider the case $W=1$. Note
that the rescaling enlarges the size of the support of $a$, but we can simply
decompose the support into $O(W^{C})$ many pieces of diameter $1$ and then
translate each piece back to the origin.

It remains to track the dependency on $\gamma$ in the proof of Theorem 2.8 II)
from [10]. The Hörmander condition (1.13) appears in an application of the
‘universal’ oscillatory integral estimate of Stein [28] (see [10, Theorem
1.3], which is used in the proof of [10, Proposition 9.1]), the ‘universal’
geometric estimates of Wisewell [32] (see [10, Proposition 1.4], which is used
in the proof of [10, Proposition 6.4]) and a direct application in [10,
Corollary 6.7]. It is easy to verify the desired polynomial dependence on
$\gamma$ from these steps.

The Nikodym non-compression hypothesis is used in the proof of [10,
Proposition 5.3] to prove a crucial sublevel set estimate. The following
quantified variant can be established via a straightforward adaptation of the
proof of [10, Proposition 5.3].

######  Proposition A.1.

There exists some $M\in\mathbb{N}$ such that the following holds. Suppose
$g_{j}\colon(-1,1)\times\mathbb{B}^{N}\to\mathbb{R}$, $1\leq j\leq 3$, are
real analytic and let $I_{\circ}\subseteq(-1,1)$ and
$Y_{\circ}\subset\mathbb{B}^{N}$ be compact sets containing the origin. For
$\gamma>0$, $W\geq 1$ suppose that matrix

$B(s;y):=(b_{i,j}(s;y))_{1\leq i,j\leq 3}\quad\textrm{where}\quad
b_{i,j}(s;y):=\frac{(\partial_{t}^{i}g_{j})(s;y)}{i!},\quad 0\leq i,\,1\leq
j\leq 3,$

satisfies $|\det B(s;y)|\geq\gamma>0$ and

$\big{(}\sum_{\begin{subarray}{c}0\leq i\leq 3\\\ 1\leq j\leq
3\end{subarray}}|b_{i,j}(s;y)|^{2}\big{)}^{1/2}\leq W,\qquad|b_{i,j}(s;y)|\leq
W^{i+1}\qquad\textrm{for all $i\in\mathbb{N}_{0}$, $1\leq j\leq 3$,}$

for all $(s;y)\in I_{\circ}\times Y_{\circ}$. Then the sublevel set estimate

$\Big{|}\Big{\\{}t\in
I_{\circ}:\big{|}1-\sum_{j=1}^{3}\mu_{j}g_{j}(t;y)\big{|}<\sigma\Big{\\}}\Big{|}\lesssim(W\gamma^{-1})^{M}\sigma^{1/3}$
(A.1)

holds uniformly over all $\sigma>0$, $y\in Y_{\circ}$ and all scalars
$\mu_{j}\in\mathbb{R}$, $1\leq j\leq m$.

The Nikodym non-compression hypothesis is only used in [10] in the proof of
the sublevel set estimate in [10, Proposition 5.3]. Moreover, the sublevel set
estimate in [10, Proposition 5.3] is applied precisely once in the proof of
[10, Theorem 2.8 II)]. Thus, once the sublevel set estimate is quantified as
in (A.1), we can guarantee quantification of the final local smooth estimate
in terms of the Nikodym non-compression parameter.

We remark that the H2) condition and Nikodym non-compression conditions do not
play a role in the iteration scheme in the proof of [10, Theorem 6.12].

##  Appendix B Calculus computations

For the reader’s convenience, we present the steps of some of the more
involved calculus computations appearing in §6.

###  B.1. Derivative dictionary

Applying the chain rule to the definition of $\phi_{\pm}$, together with the
fact that the $t_{\pm}(w;\xi,\eta)$ are roots of
$(\phi_{\xi,\eta}^{w})^{\prime}$, we see that

$\partial_{\xi}\phi_{\pm}=-t_{\pm},\qquad\partial_{\eta}\phi_{\pm}=-t_{\pm}^{2},\qquad\partial_{w}\phi_{\pm}=t_{\pm}^{3}$

and so

$\phi_{\xi\xi}^{2}\phi_{\pm}=-\partial_{\xi}t_{\pm},\qquad\phi_{\xi\eta}^{2}\phi_{\pm}=-\partial_{\eta}t_{\pm},\qquad\phi_{\eta\eta}^{2}\phi_{\pm}=-2(\partial_{\eta}t_{\pm})t_{\pm}.$

By (simple) direct computation,

$\partial_{\xi}t_{\pm}=\pm\frac{1}{2}\cdot\frac{1}{\Delta^{1/2}},\qquad\partial_{\eta}t_{\pm}=\pm\frac{t_{\pm}}{\Delta^{1/2}},\qquad\partial_{w}t_{\pm}=\mp\frac{3}{2}\cdot\frac{t_{\pm}^{2}}{\Delta^{1/2}}$
(B.1)

and, by combining the above displays, we see that

$\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\phi_{\pm}=\mp\frac{1}{2\Delta^{1/2}}\begin{bmatrix}1&2t_{\pm}\\\
2t_{\pm}&4t_{\pm}^{2}\end{bmatrix}.$ (B.2)

We use these identities to verify the various differential conditions on
$\phi_{\pm}$.

###  B.2. Hörmander condition

Here we verify (6.6). From (B.2), we see that
$\partial_{\xi\eta}^{2}\phi_{\pm}=2t_{\pm}\partial_{\xi\xi}^{2}\phi_{\pm}$ and
$\partial_{\eta\eta}^{2}\phi_{\pm}=4t_{\pm}^{2}\partial_{\xi\xi}^{2}\phi_{\pm}$.
Applying the product rule to these identities,

$\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\partial_{w}\phi_{\pm}=\begin{bmatrix}\partial_{w}\partial_{\xi\xi}^{2}\phi_{\pm}&2(\partial_{w}t_{\pm})\partial_{\xi\xi}^{2}\phi_{\pm}+2t_{\pm}\partial_{w}\partial_{\xi\xi}^{2}\phi_{\pm}\\\
2(\partial_{w}t_{\pm})\partial_{\xi\xi}^{2}\phi_{\pm}+2t_{\pm}\partial_{w}\partial_{\xi\xi}^{2}\phi_{\pm}&8(\partial_{w}t_{\pm})t_{\pm}\partial_{\xi\xi}^{2}\phi_{\pm}+4t_{\pm}^{2}\partial_{w}\partial_{\xi\xi}^{2}\phi_{\pm}\end{bmatrix}.$

Taking the determinant, we may simplify the matrix by performing the
operations

$\textrm{Row 2}\to\textrm{Row 2}-2t_{\pm}\times\textrm{Row
1}\quad\textrm{and}\quad\textrm{Col 2}\to\textrm{Col
2}-2t_{\pm}\times\textrm{Col 1}$

to obtain

$\det\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\partial_{w}\phi_{\pm}=\det\begin{bmatrix}\partial_{w}\partial_{\xi\xi}^{2}\phi_{\pm}&2(\partial_{w}t_{\pm})\partial_{\xi\xi}^{2}\phi_{\pm}\\\
2(\partial_{w}t_{\pm})\partial_{\xi\xi}^{2}\phi_{\pm}&0\end{bmatrix}=-4(\partial_{w}t_{\pm})^{2}(\partial^{2}_{\xi\xi}\phi_{\pm})^{2}.$

Applying the formula for $\partial^{2}_{\xi\xi}\phi_{\pm}$ from (B.2) and for
$\partial_{w}t_{\pm}$ from (B.1), the desired result follows.

###  B.3. Nikodym non-compression

By computing the determinant in (B.2), we see that
$\det\partial_{\boldsymbol{\xi}\boldsymbol{\xi}}^{2}\phi_{\pm}\equiv 0$.

We now verify (6.11). Again note that, by (B.2), we have
$\partial_{\xi\eta}^{2}\phi_{\pm}=2t_{\pm}\partial_{\xi\xi}^{2}\phi_{\pm}$ and