# Localised variants of multilinear restriction

David Beltran  David Beltran: Departament d’Anàlisi Matemàtica, Universitat de
València, Dr. Moliner 50, 46100 Burjassot, Spain<EMAIL_ADDRESS>,
Jennifer Duncan  Jennifer Duncan: Instituto de Ciencias Matemáticas CSIC-UAM-
UC3M-UCM, C. de Nicolas Cabrera 13-15, 28049 Madrid, Spain
<EMAIL_ADDRESS>and  Jonathan Hickman  Jonathan Hickman: University
of Edinburgh, School of Mathematics, James Clerk Maxwell Building, The King’s
Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK.
<EMAIL_ADDRESS>
###### Abstract.

We revisit certain localised variants of the Bennett–Carbery–Tao multilinear
restriction theorem, recently proved by Bejenaru. We give a new proof of
Bejenaru’s theorem, relating the estimates to the theory of
Kakeya–Brascamp–Lieb inequalities. Moreover, the new proof allows for a
substantial generalisation, exploiting the full power of the
Kakeya–Brascamp–Lieb theory.

D.B. supported by the AEI grants RYC2020-029151-I and PID2022-140977NA-I00.

J.D. supported by ERC grant 834728 and Severo Ochoa grant CEX2019-000904-S

##  1\. Introduction

###  1.1. Background

For $1\leq d\leq n$, let $\Sigma\colon U\to\mathbb{R}^{n}$ be a
parametrisation of a smooth $d$-dimensional submanifold $S$ of
$\mathbb{R}^{n}$. That is, $U\subset\mathbb{R}^{d}$ is an open, connected
neighbourhood of the origin, $\Sigma\colon U\to\mathbb{R}^{n}$ is a smooth,
injective, regular map111In particular, $\Sigma\in C^{\infty}(U)$ and
$\bigwedge_{j=1}^{d}\tfrac{\partial\Sigma}{\partial u_{j}}(u)\neq 0$ for all
$u\in U$. and $S=\Sigma(U)$. By an abuse of notation, we shall often simply
refer to $S$, with the tacit understanding that our analysis depends on a
choice of parametrisation.

Given $a\in C^{\infty}_{c}(U)$ satisfying $0\leq a(u)\leq 1$ for all $u\in U$,
we define the extension operator $E_{S}$ associated to $S$ (or, more
precisely, $\Sigma\colon U\to\mathbb{R}^{n}$) by

$E_{S}f(x):=\int_{U}e^{ix\cdot\Sigma(u)}f(u)\,a(u)\mathrm{d}u,\qquad f\in
L^{1}(S),\,\,x\in\mathbb{R}^{n}.$ (1.1)

Here and below, again by an abuse of notation, we write $L^{p}(S)$ in place of
$L^{p}(U)$ for $1\leq p\leq\infty$. We refer to $a$ as the amplitude of
$E_{S}$ and let $\rho(E_{S}):=\mathrm{diam}\,\mathrm{supp}\,a>0$.

For $\xi=\Sigma(u)\in S$, where $u\in U$, the tangent space $T_{\xi}S$ is the
$d$-dimensional subspace of $\mathbb{R}^{n}$ spanned by the vectors
$\tfrac{\partial\Sigma}{\partial
u_{1}}(u),\dots,\tfrac{\partial\Sigma}{\partial u_{d}}(u)$, and the normal
space $N_{\xi}S$ is the orthogonal complement of $T_{\xi}S$ in
$\mathbb{R}^{n}$.

######  Definition 1.1.

Fix $2\leq k\leq n$ and for $1\leq j\leq k$ let $S=\Sigma_{j}(U_{j})$ be a
smooth hypersurface in $\mathbb{R}^{n}$. Further, let
$\mathbf{q}_{k}:=(q_{k})_{j=1}^{k}$ where $q_{k}:=\tfrac{2}{k-1}$. We say
$\mathscr{E}=((S_{j})_{j=1}^{k},\mathbf{q}_{k})$ is a transverse ensemble if
there exists some $\nu>0$ such that222For $1\leq j\leq k$, let $V_{j}$ be a
vector subspace of $\mathbb{R}^{n}$ of dimension $d_{j}$ and
$\\{v_{j,1},\dots,v_{j,d_{j}}\\}$ be a choice of orthonormal basis of $V_{j}$.
We define  $|V_{1}\wedge\cdots\wedge
V_{k}|:=\Big{|}\bigwedge_{j=1}^{k}\bigwedge_{k=1}^{d_{j}}v_{j,k}\Big{|},$
noting that this quantity is independent of the choice of orthonormal bases.

$\Big{|}\bigwedge_{j=1}^{k}N_{\xi_{j}}S_{j}\Big{|}\geq\nu>0\qquad\textrm{where
$\xi_{j}:=\Sigma_{j}(0)\in S_{j}$ for $1\leq j\leq k$.}$

A fundamental result in modern Fourier analysis is the celebrated
Bennett–Carbery–Tao theorem. Here and below, $Q_{R}:=[-R,R]^{n}$ for $R>0$.

######  Theorem 1.2 (Bennett–Carbery–Tao [9, §5]).

Let $2\leq k\leq n$ and $\mathscr{E}=((S_{j})_{j=1}^{k},\mathbf{q}_{k})$ be a
transverse ensemble in $\mathbb{R}^{n}$. There exists $\rho_{\mathscr{E}}>0$
such that for $\max_{j}\rho(E_{S_{j}})<\rho_{\mathscr{E}}$ the following
holds. For all $\varepsilon>0$, there exists a constant $C_{\varepsilon}\geq
1$ such that

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j}}f_{j}|^{q_{k}}\leq
C_{\varepsilon}R^{\varepsilon}\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(S_{j})}^{q_{k}}$
(1.2)

holds for all $R\geq 1$ and $f_{j}\in L^{2}(S_{j})$, $1\leq j\leq k$.

Here we investigate localised variants of Theorem 1.2 where some of the
functions $f_{j}$ are assumed to be supported in thin sets. Under such
hypotheses, we can hope to gain in the size of the constant appearing on the
right-hand side of (1.2). This phenomenon was observed in recent papers of
Bejenaru [1, 3], and we begin by describing the existing results.

Let $S=\Sigma(U)$ be a $d$-dimensional submanifold of $\mathbb{R}^{n}$, as
above. For $1\leq d^{\prime}\leq d$, we say $S^{\prime}\subseteq S$ is a
$d^{\prime}$-dimensional submanifold of $S$ (or a codimension $d-d^{\prime}$
submanifold of $S$) if $S^{\prime}=\Sigma(M)$ for $M=\gamma(U^{\prime})$ a
$d^{\prime}$-dimensional submanifold of $\mathbb{R}^{d}$ with parametrisation
$\gamma\colon U^{\prime}\to U$, where we always assume that
$\gamma(U^{\prime})$ is compactly contained in $U$ and $\gamma(0)=0$. In this
case, we write $\mathrm{codim}(S^{\prime}\,|\,S):=d-d^{\prime}$. Note that
$S^{\prime}$ is also a $d^{\prime}$-dimensional submanifold of
$\mathbb{R}^{n}$, with $S^{\prime}=\Sigma^{\prime}(U^{\prime})$ for
$\Sigma^{\prime}:=\Sigma\circ\gamma$ satisfying

$\Sigma(0)=\Sigma^{\prime}(0).$ (1.3)

Continuing to abuse notation, given $\mu>0$ we say $f\in L^{2}(S)$ is
supported in $\mathcal{N}_{\mu}S^{\prime}$, or
$\mathrm{supp}\,f\subseteq\mathcal{N}_{\mu}S^{\prime}$, if $f$ is essentially
supported in $\mathcal{N}_{\mu}M$, where
$\mathcal{N}_{\mu}M\subseteq\mathbb{R}^{d}$ is the (open Euclidean)
$\mu$-neighbourhood of $M$.

######  Definition 1.3.

Fix $2\leq k\leq n$ and for $1\leq j\leq k$ let $S_{j}$ be a smooth
hypersurface and $S_{j}^{\prime}=\Sigma_{j}^{\prime}(U_{j}^{\prime})$ be a
smooth submanifold of $S_{j}$. Further let $\mathbf{q}_{k}:=(q_{k})_{j=1}^{k}$
where $q_{k}:=\tfrac{2}{k-1}$. We say
$\mathscr{E}=((S_{j},S_{j}^{\prime})_{j=1}^{k},\mathbf{q}_{k})$ is a
transverse ensemble in $\mathbb{R}^{n}$ if for some $\nu>0$ we have

$\Big{|}\bigwedge_{j=1}^{k}N_{\xi_{j}^{\prime}}S_{j}^{\prime}\Big{|}\geq\nu\qquad\textrm{where
$\xi_{j}^{\prime}:=\Sigma_{j}^{\prime}(0)\in S_{j}^{\prime}$ for $1\leq j\leq
k$.}$ (1.4)

We say $\mathscr{E}$ has codimension $(m_{j})_{j=1}^{k}$, where
$m_{j}:=\mathrm{codim}(S_{j}^{\prime}\,|\,S_{j})$ for $1\leq j\leq k$.

With the above definition, the local version of Theorem 1.2 reads as follows.

######  Theorem 1.4 (Bejenaru [3]).

Let $2\leq k\leq n$ and
$\mathscr{E}=((S_{j},S_{j}^{\prime})_{j=1}^{k},\mathbf{q}_{k})$ be a
transverse ensemble in $\mathbb{R}^{n}$ of codimension $(m_{j})_{j=1}^{k}$.
There exists a constant $\rho_{\mathscr{E}}>0$ such that for
$\max_{j}\rho(E_{S_{j}})<\rho_{\mathscr{E}}$ the following holds. For all
$\varepsilon>0$, there exists a constant $C_{\varepsilon}\geq 1$ such that

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j}}f_{j}|^{q_{k}}\leq
C_{\varepsilon}R^{\varepsilon}\prod_{j=1}^{k}\mu_{j}^{m_{j}q_{k}/2}\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(S_{j})}^{q_{k}},$
(1.5)

holds for all $R\geq 1$ and $f_{j}\in L^{2}(S_{j})$ satisfying
$\mathrm{supp}\,f_{j}\subseteq\mathcal{N}_{\mu_{j}}S_{j}^{\prime}$ for some
$0<\mu_{j}<1$.

An earlier (and more restricted) version of this result appeared in [1]. We
remark that it is possible to take $m_{j}=0$ in Theorem 1.4; in particular, if
$m_{1}=\dots=m_{k}=0$, then we recover Theorem 1.2. Note that in this case the
support condition is vacuous.

Theorem 1.4 has recently found two distinct applications:

  * •

In [2], Theorem 1.4 is combined with a variant of Guth’s polynomial
partitioning method [11] to establish a sharp $(n-1)$-linear restriction
estimate under a curvature hypothesis (see also [16]).

  * •

More recently, in [4], a special case of Theorem 1.4 (under a low regularity
hypothesis) was used to prove $L^{p}\to L^{q}$ estimates for the maximal
function associated to dilates of a helix in $\mathbb{R}^{3}$.

One aim of this note is to give a new proof of Theorem 1.4. For this, we
follow the standard induction-on-scale framework of [6, 9] which perhaps
provides a more contextualised approach than that of [1, 3]. In particular, we
clarify the relationship between (1.5) and Kakeya-type inequalities. In [1,
p.1588] the author raises the question of whether (1.5) has a multilinear
Kakeya counterpart. Here we show that the corresponding geometric estimates
are Kakeya–Brascamp–Lieb inequalities of the type studied in, for instance,
[6, 19] and their discretised/regularised variants, as studied in [14, 20].
Moreover, the new approach leads to our main result (Theorem 1.6), which is a
substantial generalisation of Theorem 1.4, making full use of the
Kakeya–Brascamp–Lieb theory.

###  1.2. Regularised Brascamp–Lieb inequalities

In order to state the main result, we briefly recall some elements of
Brascamp–Lieb theory. For $2\leq k\leq n$ and $1\leq n_{j}\leq n$ for $1\leq
j\leq k$, a Brascamp–Lieb datum is a pair
$(\mathbf{L},\mathbf{p}):=((L_{j})_{j=1}^{k},(p_{j})_{j=1}^{k})$ where
$L_{j}\colon\mathbb{R}^{n}\to\mathbb{R}^{n_{j}}$ are linear surjective maps
and $p_{j}\in(0,1]$. We let
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})$ denote the best constant
$C>0$ for which the inequality

$\displaystyle\int_{\mathbb{R}^{n}}\prod_{j=1}^{k}(f_{j}\circ
L_{j})^{p_{j}}\leq
C\prod_{j=1}^{k}\Big{(}\int_{\mathbb{R}^{n_{j}}}f_{j}\Big{)}^{p_{j}}$ (1.6)

holds for all non-negative functions $f_{j}\in L^{1}(\mathbb{R}^{n_{j}})$ that
are constant on cubes in the unit cube lattice
$\mathcal{Q}^{n_{j}}:=[0,1)^{n_{j}}+\mathbb{Z}^{n_{j}}$. The inequalities
(1.6) are typically referred to as regularised Brascamp–Lieb inequalities and
were first introduced in [8]. Moreover, in [8] it was shown that333Here the
notation $V\leq\mathbb{R}^{n}$ denotes that $V$ is a linear subspace of
$\mathbb{R}^{n}$.

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})<\infty\quad\Longleftrightarrow\quad\dim(V)\leq\sum_{j=1}^{k}p_{j}\dim(L_{j}V)\quad\text{for
all $V\leq\mathbb{R}^{n}$.}$ (1.7)

An advantage of regularised Brascamp–Lieb inequalities is that the
characterisation (1.7) for the finiteness of
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})$ does not involve the
scaling condition $\sum_{j=1}^{k}p_{j}n_{j}=n$, which is necessary in the
standard Brascamp–Lieb theory [7].

###  1.3. Main result

Our main theorem generalises Theorem 1.4 in three directions. First, rather
than just considering hypersurfaces $S_{j}$, we allow the $S_{j}$ to be of
arbitrary dimension. Secondly, rather than considering a single submanifold
$S_{j}^{\prime}\subseteq S_{j}$ for each $1\leq j\leq k$, we consider nested
families. The nested setup naturally arises in recent polynomial partitioning
approaches to multilinear restriction and Kakeya: see [10, 12, 13, 18].
Finally, rather than work with the transversality condition (1.4), we
formulate our transversality hypothesis in terms of regularised Brascamp–Lieb
constants, as defined above. This Brascamp–Lieb formulation greatly relaxes
the constraints on the permissible codimensions.

We first explain the nested setup. Suppose $S_{1}$ is a $d_{1}$-dimensional
submanifold of a $d_{0}$-dimensional submanifold $S_{0}=\Sigma_{0}(U_{0})$ of
$\mathbb{R}^{n}$, as in §1.1. Recall that $S_{1}$ is also a submanifold of
$\mathbb{R}^{n}$ with parametrisation $\Sigma_{1}:=\Sigma_{0}\circ\gamma_{1}$
for some map $\gamma_{1}\colon U_{1}\to U_{0}$, where
$U_{1}\subseteq\mathbb{R}^{d_{1}}$. It therefore makes sense to consider a
$d_{2}$-dimensional submanifold $S_{2}$ of $S_{1}$, which itself is a
submanifold of $\mathbb{R}^{n}$ with parameterisation
$\Sigma_{2}:=\Sigma_{1}\circ\gamma_{2}=\Sigma_{0}\circ\gamma_{1}\circ\gamma_{2}$
for some map $\gamma_{2}\colon U_{2}\to U_{1}$, where
$U_{2}\subseteq\mathbb{R}^{d_{2}}$. Continuing in this way, we can construct a
whole nested sequence of submanifolds.

To describe the setup precisely, we say
$\mathcal{S}=(S_{0},(S_{\ell})_{\ell=1}^{r})$ is a nested family of
submanifolds of $\mathbb{R}^{n}$ if
$S_{0}=\Sigma_{0}(U_{0})\subseteq\mathbb{R}^{n}$ is a submanifold of
$\mathbb{R}^{n}$ and $S_{\ell}$ is a submanifold of $S_{\ell-1}$ for
$1\leq\ell\leq r$. Thus, if $d_{\ell}=\dim(S_{\ell})$ for $1\leq\ell\leq r$,
then we have a family of open neighbourhoods
$U_{\ell}\subseteq\mathbb{R}^{d_{\ell}}$ of the origin and parametrisations
$\gamma_{\ell}\colon U_{\ell}\to U_{\ell-1}$ with $\gamma_{\ell}(U_{\ell})$
compactly contained in $U_{\ell-1}$ and $\gamma_{\ell}(0)=0$ such that

$S_{\ell}=\Sigma_{\ell}(U_{\ell})\quad\textrm{where}\quad\Sigma_{\ell}:=\Sigma_{0}\circ\sigma_{\ell}\quad\textrm{for}\quad\sigma_{\ell}:=\gamma_{1}\circ\cdots\circ\gamma_{\ell}.$
(1.8)

In this case, each set $M_{\ell}:=\sigma_{\ell}(U_{\ell})$ is a
$d_{\ell}$-dimensional submanifold of $\mathbb{R}^{d_{0}}$ and
$S_{\ell}=\Sigma_{0}(M_{\ell})$. Given
$\boldsymbol{\mu}=(\mu_{\ell})_{\ell=1}^{r}\subset(0,1]$, we shall say $f\in
L^{2}(S)$ is supported in $\mathcal{N}_{\boldsymbol{\mu}}\mathcal{S}$, or
$\mathrm{supp}\,f\subseteq\mathcal{N}_{\boldsymbol{\mu}}\mathcal{S}$, if
$\mathrm{supp}\,f\subseteq\mathcal{N}_{\mu_{\ell}}S_{\ell}$ for all
$1\leq\ell\leq r$, in the sense defined above: that is,
$\mathrm{supp}\,f\subseteq\mathcal{N}_{\mu_{\ell}}M_{\ell}$, where
$\mathcal{N}_{\mu_{\ell}}M_{\ell}\subseteq\mathbb{R}^{d}$ is the (open
Euclidean) $\mu_{\ell}$-neighbourhood of $M_{\ell}$. We also assume that
$d_{r}<\cdots<d_{1}<d_{0}$ since this is the configuration of interest for our
main theorem; in particular, $r\leq n$.

######  Definition 1.5.

Fix $2\leq k\leq n$ and suppose
$\mathscr{E}=((\mathcal{S}_{j})_{j=1}^{k},\mathbf{q})$ where the
$\mathcal{S}_{j}=(S_{j,0},(S_{j,\ell})_{\ell=1}^{r_{j}})$ are nested families
of submanifolds and $\mathbf{q}=(q_{j})_{j=1}^{k}\subset(0,2]$ is a sequence
of exponents. Define the Brascamp–Lieb datum

$(\mathbf{L}(\mathscr{E}),\mathbf{p})=((L_{j}(\mathscr{E}))_{j=1}^{k},(p_{j})_{j=1}^{k})$
(1.9)

by setting

$L_{j}(\mathscr{E}):=\partial_{s}\Sigma_{j,r_{j}}(0)^{\top}\colon\mathbb{R}^{n}\to\mathbb{R}^{d_{j,r_{j}}}\quad\textrm{and}\quad
p_{j}:=q_{j}/2\quad\textrm{for $1\leq j\leq k$,}$

where $d_{j,r_{j}}:=\dim(S_{j,r_{j}})$ and
$S_{j,r_{j}}=\Sigma_{j,r_{j}}(U_{j,r_{j}})$, with $\Sigma_{j,r_{j}}$
parametrisations constructed as in (1.8).

  1. a)

We say $\mathscr{E}$ is a transverse ensemble in $\mathbb{R}^{n}$ if

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p})<\infty.$

  2. b)

We say $\mathscr{E}$ has codimension $(m(j,\ell))_{j,\ell}$, where
$m(j,\ell):=\mathrm{codim}(S_{j,\ell}\,|\,S_{j,\ell-1})$ for $1\leq\ell\leq
r_{j}$ and $1\leq j\leq k$.

  3. c)

Finally, we say $(\boldsymbol{\mu}_{j})_{j=1}^{k}$ where
$\boldsymbol{\mu}_{j}=(\mu_{j,\ell})_{\ell=1}^{r_{j}}\subset(0,1]$ is a
compatible family of scales for $\mathscr{E}$ if
$0<\mu_{j,1}\leq\dots\leq\mu_{j,r_{j}}<1$ for $1\leq j\leq k$.

With the above definitions, the main theorem reads as follows.

######  Theorem 1.6.

Let $2\leq k\leq n$ and $\mathscr{E}=((\mathcal{S}_{j})_{j=1}^{k},\mathbf{q})$
be a transverse ensemble in $\mathbb{R}^{n}$ of codimension
$(m(j,\ell))_{j,\ell}$, with
$\mathcal{S}_{j}=(S_{j},(S_{j,\ell})_{\ell=1}^{r_{j}})$. There exists
$\rho_{\mathscr{E}}>0$ such that for
$\max_{j}\rho(E_{S_{j}})<\rho_{\mathscr{E}}$ the following holds. For all
$\varepsilon>0$, there exists a constant $C_{\varepsilon}\geq 1$ such that

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j}}f_{j}|^{q_{j}}\leq
C_{\varepsilon}R^{\varepsilon}\prod_{j=1}^{k}\prod_{\ell=1}^{r_{j}}\mu_{j,\ell}^{m(j,\ell)q_{j}/2}\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(S_{j})}^{q_{j}},$

for all $R\geq 1$ and all $f_{j}\in L^{2}(S_{j})$ satisfying
$\mathrm{supp}\,f_{j}\subseteq\mathcal{N}_{\boldsymbol{\mu}_{j}}\mathcal{S}_{j}$
for $(\boldsymbol{\mu}_{j})_{j=1}^{k}$ a compatible family of scales for
$\mathscr{E}$.

Recall that we are tacitly working with a fixed choice of parametrisations for
the $S_{j}$ and $S_{j,\ell}$, and the constant $C_{\varepsilon}$ will depend
(amongst other things) on this choice. If $\Sigma_{j}\colon
U_{j}\to\mathbb{R}^{n}$ and $\Sigma_{j,r_{j}}\colon
U_{j,r_{j}}\to\mathbb{R}^{n}$ are our fixed parametrisations for $S_{j}$ and
$S_{j,r_{j}}$ as submanifolds of $\mathbb{R}^{n}$ then, as discussed in (1.8)
and (1.3), we always assume
$\Sigma_{j,r_{j}}:=\Sigma_{j}\circ\gamma_{j,1}\circ\cdots\circ\gamma_{j,r_{j}}$
for intermediate parametrisations $\gamma_{j,\ell}$, and
$\Sigma_{j,r_{j}}(0)=\Sigma_{j}(0)$.

######  Remark 1.7.

To contextualise the result, we make the following remarks.

a) Let $2\leq k<n$. A prototypical application of our theorem is a $k$-linear
estimate for the $n$-dimensional paraboloid in which one of the functions is
localised to an anisotropic box. More precisely, let $U:=(-1,1)^{n-1}$ and
$S:=\Sigma(U)$, where $\Sigma(u):=(u,|u|^{2})$. Let $0<\delta\ll 1$ and
$B_{\delta}\subseteq\mathbb{R}^{n-1}$ be the $1\times\cdots\times
1\times\delta\times\delta^{2}\times\cdots\times\delta^{n-k}$ box centered at
the origin pointing along the directions of the standard basis in
$\mathbb{R}^{n-1}$. Let $u_{1},\dots,u_{k-1}\in U$ satisfy

$\Big{|}\bigwedge_{j=1}^{k-1}\frac{(2u_{j},-1)}{|(2u_{j},-1)|}\wedge
e_{k}\wedge\cdots\wedge e_{n}\Big{|}\geq 1/2.$ (1.10)

If $U_{1},\dots U_{k-1}$ are sufficiently small neighbourhoods of the points
$u_{1},\dots,u_{k-1}$ and $U_{k}$ is a sufficiently small neighbourhood of the
origin, then for all $\varepsilon>0$ there exists a constant
$C_{\varepsilon}\geq 1$ such that

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S}f_{j}|^{\frac{2}{k-1}}\leq
C_{\varepsilon}R^{\varepsilon}\delta^{\frac{(n-k)(n-k+1)}{2(k-1)}}\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(U_{j})}^{\frac{2}{k-1}}$
(1.11)

for all $R\geq 1$ and all $f_{j}\in L^{2}(U_{j})$ for $1\leq j\leq k$ with
$\mathrm{supp}\,f_{k}\subseteq B_{\delta}$.

To see how this follows from Theorem 1.6, define
$U_{k,\ell}:=(-1,1)^{n-1-\ell}$ and $\gamma_{k,\ell}:U_{k,\ell}\to
U_{k,\ell-1}$ by $\gamma_{k,\ell}(s):=(s,0)$ for $1\leq\ell\leq n-k$, with
$U_{k,0}:=U$. It is then clear that
$\mathrm{supp}\,f_{k}\subseteq\mathcal{N}_{\boldsymbol{\mu}_{k}}\mathcal{S}_{k}$,
where $\boldsymbol{\mu}_{k}=(\delta^{n-k+1-\ell})_{\ell=1}^{n-k}$ and
$\mathcal{S}_{k}=(S,(S_{k,\ell})_{\ell=1}^{n-k})$ for
$S_{k,\ell}=\Sigma_{k,\ell}(U_{k,\ell})$ and
$\Sigma_{k,\ell}:=\Sigma\circ\gamma_{k,1}\circ\cdots\circ\gamma_{k,\ell}$. For
all $1\leq j\leq k-1$, one can centre $U_{j}$ at the origin by a change of
variables, giving rise to a neighbourhood $U_{j}^{*}$ and an extension
operator $E_{S_{j}}$ with $S_{j}=\Sigma_{j}(U_{j}^{*})$ and
$\Sigma_{j}(u)=(u+u_{j},|u+u_{j}|^{2})$. Considering the ensemble
$\mathscr{E}=((\mathcal{S}_{j})_{j=1}^{k},\mathbf{q}_{k})$, where
$\mathcal{S}_{j}=(S_{j})$ for $1\leq j\leq k-1$ and
$\mathbf{q}_{k}=(q_{k})_{j=1}^{k}$ for $q_{k}=\frac{2}{k-1}$, we have that
$\ker L_{j}(\mathscr{E})=(2u_{j},-1)^{\top}$ for all $1\leq j\leq k-1$ and
$\ker L_{k}(\mathscr{E})=\mathrm{span}\\{e_{k},\dots,e_{n}\\}$. One can show
that (1.10) implies that the condition (1.7) holds with $p_{j}=\frac{1}{k-1}$
(see, for instance, Corollary 2.3), and thus $\mathscr{E}$ is a transverse
ensemble. Consequently, the claimed multilinear inequality (1.11) follows from
Theorem 1.6.

b) Taking $\mathcal{S}_{j}=(S_{j},S_{j}^{\prime})$ for
$S_{j}\subset\mathbb{R}^{n}$ a hypersurface and $S_{j}^{\prime}$ a submanifold
of $S_{j}$, for $1\leq j\leq k$, and $\mathbf{p}_{k}=(p_{k})_{j=1}^{k}$ for
$p_{k}=\tfrac{1}{k-1}$, we recover the localised multilinear estimates of
Theorem 1.4. Indeed, this follows from the observation

$\Big{|}\bigwedge_{j=1}^{k}N_{\xi_{j}^{\prime}}S_{j}^{\prime}\Big{|}>0\quad\iff\quad\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p}_{k})<\infty,$

where the $\xi_{j}^{\prime}$ are as defined in (1.4): see §2 for details.

c) Suppose $r_{j}=0$ for all $1\leq j\leq k$, so that
$\mathcal{S}_{j}=(S_{j})$ and there is no nesting. If we further suppose that
the scaling condition $\sum_{j=1}^{k}p_{j}\dim(S_{j})=n$ is satisfied, then
Theorem 1.6 amounts to the Restriction–Brascamp–Lieb estimates of [6, Theorem
1.3]. If we drop the scaling condition, then the corresponding estimates are
not documented in the literature, but can be deduced from the
Kakeya–Brascamp–Lieb estimates in [14, 20] (see Theorem 2.6 below) by the
argument in [6, 9].

d) As discussed in §2.3, the hypothesis
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p})<\infty$ of
Theorem 1.6 implies

$\mathrm{BL}_{\mathrm{reg}}((\partial_{s}\Sigma_{j}(0)^{\top})_{j=1}^{k},\mathbf{p})<\infty.$
(1.12)

By the argument of [6] (cf. item c) above), the condition (1.12) implies the
multilinear estimates

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j}}f_{j}|^{q_{j}}\leq
C_{\varepsilon}R^{\varepsilon}\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(S_{j})}^{q_{j}}.$

The relevance of our theorem is therefore the improvement given by the factor
$\prod_{j=1}^{k}\prod_{\ell=1}^{r_{j}}\mu_{j,\ell}^{m(j,\ell)q_{j}/2}$ under
the hypotheses
$\mathrm{supp}\,f_{j}\subseteq\mathcal{N}_{\boldsymbol{\mu}_{j}}\mathcal{S}_{j}$.

###  1.4. Proof sketch

We describe the main ideas in the simpler setting of Theorem 1.4. We remark
that, due to the local nature of the desired estimates, we may assume without
loss of generality that $\max_{j}\mu_{j}<\mu_{\mathscr{E}}$ for some
$\mu_{\mathscr{E}}>0$ depending only on $\mathscr{E}$. Under the hypotheses of
Theorem 1.4, it is possible to bound

$|E_{S_{j}}f_{j}(x)|\leq\int_{[-\mu_{j},\mu_{j}]^{m}}|E_{S_{j}^{\prime}(\boldsymbol{\eta}_{j})}f_{j,\boldsymbol{\eta}_{j}}(x)|\,\mathrm{d}\boldsymbol{\eta}_{j}$
(1.13)

where each $E_{S_{j}^{\prime}(\boldsymbol{\eta}_{j})}$ is an extension
operator associated to a submanifold $S_{j}^{\prime}(\boldsymbol{\eta}_{j})$
of $S_{j}$ of codimension $m_{j}$: see §3.2 for details. Moreover, (1.4) and
continuity imply that, for $\mu_{j}$ sufficiently small, the submanifolds
$(S_{j}^{\prime}(\boldsymbol{\eta}_{j}))_{\ell=1}^{k}$ satisfy certain
transversality conditions uniformly over all choices of
$\boldsymbol{\eta}_{j}\in[-\mu_{j},\mu_{j}]^{m_{j}}$. For each index $j$, we
plug the bound (1.13) into the left-hand side of (1.5). If one were then able
to switch the order of the $L^{q_{k}}_{x}$-norm and the
$L^{1}_{\boldsymbol{\eta}_{j}}$-norms, for instance using Minkowski’s integral
inequality, then the multilinear restriction estimates of [6] (or certain
regularised variants) would apply directly to give, together with an
application of the Cauchy–Schwarz inequality, Theorem 1.4. However,
Minkowski’s integral inequality requires $q_{k}\geq 1$, which is only valid
for $k=2$ or $k=3$. See [4, Appendix A] for an example of this strategy in the
$k=3$ case.

A similar issue with the exponent $q_{k}<1$ arises when passing between the
equivalent ‘neighbourhood’ and ‘extension’ formulations of the
Bennett–Carbery–Tao restriction theorem in [9]. One strategy to deal with this
is to use a vector-valued extension of the multilinear restriction estimates,
derived from the Marcinkiewicz–Zygmund theorem: see [17, Appendix] for
details. The present situation is somewhat more complicated than that in [9,
17], since the operators $E_{S_{j}^{\prime}(\boldsymbol{\eta}_{j})}$ typically
have a non-trivial dependence on $\boldsymbol{\eta}_{j}$ (in the context of
[9, 17], the analogous extension operators are independent of
$\boldsymbol{\eta}_{j}$). This means that the Marcinkiewicz–Zygmund theorem
cannot be applied directly to obtain the desired vector-valued extension.

The above strategy does work, however, when $\mu_{j}<R^{-1}$. Since the
$x$-integration is localised to $Q_{R}$, in this case uncertainty principle
considerations ensure that
$E_{S_{j}^{\prime}(\boldsymbol{\eta}_{j})}f_{j,\boldsymbol{\eta}_{j}}$ is
essentially constant in $\boldsymbol{\eta}_{j}\in[-\mu_{j},\mu_{j}]^{m_{j}}$.
This allows for an application of the Marcinkiewicz–Zygmund theorem as in [17,
Appendix]: see Lemma 4.1 for details. On the other hand, when $\mu_{j}\geq
R^{-1}$ it is possible to apply Kakeya–Brascamp–Lieb inequalities to reduce
the scale $R$, following an induction-on-scale procedure similar to that used
in [9, 6] (but adapted to take into account the localisation of the supports
of the $f_{j}$). Combining these two observations, one may formulate a refined
induction-on-scale procedure, which takes into account the relative sizes of
the $\mu_{j}$ and $R^{-1}$: see Lemma 4.2. As in [6, 9], the induction-on-
scale procedure yields the desired estimate.

###  1.5. Notational conventions

We shall say a constant is admissible if it depends either only on a choice of
dimension $n$ or a choice of ensemble $\mathscr{E}$. In particular, any
admissible constant is independent of parameters such as $R\geq 1$ or the
$\mu_{j,\ell}>0$. We will frequently let $C$ denote a choice of admissible
constant, whose value may change from line to line. Given a list of objects
$L$ and real numbers $A$, $B\geq 0$, 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 $L$ and either a choice of dimension $n$ or a
choice of ensemble $\mathscr{E}$. We write $A\sim_{L}B$ to indicate
$A\lesssim_{L}B$ and $B\lesssim_{L}A$. Given $d\in\mathbb{N}$ we let $I_{d}$
denote the $d\times d$ identity matrix.

##  2\. Kakeya–Brascamp–Lieb theory

###  2.1. Regularised Brascamp–Lieb revisited

Let $2\leq k\leq n$, $1\leq n_{j}\leq n$ for $1\leq j\leq k$ and
$(\mathbf{L},\mathbf{p})$ a Brascamp–Lieb datum as in §1.2. Given $R\geq 1$
and $0\leq\lambda<R$ dyadic, we let
$\mathrm{BL}_{\lambda,R}(\mathbf{L},\mathbf{p})$ denote the best constant
$C>0$ for which the associated generalised Brascamp–Lieb inequality

$\displaystyle\int_{Q_{R}}\prod_{j=1}^{k}(f_{j}\circ L_{j})^{p_{j}}\leq
C\prod_{j=1}^{k}\Big{(}\int_{\mathbb{R}^{n_{j}}}f_{j}\Big{)}^{p_{j}}$ (2.1)

holds for all non-negative functions $f_{j}\in L^{1}(\mathbb{R}^{n_{j}})$ that
are constant on cubes of sidelength $\lambda$ in the fixed dyadic grid
$[0,\lambda)^{n_{j}}+\lambda\mathbb{Z}^{n_{j}}$. Inequalities of this form
were first introduced in [7] and can be seen as a generalisation of the
classical scale-invariant Brascamp–Lieb inequalities, which correspond to
taking $\lambda\to 0$, $R\to\infty$. Regarding the latter, we let
$\mathrm{BL}(\mathbf{L},\mathbf{p})$ denote the classical Brascamp–Lieb
constant, which is the best constant $C>0$ for which (2.1) holds for all
$R\geq 1$ and all non-negative functions $f_{j}\in L^{1}(\mathbb{R}^{n_{j}})$
(without any local constancy hypothesis).

The case $\lambda=1$ of the above is often referred to as regularised.444Note
that if $0<\lambda\leq R<\infty$, one can always reduce to the case
$\lambda=1$ by the scaling relation
$\mathrm{BL}_{\lambda,R}(\mathbf{L},\mathbf{p})=\lambda^{n-\sum_{j=1}^{k}p_{j}n_{j}}\mathrm{BL}_{1,R\lambda^{-1}}(\mathbf{L},\mathbf{p}).$
Maldague [14, Theorem 1.1] recently showed that

$\mathrm{BL}_{1,R}(\mathbf{L},\mathbf{p})\sim_{\mathbf{L},\mathbf{p}}R^{\alpha(\mathbf{L},\mathbf{p})}\quad\textrm{for}\quad\alpha(\mathbf{L},\mathbf{p}):=\sup_{V\leq\mathbb{R}^{n}}\Big{(}\dim(V)-\sum_{j=1}^{k}p_{j}\dim(L_{j}V)\Big{)}.$
(2.2)

Note that the constant $\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})$
introduced in §1.2 satisfies

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})=\sup_{R>1}\mathrm{BL}_{1,R}(\mathbf{L},\mathbf{p});$

in particular, one can recover (1.7) from the characterisation (2.2) for
$\mathrm{BL}_{1,R}(\mathbf{L},\mathbf{p})$. We remark that the classical
Brascamp–Lieb constant $\mathrm{BL}(\mathbf{L},\mathbf{p})$ is finite if and
only if $\alpha(\mathbf{L},\mathbf{p})=0$ and $\sum_{j=1}^{k}p_{j}n_{j}=n$:
see [7]. In this case, we in fact have

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})=\mathrm{BL}(\mathbf{L},\mathbf{p})$

as a consequence of a simple limiting argument.

###  2.2. Regularised Brascamp–Lieb and transversality conditions

Here we compare the hypotheses of Theorem 1.6 and Theorem 1.4 and, in
particular, address the comments made in Remark 1.7 b).

######  Proposition 2.1.

Let $\mathbf{p}_{k}=(p_{k})_{j=1}^{k}$ where $p_{k}:=q_{k}/2=\frac{1}{k-1}$
and suppose that $L_{j}\colon\mathbb{R}^{n}\to\mathbb{R}^{n_{j}}$, $1\leq
j\leq k$, are linear surjections. Then the following are equivalent:

  1. a)

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p}_{k})<\infty$;

  2. b)

$\displaystyle\Big{|}\bigwedge_{j=1}^{k}\ker(L_{j})\Big{|}>0$.

###### Proof.

For $V\leq\mathbb{R}^{n}$, we first note that

$\dim(L_{j}(V))=\dim(V)-\dim(V\cap\ker(L_{j})),$ (2.3)

by the rank–nullity theorem applied to the restriction $L_{j}|_{V}$, $1\leq
j\leq k$. We define the subspace $W:=\ker(L_{1})+\cdots+\ker(L_{k})$.

a) $\Rightarrow$ b). Assume b) fails. Applying (2.3) to $W$ as defined above,
we deduce that

$\dim(L_{j}(W))=\dim(W)-\dim(\ker(L_{j})),$ (2.4)

since $\ker(L_{j})\leq W$ for all $1\leq j\leq k$. The failure of b) implies
that $\dim(W)<\sum_{j=1}^{k}\dim(\ker(L_{j}))$ and applying this to (2.4)
yields

$\sum_{j=1}^{k}\dim(L_{j}(W))=k\dim(W)-\sum_{j=1}^{k}\dim(\ker(L_{j}))<(k-1)\dim(W).$

This means that $W$ violates the finiteness characterisation (1.7) for the
datum $(\mathbf{L},\mathbf{p}_{k})$, implying that a) fails.

b) $\Rightarrow$ a). Assume that b) holds, so that
$W=\bigoplus_{j=1}^{k}\ker(L_{j})$. Given $V\leq\mathbb{R}^{n}$, we have
$\bigoplus_{j=1}^{k}(V\cap\ker(L_{j}))\leq V\cap W$ and therefore

$\sum_{j=1}^{k}\dim(V\cap\ker(L_{j}))=\dim\bigg{(}\bigoplus_{j=1}^{k}V\cap\ker(L_{j})\bigg{)}\leq\dim(V\cap
W).$

Combining the above inequality with (2.3), we obtain

$\displaystyle\sum_{j=1}^{k}\dim(L_{j}V)$
$\displaystyle=\sum_{j=1}^{k}(\dim(V)-\dim(V\cap\ker(L_{j}))$
$\displaystyle\geq k\dim(V)-\dim(V\cap W)$ $\displaystyle\geq(k-1)\dim(V).$

Since $V\leq\mathbb{R}^{n}$ was arbitrary, the characterisation (1.7) implies
that a) holds. ∎

######  Remark 2.2.

Suppose the maps $L_{j}$ are orthogonal projections. By invoking [5,
Proposition 1.2], one may deduce that

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p}_{k})\sim\Big{|}\bigwedge_{j=1}^{k}\ker(L_{j})\Big{|}^{-\frac{1}{k-1}},$

which is a quantitative variant of Proposition 2.1. Since in this paper we
only need finiteness of such transversality quantites, and not uniform
boundedness, we leave the proof of this more quantitative result to the
interested reader.

######  Corollary 2.3.

Let $2\leq k\leq n$,
$\mathscr{E}=((S_{j},S_{j}^{\prime})_{j=1}^{k},\mathbf{q}_{k})$ be a
transverse ensemble in $\mathbb{R}^{n}$. Then

$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p}_{k})<\infty\quad\Longleftrightarrow\quad\Big{|}\bigwedge_{j=1}^{k}N_{\xi_{j}^{\prime}}S_{j}^{\prime}\Big{|}>0.$
(2.5)

Here the $\xi_{j}^{\prime}$ are as defined in (1.4). In particular, Theorem
1.6 implies Theorem 1.4.

###### Proof.

For each $1\leq j\leq k$, let $\Sigma_{j}\colon
U_{j}\rightarrow\mathbb{R}^{n}$ be the parametrisation of $S_{j}$, and let
$M_{j}\subset U_{j}$ be a submanifold of $\mathbb{R}^{\dim(S_{j})}$,
parametrised by $\gamma_{j}:U_{j}^{\prime}\rightarrow U_{j}$, such that
$\Sigma_{j}(M_{j})=S_{j}^{\prime}$. Define
$\Sigma_{j}^{\prime}:=\Sigma_{j}\circ\gamma_{j}:U_{j}^{\prime}\rightarrow\mathbb{R}^{n}$.
Then, by the definition of $\mathbf{L}(\mathscr{E})$ we have that
$L_{j}(\mathscr{E})=\partial_{s}\Sigma_{j}^{\prime}(0)^{\top}$. Since
$\ker(\partial_{s}\Sigma_{j}^{\prime}(0)^{\top})=N_{\xi_{j}^{\prime}}S_{j}^{\prime}$,
we then see that the equivalence (2.5) is merely a rephrasing of Proposition
2.1 for this particular choice of Brascamp–Lieb datum. ∎

###  2.3. Brascamp–Lieb constants and subensembles

Fix $2\leq k\leq n$ and let
$\mathscr{E}=((\mathcal{S}_{j})_{j=1}^{k},\mathbf{q})$ be a transverse
ensemble in the sense of Definition 1.5, with
$\mathcal{S}_{j}=(S_{j},(S_{j,\ell})_{\ell=1}^{r_{j}})$ and
$\mathbf{q}=(q_{j})_{j=1}^{k}\subset(0,2]$. We say that
$\mathscr{F}=((\widetilde{\mathcal{S}}_{j})_{j=1}^{k},\mathbf{q})$ is a
subensemble of $\mathscr{E}$ if
$\widetilde{\mathcal{S}}_{j}=(S_{j},(S_{j,\ell})_{\ell=1}^{\tilde{r}_{j}})$
with $\tilde{r}_{j}\leq r_{j}$. Recall we allow values $\tilde{r}_{j}=0$, in
which case $\widetilde{\mathcal{S}}_{j}$ consists only of $S_{j}$.

######  Proposition 2.4.

Let $\mathscr{E}$ be a transverse ensemble in $\mathbb{R}^{n}$. Then any
subensemble $\mathscr{F}$ of $\mathscr{E}$ is also a transverse ensemble.

###### Proof.

It suffices to verify that
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{F}),\mathbf{p})<\infty$ with
$\mathbf{L}(\mathscr{F})$ as in (1.9). In view of (1.8), for each $1\leq j\leq
k$ we have that

$\Sigma_{j,r_{j}}=\Sigma_{j,\tilde{r}_{j}}\circ\gamma_{j,\tilde{r}_{j},r_{j}}\quad\text{
for
}\,\,\gamma_{j,\tilde{r}_{j},r_{j}}:=\gamma_{j,\tilde{r}_{j}+1}\circ\cdots\circ\gamma_{j,r_{j}};$

here, for the case $\tilde{r}_{j}=r_{j}$, we simply have
$\gamma_{j,r_{j},r_{j}}:=\mathrm{Id}$. Thus,

$L_{j}(\mathscr{E})=\partial_{s}\Sigma_{j,r_{j}}(0)^{\top}=(\partial_{s}\gamma_{j,\tilde{r}_{j},r_{j}}(0))^{\top}(\partial_{\tilde{s}}\Sigma_{j,\tilde{r}_{j}}(0))^{\top}=(\partial_{s}\gamma_{j,\tilde{r}_{j},r_{j}}(0))^{\top}L_{j}(\mathscr{F})$

where we have used that $\gamma_{j,\ell}(0)=0$ for all $1\leq\ell\leq r_{j}$.
It is then clear that $\dim(L_{j}(\mathscr{E})V)\leq\dim(L_{j}(\mathscr{F})V)$
for any $V\leq\mathbb{R}^{n}$ and, in view of the characterisation (1.7) for
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})$, we conclude that the
finiteness of $\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p})$
implies that of
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{F}),\mathbf{p})$. ∎

######  Remark 2.5.

Similarly to Remark 2.2, if the linear surjections $L_{j}(\mathscr{E})$ and
$L_{j}(\mathscr{F})$ are orthogonal projections for $1\leq j\leq k$, then one
can show that
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{F}),\mathbf{p})\lesssim\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p})$,
where the implicit constant does not depend on the choice of ensemble. This
refined quantitative result is not required for our purposes, and so we leave
the proof to the interested reader.

###  2.4. Kakeya–Brascamp–Lieb inequalities

Here we are interested in a class of Kakeya–Brascamp–Lieb inequalities, which
can be thought of as perturbed variants of the regularised Brascamp–Lieb
inequalities introduced above. The following result is due to Maldague [14].

######  Theorem 2.6 (Kakeya–Brascamp–Lieb [14, Theorem 1.2]).

Let $(\mathbf{L},\mathbf{p})$ be a Brascamp–Lieb datum such that
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})<\infty$ . Then there exists
$\nu>0$ such that the following holds. For all $\varepsilon>0$, there exists a
constant $C_{\varepsilon}\geq 1$ such that the inequality

$\int_{Q_{R}}\prod_{j=1}^{k}\Big{|}\sum_{T_{j}\in\mathbb{T}_{j}}c_{T_{j}}\chi_{T_{j}}\Big{|}^{p_{j}}\leq
C_{\varepsilon}R^{\varepsilon}\lambda^{n-\varepsilon}\prod_{j=1}^{k}\Big{(}\sum_{T_{j}\in\mathbb{T}_{j}}|c_{T_{j}}|\Big{)}^{p_{j}}$

holds for all $R\geq 1$ and $0<\lambda<R$, whenever the $\mathbb{T}_{j}$ are
countable collections of infinite slabs of width $\lambda$ whose core
$n_{j}$-dimensional planes are, modulo translations, within a distance $\nu>0$
(in the grassmannian sense) from the fixed subspace $X_{j}:=\ker L_{j}$, and
$(c_{T_{j}})_{T_{j}\in\mathbb{T}_{j}}\in\ell^{1}(\mathbb{T}_{j})$ for $1\leq
j\leq k$.

Here we say $T\subseteq\mathbb{R}^{n}$ is an infinite slab of width $r>0$
if555Here and below we let $|x|_{\infty}:=\max_{1\leq\ell\leq m}|x_{\ell}|$
for $x=(x_{1},\dots,x_{m})\in\mathbb{R}^{m}$.

$T=\\{x\in\mathbb{R}^{n}:|Lx-v|_{\infty}<r\\}$

for some linear surjective map $L\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ and
$v\in\mathbb{R}^{m}$. In this case, the affine subspace
$V:=\\{x\in\mathbb{R}^{n}:Lx=v\\}$ is called the core plane of $T$. We note
that endpoint variants of Theorem 2.6 can be found in [19, 20]. The case in
which $\dim\ker L_{j}=1$ for all $1\leq j\leq k$ corresponds to the $k$-linear
Kakeya inequality from [9, Theorem 5.1].

We remark that there are some differences between the statement of Theorem 2.6
and that of [14, Theorem 1.2].

  * •

Theorem 2.6 is a rescaled version of [14, Theorem 1.2]. The parameter $\delta$
in [14, Theorem 1.2] corresponds to $\lambda/R$ in Theorem 2.6.

  * •

The coefficients $(c_{T_{j}})_{T_{j}\in\mathbb{T}_{j}}$ are taken to all be
$1$ in [14, Theorem 1.2]. However, the result for general coefficients follows
from a standard approximation argument: see [9].

  * •

Theorem 2.6 is stated in terms of
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})$ whilst [14, Theorem 1.2]
is stated in terms of the exponent

$\tilde{\alpha}(\mathbf{L},\mathbf{p}):=\sup_{V\leq\mathbb{R}^{n}}\Big{(}\dim(V)-\sum_{j=1}^{k}p_{j}\dim(\pi_{j}V)\Big{)}$

where $\pi_{j}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the orthogonal
projection onto the subspaces $(X_{j})^{\perp}$ for $1\leq j\leq n$. To relate
the former from the latter, one observes that the hypothesis
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L},\mathbf{p})<\infty$ implies
$\tilde{\alpha}(\mathbf{L},\mathbf{p})=0$. Indeed, $\ker\pi_{j}=X_{j}=\ker
L_{j}$ and, given any $V\leq\mathbb{R}^{n}$, it follows from rank-nullity as
in (2.3) that $\dim\pi_{j}(V)=\dim L_{j}(V)$ for all $1\leq j\leq n$.
Consequently,
$\tilde{\alpha}(\mathbf{L},\mathbf{p})=\alpha(\mathbf{L},\mathbf{p})$ and the
desired implication follows from the characterisation (2.2).

##  3\. Fourier extension preliminaries

###  3.1. Parametrising the neighbourhoods

Let $\mathcal{S}=(S_{0},(S_{\ell})_{\ell=1}^{r})$ be a nested family of
submanifolds of $\mathbb{R}^{n}$, with $S_{0}=\Sigma_{0}(U_{0})$ a
$d_{0}$-dimensional submanifold of $\mathbb{R}^{n}$. As in §1.3, we have a
family of open neighbourhoods $U_{\ell}\subseteq\mathbb{R}^{d_{\ell}}$ of the
origin, with $d_{\ell}=\dim(S_{\ell})$ satisfying $d_{r}<\cdots<d_{1}<d_{0}$,
and parametrisations $\gamma_{\ell}\colon U_{\ell}\to U_{\ell-1}$ for
$1\leq\ell\leq r$ satisfying $\gamma_{\ell}(0)=0$ such that

$S_{\ell}=\Sigma_{\ell}(U_{\ell})\quad\textrm{where}\quad\Sigma_{\ell}:=\Sigma_{0}\circ\sigma_{\ell}\quad\textrm{for}\quad\sigma_{\ell}:=\gamma_{1}\circ\cdots\circ\gamma_{\ell}.$

Each set $M_{\ell}:=\sigma_{\ell}(U_{\ell})$ is a $d_{\ell}$-dimensional
submanifold of $\mathbb{R}^{d_{0}}$ and $S_{\ell}=\Sigma_{0}(M_{\ell})$. We
define

$m_{\ell}:=\mathrm{codim}(S_{\ell}\,|\,S_{\ell-1})=d_{\ell-1}-d_{\ell}\qquad\textrm{and}\qquad
c_{\ell}:=\mathrm{codim}(S_{\ell}\,|\,S_{0})=d_{0}-d_{\ell}$

for $1\leq\ell\leq r$. Often we drop the subscript $0$ and write $S:=S_{0}$,
$d:=d_{0}$, $\Sigma:=\Sigma_{0}$ and $U:=U_{0}$.

We begin with some (unfortunately rather technical) definitions which allow us
to work inductively with the nested framework of Theorem 1.6. With the above
setup, we may define

$M_{k,\ell}:=\gamma_{k,\ell}(U_{\ell})\subseteq
U_{k}\quad\textrm{where}\quad\gamma_{k,\ell}:=\gamma_{k+1}\circ\cdots\circ\gamma_{\ell}\qquad\textrm{for
$k+1\leq\ell\leq r$,}$

so that each $M_{k,\ell}$ is a smooth $d_{\ell}$-dimensional submanifold of
$\mathbb{R}^{d_{k}}$. Recall that the support conditions in Theorem 1.6 are
defined in terms of neighbourhoods of the submanifolds $M_{\ell}$, and note
that $M_{\ell}=M_{0,\ell}$ with the above notation. In order to argue
inductively, we shall work more generally with neighbourhoods of the
$M_{k,\ell}$ for $0\leq k<\ell$.

To describe the neighbourhoods of the $M_{k,\ell}$, we introduce a family of
parametrising maps $\Phi_{k,\ell}$. Given $1\leq\ell\leq r$, we let
$G_{\ell}\colon U_{\ell}\rightarrow\mathrm{Mat}(\mathbb{R},d_{\ell-1}\times
m_{\ell})$ denote a $(d_{\ell-1}\times m_{\ell})$-matrix-valued function such
that the columns of $G_{\ell}(s_{\ell})$ form an orthogonal basis for
$N_{\gamma_{\ell}(s_{\ell})}M_{\ell-1,\ell}\subset U_{\ell-1}$. Let
$\bar{\mu}_{\circ}>0$ be an admissible parameter, chosen sufficiently small
for the purposes of the forthcoming argument. Given $1\leq k<\ell\leq r$,
throughout the following we write

$V_{k,\ell}:=\prod_{t=k+1}^{\ell}(-\bar{\mu}_{\circ},\bar{\mu}_{\circ})^{m_{t}}\quad\textrm{and}\quad\boldsymbol{\eta}_{k,\ell}=(\eta_{k+1},\dots,\eta_{\ell})\quad\textrm{for
$\boldsymbol{\eta}_{k,\ell}\in V_{k,\ell}$.}$

For fixed $1\leq\ell\leq r$, we recursively define maps $\Phi_{k,\ell}\colon
U_{\ell}\times V_{k,\ell}\rightarrow U_{k}$ for $0\leq k\leq\ell$, where we
interpret $U_{\ell}\times V_{\ell,\ell}$ as $U_{\ell}$, by setting
$\Phi_{\ell,\ell}\colon U_{\ell}\to U_{\ell}$ to be the identity map and then
defining

$\Phi_{k-1,\ell}(s_{\ell};\boldsymbol{\eta}_{k-1,\ell}):=\gamma_{k}\circ\Phi_{k,\ell}(s_{\ell};\boldsymbol{\eta}_{k,\ell})+G_{k}\circ\Phi_{k,\ell}(s_{\ell};\boldsymbol{\eta}_{k,\ell})\eta_{k}^{\top}$
(3.1)

for $1\leq k\leq\ell$, where here
$\boldsymbol{\eta}_{k-1,\ell}=(\eta_{k},\boldsymbol{\eta}_{k,\ell})$ and
$\boldsymbol{\eta}_{\ell,\ell}$ is a null variable. Since each set
$\gamma_{k}(U_{k})$ is, by hypothesis, compactly contained in $U_{k-1}$,
provided $\bar{\mu}_{\circ}>0$ is chosen sufficiently small, the maps
$\Phi_{k,\ell}$ are well-defined and indeed map into $U_{k}$. By a simple
induction argument,

$\Phi_{k,\ell}(s_{\ell};\mathbf{0})=\gamma_{k,\ell}(s_{\ell}),$ (3.2)

where $\gamma_{\ell,\ell}:=\mathrm{Id}_{U_{\ell}}$.

We may equivalently write (3.1) as

$\Phi_{k-1,\ell}(s_{\ell};\boldsymbol{\eta}_{k-1,\ell})=\Phi_{k-1,k}(\Phi_{k,\ell}(s_{\ell};\boldsymbol{\eta}_{k,\ell});\eta_{k}).$
(3.3)

Iterating the above observation, we shall derive the following formula.

######  Lemma 3.1.

For all $0\leq k\leq\ell\leq r$, $s_{r}\in W_{r}$, and
$\boldsymbol{\eta}_{k,r}\in V_{k,r}$ we have

$\Phi_{k,r}(s_{r};\boldsymbol{\eta}_{k,r})=\Phi_{k,\ell}(\Phi_{\ell,r}(s_{r};\boldsymbol{\eta}_{\ell,r});\boldsymbol{\eta}_{k,\ell}),$
(3.4)

where we have written
$\boldsymbol{\eta}_{k,r}=(\boldsymbol{\eta}_{k,\ell},\boldsymbol{\eta}_{\ell,r})$.

###### Proof.

The case when $k=\ell=r$ is trivial. Now assume, by way of inductive
hypothesis, that (3.4) holds for some $1\leq k\leq r$ and all
$\ell\in\mathbb{N}$ such that $k\leq\ell\leq r$.

Using the induction hypothesis, we shall show

$\Phi_{k-1,r}(s_{r};\boldsymbol{\eta}_{k-1,r})=\Phi_{k-1,\ell}(\Phi_{\ell,r}(s_{r};\boldsymbol{\eta}_{\ell,r});\boldsymbol{\eta}_{k-1,\ell})$
(3.5)

holds for all $k-1\leq\ell\leq r$. If $\ell=k-1$, then this result is a
trivial consequence of the definitions, so we assume $k\leq\ell\leq r$. By
(3.3), we deduce that

$\Phi_{k-1,r}(s_{r};\boldsymbol{\eta}_{k-1,r})=\Phi_{k-1,k}(\Phi_{k,r}(s_{r};\boldsymbol{\eta}_{k,r});\eta_{k}).$
(3.6)

Our inductive hypothesis then implies that

$\Phi_{k-1,r}(s_{r};\boldsymbol{\eta}_{k-1,r})=\Phi_{k-1,k}(\Phi_{k,\ell}(\Phi_{\ell,r}(s_{r};\boldsymbol{\eta}_{\ell,r});\boldsymbol{\eta}_{k,\ell});\eta_{k}).$
(3.7)

Applying (3.3) again, here with
$s_{\ell}=\Phi_{\ell,r}(s_{r};\boldsymbol{\eta}_{\ell,r})$, then yields

$\Phi_{k-1,k}(\Phi_{k,\ell}(\Phi_{\ell,r}(s_{r};\boldsymbol{\eta}_{\ell,r});\boldsymbol{\eta}_{k,\ell});\eta_{k})=\Phi_{k-1,\ell}(\Phi_{\ell,r}(s_{r};\boldsymbol{\eta}_{\ell,r});\boldsymbol{\eta}_{k-1,\ell}).$
(3.8)

Combining (3.6), (3.7) and (3.8), we obtain (3.5), which closes the induction.
∎

For each $1\leq k\leq r$, the derivative
$\partial_{s,\eta_{k}}\Phi_{k-1,k}(s;\eta_{k})|_{\eta_{k}=0}=[\partial_{s}\gamma_{k}(s)\,\,\,G_{k}(s)]$
has full rank over all of $U_{k}$. By the inverse function theorem, provided
$\bar{\mu}_{\circ}>0$ is sufficiently small, there exist open neighbourhoods
of the origin $W_{k}\subseteq U_{k}$, $1\leq k\leq r$, such that the
restricted maps $\Phi_{k-1,k}\colon
W_{k}\times(-\bar{\mu}_{\circ},\bar{\mu}_{\circ})^{m_{k}}\rightarrow W_{k-1}$
are diffeomorphisms.666Note that the sets $W_{j}$ appear in both the domain
and codomain of these restricted mappings. This ‘compatibility’ can be
achieved by repeatedly pruning the sets arising from direct application of the
inverse function theorem; we leave the details to the dedicated reader. By
iterating (3.3), the same is true of the restrictions

$\Phi_{k,\ell}\colon W_{\ell}\times V_{k,\ell}\rightarrow W_{k},\qquad 0\leq
k\leq\ell\leq r.$ (3.9)

We later refer to these restricted mappings as the diffeomorphism chain for
$\mathcal{S}$.

Let $C_{\circ}\geq 1$ be an admissible constant, chosen sufficiently large for
the forthcoming purposes of the argument and define
$\mu_{\circ}:=C_{\circ}^{-9/8}\bar{\mu}_{\circ}$. We later refer to this
quantity as the threshold width for $\mathcal{S}$. Given
$\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{\ell})$ with $\max_{1\leq\ell\leq
r}\mu_{\ell}\leq C_{\circ}\mu_{\circ}$, define the set

$\Omega_{k,\ell}(\boldsymbol{\mu}):=\Phi_{k,\ell}(W_{\ell}\times
P_{k,\ell}(\boldsymbol{\mu}))\subseteq W_{k}\quad\textrm{where}\quad
P_{k,\ell}(\boldsymbol{\mu}):=\prod_{t=k+1}^{\ell}[-C_{\circ}^{1/8}\mu_{t},C_{\circ}^{1/8}\mu_{t}]^{m_{t}}.$
(3.10)

The following lemma tells us that the $\Phi_{k,\ell}$ can indeed be thought of
as parametrising the (anisotropic) neighbourhoods of the submanifolds
$M_{k,\ell}$.

######  Lemma 3.2.

Provided $C_{\circ}\geq 1$ is chosen sufficiently large,

$\bigcap_{t=k+1}^{\ell}\mathcal{N}_{\mu_{t}}M_{k,t}\cap
W_{k}\subseteq\Omega_{k,\ell}(\boldsymbol{\mu})\subseteq\bigcap_{t=k+1}^{\ell}\mathcal{N}_{C_{\circ}\mu_{t}}M_{k,t}\cap
W_{k}$ (3.11)

holds for $1\leq k\leq\ell\leq r$ and
$\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{\ell})$ such that
$0<\mu_{1}\leq\cdots\leq\mu_{r}\leq C_{\circ}\mu_{\circ}$.

When $k=\ell$, the left- and right-hand sides of (3.11) are understood to
equal $W_{\ell}$. For $\rho>0$, it is convenient to write

$\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};\rho):=\bigcap_{t=k+1}^{\ell}\mathcal{N}_{\rho\mu_{t}}M_{k,t}\cap
W_{k},$ (3.12)

so that the left and right-hand sets in (3.11) correspond to
$\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};1)$ and
$\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};C_{\circ})$.

###### Proof (of Lemma 3.2).

For the purpose of this proof, we shall rescale $\boldsymbol{\mu}$ and
temporarily redefine
$P_{k,\ell}(\boldsymbol{\mu}):=\prod_{t=k+1}^{\ell}[-\mu_{t},\mu_{t}]^{m_{t}}$.
Let $C\geq 1$ be a fixed constant, chosen to satisfy the forthcoming
requirements of the proof. Fixing $1\leq\ell\leq r$, we shall show, under the
modified definition,

$\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};\rho_{k}^{-1})\subseteq\Omega_{k,\ell}(\boldsymbol{\mu})\subseteq\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};\rho_{k})\qquad\textrm{for}\qquad\rho_{k}:=C^{\ell-k}$
(3.13)

holds for all $1\leq k\leq\ell$, using induction on $k$. Once this is
established, we may simply take $C_{\circ}\geq C^{8n}$ and replace
$\boldsymbol{\mu}$ with $C_{\circ}^{1/8}\boldsymbol{\mu}$ to conclude the
desired result. The claim holds vacuously for $k=\ell$, which acts as the base
case. Now let $2\leq k\leq\ell$ and suppose (3.13) holds for this value of
$k$. It suffices to show

$\mathfrak{N}_{k-1,\ell}(\boldsymbol{\mu};C^{-1}\rho_{k}^{-1})\subseteq\Omega_{k-1,\ell}(\boldsymbol{\mu})\subseteq\mathfrak{N}_{k-1,\ell}(\boldsymbol{\mu};C\rho_{k}).$
(3.14)

For $\rho>0$, define the auxiliary sets

$\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};\rho):=M_{k-1,k}\cap\bigcap_{t=k+1}^{\ell}\mathcal{N}_{\rho\mu_{t}}M_{k-1,t}\cap
W_{k-1}.$

The proof of (3.14) can be reduced to showing

$\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2C^{-1}\rho_{k}^{-1})\subseteq\gamma_{k}(\Omega_{k,\ell}(\boldsymbol{\mu}))\subseteq\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2^{-1}C\rho_{k}).$
(3.15)

We establish this reduction in two stages.

Assuming the first inclusion in (3.15), we show the first inclusion in (3.14).
Let $x\in\mathfrak{N}_{k-1,\ell}(\boldsymbol{\mu};C^{-1}\rho_{k}^{-1})$ and
note, since $x\in W_{k-1}$, that there exists $s\in W_{\ell}$ and
$\boldsymbol{\eta}_{k-1,\ell}=(\eta_{k},\boldsymbol{\eta}_{k,\ell})\in
V_{k-1,\ell}$ such that

$x=\Phi_{k-1,\ell}(s;\boldsymbol{\eta}_{k-1,\ell})=\gamma_{k}\circ\Phi_{k,\ell}(s;\boldsymbol{\eta}_{k,\ell})+G_{k}\circ\Phi_{k,\ell}(s;\boldsymbol{\eta}_{k,\ell})\eta_{k}^{\top}.$
(3.16)

where we have used the recursive definition (3.1). Observe that:

  * •

$x_{0}:=\gamma_{k}\circ\Phi_{k,\ell}(s;\boldsymbol{\eta}_{k,\ell})\in
M_{k-1,k}$ and $x_{0}=\Phi_{k-1,\ell}(s;(0,\boldsymbol{\eta}_{k,\ell}))\in
W_{k-1}$;

  * •

$x\in\mathcal{N}_{C^{-1}\rho_{k}^{-1}\mu_{t}}M_{k-1,t}$ for $k\leq t\leq\ell$;

  * •

$x-x_{0}=G_{k}\circ\Phi_{k,\ell}(s;\boldsymbol{\eta}_{k,\ell})\eta_{k}^{\top}\in
N_{x_{0}}M_{k-1,k}$.

We therefore conclude that $|x-x_{0}|=|\eta_{k}|\leq
C^{-1}\rho_{k}^{-1}\mu_{k}$. By the ordering $\mu_{1}\leq\dots\leq\mu_{r}$ and
the triangle inequality,
$x_{0}\in\mathcal{N}_{2C^{-1}\rho_{k}^{-1}\mu_{t}}M_{k-1,t}$ for each $k\leq
t\leq\ell$, and hence
$x_{0}\in\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2C^{-1}\rho_{k}^{-1})$.
The first inclusion in (3.15) therefore implies that
$x_{0}\in\gamma_{k}(\Omega_{k,\ell}(\boldsymbol{\mu}))$. By the definition of
$x_{0}$ and the injectivity of $\gamma_{k}$, we therefore deduce that
$\Phi_{k,\ell}(s;\boldsymbol{\eta}_{k,\ell})\in\Omega_{k,\ell}(\boldsymbol{\mu})$.
By the injectivity of $\Phi_{k,\ell}$ and the definition of
$\Omega_{k,\ell}(\boldsymbol{\mu})$, we further deduce that
$\boldsymbol{\eta}_{k,\ell}\in P_{k,\ell}(\boldsymbol{\mu})$. Since we have
already shown $|\eta_{k}|\leq\mu_{k}$, we conclude that
$\boldsymbol{\eta}_{k-1,\ell}\in P_{k-1,\ell}(\boldsymbol{\mu})$. Thus, (3.16)
implies $x\in\Omega_{k-1,\ell}(\boldsymbol{\mu})$, as required.

Assuming the second inclusion in (3.15), we show the second inclusion in
(3.14). Let $x\in\Omega_{k-1,\ell}(\boldsymbol{\mu})$, so that
$x=\Phi_{k-1,\ell}(s;\boldsymbol{\eta}_{k-1,\ell})$ for some $s\in W_{\ell}$
and $\boldsymbol{\eta}_{k-1,\ell}\in P_{k-1,\ell}(\boldsymbol{\mu})$. For
$x_{0}:=\gamma_{k}\circ\Phi_{k,\ell}(s;\boldsymbol{\eta}_{k,\ell})$ as before,
we automatically have $|x-x_{0}|=|\eta_{k}|\leq\mu_{k}$. Furthermore,
$x_{0}\in\gamma_{k}(\Omega_{k,\ell})$ and so (3.15) implies that
$x_{0}\in\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2^{-1}C\rho_{k})$.
By the ordering of $\boldsymbol{\mu}$ and the triangle inequality, provided
$C$ is chosen sufficiently large we have
$x\in\mathfrak{N}_{k-1,\ell}(\boldsymbol{\mu};C\rho_{k})$, as required.

It remains to verify (3.15). Let
$x\in\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2C^{-1}\rho_{k}^{-1})$.
Since $x\in M_{k-1,k}\cap W_{k-1}$, there exists some $s_{k}\in W_{k}$ such
that $x=\gamma_{k}(s_{k})$. On the other hand, fixing $k+1\leq t\leq\ell$,
there exists some $\tilde{x}\in M_{k-1,t}$ such that
$|x-\tilde{x}|<2C^{-1}\rho_{k}^{-1}\mu_{t}$. Moreover,
$\tilde{x}=\gamma_{k}(\tilde{s}_{k})$ for some $\tilde{s}_{k}\in M_{k,t}$. We
therefore have
$|\gamma_{k}(s_{k})-\gamma_{k}(\tilde{s}_{k})|<2C^{-1}\rho_{k}^{-1}\mu_{t}$.
Since the parametrisations $\gamma_{k}$ are regular, provided that $C\geq 1$
is chosen sufficiently large, it follows that
$|s_{k}-\tilde{s}_{k}|<\rho_{k}^{-1}\mu_{t}$. This implies that
$s_{k}\in\mathcal{N}_{\rho_{k}^{-1}\mu_{t}}M_{k,t}$. Since this holds for all
$k+1\leq t\leq\ell$, we have shown

$\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2C^{-1}\rho_{k}^{-1})\subseteq\gamma_{k}\big{(}\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};\rho_{k}^{-1})\big{)}\subseteq\gamma_{k}\big{(}\Omega_{k,\ell}(\boldsymbol{\mu})\big{)},$

where the last inequality is due to the induction hypothesis (3.13).

Now suppose $x\in\gamma_{k}\big{(}\Omega_{k,\ell}(\boldsymbol{\mu})\big{)}$.
By the induction hypothesis (3.13), there exists some
$s_{k}\in\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};C\rho_{k+1})=\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};\rho_{k})$
such that $x=\gamma_{k}(s_{k})$. Fixing $k+1\leq t\leq\ell$, it follows that
$s_{k}\in\mathcal{N}_{\rho_{k}\mu_{t}}M_{k,t}\cap W_{k}$ and so there exists
$\tilde{s}_{k}\in M_{k,t}$ such that $|s_{k}-\tilde{s}_{k}|<\rho_{k}\mu_{t}$.
By the mean value theorem, provided $C\geq 1$ is sufficiently large,
$|\gamma_{k}(s_{k})-\gamma_{k}(\tilde{s}_{k})|<2^{-1}C\rho_{k}\mu_{t}$. Since
$\gamma_{k}(\tilde{s}_{k})\in M_{k-1,t}\cap W_{k-1}$, it therefore follows
that $x\in\mathcal{N}_{2^{-1}C\rho_{k}\mu_{t}}M_{k-1,t}\cap W_{k-1}$. Since
this holds for all $k+1\leq t\leq\ell$, we have shown

$\gamma_{k}\big{(}\Omega_{k,\ell}(\boldsymbol{\mu})\big{)}\subseteq\gamma_{k}\big{(}\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};\rho_{k})\big{)}\subseteq\widetilde{\mathfrak{N}}_{k-1,\ell}(\boldsymbol{\mu};2^{-1}C\rho_{k}),$

which concludes the proof of (3.15). ∎

Define
$\mathfrak{N}_{k,\ell}(\boldsymbol{\mu}):=\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};1)$
and
$\mathfrak{N}_{k,\ell}^{+}(\boldsymbol{\mu}):=\mathfrak{N}_{k,\ell}(\boldsymbol{\mu};C_{\circ})$,
where $C_{\circ}$ is the constant appearing in the statement of Lemma 3.2.

######  Corollary 3.3.

Let $0\leq k\leq\ell\leq r$, and define the sets

$\Omega_{k,\ell,r}(\boldsymbol{\mu}):=\Phi_{k,\ell}\big{(}\mathfrak{N}_{\ell,r}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}\quad\textrm{and}\quad\Omega_{k,\ell,r}^{+}(\boldsymbol{\mu}):=\Phi_{k,\ell}\big{(}\mathfrak{N}_{\ell,r}^{+}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}.$

Then

$\mathfrak{N}_{k,r}(\boldsymbol{\mu})\subseteq\Omega_{k,\ell,r}^{+}(\boldsymbol{\mu})\qquad\textrm{and}\qquad\Omega_{k,\ell,r}(\boldsymbol{\mu})\subseteq\mathfrak{N}_{k,r}^{+}(\boldsymbol{\mu})$
(3.17)

for all $\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{\ell})$ such that
$0<\mu_{1}\leq\cdots\leq\mu_{r}\leq C_{\circ}\mu_{\circ}$.

###### Proof.

The formula (3.4) implies that

$\Omega_{k,r}(\boldsymbol{\mu})=\Phi_{k,\ell}\big{(}\Omega_{\ell,r}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}.$

The set inclusions (3.17) then follow from several applications of Lemma 3.2.
In particular,

$\mathfrak{N}_{k,r}(\boldsymbol{\mu})\subseteq\Omega_{k,r}(\boldsymbol{\mu})=\Phi_{k,\ell}\big{(}\Omega_{\ell,r}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}\subseteq\Phi_{k,\ell}\big{(}\mathfrak{N}_{\ell,r}^{+}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}=\Omega_{k,\ell,r}^{+}(\boldsymbol{\mu}),$

and similarly

$\Omega_{k,\ell,r}(\boldsymbol{\mu})=\Phi_{k,\ell}\big{(}\mathfrak{N}_{\ell,r}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}\subseteq\Phi_{k,\ell}\big{(}\Omega_{\ell,r}(\boldsymbol{\mu})\times
P_{k,\ell}(\boldsymbol{\mu})\big{)}=\Omega_{k,r}(\boldsymbol{\mu})\subseteq\mathfrak{N}_{k,r}^{+}(\boldsymbol{\mu}),$

as required. ∎

###  3.2. Slice formula

Continuing with the setup from §3.1, let $E_{S}$ be an extension operator
associated to $S$ with amplitude $a\in C^{\infty}_{c}(U)$ chosen to have
support in the open set $W:=W_{0}$. Suppose $f\in L^{2}(S)$ satisfies
$\mathrm{supp}\,f\subseteq\mathcal{N}_{\boldsymbol{\mu}}\mathcal{S}$ so that,
by Lemma 3.2, we have $\mathrm{supp}\,(f\cdot
a)\subseteq\mathfrak{N}_{0,\ell}(\boldsymbol{\mu})\subseteq\Omega_{0,\ell}(\boldsymbol{\mu})$
for $1\leq\ell\leq r$. Applying the change of variables

$\Phi_{\ell}:=\Phi_{0,\ell}:W_{\ell}\times V_{0,\ell}\rightarrow W,$

and setting $P_{\ell}(\boldsymbol{\mu}):=P_{0,\ell}(\boldsymbol{\mu})$, we
obtain

$E_{S}f(x)=C\int_{P_{\ell}(\boldsymbol{\mu})}e^{ix\cdot\Sigma\circ\Phi_{\ell}(0;\boldsymbol{\eta})}E_{S_{\ell}(\boldsymbol{\eta})}f_{\ell,\boldsymbol{\eta}}(x)\,\mathrm{d}\boldsymbol{\eta}$
(3.18)

where $f_{\ell,\boldsymbol{\eta}}(s):=f\circ\Phi_{\ell}(s;\boldsymbol{\eta})$
and $E_{S_{\ell}(\boldsymbol{\eta})}$ is an extension operator associated to
the $d_{\ell}$-dimensional submanifold
$S_{\ell}(\boldsymbol{\eta}):=\Sigma_{\ell,\boldsymbol{\eta}}(W_{\ell})$ for

$\Sigma_{\ell,\boldsymbol{\eta}}(s):=\Sigma\circ\Phi_{\ell}(s;\boldsymbol{\eta})-\Sigma\circ\Phi_{\ell}(0;\boldsymbol{\eta})\quad\textrm{for
$s\in W_{\ell}$ and $\boldsymbol{\eta}\in P_{\ell}(\boldsymbol{\mu})$.}$

In particular,

$E_{S_{\ell}(\boldsymbol{\eta})}g(x)=\int_{W_{\ell}}e^{ix\cdot\Sigma_{\ell,\boldsymbol{\eta}}(s)}g(s)a_{\ell,\boldsymbol{\eta}}(s)\,\mathrm{d}s$
(3.19)

for
$a_{\ell,\boldsymbol{\eta}}(s):=a\circ\Phi_{\ell}(s;\boldsymbol{\eta})C^{-1}J_{\boldsymbol{\eta}}(s)$,
where $J_{\boldsymbol{\eta}}(s)$ denotes the Jacobian factor arising from the
change of variables and $C\geq 1$ is an admissible constant, chosen to ensure
$0\leq a_{\ell,\boldsymbol{\eta}}(s)\leq 1$ for all $s\in W_{\ell}$.

By Corollary 3.3, we have $\mathrm{supp}\,(f\cdot
a)\subseteq\mathfrak{N}_{0,\ell}(\boldsymbol{\mu})\subseteq\Omega_{0,\ell,r}^{+}(\boldsymbol{\mu})$,
and so

$\mathrm{supp}\,(f_{\ell,\boldsymbol{\eta}}\cdot
a_{\ell,\boldsymbol{\eta}})\subseteq\mathfrak{N}_{\ell,r}^{+}(\boldsymbol{\mu})\qquad\textrm{for
each}\qquad\boldsymbol{\eta}\in P_{\ell}(\boldsymbol{\mu}).$ (3.20)

We refer to (3.18) as the slice formula for $E_{S}f$. We observe the following
properties of the slice formula:

  * •

$S_{\ell}(0)=\Sigma\circ\sigma_{\ell}(W_{\ell})\subseteq S_{\ell}$ is a
codimension $0$ submanifold of $S_{\ell}$.

  * •

Since $f\in L^{2}(S)$, we have that $f_{\ell,\boldsymbol{\eta}}\in
L^{2}(S_{\ell})$ for almost every $\boldsymbol{\eta}\in
P_{\ell}(\boldsymbol{\mu})$. Furthermore, by (3.20), we have

$\mathrm{supp}\,(f_{\ell,\boldsymbol{\eta}}\cdot
a_{\ell,\boldsymbol{\eta}})\subseteq\mathcal{N}_{\min\\{C_{\circ}\mu_{k},\mu_{\circ}\\}}S_{k}\quad\text{
for all $\ell+1\leq k\leq r$,}$ (3.21)

where the minimum is guaranteed provided $\rho(E_{S})$ is sufficiently small.

Here we are using the support (abuse of) notation introduced in §1.1.

###  3.3. Local constancy of the slices

Inequalities such as (1.5) are local in the sense that the left-hand
$L^{q_{k}}$-norm is localised to the cube $Q_{R}:=[-R,R]^{n}$. Consequently,
by the uncertainty principle, we should expect our functions to be locally
constant at scale $R^{-1}$ in frequency space. In particular, continuing with
the setup of the previous subsection, it should be possible to remove the
dependence on $\boldsymbol{\eta}$ in the $S_{\ell}(\boldsymbol{\eta})$ when
$\boldsymbol{\eta}\in P_{\ell}(\boldsymbol{\mu})$ for
$\boldsymbol{\mu}=(\mu_{i})_{i=1}^{r}$ with
$0<\mu_{1}\leq\dots\leq\mu_{\ell}<R^{-1}$.

To implement the locally constant property rigorously, we go back to our
formula for $E_{S_{\ell}(\boldsymbol{\eta})}$ from (3.19) and suppose
$\mu_{\ell}=\max_{i\leq\ell}\mu_{i}<\min\\{R^{-1},C_{\circ}\mu_{\circ}\\}$.
The phase is given by

$x\cdot\Sigma_{\ell,\boldsymbol{\eta}}(s)=x\cdot\Sigma_{\ell,0}(s)+x\cdot\mathcal{E}(s;\boldsymbol{\eta})$

where, by the mean value theorem, $\mathcal{E}(s,\boldsymbol{\eta})$ satisfies
$|\mathcal{E}(s;\boldsymbol{\eta})|\lesssim|\boldsymbol{\eta}|_{\infty}\lesssim
R^{-1}$ for all $s\in\mathrm{supp}\,a_{\ell,\boldsymbol{\eta}}$ and all
$\boldsymbol{\eta}\in P_{\ell}(\boldsymbol{\mu})$. Thus, for $x\in Q_{R}$, the
function $e^{ix\cdot\mathcal{E}(s;\boldsymbol{\eta})}$ is essentially non-
oscillatory and can therefore be removed. More precisely, letting $C\geq 1$
denote a large, admissible constant, by power series expansion

$e^{ix\cdot\mathcal{E}(s;\boldsymbol{\eta})}=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\prod_{j=1}^{n}\frac{(iCR^{-1}x_{j})^{\alpha_{j}}}{\alpha_{j}!}(C^{-1}R\mathcal{E}_{j}(s;\boldsymbol{\eta}))^{\alpha_{j}}$

For $x\in Q_{R}$, we may therefore write777Here the extension operator
$E_{S_{\ell}}$ is defined as in (1.1), with an implicit amplitude $a$ chosen
so that (3.22) holds. In particular, we choose this amplitude such that it is
identically $1$ on the support of $a_{\ell,\boldsymbol{\eta}}$ and with
support contained in $W_{\ell}$.

$E_{S_{\ell}(\boldsymbol{\eta})}g(x)=\sum_{\alpha\in\mathbb{N}_{0}^{n}}B_{\alpha}(x)E_{S_{\ell}}\big{[}a_{\boldsymbol{\eta}}^{\alpha}g\big{]}(x)$
(3.22)

where, provided $C\geq 1$ is chosen sufficiently large,

$a_{\boldsymbol{\eta}}^{\alpha}(s):=a_{\ell,\boldsymbol{\eta}}(s)\prod_{j=1}^{n}(C^{-1}R\mathcal{E}_{j}(s;\boldsymbol{\eta}))^{\alpha_{j}}$

satisfies $|a_{\boldsymbol{\eta}}^{\alpha}(s)|\leq 1$, and

$B_{\alpha}(x):=\prod_{j=1}^{n}\frac{(iCR^{-1}x_{j})^{\alpha_{j}}}{\alpha_{j}!}\quad\textrm{and}\quad
b_{\alpha}:=\prod_{j=1}^{n}\frac{C^{\alpha_{j}}}{\alpha_{j}!}$

satisfy $|B_{\alpha}(x)|\leq b_{\alpha}$, so that
$\sum_{\alpha\in\mathbb{N}_{0}^{n}}|B_{\alpha}(t)|\leq e^{nC}$. Thus, by
Cauchy–Schwarz,

$|E_{S_{\ell}(\boldsymbol{\eta})}g(x)|^{2}\lesssim\sum_{\alpha\in\mathbb{N}_{0}^{n}}b_{\alpha}|E_{S_{\ell}}\big{[}a_{\boldsymbol{\eta}}^{\alpha}g\big{]}(x)|^{2},\qquad
x\in Q_{R}.$ (3.23)

Since the sequence $b_{\alpha}$ decreases rapidly, (3.23) effectively bounds
$|E_{S_{\ell}(\boldsymbol{\eta})}g(x)|$ by $|E_{S_{\ell}}g(x)|$, and therefore
is a rigorous interpretation of the locally constant property.

###  3.4. Wavepacket decomposition

Here we construct a variant of the classical wavepacket decomposition adapted
to our nested geometry.

The derivative of $\Phi_{\ell}$. We first compute the differential
$D\Phi_{\ell}|_{(s;\mathbf{0})}$, which is a $d\times d$ matrix. By first
applying the chain rule to the recursive formula (3.1), we have

$D\Phi_{k-1,\ell}|_{(s;\mathbf{0}_{k-1,\ell})}\begin{pmatrix}v_{\ell}\\\
\mathbf{w}_{k-1,\ell}\end{pmatrix}=D\gamma_{k}|_{\gamma_{k,\ell}(s)}D\Phi_{k,\ell}|_{(s;\mathbf{0}_{k,\ell})}\begin{pmatrix}v_{\ell}\\\
\mathbf{w}_{k,\ell}\end{pmatrix}+G_{k}\big{(}\gamma_{k,\ell}(s)\big{)}w_{k}$
(3.24)

for all $1\leq k\leq\ell\leq r$, $s\in U_{\ell}$,
$v_{\ell}\in\mathbb{R}^{d_{\ell}}$ and
$\mathbf{w}_{k-1,\ell}=(w_{k},\mathbf{w}_{k,\ell})\in\mathbb{R}^{m_{k}}\times\mathbb{R}^{c_{\ell}-c_{k}}$.
Here we have used (3.2).

Let $\partial_{s}\sigma_{\ell}(s)$ denote the Jacobian matrix of
$\sigma_{\ell}$ at $s$ and $B_{k,\ell}(s)\in\mathrm{Mat}(\mathbb{R},d\times
m_{k})$ denote the composition of matrices

$\displaystyle B_{k,\ell}(s):$
$\displaystyle=D\gamma_{1}|_{\gamma_{1,\ell}(s)}\circ\cdots\circ
D\gamma_{k-1}|_{\gamma_{k-1,\ell}(s)}\circ
G_{k}\big{(}\gamma_{k,\ell}(s)\big{)},$
$\displaystyle=\partial_{s}\sigma_{k-1}\big{(}\gamma_{k-1,\ell}(s)\big{)}G_{k}\big{(}\gamma_{k,\ell}(s)\big{)},$

for $1\leq k\leq\ell$, where here
$B_{1,\ell}(s):=G_{1}\circ\gamma_{1,\ell}(s)$ and the second equality is due
to the chain rule. Furthermore, let
$\mathbf{B}_{\ell}(s)\in\mathrm{Mat}(\mathbb{R},d\times c_{\ell})$ denote the
block matrix

$\mathbf{B}_{\ell}(s):=\begin{bmatrix}B_{1,\ell}(s)&B_{2,\ell}(s)&\cdots&B_{\ell,\ell}(s)\end{bmatrix}.$
(3.25)

With these definitions, repeated application of (3.24) yields

$D\Phi_{\ell}|_{(s;\mathbf{0})}\begin{pmatrix}v_{\ell}\\\
\mathbf{w}_{\ell}\end{pmatrix}=\partial_{s}\sigma_{\ell}(s)v_{\ell}+\sum_{k=1}^{\ell}B_{k,\ell}(s)w_{k}=\partial_{s}\sigma_{\ell}(s)v_{\ell}+\mathbf{B}_{\ell}(s)\mathbf{w}_{\ell}$
(3.26)

for $s\in U_{\ell}$, $v_{\ell}\in\mathbb{R}^{d_{\ell}}$ and
$\mathbf{w}_{\ell}\in\mathbb{R}^{d-d_{\ell}}$. Let $b_{j,\ell}(s)$ denote the
$j$th column of $\mathbf{B}_{\ell}(s)$ and
$V_{k,\ell}(s):=\mathrm{span}\,\\{b_{1,\ell}(s),\dots,b_{c_{k},\ell}(s)\\}$
for $1\leq k\leq\ell$. By the non-degeneracy of the parametrisations,

$\bigg{|}\bigwedge_{j=1}^{c_{\ell}}b_{j,\ell}(s)\bigg{|}\gtrsim
1,\qquad|b_{j,\ell}(s)|\lesssim 1,\qquad\big{|}V_{k,\ell}(s)\wedge
T_{\sigma_{\ell}(s)}M_{k}\big{|}\gtrsim 1$ (3.27)

for all $1\leq k\leq\ell\leq r$, $1\leq j\leq c_{\ell}$, and $s\in U_{\ell}$.

Anisotropic decomposition. We continue with the nested family
$\mathcal{S}=(S,(S_{\ell})_{\ell=1}^{r})$ from §3.1. Let $R\geq 1$ and suppose
$\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{r})$ with
$R^{-1}\leq\mu_{1}\leq\dots\leq\mu_{r}\leq\mu_{\circ}$. Note that the upper
bound is slightly stronger than that appearing in the setup in previous
subsections, whilst the lower bound corresponds to the regime in which we do
not have access to the local constancy property (3.23). Set

$\mathcal{L}_{R^{1/2},\boldsymbol{\mu}}:=\big{\\{}1\leq\ell\leq
r:\mu_{\ell}>R^{-1/2}\big{\\}}\quad\textrm{and}\quad\ell_{R^{1/2},\boldsymbol{\mu}}:=\begin{cases}\min\mathcal{L}_{R^{1/2},\boldsymbol{\mu}}-1&\textrm{if
$\mathcal{L}_{R^{1/2}}\neq\emptyset$,}\\\ r&\textrm{otherwise},\end{cases}$

so that if $\mathcal{L}_{R^{1/2},\boldsymbol{\mu}}\neq\emptyset$, then
$\mathcal{L}_{R^{1/2},\boldsymbol{\mu}}=\\{\ell_{R^{1/2},\boldsymbol{\mu}}+1,\dots,r\\}$.
For the ease of notation in this section, we set
$\ell_{\star}:=\ell_{R^{1/2},\boldsymbol{\mu}}$. We define a $d\times d$
diagonal matrix

$D_{R^{1/2},\boldsymbol{\mu}}:=\begin{bmatrix}\mu_{1}I_{m_{1}}&\cdots&\mathbf{0}&\mathbf{0}\\\
\vdots&\ddots&\vdots&\vdots\\\
\mathbf{0}&\cdots&\mu_{\ell_{\star}}I_{m_{\ell_{\star}}}&\mathbf{0}\\\
\mathbf{0}&\cdots&\mathbf{0}&R^{-1/2}I_{d_{\star}}\end{bmatrix},$

where
$d_{\star}:=d-\sum_{\ell=1}^{\ell_{\star}}m_{\ell}=\dim(S_{\ell_{\star}})$.
Finally, for $\mathbf{B}_{\ell}(s)$ as in (3.25), define the $d\times d$
matrices

$\Lambda_{\ell}(s):=\begin{bmatrix}\mathbf{B}_{\ell}(s)&\displaystyle\frac{\partial\sigma_{\ell}}{\partial
s}(s)\end{bmatrix},\qquad s\in W_{\ell},$ (3.28)

which, by (3.27), are invertible for $1\leq\ell\leq r$, and let

$\Lambda_{R,\boldsymbol{\mu}}(s):=\begin{cases}C_{\circ}^{1/2}\Lambda_{\ell_{\star}}(s)\circ
D_{R^{1/2},\boldsymbol{\mu}}&\textrm{if $1\leq\ell_{\star}\leq r$,}\\\
C_{\circ}^{1/2}R^{-1/2}I_{d}&\textrm{if $\ell_{\star}=0$,}\end{cases}$ (3.29)

noting that $D_{R^{1/2},\boldsymbol{\mu}}=R^{-1/2}I_{d}$ in the
$\ell_{\star}=0$ case.

Define a grid of points

$\Gamma_{R,\boldsymbol{\mu}}(\mathcal{S}):=R^{-1/2}\mathbb{Z}^{d_{\star}}\cap\mathfrak{N}_{\ell_{\star},r}(\boldsymbol{\mu};2C_{\circ})$

where $\mathfrak{N}_{\ell_{\star},r}(\boldsymbol{\mu};2C_{\circ})$ is as
defined in (3.12) and $C_{\circ}\geq 1$ is an admissible constant, chosen
large enough so that the conclusion of Lemma 3.2 holds. We form a covering of

$\mathfrak{N}_{r}(\boldsymbol{\mu}):=\mathfrak{N}_{0,r}(\boldsymbol{\mu})=\mathcal{N}_{\mu_{1}}M_{1}\cap\cdots\cap\mathcal{N}_{\mu_{r}}M_{r}\cap
W$

by parallelepipeds. Let $\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$ denote the
family of all parallelepipeds of the form

$\theta=u_{\theta}+\Lambda_{\theta}([-1,1]^{d})\quad\textrm{where
$u_{\theta}:=\sigma_{\ell_{\star}}(s_{\theta})$ and
$\Lambda_{\theta}:=\Lambda_{R,\boldsymbol{\mu}}(s_{\theta})$}$ (3.30)

for $s_{\theta}\in\Gamma_{R,\boldsymbol{\mu}}(\mathcal{S})$. Here
$\sigma_{0}:=\mathrm{Id}$.

######  Lemma 3.4.

Provided $C_{\circ}\geq 1$ is sufficiently large, the following properties
hold:

  1. i)

The collection $\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$ forms a cover of
$\mathfrak{N}_{r}(\boldsymbol{\mu})$.

  2. ii)

The scaled parallelepipeds
$\\{4\cdot\theta:\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})\\}$ are
finitely-overlapping.

  3. iii)

$4\cdot\theta\cap W\subseteq\mathfrak{N}_{r}(C_{\circ}^{2}\boldsymbol{\mu})$
for all $\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$.

###### Proof.

If $\ell_{\star}=0$, then the $\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$ is a
standard covering by $R^{1/2}$ cubes, and the desired properties are
immediate. We therefore assume $1\leq\ell_{\star}\leq r$.

i) By Corollary 3.3, it suffices to show that
$\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$ forms a cover of
$\Omega_{0,\ell_{\star},r}^{+}(\boldsymbol{\mu})$. Fix
$x\in\Omega_{0,\ell_{\star},r}^{+}(\boldsymbol{\mu})$ so that

$x=\Phi_{0,\ell_{\star}}(s;\boldsymbol{\eta})=\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})\quad\textrm{for
some $s\in\mathfrak{N}_{\ell_{\star},r}^{+}(\boldsymbol{\mu})$ and
$\boldsymbol{\eta}\in
P_{0,\ell_{\star}}(\boldsymbol{\mu})=P_{\ell_{\star}}(\boldsymbol{\mu})$.}$

Since
$\mathfrak{N}_{\ell_{\star},r}^{+}(\boldsymbol{\mu})=\mathfrak{N}_{\ell_{\star},r}(\boldsymbol{\mu};C_{\circ})$,
there exists some $s_{\theta}\in\Gamma_{R,\boldsymbol{\mu}}(\mathcal{S})$ such
that $|s-s_{\theta}|_{\infty}<R^{-1/2}$. The problem is therefore reduced to
showing $\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})\in\theta$, provided
$C_{\circ}\geq 1$ is sufficiently large.

For
$u_{\theta}:=\sigma_{\ell_{\star}}(s_{\theta})=\Phi_{\ell_{\star}}(s_{\theta};\mathbf{0})$,
write

$\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})-u_{\theta}=\mathbf{B}_{\ell_{\star}}(s_{\theta})\boldsymbol{\eta}+\partial_{s}\sigma_{\ell_{\star}}(s_{\theta})(s-s_{\theta})+\mathcal{E}_{\theta}(s;\boldsymbol{\eta})$

where

$\mathcal{E}_{\theta}(s;\boldsymbol{\eta}):=\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})-\sigma_{\ell_{\star}}(s_{\theta})-\partial_{s}\sigma_{\ell_{\star}}(s_{\theta})(s-s_{\theta})-\mathbf{B}_{\ell_{\star}}(s_{\theta})\boldsymbol{\eta}.$
(3.31)

By combining the definitions (3.28), (3.29) and (3.30), we have

$\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})-u_{\theta}=\Lambda_{\theta}(\boldsymbol{\xi}+\boldsymbol{\zeta})\qquad\textrm{where}\qquad\boldsymbol{\zeta}:=\big{(}\Lambda_{\theta}\big{)}^{-1}\mathcal{E}_{\theta}(s;\boldsymbol{\eta})$

and

$\boldsymbol{\xi}:=C_{\circ}^{-1/2}\big{(}D_{R^{1/2},\boldsymbol{\mu}}\big{)}^{-1}\begin{bmatrix}\boldsymbol{\eta}\\\
s-s_{\theta}\end{bmatrix}\in[-1/2,1/2]^{d};$

here we are assuming that $C_{\circ}^{3/8}\geq 2$ and using the definition of
$P_{\ell_{\star}}(\boldsymbol{\mu})$. Since $\mu_{k}\leq R^{-1/2}$ for $1\leq
k\leq\ell_{\star}$, by (3.26), (3.2) and Taylor’s theorem, we have

$|\mathcal{E}_{\theta}(s;\boldsymbol{\eta})|\lesssim|s-s_{\theta}|_{\infty}^{2}+|\boldsymbol{\eta}|_{\infty}^{2}\lesssim
C_{\circ}^{1/4}R^{-1}.$ (3.32)

Thus, since we also have $\mu_{k}\geq R^{-1}$ for $1\leq k\leq\ell_{\star}$,
we can ensure that $\boldsymbol{\zeta}\in[-1/2,1/2]^{d}$ provided that
$C_{\circ}\geq 1$ is sufficiently large.

Combining the above observations,
$\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})=u_{\theta}+\Lambda_{\theta}(\boldsymbol{\xi}+\boldsymbol{\zeta})$
where $\boldsymbol{\xi}+\boldsymbol{\zeta}\in[-1,1]^{d}$ and so
$\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})\in\theta$, as required.

ii) The finite overlap property is immediate. Indeed, if $\theta_{1}$,
$\theta_{2}\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$ are such that
$4\cdot\theta_{1}\cap 4\cdot\theta_{2}\neq\emptyset$, then it follows that the
corresponding centres $u_{\theta_{i}}=\sigma_{\ell_{\star}}(s_{\theta_{i}})$
satisfy $|u_{\theta_{1}}-u_{\theta_{2}}|_{\infty}\lesssim R^{-1/2}$. Since
$\sigma_{\ell_{\star}}$ maps diffeomorphically onto its image, we conclude
that $|s_{\theta_{1}}-s_{\theta_{2}}|_{\infty}\lesssim R^{-1/2}$. But for
$\theta_{1}$ fixed, this is only possible for $O(1)$ choices of $\theta_{2}$
since the $s_{\theta_{2}}$ lie on $R^{-1/2}$ separated points.

iii) Fix $\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$ with centre
$u_{\theta}=\sigma_{\ell_{\star}}(s_{\theta})$ for some
$s_{\theta}\in\Gamma_{R,\boldsymbol{\mu}}(\mathcal{S})$.

Let $\ell_{\star}+1\leq t\leq r$ so that
$s_{\theta}\in\mathcal{N}_{2C_{\circ}\mu_{t}}M_{\ell_{\star},t}$. Thus there
exists some $s_{\theta,t}\in M_{\ell_{\star},t}$ with
$|s_{\theta}-s_{\theta,t}|<2C_{\circ}\mu_{t}$. Consequently,
$|u_{\theta}-u_{\theta,t}|\lesssim C_{\circ}\mu_{t}$ where
$u_{\theta,t}:=\sigma_{\ell_{\star}}(s_{\theta,t})\in M_{t}$ and so
$\mathrm{dist}(u_{\theta},M_{t})\lesssim C_{\circ}\mu_{t}$. On the other hand,
it is clear from the definition that $4\cdot\theta$ lies in a ball of radius
$O(C_{\circ}^{1/2}R^{-1/2})$ centred at $u_{\theta}$. Since
$\mu_{t}>R^{-1/2}$, we conclude that

$4\cdot\theta\subseteq\bigcap_{t=\ell_{\star}+1}^{r}\mathcal{N}_{C_{\circ}^{2}\mu_{t}}M_{t},$
(3.33)

provided $C_{\circ}\geq 1$ is chosen sufficiently large.

We now consider $1\leq k\leq\ell_{\star}$. Suppose $u\in 4\cdot\theta\cap W$,
so that, by the definitions (3.28), (3.29) and (3.30), we can write

$u=\mathbf{B}_{\ell_{\star}}(s_{\theta})\boldsymbol{\eta}+\partial_{s}\sigma_{\ell_{\star}}(s_{\theta})(s-s_{\theta})+\sigma_{\ell_{\star}}(s_{\theta})$

for some $\boldsymbol{\eta}\in
P_{\ell_{\star}}(4C_{\circ}^{3/8}\boldsymbol{\mu})$ and
$s-s_{\theta}\in[-4C_{\circ}^{1/2}R^{-1/2},4C_{\circ}^{1/2}R^{-1/2}]^{d_{\star}}$.
Thus,

$u=\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})-\mathcal{E}_{\theta}(s;\boldsymbol{\eta}),$

where $\mathcal{E}_{\theta}(s;\boldsymbol{\eta})$ is as defined in (3.31).
Arguing as in (3.32), we have
$|\mathcal{E}_{\theta}(s;\boldsymbol{\eta})|\lesssim C_{\circ}R^{-1}$.

Since $\Phi_{\ell_{\star}}\colon W_{\ell_{\star}}\times V_{\ell_{\star}}\to W$
is a bijection, we can write
$u=\Phi_{\ell_{\star}}(\tilde{s};\tilde{\boldsymbol{\eta}})$ for some
$\tilde{s}\in W_{\ell_{\star}}$ and $\tilde{\boldsymbol{\eta}}\in
V_{\ell_{\star}}:=V_{0,\ell_{\star}}$. Taking inverses,

$(\tilde{s}-s;\tilde{\boldsymbol{\eta}}-\boldsymbol{\eta})=\Phi_{\ell_{\star}}^{-1}\big{(}\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})-\mathcal{E}_{\theta}(s;\boldsymbol{\eta})\big{)}-\Phi_{\ell_{\star}}^{-1}\big{(}\Phi_{\ell_{\star}}(s;\boldsymbol{\eta})\big{)},$

and therefore, by the mean value theorem,
$|\tilde{\boldsymbol{\eta}}-\boldsymbol{\eta}|_{\infty}\lesssim
C_{\circ}R^{-1}$. Since $\mu_{k}\geq R^{-1}$ for $1\leq k\leq r$ and
$\boldsymbol{\eta}\in P_{\ell_{\star}}(4C_{\circ}^{3/8}\boldsymbol{\mu})$, we
have $\tilde{\boldsymbol{\eta}}\in
P_{\ell_{\star}}(C_{\circ}\boldsymbol{\mu})$, provided $C_{\circ}\geq 1$ is
chosen sufficiently large. By the definition (3.10), we have
$u=\Phi_{\ell_{\star}}(\tilde{s},\tilde{\boldsymbol{\eta}})\in\Omega_{0,\ell_{\star}}(C_{\circ}\boldsymbol{\mu})$.
Our hypotheses ensure that $C_{\circ}\mu_{1}\leq\dots\leq C_{\circ}\mu_{r}\leq
C_{\circ}\mu_{\circ}$, and so we may apply Lemma 3.2 to deduce that
$u\in\mathfrak{N}_{\ell_{\star}}^{+}(C_{\circ}\boldsymbol{\mu})=\mathfrak{N}_{\ell_{\star}}(C_{\circ}^{2}\boldsymbol{\mu})$.
Thus,

$4\cdot\theta\cap
W\subseteq\mathfrak{N}_{\ell_{\star}}(C_{\circ}^{2}\boldsymbol{\mu})$ (3.34)

and combining (3.33) and (3.34) concludes the proof. ∎

Definition of the wave packets. Let $\psi\in C^{\infty}_{c}(\mathbb{R}^{d})$
satisfy $0\leq\psi\leq 1$; $\psi(u)=1$ if $|u|_{\infty}\leq 1$ and $\psi(u)=0$
if $|u|_{\infty}\geq 2$. Define $\tilde{\psi}(u):=\psi(u/2)$, so that
$\psi\tilde{\psi}=\psi$. Given
$\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$, let

$\psi_{\theta}(u):=\psi\big{(}\Lambda_{\theta}^{-1}(u-u_{\theta})\big{)}\quad\textrm{and}\quad\tilde{\psi}_{\theta}(u):=\tilde{\psi}\big{(}\Lambda_{\theta}^{-1}(u-u_{\theta})\big{)}.$

Thus, $\psi_{\theta}(u)=1$ if $u\in\theta$,
$\mathrm{supp}\,\psi_{\theta}\subseteq 2\cdot\theta$,
$\mathrm{supp}\,\tilde{\psi}_{\theta}\subseteq 4\cdot\theta$ and
$\psi_{\theta}\tilde{\psi}_{\theta}=\psi_{\theta}$. By Lemma 3.4 i) and ii),
we have

$1\leq\sum_{\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})}\psi_{\theta}(u)\lesssim
1\qquad\textrm{for all $u\in\mathfrak{N}_{r}(\boldsymbol{\mu})$.}$

Now suppose $f\in L^{2}(S)$ is smooth and satisfies
$\mathrm{supp}\,f\subseteq\mathcal{N}_{\boldsymbol{\mu}}\mathcal{S}$. This
abuse of notation translates as $f\in C^{\infty}_{c}(U)$ with
$\mathrm{supp}\,f\subseteq\mathfrak{N}_{r}(\boldsymbol{\mu})$, provided we
also assume $\mathrm{supp}\,f\subseteq W$. Under these conditions, we may
write

$f=\sum_{\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})}f_{\theta}\psi_{\theta}\qquad\textrm{where
$f_{\theta}\in C^{\infty}_{c}(U)$ satisfy $|f_{\theta}(u)|\lesssim|f(u)|$ for
all $u\in U$.}$

Letting
$\Lambda_{\theta}^{-\top}(\mathbb{Z}^{d}):=\\{\Lambda_{\theta}^{-\top}m:m\in\mathbb{Z}^{d}\\}$,
define

$\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S}):=\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})\times\Lambda_{\theta}^{-\top}(\mathbb{Z}^{d})\quad\textrm{and}\quad\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S};\theta):=\\{\theta\\}\times\Lambda_{\theta}^{-\top}(\mathbb{Z}^{d})$

for all $\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$. By rescaling and
applying a Fourier series decomposition,

$f(u)=\sum_{(\theta,v)\in\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S})}f_{\theta,v}(u)\quad\textrm{and}\quad(f_{\theta}\psi_{\theta})(u)=\sum_{(\theta,v)\in\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S};\theta)}f_{\theta,v}(u)$

for each $\theta\in\Theta_{R,\boldsymbol{\mu}}(\mathcal{S})$, where

$f_{\theta,v}(u):=(2\pi)^{-d}|\det\Lambda_{\theta}|^{-1}(f_{\theta}\psi_{\theta})\;\widehat{}\;(v)e^{iv\cdot
u}\tilde{\psi}_{\theta}(u).$

Here
$(f_{\theta}\psi_{\theta})\;\widehat{}\;(v)=\int_{\mathbb{R}^{d}}e^{-iv\cdot
u}(f_{\theta}\psi_{\theta})(u)\,\mathrm{d}u$ is the Fourier transform of
$f_{\theta}\psi_{\theta}$ evaluated at $v$.

Basic properties of the wave packets. We identify three key properties of the
above decomposition:

Orthogonality. Parseval’s identity for Fourier series and Lemma 3.4 ii) give

$\big{\|}\sum_{(\theta,v)\in\mathcal{T}}f_{\theta,v}\big{\|}_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\sum_{(\theta,v)\in\mathcal{T}}\|f_{\theta,v}\|_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\|f\|_{L^{2}(\mathbb{R}^{d})}^{2}$

for any collection
$\mathcal{T}\subseteq\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S})$.

Spatial localisation. Recall
$\Sigma_{\ell_{\star}}:=\Sigma\circ\sigma_{\ell_{\star}}$ and
$S_{\ell_{\star}}=\Sigma_{\ell_{\star}}(U_{\ell_{\star}})$. Given
$\varepsilon>0$, let $\varepsilon_{\circ}:=\varepsilon/(100n)$ and define the
slabs

$T_{\theta,v}:=\big{\\{}x\in\mathbb{R}^{n}:\big{|}(\partial_{s}\Sigma_{\ell_{\star}})(s_{\theta})^{\top}x+v_{\theta}^{\star}\big{|}_{\infty}<R^{1/2+\varepsilon_{\circ}}\big{\\}},\quad
v_{\theta}^{\star}:=(\partial_{s}\sigma_{\ell_{\star}})(s_{\theta})^{\top}v$
(3.35)

for all $(\theta,v)\in\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S})$. Let
$u_{\theta}$ be as defined in (3.30). Observe that, by the chain rule formula

$(\partial_{s}\Sigma_{\ell_{\star}})(s_{\theta})=(\partial_{u}\Sigma)(u_{\theta})\circ(\partial_{s}\sigma_{\ell_{\star}})(s_{\theta})$

and the definition of $\Lambda_{\theta}$, we have
$\widetilde{T}_{\theta,v}\subseteq T_{\theta,v}$ where

$\widetilde{T}_{\theta,v}:=\big{\\{}x\in
Q_{R}:\big{|}\Lambda_{\theta}^{\top}\big{(}(\partial_{u}\Sigma)(u_{\theta})^{\top}x+v\big{)}\big{|}_{\infty}<R^{\varepsilon_{\circ}}\big{\\}}.$
(3.36)

Let
$\mathbf{T}_{R,\boldsymbol{\mu}}(\mathcal{S}):=\\{T_{\theta,v}:(\theta,v)\in\mathcal{T}_{R,\boldsymbol{\mu}}(\mathcal{S})\\}$
denote the collection of all slabs of the form (3.35) and, for each
$T=T_{\theta,v}\in\mathbf{T}_{R,\boldsymbol{\mu}}(\mathcal{S})$, write
$f_{T}:=f_{\theta,v}$. A standard integration-by-parts argument then shows
that

$|E_{S}f_{T}(x)|\lesssim_{N,\varepsilon}|E_{S}f_{T}(x)|\chi_{T}(x)+R^{-N}\|f\|_{L^{2}(U)}$

for all $x\in Q_{R}$ and all $N\in\mathbb{N}_{0}$, where $E_{S}$ has amplitude
$a$ supported in the open set $W$. Indeed, we have a stronger estimate with
$\chi_{T}$ replaced with $\chi_{\widetilde{T}}$ for $\widetilde{T}$ as defined
in (3.36), but, for our purposes, we only require localisation to the larger
slab $T$.

Preservation of support. With the above setup, for
$C_{\circ}^{2}\boldsymbol{\mu}=(C_{\circ}^{2}\mu_{\ell})_{\ell=1}^{r}$, we
have

$\mathrm{supp}\,f_{T}\cap
W\subseteq\mathfrak{N}_{r}(C_{\circ}^{2}\boldsymbol{\mu})\qquad\textrm{for all
$T\in\mathbf{T}_{R,\boldsymbol{\mu}}(\mathcal{S})$.}$ (3.37)

Indeed, this follows from Lemma 3.4 iii) after unravelling the notation and
noting that $\mathrm{supp}\,\tilde{\psi}_{\theta}\subseteq 4\cdot\theta$. The
localisation (3.37) is the key advantage of the above anisotropic wave packet
decomposition, as opposed to the regular wave packet decomposition used, for
instance, in [9, 6], and is crucial for the forthcoming induction-on-scales
argument.

##  4\. The recursive scheme

Throughout this section, we fix $2\leq k\leq n$ and

$\mathscr{E}=((\mathcal{S}_{j})_{j=1}^{k},\mathbf{q})\quad\textrm{ with
}\quad\mathcal{S}_{j}=(S_{j},(S_{j,\ell})_{\ell=1}^{r_{j}})$

a transverse ensemble in $\mathbb{R}^{n}$ of codimension
$(m(j,\ell))_{j,\ell}$. We apply the observations of §3 to each of the nested
families of submanifolds $\mathcal{S}_{j}$. In particular, for $1\leq j\leq k$
we let $\Phi_{k,\ell}^{(j)}\colon W_{\ell}^{(j)}\times V_{k,\ell}^{(j)}\to
W_{k}^{(j)}$ denote the diffeomorphism chain and $\mu_{\circ}^{(j)}>0$ be the
threshold width for $\mathcal{S}_{j}$, as defined in (3.9) and on page 3.1,
respectively. We let
$\mu_{\circ}:=\min\\{\mu_{\circ}^{(1)},\dots,\mu_{\circ}^{(k)}\\}>0$.

###  4.1. A reformulation

Given $R\geq 1$ and $\boldsymbol{\mu}=(\boldsymbol{\mu}_{j})_{j=1}^{k}$ with
$\boldsymbol{\mu}_{j}:=(\mu_{j,\ell})_{\ell=1}^{r_{j}}$ a compatible family of
scales for $\mathscr{E}$, set

$\mathcal{L}_{R,\boldsymbol{\mu}}(j):=\big{\\{}1\leq\ell\leq
r_{j}:\mu_{j,\ell}>R^{-1}\big{\\}}$

and

$\ell_{R,\boldsymbol{\mu}}(j):=\begin{cases}\min\mathcal{L}_{R,\boldsymbol{\mu}}(j)-1&\textrm{if
$\mathcal{L}_{R,\boldsymbol{\mu}}(j)\neq\emptyset$,}\\\
r_{j}&\textrm{otherwise},\end{cases}$

so that if $\mathcal{L}_{R,\boldsymbol{\mu}}(j)\neq\emptyset$, then
$\mathcal{L}_{R,\boldsymbol{\mu}}(j)=\\{\ell_{R,\boldsymbol{\mu}}(j)+1,\dots,r_{j}\\}$.
We then set

$S_{j,R}:=S_{j,\ell_{R,\boldsymbol{\mu}}(j)}\quad\textrm{and}\quad
U_{j,R}:=U_{j,\ell_{R,\boldsymbol{\mu}}(j)},\quad 1\leq j\leq k,$ (4.1)

where $S_{j,0}:=S_{j}$ and $U_{j,0}:=U_{j}$; note that $S_{j,R}$ and $U_{j,R}$
also depend on $\boldsymbol{\mu}$, but we suppress this for notational
simplicity. Consider also the restricted nested family of submanifolds
$\mathcal{S}_{j,R}:=(S_{j,R},(S_{j,\ell})_{\ell\in\mathcal{L}_{R,\boldsymbol{\mu}}(j)})$
for $1\leq j\leq k$.

We let $\rho_{\circ}>0$ be a constant, chosen sufficiently small for the
purposes of the forthcoming argument. In particular, we assume
$B(0,\rho_{\circ})\subseteq W_{0}^{(j)}$ for $1\leq j\leq k$. For all $R\geq
1$ and $\boldsymbol{\mu}$ as above, let $\mathfrak{R}(R;\boldsymbol{\mu})$
denote the smallest constant $C\geq 1$ for which the inequality

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}\leq
C\mathbf{C}_{R}(\boldsymbol{\mu})\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(U_{j,R})}^{q_{j}}$
(4.2)

holds for

$\mathbf{C}_{R}(\boldsymbol{\mu}):=\prod_{j=1}^{k}\prod_{\ell\in\mathcal{L}_{R,\boldsymbol{\mu}}(j)}(\mu_{j,\ell})^{m(j,\ell)q_{j}/2}$
(4.3)

whenever $\rho(E_{S_{j,R}})<\rho_{\circ}^{n-\ell_{R,\boldsymbol{\mu}}(j)}$ for
$1\leq j\leq k$ and the $f_{j}\in L^{2}(S_{j,R})$ are smooth, compactly
supported and satisfy888Since $f_{j}\in L^{2}(S_{j,R})$, the reference
manifolds are $S_{j,R}$, and we understand
$\mathcal{N}_{\mu_{j,\ell}}(S_{j,\ell})$ as the neighbourhood in
$\mathbb{R}^{\dim(S_{j,R})}$ of the submanifold
$\gamma_{j,\ell_{R,\boldsymbol{\mu}(j)}+1}\circ\cdots\circ\gamma_{j,\ell}(U_{j,\ell})$.

$\mathrm{supp}\,f_{j}\subseteq\mathcal{N}_{\mu_{j,\ell}}S_{j,\ell}\quad\textrm{for
all $\ell\in\mathcal{L}_{R,\boldsymbol{\mu}}(j)$.}$

If $\mathcal{L}_{R,\boldsymbol{\mu}}(j)=\emptyset$, then the product over
$\ell\in\mathcal{L}_{R,\boldsymbol{\mu}}(j)$ in (4.3) is interpreted as equal
to $1$. We also define

$\mathfrak{R}(R):=\sup_{\boldsymbol{\mu},\,\mu_{j,\ell}\leq\mu_{\circ}}\mathfrak{R}(R;\boldsymbol{\mu}),$

where the supremum is taken over all compatible families of scales
$\boldsymbol{\mu}=(\boldsymbol{\mu}_{j})_{j=1}^{k}$ for $\mathscr{E}$ with
$\boldsymbol{\mu}_{j}=(\mu_{j,\ell})_{\ell=1}^{r_{j}}$ satisfying
$0<\mu_{j,\ell}\leq\mu_{\circ}$ for $1\leq\ell\leq r_{j},1\leq j\leq k$.

Our goal is to show that, for all $\varepsilon>0$, the bound
$\mathfrak{R}(R)\lesssim_{\varepsilon}R^{\varepsilon}$ holds for all $R\geq
1$. From this we shall deduce Theorem 1.6: see §4.3. We further reformulate
our goal as follows. Let $\boldsymbol{\mu}$ be a compatible family of scales
for $\mathscr{E}$. We first observe a trivial estimate for
$\mathfrak{R}(R;\mu)$, by bounding the left-hand side of (4.2) in terms of the
$L^{\infty}$-norms of the $|E_{S_{j,R}}f_{j}|$ and applying a
Riemann–Lebesgue-type argument to arrive at an expression involving the
$L^{1}$ norms of the $f_{j}$. We can then pass to $L^{2}$ norms using
Cauchy–Schwarz, taking advantage of the localisation of the
$\mathrm{supp}\,f_{j}$, to deduce that

$\mathfrak{R}(R;\boldsymbol{\mu})\lesssim R^{n}\qquad\textrm{for all
$\boldsymbol{\mu}$ compatible for $\mathscr{E}$, and
so}\qquad\mathfrak{R}(R)\lesssim R^{n}.$ (4.4)

Thus, the upper exponent

$\eta_{\mathrm{exp}}:=\limsup_{R\to\infty}\frac{\log\mathfrak{R}(R)}{\log R}$
(4.5)

is a well defined real number which satisfies $\eta_{\mathrm{exp}}\leq n$. To
prove Theorem 1.6, it shall suffice to show $\eta_{\mathrm{exp}}\leq 0$.

###  4.2. Uncertainty principle reduction

As a manifestation of the uncertainty principle,
$\mathfrak{R}(R;\boldsymbol{\mu})$ automatically bounds a wider variety of
multilinear expressions. In what follows, we let $C_{\circ}\geq 1$ be an
admissible constant, chosen sufficiently large for the purposes of the
forthcoming argument and so that the conclusions of §3 hold. Let
$\boldsymbol{\mu}=(\boldsymbol{\mu}_{j})_{j=1}^{k}$ be a compatible family of
scales for $\mathscr{E}$, with
$\boldsymbol{\mu}_{j}=(\mu_{j,\ell})_{\ell=1}^{r_{j}}$ and
$0<\mu_{j,\ell}\leq\mu_{\circ}$ for $1\leq\ell\leq r_{j}$, $1\leq j\leq k$. We
define $\boldsymbol{\tilde{\mu}}=(\boldsymbol{\tilde{\mu}}_{j})_{j=1}^{k}$
with $\boldsymbol{\tilde{\mu}}_{j}=(\tilde{\mu}_{j,\ell})_{\ell=1}^{r_{j}}$
for $\tilde{\mu}_{j,\ell}:=\min\\{C_{\circ}^{2}\mu_{j,\ell},\mu_{\circ}\\}$,
$1\leq\ell\leq r_{j}$, $1\leq j\leq k$.

######  Lemma 4.1.

Let $R\geq 1$ and
$\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)\subseteq\mathcal{L}^{\star}(j)\subseteq\\{1,\dots,r_{j}\\}$
for $1\leq j\leq k$. Suppose that whenever $\ell\in\mathcal{L}^{\star}(j)$ and
$\ell\leq\ell^{\prime}\leq r_{j}$, we have
$\ell^{\prime}\in\mathcal{L}^{\star}(j)$. Define

$\ell^{\star}(j):=\begin{cases}\min\mathcal{L}^{\star}(j)-1&\textrm{if
$\mathcal{L}^{\star}(j)\neq\emptyset$,}\\\
r_{j}&\textrm{otherwise,}\end{cases}$

so that if $\mathcal{L}^{\star}(j)\neq\emptyset$, then
$\mathcal{L}^{\star}(j):=\\{\ell^{\star}(j)+1,\dots,r_{j}\\}$. Set

$S_{j}^{\star}:=S_{j,\ell^{\star}(j)}\quad\textrm{and}\quad
U_{j}^{\star}:=U_{j,\ell^{\star}(j)}\quad\textrm{for $1\leq j\leq k$.}$

If $\rho(E_{S_{j}^{\star}})<\rho_{\circ}^{n-\ell^{\star}(j)}$ for $1\leq j\leq
k$, then

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j}^{\star}}f_{j}|^{q_{j}}\lesssim\mathfrak{R}(R;\boldsymbol{\tilde{\mu}})\prod_{j=1}^{k}\prod_{\ell\in\mathcal{L}^{\star}(j)}(\mu_{j,\ell})^{m(j,\ell)q_{j}/2}\prod_{j=1}^{k}\|f_{j}\|_{L^{2}(U_{j}^{\star})}^{q_{j}}$
(4.6)

holds whenever the $f_{j}\in L^{2}(S_{j}^{\star})$ are smooth, compactly
supported and

$suppf_{j}\subseteq\mathcal{N}_{\mu_{j,\ell}}S_{j,\ell}\quad\textrm{for all
$\ell\in\mathcal{L}^{\star}(j)$.}$ (4.7)

###### Proof.

Let $\mathcal{J}_{R}:=\\{1\leq j\leq
k:\ell^{\star}(j)=\ell_{R,\boldsymbol{\tilde{\mu}}}(j)\\}$ and
$\mathcal{J}^{\star}:=\\{1,\dots,k\\}\setminus\mathcal{J}_{R}$, so that
$S_{j}^{\star}=S_{j,R}$ for all $j\in\mathcal{J}_{R}$ and
$\ell^{\star}(j)<\ell_{R,\boldsymbol{\tilde{\mu}}}(j)$ for
$j\in\mathcal{J}^{\star}$. Provided $\rho_{\circ}>0$ is chosen sufficiently
small, we can apply the slice formula (3.18) to the pairs $S_{j}^{\star}$ and
$S_{j,R}(\boldsymbol{\eta})$ for $j\in\mathcal{J}^{\star}$ and we have, by
Cauchy–Schwarz,

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j}^{\star}}f_{j}|^{q_{j}}\lesssim\prod_{j\in\mathcal{J}^{\star}}\prod_{\ell\in\mathfrak{m}_{j}}(\mu_{j,\ell})^{m(j,\ell)q_{j}/2}I(R;\boldsymbol{\mu})$
(4.8)

where
$\mathfrak{m}_{j}:=\\{\ell^{\star}(j)+1,\dots,\ell_{R,\boldsymbol{\tilde{\mu}}}(j)\\}$
and

$I(R;\boldsymbol{\mu}):=\int_{Q_{R}}\prod_{j\in\mathcal{J}_{R}}|E_{S_{j,R}}f_{j}(x)|^{q_{j}}\prod_{j\in\mathcal{J}^{\star}}\Big{(}\int_{P_{j}(\boldsymbol{\mu}_{j})}|E_{S_{j,R}(\boldsymbol{\eta})}f_{j,\boldsymbol{\eta}}(x)|^{2}\,\mathrm{d}\boldsymbol{\eta}\Big{)}^{q_{j}/2}\,\mathrm{d}x,$

for

$P_{j}(\boldsymbol{\mu}_{j}):=\prod_{\ell\in\mathfrak{m}_{j}}[-C_{\circ}^{1/8}\mu_{j,\ell},C_{\circ}^{1/8}\mu_{j,\ell}]^{m(j,\ell)}\qquad\textrm{for
$j\in\mathcal{J}^{\star}$.}$

Here, for an appropriate map $\Phi_{j}^{\star}$ satisfying
$\Phi_{j}^{\star}(0;\mathbf{0})=0$,999Explicitly,
$\Phi_{j}^{\star}:=\Phi_{\ell^{\star}(j),\ell_{R,\boldsymbol{\tilde{\mu}}}(j)}^{(j)}\colon
W^{(j)}_{\ell_{R,\boldsymbol{\tilde{\mu}}}(j)}\times
V_{\ell^{\star}(j),\ell_{R,\boldsymbol{\tilde{\mu}}}(j)}^{(j)}\to
W_{\ell^{\star}(j)}^{(j)}$. we have

$f_{j,\boldsymbol{\eta}}(s):=f_{j}\circ\Phi_{j}^{\star}(s;\boldsymbol{\eta})\qquad\textrm{for
$j\in\mathcal{J}^{\star}$.}$

Observe that
$\mathcal{L}^{\star}(j)=\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)\cup\mathfrak{m}_{j}$
for $1\leq j\leq k$, where the union is disjoint and
$\mathfrak{m}_{j}=\emptyset$ whenever $j\in\mathcal{J}_{R}$.

Fix $j\in\mathcal{J}^{\star}$ so that
$\ell^{\star}(j)<\ell_{R,\boldsymbol{\tilde{\mu}}}(j)$. By the definition and
the ordering of the $(\mu_{j,\ell})_{\ell=1}^{r_{j}}$, we have

$\\{1\leq\ell\leq r_{j}:\tilde{\mu}_{j,\ell}\leq
R^{-1}\\}=\\{1,\dots,\ell_{R,\boldsymbol{\tilde{\mu}}}(j)\\}$

and so $\mu_{j,\ell}\leq\tilde{\mu}_{j,\ell}\leq R^{-1}$ for all
$\ell\in\mathfrak{m}_{j}$. Consequently, given $g\in C^{\infty}_{c}(U_{j,R})$,
from (3.23) we have

$|E_{S_{j,R}(\boldsymbol{\eta})}g(x)|^{2}\lesssim\sum_{\alpha[j]\in\mathbb{N}_{0}^{n}}b_{\alpha[j]}|E_{S_{j,R}}\big{[}a_{j,\boldsymbol{\eta}}^{\alpha[j]}g\big{]}(x)|^{2},\quad\boldsymbol{\eta}\in
P_{j}(\boldsymbol{\mu}_{j}),\,x\in Q_{R},$ (4.9)

where the $a_{j,\boldsymbol{\eta}}^{\alpha[j]}\in C^{\infty}(U_{j,R})$ satisfy
$|a_{j,\boldsymbol{\eta}}^{\alpha[j]}(s)|\leq 1$ and
$b_{\alpha[j]}:=\prod_{i=1}^{n}\tfrac{C^{\alpha[j]_{i}}}{\alpha[j]_{i}!}$ for
an admissible constant $C\geq 1$.

For each $1\leq j\leq k$, define the Hilbert space
$H_{j}:=L^{2}(P_{j}(\boldsymbol{\mu}_{j}))$ and consider the $H_{j}$-valued
operator $\mathbf{E}_{S_{j,R}}$ given by the mapping

$\mathbf{E}_{S_{j,R}}\mathbf{g}_{j}(x):=(E_{S_{j,R}}g_{j,\boldsymbol{\eta}}(x))_{\boldsymbol{\eta}},\qquad\boldsymbol{\eta}\in
P_{j}(\boldsymbol{\mu}_{j}),\,x\in\mathbb{R}^{n},$

where $\mathbf{g}_{j}:=(g_{j,\boldsymbol{\eta}})_{\boldsymbol{\eta}}\in
L^{2}(U_{j,R};H_{j})$. Observe that, by Fubini’s theorem, for any
$x\in\mathbb{R}^{n}$ and $h_{j}\in H_{j}$, we have

$\langle\mathbf{E}_{S_{j,R}}\mathbf{g}_{j}(x),h_{j}\rangle_{H_{j}}=E_{S_{j,R}}(\langle\mathbf{g}_{j}(\cdot),h_{j}\rangle_{H_{j}})(x).$

Hence $\mathbf{E}_{S_{j,R}}$ coincides with the $H_{j}$-valued extension of
$E_{S_{j,R}}$, as described in the appendix. Letting
$\kappa:=\\#\mathcal{J}^{\star}$, we write
$\boldsymbol{\alpha}=(\alpha[j])_{j\in\mathcal{J}^{\star}}$ for
$\boldsymbol{\alpha}\in(\mathbb{N}_{0}^{n})^{\kappa}$,
$\mathbf{f}_{j}^{\boldsymbol{\alpha}}:=(f_{j,\boldsymbol{\eta}}^{\boldsymbol{\alpha}})_{\boldsymbol{\eta}}$
with
$f_{j,\boldsymbol{\eta}}^{\boldsymbol{\alpha}}:=a_{j,\boldsymbol{\eta}}^{\alpha[j]}f_{j,\boldsymbol{\eta}}$.
Using this notation, and applying (4.9), we have

$\displaystyle I(R;\boldsymbol{\mu})$
$\displaystyle\lesssim\int_{Q_{R}}\prod_{j\in\mathcal{J}_{R}}|E_{S_{j,R}}f_{j}(x)|^{q_{j}}\prod_{j\in\mathcal{J}^{\star}}\Big{(}\sum_{\alpha[j]\in\mathbb{N}_{0}^{n}}b_{\alpha[j]}\|\mathbf{E}_{S_{j,R}}\mathbf{f}_{j}^{\boldsymbol{\alpha}}(x)\|_{H_{j}}^{2}\Big{)}^{q_{j}/2}\,\mathrm{d}x$
$\displaystyle\lesssim\sum_{\boldsymbol{\alpha}\in(\mathbb{N}_{0}^{n})^{\kappa}}\prod_{j\in\mathcal{J}^{\star}}b_{\alpha[j]}^{q_{j}/2}\int_{Q_{R}}\prod_{j\in\mathcal{J}_{R}}|E_{S_{j,R}}f_{j}(x)|^{q_{j}}\prod_{j\in\mathcal{J}^{\star}}\|\mathbf{E}_{S_{j,R}}\mathbf{f}_{j}^{\boldsymbol{\alpha}}(x)\|_{H_{j}}^{q_{j}}\,\mathrm{d}x.$
(4.10)

Here the last inequality follows by the nesting of the $\ell^{2/q_{j}}$ and
$\ell^{1}$ norms for $2/q_{j}\geq 1$.

If $j\in\mathcal{J}^{\star}$, then, provided $\rho_{\circ}>0$ is chosen
sufficiently small, we have
$\rho(E_{S_{j,R}})<\rho_{\circ}^{-1}\rho(E_{S_{j}^{\star}})$. Our hypotheses
therefore ensure

$\rho(E_{S_{j,R}})<\rho_{\circ}^{-1}\rho_{\circ}^{n-\ell^{\star}(j)}\leq\rho_{\circ}^{n-\ell_{R,\boldsymbol{\tilde{\mu}}}(j)}\qquad\textrm{for
$j\in\mathcal{J}^{\star}$,}$

since $0<\rho_{\circ}<1$ and
$\ell_{R,\boldsymbol{\tilde{\mu}}}(j)\geq\ell^{\star}(j)+1$. Note also that
$\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)\subseteq\mathcal{L}^{\star}(j)$
and that, by (3.21), the functions $f_{j,\boldsymbol{\eta}}$ are in
$L^{2}(S_{j,R})$ for almost every $\boldsymbol{\eta}\in
P_{j}(\boldsymbol{\mu}_{j})$ and are supported in
$\mathcal{N}_{\tilde{\mu}_{j,\ell}}S_{j,\ell}$ for all
$\ell\in\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)$.

In light of the preceding discussion, we can apply the inequality (4.2)
featured in the definition of $\mathfrak{R}(R;\boldsymbol{\tilde{\mu}})$ to
the extension operators $E_{S_{j,R}}$ and the functions
$(f_{j})_{j\in\mathcal{J}_{R}}$,
$(f_{j,\boldsymbol{\eta}_{j}}^{\boldsymbol{\alpha}})_{j\in\mathcal{J}^{\star}}$
as defined above. Moreover, the inequality (4.2) can be lifted to a vector-
valued variant using a multilinear extension of the Marcinkiewicz–Zygmund
theorem (see Proposition A.1). Combining these observations,

$\displaystyle\int_{Q_{R}}$
$\displaystyle\prod_{j\in\mathcal{J}_{R}}|E_{S_{j,R}}f_{j}(x)|^{q_{j}}\prod_{j\in\mathcal{J}^{\star}}\|\mathbf{E}_{S_{j,R}}\mathbf{f}_{j}^{\boldsymbol{\alpha}}(x)\|_{H_{j}}^{q_{j}}\,\mathrm{d}x$
(4.11)
$\displaystyle\lesssim\mathfrak{R}(R;\boldsymbol{\tilde{\mu}})\prod_{j=1}^{k}\prod_{\ell\in\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)}(\mu_{j,\ell})^{m(j,\ell)q_{j}/2}\prod_{j\in\mathcal{J}_{R}}\|f_{j}\|_{2}^{q_{j}}\prod_{j\in\mathcal{J}^{\star}}\|\|\mathbf{f}_{j}^{\boldsymbol{\alpha}}\|_{H_{j}}\|_{2}^{q_{j}}.$

Here we have used the fact that $\tilde{\mu}_{j,\ell}\leq
C_{\circ}^{2}\mu_{j,\ell}$ for all $1\leq j\leq k$ and
$\ell\in\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)$.

By the pointwise bound for the $a_{j,\boldsymbol{\eta}}^{\alpha[j]}$ and a
change of variables,

$\|\|\mathbf{f}_{j}^{\boldsymbol{\alpha}}\|_{H_{j}}\|_{2}=\Big{(}\int_{P_{j}(\boldsymbol{\mu}_{j})}\|f_{j,\boldsymbol{\eta}_{j}}^{\boldsymbol{\alpha}}\|_{2}^{2}\,\mathrm{d}\boldsymbol{\eta}_{j}\Big{)}^{1/2}\lesssim\|f_{j}\|_{2}\qquad\textrm{for
$j\in\mathcal{J}^{\star}$.}$

Combining this with (4.11), we obtain

$\displaystyle\int_{Q_{R}}\prod_{j\in\mathcal{J}_{R}}$
$\displaystyle|E_{S_{j,R}}f_{j}(x)|^{q_{j}}\prod_{j\in\mathcal{J}^{\star}}\|\mathbf{E}_{S_{j,R}}\mathbf{f}_{j}^{\boldsymbol{\alpha}}(x)\|_{H_{j}}^{q_{j}}\,\mathrm{d}x$
(4.12)
$\displaystyle\lesssim\mathfrak{R}(R;\boldsymbol{\tilde{\mu}})\prod_{j=1}^{k}\prod_{\ell\in\mathcal{L}_{R,\boldsymbol{\tilde{\mu}}}(j)}(\mu_{j,\ell})^{m(j,\ell)q_{j}/2}\prod_{j=1}^{k}\|f_{j}\|_{2}^{q_{j}}.$

We apply (4.12) to bound the right-hand side of (4.10), noting that the
sequence $(b_{\alpha}^{q_{j}/2})_{\alpha\in\mathbb{N}_{0}^{n}}$ is
(absolutely) summable. Combining the resulting inequality with (4.8), we
deduce the desired bound. ∎

###  4.3. Recursive step

Let $p_{j}:=q_{j}/2$ and $1\leq j\leq k$. For $R\geq 1$, let $\mathfrak{K}(R)$
be the smallest constant $C\geq 1$ for which the inequality

$\int_{Q_{R}}\prod_{j=1}^{k}\Big{|}\sum_{T_{j}\in\mathbf{T}_{j}}c_{T_{j}}\chi_{T_{j}}\Big{|}^{p_{j}}\leq
CR^{n/2}\prod_{j=1}^{k}\Big{(}\sum_{T_{j}\in\mathbf{T}_{j}}|c_{T_{j}}|\Big{)}^{p_{j}}$

holds whenever, for each $1\leq j\leq k$, there exists some $1\leq\ell_{j}\leq
r_{j}$ such that $\mathbf{T}_{j}$ is a countable set of slabs of the form

$T_{j}=\big{\\{}x\in\mathbb{R}^{n}:\big{|}(\partial_{s}\Sigma_{j,\ell_{j}})(s)^{\top}x-v\big{|}<R^{1/2}\big{\\}}$
(4.13)

where $s$, $v\in\mathbb{R}^{d_{j}^{\prime}}$ with $|s|_{\infty}<\rho_{\circ}$
for $d_{j}^{\prime}:=\dim(S_{j,\ell_{j}})$ and
$(c_{T_{j}})_{T_{j}\in\mathbf{T}_{j}}\in\ell^{1}(\mathbf{T}_{j})$.

Let $\boldsymbol{\mu}=(\boldsymbol{\mu}_{j})_{j=1}^{k}$ be a compatible family
of scales for $\mathscr{E}$, with
$\boldsymbol{\mu}_{j}=(\mu_{j,\ell})_{\ell=1}^{r_{j}}$ and
$0<\mu_{j,\ell}\leq\mu_{\circ}$ for $1\leq\ell\leq r_{j}$, $1\leq j\leq k$.
Recall the constant $C_{\circ}\geq 1$ and the definition of
$\boldsymbol{\tilde{\mu}}$ from §4.2. Let
$\boldsymbol{\accentset{\approx}{\mu}}$ denote the compatible family of scales
which result from applying the operation
$\boldsymbol{\mu}\mapsto\boldsymbol{\tilde{\mu}}$ twice to $\boldsymbol{\mu}$.
Provided $C_{\circ}\geq 1$ is chosen sufficiently large, the following holds.

######  Lemma 4.2.

For all $\varepsilon>0$, $R\geq C_{\circ}^{8}$ and $\boldsymbol{\mu}$
compatible for $\mathscr{E}$, we have

$\mathfrak{R}(R;\boldsymbol{\mu})\lesssim_{\varepsilon}R^{\varepsilon}\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\mathfrak{K}(R).$

Once Lemma 4.2 is established, Theorem 1.6 may be deduced as follows.

###### Proof (of Theorem 1.6).

We work with the fixed ensemble $\mathscr{E}$ introduced at the beginning of
the section. Without loss of generality, we may assume our compatible family
of scales $\boldsymbol{\mu}=(\boldsymbol{\mu}_{j})_{j=1}^{k}$ with
$\boldsymbol{\mu}_{j}=(\mu_{j,\ell})_{\ell=1}^{r_{j}}$ satisfies
$0<\mu_{j,\ell}\leq\mu_{\circ}$ for $1\leq\ell\leq r_{j}$, $1\leq j\leq k$.
Indeed, otherwise we can pass to a (strict) subensemble $\mathscr{F}$ of
$\mathscr{E}$ (consisting precisely of those $S_{j,\ell}$ for which
$0<\mu_{j,\ell}\leq\mu_{\circ}$ holds) and the analysis is simpler. This
reduction can be formalised using an induction on the number of submanifolds
in the ensemble.

In view of (4.4), we can assume $R$ is sufficiently large, namely $R\geq
C_{\circ}^{8}$. By the hypothesis
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{E}),\mathbf{p})<\infty$ and
Proposition 2.4, we have that
$\mathrm{BL}_{\mathrm{reg}}(\mathbf{L}(\mathscr{F}),\mathbf{p})<\infty$ for
all subensembles $\mathscr{F}$ of $\mathscr{E}$. For $\varepsilon>0$, let
$\nu_{\mathscr{F}}>0$ and $C_{\mathscr{F},\varepsilon}\geq 1$ be the constants
appearing in Theorem 2.6 for each datum
$(\mathbf{L}(\mathscr{F}),\mathbf{p})$, and set

$\displaystyle\overline{\nu}_{\mathscr{E}}$
$\displaystyle:=\min\\{\nu_{\mathscr{F}}:\mathscr{F}\text{ subensemble of
}\mathscr{E}\\}>0$ $\displaystyle\overline{C}_{\mathscr{E},\varepsilon}$
$\displaystyle:=\max\\{C_{\mathscr{F},\varepsilon}:\mathscr{F}\text{
subensemble of }\mathscr{E}\\}<\infty.$

Provided $\rho_{\circ}>0$ is sufficiently small, the slabs (4.13) have core
planes which are, modulo translations, within distance
$\overline{\nu}_{\mathscr{E}}$ of the fixed subspaces
$\ker\partial_{s}\Sigma_{j,\ell_{j}}(0)^{\top}$. Since, by (3.2), any
$(\partial_{s}\Sigma_{j,\ell_{j}}(0)^{\top})_{j=1}^{k}=((\partial_{s}\sigma_{\ell_{j}}(0))^{\top}(\partial_{s}\Sigma_{j}(0))^{\top})_{j=1}^{k}$
corresponds to $\mathbf{L}(\mathscr{F})$ for some subensemble $\mathscr{F}$ of
$\mathscr{E}$, it then follows from Theorem 2.6, taking $\lambda=R^{1/2}$,
that for every $\varepsilon>0$, we have
$\mathfrak{K}(R)\leq\overline{C}_{\mathscr{E},\varepsilon}R^{\varepsilon}$ for
all $R\geq 1$. Combining this with Lemma 4.2, for all $\varepsilon>0$ we have

$\mathfrak{R}(R;\boldsymbol{\mu})\lesssim_{\varepsilon}R^{\varepsilon}\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\qquad\textrm{for
all $R\geq C_{\circ}^{8}$. }$ (4.14)

Recall the definition of the upper exponent $\eta_{\mathrm{exp}}$ from (4.5)
which, by (4.4), is a well-defined real number. The inequality (4.14) implies
that $\eta_{\mathrm{exp}}\leq\eta_{\mathrm{exp}}/2$ and so
$\eta_{\mathrm{exp}}\leq 0$. Thus, for all $\varepsilon>0$, we have
$\mathfrak{R}(R)\lesssim_{\varepsilon}R^{\varepsilon}$. Let
$\mathcal{L}^{\star}(j)=\\{1,\dots,r_{j}\\}$, so that $\ell^{\star}(j)=0$, for
$1\leq j\leq k$. Applying Lemma 4.1 with these $\mathcal{L}^{\star}(j)$
together with the above estimate for $\mathfrak{R}(R)$, the desired bound
follows with $\rho_{\mathscr{E}}:=\rho_{\circ}^{n}$. ∎

We turn to the proof of Lemma 4.2, which is a variant of an argument from [9,
6].

###### Proof (of Lemma 4.2).

Let $\varepsilon>0$ and, recalling the definitions from §3.4, set

$\mathbf{T}_{j}:=\mathbf{T}_{R,\boldsymbol{\mu}}(\mathcal{S}_{j,R}),\qquad
1\leq j\leq k.$

Using the notation in (4.1), the slabs in $\mathbf{T}_{j}$ have width
$R^{1/2+\varepsilon_{\circ}}$, where
$\varepsilon_{\circ}:=\varepsilon/(100n)$, and their core planes are normal to
the tangent plane to $S_{j,R^{1/2}}=\Sigma_{j,R^{1/2}}(U_{j,R^{1/2}})$ at
$\Sigma_{j,R^{1/2}}(s)$ for some $s\in U_{j,R^{1/2}}$: that is, they are of
the type (4.13). Provided $\rho_{\circ}>0$ is sufficiently small,

$|s|_{\infty}\leq\rho_{\circ}^{-1}\rho(E_{S_{j,R}})<\rho_{\circ}^{n-\ell_{R,\boldsymbol{\mu}}(j)-1}\leq\rho_{\circ},\qquad\textrm{since
$\ell_{R,\boldsymbol{\mu}}(j)+1\leq r\leq n-1$.}$

This observation will allow for an application of the bound $\mathfrak{K}(R)$.

Let $\mathcal{Q}_{R^{1/2}}$ denote a cover of $Q_{R}:=[-R,R]^{n}$ by
essentially disjoint cubes of sidelength $R^{1/2}$, so that

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}\leq\sum_{Q\in\mathcal{Q}_{R^{1/2}}}\int_{Q}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}.$
(4.15)

Fix $Q\in\mathcal{Q}_{R^{1/2}}$. Provided $\rho_{\circ}>0$ is chosen
sufficiently small, we can perform the wave packet decomposition from §3.4. By
the spatial localisation property,

$|E_{S_{j,R}}f_{j}(x)|\lesssim_{N,\varepsilon}|E_{S_{j,R}}f_{j,Q}(x)|+R^{-N}\|f_{j}\|_{2}\qquad\textrm{for
all $N\in\mathbb{N}_{0}$ and $x\in Q$,}$

where

$f_{j,Q}:=\sum_{T_{j}\in\mathbf{T}_{j}[Q]}f_{j,T}\qquad\textrm{for}\qquad\mathbf{T}_{j}[Q]:=\\{T\in\mathbf{T}_{j}:T\cap
Q\neq\emptyset\\}.$

Thus,

$\int_{Q}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}\lesssim_{\varepsilon}\int_{Q}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j,Q}|^{q_{j}}+R^{-100n}\prod_{j=1}^{k}\|f_{j}\|_{2}^{q_{j}}.$

For each $T\in\mathbf{T}_{j}$, we have $\mathrm{supp}\,f_{j,T_{j}}\subseteq
B(0,\rho_{\circ}+4\cdot R^{-1/2})$. Thus, for large $R$, we can guarantee that
$\mathrm{supp}\,f_{j,Q}\subseteq W_{0}^{(j)}$. By the preservation of support
property (3.37), provided $C_{\circ}\geq 1$ is chosen sufficiently large, each
$f_{j,Q}$ satisfies
$\mathrm{supp}\,f_{j,Q}\subseteq\mathcal{N}_{C_{\circ}^{2}\boldsymbol{\mu}_{j}}\mathcal{S}_{j}$.
Moreover, by the support properties of our amplitude functions, we may assume
without loss of generality that
$\mathrm{supp}\,f_{j,Q}\subseteq\mathcal{N}_{\boldsymbol{\tilde{\mu}}_{j}}\mathcal{S}_{j}$
for $\boldsymbol{\tilde{\mu}}_{j}$ as defined in §4.2.

Define $\mathcal{L}^{\star}(j):=\mathcal{L}_{R,\boldsymbol{\mu}}(j)$ and,
since $\mu_{j,\ell}\geq C_{\circ}^{-4}\accentset{\approx}{\mu}_{j,\ell}\geq
R^{-1/2}\accentset{\approx}{\mu}_{j,\ell}$, note that

$\mathcal{L}_{R^{1/2},\boldsymbol{\accentset{\approx}{\mu}}}(j)\subseteq\mathcal{L}^{\star}(j)\subseteq\\{1,\dots,r_{j}\\}\qquad\textrm{for
$1\leq j\leq k$.}$

From the observations of the previous paragraph, the $f_{j,Q}$ satisfy the
hypothesis (4.7) of Lemma 4.1 with $\boldsymbol{\mu}$ replaced with
$\boldsymbol{\tilde{\mu}}$. We may therefore apply Lemma 4.1 to deduce that

$\int_{Q}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j,Q}|^{q_{j}}\lesssim\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\mathbf{C}_{R}(\boldsymbol{\mu})\prod_{j=1}^{k}\|f_{j,Q}\|_{2}^{q_{j}},$
(4.16)

where $\mathbf{C}_{R}(\boldsymbol{\mu})$ is as defined in (4.3).101010Observe
that, for our choice of $\mathcal{L}^{\star}(j)$ and $\boldsymbol{\mu}$
replaced with $\boldsymbol{\tilde{\mu}}$, the factors on the right-hand side
of (4.6) satisfy
$\prod_{\ell\in\mathcal{L}^{\star}(j)}(\tilde{\mu}_{j,\ell})^{m(j,\ell)q_{j}/2}\lesssim\prod_{\ell\in\mathcal{L}_{R,\boldsymbol{\mu}}(j)}(\mu_{j,\ell})^{m(j,\ell)q_{j}/2},$
justifying the appearance of $\mathbf{C}_{R}(\boldsymbol{\mu})$ in (4.16).
Here we have also used the translation invariance of the estimates to recentre
$Q$ at the origin.

Taking the sum over $\mathcal{Q}_{R^{1/2}}$ on both sides of (4.16) and
combining the resulting estimate with (4.15), we deduce that

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}\lesssim_{\varepsilon}\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\mathbf{C}_{R}(\boldsymbol{\mu})\sum_{Q\in\mathcal{Q}_{R^{1/2}}}\prod_{j=1}^{k}\|f_{j,Q}\|_{2}^{q_{j}}$
(4.17)

holds up to the inclusion of a rapidly decaying error term. It remains to
estimate the right-hand sum in (4.17).

By the orthogonality properties of the wave packets,

$\prod_{j=1}^{k}\|f_{j,Q}\|_{2}^{2}\lesssim\prod_{j=1}^{k}\Big{(}\sum_{T_{j}\in\mathbf{T}_{j}[Q]}\|f_{j,T_{j}}\|_{2}^{2}\Big{)}.$

Furthermore, if $T_{j}\in\mathbf{T}_{j}[Q]$, then it follows that $Q\subset
T_{j}^{\,\circ}$, where $T_{j}^{\,\circ}$ denotes the slab with the same core
plane as $T_{j}$ but with width scaled by a factor of $2$. Consequently,

$\prod_{j=1}^{k}\|f_{j,Q}\|_{2}^{q_{j}}\lesssim\frac{1}{|Q|}\int_{Q}\prod_{j=1}^{k}\big{|}\sum_{T_{j}\in\mathbf{T}_{j}}\|f_{j,T_{j}}\|_{2}^{2}\cdot\chi_{T^{\,\circ}_{j}}(x)\big{|}^{q_{j}/2}\,\mathrm{d}x.$

Summing this inequality over all $Q\in\mathcal{Q}_{R^{1/2}}$ and combining the
resulting estimate with (4.17), we obtain

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}\lesssim_{\varepsilon}\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\mathbf{C}_{R}(\boldsymbol{\mu})R^{-n/2}\int_{Q_{R}}\prod_{j=1}^{k}\big{|}\sum_{T_{j}\in\mathbf{T}_{j}}\|f_{j,T_{j}}\|_{2}^{2}\cdot\chi_{T^{\,\circ}_{j}}\big{|}^{p_{j}},$
(4.18)

up to the inclusion of a rapidly decaying error term, where we recall
$p_{j}:=q_{j}/2$ for $1\leq j\leq k$.

By the definition of the constant $\mathfrak{K}(R)$ and the orthogonality of
the wave packets,

$\displaystyle\int_{Q_{R}}\prod_{j=1}^{k}\big{|}\sum_{T_{j}\in\mathbf{T}_{j}}\|f_{j,T_{j}}\|_{2}^{2}\cdot\chi_{T^{\,\circ}}\big{|}^{p_{j}}$
$\displaystyle\lesssim_{\varepsilon}\mathfrak{K}(R)R^{n/2+\varepsilon}\prod_{j=1}^{k}\big{(}\sum_{T_{j}\in\mathbf{T}_{j}}\|f_{j,T_{j}}\|_{2}^{2}\big{)}^{p_{j}}$
$\displaystyle\lesssim\mathfrak{K}(R)R^{n/2+\varepsilon}\prod_{j=1}^{k}\|f_{j}\|_{2}^{q_{j}}.$
(4.19)

Note that the slabs featured in the above displays (whose geometry coincides
with that of the slabs defined in (3.35)), are slightly wider (by a factor of
$R^{\varepsilon_{\circ}}$ where $\varepsilon_{\circ}:=\varepsilon/(100n)$)
than those appearing in the definition of $\mathfrak{K}(R)$. This discrepancy
can be dealt with using a simple covering argument, which accounts for the
additional $R^{\varepsilon}$ factor on the right-hand side of the above
inequality.

Combining (4.19) with (4.18), we obtain

$\int_{Q_{R}}\prod_{j=1}^{k}|E_{S_{j,R}}f_{j}|^{q_{j}}\lesssim_{\varepsilon}R^{\varepsilon}\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\mathfrak{K}(R)\mathbf{C}_{R}(\boldsymbol{\mu})\prod_{j=1}^{k}\|f_{j}\|_{2}^{q_{j}}$

and therefore, by definition,
$\mathfrak{R}(R;\boldsymbol{\mu})\lesssim_{\varepsilon}R^{\varepsilon}\mathfrak{R}(R^{1/2};\boldsymbol{\accentset{\approx}{\mu}})\mathfrak{K}(R)$,
as required. ∎

##  Appendix A A multilinear Marcinkiewicz–Zygmund theorem

Let $H$ be a separable complex Hilbert space and
$\\{e_{a}\\}_{a\in\mathcal{A}}$ be a choice of orthonormal basis for $H$,
where $\mathcal{A}$ is a countable set. Given any $g\in H$, we may write

$g=\sum_{a\in\mathcal{A}}\langle
g,e_{a}\rangle\quad\textrm{where}\quad\|g\|_{H}=\big{(}\sum_{a\in\mathcal{A}}|\langle
g,e_{a}\rangle|^{2}\big{)}^{1/2}.$

Let $(\Omega,\Sigma,\mu)$ and $(\Omega^{\prime},\Sigma^{\prime},\mu^{\prime})$
be measure spaces. We say that $\mathbf{f}\colon\Omega^{\prime}\to H$ is
$\Sigma^{\prime}$-measurable if $\langle\mathbf{f},g\rangle$ is a
$\Sigma^{\prime}$-measurable function for all $g\in H$. Let $T$ be a linear
mapping sending $\Sigma^{\prime}$-measurable functions on $\Omega^{\prime}$ to
$\Sigma$-measurable functions on $\Omega$. We define the vector-valued
extension $\mathbf{T}$ to be the operator mapping $H$-valued measurable
functions on $\Omega^{\prime}$ to $H$-valued measurable functions on $\Omega$
given by

$\langle\mathbf{T}\mathbf{f}(x),e_{a}\rangle:=T(\langle\mathbf{f},e_{a}\rangle)(x)\qquad\textrm{for
all $\mathbf{f}\colon\Omega^{\prime}\to H$ measurable and all
$a\in\mathcal{A}$.}$

This uniquely defines the operator $\mathbf{T}$, which is linear.

The classical Marcinkiewicz–Zygmund inequality [15] for separable Hilbert
spaces has the following multilinear counterpart.

######  Proposition A.1 (Multilinear Marcinkiewicz–Zygmund Theorem).

Let $(\Omega,\Sigma,\mu)$ and $(\Omega_{j},\Sigma_{j},\mu_{j})$ for $1\leq
j\leq m$ be $\sigma$-finite measure spaces. For $1\leq j\leq m$, let $T_{j}$
be linear mappings sending $\Sigma_{j}$-measurable functions on $\Omega_{j}$
to $\Sigma$-measurable functions on $\Omega$. Let and $0<p_{j},q_{j}<\infty$
for $1\leq j\leq k$ and further suppose that there exists some $M\geq 0$ such
that

$\int_{\Omega}\prod_{j=1}^{k}|T_{j}f_{j}(x)|^{q_{j}}\,\mathrm{d}\mu(x)\leq
M\prod_{j=1}^{k}\|f_{j}\|_{L^{p_{j}}(\Omega_{j},\mu_{j})}^{q_{j}}$

holds for all $\Sigma_{j}$-measurable functions $f_{j}$, $1\leq j\leq k$.

Let $H_{j}$ be separable complex Hilbert spaces and let $\mathbf{T}_{j}$
denote the vector-valued extensions of $T_{j}$, $1\leq j\leq k$. Then the
inequality

$\int_{\Omega}\prod_{j=1}^{k}\|\mathbf{T}_{j}\mathbf{f}_{j}(x)\|_{H_{j}}^{q_{j}}\,\mathrm{d}\mu(x)\lesssim_{p,q}M\prod_{j=1}^{k}\|\|\mathbf{f}_{j}\|_{H_{j}}\|_{L^{p_{j}}(\Omega_{j},\mu_{j})}^{q_{j}}$

holds for all $\Sigma_{j}$-measurable functions
$\mathbf{f}_{j}\colon\Omega_{j}\to H_{j}$, $1\leq j\leq k$.

If the Hilbert spaces $H_{j}$ are finite dimensional, the result follows from
a standard Khintchine’s inequality argument and the Fubini–Tonelli theorem.
The extension to separable spaces follows from Fatou’s lemma. We omit the
details.

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