# Singular integrals along variable codimension one subspaces

Odysseas Bakas  BCAM - Basque Center for Applied Mathematics, 48009 Bilbao,
Spain<EMAIL_ADDRESS>,  Francesco Di Plinio  Dipartimento di Matematica e
Applicazioni “R. Caccioppoli”

Università di Napoli “Federico II”, Via Cintia, Monte S. Angelo 80126 Napoli,
Italy<EMAIL_ADDRESS>,  Ioannis Parissis  Departamento de
Matemáticas, Universidad del Pais Vasco, Aptdo. 644, 48080 Bilbao, Spain and
Ikerbasque, Basque Foundation for Science, Bilbao, Spain
<EMAIL_ADDRESS>and  Luz Roncal  BCAM - Basque Center for Applied
Mathematics 48009 Bilbao, Spain,
Ikerbasque, Basque Foundation for Science, Bilbao, Spain,
and Departamento de Matemáticas, Universidad del Pais Vasco, Aptdo. 644, 48080
Bilbao, Spain<EMAIL_ADDRESS>
###### Abstract.

This article deals with maximal operators on $\mathbb{R}^{n}$ formed by taking
arbitrary rotations of tensor products of a $d$-dimensional Hörmander–Mihlin
multiplier with the identity in $n-d$ coordinates, in the particular
_codimension 1_ case $d=n-1$. These maximal operators are naturally connected
to differentiation problems and maximally modulated singular integrals such as
Sjölin’s generalization of Carleson’s maximal operator. Our main result, a
weak-type $L^{2}(\mathbb{R}^{n})$-estimate on band-limited functions, leads to
several corollaries. The first is a sharp $L^{2}(\mathbb{R}^{n})$ estimate for
the maximal operator restricted to a finite set of rotations in terms of the
cardinality of the finite set. The second is a version of the Carleson–Sjölin
theorem. In addition, we obtain that functions in the Besov space
$B_{p,1}^{0}(\mathbb{R}^{n})$, $2\leq p<\infty$, may be recovered from their
averages along a measurable choice of codimension $1$ subspaces, a form of
Zygmund’s conjecture in general dimension $n$.

###### Key words and phrases:

Directional operators, Zygmund’s conjecture, Stein’s conjecture, maximally
rotated singular integrals, time-frequency analysis

###### 2010 Mathematics Subject Classification:

Primary: 42B25. Secondary: 42B20

O. Bakas is partially supported by the projects SEV-2017-0718,
RYC2018-025477-I, PID2021-122156NB-I00/AEI/10.13039/501100011033 funded by
Agencia Estatal de Investigación and acronym “HAMIP”, Juan de la Cierva
Incorporación IJC2020-043082-I and grant BERC 2022-2025 of the Basque
Government.

F. Di Plinio has been partially supported by the National Science Foundation
under the grants NSF-DMS-2000510, NSF-DMS-2054863, as well as NSF-DMS-1929284
while in residence at the Institute for Computational and Experimental
Research in Mathematics in Providence, RI, during the _Harmonic Analysis and
Convexity_ program in Fall 2022.

I. Parissis is partially supported by the project
PID2021-122156NB-I00/AEI/10.13039/501100011033 funded by Agencia Estatal de
Investigación and acronym “HAMIP”, grant T1615-22 of the Basque Government and
IKERBASQUE

L. Roncal is partially supported by the projects SEV-2017-0718,
RYC2018-025477-I and PID2020-113156GB-I00/AEI/10.13039/501100011033 with
acronym “HAPDE” funded by Agencia Estatal de Investigación, grant BERC
2022-2025 of the Basque Government and IKERBASQUE

##  1\. Introduction

Let $n\geq 2$ be the linear dimension of the Euclidean space $\mathbb{R}^{n}$
endowed with the canonical basis $\\{e_{1},\ldots,e_{n}\\}$ and let
$d\in\\{1,\ldots,n-1\\}$. Singular integrals in $d$ variables embed in
$\mathbb{R}^{n}$ when made to act trivially in the $n-d$ variables
perpendicular to a chosen $d$-dimensional subspace. More specifically, each
pair $(U,\mathscr{K})$ of orientation preserving rotation $U\in\mathrm{SO}(n)$
and tempered distribution $\mathscr{K}\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$
induces a map
$T:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{S}^{\prime}(\mathbb{R}^{n})$ by

(1.1) $\langle Tf,g\rangle\coloneqq\int_{\mathbb{R}^{n-d}}\left\langle f\circ
U(\cdot,y)*\mathscr{K},g\circ U(\cdot,y)\right\rangle{\rm d}y=\left\langle
f\circ U*\left(\mathscr{K}\otimes\delta\right),g\circ U\right\rangle,$

with $\delta\in\mathcal{S}^{\prime}(\mathbb{R}^{n-d})$ denoting the Dirac
distribution supported at $0\in\mathbb{R}^{n-d}$. If the Fourier transform of
$\mathscr{K}$ is a bounded function $m\in L^{\infty}(\mathbb{R}^{d})$ then
$Tf$ coincides with the continuous bounded function

(1.2) $Tf(x)=\int_{\mathbb{R}^{n}}m(V\Pi_{\sigma}\xi)\widehat{f}(\xi)e^{2\pi
i\langle x,\xi\rangle}\,{\rm d}\xi,\qquad x\in\mathbb{R}^{n},$

where $V=U^{-1}$, $\sigma$ is the preimage through $V$ of the subspace
$\mathbb{R}^{d}\coloneqq\mathrm{span}\\{e_{1},\ldots,e_{d}\\}$, and
$\Pi_{\sigma}$ denotes orthogonal projection on $\sigma$. Our interest lies in
the further particular case where $m$ is a Hörmander–Mihlin multiplier on
$\mathbb{R}^{d}$, namely

(1.3) $\|m\|_{\mathcal{M}_{A}(d)}\coloneqq\sup_{0\leq|\alpha|\leq
A}\,\sup_{\eta\in\mathbb{R}^{d}}|\eta|^{|\alpha|}\left|D^{\alpha}m(\eta)\right|$

is finite. If $A\geq d+1$ say, an application of the Hörmander–Mihlin theorem
on each $\sigma$-fiber of $\mathbb{R}^{n}$ and Fubini’s theorem tell us that
(1.2) extends to a bounded operator on $L^{p}(\mathbb{R}^{n}),1<p<\infty$.

The broad object of interest of this article are maximal versions of (1.2)
where the rotation $V\in\mathrm{SO}(n)$ depends measurably, or in some other
specific fashion, on the point $x\in\mathbb{R}^{n}$. Note that the multiplier
(1.2) is allowed to exhibit singularities on the $(n-d)$-dimensional subspace
$\sigma^{\perp}=(V^{-1}\mathbb{R}^{d})^{\perp}$. In light of this fact and of
the invariance of the Mihlin norms (1.3), an efficient parametrization of the
multipliers (1.2) must involve the corresponding oriented Grassmanian
$\mathrm{\bf{Gr}}(d,n)$, namely the space of oriented $d$-dimensional
subspaces of $\mathbb{R}^{n}$. When equipped with the canonical distance
(2.1), recalled below, $\mathrm{\bf{Gr}}(d,n)$ is a compact metric space which
is isometrically isomorphic to the quotient group

$\frac{\mathrm{SO}(n)}{\mathrm{SO}(d)\otimes\mathrm{SO}(n-d)}.$

The description of the isomorphism is useful to us and can be loosely given as
follows. Thinking again
$\mathbb{R}^{d}=\mathrm{span}\\{e_{1},\ldots,e_{d}\\}\in\mathrm{\bf{Gr}}(d,n)$,
a subspace $\sigma\in\mathrm{\bf{Gr}}(d,n)$ is identified with the class of
matrices $\mathcal{V}(\sigma)\subset\mathrm{SO}(n)$ which map $\sigma$ to
$\mathbb{R}^{d}$. Any two elements $V,O\in\mathcal{V}(\sigma)$ are related by

(1.4) $V=(Q\otimes P)O,\qquad Q\in\mathrm{SO}(d),\,P\in\mathrm{SO}(n-d).$

For this reason, fixing in the remainder of our discussion a family

(1.5)
$\\{{O_{\sigma}}\in\mathcal{V}(\sigma):\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$

of representatives, the class of rotated singular integrals (1.2) associated
to a single multiplier $m\in\mathcal{M}_{A}(d)$ may be parametrized as

(1.6)
$T_{m}f(x;\sigma,Q)\coloneqq\int_{\mathbb{R}^{n}}m(Q{O_{\sigma}}\Pi_{\sigma}\xi)\widehat{f}(\xi)e^{2\pi
i\langle x,\xi\rangle}\,{\rm d}\xi,\qquad x\in\mathbb{R}^{n},$

with parameters $\sigma\in\mathrm{\bf{Gr}}(d,n)$ and $Q\in\mathrm{SO}(d)$.
Note that (1.6) does not depend on the component $P\in\mathrm{SO}(n-d)$ in
(1.4), reflecting the trivial action of $T$ in the $\sigma^{\perp}$-variables,
cf. (1.2). Also note that the family in (1.5) can be chosen arbitrarily. A
specific family is explicitly constructed in Lemma 2.3 with the additional
property that the map $\sigma\mapsto O_{\sigma}$ is $C^{1}$; see Remark 2.6.
Finally, the directional operator (1.6) may be further interpreted by
rewriting formally

(1.7)
${}T_{m}f(x;\sigma,Q)=\int_{\mathbb{R}^{d}}f(x-V^{-1}t)\,K_{\sigma}(t){\rm
d}t,\qquad V=Q{O_{\sigma}},\quad x\in\mathbb{R}^{n},$

with $K_{\sigma}(t)$ the kernel of
$\mathscr{K}_{\sigma}=\widehat{m_{\sigma}}$.

The maps (1.6), or alternatively (1.7), are termed _$\mathrm{\bf{Gr}}(d,n)$
-subspace singular integral operators._ For instance, they encompass
$\sigma$-subspace averaging operators, corresponding to the choice
$\mathscr{K}\in L^{1}(\mathbb{R}^{d})$ in (1.1), as well as the
$d$-dimensional Riesz transform in the directions of $\sigma$, obtained by
choosing the vector distribution

$\mathscr{K}(y)\coloneqq\mathrm{p.v.}\frac{y}{|y|^{d+1}}\in\mathcal{S^{\prime}}(\mathbb{R}^{d})^{d}.$

When $d=1$, in particular, these recover the familiar directional maximal
averaging operator, and the maximal directional Hilbert transform
respectively.

Subspace singular integrals, along _variable choices_ of $d$-dimensional
subspaces $\sigma$ are connected to several deep lines of investigations in
harmonic analysis and partial differential equations. This article focuses on
the _codimension 1_ case $n=d+1$. Higher codimension cases are more singular
and very few instances of operators have been treated, none beyond the $L^{2}$
threshold. A full account of the history of this circle of problems is
postponed to the end of this introduction.

Below, we consider the maximal operator associated to a family of multipliers
$\mathscr{m}=\\{m_{\sigma}\in\mathcal{M}_{A}(d):\,\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$,

(1.8)
$T^{\star}_{\mathscr{m}}f(x)\coloneqq\sup_{\sigma\in\mathrm{\bf{Gr}}(d,n)}\sup_{Q\in\mathrm{SO(d)}}\left|T_{m_{\sigma}}f(x;\sigma,Q)\right|,\qquad
x\in\mathbb{R}^{n},$

or linearized variants thereof where the variable choices of $\sigma,Q$ are
expressed by measurable functions
$\sigma(\cdot):\mathbb{R}^{n}\to\mathrm{\bf{Gr}}(d,n)$ and
$Q(\cdot):\mathbb{R}^{n}\to\mathrm{SO}(d)$, possibly under additional
regularity assumptions on these functions. Concerning (1.8), its maximal
nature shows how the choice of the family ${O_{\sigma}}$ in (1.5) is
immaterial.

Our main result concerns the action of (1.8) on frequency band-limited
functions. In order to state it we fix a smooth radial function
$\zeta:\mathbb{R}^{n}\to\mathbb{R}_{+}$ such that
$\mathrm{supp}(\zeta)\subset\\{\xi\in\mathbb{R}^{n}:\,1<|\xi|<\frac{3}{2}\\}$
and define for $k\in\mathbb{R}$

(1.9)
$P_{k}f(x)\coloneqq\int_{\mathbb{R}^{n}}\zeta\left(2^{-k}|\xi|\right)\widehat{f}(\xi)e^{2\pi
i{\langle x,\xi\rangle}}\,{\rm d}\xi,\qquad x\in\mathbb{R}^{n}.$

######  Theorem A.

There exists $A=A(d)$ such that the following holds. Suppose that the family
$\mathscr{m}\coloneqq\\{m_{\sigma}\in
L^{\infty}(\mathbb{R}^{d}):\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ is such that

(1.10)
$\begin{split}\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\coloneqq\sup_{\sigma,\tau\in\mathrm{\bf{Gr}}(d,n)}\left[\|m_{\sigma}\|_{\mathcal{M}_{A}(d)}+\log\left({\rm
e}+{\textstyle\frac{1}{{\rm
dist}(\sigma,\tau)}}\right)\|m_{\tau}-m_{\sigma}\|_{\mathcal{M}_{A}(d)}\right]&\leq
1.\end{split}$

Referring to (1.8), there holds

(1.11) $\left\|T^{\star}_{\mathscr{m}}\circ P_{0}:L^{2}(\mathbb{R}^{n})\to
L^{2,\infty}(\mathbb{R}^{n})\right\|\lesssim
1,\qquad\left\|T^{\star}_{\mathscr{m}}\circ
P_{0}:L^{p}(\mathbb{R}^{n})\right\|\lesssim_{p}1,\quad 2<p<\infty.$

######  Remark 1.1.

The norm (1.3) is invariant under isotropic scaling of $\mathbb{R}^{d}$.
Condition (1.10) inherits such invariance, whence (1.11) holds with the same
constants for $T^{\star}_{\mathscr{m}}\circ P_{k}$ as well, $k\in\mathbb{R}$.

######  Remark 1.2.

A sufficient condition for assumption (1.10) to hold is the following. Let

$\|\mathscr{m}\|_{\mathcal{M}_{A,1}(\mathrm{\bf{Gr}}(d,n))}\coloneqq\sup_{\sigma\in\mathrm{\bf{Gr}}(d,n)}\sup_{0\leq|\beta|\leq
1}\|{\mathscr{D}}_{\sigma}^{\beta}m_{\sigma}\|_{\mathcal{M}_{A}(d)}$

where the differential operator $\mathscr{D}_{\sigma}$ is the natural
invariant vector of $\sigma$-derivatives; an explicit description is given in
Lemma 2.4. Since

$\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\lesssim\|\mathscr{m}\|_{\mathcal{M}_{A,1}(\mathrm{\bf{Gr}}(d,n))}$

it follows that the conclusion of Theorem A holds under the assumption
$\|\mathscr{m}\|_{\mathcal{M}_{A,1}(\mathrm{\bf{Gr}}(d,n))}\leq 1$.

In order to further emphasize the interpretation of our operators as
directional multipliers and draw a formal parallel to the two-dimensional case
of [LacLi:tams], let us rewrite (1.7) in yet another form. Equip
$\mathbb{R}^{n}$ with the standard basis $(e_{1},\ldots,e_{d},e_{n})$ and
define an orthonormal basis on $\sigma$ by setting

$v_{j}^{\sigma}\coloneqq V^{-1}e_{j},\qquad
j\in\\{1,\ldots,d\\},\quad\mathrm{span}(v_{1}^{\sigma},\ldots,v_{d}^{\sigma})=\sigma,\qquad
V\in\mathcal{V}(\sigma).$

Then, calculate

$V\Pi_{\sigma}\xi=V\sum_{j=1}^{d}\langle\xi,v_{j}^{\sigma}\rangle
v_{j}^{\sigma}=\sum_{j=1}^{d}\langle\xi,v_{j}^{\sigma}\rangle
e_{j}=\left(\langle\xi,v_{1}^{\sigma}\rangle,\ldots,\langle\xi,v_{d}^{\sigma}\rangle\right)\in\mathbb{R}^{d}$

so that (1.6) takes the form

$\begin{split}T_{m}f(x;\sigma,Q)=\int_{\mathbb{R}^{n}}m\left(\langle\xi,v_{1}^{\sigma}\rangle,\ldots,\langle\xi,v_{d}^{\sigma}\rangle\right)\widehat{f}(\xi)e^{2\pi
i{\langle x,\xi\rangle}}\,{\rm d}\xi,\qquad x\in\mathbb{R}^{n}.\end{split}$

Then Theorem A tells us that the operator

$f\mapsto\sup_{\sigma\in\mathrm{\bf{Gr}}(d,n)}\sup_{v_{1}^{\sigma},\ldots,v_{d}^{\sigma}}\left|\int_{\mathbb{R}^{n}}m_{\sigma}\left(\langle\xi,v_{1}^{\sigma}\rangle,\ldots,\langle\xi,v_{d}^{\sigma}\rangle\right)\widehat{f}(\xi)e^{2\pi
i{\langle\cdot,\xi\rangle}}\,{\rm d}\xi\right|$

is of weak-type $(2,2)$ and strong type $(p,p)$, $p>2$, whenever
$\mathscr{m}=\\{m_{\sigma}\in
L^{\infty}(\mathbb{R}^{d}):\,\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ satisfies
$\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\leq 1$; above, the
inner supremum is over all orthonormal bases of $\sigma$. Specializing to a
single multiplier $m\in\mathcal{M}_{A}(d)$ we record below an immediate
corollary.

######  Corollary A.1.

Let $m\in\mathcal{M}_{A}(d)$ and consider the maximal directional multiplier

$T^{\star}_{m}f(x)\coloneqq\sup_{(v_{1},\ldots,v_{d})}\left|\int_{\mathbb{R}^{n}}m\left(\langle\xi,v_{1}\rangle,\ldots,\langle\xi,v_{d}\rangle\right)\widehat{f}(\xi)e^{2\pi
i\langle x,\xi\rangle}\,{\rm d}\xi\right|,\qquad x\in\mathbb{R}^{n},$

where the supremum is taken over all orthonormal $d$-tuples
$(v_{1},\ldots,v_{d})\in(\mathbb{R}^{n})^{d}$. Then $T^{\star}_{m}\circ P_{0}$
maps $L^{2}(\mathbb{R}^{n})$ to $L^{2,\infty}(\mathbb{R}^{n})$ and
$L^{p}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for all $2<p<\infty$.

We encourage the reader to compare with the very familiar two-dimensional
directional multipliers, [Dem, LacLi:tams], given in the form

$Tf(x,v)\coloneqq\int_{\mathbb{R}^{2}}m\left(\langle\xi,v\rangle\right)\widehat{f}(\xi)e^{2\pi
i{\langle x,\xi\rangle}}\,{\rm
d}\xi,\qquad(x,v)\in\mathbb{R}^{2}\times\mathbb{S}^{1},$

when $m\in\mathcal{M}_{A}(1)$ is a one-dimensional Hörmander–Mihlin
multiplier. In the two-dimensional case, $\mathrm{SO}(1)$ is trivial, so that
such dependence may be omitted.

When the band limited restriction imposed by precomposing with $P_{k}$ is
lifted and no regularity assumptions are placed on the subspace choice
function, such function $\sigma(\cdot)$ is allowed to be oriented along a
suitable Kakeya–Nikodym set and the subspace singular integrals (1.8) are in
general unbounded on $L^{p}(\mathbb{R}^{n})$. One particularly deep line of
investigation is seeking for suitable regularity assumptions on $\sigma$
bypassing the above mentioned counterexample and ultimately leading to
$L^{p}$-bounds. For instance, if $\sigma(\cdot)$ is $1$-Lipschitz and
$m_{\sigma}(\cdot)$ is supported on frequency scales $\gg 1$, then Kakeya-type
counterexamples are avoided. Zygmund suggested that — in this context — a
suitably truncated version of the averaging directional operator along a
Lipschitz choice of subspace should be bounded, at least above a critical
$L^{p}$-space which can be identified for each dimension $n$ and codimension
$n-d$, see [DPP_adv]. The corresponding version of this conjecture for
multipliers $m_{\sigma}$, allowed to be singular along $\sigma^{\perp}$ is
usually attributed to Stein. These are the Zygmund and Stein conjectures
alluded to above and underlying the investigations in this paper.

For the case of directional singular integrals, namely when $m_{\sigma}$ is
allowed to be singular along $\sigma^{\perp}$, further counterexamples of non-
Kakeya type do exist even when the range of $\sigma(\cdot)$ has special (e.g.
lacunary) structure as has been exhibited in [Karag, LMP]. For this reason it
is customary, at least in the case $d=1$, to study $\sigma$ with finite range
and prove optimal bounds in terms of the cardinality of such range. Optimal
cardinality bounds for directional singular integrals are mostly known in the
two-dimensional case, see [Dem, DeDP], while in higher dimensions $n\geq 2$
and $d=1$ optimal bounds for directional multipliers and $\sigma$ with
lacunary range are contained in [ADP, DPP2].

In the present paper we prove the first such result in higher dimensions
$n\geq 2$ and codimension $n-d=1$. It is worth mentioning that while most of
the results in the literature deal with maximal singular integrals generated
by a single multiplier, the statement below allows for a log-Hölder dependence
of the family $m_{\sigma}$ in $\sigma\in\mathrm{\bf{Gr}}(d,n)$. A two-
dimensional partial analogue has appeared in [KaLa], where the authors even
allow for a measurable with respect to $\sigma$ choice of multipliers, albeit
with sub-optimal bounds in terms of the cardinality.

######  Corollary A.2.

Suppose that the family $\mathscr{m}\coloneqq\\{m_{\sigma}\in
L^{\infty}(\mathbb{R}^{d}):\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ satisfies
$\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\leq 1$. With
reference to (1.6), the maximal operator

(1.12)
$T^{\star}_{\Sigma,\mathscr{m}}f(x)\coloneqq\sup_{\sigma\in\Sigma}\sup_{Q\in\mathrm{SO}(d)}\left|T_{m_{\sigma}}f(x;\sigma,Q)\right|,\qquad
x\in\mathbb{R}^{n},\qquad\Sigma\subset\mathrm{\bf{Gr}}(d,n),$

satisfies the norm inequality

$\sup_{\begin{subarray}{c}\Sigma\subset\mathrm{\bf{Gr}}(d,n)\\\ \\#\Sigma\leq
N\end{subarray}}\big{\|}T^{\star}_{\Sigma,\mathscr{m}}\big{\|}_{L^{2}(\mathbb{R}^{n})}\lesssim_{n}\log
N$

and this is best possible up to the implicit dimensional constant.

From the proof of Theorem A, it is also possible to deduce weak $(2,2)$ and
strong $(p,p)$ bounds, with $2<p<\infty$, for maximally truncated versions of
(1.8). When considering the multiplier $m_{\sigma}\equiv 1$, this corresponds
to a maximal subspace averaging operator. The obtained mapping properties can
then be used to deduce subspace Lebesgue differentiation theorems for
functions in the Besov spaces $B_{p,1}^{s}(\mathbb{R}^{n})$, $s\geq 0$. These
recover past results due to Murcko [Mur], see also [AFN, Naibo], for the case
$s>0,n=2$, and appear to be new in the cases $s=0$ or $n>2$. A detailed
statement is given in Theorem C, Subsection 7.1.

Further motivation, as well as influence on the proof techniques, for Theorem
A comes from the connection with maximally modulated singular integrals. In
the two-dimensional case, $n=2=d+1$, there are well-explored [BaTh,
LacLi:tams] ties between the band-limited behavior of the maximal directional
Hilbert transform and the boundedness of the Carleson maximal partial inverse
Fourier transform operator [car]. In a similar fashion, Theorem A implies
$L^{p}$-estimates for maximally modulated Hörmander–Mihlin multipliers in the
spirit of Sjölin [sjolin].

######  Theorem B.

Suppose $m\in\mathcal{M}_{A}(d)$. The conclusion of Theorem A implies that the
Carleson–Sjölin operator

$\mathsf{CS}[f](x)\coloneqq\sup_{N\in\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}\widehat{f}(\eta)m(\eta+N)e^{2\pi
i\langle\eta,x\rangle}\,{\rm d}\eta\right|,\qquad x\in\mathbb{R}^{d},$

maps $L^{2}(\mathbb{R}^{d})$ to $L^{2,\infty}(\mathbb{R}^{d})$ and
$L^{p}(\mathbb{R}^{d})$ to $L^{p}(\mathbb{R}^{d})$ for all $2<p<\infty$.

The transference type argument leading to the proof of Theorem B is a modified
version of the observation, commonplace in the literature, that any horizontal
frequency cutoff to a half-line of a function supported on a thin vertical
frequency strip may be obtained by applying a directional Hilbert transform
with a suitably chosen slope. The connection to Carleson-type theorems and,
more generally, modulation invariant operators, is in fact also apparent from
the proof of Theorem A itself, which, much like its predecessors in
[LacLi:mem, BatRMI], borrows from the Lacey–Thiele argument for Carleson’s
maximal operator, [LTC].

Due to the higher dimensional nature of the problem considered, several new
conceptual and technical difficulties arise. A first element of proof, which
is only relevant in ambient dimension $n\geq 3$, addresses the possibility of
choosing several rotations mapping $\mathbb{R}^{d}$ to a given
$\sigma\in\mathrm{\bf{Gr}}(d,n)$. The underlying rotation invariance allows
for arbitrary choices of coordinates on each $\sigma$, introducing aritificial
discontinuities to the problem in hand and is readily appreciated in formula
(1.6). The necessary reduction which allows us to dispose of the arbitrary
rotation $Q$ in (1.6) and instead use canonical smooth rotations is contained
in §3; the study of the model case of maximally rotated translation invariant
Calderón–Zygmund operators is instructive and is also presented in §3.
Additionally, the higher dimensional time-frequency analysis of the current
paper combined with the directional nature of the operator under study leads
to a novel model sum in terms of wave-packets conforming to the geometry of
the operator, and requiring us to work with a choice function, dictating the
measurable choice of subspaces from $\mathrm{\bf{Gr}}(d,n)$, which localizes
in higher dimensional frequency caps.

The final paragraph of this introduction serves as a more comprehensive
historical account of past progress on maximal directional singular and
averaging operators. This subject has attracted considerable attention in the
last fifty years, mainly in connection to the Kakeya maximal conjecture. In
this context, the dimension $d$ of the averaging manifold is 1, and the
connection may be described through the $L^{p}$-bounds, $1<p<\infty$, for the
single and multi-scale directional maximal functions

${\mathrm{M}}_{V,1}f(x)\coloneqq\sup_{v\in
V}\int_{-\frac{1}{2}}^{\frac{1}{2}}|f(x-vt)|\,{\rm
d}t,\quad{\mathrm{M}}_{V}f(x)\coloneqq\sup_{r>0}\sup_{v\in V}\
\frac{1}{2r}\int_{-r}^{r}|f(x-vt)|\,{\rm d}t,\quad x\in\mathbb{R}^{n}$

associated to a generic $V\subseteq\SS^{n-1}$. The existence of Kakeya sets
implies that these bounds blow up as $\delta\to 0^{+}$ when $V$ is a
$\delta$-net on $\SS^{n-1}$. When $n=2$, tight upper bounds for $\delta$-nets
are known and due by Strömberg, [Stromberg], and Córdoba, [Cor77], who e.g.
prove sharp logarithmic blowup in terms of $\delta$ above
$L^{2}(\mathbb{R}^{2})$, which is the critical space for the Kakeya maximal
function in $\mathbb{R}^{2}$. These results imply the maximal Kakeya
conjecture in two dimensions and for example we have

$\|M_{V,1}\|_{L^{2}(\mathbb{R}^{2})}\eqsim\sqrt{|\log\delta|},\qquad\|{\mathrm{M}}_{V}\|_{L^{2}(\mathbb{R}^{2})}\eqsim|\log\delta|.$

In parallel it was known that if $V$ has additional structure, namely if it is
a _lacunary set of directions_ , then ${\mathrm{M}}_{V}$ is bounded on all
$L^{p}(\mathbb{R}^{2})$ spaces, $1<p\leq\infty$, as was shown collectively by
[CorFeflac, NSW, SS]. Bateman [Bat] showed that in fact ${\mathrm{M}}_{V}$ is
bounded on some (equivalently all) $L^{p}(\mathbb{R}^{2})$ spaces,
$1<p\leq\infty$, if and only if $V$ is a lacunary set. This means that if $V$
is an arbitrary infinite set of directions then ${\mathrm{M}}_{V}$ will be
generically unbounded so that one can assume that $\\#V<+\infty$ and seek for
best possible bounds in terms of the cardinality $\\#V$. This was done by Katz
who showed the best possible bounds for ${\mathrm{M}}_{V,1}$ in [KB], and for
${\mathrm{M}}_{V}$ in [Katz]. These bounds match the optimal bounds for
$\delta$-nets stated above.

The maximal directional Hilbert transform

$H_{V}f(x)\coloneqq\sup_{v\in V}|H_{v}f(x)|,\qquad
H_{v}f(x)\coloneqq\mathrm{p.v.}\int_{\mathbb{R}}f(x-vt)\frac{{\rm
d}t}{t},\qquad x\in\mathbb{R}^{2},$

is always unbounded if $\\#V=\infty$; this was shown for
$L^{2}(\mathbb{R}^{2})$ by Karagulyan in [Karag] and for all
$L^{p}(\mathbb{R}^{n})$ by Łaba, Marinelli and Pramanik in [LMP]. The upper
bounds for $H_{V}$ and $V$ lacunary were shown by Demeter and one of the
authors, [DeDP]; see also [DPP], resulting to the sharp estimate

$\sup_{\begin{subarray}{c}V\,\textrm{lacunary}\\\ \\#V\leq
N\end{subarray}}\|H_{V}\|_{L^{2}(\mathbb{R}^{n})}\eqsim\sqrt{\log N}.$

The contributions [Dem, DeDP], containing what is essentially the $n=2$ case
of Corollary A.2, have already been recalled. Finally the $n=2$ case of
Theorem A has a partial analogue in a result of Lacey and X. Li [LacLi:tams].
Therein, the authors pursued this question as an intermediate step towards the
two-dimensional version of the Stein conjecture. Further results are available
under other specific structural assumptions on the linearizing choice function
for the directions $v$. The articles [BatRMI] by Bateman and [BaTh] by Bateman
and Thiele deal with the case where $v$ is almost horizontal and does not
depend on the vertical variable. In particular, the latter article showed that
this structure entails $L^{p}(\mathbb{R}^{2})$-bounds for all $3/2<p<\infty$.
Variants of these assumptions have been dealt with by S. Guo [Guo, Guo2]; see
also [DPGTZK] for a comprehensive result of this type. In a different
direction Bourgain [Bourgain] has proved that a suitably truncated version of
$M_{v(\cdot)}$ is bounded on $L^{2}(\mathbb{R}^{2})$ when $v$ satisfies a
certain curvature condition, which in particular is verified by real analytic
vector fields; see [Guo3] for a geometric proof. Analogous results for the
corresponding truncated directional Hilbert transform along $v$ real analytic
fields are due to obtained by Stein and Street, [StStr].

In higher dimensions, $n\geq 3$ most of the known results concern the case of
codimension $n-1$ so that $d=1$. Parcet and Rogers [PR2] extended the notion
of lacunarity to any dimension and showed that ${\mathrm{M}}_{V}$ is bounded
on all $L^{p}(\mathbb{R}^{n})$-spaces under the assumption that $V$ is
lacunary. A partial converse in the spirit of Bateman is also proved in [PR2]
but a full characterization is still pending; for $d>1$ there is currently no
notion of lacunary subsets of $\mathrm{\bf{Gr}}(d,n)$ in the literature. For
arbitrary $V\subset\SS^{n-1}$ sharp $L^{2}(\mathbb{R}^{n})$-bounds were shown
for ${\mathrm{M}}_{V,1}$ by two of the authors using the polynomial method,
[DPPalg]. Regarding ${\mathrm{M}}_{V}$, the special case where
$V\subset\SS^{n-1}$ is a $\delta$-net is considered in [Kim]. Sharp
$L^{2}(\mathbb{R}^{n})$-bounds for maximal single-scale subspace averages are
proved in [DPP_adv] for any $1\leq d\leq n-1$; these include the case of
codimension $n-d=1$ where $L^{2}(\mathbb{R}^{n})$ is the critical space.

Before this work, maximal directional singular multipliers had only been
studied for $d=1$. When $V$ is lacunary and $n\geq 3$, Accomazzo and two of
the authors have proved sharp $L^{p}(\mathbb{R}^{n})$-bounds in [ADP]. In
[KP], Kim and Pramanik prove the sharp $L^{2}(\mathbb{R}^{n})$-bounds for
$V\subset\SS^{n-1}$ arbitrary with $\\#V<+\infty$. It should be noted that
lacunary sets yield the same bound of the order $\sqrt{\log\\#V}$ for all
$L^{p}$-norms of the maximal directional singular integral in all dimensions.
For $V\subset\SS^{n-1}$ arbitrary this is no longer the case and logarithmic
cardinality bounds can only be expected above the critical exponent $p=n$.
This problem would however entail the resolution of the Kakeya conjecture and
is currently wide open. Note that Corollary A.2 above provides the first sharp
bound for maximal directional multipliers and $d>1$; since in our case the
codimension $n-d=1$ the critical space for this problem is still
$L^{2}(\mathbb{R}^{n})$, as in the case $n=2=d+1$.

### Structure of the paper

Section 2 contains a few geometric preliminaries on the metric and
differential structure of $\mathrm{\bf{Gr}}(d,n)$. In Section 3, Theorem A is
reduced to a simpler version by removing the $\mathrm{SO}(d)$ maximality, see
Proposition 3.3 for a precise statement. Section 4 is devoted to the deduction
of Corollary A.2 and Theorem B. Section 5 further reduces Proposition 3.3 to
the estimation of a time-frequency model operator, which is finally performed
in Section 6. The concluding Section 7 is devoted to complementary results and
questions on maximally truncated and bi-parametric variable
$\mathrm{\bf{Gr}}(d,n)$-subspace singular integrals.

##  2\. Geometric preliminaries

###  2.1. $\mathrm{\bf{Gr}}(d,n)$ as a metric space and notational remarks

Recall that $\mathrm{\bf{Gr}}(d,n)$, $1\leq d<n$ is a compact metric space
when endowed with

(2.1) ${\rm
dist}^{\prime}(\sigma,\tau)\coloneqq\sup_{\omega\in\SS^{n-1}}|\Pi_{\sigma}\omega-\Pi_{\tau}\omega|,\qquad\sigma,\tau\in\mathrm{\bf{Gr}}(d,n),$

and the map

$\sigma\mapsto\sigma^{\perp},\qquad\mathrm{\bf{Gr}}(d,n)\to\mathrm{\bf{Gr}}(n-d,n),$

is an isometric isomorphism. The remainder of this article is concerned with
the codimension $1$ case, $d=n-1$. In this case $\mathrm{\bf{Gr}}(d,n)$ is
naturally identified with $\mathbb{S}^{d}$, and this identification is
exploited with the notation $v_{\sigma}\in\mathbb{S}^{d}$ as the unit normal
vector to $\sigma$. Indeed, elementary computations show

(2.2) ${\rm
dist}^{\prime}(\sigma,\tau)=|\sin[\theta(v_{\sigma},v_{\tau})]|\eqsim\sqrt{2}\sqrt{1-\cos[\theta(v_{\sigma},v_{\tau})]}=|v_{\sigma}-v_{\tau}|$

where $\theta(u,v)$ stands for the convex angle between
$u,v\in\mathbb{S}^{d}$. The equivalence constant implied by $\eqsim$ is
obviously independent of dimension. Because of (2.2) above it will be
convenient for us to redefine the distance on $\mathrm{\bf{Gr}}(d,n)$ for the
special case $d=n-1$ in hand as follows

(2.3) ${\rm
dist}(\sigma,\tau)\coloneqq|v_{\sigma}-v_{\tau}|,\qquad\sigma,\tau\in\mathrm{\bf{Gr}}(d,n),\qquad
d=n-1.$

We set up a unified notation for balls in metric spaces. The standing
convention is that $\xi\in X$ then $B(\xi,r)$ is the ball in the metric space
$X$, centered at $\xi$ and of radius $r>0$. The default metric on linear
subspaces of $\mathbb{R}^{n}$ and $\mathbb{S}^{d}$ is the Euclidean metric and
we write $B^{m}(R)$ for the Euclidean ball in $\mathbb{R}^{m}$, centered at
$0$ and having radius $R>0$. In $\mathrm{\bf{Gr}}(d,n)$, the metric employed
is defined above in (2.3). We also adopt the following notation for annuli in
a generic metric space $X$

$\mathrm{Ann}(x,r,R)\coloneqq B(x,R)\setminus\overline{B(x,r)},\qquad
0<r<R,\quad x\in X.$

When $X$ is a linear space the shorthand $\mathrm{Ann}(r,R)$ is also used in
place of $\mathrm{Ann}(0,r,R)$ and $B(R)$ is used in place of $B(0,R)$.

###  2.2. Smooth families of rotations

The next lemma constructs a typical family of Lipschitz rotations of
$\mathbb{R}^{n}$. These are essentially two-dimensional rotations between
$\sigma$ and $\tau$ acting in the plane perpendicular to the common
$(d-1)$-dimensional subspace
$\sigma\cap\tau=(\mathrm{span}\\{v_{\sigma},v_{\tau}\\})^{\perp}$.

######  Lemma 2.3.

Let $\sigma,\tau\in\mathrm{\bf{Gr}}(d,n)$. There exist
$O_{\tau,\sigma}\in\mathrm{SO}(n)$ such that

$O_{\tau,\sigma}\sigma=\tau,\qquad
O_{\tau,\sigma}v_{\sigma}=v_{\tau},\qquad\left\|O_{\tau,\sigma}-\mathrm{Id}\right\|={\rm
dist}(\sigma,\tau).$

###### Proof.

If $\sigma=\tau$ take $O_{\tau,\sigma}=\mathrm{Id}$. Otherwise define
$\zeta\coloneqq\sigma\cap\tau\in\mathrm{\bf{Gr}}(d-1,n)$; see Figure 2.1.
Since $\zeta^{\perp}=\mathrm{span}\\{v_{\sigma},v_{\tau}\\}$ and
$\sigma\cap\zeta^{\perp}$ is a one-dimensional subspace of $\zeta^{\perp}$, we
may pick the unique unit vector $u_{\sigma}\in\sigma\cap\zeta^{\perp}$
satisfying the equation

$v_{\tau}=-(\sin\theta)u_{\sigma}+(\cos\theta)v_{\sigma},\qquad\theta\coloneqq\arccos(\langle
v_{\sigma},v_{\tau}\rangle).$

Let $O_{\tau,\sigma}$ be the rotation that acts as the identity on $\zeta$ and
such that

$O_{\tau,\sigma}u_{\sigma}=u_{\tau}\coloneqq(\cos\theta)u_{\sigma}+(\sin\theta)v_{\sigma},\qquad
O_{\tau,\sigma}v_{\sigma}=v_{\tau}.$

$x$$y$$z$$\sigma$$\tau$$\zeta^{\perp}$$\zeta=\sigma\cap\tau$ Figure 2.1. A
figure for the proof of Lemma 2.3

Note that $u_{\tau}$ is a unit vector orthogonal to $\zeta$ and $v_{\tau}$,
therefore $\tau=\textrm{span}\,\\{\zeta,u_{\tau}\\}=O_{\tau,\sigma}\sigma$. As
$O_{\tau,\sigma}-\mathrm{Id}=[O_{\tau,\sigma}-\mathrm{Id}]\Pi_{\zeta^{\perp}}$,
the equality

$\left\|O_{\tau,\sigma}-\mathrm{Id}\right\|=\sqrt{2}\sqrt{1-\langle
v_{\sigma},v_{\tau}\rangle}=|v_{\sigma}-v_{\tau}|={\rm dist}(\sigma,\tau)$

follows by immediate two-dimensional trigonometry. ∎

We give below an explicit description of the tangent bundle
$\mathrm{T}\mathrm{\bf{Gr}}(d,n)$. This is used in the sequel to give a
concrete notion of derivatives of functions defined on
$\mathrm{\bf{Gr}}(d,n)$. In turn this will help us establish the smoothness of
the map $\mathrm{\bf{Gr}}(d,n)\ni\sigma\mapsto O_{\rho,\sigma}$ for fixed
$\rho\in\mathrm{\bf{Gr}}(d,n)$.

######  Lemma 2.4.

Let $\sigma\in\mathrm{\bf{Gr}}(d,n)$ and let
$\\{v^{\sigma}_{1},\cdots,v^{\sigma}_{d}\\}$ be an orthonormal basis of
$\sigma$ and as usual $\sigma^{\perp}=\mathbb{R}v_{\sigma}$. The tangent space
$\mathrm{T}_{\sigma}\mathrm{\bf{Gr}}(d,n)$ is the linear span of

$X_{v_{j}^{\sigma}}\in\mathfrak{so}(n),\quad j=1,\ldots,d,\qquad
X_{v_{j}^{\sigma}}[v_{k}^{\sigma}]=\begin{cases}\vec{0},&k\neq
j\vspace{.8em}\\\ v_{\sigma},&k=j\end{cases},\quad 1\leq k\leq d,\qquad
X_{v_{j}^{\sigma}}v_{\sigma}=-v^{\sigma}_{j},$

where $\mathfrak{so}(n)$ stands for $n\times n$ real skew-symmetric matrices.
Furthermore, if $\xi\in\mathbb{R}^{n}$ is a fixed vector then the following
hold

$\partial_{{v_{j}^{\sigma}}}(\Pi_{\sigma}\xi)=X_{v_{j}^{\sigma}}\xi,\qquad\partial_{{v_{j}^{\sigma}}}(\Pi_{\sigma^{\perp}}\xi)=-X_{v_{j}^{\sigma}}\xi,\qquad\,j=1,\ldots,d,$

and the vector of derivatives $\mathscr{D}_{\sigma}$ can be described as

$\mathscr{D}_{\sigma}=\big{(}\partial_{v_{1}^{\sigma}},\ldots,\partial_{{v_{d}^{\sigma}}}\big{)},\qquad\sigma\in\mathrm{\bf{Gr}}(d,n).$

###### Proof.

A basis of the tangent space $\mathrm{T}_{\sigma}\mathrm{\bf{Gr}}(d,n)$ at
$\sigma\in\mathrm{\bf{Gr}}(d,n)$ can be given by
$(X_{v_{1}^{\sigma}},\ldots,X_{v_{d}^{\sigma}})$ where we define
$X_{v_{j}^{\sigma}}$ for $1\leq j\leq d$ to be the tangent vector at $t=0$ to
the $\mathrm{\bf{Gr}}(d,n)$-valued curve

$\sigma(t;j)\coloneqq\mathrm{span}\,\\{v^{\sigma}_{1},\ldots,(\cos
t)v^{\sigma}_{j}+(\sin t)v_{\sigma},\ldots,v^{\sigma}_{d}\\}.$

For the second claim we compute

$\begin{split}\Pi_{\sigma(t;j)}\xi-\Pi_{\sigma}\xi&=\langle\xi,(\cos
t)v^{\sigma}_{j}+(\sin t)v_{\sigma}\rangle((\cos t)v^{\sigma}_{j}+(\sin
t)v_{\sigma})-\langle\xi,v_{j}^{\sigma}\rangle v_{j}^{\sigma}\\\ &=(\sin t\cos
t)\left[\langle\xi,v^{\sigma}_{j}\rangle
v_{\sigma}+\langle\xi,v_{\sigma}\rangle v^{\sigma}_{j}\right]+[(\cos
t)^{2}-1]\langle\xi,v^{\sigma}_{j}\rangle v^{\sigma}_{j}+(\sin
t)^{2}\langle\xi,v_{\sigma}\rangle v_{\sigma}\\\ &=(\sin t\cos
t)\left[\langle\xi,v^{\sigma}_{j}\rangle
v_{\sigma}+\langle\xi,v_{\sigma}\rangle
v^{\sigma}_{j}\right]+O(t^{2})\end{split}$

so that

$\partial_{v_{j}^{\sigma}}(\Pi_{\sigma}\xi)=\lim_{t\to
0}\frac{\Pi_{\sigma(t;j)}\xi-\Pi_{\sigma}\xi}{t}=\langle\xi,v^{\sigma}_{j}\rangle
v_{\sigma}+\langle\xi,v_{\sigma}\rangle v^{\sigma}_{j}=X_{v_{j}^{\sigma}}\xi$

as claimed. Finally,

$0=\partial_{{v_{j}^{\sigma}}}\xi=\partial_{{v_{j}^{\sigma}}}(\Pi_{\sigma}\xi)+\partial_{{v_{j}^{\sigma}}}(\Pi_{\sigma^{\perp}}\xi)$

whence the corresponding conclusion. ∎

The last lemma of this short section contains a computation of the
$\sigma$-derivatives of the $\mathrm{SO}(n)$-valued map

$\mathrm{\bf{Gr}}(d,n)\ni\sigma\mapsto R_{\sigma}\coloneqq
O_{\sigma,\rho}\qquad\sigma\in\mathrm{Gr}(d,n),$

where $O_{\sigma,\rho}$ references Lemma 2.3 and $\rho\in\mathrm{Gr}(d,n)$ is
kept fixed and will henceforth be omitted from the notation. In general, for
$\rho\in\mathrm{\bf{Gr}}(d,n)$ fixed we consider the following mappings:

  * $\cdot$

$\sigma\mapsto R_{\sigma}$ is the rotation on the plane
$\mathrm{span}\\{v_{\sigma},v\\}$ with $R_{\sigma}v=v_{\sigma}$; remember that
$v=v_{\rho}$.

  * $\cdot$

$\sigma\mapsto\theta_{\sigma}\coloneqq\arccos(\langle v_{\sigma},v\rangle).$

  * $\cdot$

$\sigma\mapsto u_{\sigma}$, where $u_{\sigma}$ is the unit vector of $\rho$
which is perpendicular to $\sigma\cap\rho$ with $\langle
u_{\sigma},v_{\sigma}\rangle<0$, and therefore lies in the plane
$\mathrm{span}\\{v_{\sigma},v\\}$.

  * $\cdot$

$\sigma\mapsto U_{\sigma}\coloneqq R_{\sigma}u_{\sigma}$.

$\theta_{\sigma}$
---
$U_{\sigma}$
---
$v=v_{\rho}$
---
$v_{\sigma}$
---
$\sigma\cap\rho$
---
$\sigma$
---
$u_{\sigma}$
---
$\rho$
---
Figure 2.2. A figure for the proof of Lemma 2.5.

With these notations, depicted in Figure 2.2, there holds

(2.4) $U_{\sigma}=(\cos\theta_{\sigma})u_{\sigma}+(\sin\theta_{\sigma})v,\quad
R_{\sigma}v=v_{\sigma},\quad R_{\sigma}w=w\quad\forall w\in\sigma\cap\rho.$

For the statement and proof of the lemma below it will be convenient to write
$\mathbb{R}^{n}$ as the direct sum

$\mathbb{R}^{n}=(\sigma\cap\rho)\oplus\mathbb{R}u_{\sigma}\oplus\mathbb{R}v.$

Note then that $\sigma$ can be described in the form
$\sigma=\sigma\cap\rho\oplus\mathbb{R}U_{\sigma}$. Now Lemma 2.4 shows that in
order to determine $\mathscr{D}_{\sigma}^{\beta}O_{\rho}(\sigma)$ for
$|\beta|\leq 1$ it suffices to determine $\partial_{v_{j}^{\sigma}}R_{\sigma}$
for each vector from a suitably chosen basis of $\sigma$. This is effectively
accomplished by the next lemma, as one may complete $U_{\sigma}$ to an
orthonormal basis of $\sigma$ by choosing $d-1$ orthonormal vectors in
$\sigma\cap\rho$.

######  Lemma 2.5.

Let $a,b,c\in\mathbb{R}$ and $\rho\in\mathrm{\bf{Gr}}(d,n)$ be fixed. For
every $\sigma\in\mathrm{\bf{Gr}}(d,n)$ we have

$\partial_{U_{\sigma}}R_{\sigma}[w+bu_{\sigma}+cv]=bv_{\sigma}-cU_{\sigma},\qquad\forall
w\in\sigma\cap\rho.$

Furthermore for all $w\in\sigma\cap\rho$ there holds

$\partial_{w}R_{\sigma}[z+aw+bu_{\sigma}+cv]=[b\varpi(\theta_{\sigma})-c]w+a\varpi(\theta_{\sigma})u_{\sigma}+av\qquad\forall
z\in\sigma\cap\rho\cap w^{\perp},$

for the bounded function
$\varpi(\beta)\coloneqq\frac{\cos\beta-1}{\sin\beta}$.

######  Remark 2.6.

In particular, $\sup_{0\leq|\beta|\leq
1}\|\mathscr{D}^{\beta}_{\sigma}R_{\sigma}\|\lesssim
1+|\varpi(\theta)|\lesssim_{d}1.$ Integrating along the geodesic connecting
$\sigma,\tau\in\mathrm{\bf{Gr}}(d,n)$, we obtain the operator norm estimate

$\left\|R_{\tau}-R_{\sigma}\right\|=\left\|R_{\tau}R_{\sigma}^{-1}-\mathrm{Id}\right\|\lesssim{\rm
dist}(\tau,\sigma).$

Throughout the paper we reserve the notation
$O_{\sigma}\in\mathcal{V}(\sigma)$ for the rotations

$O_{\sigma}=O_{\mathbb{R}^{d},\sigma}\in\mathrm{SO}(n),\qquad
O_{\sigma}\sigma=\mathbb{R}^{d},\qquad O_{\sigma}v_{\sigma}=e_{n}.$

Then, if the family
$\\{O_{\sigma}\in\mathcal{V}(\sigma):\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ is
constructed via Lemma 2.3, we will have

(2.5) $\sup_{0\leq|\beta|\leq
1}\|\mathscr{D}^{\beta}_{\sigma}O_{\sigma}\|\lesssim
1,\qquad\|O_{\sigma}-O_{\tau}\|\lesssim{\rm dist}(\sigma,\tau),$

and these rotations will be used as the canonical family in (1.4).

###### Proof of Lemma 2.5.

In order to compute $\partial_{U_{\sigma}}R_{\sigma}$, we need to compute the
limit

(2.6) $\partial_{U_{\sigma}}R_{\sigma}=\lim_{t\to
0}\frac{R_{\tau(t)}-R_{\sigma}}{t}$

where $\tau:[0,\pi/2]\to\mathrm{\bf{Gr}}(d,n)$ is the
$\mathrm{\bf{Gr}}(d,n)$-valued curve

$\begin{split}\tau=\tau(t)\coloneqq(\sigma\cap\rho)\oplus&\mathrm{span}\\{U_{\tau}\\},\quad
U_{\tau}\coloneqq(\cos t)U_{\sigma}+(\sin t)v_{\sigma}=R_{\tau}u_{\sigma},\\\
&v_{\tau}=-(\sin t)U_{\sigma}+(\cos t)v_{\sigma},\end{split}$

and note that

$\tau(0)=\sigma,\qquad U_{\tau(0)}=U_{\sigma},\qquad v_{\tau(0)}=v_{\sigma},$

and that $R_{\tau}$ is a rotation by an angle $\arccos(\langle
v_{\tau},v\rangle)=\theta_{\sigma}+t$ in the plane
$\mathrm{span}\\{U_{\sigma},v_{\sigma}\\}=\mathrm{span}\\{v_{\tau},v\\}$, and
$v_{\tau}=R_{\tau}v$. It is then quite easy to see that

$\partial_{U_{\sigma}}R_{\sigma}v=-U_{\sigma},\qquad\partial_{U_{\sigma}}R_{\sigma}u_{\sigma}=v_{\sigma},\qquad\partial_{U_{\sigma}}R_{\sigma}w=0\quad\forall
w\in\sigma\cap\rho.$

This proves the first identity in the statement of the lemma.

In order to prove the second claim of the statement it will suffice to compute
$\partial_{w}R_{\sigma}$ for a generic vector $w\in\sigma\cap\rho$ of unit
norm. Fix one such $w$ and set $\zeta\coloneqq\sigma\cap\rho\cap w^{\perp}$.
The derivative $\partial_{w}R_{\sigma}$ will be calculated by repeating the
limiting procedure (2.6) for

$\tau=\tau(t)\coloneqq\zeta\oplus\mathrm{span}\\{w_{\tau(t)},U_{\sigma}\\},\quad
w_{\tau}\coloneqq(\cos t)w+(\sin t)v_{\sigma},\quad v_{\tau}\coloneqq-(\sin
t)w+(\cos t)v_{\sigma}.$

Note again that we have

$\tau(0)=\sigma,\qquad w_{\tau(0)}=w,\qquad v_{\tau(0)}=v_{\sigma}.$

As $R_{\sigma},R_{\tau}$ both act as the identity on $\zeta$, one has

$\partial_{w}R_{\sigma}z=0\qquad\forall z\in\zeta$

and it suffices to test the action of $\partial_{w}R_{\sigma}$ on the basis
vectors of $\zeta^{\perp}=\mathrm{span}\\{u_{\sigma},w,v\\}$. The vectors
$\\{\widetilde{w},\widetilde{u}\\}$ given by

(2.7) $\begin{split}&\widetilde{w}\coloneqq\frac{1}{\beta(t)}\left[(\cos
t)w-\frac{\sin
t}{\sin\theta_{\sigma}}u_{\sigma}\right],\qquad\widetilde{u}\coloneqq\frac{1}{\beta(t)}\left[\frac{\sin
t}{\sin\theta_{\sigma}}w+(\cos t)u_{\sigma}\right],\\\
&\beta(t)\coloneqq\sqrt{1+(\sin t\cot\theta_{\sigma})^{2}}\end{split}$

form an orthonormal basis of
$\zeta^{\perp}\cap\rho=\mathrm{span}\\{u_{\sigma},w\\}$ with, in addition,
$\mathrm{span}\\{\widetilde{w}\\}=\zeta^{\perp}\cap\tau(t)\cap\rho$. Therefore
$R_{\tau}$ acts as a rotation by $\theta_{\tau}$ in the plane spanned by
$\widetilde{u},v$, where

$\cos\theta_{\tau}=\langle v_{\tau},v\rangle=\cos
t\cos\theta_{\sigma},\qquad\sin\theta_{\tau}=\sqrt{1-(\cos t\cos\theta)^{2}},$

namely,

$R_{\tau}\widetilde{w}=\widetilde{w},\qquad
R_{\tau}\widetilde{u}=\cos[\theta_{\tau}]\widetilde{u}+\sin\theta_{\tau}v,\qquad
R_{\tau}v=v_{\tau}.$

Recalling our notation this means that $u_{\tau}=\widetilde{u}$ and
$U_{\tau}=R_{\tau}u_{\tau}=R_{\tau}\widetilde{u}$.

#### Computing $\partial_{w}R_{\sigma}u_{\sigma}$

The system (2.7) may be easily inverted, giving

(2.8) $\begin{split}&w=\frac{1}{\beta(t)}\left[(\cos
t)\widetilde{w}+\frac{\sin t}{\sin\theta}\widetilde{u}\right],\qquad
u_{\sigma}=\frac{1}{\beta(t)}\left[-\frac{\sin
t}{\sin\theta}\widetilde{w}+(\cos t)\widetilde{u}\right].\end{split}$

Also recalling (2.4), (2.7),

$\begin{split}R_{\tau}u_{\sigma}&=\frac{1}{\beta(t)}\left[-\frac{\sin
t}{\sin\theta_{\sigma}}\widetilde{w}+((\cos
t)^{2}\cos\theta_{\sigma})\widetilde{u}+(\cos t)\sqrt{1-(\cos
t\cos\theta_{\sigma})^{2}}v\right],\\\
R_{\sigma}u_{\sigma}&=\frac{1}{\beta(t)}\left[-\frac{\cos\theta_{\sigma}\sin
t}{\sin\theta_{\sigma}}\widetilde{w}+(\cos
t\cos\theta_{\sigma})\widetilde{u}+\beta(t)(\sin\theta_{\sigma})v\right].\end{split}$

Taking the difference and using some calculus

$[R_{\tau}-R_{\sigma}]u_{\sigma}=\frac{(\sin
t)\varpi(\theta_{\sigma})}{\beta(t)}\widetilde{w}+{o(t)\widetilde{u}+o(t)v}.$

Therefore, dividing by $t$ and taking the limit

$\partial_{w}R_{\sigma}u_{\sigma}=\varpi(\theta_{\sigma})w=\frac{\langle
v_{\sigma},v\rangle-1}{\sqrt{1-\langle v_{\sigma},v\rangle^{2}}}w.$

#### Computing $\partial_{w}R_{\sigma}w$

Similarly

$\begin{split}R_{\tau}w&=\frac{1}{\beta(t)}\left[(\cos
t)\widetilde{w}+\frac{\sin t\cos
t\cos\theta_{\sigma}}{\sin\theta_{\sigma}}\widetilde{u}+\frac{\sin
t}{\sin\theta_{\sigma}}\sqrt{1-(\cos t\cos\theta_{\sigma})^{2}}v\right],\\\
R_{\sigma}w&=w=\frac{1}{\beta(t)}\left[(\cos t)\widetilde{w}+\frac{\sin
t}{\sin\theta_{\sigma}}\widetilde{u}\right].\end{split}$

Taking the difference and using some more calculus

$[R_{\tau}-R_{\sigma}]w=\frac{(\sin
t)\varpi(\theta_{\sigma})}{\beta(t)}\widetilde{u}+(\sin t)v+{o(t)v(t)}.$

Therefore, dividing by $t$ and taking the limit

$\partial_{w}R_{\sigma}w=\varpi(\theta_{\sigma})u_{\sigma}+v=\frac{\langle
v_{\sigma},v\rangle-1}{\sqrt{1-\langle v_{\sigma},v\rangle^{2}}}u_{\sigma}+v.$

#### Computing $\partial_{w}R_{\sigma}v_{\rho}$

This one is the easiest, as

$\partial_{w}R_{\sigma}v=\lim_{t\to 0}\frac{[(\cos t)-1]v_{\sigma}-(\sin
t)w}{t}=-w$

and the proof is complete. ∎

##  3\. Removing the $\mathrm{SO}(d)$ invariance

The maximal operator (1.8) is invariant under precompositions of the family
$m_{\sigma}$ with elements $Q_{\sigma}\in\mathrm{SO}(d)$. In other words,
$T^{\star}$ does not change if each $m_{\sigma}$ is replaced by
$m_{\sigma}\circ Q_{\sigma}$ for some $Q_{\sigma}\in\mathrm{SO}(d)$ for each
$\sigma\in\mathrm{\bf{Gr}}(d,n)$. In this section, Theorem A is deduced from
an analogous estimate for the formally weaker maximal operator (3.3) where
$Q_{\sigma}\equiv$ the identity in $\mathrm{SO}(d)$. This deduction is
summarized in Proposition 3.2 below. The essential idea of the argument comes
from an averaging argument showing the $L^{p}(\mathbb{R}^{d})$-boundedness of
maximally rotated $d$-dimensional Hörmander–Mihlin multipliers, which is
presented below as a warmup.

######  Proposition 3.1.

Let $m\in\mathcal{M}_{A}(d)$ for $A\geq d+1+\frac{d(d-1)}{2}$. Then

$T^{\star}f(x)\coloneqq\sup_{Q\in\mathrm{SO}(d)}\left|\int_{\mathbb{R}^{d}}\widehat{f}(\eta)m(Q\eta)e^{2\pi
ix\cdot\eta}\,{\rm d}\eta\right|,\qquad x\in\mathbb{R}^{d},$

maps $L^{p}(\mathbb{R}^{d})$ into itself for all $1<p<\infty$.

###### Proof.

The idea of the proof is the same in all dimensions $d>1$. In order to make
the argument more concrete we first prove the case $d=2$. A generic element
$O\in\mathrm{SO}(2)$ may be written in the canonical basis of $\mathbb{R}^{2}$
as

$O=\left[\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right]$

for some $\theta\in[0,2\pi)$. Let $O(\theta)$ be the element corresponding to
$\theta\in[0,2\pi)$. We note that the coordinates of $O(\theta)$ are smooth
functions of $\theta$ and we adopt the notation

$O^{\prime}(\theta)=\left[\begin{array}[]{cc}-\sin\theta&-\cos\theta\\\
\cos\theta&-\sin\theta\end{array}\right]$

for the derivative. It is then useful to introduce the family of multipliers

(3.1) $m_{\theta}(\eta)\coloneqq\langle\nabla
m(O(\theta)\eta),O^{\prime}(\theta)\eta\rangle,\qquad\eta\in\mathbb{R}^{2}.$

It is immediate to see that there exists a constant $c$ independent of
$\theta$ such that $cm_{\theta}\in\mathcal{M}_{A-1}(2)$. Defining

$Tf(x,\theta)\coloneqq\int_{\mathbb{R}^{2}}\widehat{f}(\eta)m(O(\theta)\eta)e^{2\pi
ix\cdot\eta}\,{\rm d}\eta,\quad
Sf(x,\theta)\coloneqq\partial_{\theta}Tf(x,\theta)=\int_{\mathbb{R}^{2}}\widehat{f}(\eta)m_{\theta}(\eta)e^{2\pi
ix\cdot\eta}\,{\rm d}\eta,$

one has the equality

$Tf(x,\theta)=Tf(x,0)+\int_{0}^{\theta}Sf(x,\tau)\,{\rm d}\tau,\qquad
0\leq\theta<2\pi,$

whence

$|Tf(\cdot,\theta)|\leq|Tf(\cdot,0)|+\int_{0}^{2\pi}|Sf(\cdot,\tau)|\,{\rm
d}\tau,\qquad 0\leq\theta<2\pi.$

Therefore, using Minkowski’s inequality

$\begin{split}\left\|T^{*}f\right\|_{p}&=\left\|\sup_{\theta\in[0,2\pi)}\left|Tf(\cdot,\theta)\right|\right\|_{p}\leq\|Tf(\cdot,0)\|_{p}+\left\|\int_{0}^{2\pi}|Sf(\cdot,\tau)|\,{\rm
d}\tau\right\|_{p}\\\
&\leq\|Tf(\cdot,0)\|_{p}+2\pi\sup_{\tau\in[0,2\pi)}\|Sf(\cdot,\tau)\|_{p}\lesssim\|f\|_{p}\end{split}$

using that $m\in\mathcal{M}_{A}(2)$, $cm_{\theta}\in\mathcal{M}_{A-1}(2)$
uniformly, and the Hörmander–Mihlin theorem.

We then sketch the argument for $d>2$. Recall that $\mathrm{SO}(d)$ is a
compact Lie group of dimension $D=\frac{d(d-1)}{2}$. Hence, there exist $C>0$
and $0<\varepsilon<1$ so that $\mathrm{SO}(d)$ may be covered by smooth charts

$Q_{j}=Q_{j}(\theta)=Q_{j}(\theta_{1},\ldots,\theta_{D}):(-\varepsilon,\varepsilon)^{D}\to\mathrm{SO}(d),\qquad
1\leq j\leq C.$

For example, if $d=3$ one may use axis-angle pairs to parametrize rotations.
Namely, $O_{j}(\theta_{1},\theta_{2},\theta_{3})$ is the rotation by angle
$\theta_{3}$ in the plane perpendicular to the unit vector

$N(\theta_{1},\theta_{2})=(\cos\theta_{1}\sin\theta_{2},\sin\theta_{1}\sin\theta_{2},\cos\theta_{2}).$

Then $\mathrm{SO}(3)$ may be covered by $O(1)$ smooth axis angle charts with
parameter $\varepsilon=2^{-3}\pi$.

By finite splitting it thus suffices to bound the maximal operator

$\sup_{\theta\in[0,\varepsilon)^{D}}|Tf(x,\theta)|\,\,\,\text{ where
}\,\,\,Tf(x,\theta)\coloneqq\int_{\mathbb{R}^{d}}\widehat{f}(\eta)m(Q_{j}(\theta)\eta)e^{2\pi
ix\cdot\eta}\,{\rm d}\eta,\quad x\in\mathbb{R}^{n}$

for each $j=1,\ldots,C$ fixed. For
$S=\\{k_{1}<\ldots<k_{\\#S}\\}\subset\\{1,\ldots,D\\}$ write
$\partial_{S}=\partial_{k_{1}}\cdots\partial_{k_{\\#S}}$. Let also $\Pi_{S}$
be the orthogonal projection on
$\mathbb{R}^{S}\coloneqq\mathrm{span}\\{e_{k}:\,k\in S\\}$, so that
$\theta=\Pi_{S}\theta\oplus\Pi_{S^{\mathsf{c}}}\theta.$ If
$\tau\in\mathbb{R}^{S}$, it makes sense to write $\theta=\tau\oplus
0_{S^{\mathsf{c}}}\in\mathbb{R}^{D}$ with the meaning that
$\Pi_{S}\theta=\tau$ and $\Pi_{S^{\mathsf{c}}}\theta=0_{S^{\mathsf{c}}}$. The
key of our argument is again that the multiplier operators

(3.2)
$T_{S}f(x,\theta)=\int_{\mathbb{R}^{d}}\widehat{f}(\eta)m_{\theta,S}(\eta)e^{2\pi
ix\cdot\eta}\,{\rm d}\eta,\qquad
m_{\theta,S}\coloneqq\partial_{S}m(Q_{j}(\theta)\cdot),$

satisfy $cm_{\theta,S}\in\mathcal{M}_{A-\\#S}(d)\subset\mathcal{M}_{A-D}(d)$
uniformly. Then one has the equality

$Tf(x,\theta)=Tf(x,0)+\sum_{\varnothing\subsetneq
S\subseteq\\{1,\ldots,D\\}}\,(-1)^{\\#S}\int_{\prod_{k\in
S}[0,\theta_{k}]}T_{S}f(x,\tau\oplus 0_{S^{\mathsf{c}}}){\rm
d}\tau,\qquad\theta\in[0,\varepsilon)^{D},$

and similarly to what we have done before,

$\left\|\sup_{\theta\in[0,\varepsilon)^{D}}\left|Tf(\cdot,\theta)\right|\right\|_{p}\leq\left\|Tf(\cdot,0)\right\|_{p}+2^{D}\sup_{\varnothing\subsetneq
S\subset\\{1,\ldots,D\\}}\sup_{\tau\in[0,\varepsilon]^{\\#S}}\left\|T_{S}f(\cdot,\tau\oplus
0_{S^{\mathsf{c}}})\right\|_{p}$

so that we end up with the latter supremum being controlled by $C\|f\|_{p}$
via an application of the Hörmander–Mihlin theorem. The proof is complete. ∎

We are now ready to state and prove the anticipated reduction of Theorem A.

######  Proposition 3.2.

Let $\Sigma\subset\mathrm{\bf{Gr}}(d,n)$ be arbitrary. Let
$O_{\sigma}=O_{\mathbb{R}^{d},\sigma}$, with reference to the family of
rotations constructed in Lemma 2.3; cf. Remark 2.6. Consider the maximal
operator

(3.3)
$U^{\star}_{\Sigma,\mathscr{m}}f(x)\coloneqq\sup_{\sigma\in\Sigma}|T_{m_{\sigma}}f(x,\sigma)|$

where

$T_{m_{\sigma}}f(x,\sigma)\coloneqq\int_{\mathbb{R}^{n}}m_{\sigma}(O_{\sigma}\Pi_{\sigma}\xi)\widehat{f}(\xi)e^{2\pi
i\langle x,\xi\rangle}\,{\rm d}\xi,\quad x\in\mathbb{R}^{n}.$

For $1<p<\infty,1\leq q\leq\infty$, $D=\frac{d(d-1)}{2}$ and
$T^{\star}_{\Sigma,\mathscr{m}}$ from (1.12), there holds

$\sup\\{\|T^{\star}_{\Sigma,\mathscr{m}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{p,q}(\mathbb{R}^{n})}:\,\|\mathscr{m}\|_{\mathcal{M}_{A+D}(\mathrm{\bf{Gr}}(d,n))}\leq
1\\}\\\
\lesssim\sup\\{\|U^{\star}_{\Sigma,\mathscr{m}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{p,q}(\mathbb{R}^{n})}:\,\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\leq
1\\}$

with implicit constant depending upon dimension only. The same statement holds
if $U^{\star}_{\Sigma,\mathscr{m}}$ is replaced by
$U^{\star}_{\Sigma,\mathscr{m}}\circ P_{0}$ in the right hand side and
$T^{\star}_{\Sigma,\mathscr{m}}$ is replaced by
$T^{\star}_{\Sigma,\mathscr{m}}\circ P_{0}$ in the left hand side of the
estimate above.

###### Proof.

Fix a family $\mathscr{m}$ with
$\|\mathscr{m}\|_{\mathcal{M}_{A+D}(\mathrm{\bf{Gr}}(d,n))}\leq 1$. By finite
splitting, we obtain the claimed estimate for (1.12) from the same claim on
the restricted maximal operator

$T^{\star}_{\Sigma,\mathscr{m}}f(x)\coloneqq\sup_{\sigma\in\Sigma}\sup_{Q\in\mathcal{Q}}|T_{m_{\sigma}}f(x;\sigma,Q)|$

with

$T_{m_{\sigma}}f(x;\sigma,Q)\coloneqq\int_{\mathbb{R}^{n}}m_{\sigma}(QO(\sigma)\Pi_{\sigma}\xi)\widehat{f}(\xi)e^{2\pi
i\langle x,\xi\rangle}\,{\rm d}\xi,\quad x\in\mathbb{R}^{n},$

where $\mathcal{Q}$ is range of the chart $Q_{j}$ for some $j=1,\ldots,C$; the
overloading of the symbol $T^{\star}$ creates no confusion. At this point, for
$Q\in\mathcal{Q}$, let $\theta(Q)$ be such that $Q=Q_{j}(\theta)$. Define the
families

$\mathscr{m}_{\theta,S}=\\{m_{\sigma,\theta,S}:\sigma\in\mathrm{\bf{Gr}}(d,n)\\},\quad
m_{\sigma,\theta,S}(\eta)\coloneqq\partial^{S}m(Q_{j}(\vec{\theta})\eta,\sigma),\quad\theta\in[0,\varepsilon)^{D},\,S\subset\\{1,\ldots,D\\}.$

With this definition, observe that

$T_{m_{\sigma}}f(x;\sigma,Q)=T_{m_{\sigma,\theta(Q),\varnothing}}f(x;\sigma,\mathrm{Id}_{\mathbb{R}^{d}})=T_{m_{\sigma,\theta(Q),\varnothing}}f(x,\sigma).$

It is then routine to verify that

(3.4)
$\sup_{\theta\in[0,\varepsilon)^{D}}\sup_{S\subset\\{1,\ldots,d\\}}\left\|\mathscr{m}_{\theta,S}\right\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\lesssim\left\|\mathscr{m}\right\|_{\mathcal{M}_{A+D}(\mathrm{\bf{Gr}}(d,n))}\leq
1.$

Arguing as in the previous proof we then have

$T^{\star}_{\mathscr{m},\Sigma}f(x)\leq
U^{\star}_{\mathscr{m}_{0,\varnothing},\Sigma}f(x)+\sum_{\varnothing\subsetneq
S\subseteq\\{1,\ldots,D\\}}\,(-1)^{\\#S}\int_{\prod_{k\in
S}[0,\varepsilon]}U^{\star}_{\Sigma,\mathscr{m}_{\tau\oplus
0_{S^{\mathsf{c}}},S}}f(x){\rm
d}\tau,\qquad\vec{\theta}\in[0,\varepsilon)^{D},$

which turns into the norm inequality

$\|T^{\star}_{\mathscr{m}}f\|_{p,q}\lesssim_{p,q}\|U^{\star}_{\mathscr{m}_{0,\varnothing},\Sigma}f\|_{p,q}+2^{D}\sup_{\varnothing\subsetneq
S\subset\\{1,\ldots,D\\}}\sup_{\tau\in[0,\varepsilon]^{\\#S}}\left\|U^{\star}_{\Sigma,\mathscr{m}_{\tau\oplus
0_{S^{\mathsf{c}}},S}}f\right\|_{p,q}.$

The norms above are then controlled by $\|f\|_{p}$ by assumption in view of
estimate (3.4), and the proof of the first claim is complete. For the second
claim, it suffices to apply the above argument with $P_{0}f$ in place of $f$.
∎

Because of the reduction devised in Proposition 3.2 above, Theorem A with
$A=A(d)+\frac{d(d-1)}{2}$ is obtained from the following proposition.

######  Proposition 3.3.

Let $U^{\star}_{\Sigma,\mathscr{m}}$ be defined as in the statement of
Proposition 3.2. There exists $A=A(d)$ such that the operator
$U^{\star}_{\Sigma,\mathscr{m}}\circ P_{0}$ for $\Sigma=\mathrm{\bf{Gr}}(d,n)$
maps $L^{2}(\mathbb{R}^{n})$ to $L^{2,\infty}(\mathbb{R}^{n})$ and
$L^{p}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$, $p>2$, uniformly over all
families $\mathscr{m}$ satisfying
$\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\leq 1$.

The main line of proof of Proposition 3.3 occupies §5 and §6.

##  4\. Maximal codimension-$1$ multipliers and the Carleson–Sjölin theorem

In this section we discuss the proofs of some of the consequences of our main
result.

###  4.1. Maximal codimension-$1$ multipliers and the proof of Corollary A.2

Let us recall our setup. Given $\Sigma\subset\mathrm{\bf{Gr}}(d,n)$ with
$\\#\Sigma=N$, consider the maximal directional multiplier operator

$f\mapsto
T^{\star}_{\Sigma,\mathscr{m}}f(x)\coloneqq\sup_{\sigma\in\Sigma}\sup_{Q\in\rm{SO}(d)}|T_{m_{\sigma}}f(x;\sigma,Q)|,\qquad
x\in\mathbb{R}^{n},\qquad\Sigma\subset\mathrm{\bf{Gr}}(d,n),$

where $T_{m_{\sigma}}f(x,\sigma,Q)$ is given by (1.6) and the family of
multipliers
$\mathscr{m}=\\{m_{\sigma}\in\mathcal{M}_{A}(d):\,\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$
satisfies $\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\leq 1$. We
will prove that

$\sup_{\begin{subarray}{c}\Sigma\subset\mathrm{\bf{Gr}}(d,n)\\\ \\#\Sigma\leq
N\end{subarray}}\left\|U_{\Sigma,\mathscr{m}}^{\star}f\right\|_{L^{2}(\mathbb{R}^{n})}\lesssim\log
N\|f\|_{L^{2}(\mathbb{R}^{n})}$

which implies the conclusion of Corollary A.2 by Proposition 3.2.

A first well known consequence of Theorem A is that, in the case of
$N$-multipliers, we can upgrade the single-annulus weak $(2,2)$ bound to a
strong $(2,2)$ single-annulus bound, at a cost of $\sqrt{\log N}$.

######  Lemma 4.2.

We have the bound

$\sup_{\begin{subarray}{c}\Sigma\subset\mathrm{\bf{Gr}}(d,n)\\\ \\#\Sigma\leq
N\end{subarray}}\sup_{k\in{\mathbb{Z}}}\|U^{\star}_{\Sigma,\mathscr{m}}\circ
P_{k}f\|_{L^{2}(\mathbb{R}^{n})}\lesssim\sqrt{\log
N}\|f\|_{L^{2}(\mathbb{R}^{n})},$

with implicit constant depending only upon dimension.

The lemma follows easily by the weak $(2,2)$ bound

$\sup_{k\in\mathbb{R}}\|U^{\star}_{\Sigma,\mathscr{m}}(P_{k}f)\|_{L^{2,\infty}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})},$

which is a special case of Theorem A, together with trivial weak $(1,1)$ and
strong $(p,p)$ estimates for $U^{\star}_{\Sigma,\mathscr{m}}\circ P_{k}$ for
$2<p<\infty$, with bounds which are polynomial in $N$, and a Marcinkiewicz
interpolation-type argument. The details can be found for example in
[Dem]*Lemma 3.1, but essentially the same argument has been used in several
places, for example in [Katz, Stromberg].

With a strong $(2,2)$ bound in hand the proof of Corollary A.2 follows a well
know reduction, based on the Chang–Wilson–Wolff inequality [CWW], which allows
us to essentially commute the supremum over $N$ Fourier multipliers with a
Littlewood–Paley square function. The argument leading to this reduction was
introduced in [GHS], while in the context of directional multipliers it has
been extensively used; see for example [ADP, Dem]. The punchline proposition
is stated below. Here, we use a smooth Littlewood–Paley partition of
$\mathbb{R}^{n}$ given by means of

$(S_{t}f)^{\wedge}(\xi)\coloneqq\Psi\left(\frac{|\xi|}{t}\right)\widehat{f}(\xi),\qquad\xi\in\mathbb{R}^{n},\quad
t\in 2^{\mathbb{Z}},$

where $\Psi$ is a smooth radial function with compact support
$\mathrm{supp}\Psi\subset\\{\frac{1}{2}<|\xi|<2\\}$ with the additional
requirement that

$\sum_{t\in
2^{\mathbb{Z}}}\Psi\left(\frac{|\xi|}{t}\right)=1,\qquad\xi\in\mathbb{R}^{n}\setminus\\{0\\}.$

######  Proposition 4.3.

Let $\\{T_{1},\ldots,T_{N}\\}$ be Fourier multiplier operators in
$\mathbb{R}^{n}$ with uniform bound

$\sup_{1\leq\nu\leq N}\|T_{\nu}\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}\leq 1.$

If $\\{S_{t}\\}$ is a smooth Littlewood–Paley decomposition as described above
then for $1<p<\infty$ there holds

$\left\|\sup_{1\leq\nu\leq
N}|T_{\nu}f|\right\|_{L^{p}(\mathbb{R}^{n})}\lesssim_{p,n}\|f\|_{L^{p}(\mathbb{R}^{n})}+\sqrt{\log(N+10)}\left\|\left(\sum_{t\in
2^{\mathbb{Z}}}\,\sup_{1\leq\nu\leq N}|(T_{\nu}\circ
S_{t})f|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\mathbb{R}^{n})}.$

The proof of this proposition can be found within [DPGTZK]*Proof of Corollary
1.14; see also [Dem]. Now note that our main result, Theorem A, readily
implies that

$\sup_{t\in 2^{{\mathbb{Z}}}}\left\|U^{\star}_{\Sigma,\mathscr{m}}\circ
S_{t}f\right\|_{L^{2,\infty}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}$

by splitting the support of $\Psi$ into $O(1)$ pieces that match the size of
the support of the auxiliary function $\zeta$ in the statement of Theorem A,
applying the conclusion of the theorem to each one of them, and adding them
appropriately. Thus Lemma 4.2 yields the strong $L^{2}(\mathbb{R}^{n})$-bound

$\sup_{t\in 2^{{\mathbb{Z}}}}\left\|U^{\star}_{\Sigma,\mathscr{m}}\circ
S_{t}f\right\|_{L^{2}(\mathbb{R}^{n})}\lesssim(\log\\#\Sigma)^{\frac{1}{2}}\|f\|_{L^{2}(\mathbb{R}^{n})}$

for $N=\\#\Sigma$ large. The conclusion of Corollary A.2 now follows
immediately by inserting the $L^{2}$-bound above in the conclusion of
Proposition 4.3 for $p=2$. Note that the uniform $L^{2}$-bound, which is
necessary for the application of Proposition 4.3, is an immediate consequence
of the standing assumption
$\|\mathscr{m}\|_{\mathcal{M}_{A}(\mathrm{\bf{Gr}}(d,n))}\leq 1$ for the
family of multipliers $\mathscr{m}$.

###  4.4. The Carleson–Sjölin theorem

The purpose of this subsection is to provide the connection of our main
theorem, Theorem A, with the Carleson–Sjölin theorem, as formulated in Theorem
B. We begin with a simple but useful geometric lemma.

######  Lemma 4.5.

Let $\varepsilon>0$ be a small parameter, $\Sigma\subset\mathrm{\bf{Gr}}(d,n)$
be a $\varepsilon$-neighborhood of $\mathbb{R}^{d}=e_{n}^{\perp}$. For all
$R>0$ the map

$N:\Sigma\to\mathbb{R}^{d},\qquad N(\sigma)\coloneqq R\left[\langle
v_{\sigma},e_{n}\rangle e_{n}-v_{\sigma}\right]$

is onto the ball $B^{d}(c_{o}\varepsilon R)\subset\mathbb{R}^{d}$, for some
dimensional constant $c_{o}>0$. Furthermore

$|N(\sigma)-N(\tau)|\lesssim R{\rm
dist}(\sigma,\tau),\qquad\sigma,\tau\in\Sigma$

with implicit absolute constant.

###### Proof.

Clearly $N(\mathbb{R}^{d})=0$. If $N\neq 0$ belongs to the ball
$B^{d}(c_{o}\varepsilon R)$, set

$v_{\sigma}\coloneqq-\frac{N}{R}+\sqrt{1-\frac{|N|^{2}}{R^{2}}}e_{n},\qquad\sigma\coloneqq
v_{\sigma}^{\perp}.$

As ${\rm dist}(\sigma,\mathbb{R}^{d})\sim\arccos\left(\langle
v_{\sigma},e_{n}\rangle\right)\sim\frac{|N|}{R}$, and $N\leq c_{o}\varepsilon
R$, it follows that $\sigma\in\Sigma$ if $c_{o}$ is suitably chosen. Also, it
is immediate to verify that $N(\sigma)=N$. The proof of the surjectivity claim
is thus complete. The Lipschitz estimate is immediate, and its proof is thus
omitted. ∎

Note that the unit vector

$U_{\sigma}\coloneqq\frac{\left[e_{n}-\langle v_{\sigma},e_{n}\rangle
v_{\sigma}\right]}{\sqrt{1-\langle v_{\sigma},e_{n}\rangle^{2}}},\qquad
v_{\sigma}\neq e_{n}$

spans $\sigma\cap\mathrm{span}\\{v_{\sigma},e_{n}\\}$. For $\sigma\neq e_{n}$,
it is also convenient to define the unit vector
$u_{\sigma}\coloneqq{N(\sigma)}/{|N(\sigma)|}$ spanning
$\mathbb{R}^{d}\cap\mathrm{span}\\{v_{\sigma},e_{n}\\}$. Below, let
$\\{O_{\sigma}:\,\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ be the $C^{1}$ family of
rotations of Remark 2.6; we remember that $O_{\mathbb{R}^{d}}\equiv\rm{Id}$
and each $O_{\sigma}$ acts as the identity on $\sigma\cap\mathbb{R}^{d}$ and,
in the orthogonal complement $\mathrm{span}\\{v_{\sigma},e_{n}\\}$ as the
rotation mapping $U_{\sigma}$ to $u_{\sigma}$, and consequently $v_{\sigma}$
to $e_{n}$. With the notation of Lemma 4.5 we have that, for $N\leq
c_{o}\varepsilon R$, $\cos(|N|/R)=\langle v_{\sigma},e_{n}\rangle$ and
$\sin(|N|/R)\sim|N|/R$. A direct calculation entails the equality

(4.1) $O_{\sigma}\Pi_{\sigma}\xi=\Pi_{\mathbb{R}^{d}}\xi+\left[\langle
v_{\sigma},e_{n}\rangle-1\right]\langle\xi,u_{\sigma}\rangle
u_{\sigma}+\langle\xi,e_{n}\rangle\frac{N(\sigma)}{R}.$

The following definitions and observations readily yield the following lemma.

######  Lemma 4.6.

Let $m\in\mathcal{M}_{A}(d)$ and consider the $\varepsilon$-neighborhood
$\Sigma$ of $\,\mathbb{R}^{d}$ in $\mathrm{\bf{Gr}}(d,n)$, constructed in
Lemma 4.5 with $\varepsilon$ sufficiently small depending upon dimension only.
Let also $\\{O_{\sigma}:\,\sigma\in\Sigma\\}$ be the family defined in Remark
2.6. Suppose $F$ is a Schwartz function on $\mathbb{R}^{n}$ with
${\mathrm{supp}\,}\widehat{F}\subset\\{\xi\in\mathbb{R}^{n}:\,R/2<|\xi_{n}|<2R\\}$.
Define

$M(\xi,\sigma)\coloneqq m\left(\Pi_{\mathbb{R}^{d}}\xi+\left[\langle
v_{\sigma},e_{n}\rangle-1\right]\langle\xi,u_{\sigma}\rangle
u_{\sigma}+\xi_{n}\frac{N(\sigma)}{R}\right),\qquad(\xi,\sigma)\in\mathbb{R}^{n}\times\Sigma,$

and let $TF(x,\sigma)\coloneqq(M(\cdot,\sigma)\widehat{F})^{\vee}(x)$. There
holds

$\left\|\sup_{\sigma\in\Sigma}|TF(\cdot,\sigma)|\,\right\|_{L^{2,\infty}(\mathbb{R}^{n})}\lesssim\|F\|_{L^{2}(\mathbb{R}^{n})},\quad\left\|\sup_{\sigma\in\Sigma}|TF(\cdot,\sigma)|\,\right\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|F\|_{L^{p}(\mathbb{R}^{n})},\quad
2<p<\infty.$

###### Proof.

Let $\Gamma\in\mathcal{S}(\mathbb{R})$ have compact support in
$[-2^{-10},2^{-10}]$ and $\Gamma=1$ in $[-2^{-15},2^{-15}]$. By virtue of the
support condition on $\widehat{F}$ we will have that

$\left|\int_{\mathbb{R}^{n}}\widehat{F}(\xi)m(O_{\sigma}^{-1}\Pi_{\sigma}\xi)\left[1-\Gamma\left({\textstyle\frac{|\Pi_{\mathbb{R}^{d}}\xi|}{R}}\right)\right]{\rm
e}^{2\pi i\langle\xi,x\rangle}\,{\rm
d}\xi\right|\lesssim\mathrm{M}_{\rm{S}}F(x)$

where $\mathrm{M}_{\rm{S}}$ stands for the strong maximal function, provided
that $\varepsilon>0$ was chosen sufficiently small in Lemma 4.5 depending upon
dimension only. Therefore we may assume that $\widehat{F}$ is supported on the
cone $\\{|\Pi_{\mathbb{R}^{d}}\xi|\lesssim|\xi|\\}$. Again by virtue of the
support of $\widehat{F}$ it is clear that $T_{\Sigma}F=T_{\Sigma}S_{R}F$, with
$S_{R}$ a smooth frequency projection onto the annulus $\\{R/3<|\xi|<3R\\}$
whose symbol is identically one on the support of $\widehat{F}$. Because of
(4.1) the lemma follows by a suitable application of Theorem A. ∎

###### Proof of Theorem B.

Let us fix a Hörmander–Mihlin multiplier $m:\mathbb{R}^{d}\to\mathbb{C}$ as in
the statement. We will make the qualitative assumption that $m$ is a smooth
function with compact support in $\mathbb{R}^{d}$, which is always possible by
a suitable approximation; this assumption will be removed at the end of the
proof. Our goal is to show that the operator

$\mathsf{CS}[f](x)\coloneqq\sup_{N\in\mathbb{R}^{d}}|\mathsf{CS}_{N}[f](x)|,\qquad\mathsf{CS}_{N}[f](x)\coloneqq\int_{\mathbb{R}^{d}}\widehat{f}(\eta)m(\eta+N)e^{2\pi
i\langle\eta,x\rangle}\,{\rm d}\eta,\qquad x\in\mathbb{R}^{d},$

maps $L^{2}(\mathbb{R}^{d})$ to $L^{2,\infty}(\mathbb{R}^{d})$ and
$L^{p}(\mathbb{R}^{d})$ to itself for all $2<p<\infty$. To that end let us
also fix $p\in[2,\infty)$ and a function $f\in\mathcal{S}(\mathbb{R}^{d})$
with compact frequency support. By our qualitative assumptions there exists
$R_{o}>0$ such that

${\mathrm{supp}\,}(\widehat{f})\subset
B^{d}(R_{o}),\qquad{\mathrm{supp}\,}(m)\subset B^{d}(R_{o}),$

and thus only values $N\in B^{d}(2R_{o})$ need be considered in the definition
of $\mathsf{CS}[f]$. We now fix $\varepsilon>0$ sufficiently small, as coming
from the statement of Lemma 4.6, but also satisfying the smallness condition

$\varepsilon<\left[\max\left(100R_{o}^{d}\|m\|_{C^{20d}},(100R_{o})\right)\right]^{-1}.$

Above we have denoted for positive integers $\nu$

$\|g\|_{C^{\nu}}\coloneqq\sum_{0\leq|\alpha|\leq\nu}\|\partial^{\alpha}g\|_{L^{\infty}(\mathbb{R}^{d})}.$

We will use an auxiliary function $\psi$ to lift $f$ from $\mathbb{R}^{d}$ to
$\mathbb{R}^{n}$. More precisely we let $\psi:\mathbb{R}\to\mathbb{R}$ be a
smooth function with compact frequency support
$\mathrm{supp}(\widehat{\psi})\subset(-1,1)$ and such that $\psi\geq 0$,
$\psi\geq 1$ on $(-2^{-3},2^{-3})$, and $\widehat{\psi}\geq 0$, and define

$\psi_{R}(t)\coloneqq\int_{\mathbb{R}}\widehat{\psi}\left(u-R\right)e^{2\pi
itu}\,{\rm d}u,\qquad t\in\mathbb{R},$

with $R>0$ to be chosen momentarily. We readily see that
$\mathrm{supp}(\widehat{\psi}_{R})\subset(R-1,R+1)$. We then define the smooth
function with compact frequency support

$F(x)\coloneqq f(y)\psi_{R}(x_{n}),\qquad
x=(y,x_{n})\in\mathbb{R}^{n}=\mathbb{R}^{d}\times\mathbb{R},$

where we use the conventions
$x=(\Pi_{\mathbb{R}^{d}}x,x_{n})\eqqcolon(y,x_{n})\in\mathbb{R}^{n}$ and
$\xi=(\Pi_{\mathbb{R}^{d}}\xi,\xi_{n})\eqqcolon(\eta,\xi_{n})\in\mathbb{R}^{n}$
in order to simplify our formulas.

For any $N\in B^{d}(2R_{o})$ we apply Lemma 4.5 with parameters
$\varepsilon,R$ such that $2R_{o}<c_{o}\varepsilon R$. We then know that there
exists $\sigma=\sigma(N)\in\Sigma$ such that $N(\sigma)=N$. We fix such a
sufficiently large value of $R$ and note that

$\operatorname{supp}(\widehat{\psi}_{R})\subset\\{t:\,R/2<t<2R\\}.$

Now set

$\mathcal{E}(\xi,\sigma)\coloneqq\left[\langle
v_{\sigma},e_{n}\rangle-1\right]\langle\xi,u_{\sigma}\rangle
u_{\sigma}+\left(\frac{\xi_{n}}{R}-1\right)N(\sigma),\qquad(\xi,\sigma)\in\mathbb{R}^{n}\times\Sigma,$

and immediately observe that for $\xi\in\mathrm{supp}(\widehat{F})$ we have,
taking into account the facts in Lemma 4.5,

(4.2) $\begin{split}|\mathcal{E}(\xi,\sigma)|&\lesssim{\rm
dist}(\sigma,\mathbb{R}^{d})^{2}R_{o}+{\rm dist}(\sigma,\mathbb{R}^{d}),\\\
|\partial_{\xi_{j}}\mathcal{E}(\xi,\sigma)|&\lesssim{\rm
dist}(\sigma,\mathbb{R}^{d})^{2},\qquad 1\leq j\leq d,\\\
|\partial_{\xi_{n}}\mathcal{E}(\xi,\sigma)|&\lesssim{\rm
dist}(\sigma,\mathbb{R}^{d}),\\\
|\nabla^{\alpha}\mathcal{E}(\xi,\sigma)|&=0\quad\text{if}\quad|\alpha|>1.\end{split}$

Defining $TF(x,\sigma)$ as in the statement of Lemma 4.6, with the choice of
$\sigma\in\Sigma$ as above, we calculate

$\begin{split}TF(x,\sigma)&=\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}m\left(\eta+N(\sigma)+\mathcal{E}(\xi,\sigma)\right)\widehat{f}(\eta)\widehat{\psi}_{R}(\xi_{n})e^{2\pi
i\langle\eta,y\rangle}e^{2\pi ix_{n}\xi_{n}}\,{\rm d}\eta\,{\rm d}\xi_{n}\\\
&=\mathsf{CS}_{N}[f](y)\psi_{R}(x_{n})+\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\widehat{f}(\eta)\widehat{\psi}_{R}(\xi_{n})\Delta(\xi,\sigma)e^{2\pi
i\langle\eta,y\rangle}e^{2\pi ix_{n}\xi_{n}}\,{\rm d}\eta\,{\rm
d}\xi_{n},\end{split}$

where

$\Delta(\xi,\sigma)\coloneqq
m(\eta+N(\sigma)+\mathcal{E}(\xi,\sigma))-m(\eta+N(\sigma)),\qquad(\xi,\sigma)\in\mathbb{R}^{n}\times\Sigma.$

Note that $\mathrm{supp}(\widehat{F})\subseteq
B^{d}(R_{o})\times\\{R-1<|\xi_{n}|<R+1\\}$ which we want to keep memory of.
For that reason we let $\lambda\in\mathcal{S}(\mathbb{R}^{n})$ be identically
one on $\mathrm{supp}(\widehat{F})$ and vanish outside
$B^{d}(10R_{o})\times\\{R-10<|\xi_{n}|<R+10\\}$, with the natural derivative
bounds. Define the operator

$\mathrm{Err}_{\sigma}F(x)\coloneqq\int_{\mathbb{R}^{n}}\widehat{F}(\xi)\lambda(\xi)\Delta(\xi,\sigma)e^{2\pi
i\langle\xi,x\rangle}\,{\rm d}\xi,\qquad x\in\mathbb{R}^{n}.$

We then have

$|\mathsf{CS}[f](y)\psi_{R}(x_{n})|\leq\sup_{\sigma\in\Sigma}|TF(x,\sigma)|+\sup_{\sigma\in\Sigma}|\mathrm{Err}_{\sigma}F(x)|$

and the proof of the theorem will be complete once we control the error term
$\mathrm{Err}_{\sigma}F$ in $L^{p}(\mathbb{R}^{n})$. We will do so by proving

(4.3)
$\sup_{\sigma\in\Sigma}|\mathrm{Err}_{\sigma}F(x)|\lesssim{\mathrm{M}}F(x),\qquad
x\in\mathbb{R}^{n};$

here ${\mathrm{M}}$ denotes the $n$-dimensional Hardy–Littlewood maximal
function. To that end let us define

$E(x,\sigma)\coloneqq\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\lambda(\xi)\Delta(\xi,\sigma)e^{2\pi
i\langle\eta,y\rangle}e^{2\pi ix_{n}\xi_{n}}\,{\rm d}\eta\,{\rm
d}\xi_{n},\qquad x\in\mathbb{R}^{n},$

and note that (4.2) remains valid for $\xi\in\mathrm{supp}(\lambda)$.
Estimates (4.2) combined with a tedious calculation involving the chain rule
and the derivatives of $m$, $\mathcal{E}(\xi,\sigma)$ and $\lambda$, imply

$\begin{split}|\mathrm{supp}(\lambda(\cdot)\Delta(\cdot,\sigma))|&\lesssim
R_{o}^{d}\quad(\text{using that }\operatorname{supp}m\subset B^{d}(R_{o})),\\\
|\nabla^{\gamma}(\lambda(\xi)\Delta(\xi,\sigma))|&\lesssim\|m\|_{C^{|\gamma|+1}}\left[{\rm
dist}(\sigma,\mathbb{R}^{d})^{2}R_{o}+{\rm
dist}(\sigma,\mathbb{R}^{d})\right].\\\ \end{split}$

Using these bounds and integrating by parts inside the integral defining
$E(x,\sigma)$ we gather

$|E(x,\sigma)|\lesssim\frac{\|m\|_{C^{|\gamma|+1}}R_{o}^{d}\left[{\rm
dist}(\sigma,\mathbb{R}^{d})R_{o}+1\right]{\rm
dist}(\sigma,\mathbb{R}^{d})}{(1+|y|)^{10d}(1+|x_{n}|)^{10d}},\qquad
x=(y,x_{n})\in\mathbb{R}^{d}\times\mathbb{R}.$

Remembering our choice of $\varepsilon>0$ and that ${\rm
dist}(\sigma,\mathbb{R}^{d})\lesssim\varepsilon$, this proves (4.3) with
implicit constant depending only upon dimension. Thus the error term is under
control

$\|\sup_{\sigma\in\Sigma}|\mathrm{Err}_{\sigma}F|\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|F\|_{L^{p}(\mathbb{R}^{n})}\eqsim\|f\|_{L^{p}(\mathbb{R}^{d})}.$

We have proved the estimates

$\|\mathsf{CS}[f]\|_{L^{2,\infty}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{2}(\mathbb{R}^{d})},\qquad\|\mathsf{CS}[f]\|_{L^{p}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{d})},\qquad
2<p<\infty,$

whenever $f\in\mathcal{S}(\mathbb{R}^{d})$ has compact Fourier support and $m$
is a $C^{20d}(\mathbb{R}^{d})$-function with compact support. One easily
removes these assumptions on $m$ by approximating with a sequence $m_{k}$
which is smooth and has compact Fourier support, and converges pointwise to
$m$ a.e. in $\mathbb{R}^{d}$, and satisfies
$\|m_{k}\|_{\mathcal{M}_{A}(d)}\lesssim\|m\|_{\mathcal{M}_{A}(d)}$ uniformly
in $k$. Then dominated convergence shows that for any measurable map
$\mathbb{R}^{d}\ni y\mapsto N(y)$ we have

$\int_{\mathbb{R}^{d}}m(\xi+N(y))\widehat{f}(\xi)e^{2\pi i\langle
y,\eta\rangle}\,{\rm
d}\eta=\lim_{k}\int_{\mathbb{R}^{d}}m_{k}(\xi+N(y))\widehat{f}(\xi)e^{2\pi
i\langle y,\eta\rangle}\,{\rm d}\eta$

whenever $f\in\mathcal{S}(\mathbb{R}^{d})$ has compact Fourier support. Taking
absolute values and applying Fatou’s lemma inside the
$L^{p}(\mathbb{R}^{d})$-norm yields the conclusion of the theorem for
$f\in\mathcal{S}(\mathbb{R}^{d})$ with compact Fourier support. As such
functions are dense in $L^{p}(\mathbb{R}^{d})$ the proof is complete. ∎

##  5\. Tiles, adapted functions, and model operator

This and the following sections are dedicated to the proof of Proposition 3.3
which, by the reductions of §3, implies the main theorem, Theorem A. In the
remaining of the paper we use the $C^{1}$ family of rotations
$\\{O_{\sigma}:\,\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ of Remark 2.6, underlying
the definition of the operator $U^{\star}_{\Sigma,\mathscr{m}}$ in (3.3).

###  5.1. Preliminaries

The family $\mathscr{m}=\\{m_{\sigma}\in
L^{\infty}(\mathbb{R}^{d}):\,\sigma\in\mathrm{\bf{Gr}}(d,n)\\}$ satisfying
(1.10) is fixed for the rest of the paper. Recall, cf. (3.3), that the
definition of $U^{\star}_{\Sigma,\mathscr{m}}$ involves the singular integrals

(5.1)
$T_{m_{\sigma}}f(x,\sigma)\coloneqq\int_{\mathbb{R}^{n}}m_{\sigma}(O_{\sigma}\Pi_{\sigma}\xi)\widehat{f}(\xi)e^{2\pi
i\langle x,\xi\rangle}\,{\rm d}\xi,\qquad x\in\mathbb{R}^{n}.$

The subscript $m_{\sigma}$ in (5.1) will be omitted from now on. Let
${\upalpha}\in 3^{-\mathbb{N}}$ and denote by

$\Sigma_{\alpha}\coloneqq\\{\sigma\in\mathrm{\bf{Gr}}(d,n):\,|v_{\sigma}-e_{n}|={\rm
dist}(\sigma,\mathbb{R}^{d}\otimes\\{0\\})<{\upalpha}\\},$

a small neighborhood of $e_{n}^{\perp}$. The small constant ${\upalpha}$ will
depend on dimension only and its choice will be explained below. With such a
choice, the parameter $\upalpha$ will be fixed along the paper, so we will
drop the subindex $\upalpha$ and we will write $\Sigma$ from now on. Consider
the maximal operator

(5.2) $f\mapsto\sup_{\sigma\in\Sigma}|T[P_{0}f](\cdot,\sigma)|$

with $Tf(\cdot,\sigma)$ as defined in (5.1) and $P_{0}$ given by (1.9). Note
that the operator $U_{\mathrm{\bf{Gr}}(d,n),\mathscr{m}}^{\star}$ of
Proposition 3.2 is controlled by $O_{\upalpha}(1)$ rotated copies of (5.2). In
order to prove Proposition 3.3, it thus suffices to obtain the corresponding
estimate for (5.2). Restricting the choice of variable subspace to $\Sigma$
allows us to precompose $T\circ P_{0}$ in (5.2) with a smooth restriction to a
small frequency cone about $e_{n}$ as follows. Let
$\Psi\in\mathcal{S}(\mathbb{R}^{n})$ satisfying

(5.3) ${}\begin{split}\Psi\equiv
1\;\textrm{on}\;\,\Gamma_{0}\cap\mathrm{Ann}\left(\textstyle
1,\frac{3}{2}\right),\qquad&\Gamma_{0}\coloneqq\\{\xi\in\mathbb{R}^{n}\setminus\\{0\\}:\,|\xi^{\prime}-e_{n}|<3^{4}{\upalpha}\\},\\\
{\mathrm{supp}\,}\Psi\subset\Gamma_{1}\cap\mathrm{Ann}\left(\textstyle\frac{1}{2},2\right),\qquad&\Gamma_{1}\coloneqq\\{\xi\in\mathbb{R}^{n}\setminus\\{0\\}:\,|\xi^{\prime}-e_{n}|<3^{5}{\upalpha}\\},\end{split}$

where $\xi^{\prime}\coloneqq\xi/|\xi|$, and let ${P}_{\mathsf{cn}}f$ be the
corresponding frequency cutoff
${P}_{\mathsf{cn}}f\coloneqq(\Psi\widehat{f})^{\vee}$. Then

(5.4) $\sup_{\sigma\in\Sigma}|T[(\mathrm{Id}-P_{\mathsf{cn}})\circ
P_{0}f](\cdot,\sigma)|\lesssim{\mathrm{M}}f$

where the ${\mathrm{M}}$ denotes the Hardy–Littlewood maximal operator;
indeed, the singularities of the symbols of the operators $f\mapsto
Tf(\cdot,\sigma)$ for $\sigma\in\Sigma$ are all contained in $\Gamma_{0}$,
namely away from the frequency support of $(\mathrm{Id}-P_{\mathsf{cn}})\circ
P_{0}f$. This reduces the proof of Proposition 3.3 to the proof of the
corresponding bound for the maximal operator

(5.5) $U^{\star}f\coloneqq\sup_{\sigma\in\Sigma}\left|T[P_{0}\circ
P_{\mathsf{cn}}f](\cdot,\sigma)\right|.$

The multiplier operators $g\mapsto T[P_{0}\circ
P_{\mathsf{cn}}g](\cdot,\sigma)$ have Fourier support in the truncated cone

(5.6) $\Delta\coloneqq\Gamma_{1}\cap\mathrm{Ann}\left(1,\frac{3}{2}\right).$

In our analysis, in accordance with the uncertainty principle and the Fourier
support of the restriction to $\Delta$, the relevant spatial scales for each
$\sigma\in\Sigma$ will then be

(5.7) $s\in\mathbb{S}\coloneqq\left\\{s\in
3^{\mathbb{Z}},\,3^{6}s\cdot{\upalpha}\geq 1\right\\}$

in the $\sigma$ variables, while the spatial scale will always be $\sim 1$ in
the coordinate $\sigma^{\perp}$. Note that by taking $\upalpha$ sufficiently
small, depending on dimension only, we can and will always assume that $s\geq
3^{10d}$ for all relevant scales $s\in\mathbb{S}$. The rest of the section is
devoted to construction of a model sum representation for $U^{\star}$.

###  5.2. Grids and tiles

One of the ways in which the simultaneous localization to $\sigma\in\Sigma$
and to the homogeneous Fourier region
$\Delta^{\prime}=\\{\xi^{\prime}:\xi\in\Delta\\}\subset\mathbb{S}^{d}$ is
exploited below is to conflate either region with a copy of $\mathbb{R}^{d}$
as follows. Let $\widetilde{\Delta^{\prime}}\subset\mathbb{S}^{d}$ be the
$3^{6}\sqrt{d}$-neighborhood of $\Delta^{\prime}$. Then, provided the constant
${\upalpha}$ is chosen sufficiently small depending on dimension, there is an
approximate isometry

$\xi\in\widetilde{\Delta^{\prime}}\mapsto\Pi_{e_{n}^{\perp}}\xi\in
e_{n}^{\perp}\equiv\mathbb{R}^{d}$

in the sense that

(5.8)
${\frac{\theta(\xi,\eta)}{\left|\Pi_{e_{n}^{\perp}}\xi-\Pi_{e_{n}^{\perp}}\eta\right|}},\,\frac{|\xi-\eta|}{\left|\Pi_{e_{n}^{\perp}}\xi-\Pi_{e_{n}^{\perp}}\eta\right|}\in\left[\textstyle\frac{1}{3},3\right],\qquad\xi,\eta\in\widetilde{\Delta^{\prime}},\,\,\,\xi\neq\eta,$

where we remember that $\theta(\xi,\eta)$ denotes the convex angle between
$\xi,\eta\in\SS^{d}$. This together with our previous restriction in the
remark following (5.7) fixes the choice of ${\upalpha}$ to be a small
dimensional constant. When $\sigma\in\Sigma$ we have that
$v_{\sigma}\in\Delta^{\prime}$ and the approximate isometry extends to
$\Sigma$ as well. This allows us to construct a grid structure localizing the
$\Sigma$ and $\Delta^{\prime}$ components by pulling back the corresponding
Euclidean structure. Rotated Euclidean grids will also localize the spatial
component: relevant definitions follow.

A standard triadic grid $\mathcal{G}$ on $\mathbb{R}^{d}$ of pace $(K,m)$,
where $K\geq 1$ is an integer and $m\in[0,K)$ is _not necessarily an integer_
, is a collection of cubes $Q\subset\mathbb{R}^{d}$ satisfying the properties

  * g1.

(quantized length) $\mathcal{G}=\bigcup_{j\in\mathbb{Z}}\mathcal{G}_{Kj+m}$,
where $\mathcal{G}_{u}=\\{Q\in\mathcal{G}:\,\ell(Q)=3^{u}\\}$,

  * g2.

(partition)
$\displaystyle\mathbb{\mathbb{R}}^{d}=\bigcup_{Q\in\mathcal{G}_{Kj+m}}Q$ for
all $j\in\mathbb{Z}$,

  * g3.

(grid) $L\cap Q\in\\{L,Q,\varnothing\\}$ for every $L,Q\in\mathcal{G}$.

In particular, $\mathcal{G}$ is partially ordered by inclusion and we may
define for each $Q\in\mathcal{G}$

$Q^{(1)}\coloneqq\textrm{minimal }L\in\mathcal{G}\textrm{ with }Q\subsetneq
L,$

and inductively $Q^{(j)}\coloneqq[Q^{(j-1)}]^{(1)}$ for all $j>1$ to be the
$j$-th parent of $Q$ in the grid $\mathcal{G}$. With these definitions,
properties g2. and g3. yield for any positive integer $\kappa$ the partition

$L=\bigsqcup_{Q\in\mathsf{ch}_{\kappa}(L)}Q,\qquad\mathsf{ch}_{\kappa}(L)\coloneqq\\{Q\in\mathcal{G}:\,Q^{(\kappa)}=L\\}$

with the collection $\mathsf{ch}_{\kappa}(L)$ being referred to as the
$\kappa$-th generation children of $L\in\mathcal{G}$. Triadic grids have been
chosen because of the following convenient property. For each
$L\in\mathcal{G}$ there is a unique
$Q=L^{\circ,\kappa}\in\mathsf{ch}_{\kappa}(L)$ which is concentric with $L$,
which is referred to as the $\kappa$-_center_ of $L$. The next definition
highlights a different role for certain other elements of
$\mathsf{ch}_{\kappa}(L)$. Define

$\begin{split}&\mathsf{ch}_{\kappa,\square}(L)\coloneqq\\{Q\in\mathsf{ch}_{\kappa}(L):\,\mathrm{dist}(Q,L^{\circ,\kappa})\geq
3^{3-\kappa}\ell(L)\\}\end{split}$

referring to this collection as the _peripheral_ $\kappa$-th children of $L$.
In the following lemma and hereafter we fix a large value of $\kappa\geq
9+\log_{3}d$.

######  Lemma 5.3.

Let $\kappa\geq 9+\log_{3}d$. There exist standard triadic grids
$\mathcal{G}_{1},\ldots,\mathcal{G}_{C}$ of pace $(1,m_{j})$ on
$\mathbb{R}^{d}\equiv e_{n}^{\perp}$, $1\leq j\leq C$, $C=C_{\kappa,d}$, with
the following property: For each $\beta\in\Delta^{\prime}$, $s\in\mathbb{S}$,
there exists $j=j(\beta,s)\in\\{1,\ldots,C\\}$ and a cube
$L=L(\beta,s)\in\mathcal{G}_{j}$ such that

  * _1._

$3^{3}s^{-1}\leq\ell(L)\leq 3^{4}s^{-1}$,

  * _2._

$B\left(\beta,3^{-\kappa}s^{-1}\right)\subset\Pi_{e_{n}^{\perp}}^{-1}(L^{\circ,\kappa})$,

  * _3._

$\mathrm{Ann}\left(\beta,\textstyle
3^{-1}s^{-1},3^{2}s^{-1}\right)\subset\bigcup\left\\{\Pi_{e_{n}^{\perp}}^{-1}(Q):\,Q\in\mathsf{ch}_{\kappa,\square}(L)\right\\}$.

######  Remark 5.4.

With reference to the previous lemma, say that the pair
$(\beta,s)\in\Delta^{\prime}\times\mathbb{S}$ is of type
$j\in\\{1,\ldots,C\\}$ if $j(\beta,s)=j$. The restriction $s\in\mathbb{S}$,
see (5.7), and properties 1. and 2. ensure the inclusion

$L(\beta,s)\subset
B\left(\Pi_{e_{n}^{\perp}}\beta,3^{5}\sqrt{d}{s}^{-1}\right)\subset\Pi_{e_{n}^{\perp}}\widetilde{\Delta^{\prime}}.$

###### Proof of Lemma 5.3.

For this proof we denote by $Q(\xi,r)$ the cube with center
$\xi\in\mathbb{R}^{d}$ and sidelength $2r$ and by $\|\cdot\|_{\infty}$ the
$\ell^{\infty}$-norm on $\mathbb{R}^{d}$. First of all, using standard grid
techniques as in [LeNa] we may find triadic grids
$\mathcal{G}_{1},\ldots,\mathcal{G}_{C}$ with the property that for each cube
$P\subset\mathbb{R}^{d}$ there is $j\in\\{1,\ldots,C\\}$ and
$L(P)\in\mathcal{G}_{j}$ with $P\subset L(P)\subset(1+3^{-(\kappa+9)})P$. As a
particular consequence

$\begin{split}|c(P)-c(L(P))|&\leq\sqrt{d}3^{-(\kappa+9)}\ell(P)\leq\sqrt{d}3^{-(\kappa+9)}\ell(L(P)),\\\
\|c(P)-c(L(P))\|_{\infty}&\leq 3^{-(\kappa+9)}\ell(P)\leq
3^{-(\kappa+9)}\ell(L(P)).\end{split}$

We have that $\Pi_{e_{n}^{\perp}}B(\beta,3^{2}s^{-1})\subset
B(\Pi_{e_{n}^{\perp}}\beta,3^{3}s^{-1})\subset
Q(\Pi_{e_{n}^{\perp}}\beta,3^{3}s^{-1})$. There exists
$L=L(\beta,s)\in\mathcal{G}_{j}$for some $j\in\\{1,\ldots,C\\}$ such that

$Q(\Pi_{e_{n}^{\perp}}\beta,3^{3}s^{-1})\subseteq
L\subseteq(1+3^{-(\kappa+9)})Q(\Pi_{e_{n}^{\perp}}\beta,3^{3}s^{-1})$

from which 1. follows immediately. In order to see 2. let $\eta\in
B(\beta,3^{-\kappa}s^{-1})$ so that $\Pi_{e_{n}^{\perp}}\eta\in
B(\Pi_{e_{n}^{\perp}}\beta,3^{-\kappa+1}s^{-1})$. Then we have the chain of
inequalities

$\|\Pi_{e_{n}^{\perp}}\eta-c(L)\|_{\infty}\leq|\Pi_{e_{n}^{\perp}}\beta-\Pi_{e_{n}^{\perp}}\eta|+\|\Pi_{e_{n}^{\perp}}\beta-c(L)\|_{\infty}\leq
3^{-\kappa+1}s^{-1}+3^{-(\kappa+9)}\ell(L)\leq 3^{-\kappa-1}\ell(L).$

This shows that $\Pi_{e_{n}^{\perp}}\eta\in
Q(c(L),\,3^{-\kappa-1}\ell(L))=3^{-1}L^{\circ,\kappa}$ and so
$\eta\in\Pi_{e_{n}^{\perp}}^{-1}(L^{\circ,\kappa})$ as desired.

In order to complete the proof it suffices to show that if $\eta\in
B(\beta,3^{2}s^{-1})\setminus B(\beta,3^{-1}s^{-1})$ then
$\Pi_{e_{n}^{\perp}}\eta\in Q$ for some $Q\in\mathsf{ch}_{\kappa,\square}(L)$.
To that end let $\eta\notin B(\beta,3^{-1}s^{-1})$; we will have that
$\Pi_{e_{n}^{\perp}}\eta\notin B(\Pi_{e_{n}^{\perp}}\beta,3^{-2}s^{-1})$ and
so

$\|\Pi_{e_{n}^{\perp}}\eta-c(L)\|_{\infty}\geq\frac{1}{\sqrt{d}}|\Pi_{e_{n}^{\perp}}\eta-\Pi_{e_{n}^{\perp}}\beta|-\|\Pi_{e_{n}^{\perp}}\beta-c(L)\|_{\infty}\geq\frac{3^{-4}}{\sqrt{d}}\ell(L)\geq\sqrt{d}3^{5-\kappa}\ell(L).$

Since $\Pi_{e_{n}^{\perp}}\eta\in
B(\Pi_{e_{n}^{\perp}}\beta,3^{3}s^{-1})\subseteq L$ there exists
$Q\in\mathsf{ch}_{\kappa,\square}(L)$ such that $\Pi_{e_{n}^{\perp}}\eta\in
Q$. Then for all $y_{1}\in Q$ and $y_{2}\in L^{\circ,\kappa}$ we calculate

$\|y_{1}-y_{2}\|_{\infty}\geq\sqrt{d}3^{5-\kappa}\ell(L)-\|y_{1}-\Pi_{e_{n}^{\perp}}\eta\|_{\infty}-\|y_{2}-c(L)\|_{\infty}\geq\sqrt{d}3^{5-\kappa}\ell(L)-23^{-\kappa}\ell(L)\geq\sqrt{d}3^{4-\kappa}\ell(L)$

which in turn implies that ${\rm dist}(Q,L^{\circ,\kappa})\geq
3^{4-\kappa}\ell(L)$. We have showed that
$Q\in\mathsf{ch}_{\kappa,\square}(L)$ so the proof is complete. ∎

More generally, if $X$ is any set, any collection
$\mathcal{G}\subset\mathcal{P}(X)$ satisfying the grid property g3. is
referred to as a _grid_. The forthcoming definitions construct suitable
rotated grid systems in $\mathbb{R}^{n}$ whose purpose is to provide spatial
localization.

####  5.4.1. Spatial grids

Fix a standard triadic grid $\mathcal{L}$ of pace $(1,0)$ in $\mathbb{R}^{d}$
and define the subsets of $\mathbb{R}^{n}$

$\mathcal{R}\coloneqq\left\\{L\times[j,j+1)):\;L\in\mathcal{L},\,\ell(L)\geq
1,j\in\mathbb{Z}\right\\}.$

The collection $\mathcal{R}=\mathcal{R}_{e_{n}}$ of _plates parallel to
$e_{n}^{\perp}\in\mathrm{\bf{Gr}}(d,n)$_ obviously inherits property g3. from
$\mathcal{L}$ and the unit length partition of $\mathbb{R}$. If
$\mathcal{R}\ni R=L\times J$, call $L$ the _horizontal component_ and $J$ the
_vertical component_ of $R$. The condition $\ell(L)\geq 1$ in the definition
of $\mathcal{R}$ guarantees that the elements $L\in\mathcal{R}$ have
horizontal components of larger scale than their vertical components, so they
can be in general understood as plates in $\mathbb{R}^{d+1}$ of _scale_
$\mathrm{scl}(R)\coloneqq\ell(L)=3^{k}$, $k\in\mathbb{N}$, and thickness $1$
along the direction $e_{n}$. For $\beta\in\mathbb{S}^{d}$ the rotated grid
$\mathcal{R}_{\beta}$ of _plates parallel to
$\beta^{\perp}\in\mathrm{\bf{Gr}}(d,n)$_ is then defined by

$\mathcal{R}_{\beta}\coloneqq\left\\{O_{\beta^{\perp},e_{n}^{\perp}}Q:\,Q\in\mathcal{R}\right\\}.$

It is convenient to associate to an element $R\in\mathcal{R}_{\beta}$ the
translation and anisotropic rescaling operator

$\mathsf{Sy}_{R}^{p}f(x)\coloneqq\frac{1}{|R|^{\frac{1}{p}}}f\left(\frac{\Pi_{\beta^{\perp}}(x-c(R))}{{|\Pi_{\beta^{\perp}}R|}}+\frac{\Pi_{\beta}(x-c(R))}{{|\Pi_{\beta}(R)|}}\right),\qquad
x\in\mathbb{R}^{n}.$

The sense of this definition is that if $f$ is a bump function around the
origin of $\mathbb{R}^{n}$ then $\mathsf{Sy}_{R}^{p}f$ is an
$L^{p}$-normalized bump adapted to the plate $R\in\mathcal{R}_{\beta}$.

####  5.4.2. Tiles

Let $\mathcal{G}$ be any standard triadic grid on $\mathbb{R}^{d}$. Define the
_admissible tiles_ $t\in\mathcal{T}_{\mathcal{G}}$ generated by $\mathcal{G}$
as the collection of cartesian products $t=R_{t}\times Q_{t}$ where

  * 1.

$Q_{t}\in\mathcal{G}$,
$Q_{t}\subset\Pi_{e_{n}^{\perp}}\widetilde{\Delta^{\prime}}$,

  * 2.

$R_{t}=L_{t}\times I_{t}\in\mathcal{R}_{v_{t}}$, where
$v_{t}\coloneqq\Pi^{-1}_{e_{n}}c(Q_{t})\in\mathbb{S}^{d}$,

  * 3.

$\mathrm{scl}(R_{t})\ell(Q_{t})\in[1,3)$.

The _scale_ of $t\in\mathcal{T}_{\mathcal{G}}$ is
$\mathrm{scl}(t)\coloneqq\mathrm{scl}(R_{t})$. Note that $v_{t}$ is correctly
uniquely defined because of property 1. and the fact that
$\Pi_{e_{n}^{\perp}}$ is a quasi-isometry on $\widetilde{\Delta^{\prime}}$.
The _frequency support_ $\omega_{t}\subset\mathbb{R}^{n}$ of a tile $t$ is
given by

$\omega_{t}\coloneqq\left\\{\xi\in\mathrm{Ann}\textstyle(\frac{1}{2},2):\,\Pi_{e_{n}^{\perp}}\xi^{\prime}\in
Q_{t}^{\circ,\kappa}\right\\}\subset\mathbb{R}^{n}.$

Namely, $\omega_{t}$ consists of those points of
$\mathrm{Ann}\textstyle(\frac{1}{2},2)$ whose projection on the sphere is
contained in the preimage of the $\kappa$-th center of $Q_{t}$, denoted now by
$Q_{t}^{\circ,\kappa}$; see Figure 5.1. Note that the collection
$\Omega_{\mathcal{G}}\coloneqq\\{\omega_{t}:\,t\in\mathcal{T}_{\mathcal{G}}\\}$
is a grid on $\mathbb{R}^{n}$, as it inherits property g3. from $\mathcal{G}$.

Figure 5.1. The frequency component of a tile and its $\kappa$-children

Let $Q_{0}=[0,1)^{d}$ be considered as a triadic cube and let us fix an
arbitrary enumeration of the elements of $\mathsf{ch}_{\kappa,\square}(Q_{0})$

$\mathsf{ch}_{\kappa,\square}(Q_{0})=\bigsqcup_{\tau=1}^{N}Q_{0,\tau},\qquad
N=c_{d}3^{\kappa d},$

where $c_{d}$ is a dimensional constant. Note that this indexing of the
peripheral $\kappa$-children of $Q_{0}$ induces an indexing of the peripheral
$\kappa$-children in $\mathsf{ch}_{\kappa,\square}(Q_{t})$ for any tile $t$,
in a way that is independent of the particular choice of tile $t$ and grid
$\mathcal{G}$. The _directional support_ $\alpha_{t}\subset\mathbb{S}^{d}$ of
a tile $t$ is given by

(5.9)
$\begin{split}&\alpha_{t}\coloneqq\bigsqcup_{\tau=1,\ldots,N}\alpha_{t,\tau},\\\
&\alpha_{t,\tau}\coloneqq\Pi^{-1}_{e_{n}^{\perp}}(Q_{t;\tau}),\qquad
Q_{t;\tau}\in\mathsf{ch}_{\kappa,\square}(Q_{t}),\qquad\tau\in\\{1,\ldots,N\\}.\end{split}$

The collections
$A_{\mathcal{G},\tau}\coloneqq\\{\alpha_{t,\tau}:\,t\in\mathcal{T}_{\mathcal{G}}\\}$
are grids on $\mathbb{S}^{d}$, for each $1\leq\tau\leq N$.

####  5.4.3. Adapted classes and model sums

Hereafter, let $\Theta_{M}$ stand for the unit ball of the Banach space of
functions

$u\in\mathcal{C}^{M}(\mathbb{R}^{n}),\qquad\|u\|_{\star,M}\coloneqq\sup_{0\leq|\alpha|\leq
M}\left\|\left\langle
x\right\rangle^{M}\partial_{x}^{\alpha}u(x)\right\|_{\infty}<\infty,$

where $\langle x\rangle\coloneqq(1+|x|^{2})^{\frac{1}{2}}$. We will often use
the special bump function $\chi_{M}\in\mathcal{C}^{\infty}$ given by

$\chi_{M}(x)\coloneqq\langle x\rangle^{-M}.$

Given a tile $t=R_{t}\times Q_{t}$ and a large positive integer $M\gg 1$ we
define $\mathcal{F}_{t}^{M}$ as the collection of complex-valued functions
$\varphi$ on $\mathbb{R}^{n}$ with the properties ($\varphi$ are
$L^{2}$-normalized bump functions adapted to $R_{t}$)

  * (i)

$\varphi\in\left\\{\mathrm{Mod}_{v_{t}}\mathsf{Sy}_{R_{t}}^{2}\Phi:\Phi\in\Theta_{M}\right\\},\qquad\mathrm{Mod}_{v}g(x)\coloneqq
e^{2\pi i\langle x,v\rangle}g(x),\qquad x\in\mathbb{R}^{n},\quad v\in\SS^{d}$;

  * (ii)

$\mathrm{supp}\,\widehat{\varphi}\subseteq\omega_{t}$.

Further, define $\mathcal{A}_{t}^{M}$ as the collection of complex-valued
functions ${\vartheta}={\vartheta}(x,\sigma)$ on $\mathbb{R}^{n}\times\Sigma$
satisfying

  * (iii)

${\vartheta}(\cdot,\sigma)\in\mathcal{F}_{t}^{M}$ for all fixed
$\sigma\in\Sigma$;

  * (iv)

${\vartheta}(\cdot,\sigma)=0$ for all $\sigma\in\Sigma$ such that
$v_{\sigma}\notin\alpha_{t}$;

  * (v)

$\vartheta$ verifies the adaptedness condition

$|{\vartheta}(\cdot,\sigma)-{\vartheta}(\cdot,\rho)|\leq\max\left\\{\mathrm{scl}(t){\rm
dist}(\sigma,\rho),\frac{1}{\log({\rm e}+\left[{\rm
dist}(\sigma,\rho)\right]^{-1})}\right\\}\mathsf{Sy}_{R_{t}}^{2}\chi_{M}$

for all $\sigma,\rho\in\mathrm{\bf{Gr}}(d,n)$ with $\mathrm{scl}(t){\rm
dist}(\sigma,\rho)\lesssim 1$.

Note that both classes are $L^{2}$-normalized.

######  Remark 5.5.

The point of condition (v) is that in a tree, cf. Definition 6.1, with top
direction $\sigma$ there are $\lesssim|\log({\rm dist}(\sigma(x),\sigma))|$
frequency scales contributing at each point, and the total contribution of the
difference $|{\vartheta}_{t}(x,\sigma(x))-{\vartheta}_{t}(x,\sigma)|$ is
summable.

At this point, define the tile coefficient maps

(5.10) $F_{M}[f](t)\coloneqq\sup_{\varphi\in\mathcal{F}_{t}^{M}}\left|\langle
f,\varphi\rangle\right|$

and, for a fixed measurable choice
$\sigma:\mathbb{R}^{n}\to\mathrm{\bf{Gr}}(d,n)$ with the property that
$\sigma(x)\in\Sigma$ for all $x\in\mathbb{R}^{n}$, and $1\leq\tau\leq N$

(5.11)
$A_{\sigma,\tau,M}[g](t)\coloneqq\sup_{\vartheta\in\mathcal{A}_{t}^{M}}\left|\left\langle
g,\vartheta\left(\cdot,\sigma(\cdot)\right)\bm{1}_{\alpha_{t,\tau}}\left(v_{\sigma(\cdot)}\right)\right\rangle\right|.$

The model bisublinear operator

(5.12)
$\Lambda_{\mathbb{P};\sigma,\tau,M}(f,g)=\sum_{t\in\mathbb{P}}F_{M}[f](t)A_{\sigma,\tau,M}[g](t)$

is then associated to each subset
$\mathbb{P}\subset\mathcal{T}_{\mathcal{G}}$. Momentarily, it will be shown
that the duality form for the maximal operator (5.5) is controlled by a
suitable combination of finitely many forms (5.12).

###  5.6. A homogeneous partition of unity

Our goal here is to produce a single scale Gabor decomposition of the
frequency region $\Delta$ defined in (5.6) which contains the Fourier support
of the multiplier $P_{0}\circ P_{\mathsf{cn}}$.

Fix a (spatial) scale $s\in\mathbb{S}$, cf. (5.7), and recall that if
$\beta\in\Delta^{\prime}$, this choice guarantees that
$B(\beta,3^{6}s^{-1})\subset\widetilde{\Delta^{\prime}}$. We remember that
$\kappa\geq 9+\log_{3}d$ is a fixed large positive integer. Let
$\mathcal{B}_{s^{-1}}$ be a $3^{-\kappa}s^{-1}$-net on $\Delta^{\prime}$, in
the sense that

  * -

the sets
$\left\\{B(\beta,3^{-\kappa}s^{-1}):\beta\in\mathcal{B}_{s^{-1}}\right\\}$
form a finitely overlapping cover of $\Delta^{\prime}$;

  * -

the sets
$\left\\{B(\beta,3^{-(\kappa+1)}s^{-1}):\beta\in\mathcal{B}_{s^{-1}}\right\\}$
are pairwise disjoint.

Now let $\\{\theta_{\beta}\\}_{\beta\in\mathcal{B}_{s^{-1}}}$ be a smooth,
$0$-homogeneous partition of unity on $\Gamma_{1}$, consisting of nonnegative
real-valued functions, which is subordinate to
$\\{{B(\beta,{3^{-\kappa}s^{-1}})}:\beta\in\mathcal{B}_{s^{-1}}\\}$. By this,
we mean

(5.13)
${\mathrm{supp}\,}\theta_{\beta}\subset{B(\beta,{3^{-\kappa}s^{-1}})}\qquad{\text{and}}\qquad\sum_{\beta\in\mathcal{B}_{s^{-1}}}\left[\theta_{\beta}(\eta^{\prime})\right]^{2}=1\qquad\forall\eta\in\Gamma_{1},\quad\eta^{\prime}\coloneqq\frac{\eta}{|\eta|},$

and that the $k$-th order derivatives of $\theta_{\beta}$ are of the order
$O_{k,n,\delta}(s^{k})$ uniformly in $\beta\in\mathcal{B}_{s^{-1}}$. Note also
that (5.13) implies that $\theta_{\beta}\equiv 1$ on
$B(\beta,{3^{-(\kappa+1)}s^{-1}})$. For $\beta\in\mathcal{B}_{s^{-1}}$,
consider the function $\phi_{\beta}\in\mathcal{S}(\mathbb{R}^{n})$ whose
Fourier transform is given by

(5.14)
$\widehat{{\phi}_{\beta}}(\xi)\coloneqq\theta_{\beta}\left(\xi^{\prime}\right)\zeta(|\xi|),\qquad\xi\in\mathbb{R}^{n}\setminus\\{0\\}.$

By construction,

(5.15)
${\mathrm{supp}\,}\widehat{\phi_{\beta}}\subset\left\\{\xi\in\mathrm{Ann}\left(\textstyle\frac{1}{2},2\right):\xi^{\prime}\in
B\left(\beta,{3^{-\kappa}s^{-1}}\right)\right\\}.$

and (5.13) then ensures

$(P_{0}\circ P_{\mathsf{cn}})f=\sum_{\beta\in\mathcal{B}_{s^{-1}}}P_{0}\circ
P_{\mathsf{cn}}f\ast\phi_{\beta}\ast\phi_{\beta}.$

For each $\beta\in\mathcal{B}_{s^{-1}}$, consider the lattice
$Z(\beta)\coloneqq
O_{\beta^{\perp},e_{n}^{\perp}}\left[s\mathbb{Z}^{d}\times\mathbb{Z}\right]$.
Using Fourier series on the support of each $\widehat{\phi_{s,\beta}}$ we
obtain

(5.16) $P_{0}\circ
P_{\mathsf{cn}}f=\lim_{T\to+\infty}\frac{1}{T^{n}}\int_{[0,T]^{n}}s^{d}\sum_{\beta\in\mathcal{B}_{s^{-1}}}\sum_{z\in
Z(\beta)}\langle(P_{0}\circ
P_{\mathsf{cn}})f,\mathrm{Tr}_{z+y}\phi_{\beta}\rangle\mathrm{Tr}_{z+y}\phi_{\beta}\,{\rm
d}y,$

for all $f\in\mathcal{S}(\mathbb{R}^{n})$, where $\mathrm{Tr}_{u}h(x)\coloneqq
h(x-u)$ for $x,u\in\mathbb{R}^{n}$.

###  5.7. Discretization of the kernel

We turn to the discretization of the variable kernels in (5.5). Choose a non-
negative, smooth radial function $\Psi$ on $\mathbb{R}^{n}$ with

${\mathrm{supp}\,}{\Psi}\subset\mathrm{Ann}\left({\textstyle\frac{8}{9},\frac{28}{9}}\right),\qquad\sum_{s\in
3^{\mathbb{Z}}}{\Psi}(s\xi)=1\qquad\forall\xi\in\mathbb{R}^{n}\setminus\\{0\\},$

while $\Phi$ is a smooth radial cutoff equal to $1$ on
$\mathrm{Ann}(1,\frac{3}{2})$ and vanishing off $\mathrm{Ann}(\frac{1}{2},2)$.
These definitions entail

(5.17) $\begin{split}&T[P_{0}\circ P_{\mathsf{cn}}f](\cdot,\sigma)=\sum_{s\in
3^{\mathbb{Z}}}P_{0}\circ P_{\mathsf{cn}}f*\psi_{s}(\cdot,\sigma),\\\
&\psi_{s}(x,\sigma)\coloneqq\int_{\mathbb{R}^{n}}m_{\sigma}(O_{\sigma}\Pi_{\sigma}\xi){\Psi}(s\Pi_{\sigma}\xi)\Phi(\xi)e^{2\pi
i{\langle x,\xi\rangle}}\,{\rm
d}\xi,\qquad(x,\sigma)\in\mathbb{R}^{n}\times\Sigma.\end{split}$

First, we use condition (5.3) on the frequency support of $P_{\mathsf{cn}}f$
to deduce that $P_{\mathsf{cn}}f*\psi_{s}=0$ unless $s\in\mathbb{S}$, cf.
(5.7), and consequently restrict the sum in (5.17) to $s\in\mathbb{S}$.
Subsequently, use (5.16) and (5.4) to estimate

$|T_{\sigma}\circ P_{0}\circ
P_{\mathsf{cn}}f|\lesssim\lim_{T\to\infty}\frac{1}{T^{n}}\int_{[0,T]^{n}}\bigg{|}\sum_{s\in\mathbb{S}}s^{d}\sum_{\beta\in\mathcal{B}_{s^{-1}}}\sum_{z\in
Z(\beta)}\langle\mathrm{Tr}_{-y}g,\mathrm{Tr}_{z}\phi_{\beta}\rangle\mathrm{Tr}_{y}(\mathrm{Tr}_{z}\phi_{\beta}*\psi_{s}(\cdot,\sigma))\bigg{|}\,{\rm
d}y.$

The norm estimates for $U^{\star}$ of (5.5) will follow from corresponding
estimates for the maximal operator

(5.18)
$f\mapsto\sup_{\sigma\in\Sigma}\bigg{|}\sum_{{s\in\mathbb{S}}}s^{d}\sum_{\beta\in\mathcal{B}_{s^{-1}}}\sum_{z\in
Z(\beta)}\langle
f,\mathrm{Tr}_{z}\phi_{\beta}\rangle\mathrm{Tr}_{z}\phi_{\beta}*\psi_{s}(\cdot,\sigma)\bigg{|}.$

To each triplet $(\beta,s,z)$ appearing in the sum (5.18), assume $(\beta,s)$
is of type $j$ in accordance to Remark 5.4 and associate a tile

(5.19) $t=R_{t}\times Q_{t}=t(\beta,s,z)\in\mathcal{G}_{j},$

where the latter is one of the grids appearing in the conclusion of Lemma 5.3,
as follows. Firstly, $Q_{t}=L(\beta,s)$. Secondly, $R_{t}$ is the unique
element of $\mathcal{R}_{\beta}$ with $z\in R_{t}$ and
$\mathrm{scl}(R_{t})\ell(L(\beta,s))\in{[1,3)}$. Of course,
$\mathrm{scl}(t)\sim s$. Recall the definitions of the collections
$\mathcal{F}_{t}^{M}$ and $\mathcal{A}_{t}^{M}$ from §5.4.3. The following
lemma will be used in §6 for $M=50(d+1)$.

######  Lemma 5.8.

Let $M$ be a large integer. There is $A=A(M,d)$ such that if $\mathscr{m}$
satisfies assumption (1.10) the following holds. With reference to the
expressions in (5.18), if $t=t(\beta,s,z)$ define the functions

$\varphi_{t}(x)\coloneqq\mathrm{scl}(t)^{\frac{d}{2}}\mathrm{Tr}_{z}\phi_{\beta}(x),\quad{\vartheta}_{t}(x,\sigma)\coloneqq\mathrm{scl}(t)^{\frac{d}{2}}\int_{\mathbb{R}^{n}}\phi_{\beta}(y-z)\psi_{s}(x-y,\sigma)\,{\rm
d}y,\quad(x,\sigma)\in\mathbb{R}^{n}\times\Sigma.$

Then

(5.20) $\displaystyle c\varphi_{t}\in\mathcal{F}_{t}^{M},\qquad
c{\vartheta}_{t}\in\mathcal{A}_{t}^{M-1}$

where the constant $c>0$ may be chosen uniformly in $\beta,s,z$.

###### Proof.

Firstly, property (ii) in the definition of the class $\mathcal{F}_{t}^{M}$
follows by (5.15) together with condition 2. of Lemma 5.3. We now check
condition (i) for $\varphi_{t}$. By rotation and translation we can assume
that $t=t(e_{n},s,0)$ and write $\mathbb{R}^{d}\otimes\mathbb{R}\ni
x\eqqcolon(y,x_{n})$, where $s=\mathrm{scl}(t)$. Using property (ii) we
readily see that the function

$\widetilde{\mathbf{1}}_{B}(x)\coloneqq\mathrm{Mod}_{-e_{n}}s^{d/2}\varphi_{t}(sy,x_{n}),\qquad
x=(y,x_{n})\in\mathbb{R}^{d}\times\mathbb{R},$

is a $\mathcal{C}^{\infty}$ bump function with
$\|\widetilde{\mathbf{1}}_{B}\|_{1}\eqsim 1$, which is adapted of any order to
the unit ball of $\mathbb{R}^{n}$, centered at the origin. Thus
$c\widetilde{\mathbf{1}}_{B}\in\Theta_{M}$ as defined in §5.4.3 for any order
$M$, where $c>0$ is a normalization constant depending only upon dimension. We
get that

$c\varphi_{t}(x)=c\mathrm{Mod}_{e_{n}}s^{-d/2}\,\widetilde{\mathbf{1}}_{B}(y/s,x_{n}),\qquad
x=(y,x_{n})\in\mathbb{R}^{d}\times\mathbb{R},$

which shows (i). These two remarks show that
$c\varphi_{t}\in\mathcal{F}_{t}^{M}$ for any desired large positive integer
$M$.

We move to the verification of property
$c\vartheta_{t}\in\mathcal{A}_{t}^{M}$. For this let us fix $\sigma\in\Sigma$
and note that
$\vartheta_{t}(\cdot,\sigma)=\phi_{\beta}(\cdot-z)*\psi_{s}(\sigma,\cdot)$. In
view of (5.14) and (5.17), this immediately implies the Fourier support
condition

${\mathrm{supp}\,}\widehat{\vartheta_{t}(\cdot,\sigma)}\subseteq{\mathrm{supp}\,}\widehat{\phi_{t}}\subseteq\omega_{t}.$

We now check condition (iv) for $\vartheta_{t}$. Recall that the tile $t$ has
been chosen so that $\alpha_{t}$ contains $\mathrm{Ann}\left(\beta,\textstyle
3^{-1}s^{-1},3^{2}s^{-1}\right)$, and that
${\mathrm{supp}\,}\widehat{\phi}_{\beta}\subset B(\beta,{3^{-\kappa}s^{-1}})$
which tells us that whenever
$\xi\in{\mathrm{supp}\,}(\mathrm{Tr}_{z}\phi_{\beta}*\psi_{s})^{\wedge}$ the
following inequalities

$\frac{4}{9}<s|\Pi_{\sigma}\xi^{\prime}|<\frac{56}{9},\qquad|\xi^{\prime}-\beta|\leq{3^{-\kappa}s^{-1}},\qquad\xi^{\prime}\coloneqq\xi/|\xi|\in\SS^{d},$

must hold. Furthermore, since $\beta\in\SS^{d}$,

$\begin{split}|\Pi_{\sigma}\xi^{\prime}|+{3^{-\kappa}s^{-1}}\geq|\Pi_{\sigma}\xi^{\prime}|+|\xi^{\prime}-\beta|\geq|\Pi_{\sigma}\beta|\geq|\Pi_{\sigma}\xi^{\prime}|-|\xi^{\prime}-\beta|\geq|\Pi_{\sigma}\xi^{\prime}|-{3^{-\kappa}s^{-1}}\end{split}$

which, in view of the control $\kappa\geq 9+\log_{3}d$ yields

$|\Pi_{\sigma}\beta|\in s^{-1}\left[\textstyle\frac{1}{3},7\right]\iff
v_{\sigma}\in\mathrm{Ann}\left(\beta,\textstyle\frac{1}{3}s^{-1},7s^{-1}\right).$

This proves that $\vartheta_{t}(\cdot,\sigma)=0$ unless
$v_{\sigma}\in\mathrm{Ann}\left(\beta,\textstyle
3^{-1}s^{-1},3^{2}s^{-1}\right).$ Condition 3. of Lemma 5.3 then entails (iv)
for $\vartheta=\vartheta_{t}$.

We now move to our main task which is to verify that
$c\vartheta_{t}(\cdot,\sigma)$ verifies conditions (iii) and (v) in the
definition of the class $\mathcal{A}_{t}^{M}$. We can again reduce to the case
$\beta=e_{n}$ and $z=0$; we write
$x=(x^{\prime},x_{n})\in\mathbb{R}^{d}\otimes\mathbb{R}$ and $\sigma\in\Sigma$
with $v_{\sigma}\in\mathrm{Ann}\left(e_{n},\textstyle
3^{-1}s^{-1},3^{2}s^{-1}\right)$. Now, express $x=(y,x_{n})$ in terms of
$\Pi_{\sigma}x,\Pi_{\sigma^{\perp}}x$ as in Lemma 2.4,

(5.21) $\begin{split}y-\Pi_{\sigma}x&=q_{1}(\langle x,u\rangle,\langle
x,v_{\sigma}\rangle)u+q_{2}(\langle x,u\rangle,\langle
x,v_{\sigma}\rangle)v_{\sigma},\\\
x_{n}-\Pi_{\sigma^{\perp}}x&=q_{2}(\langle-x,v_{\sigma}\rangle,\langle
x,u\rangle)u+q_{1}(\langle
x,v_{\sigma}\rangle,\langle-x,u\rangle)v_{\sigma},\\\
q_{1}(t,w)&\coloneqq\left[\frac{\sin
2\theta}{2}w-(\sin\theta)^{2}t\right],\qquad
q_{2}(t,w)\coloneqq\left[\frac{\sin
2\theta}{2}t+(\sin\theta)^{2}w\right].\end{split}$

Above $\theta\coloneqq{\rm dist}(\sigma,e_{n}^{\perp})$ and $u\in\sigma$ is a
unit vector orthogonal to $\sigma\cap e_{n}^{\perp}$. By the already verified
directional support condition (iv) we know that $\theta={\rm
dist}(\sigma,e_{n}^{\perp})=|v_{\sigma}-e_{n}|\lesssim s^{-1}$, which implies
that

(5.22) $|q_{1}(y,w)|+|q_{2}(y,w)|\lesssim s^{-1}\big{(}|y|+|w|\big{)}.$

Combining (5.21) with (5.22) shows that the decay rates

$\left[\left(1+s^{-1}|\Pi_{\sigma}x|\right)\left(1+|\Pi_{\sigma^{\perp}}x|\right)\right]^{-M},\qquad\left[\left(1+s^{-1}|y|\right)\left(1+|x_{n}|\right)\right]^{-M^{\prime}}$

are equivalent, with $M\leq M^{\prime}/2\leq 2M$. Defining the dilation
operator $\mathrm{Dil}_{\sigma}^{s}$, $\sigma\in\mathrm{\bf{Gr}}(d,n)$ as

$\mathrm{Dil}_{\sigma}^{s}g(x)\coloneqq
s^{d/2}g(s\Pi_{\sigma}x,\Pi_{\sigma^{\perp}}x),\qquad
x=(\Pi_{\sigma}x,\Pi_{\sigma^{\perp}}x)\in\mathbb{R}^{n},$

it will be enough to show that for some dimensional constant $c>0$, the
rescaled function

$\mathbb{R}^{n}\ni x\mapsto
c\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}F(x)$

is a bump function adapted to the unit ball of $\mathbb{R}^{n}$, centered at
the origin, where

$F=\vartheta_{t}(\cdot,\sigma)\qquad\text{or}\qquad
F={\vartheta_{t}(\cdot,\sigma)-\vartheta_{t}(\cdot,\rho),}\qquad\rho,\sigma\in\Sigma,\quad\rho\neq\sigma.$

We begin with the first case where $F=\vartheta_{t}(\cdot,\sigma)$. Assuming
again that $t=t(e_{n},s,0)$ we note that for every multiindex $\alpha$

$\begin{split}\partial_{x}^{\alpha}[\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\vartheta_{t}(\cdot,\sigma)]&=[\partial_{x}^{\alpha}\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\varphi_{t}]*\mathrm{Mod}_{-e_{n}}s^{d/2}\mathrm{Dil}_{\sigma}^{s}\psi_{s}(\cdot,\sigma)\\\
&\eqqcolon[\partial_{x}^{\alpha}\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\varphi_{t}]*\mathrm{Mod}_{-e_{n}}G_{1}.\end{split}$

We have already seen that the function
$\mathrm{Mod}_{-e_{n}}{s^{d/2}}\varphi_{t}(sy,x_{n})$, with
$x=(y,x_{n})\in\mathbb{R}^{d}\times\mathbb{R}$ is a smooth bump adapted at
every order to the unit ball of $\mathbb{R}^{n}$ centered at the origin. By
the remark following (5.22) this is equivalent to the fact that the function
$\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\varphi_{t}(x)$ has the same
adaptedness property. Thus, in order to show that
$\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\vartheta_{t}$ is adapted to
the unit ball of $\mathbb{R}^{n}$ centered at the origin at order $M$ it
suffices to show that the function $G_{1}$ has decay of order $M$ at scale $1$
away from the origin. Then

$\begin{split}\widehat{G_{1}}(\xi)\coloneqq(s^{d/2}\mathrm{Dil}_{\sigma}^{s}\psi_{s}(\cdot,\sigma))^{\wedge}(\xi)=m_{\sigma}(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi)\Psi(\Pi_{\sigma}\xi)\Phi(\textstyle{\frac{1}{s}}\Pi_{\sigma}\xi+\Pi_{\sigma^{\perp}}\xi).\end{split}$

We record the easy estimate $|\mathrm{supp}\,\widehat{G}_{1}|\lesssim_{n}1$
which is due to the fact that on the support of the function $\widehat{G}_{1}$
we have

(5.23) $|\Pi_{\sigma}\xi|\eqsim
1\quad\text{and}\quad|\Pi_{\sigma}\xi|^{2}/s^{2}+|\Pi_{\sigma^{\perp}}\xi|^{2}\eqsim
1\implies|\Pi_{\sigma^{\perp}}\xi|\eqsim 1.$

It is then apparent that for $M\leq A$ we have

(5.24)
$\left\|\partial_{\xi}^{\gamma}\widehat{G}_{1}(\xi)\right\|_{L^{\infty}_{\xi}}\lesssim
1,\qquad 0\leq|\gamma|\leq M.$

Here, we crucially use (5.23), allowing the exploitation of the
Hörmander–Mihlin condition on $m_{\sigma}$ in the form

$\left|\partial^{\gamma}_{\xi}m_{\sigma}(s^{-1}O_{\sigma}\Pi_{\sigma}\xi)\right|\lesssim
s^{-|\gamma|}\left|\textstyle{\frac{1}{s}}\Pi_{\sigma}\xi\right|^{-|\gamma|}\|m_{\sigma}\|_{\mathcal{M}_{A}(d)}\lesssim
1,\qquad|\gamma|\leq A.$

This completes the proof of adaptation of $F=\vartheta_{t}(\sigma,\cdot)$.

Let now $F=\vartheta_{t}(\cdot,\sigma)-\vartheta_{t}(\cdot,\rho)$. Firstly
note that we can assume that $\rho,\sigma$ are such that
$v_{\sigma},v_{\rho}\in\alpha_{t}$ (which was defined in (5.9)) for
$t=t(e_{n},s,0)$, see (5.19). Indeed, if say $\rho\notin\alpha_{t}$ then
$\vartheta_{t}(\cdot,\rho)=0$ by the directional support condition (iv), which
is already verified. Then ${\rm dist}(\rho,\sigma)\gtrsim\mathrm{scl}(t)^{-1}$
and the desired adaptedness property follows from the case
$F=\vartheta_{t}(\cdot,\sigma)$ proved above.

Now, assume that $\rho,\sigma\in\alpha_{t}$. Using (5.8) and the definition of
$\alpha_{t}$ in (5.9), ${\rm dist}(\rho,\sigma)\leq
3\sqrt{d}3^{-\kappa}\mathrm{scl}(t)^{-1}\leq 3^{-8}s^{-1}$. We can apply the
same argument as above to write

$\begin{split}\partial_{x}^{\alpha}[\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}F]&=\partial_{x}^{\alpha}[\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\varphi_{t}]*\mathrm{Mod}_{-e_{n}}s^{d/2}\mathrm{Dil}_{\sigma}^{s}[\psi_{s}(\cdot,\sigma)-\psi_{s}(\cdot,\rho)]\\\
&\eqqcolon\partial_{x}^{\alpha}[\mathrm{Mod}_{-e_{n}}\mathrm{Dil}_{\sigma}^{s}\varphi_{t}]*\mathrm{Mod}_{-e_{n}}G_{2}.\end{split}$

As before it will be enough to show that the function $G_{2}$ decays with
order $M$ at scale $1$ away from the origin; more precisely we will show that
$G_{2}\lesssim c(\sigma,\rho,s)\chi_{M}$, with $c(\sigma,\rho,s)$ the
multiplicative factor appearing in condition (v) of §5.4.3. We will do so by
estimating derivatives of sufficiently high order of the Fourier transform of
$G_{2}$. To calculate the Fourier transform of $G_{2}$ it is convenient to
first note that the anisotropic rescaling of $\psi_{s}(\rho,\cdot)$ in the
directions of $\sigma$ creates a small error that will be kept under control
by the closeness of $\sigma$ and $\rho$. Setting

$B(\xi)\coloneqq\Pi_{\rho}\Pi_{\sigma}\xi+s\Pi_{\rho}\Pi_{\sigma^{\perp}}\xi-\Pi_{\rho}\xi=(s-1)\Pi_{\rho}\Pi_{\sigma^{\perp}}\xi,\quad|B(\xi)|\lesssim
s{\rm dist}(\sigma,\rho)|\xi|\ll|\xi|,$

an explicit computation tells us that

$\widehat{G_{2}}(\xi)=\left[m_{\sigma}\left(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi\right)\Psi(\Pi_{\sigma}\xi)-m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\rho}(\Pi_{\rho}\xi+B(\xi))\right)\Psi(\Pi_{\rho}\xi+B(\xi))\right]\\\
\times\Phi\left(\textstyle{\frac{1}{s}}\Pi_{\sigma}\xi+\Pi_{\sigma^{\perp}}\xi\right).$

Note that since $\sigma\in\Sigma$ we always have that
$|\xi|\eqsim|\Pi_{\sigma^{\perp}}\xi|\eqsim 1$ for
$s^{-1}\Pi_{\sigma}\xi+\Pi_{\sigma^{\perp}}\xi\in\rm{supp}(\Phi)$ and in
particular we recover that
$|\Pi_{\sigma}\xi|\eqsim|\Pi_{\rho}\xi|\eqsim|\xi|\eqsim 1$ on the support of
$\widehat{G_{2}}$. Three mean value estimates combined with (1.10) and (2.5)
tell us that

(5.25)
$\begin{split}&\left|\partial^{\gamma}_{\xi}\left[\Psi(\Pi_{\rho}\xi+B(\xi))-\Psi(\Pi_{\sigma}\xi)\right]\right|\lesssim|B(\xi)|+|\Pi_{\rho}\xi-\Pi_{\sigma}\xi|\lesssim
s{\rm dist}(\sigma,\rho),\\\
&\left|\partial^{\gamma}_{\xi}\left[m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi\right)-\
m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\rho}\Pi_{\sigma}\xi\right)\right]\right|\lesssim\|O_{\rho}-O_{\sigma}\|\lesssim{\rm
dist}(\sigma,\rho),\\\
&\left|\partial^{\gamma}_{\xi}\left[m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\rho}\Pi_{\sigma}\xi\right)-\
m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\rho}(\Pi_{\rho}\xi+B(\xi))\right)\right]\right|\lesssim|B(\xi)|+|\Pi_{\rho}\xi-\Pi_{\sigma}\xi|\lesssim
s{\rm dist}(\sigma,\rho).\end{split}$

Hence, we may replace $\widehat{G_{2}}$ by

$\begin{split}\widehat{G_{3}}(\xi)&\coloneqq\left[m_{\sigma}\left(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi\right)-m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi\right)\right]\Psi(\Pi_{\rho}\xi+B(\xi))\Phi(s^{-1}\Pi_{\sigma}\xi+\Pi_{\sigma^{\perp}}\xi)\end{split}$

controlling the difference by means of (5.25). Finally, the main assumption
(1.10) on the family $\mathscr{m}$ tells us that

$\partial^{\gamma}_{\xi}\left[m_{\sigma}\left(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi\right)-m_{\rho}\left(\textstyle{\frac{1}{s}}O_{\sigma}\Pi_{\sigma}\xi\right)\right]\lesssim\frac{1}{\log\left({\rm
e}+[{\rm dist}(\sigma,\rho)]^{-1}\right)}$

on the support of $\widehat{G_{3}}$; collecting the above inequalities leads
to the conclusion

(5.26)
$\left\|\partial_{\xi}^{\gamma}\widehat{G}_{2}(\xi)\right\|_{L^{\infty}_{\xi}}\lesssim(1+s){\rm
dist}(\sigma,\rho)+\frac{1}{\log\left({\rm e}+[{\rm
dist}(\sigma,\rho)]^{-1}\right)},\qquad 0\leq|\gamma|\leq A-1,$

which completes the proof of (v). ∎

With the notation introduced in Lemma 5.8, the right hand side of (5.18) may
be rewritten as, and subsequently bounded by, as follows:

(5.27)
$\begin{split}&\sup_{\sigma\in\Sigma}\bigg{|}\sum_{{s\in\mathbb{S}}}\sum_{\beta\in\mathcal{B}_{s^{-1}}}\sum_{z\in
Z(\beta)}\langle
f,\varphi_{t(\beta,s,z)}\rangle{\vartheta}_{t(\beta,s,z)}(\cdot,\sigma)\bigg{|}\leq\sum_{j=1}^{C}\sum_{\tau=1}^{N}\mathsf{T}_{j,\tau}f,\\\
&\mathsf{T}_{j,\tau}f\coloneqq\sup_{\sigma\in\Sigma}\left|\sum_{\begin{subarray}{c}(\beta,s,z)\\\
j(\beta,s)=j\end{subarray}}\langle
f,\varphi_{t(\beta,s,z)}\rangle{\vartheta}_{t(\beta,s,z)}(\cdot,\sigma)\bm{1}_{\alpha_{t(\beta,s,z),\tau}}(v_{\sigma})\right|.\end{split}$

Property (iv) of the $\mathcal{A}_{t}$ class was used above. It thus suffices
to control one of the $\mathsf{T}_{j,\tau}f$ summands for an arbitrary but
fixed value of $(j,\tau)\in\\{1,\ldots,C\\}\times\\{1,\ldots,N\\}$. Choose a
measurable function $\sigma=\sigma(x):\mathbb{R}^{n}\to\Sigma$, linearizing
the corresponding supremum in the definition of $\mathsf{T}_{j,\tau}f$, and
$g\in\mathcal{S}(\mathbb{R}^{n})$ of unit norm in
$L^{p^{\prime},q^{\prime}}(\mathbb{R}^{n})$ so that

$\begin{split}\|\mathsf{T}_{j,\tau}f\|_{L^{p,q}(\mathbb{R}^{n})}&\leq
2\int\sum_{{s\in\mathbb{S}}}\sum_{\beta\in\mathcal{B}_{s^{-1}}}\sum_{z\in
Z(\beta)}\langle
f,\varphi_{t(\beta,s,z)}\rangle{\vartheta}_{t(\beta,s,z)}(x,\sigma(x))\bm{1}_{\alpha_{t(\beta,s,z),\tau}}(v_{\sigma(x)})g(x)\,{\rm
d}x\\\
&\leq\Lambda_{\mathcal{T}_{\mathcal{G}_{j},M};\sigma,\tau}(f,g)\end{split}$

By a limiting argument and restricted weak-type interpolation, Proposition 3.3
is reduced to proving the two estimates

(5.28)
$\displaystyle\Lambda_{\mathbb{P};\sigma,\tau,M}(f,g\bm{1}_{E})\lesssim\|f\|_{2}|E|^{\frac{1}{2}}$
(5.29)
$\displaystyle\Lambda_{\mathbb{P};\sigma,\tau,M}(\bm{1}_{F},g\bm{1}_{G})\lesssim|F|^{\frac{1}{p}}|E|^{1-\frac{1}{p}}$

uniformly over all $f,g\in L^{\infty}_{0}(\mathbb{R}^{n})$ with the
normalization $\|g\|_{\infty}=1$, all sets $F,E\subset\mathbb{R}^{d}$ of
finite measure $(j,\tau)\in\\{1,\ldots,C\\}\times\\{1,\ldots,N\\}$, a fixed
choice of triadic grid $\mathcal{G}=\mathcal{G}_{j}$ and
$\mathbb{P}\subset\mathcal{T}_{\mathcal{G}}$. We also fix $M=50n$ and $\tau$
and write $\Lambda_{\mathbb{P}}(f,g)$ for
$\Lambda_{\mathbb{P};\sigma,\tau,M}(f,g)$. The proof of (5.28) will be
sketched at the end of §6, relying upon the size, density and tree lemmas,
also proved in §6 below.

##  6\. Trees, size and density

We fix a finite collection of tiles
$\mathbb{P}\subset\mathcal{T}_{\mathcal{G}}$, for a fixed choice of triadic
grid $\mathcal{G}$. We remember that tiles are sets of the form $t=R_{t}\times
Q_{t}$ where $Q_{t}\in\mathcal{G}$ is a triadic cube in
$\mathbb{R}^{d}\times\\{0\\}$. Recall the roles of $Q_{t}^{\circ,\kappa}$ and
$\mathsf{ch}_{\kappa,\square}(Q)$; since $\kappa$ is a fixed large dimensional
constant, we will omit it from the notation from now on and write instead
$Q_{t}^{\circ}$ and $\mathsf{ch}_{\square}(Q)$ for the _center_ and the
_peripheral children_ of $Q$, respectively.

We will be sorting these arbitrary collections of tiles into _trees_ , defined
below.

######  Definition 6.1 (trees).

A _tree_ is a collection of tiles $\mathbf{T}\subseteq\mathbb{P}$ such that,
there exists a pair $(\xi_{\mathbf{T}},R_{\mathbf{T}})$ with
$\xi_{\mathbf{T}}\in\mathbb{R}^{d}$ and
$R_{\mathbf{T}}\in\\{R_{t}:\,t\in\mathcal{T}_{\mathcal{G}}\\}$ such that

  * 1.

$\xi_{\mathbf{T}}\in Q_{t}$ for all $t\in\mathbf{T}$;

  * 2.

$\mathrm{scl}(R_{t})\leq\mathrm{scl}(R_{\mathbf{T}})$ and $R_{t}\cap
R_{\mathbf{T}}\neq\varnothing$ for every $t\in\mathbf{T}$.

A _tree_ will be called _lacunary_ if $\xi_{\mathbf{T}}\in Q_{t}\setminus
Q_{t}^{\circ}$, while it will be called _overlapping_ if $\xi_{\mathbf{T}}\in
Q_{t}^{\circ}$.

The following relation of order will be useful: For two tiles
$t,t^{\prime}\in\mathcal{T}_{\mathcal{G}}$ we will define

$t\leq t^{\prime}\overset{\mathrm{def}}{\iff}Q_{t^{\prime}}\subseteq
Q_{t}\quad\text{and}\quad{R_{t}\cap R_{t^{\prime}}\neq\varnothing}$

Note that this is _not_ a partial order as it is not transitive. The following
geometric lemma will be used in several places throughout the paper.

######  Lemma 6.2.

There exists a constant $K_{n}$ depending only upon dimension, such that for
every $t,t^{\prime}\in\mathbb{P}$ with $Q_{t^{\prime}}\subseteq Q_{t}$, the
following hold.

  * -

If $R_{t}\cap R_{t^{\prime}}\neq\varnothing$ then $R_{t}\subseteq
K_{n}R_{t^{\prime}}$.

  * -

If $K_{n}R_{t}\cap K_{n}R_{t^{\prime}}\neq\varnothing$ then
$K_{n}R_{t}\subseteq K_{n}^{2}R_{t^{\prime}}$.

######  Remark 6.3.

The following facts about trees will be used without particular mention

  * -

If $\mathbf{T}$ is a tree then there exists a _top tile_
$\mathrm{top}(\mathbf{T})=R_{\mathbf{T}}\times Q_{\mathbf{T}}$ such that
$\xi_{\mathbf{T}}\in Q_{t}$ and $t\leq\mathrm{top}(\mathbf{T})$ for all
$t\in\mathbf{T}$. In particular we have that for lacunary $\mathbf{T}$ there
holds $t\leq\mathrm{top}(\mathbf{T})$ and $Q_{\mathbf{T}}\subseteq
Q_{t}\setminus Q_{t}^{\circ}$ for all $t\in\mathbf{T}$.

  * -

Because of the point above, we can and will always assume that
$\xi_{\mathbf{T}}$ has no triadic coordinates.

  * -

If $\mathbf{T}$ is a lacunary tree then the collections
$\\{Q:\,Q=Q_{t}^{\circ}\,\,\,\text{for some}\,\,\,t\in\mathbf{T}\\}$ and
$\\{\omega:\,\omega=\omega_{t}\,\,\,\text{for some}\,\,\,t\in\mathbf{T}\\}$
are pairwise disjoint.

For the following definition, $E\subset\mathbb{R}^{n}$ is a measurable set of
finite measure. This set remains fixed throughout the paper. Also for any
collection of tiles $\mathbb{T}\subset\mathbb{P}$ we define the _shadow_ of
the collection

$\mathsf{sh}(\mathbb{T})\coloneqq\bigcup_{t\in\mathbb{T}}R_{t}.$

Recall also the definition of the intrinsic coefficients $F[f](t)$ from
(5.10). The next step consists of defining two coefficient maps respectively
tied to the Carleson measure properties of the coefficients (5.10), (5.11)

######  Definition 6.4 (Size).

Let $\mathbb{P}$ be a finite collection of tiles. For fixed $f\in
L^{\infty}_{0}(\mathbb{R}^{n})$, define the map

$\mathrm{size}:\mathcal{P}(\mathbb{P})\to[0,\infty),\qquad\mathrm{size}(\mathbb{Q})\coloneqq\sup_{\begin{subarray}{c}{\mathbf{T}}\subseteq\mathbb{Q}\\\
\mathbf{T}\text{ lac.
tree}\end{subarray}}\Big{(}\frac{1}{|R_{\mathbf{T}}|}\sum_{t\in\mathbf{T}}F_{10n}[f](t)^{2}\Big{)}^{\frac{1}{2}}.$

######  Definition 6.5 (Density).

Let $\mathbb{P}$ be a finite collection of tiles and $t\in\mathbb{P}$. For a
fixed measurable set $E\subset\mathbb{R}^{n}$ of finite measure, we define

(6.1) $E_{t}\coloneqq\left\\{x\in
E:\,v_{\sigma(x)}\in\alpha_{t}\right\\},\qquad\mathrm{dense}(t)\coloneqq\sup_{\begin{subarray}{c}t^{\prime}\geq
t\\\
t^{\prime}\in\mathbb{P}\end{subarray}}\int_{E_{t}}\mathsf{Sy}_{R_{t}}^{1}\chi_{10n},\qquad$

as well as the map

$\mathrm{dense}:\mathcal{P}(\mathbb{P})\to[0,\infty),\qquad\mathrm{dense}(\mathbb{Q})\coloneqq\sup_{t\in\mathbb{Q}}\mathrm{dense}(t).$

###  6.6. Orthogonality estimates for lacunary trees

We will use different orthogonality estimates for wave packets adapted to
special collections of tiles. Most of them are rather standard in the
literature but we include them here for completeness.

######  Lemma 6.7.

Let $\mathbf{T}\subset\mathbb{P}$ be a lacunary tree. Then the following hold

  * 1.

For adapted families
$\\{\varphi_{t}:\,\varphi_{t}\in\mathcal{F}_{t}^{M},\,t\in\mathbf{T}\\}$,
$\\{\psi_{t}:\,\varphi_{t}\in\mathcal{A}_{t}^{M},\,t\in\mathbf{T}\\}$ with
$M>2n$, we have

$\sup_{|\varepsilon_{t}|=1}\left\|\sum_{t\in\mathbf{T}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\psi_{t}\right\|_{L^{2}(\mathbb{R}^{n})}\lesssim\left(\sum_{t\in\mathbf{T}}F_{M}[f](t)^{2}\right)^{\frac{1}{2}}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}.$

with implicit dimensional constants.

  * 2.

There holds

$\sum_{t\in\mathbf{T}}F_{10n}[f](t)^{2}\lesssim\|f\|_{L^{\infty}(\mathbb{R}^{n})}^{2}|R_{\mathbf{T}}|.$

###### Proof.

Let $S_{\mathbf{T}}f\coloneqq\sum_{t\in\mathbf{T}}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\psi_{t}$ and
$Q(\mathbf{T})\coloneqq\\{Q_{t}^{\circ}:\,t\in\mathbf{T}\\}$. For $Q\in
Q(\mathbf{T})$ we also write
$\mathbf{T}(Q)\coloneqq\\{t\in\mathbf{T}:\,Q_{t}^{\circ}=Q\\}$. Then

$S_{\mathbf{T}}f=\sum_{Q\in
Q(\mathbf{T})}\sum_{t\in\mathbf{T}(Q)}\varepsilon_{t}\langle
f,\varphi_{t}\rangle\psi_{t}\eqqcolon\sum_{Q\in Q(\mathbf{T})}S_{Q}(f)$

and because of Remark 6.3 we have

$\|S_{\mathbf{T}}f\|_{2}^{2}\leq\sum_{Q\in Q(\mathbf{T})}\|S_{Q}f\|_{2}^{2}.$

Thus the proof of the lemma will be complete once we show that for each $Q\in
Q(\mathbf{T})$ we have

$\|S_{Q}(g)\|_{2}^{2}\lesssim\sum_{t\in\mathbf{T}(Q)}F_{M}[g](t)^{2}.$

Expanding the square we get for each $Q\in Q(\mathbf{T})$

$\|S_{Q}(g)\|_{2}^{2}=\sum_{t,t^{\prime}\in\mathbf{T}(Q)}|\langle
g,\varphi_{t}\rangle||\langle
g,\psi_{t^{\prime}}\rangle||\langle\varphi_{t},\psi_{t^{\prime}}\rangle|\lesssim\sum_{t\in\mathbf{T}(Q)}F_{M}[g](t)^{2}\sup_{t\in\mathbf{T}(Q)}\sum_{t\in\mathbf{T}(Q)}|\langle\varphi_{t},\psi_{t^{\prime}}\rangle|.$

However
$\sup_{t^{\prime}\in\mathbf{T}(Q)}\sum_{t\in\mathbf{T}(Q)}|\langle\varphi_{t},\psi_{t^{\prime}}\rangle|\lesssim
1$ by the fast decay of the wave packets
$\\{\varphi_{t},\psi_{t}:\,t\in\mathbf{T}(Q)\\}$ and the fact that the spatial
components $\\{R_{t}:\,t\in\mathbf{T}(Q)\\}$ tile $\mathbb{R}^{n}$. This
proves the first estimate of 1.

To see the second estimate let
$\\{\varphi_{t}:\,\varphi_{t}\in\mathcal{F}_{t}^{M},\,t\in\mathbf{T}\\}$ and
write

$\sum_{t\in\mathbf{T}}\langle f,\varphi_{t}\rangle^{2}=\left\langle
f,\sum_{\in\mathbf{T}}\langle
f,\varphi_{t}\rangle\varphi_{t}\right\rangle\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}\left(\sum_{t\in\mathbf{T}}F_{M}[f](t)^{2}\right)^{\frac{1}{2}}$

where we used 1. in the last approximate inequality. This readily yields the
desired estimate.

Finally for 2. let now
$\\{\varphi_{t}:\,\varphi_{t}\in\mathcal{F}_{t}^{10n},\,t\in\mathbf{T}\\}$ and
note that $R_{t}\subseteq K_{n}R_{\mathbf{T}}$ by Lemma 6.2 and let
$t_{\mathbf{T}}\coloneqq\mathsf{Sy}_{K_{n}R_{\mathbf{T}}}^{\infty}\chi_{4n}$.
Now for $c>0$ we use 2. to estimate

$\sum_{t\in\mathbf{T}}\langle
f,\varphi_{t}\rangle^{2}=\sum_{t\in\mathbf{T}}\langle
c^{-1}t_{\mathbf{T}}f,ct_{\mathbf{T}}^{-1}\varphi_{t}\rangle^{2}\lesssim\|ft_{\mathbf{T}}\|_{L^{2}(\mathbb{R}^{n})}^{2}$

by noticing that for a suitable choice of $c$ the wave packets
$ct_{\mathbf{T}}^{-1}\varphi_{t}\in\mathcal{F}_{t}^{6n}$ for each
$t\in\mathbf{T}$. The estimate in the last display readily yields 3. ∎

###  6.8. Strongly disjoint families of trees and almost orthogonality

In what follows we will be sorting our collection of tiles $\mathbb{P}$ into
trees selected by means of a greedy algorithm. The selection process will
imply certain disjointness properties which we encode in the definition below.

######  Definition 6.9 (strongly disjoint lacunary families).

Let $\mathfrak{T}$ be a family of trees with $\mathbf{T}\subset\mathbb{P}$ for
every $\mathbf{T}\in\mathfrak{T}$. The family $\mathfrak{T}$ is called
(lacunary) strongly disjoint if

  * 1.

Every tree $\mathbf{T}\in\mathfrak{T}$ is lacunary.

  * 2.

If $t\in\mathbf{T}$ and $t^{\prime}\in\mathbf{T}^{\prime}\in\mathfrak{T}$ with
$\mathbf{T}\neq\mathbf{T}^{\prime}$ and $Q_{t}\subseteq
Q_{t^{\prime}}^{\circ}$ then $R_{t^{\prime}}\cap
K_{n}^{2}R_{\mathbf{T}}=\varnothing$

with $K_{n}$ the constant of $Lemma~{}\ref{lem:ortho}$.

The point of the definition above is that if conclusion 2. failed then the
tile $t^{\prime}$ would essentially qualify to be included in a completion of
the tree $\mathbf{T}$, if that tree was chosen first via a greedy selection
algorithm. This point will become apparent in the proof of the size lemma,
Lemma 6.17 below. Furthemore, the definition above implies the following
property.

######  Lemma 6.10.

Let $\mathbb{T}\coloneqq\cup_{\mathbf{T}\in\mathfrak{T}}\mathbf{T}$ where
$\mathfrak{T}$ is a lacunary strongly disjoint family. Then the collection

$\mathbb{T}^{\circ}\coloneqq\\{R_{t}\times Q_{t}^{\circ}:\,t\in\mathbb{T}\\}$

is pairwise disjoint.

###### Proof.

Let $t,t^{\prime}\in\mathbb{T}$. If $Q_{t}^{\circ}=Q_{t^{\prime}}^{\circ}$
then either $t\equiv t^{\prime}$ or $R_{t^{\prime}}\cap R_{t}=\varnothing$.
Assume then that $Q_{t}^{\circ}\subsetneq Q_{t^{\prime}}^{\circ}$ so that
$Q_{t}\subseteq Q_{t^{\prime}}^{\circ}$. Since the trees in the collection are
lacunary this implies that $t,t^{\prime}$ must come from different trees, say
$\mathbf{T}\neq\mathbf{T}^{\prime}$, respectively. If $R_{t}\cap
R_{t^{\prime}}\neq\varnothing$ then by consecutive applications of Lemma 6.2
we would have that $R_{t^{\prime}}\subseteq K_{n}^{2}R_{\mathbf{T}}$,
contradicting the definition of a strongly disjoint family. ∎

It is well known that strongly disjoint families of trees obey certain almost
orthogonality estimates for the corresponding tree projections. The precise
statement in the context of this paper is contained in Lemma 6.11 below.

######  Lemma 6.11.

Let $M>4n$. For a (lacunary) strongly disjoint family $\mathfrak{T}$ with
$\mathbb{T}\coloneqq\cup_{\mathbf{T}\in\mathfrak{T}}\mathbf{T}$, there holds

$\left(\sum_{\mathbf{T}\in\mathfrak{T}}\sum_{t\in\mathbf{T}}F_{M}[f](t)^{2}\right)^{\frac{1}{2}}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}+\left(\sup_{t\in\mathbb{T}}\frac{F_{M}[f](t)}{|R_{t}|^{\frac{1}{2}}}\left[\sum_{\mathbf{T}\in\mathfrak{T}}|R_{\mathbf{T}}|\right]^{\frac{1}{2}}\right)^{\frac{1}{3}}\|f\|_{L^{2}(\mathbb{R}^{n})}^{\frac{2}{3}}.$

###### Proof.

Let $\\{\varphi_{t}:\,\varphi_{t}\in\mathcal{F}_{t}^{M}\,t\in\mathbb{T}\\}$ be
any adapted family and note that

$S(f)^{2}\coloneqq\sum_{t\in\mathbb{T}}\langle
f,\varphi_{t}\rangle^{2}\leq\|f\|_{2}\left\|\sum_{t\in\mathbb{T}}\langle
f,\varphi_{t}\rangle\varphi_{t}\right\|_{2}.$

Letting $Q(\mathbb{T})\coloneqq\\{Q_{t}:\,t\in\mathbb{T}\\}$ and
$\mathbb{T}(Q)\coloneqq\\{t:\,Q_{t}=Q\\}$ for $Q\in Q(\mathbb{T})$, we have

$\begin{split}\left\|\sum_{t\in\mathbb{T}}\langle
f,\varphi_{t}\rangle\varphi_{t}\right\|_{2}^{2}&=\sum_{Q\in
Q(\mathbb{T})}\sum_{\begin{subarray}{c}t,t^{\prime}\in\mathbb{T}\\\
Q_{t}=Q_{t^{\prime}}=Q\end{subarray}}\langle f,\varphi_{t}\rangle\langle
f,\varphi_{t^{\prime}}\rangle\langle\varphi_{t},\varphi_{t^{\prime}}\rangle\\\
&\qquad+2\sum_{t\in\mathbb{T}}\sum_{\begin{subarray}{c}t^{\prime}\in\mathbb{T}\\\
Q_{t^{\prime}}\supsetneq Q_{t}\end{subarray}}\langle
f,\varphi_{t}\rangle\langle
f,\varphi_{t^{\prime}}\rangle\langle\varphi_{t},\varphi_{t^{\prime}}\rangle\\\
&\coloneqq S_{1}(f)^{2}+2S_{2}(f)^{2}.\end{split}$

As in the proof of Lemma 6.7 it is relatively easy to see that
$|S_{1}(f)|\lesssim S(f)$ since for a fixed $Q\in Q(\mathbb{T})$, the spatial
components $\\{R_{t}:\,Q_{t}=Q\\}$ tile $\mathbb{R}^{n}$. We thus focus on
$S_{2}(f)$ which can be estimated in the form

$\begin{split}S_{2}(f)^{2}&\leq\sup_{t^{\prime}\in\mathbb{T}}\frac{|\langle
f,\phi_{t^{\prime}}\rangle|}{|R_{t^{\prime}}|^{\frac{1}{2}}}\sum_{t\in\mathbb{T}}|\langle
f,\varphi_{t}\rangle|\sum_{\begin{subarray}{c}t^{\prime}\in\mathbb{T}\\\
Q_{t^{\prime}}^{\circ}\supseteq
Q_{t}\end{subarray}}|R_{t^{\prime}}|^{\frac{1}{2}}|\langle\varphi_{t},\varphi_{t^{\prime}}\rangle|.\end{split}$

Note that for fixed $t\in\mathbb{T}$ the collection
$\mathbb{T}(t)\coloneqq\\{R_{t^{\prime}}:\,t^{\prime}\in\mathbb{T},\,Q_{t}\subseteq
Q_{t^{\prime}}^{\circ}\\}$ is pairwise disjoint and
$\mathsf{sh}(\mathbb{T}(t))\subseteq(K_{n}^{2}R_{\mathbf{T}})^{\mathsf{c}}$;
this follows immediate from Lemma 6.10 since $\mathfrak{T}$ is a (lacunary)
strongly disjoint family, and the definition of a strongly disjoint family
itself. At this point we use the following estimate: For
$t,t^{\prime}\in\mathbb{P}$ such that $R_{t}\cap R_{t^{\prime}}=\varnothing$
and $Q_{t}\subseteq Q_{t^{\prime}}$ there holds

$|\langle\varphi_{t},\varphi_{t^{\prime}}\rangle|\lesssim\left(\frac{|R_{t^{\prime}}|}{|R_{t}|}\right)^{\frac{1}{2}}\inf_{x\in
R_{t^{\prime}}}(1+\rho_{R_{t}}(x))^{-M+n}$

where $\rho_{R}(x)\coloneqq\inf\\{r>0:\,x\in rR\\}$ for any rectangular
parallelepiped $R$ and $M>2n$, say. Combining these observations and estimates
with the estimate for $S_{2}(f)^{2}$ above yields

$\begin{split}S_{2}(f)^{2}&\lesssim\sup_{t^{\prime}\in\mathbb{T}}\frac{|\langle
f,\phi_{t^{\prime}}\rangle|}{|R_{t^{\prime}}|^{\frac{1}{2}}}\sum_{\mathbf{T}\in\mathfrak{T}}\sum_{t\in\mathbf{T}}|\langle
f,\varphi_{t}\rangle||R_{t}|^{-\frac{1}{2}}\sum_{\begin{subarray}{c}t^{\prime}\in\mathbb{T}\\\
Q_{t^{\prime}}^{\circ}\supseteq
Q_{t}\end{subarray}}|R_{t^{\prime}}|\inf_{R_{t^{\prime}}}(1+\rho_{R_{t}})^{-M+n}\\\
&\lesssim\sup_{t^{\prime}\in\mathbb{T}}\frac{|\langle
f,\phi_{t^{\prime}}\rangle|}{|R_{t^{\prime}}|^{\frac{1}{2}}}\sum_{\mathbf{T}\in\mathfrak{T}}\sum_{t\in\mathbf{T}}|\langle
f,\varphi_{t}\rangle||R_{t}|^{-\frac{1}{2}}\int_{(K_{n}^{2}R_{\mathbf{T}(t)})^{\mathsf{c}}}(1+\rho_{R_{t}}(x))^{-M+n}\,{\rm
d}x\\\ &\lesssim\sup_{t^{\prime}\in\mathbb{T}}\frac{|\langle