# An endpoint estimate for product singular integral operators on stratified
Lie groups

Michael G. Cowling  School of Mathematics and Statistics, University of New
South Wales, Sydney 2052, Australia<EMAIL_ADDRESS>,  Ming-Yi Lee
Department of Mathematics, National Central University, Chung-Li 320, Taiwan,
Republic of China<EMAIL_ADDRESS>,  Ji Li  Ji Li, Department of
Mathematics, Macquarie University, NSW, 2109, Australia<EMAIL_ADDRESS>and
Jill Pipher  Department of Mathematics, Brown University, Providence RI 02912,
USA<EMAIL_ADDRESS>
###### Abstract.

We establish hyperweak boundedness of area functions, square functions,
maximal operators and Calderón–Zygmund operators on products of two stratified
Lie groups.

###### Key words and phrases:

Strong maximal function, $\mathsf{L\,log\,L}$, atomic decomposition

###### 2010 Mathematics Subject Classification:

43A17, 42B20, 43A80

Cowling and Li are supported by ARC DP 220100285. Lee is supported by NSTC
112-2115-M-008-001-MY2.

##  1\. Introduction and statement of main results

Since the 1980s, the development of multiparameter harmonic analysis has
proceeded apace; recent contributions in the area include [1, 8, 9, 10, 16,
17, 19, 21, 22, 23]. Much of the product space theory on
$\mathbb{R}^{m}\times\mathbb{R}^{n}$, including the duality of
$\mathsf{H}^{1}$ with $\mathsf{BMO}$, characterisation of $\mathsf{H}^{1}$ by
square functions and atomic decompositions, and interpolation, has been
extended to more general products of spaces of homogeneous type.

In contrast to the classical theory of singular integrals, in multiparameter
harmonic analysis product singular integrals are not of weak type $(1,1)$. For
functions supported on the unit cube, the classical weak type $(1,1)$ estimate
was replaced by a $\mathsf{L\,log\,L}$ to $\mathsf{L}^{1,\infty}$ estimate by
R. Fefferman [8]. More precisely, let $\mathcal{T}$ be a product
Calderón–Zygmund operator on
$\mathsf{L}^{2}(\mathbb{R}^{m}\times\mathbb{R}^{n})$ as in [17] or [8]; then
for every $\lambda\in\mathbb{R}_{+}$,

(1.1)
$\displaystyle|\\{(x_{1},x_{2})\in\mathbb{R}^{m}\times\mathbb{R}^{n}\colon|\mathcal{T}f(x_{1},x_{2})|>\lambda\\}|\leq\frac{C}{\lambda}\|f\|_{\mathsf{L\,log\,L}(Q_{1}\times
Q_{2})}$

for all $f$ supported in the product $Q_{1}\times Q_{2}$ of the unit cubes in
$\mathbb{R}^{m}$ and $\mathbb{R}^{n}$, where

$\displaystyle\|f\|_{\mathsf{L\,log\,L}(Q_{1}\times Q_{2})}=\iint_{Q_{1}\times
Q_{2}}|f(x_{1},x_{2})|(1+\log^{+}(|f(x_{1},x_{2})|))\,\mathrm{d}x_{2}\,\mathrm{d}x_{1};$

here $\log^{+}t:=\log(\max\\{1,t\\})$. The proof in [8] relies on the
boundedness of the strong maximal function and the area function from
$\mathsf{L\,log\,L}$ to $\mathsf{L}^{1,\infty}$, the local atomic
decomposition of functions in $\mathsf{L\,log\,L}$ produced using the
$\mathsf{L\,log\,L}$ to $\mathsf{L}^{1,\infty}$ boundedness of the area
function, and the boundedness of $\mathcal{T}$ on $\mathsf{L\,log\,L}$ atoms.

A natural question arises: is (1.1) true globally for area functions, square
functions, maximal operators and singular integral operators, even in more
general product settings?

In this paper, we answer the above question positively on product spaces
$\mathbf{G}:=G_{1}\times G_{2}$, where $G_{1}$ and $G_{2}$ are stratified Lie
groups. To state our results, we need a little notation. Details may be found
later.

In this introduction, the auxiliary functions $\varphi^{[i]}$ on $G_{i}$
satisfy standard decay and smoothness conditions and have integral $1$.
Likewise, the functions $\psi^{[i]}$ on $G_{i}$ satisfy standard decay and
smoothness and have integral $0$. We write $\zeta^{[1]}_{t_{1}}$ and
$\zeta^{[2]}_{t_{2}}$ for normalised dilates of functions $\zeta^{[1]}$ on
$G_{1}$ and $\zeta^{[2]}$ on $G_{2}$, and $\varphi_{\mathbf{t}}$ and
$\psi_{\mathbf{t}}$ for the product functions
$\varphi^{[1]}_{t_{1}}\otimes\varphi^{[2]}_{t_{2}}$ and
$\psi^{[1]}_{t_{1}}\otimes\psi^{[2]}_{t_{2}}$.

We define $\mathbf{T}:=\mathbb{R}_{+}\times\mathbb{R}_{+}$. For
$\mathbf{g}:=(g_{1},g_{2})\in\mathbf{G}$, $\mathbf{t}\in\mathbf{T}$ and
$\eta\in[0,\infty)$, we denote by $P(\mathbf{g},\mathbf{t})$ the product of
open balls $B_{1}(g_{1},t_{1})\times B_{2}(g_{2},t_{2})$ and by
$\Gamma^{\eta}(\mathbf{g})$ the product cone
$\Gamma_{1}^{\eta}(g_{1})\times\Gamma_{2}^{\eta}(g_{2})$, where

$\Gamma_{i}^{\eta}(g_{i}):=\\{(h_{i},t_{i})\in
G_{i}\times\mathbb{R}_{+}\colon\rho_{i}(g_{i},h_{i})\leq\eta t_{i}\\}.$

For $\eta\in[0,\infty)$ and $f\in\mathsf{L}^{1}(\mathbf{G})$, we define the
maximal function:

$\mathcal{M}_{\varphi,\eta}(f)(\mathbf{g}):=\sup\Bigl{\\{}\bigl{|}f\ast\varphi_{\mathbf{t}}(\mathbf{h})\bigr{|}\colon(\mathbf{h},\mathbf{t})\in\Gamma^{\eta}(\mathbf{g})\Bigr{\\}}\qquad\forall\mathbf{g}\in\mathbf{G}.$

This maximal function is called radial when $\eta=0$ and nontangential when
$\eta>0$.

For $\eta\in\mathbb{R}_{+}$ and $f\in\mathsf{L}^{1}(\mathbf{G})$, we define
the Lusin area function $\mathcal{S}_{\psi,\eta}(f)$ by

$\mathcal{S}_{\psi,\eta}(f)(\mathbf{g}):=\left(\iint_{\Gamma^{\eta}(\mathbf{g})}\frac{\left|(f\ast\psi_{\mathbf{t}}(\mathbf{h})\right|^{2}}{\left|P(\mathbf{g},\eta\mathbf{t})\right|}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\right)^{1/2}\qquad\forall\mathbf{g}\in\mathbf{G}.$

When $\eta=0$ and $f\in\mathsf{L}^{1}(\mathbf{G})$, we define the
Littlewood–Paley square function $\mathcal{S}_{\psi,0}(f)$ by

$\mathcal{S}_{\psi,0}(f)(\mathbf{g}):=\left(\int_{\mathbf{T}}\left|(f\ast\psi_{\mathbf{t}}(\mathbf{g})\right|^{2}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\right)^{1/2}\qquad\forall\mathbf{g}\in\mathbf{G}.$

These are often given different symbols, but it is convenient to treat them
together.

On a stratified Lie group $G$, we may take a basis
$\\{\mathcal{X}_{1},\dots,\mathcal{X}_{d}\\}$ for the space of left-invariant
horizontal vector fields, and define the sublaplacian $\mathcal{L}$ to be
$-\sum_{j-1}^{d}\mathcal{X}_{j}^{2}$. The Riesz transformations are then the
operators $\mathcal{X}_{j}\mathcal{L}^{-1/2}$. The double Riesz
transformations $\mathcal{R}^{[1]}_{j_{1}}\otimes\mathcal{R}^{[2]}_{j_{2}}$ on
$\mathbf{G}$ are defined in the obvious way when $1\leq j_{i}\leq d_{i}$.

######  Theorem 1.1.

Let $\mathcal{T}$ be a maximal operator $\mathcal{M}_{\varphi,\eta}$ or a
Littlewood–Paley operator $\mathcal{S}_{\psi,\eta}$, where $\eta\geq 0$, or a
double Riesz transformation
$\mathcal{R}^{[1]}_{j_{1}}\otimes\mathcal{R}^{[2]}_{j_{2}}$. Then

(1.2)
$\displaystyle\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon|\mathcal{T}f(\mathbf{g})|>\lambda\big{\\}}\big{|}\lesssim
F_{\Phi}({f/\lambda})\qquad\forall\lambda\in\mathbb{R}_{+},$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$, where
$\mathsf{L\,log\,L}(\mathbf{G})$ is the Orlicz space associated to the
functional $F_{\Phi}$, given by

$\displaystyle
F_{\Phi}(f):=\iint_{\mathbf{G}}|f(\mathbf{g})|\log(\mathrm{e}+|f(\mathbf{g})|)\,\mathrm{d}\mathbf{g}.$

It is not possible to replace $F_{\Phi}(f/\lambda)$ by
$\left\|f\right\|_{\mathsf{L\,log\,L}(\mathbf{G})}/\lambda$, where
$\left\|\cdot\right\|_{\mathsf{L\,log\,L}(\mathbf{G})}$ is the Luxemburg norm
associated to $F_{\Phi}$. We explain Orlicz spaces and Luxemburg norms later.
We say that the operator $\mathcal{T}$ is hyperweakly bounded when an estimate
of the form (1.2) holds.

It is easy to iterate estimates for one-parameter maximal, area or singular
integral operators to prove a local version of this result, where the support
of the function $f$ is restricted to lie in a compact set. However, iteration
does not seem to be able to deal with the global case, and this is the main
difficulty that we need to confront in this paper.

Our presentation has the following structure. In Section 2 we review some
background on stratified Lie groups, product spaces and Orlicz spaces.

In Section 3 we show that the strong maximal operator $\mathcal{M}_{s}$ is
hyperweakly bounded, using a covering lemma, Theorem 3.2, that goes back to
[5]. We then prove Theorem 1.1 for the maximal operator
$\mathcal{M}_{\varphi,\eta}$, by using group properties to dominate the
maximal operator $\mathcal{M}_{\varphi,\eta}$ by the strong maximal operator.

In Section 4, we construct the atomic decomposition. We use the gradient
$\not{\nabla}p$ of the Poisson kernel $p$ as our auxiliary function, and then
have a key result of [6, (12) and following], namely, the good-$\lambda$
inequality:

(1.3)
$\displaystyle\bigl{|}\bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\not{\nabla}p,1}(f)(\mathbf{g})>\lambda\bigr{\\}}\bigr{|}$
$\displaystyle\lesssim\bigl{|}L_{\eta}^{c}\bigr{|}+\frac{1}{\lambda^{2}}\int_{L_{1}(\lambda)}\mathcal{M}_{p,\eta}^{2}(f)(\mathbf{g})\,\mathrm{d}\mathbf{g},$

when $\eta$ is sufficiently large; here
$L_{\eta}(\lambda):=\left\\{\mathbf{g}\in{\mathbf{G}}\colon\mathcal{M}_{p,\eta}(f)(\mathbf{g})\leq\lambda\right\\}$.
This implies that the area operator $\mathcal{S}_{\not{\nabla}p,1}$ is
hyperweakly bounded. By using this boundedness and the Calderón reproducing
formula, we can decompose global $\mathsf{L\,log\,L}$ functions into atoms.

In Section 5, we apply our atomic decomposition and a version of Journé’s
covering lemma for spaces of homogeneous type established in [12]. We prove
Theorem 1.1 for _general_ operators $\mathcal{S}_{\psi,\eta}$ and the double
Riesz transformations
$\mathcal{R}^{[1]}_{j_{1}}\otimes\mathcal{R}^{[2]}_{j_{2}}$. The same argument
holds for general product Calderón–Zygmund operators, as in Journé [17].

Most of the arguments rely only the theory of spaces of homogeneous type.
However, we need the setting of a stratified Lie group in two places. First,
it gives us the good $\lambda$ inequality (1.3). Second, it gives us the
Calderón reproducing formula for $\mathsf{L}^{2}(G)$ functions, which is
needed for the atomic decomposition.

“Constants” are positive real numbers, depending only on the geometry of
$\mathbf{G}$ unless otherwise indicated; we write $A\lesssim B$ when there
exists a constant $C$ such that $A\leq CB$. We write $\chi_{E}$ for the
indicator function of a set $E$, and $o$ denotes the identity of a group.

##  2\. Preliminaries

###  2.1. Stratified nilpotent Lie groups

Let $G$ be a (real and finite dimensional) stratified nilpotent Lie group of
step $s$ with Lie algebra $\mathfrak{g}$. This means that we may write
$\mathfrak{g}$ as a vector space direct sum
$\bigoplus_{j=1}^{s}\mathfrak{v}_{j}$, where
$[\mathfrak{v}_{1},\mathfrak{v}_{j}]=\mathfrak{v}_{j+1}$ when $1\leq j\leq s$;
here $\mathfrak{v}_{s+1}:=\\{0\\}$. Let $\nu$ denote the homogeneous dimension
of $G$; that is, $\sum_{j=1}^{s}j\dim\mathfrak{v}_{j}$. There is a one-
parameter family of automorphic dilations $\delta_{t}$ on $\mathfrak{g}$,
given by

$\displaystyle\delta_{t}(\mathcal{X}_{1}+\mathcal{X}_{2}+\dots+\mathcal{X}_{s}):=t\mathcal{X}_{1}+t^{2}\mathcal{X}_{2}+\dots+t^{s}\mathcal{X}_{s};$

here each $\mathcal{X}_{j}\in\mathfrak{v}_{j}$ and $t\in\mathbb{R}_{+}$. The
exponential mapping $\exp\colon\mathfrak{g}\to G$ is a diffeomorphism, and we
identify $\mathfrak{g}$ and $G$. The dilations extend to automorphic dilations
of $G$, also denoted by $\delta_{t}$, by conjugation with $\exp$. The Haar
measure on $G$, which is bi-invariant, is the Lebesgue measure on
$\mathfrak{g}$ lifted to $G$ using $\exp$.

By [13], the group $G$ may be equipped with a smooth subadditive homogeneous
norm $\rho$, a continuous function from $G$ to $[0,\infty)$ that is smooth on
$G\setminus\\{o\\}$ and satisfies

  1. (1)

$\rho(g^{-1})=\rho(g)$;

  2. (2)

$\rho(g^{-1}h)\leq\rho(g)+\rho(h)$;

  3. (3)

$\rho({\delta_{t}(g)})=t\rho(g)$ for all $g\in G$ and all
$t\in\mathbb{R}_{+}$;

  4. (4)

$\rho(g)=0$ if and only if $g=o$.

Abusing notation, we set $\rho(g,g^{\prime}):=\rho(g^{-1}g^{\prime})$ for all
$g,g^{\prime}\in G$; this defines a metric on $G$. We write $B(g,r)$ for the
open ball with centre $g$ and radius $r$ with respect to $\rho$:

$B(g,r)=gB(o,r)=g\\{h\in G\colon\rho(h)<1\\}.$

The metric space $(G,\rho)$ is _geometrically doubling_ ; that is, there
exists $A\in\mathbb{N}$ such that every metric ball $B(x,2r)$ may be covered
by at most $A$ balls of radius $r$.

We remind the reader that a stratified Lie group is a space of homogenous type
in the sense of Coifman and Weiss [3, 4], and analysis on stratified Lie
groups uses much from the theory of such spaces. In particular, we frequently
deal with _molecules_ , that is, functions $\zeta$ that satisfy _standard
decay and smoothness conditions_ , by which we mean that there is a parameter
$\varepsilon\in(0,1]$, which we fix once and for all, such that

(2.1)
$\begin{gathered}\left|\zeta(g)\right|\lesssim\frac{1}{(1+\rho(g))^{\nu+\varepsilon}},\\\
\left|\zeta(g)-\zeta(g^{\prime})\right|\lesssim\frac{\rho(g^{\prime}g^{-1})^{\varepsilon}}{(1+\rho(g)+\rho(g^{\prime}))^{\nu+2\varepsilon}}\end{gathered}$

for all $g,g^{\prime}\in G$. We often impose an additional _cancellation
condition_ , namely

(2.2) $\displaystyle\int_{G}\zeta(g)\,\mathrm{d}g=0.$

The _normalised dilate_ $f_{t}$ of a function $f$ on $G$ by
$t\in\mathbb{R}_{+}$ is given by $f_{t}:=t^{-\nu}f\circ\delta_{1/t}$, and the
_convolution_ $f\ast f^{\prime}$ of suitable functions $f$ and $f^{\prime}$ on
$G$ is defined by

$\displaystyle f\ast
f^{\prime}(g):=\int_{G}f(h)f^{\prime}(h^{-1}g)\,\mathrm{d}h=\int_{G}f(gh^{-1})f^{\prime}(h)\,\mathrm{d}h.$

Take left-invariant vector fields $\mathcal{X}_{1}$, …, $\mathcal{X}_{n}$ on
$G$ that form a basis of $\mathfrak{v}_{1}$, and define the _sub-Laplacian_
$\mathcal{L}:=-\sum_{j=1}^{n}(\mathcal{X}_{j})^{2}$. Observe that each
$\mathcal{X}_{j}$ is homogeneous of degree $1$ and $\mathcal{L}$ is
homogeneous of degree $2$, in the sense that

$\displaystyle\mathcal{X}_{j}\left(f\circ\delta_{t}\right)=t\left(\mathcal{X}_{j}f\right)\circ\delta_{t},$
$\displaystyle\mathcal{L}\left(f\circ\delta_{t}\right)=t^{2}\left(\mathcal{L}f\right)\circ\delta_{t}$

for all $t\in\mathbb{R}_{+}$ and all $f\in\mathsf{C}^{2}(G)$.

Associated to the sub-Laplacian, Folland [9] defined the Riesz potential
operators $\mathcal{L}^{-\alpha}$, where $\alpha\in\mathbb{R}_{+}$; these are
convolution operators with homogeneous kernels. The _Riesz transformation_
$\mathcal{R}_{j}:=\mathcal{X}_{j}\mathcal{L}^{-1/2}$ is a singular integral
operator, and is bounded on $\mathsf{L}^{p}(G)$ when $1<p<\infty$ as well as
from the Folland–Stein Hardy space $\mathsf{H}^{1}(G)$ to $\mathsf{L}^{1}(G)$.

The Hardy–Littlewood maximal operator $\mathcal{M}$ on $G$ is defined using
the metric balls:

$\mathcal{M}f(g):=\sup\left\\{\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(g^{\prime},r)}\left|f(g^{\prime\prime})\right|\,\mathrm{d}g^{\prime\prime}\colon
g\in B(g^{\prime},r)\right\\}.$

where the “average integral”
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int$
is defined by

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}f(\mathbf{g})\,\mathrm{d}\mathbf{g}:=\frac{1}{\left|B\right|}\int_{B}f(\mathbf{g})\,\mathrm{d}\mathbf{g}.$

For future use, we note that the layer cake formula implies that, if $\mu$ is
a radial decreasing function on $G$ (that is, $\mu(g)$ depends only on
$\rho(g)$ and decreases as $\rho(g)$ increases), then

(2.3)
$\left|f\right|\ast\mu_{\varepsilon}(g)\leq\left\|\mu\right\|_{\mathsf{L}^{1}(G)}\mathcal{M}f(g)\qquad\forall
g\in G.$

###  2.2. Functional calculus for the sub-Laplacian

The sub-Laplacian $\mathcal{L}$ has a spectral resolution:

$\mathcal{L}(f)=\int_{\mathbb{R}_{+}}\lambda\,\mathrm{d}\mathcal{E}_{\mathcal{L}}(\lambda)f\qquad\forall
f\in\mathsf{L}^{2}(G),$

where $\mathcal{E}_{\mathcal{L}}(\lambda)$ is a projection-valued measure on
$[0,\infty)$, the spectrum of $\mathcal{L}$. For a bounded Borel function
$m\colon[0,\infty)\to\mathbb{C}$, we define the operator $F(\mathcal{L})$
spectrally:

$m(\mathcal{L})f:=\int_{\mathbb{R}_{+}}m(\lambda)\,\mathrm{d}\mathcal{E}_{\mathcal{L}}(\lambda)f\qquad\forall
f\in\mathsf{L}^{2}(G).$

This operator is a convolution with a Schwartz distribution on $G$.

###  2.3. The heat and Poisson kernels

Let $h_{t}$ and $p_{t}$, where $t\in\mathbb{R}_{+}$, be the heat and Poisson
kernels associated to the sub-Laplacian operator $\mathcal{L}$, that is, the
convolution kernels of the operators $e^{t\mathcal{L}}$ and
$e^{t\sqrt{\mathcal{L}}}$ on $G$. We write ${{q}}_{t}$ for
$t\partial_{t}p_{t}$. We warn the reader that $p_{t}$ and ${{q}}_{t}$ are the
normalised dilates of $p_{1}$ and ${{q}}_{1}$ by the factor $t$, but $h_{t}$
is the normalised dilate of $h_{1}$ by a factor of $t^{1/2}$. Let $\nabla$
denote the subgradient on $G$ and $\not{\nabla}$ denote the gradient
$(\nabla,\partial_{t})$ on $G\times\mathbb{R}_{+}$.

######  Lemma 2.1.

The kernels $h_{t}$ and $p_{t}$ are $\mathbb{R}_{+}$-valued. Further, $h_{t}$
and $p_{t}$ have integral $1$, while ${{q}}_{t}$ has integral $0$ for all
$t\in\mathbb{R}_{+}$. Finally, there exists a constant $c$ such that

$\displaystyle h_{t}(g)$ $\displaystyle\lesssim
t^{-\nu/2}\exp\left(-{\rho^{2}(g)}/{ct}\right),$
$\displaystyle\left|\not{\nabla}h_{t}(g)\right|$ $\displaystyle\lesssim
t^{-(\nu+1)/2}\exp\left(-{\rho^{2}(g)}/{ct}\right),$ $\displaystyle p_{t}(g)$
$\displaystyle\eqsim\frac{t}{(t^{2}+\rho(g)^{2})^{(\nu+1)/2}},$
$\displaystyle\left|\not{\nabla}p_{t}(g)\right|$
$\displaystyle\lesssim\frac{t}{(t^{2}+\rho(g)^{2})^{(\nu+2)/2}}$

for all $g\in G$ and $t\in\mathbb{R}_{+}$.

###### Proof.

For the heat kernel estimates, see [25, Theorem IV.4.2]. Note that the first
estimate has a version with the opposite inequality, with a different constant
$c$.

The estimates for $p_{t}$ and ${{q}}_{t}$ follow from the subordination
formula

$e^{-t\sqrt{\mathcal{L}}}=\frac{1}{2\sqrt{\pi}}\int_{\mathbb{R}_{+}}\frac{te^{-{t^{2}}/{4v}}}{\sqrt{v}}e^{-v\mathcal{L}}\frac{\,\mathrm{d}v}{v}.$

We leave the details to the reader. ∎

This lemma shows that the heat kernel $h_{1}$ and the Poisson kernel $p_{1}$
and their derivatives satisfy the standard decay and smoothness conditions
(2.1); their derivatives also satisfy the cancellation condition (2.2).

###  2.4. Systems of pseudodyadic cubes

We use the Hytönen–Kairema [15] families of “dyadic cubes” in geometrically
doubling metric spaces. We state a version of [15, Theorem 2.2] that is
simpler, in that we work on metric spaces rather than pseudometric spaces. The
Hytönen–Kairema construction builds on seminal work of Christ [7] and of
Sawyer and Wheeden [24].

######  Theorem 2.2 ([15]).

Let $c$, $C$ and $\kappa$ be constants such that $0<c\leq C<\infty$ and
$12C\kappa\leq c$, and let $(G,\rho)$ be a metric stratified group. Then, for
all $k\in\mathbb{Z}$, there exist families $\mathscr{Q}_{k}(G)$ of
_pseudodyadic cubes_ $Q$ with _centres_ $z(Q)$, such that:

  1. (1)

$G$ is the disjoint union of all $Q\in\mathscr{Q}_{k}(G)$, for each
$k\in\mathbb{Z}$;

  2. (2)

$B(z(Q),c\kappa^{k}/3)\subseteq Q\subseteq B(z(Q),2C\kappa^{k})$ for all
$Q\in\mathscr{Q}_{k}(G)$;

  3. (3)

if $Q\in\mathscr{Q}_{k}(G)$ and $Q^{\prime}\in\mathscr{Q}_{k^{\prime}}(G)$
where $k\leq k^{\prime}$, then either $Q\cap Q^{\prime}=\emptyset$ or
$Q\subseteq Q^{\prime}$; in the second case, $B(z(Q),2C\kappa^{k})\subseteq
B(z(Q^{\prime}),2C\kappa^{k^{\prime}})$;

We write $\mathscr{Q}(G)$ for the union of all $\mathscr{Q}_{k}(G)$, and call
this a system of pseudodyadic cubes. Given a cube $Q\in\mathscr{Q}_{k}(G)$, we
denote the quantity $\kappa^{k}$ by $\operatorname{\ell}(Q)$, by analogy with
the side-length of a Euclidean cube.

A finite collection
$\\{\mathscr{Q}^{\tau}\colon{\tau}=1,2,\dots,\mathrm{T}\\}$ of systems of
pseudodyadic cubes is called a _collection of adjacent systems of pseudodyadic
cubes with parameters $C^{\prime}$, $c$, $C$ and $\kappa$_, if it has the
following properties: individually, each $\mathscr{Q}^{\tau}$ is a system of
pseudodyadic cubes with parameters $c$, $C$ and $\kappa$ as in Theorem 2.2;
collectively, for each ball $B(x,r)\subseteq G$ such that
$\kappa^{k+3}<r\leq\kappa^{k+2}$, where $k\in\mathbb{Z}$, there exist
${\tau}\in\\{1,2,\dots,\mathrm{T}\\}$ and $Q\in\mathscr{Q}^{\tau}_{k}$ with
centre $z(Q)$ such that $d(x,{}^{\tau}x_{\alpha}^{k})<2\kappa^{k}$

(2.4) $B(x,r)\subseteq Q\subseteq B(x,C^{\prime}r).$

The following construction is due to [15].

######  Theorem 2.3.

Suppose that $(G,\rho)$ is a metric stratified group. Then there exists a
finite collection $\\{\mathscr{Q}^{\tau}\colon{\tau}=1,2,\dots,\mathrm{T}\\}$
of adjacent systems of pseudodyadic cubes with parameters $C^{\prime}$, $c$,
$C$ and $\kappa$, where $\kappa:=1/100$, $c:=12^{-1}$, $C:=4$ and
$C^{\prime}:=8\times 10^{6}$. For each ${\tau}\in\\{1,2,\dots,\mathrm{T}\\}$,
the centres $z(Q)$ of the cubes $Q\in\mathscr{Q}^{\tau}_{k}$ have the
properties that

$d(z(Q),z(Q^{\prime}))\geq\kappa^{k}/4\qquad\text{when $Q\neq Q^{\prime}$}$

and

$\min\\{d(x,z(Q))\colon Q\in\mathscr{Q}^{\tau}_{k}\\}<2\kappa^{k}\qquad\forall
x\in G.$

From [18, Remark 2.8], the number $\mathrm{T}$ of adjacent systems of
pseudodyadic cubes in Theorem 2.3 may be taken to be at most
$A^{6}\kappa^{-\log_{2}(A)}$, where $A$ is the geometric doubling constant of
$G$. The constants $c$, $C$, $C^{\prime}$ and $\kappa$ do not depend on the
choice of the metric stratified group $(G,\rho)$.

###  2.5. Products of stratified groups

We equip the product of two stratified groups $G_{1}$ and $G_{2}$ with a
product structure. We carry forward the notation from Section 2.1, adding a
subscript $i$ or superscript $[i]$ to clarify that we are dealing with
$G_{i}$; the parameter $i$ is always $1$ or $2$. To shorten the formulae, we
often use bold face type to indicate a product object: thus we write
$\mathbf{G}$, $\mathbf{T}$, $\mathbf{g}$, $\mathbf{r}$ and $\mathbf{t}$ in
place of $G_{1}\times G_{2}$, $\mathbb{R}_{+}\times\mathbb{R}_{+}$,
$(g_{1},g_{2})$, $(r_{1},r_{2})$ and $(t_{1},t_{2})$. For example,
$B_{i}(g_{i},r_{i})$ denotes the open ball in $G_{i}$ with centre $g_{i}$ and
radius $r_{i}$, with respect to the homogeneous norm $\rho_{i}$, and we write
$P(\mathbf{g},\mathbf{r})$ for the product $B_{1}(g_{1},r_{1})\times
B_{2}(g_{2},r_{2})$. We write $\pi_{i}$ for the projection of $\mathbf{G}$
onto $G_{i}$.

Products of balls are basic geometric objects and we write
$\mathscr{P}(\mathbf{G})$ for the family of all such products. In addition we
deal with _rectangles_ , by which we mean products of pseudodyadic cubes. We
write $\mathscr{R}(\mathbf{G})$ for the family of all rectangles, and for
adjacent pseudodyadic systems as constructed in Theorem 2.3, we let
$\mathscr{R}^{{\tau}_{1},{\tau}_{2}}(\mathbf{G})$ be the family of rectangles
$\\{Q_{1}\times Q_{2}\colon Q_{i}\in\mathscr{Q}^{{\tau}_{i}},\
{\tau}_{i}=1,2,\dots,\mathrm{T}_{i}\\}$. We let
$\operatorname{\boldsymbol{\ell}}\colon\mathscr{R}(\mathbf{G})\to\mathbf{T}$
be the function such that $\operatorname{\ell}_{i}(Q_{1}\times
Q_{2})=\operatorname{\ell}(Q_{i})$, the “side-length” of $Q_{i}$.

The element of Haar measure on $\mathbf{G}$ is denoted
$\,\mathrm{d}\mathbf{g}$, but is often written
$\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$ for calculations. The convolution $f\ast
f^{\prime}$ of suitable functions $f$ and $f^{\prime}$ on $\mathbf{G}$ is
defined by

$\displaystyle(f\ast
f^{\prime})(\mathbf{g}):=\int_{\mathbf{G}}f(\mathbf{h})f^{\prime}(\mathbf{h}^{-1}\mathbf{g})\,\mathrm{d}\mathbf{h}.$

The strong maximal operator $\mathcal{M}_{s}$ is defined by

$\displaystyle\mathcal{M}_{s}(f)(\mathbf{g}):=\sup\left\\{\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P}\left|f(\mathbf{h})\right|\,\mathrm{d}\mathbf{h}\colon
P\in\mathscr{P}(\mathbf{G}),P\ni\mathbf{g}\right\\}.$

for all $f\in\mathsf{L}^{1}_{\mathrm{loc}}(\mathbf{G})$. It is evident that
$\mathcal{M}_{s}$ is dominated by the composition of the Hardy–Littlewood
maximal operators in the factors:

$\mathcal{M}_{s}{f}\leq\mathcal{M}_{1}\mathcal{M}_{2}(f)\qquad\text{and}\qquad\mathcal{M}_{s}{f}\leq\mathcal{M}_{2}\mathcal{M}_{1}(f).$

When $1<p\leq\infty$, the operators $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ in
the factors are $\mathsf{L}^{p}$-bounded, so the iterated maximal operators
and hence the strong maximal operator are also $\mathsf{L}^{p}$-bounded.

Given an open subset $U$ of $\mathbf{G}$ with finite measure $\left|U\right|$,
we define the enlargement $U^{*}$ of $U$ using the strong maximal operator
$\mathcal{M}_{s}$:

$\displaystyle
U^{*}:=\Bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}\chi_{U}(\mathbf{g})>\alpha\Bigr{\\}},$

for some $\alpha\in(0,1)$ that varies from instance to instance. We write
$\mathscr{M}(U)$ for the family of maximal rectangles contained in $U$.

If $\varphi^{[1]}$ on $G_{1}$ and $\varphi^{[2]}$ on $G_{2}$ both satisfy the
decay and smoothness conditions (2.1) and have integral $1$, then

(2.5)
$\left|f\ast\varphi_{\mathbf{t}}(\mathbf{g})\right|\lesssim\mathcal{M}_{s}(f)(\mathbf{g})\qquad\forall\mathbf{g}\in\mathbf{G}\quad\forall
f\in\mathsf{L}^{1}(\mathbf{G}),$

much as argued to prove (2.3), but with “biradial” in place of “radial”.

###  2.6. Journé’s covering lemma

We recall a covering lemma for product spaces. Journé [17] first established
this result on $\mathbb{R}\times\mathbb{R}$. The fourth author [23] extended
it to higher dimensional Euclidean spaces and an arbitrary number of factors
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$.
This was extended to products of spaces of homogeneous type in [12, Lemma
2.1].

Let $U$ be an open subset of $\mathbf{G}$ of finite measure and
$\mathscr{M}_{i}(U)$ denote the family of rectangles $R$ in $U$ which are
maximal (in terms of inclusion) in the $i$th “direction”. Recall that
$\mathscr{M}(U)$ is the set of all maximal subrectangles of $U$. Given
$R=Q_{1}\times Q_{2}\in\mathscr{M}_{1}(U)$, let $\tilde{Q}_{2}$ be the biggest
pseudodyadic cube containing $Q_{2}$ such that

$\big{|}\bigl{(}Q_{1}\times\tilde{Q}_{2}\bigr{)}\cap
U\big{|}>\frac{1}{2}|Q_{1}\times\tilde{Q}_{2}|.$

Similarly, given $R=Q_{1}\times Q_{2}\in\mathscr{M}_{2}(U)$, let
$\tilde{Q}_{1}$ be the biggest pseudodyadic cube containing $Q_{1}$ such that

$\big{|}\bigl{(}\tilde{Q}_{1}\times Q_{2}\bigr{)}\cap
U\big{|}>\frac{1}{2}|\tilde{Q}_{1}\times Q_{2}|.$

We then set

$\gamma_{1}(R):=\frac{\operatorname{\ell}(\tilde{Q}_{1})}{\operatorname{\ell}(Q_{1})}\qquad\text{and}\qquad\gamma_{2}(R):=\frac{\operatorname{\ell}(\tilde{Q}_{2})}{\operatorname{\ell}(Q_{2})}.$

Here is our Journé-type covering lemma on $\mathbf{G}$.

######  Lemma 2.4.

Suppose that $U$ is an open subset in $\mathbf{G}$ of finite measure and
$\delta\in\mathbb{R}_{+}$. Then

$\displaystyle\sum_{R\in\mathscr{M}_{1}(U)}\left|R\right|\gamma_{1}(R)^{-\delta}\lesssim|U|\qquad\text{and}\qquad\sum_{R\in\mathscr{M}_{2}(U)}\left|R\right|\gamma_{2}(R)^{-\delta}\lesssim|U|.$

###### Proof.

Apply the statement in [12, Lemma 2.1] to $\mathbf{G}$. ∎

###  2.7. The Orlicz space $\mathsf{L\,log\,L}(\mathbf{G})$

We recall the definition of an Orlicz space on $\mathbf{G}$. A Young function
is a continuous, convex, increasing bijective function
$\Phi\colon[0,\infty)\to[0,\infty)$. To a Young function $\Phi$, we associate
the (usually nonlinear) functional $F_{\Phi}$, given by

$F_{\Phi}(f)=\int_{\mathbf{G}}\Phi(|f(\mathbf{g})|)\,\mathrm{d}\mathbf{g}\qquad\forall
f\in\mathsf{L}^{1}_{\mathrm{loc}}(\mathbf{G}).$

The convexity of $\Phi$ implies that
$\\{f\in\mathsf{L}^{1}_{\mathrm{loc}}(\mathbf{G})\colon F_{\Phi}(f)\leq 1\\}$
is a closed convex symmetric set, and so the Luxemburg norm, given by

$\|f\|_{\mathsf{L}^{\Phi}(\mathbf{G})}:=\inf\left\\{\lambda\in\mathbb{R}_{+}\colon
F_{\Phi}(f/\lambda)\leq 1\right\\},$

is indeed a norm, and $\mathsf{L}^{\Phi}(\mathbf{G})$, the set of functions
$f$ for which $\|f\|_{\mathsf{L}^{\Phi}(\mathbf{G})}$ is finite, is a Banach
space. The sets $\\{f\in\mathsf{L}^{1}_{\mathrm{loc}}(\mathbf{G})\colon
F_{\Phi}(f)\leq 1\\}$ and
$\\{f\in\mathsf{L}^{1}_{\mathrm{loc}}(\mathbf{G})\colon\left\|f\right\|_{\mathsf{L}^{\Phi}(\mathbf{G})}\leq
1\\}$ coincide.

Suppose that $\Phi$ and $\Psi$ are Young functions such that

(2.6) $st\leq\Phi(s)+\Psi(t)\qquad\forall s,t\in\mathbb{R}_{+}.$

Then

(2.7)
$\left|\int_{\mathbf{G}}f(\mathbf{g})h(\mathbf{g})\,\mathrm{d}\mathbf{g}\right|\leq\int_{\mathbf{G}}\Phi(f(\mathbf{g}))\,\mathrm{d}\mathbf{g}+\int_{\mathbf{G}}\Psi(h(\mathbf{g}))\,\mathrm{d}\mathbf{g},$

and it follows that the corresponding Luxemburg norms are related by the
_generalized Hölder inequalities_ , namely,

(2.8) $\int_{\mathbf{G}}f(\mathbf{g})h(\mathbf{g})\,\mathrm{d}\mathbf{g}\leq
2\|f\|_{\mathsf{L}^{\Phi}(\mathbf{G})}\|h\|_{\mathsf{L}^{\bar{\Phi}}(\mathbf{G})}.$

We are particularly interested in a special pair of Young functions.
Henceforth,

(2.9)
$\Phi(s):=s[\log(\mathrm{e}+s)]\qquad\text{and}\qquad\Psi(t):=\exp(t)-1.$

By maximising in $s$, it is straightforward to show that
$1+st-s\log(\mathrm{e}+s)\leq\mathrm{e}^{t}$, so these functions satisfy
(2.6), whence (2.7) and (2.8) hold. For this choice of $\Phi$ and $\Psi$, the
corresponding spaces are denoted by $\mathsf{L\,log\,L}(\mathbf{G})$ and
$\mathrm{e}^{\mathsf{L}}(\mathbf{G})$. Clearly
$\mathsf{L\,log\,L}(\mathbf{G})\subseteq\mathsf{L}^{1}(\mathbf{G})$. Further,
$\Phi(\lambda t)\leq\Phi(\lambda)\Phi(t)$, whence

(2.10) $F_{\Phi}(\lambda
f)\leq\Phi(\lambda)F_{\Phi}(f)\qquad\forall\lambda\in\mathbb{R}_{+}\quad\forall
f\in\mathsf{L\,log\,L}(\mathbf{G}).$

This implies that $f\in\mathsf{L\,log\,L}(\mathbf{G})$ if and only if
$F_{\Phi}(f)$ is finite. Further, the hyperweak boundedness estimate (1.2)
that we work with is equivalent to an estimate of the form

$\displaystyle\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon|\mathcal{T}f(\mathbf{g})|>\lambda\big{\\}}\big{|}\lesssim\frac{\log(\mathrm{e}+1/\lambda)}{\lambda}\left\|f\right\|_{\mathsf{L\,log\,L}(\mathbf{G})}\qquad\forall\lambda\in\mathbb{R}_{+},$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$ of norm $1$, that is,
$\mathcal{T}f$ decays a little slower at infinity that would occur if we had a
weak type estimate. Finally, since $\Psi$ is a Young function,

(2.11) $\Psi(t/\lambda)\leq\Psi(t)/\lambda\qquad\forall\lambda\in[1,\infty).$

We need a density result.

######  Proposition 2.5.

$\mathsf{L}^{2}\cap\mathsf{L\,log\,L}(\mathbf{G})$ is dense in
$\mathsf{L\,log\,L}(\mathbf{G})$.

###### Proof.

For $f\in\mathsf{L\,log\,L}(\mathbf{G})$, we define, for all $N\in\mathbb{N}$,

$f_{N}:=\chi_{\\{\mathbf{g}\in\mathbf{G}\colon|f(\mathbf{g})|\leq N\\}}f.$

Then $f_{N}\in\mathsf{L\,log\,L}(\mathbf{G})$, $|f_{N}|\leq|f|$, and $f_{N}\to
f$ almost everywhere as $n\to\infty$. Next,

$\displaystyle\int_{\mathbf{G}}|f_{N}(\mathbf{g})|^{2}\,\mathrm{d}\mathbf{g}\leq\int_{\mathbf{G}}N|f(\mathbf{g})|\log(\mathrm{e}+|f(\mathbf{g})|)\,\mathrm{d}\mathbf{g}=NF_{\Phi}(f),$

which shows that $f_{N}\in\mathsf{L}^{2}(\mathbf{G})$ for all
$N\in\mathbb{Z}^{+}$. Finally, if $\lambda<1$, then

$\displaystyle F_{\Phi}((f-f_{N})/\lambda)$
$\displaystyle=\frac{1}{\lambda}\int_{\mathbf{G}}|f(\mathbf{g})-f_{N}(\mathbf{g})|\log(\mathrm{e}+|(f(\mathbf{g})-f_{N}(\mathbf{g}))/\lambda)|\,\mathrm{d}\mathbf{g}$
$\displaystyle\leq\frac{1}{\lambda}\int_{\\{\mathbf{g}\in\mathbf{G}\colon|f(\mathbf{g})|>N\\}}|f(\mathbf{g})|\left(\log(\mathrm{e}+|f(\mathbf{g})|)+\log(1/\lambda)\right)\,\mathrm{d}\mathbf{g}$
$\displaystyle\to 0$

as $N\to\infty$. Consequently,
$\left\|{f-f_{N}}\right\|_{\mathsf{L\,log\,L}(\mathbf{G})}$ tends to zero as
$N$ tend to infinity. ∎

##  3\. The strong maximal function and Proof of Theorem 1.1

Before we tackle the main topic of this section, we state and prove a covering
lemma. We begin with a geometric observation.

######  Lemma 3.1.

There is a geometric constant $C(\mathbf{G})$ with the property that

$R\subseteq\\{g\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{U})(\mathbf{g})>C(\mathbf{G})\alpha\\}$

for all measurable subsets $U$ of $\mathbf{G}$, all $\alpha\in(0,1)$ and all
rectangles $R\in\mathscr{P}$ such that

$|R\cap U|\geq\alpha\left|R\right|.$

###### Proof.

By definition, $R$ is a product of pseudodyadic cubes $Q_{1}\times Q_{2}$, and
by Theorem 2.2,

$B(z(Q_{i}),c\ell(Q_{i})/3)\subseteq Q_{i}\subseteq
B(z(Q_{i}),2C\ell(Q_{i})).$

Define $P:=B(z(Q_{1}),2C\ell_{1}(R))\times B(z(Q_{2}),2C\ell_{2}(R))$. Then
$R\subseteq P$ and

$\displaystyle|P\cap U|$ $\displaystyle\geq|R\cap U|\geq\lambda\left|R\right|$
$\displaystyle\geq\lambda|B(z(Q_{1}),c\ell(Q_{1})/3)\times
B(z(Q_{2}),c\ell(Q_{2})/3)|$
$\displaystyle=\lambda\Bigl{(}\frac{c}{6C}\Bigr{)}^{\nu_{1}+\nu_{2}}|B(z(Q_{i}),2C\ell(Q_{i}))\times
B(z(Q_{2}),2C\ell(Q_{2}))|$ $\displaystyle=:\lambda C(\mathbf{G})|P|.$

Hence $R$ is a subset of the set

$U^{*}=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}\chi_{U}(\mathbf{g})>C(\mathbf{G})\lambda\\},$

as required. ∎

######  Theorem 3.2.

[5] Let $\\{R_{j}\\}_{j\in J}$ be a family of rectangles in $\mathbf{G}$ such
that $\big{|}\bigcup_{j\in J}R_{j}\big{|}$ is finite. Then there is a sequence
of rectangles $\\{\tilde{R}_{k}\\}\subset\\{R_{j}\\}_{j\in J}$ such that

$\big{|}\bigcup_{j\in
J}R_{j}\big{|}\lesssim\big{|}\bigcup_{k}\tilde{R}_{k}\big{|}\qquad\text{and}\qquad\Bigl{\|}\sum_{k}\chi_{\tilde{R}_{k}}\Bigr{\|}_{\mathrm{e}^{\mathsf{L}}(\mathbf{G})}\lesssim\big{|}\bigcup_{j\in
J}R_{j}\big{|}.$

###### Proof.

The proof is a generalisation of that in [5]. However, the pseudodyadic case
is somewhat trickier than the dyadic case in $\mathbb{R}^{2}$, and the
argument in [5] is rather brief and not always precise, so it seems worthwhile
to provide a complete proof.

Since there are countably many rectangles, we may assume that $J=\mathbb{N}$
and $j\in\mathbb{N}$. Let $E$ be the set $\bigcup_{j}R_{j}$.

Choose a subsequence $\\{R_{\sigma(j)}\colon j\in\mathbb{N}\\}$ of
$\\{R_{j}\colon j\in\mathbb{N}\\}$, using the rules that $\sigma(1)=1$ and
$\sigma(k+1)$ is the least $j>\sigma(k)$ such that

$|R_{j}\cap(R_{\sigma(1)}\cup\dots\cup R_{\sigma(k)})|\leq\frac{1}{4}|R_{j}|;$

the construction terminates if this is not possible. Let $E_{K}=\bigcup_{1\leq
k\leq K}R_{\sigma(k)}\subseteq E$. If the construction does not terminate,
then by monotone convergence, we may choose $K$ such that
$|E_{K}|\geq|E_{\infty}|/2$; if the construction terminates, we take $K$ to be
the last index of the finite sequence. The set $E$ consists of rectangles
$R_{j}$ that are not one of the chosen rectangles $R_{\sigma(k)}$ together
with rectangles that are subsets of $E_{\infty}$ if the construction did not
terminate, or of $E_{K}$ otherwise. Suppose that the construction did not
terminate. Then by Lemma 3.1,

$R_{j}\subseteq
E_{\infty}^{*}:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}\chi_{E_{\infty}}(\mathbf{g})>{C(\mathbf{G})}/{4}\\}.$

Hence $E\subseteq E_{\infty}^{*}\cup E_{\infty}$, and the boundedness of
$\mathcal{M}_{s}$ on $\mathsf{L}^{2}(\mathbf{G})$ shows that

$|E|\leq|E_{\infty}^{*}|+|E_{\infty}|\lesssim|E_{\infty}|\leq 2|E_{K}|.$

When the construction terminates, a slightly easier argument shows that
$|E|\lesssim|E_{K}|$.

We now relabel the finite family of rectangles
$\\{R_{\sigma(1)},\dots,R_{\sigma(K)}\\}$ as $R_{K},\dots,R_{1}$ and repeat
the construction, setting $\tau(1):=1$ and choosing $\tau(l+1)$ to be the
least $l>\tau(l)$ such that

$|R_{k}\cap(R_{\tau(1)}\cup\dots\cup R_{\tau(l)})|\leq\frac{1}{4}|R_{k}|;$

we end up with $L$ terms, say. Since

$|R_{k}\cap(R_{k+1}\cup\dots\cup R_{K})|\leq\frac{1}{4}|R_{k}|$

by construction, it follows that

(3.1) $|R_{\tau(l+1)}\cap((R_{\tau(1)}\cup\dots\cup
R_{\tau(l)})\cup(R_{\tau(l+2)}\cup\dots\cup
R_{\tau(L)}))|\leq\frac{1}{2}|R_{\tau(l+1)}|.$

Write $\tilde{R}_{l}$ for $R_{\tau(l)}$ and $\tilde{E}$ for
$\bigcup_{l}R_{\tau(l)}$. Much as before,
$|E|\lesssim\bigl{|}\tilde{E}\bigr{|}$. Note that

(3.2) $\bigl{|}\tilde{E}\bigr{|}\leq\sum_{l}\bigl{|}\tilde{R}_{l}\bigr{|}\leq
2\sum_{l}\biggl{|}\tilde{R}_{l}\setminus\biggl{(}\bigcup_{l^{\prime}\neq
l}\tilde{R}_{l^{\prime}}\biggr{)}\biggr{|}=2\biggl{|}\bigcup_{l}\tilde{R}_{l}\setminus\biggl{(}\bigcup_{l^{\prime}\neq
l}\tilde{R}_{l^{\prime}}\biggr{)}\biggr{|}\leq 2\bigl{|}\tilde{E}\bigr{|}.$

We now claim that

(3.3)
$\bigl{|}\\{\mathbf{g}\in\mathbf{G}\colon\sum_{l}\chi_{\tilde{R}_{l}}\geq
n\\}\bigr{|}\lesssim 2^{-n/2}|E|.$

From this claim, it follows that

$\displaystyle\int_{\mathbf{G}}\Psi\Bigl{(}\sum_{l}\chi_{\tilde{R}_{L}}(\mathbf{g})/\lambda\Bigr{)}\,\mathrm{d}\mathbf{g}$
$\displaystyle\lesssim\sum_{n=1}^{\infty}\Psi(n/\lambda)2^{-n/2}|E|$
$\displaystyle\lesssim\sum_{n=1}^{\infty}\exp(n/\lambda)2^{-n/2}|E|$
$\displaystyle=\sum_{n=1}^{\infty}\exp(n[1/\lambda-\log(2)/2])|E|$
$\displaystyle\lesssim|E|$

as long as $1/\lambda<\log(2)/2$, and so
$\sum_{l=1}^{L}\chi_{R_{l}}\in\mathrm{e}^{\mathsf{L}}(G)$. It remains to prove
(3.3).

Consider $\mathbf{g}\in\mathbf{G}$ that lies in two distinct rectangles from
the family $\\{\tilde{R}_{l}\\}$, $R$ and $S$ say. If
$\operatorname{\ell}_{1}(S)=\operatorname{\ell}_{1}(R)$, then
$\pi_{1}(R)=\pi_{1}(S)$, since both $\pi_{1}(R)$ and $\pi_{1}(S)$ are
pseudodyadic cubes containing $\pi_{1}(\mathbf{g})$; now
$\pi_{2}(R)\subseteq\pi_{2}(S)$ or $\pi_{2}(R)\supseteq\pi_{2}(S)$, so $R\cap
S$ is either $R$ or $S$, and this contradicts (3.1). Hence we may assume,
without loss of generality, that
$\operatorname{\ell}_{1}(S)<\operatorname{\ell}_{1}(R)$. A similar argument
then shows that $\operatorname{\ell}_{2}(S)>\operatorname{\ell}_{2}(R)$. Thus,
if there are $n$ distinct rectangles in $\\{\tilde{R}_{l}\\}$ that contain
$\mathbf{g}$, then we may label them $R_{1}(\mathbf{g})$, …,
$R_{n}(\mathbf{g})$, in such a way that
$\operatorname{\ell}_{1}(R_{j}(\mathbf{g}))$ decreases with $j$ and
$\operatorname{\ell}_{2}(R_{j}(\mathbf{g}))$ increases with $j$; then

$R_{1}(\mathbf{g})\cap\dots\cap
R_{n}(\mathbf{g})=\pi_{1}(R_{n}(\mathbf{g}))\times\pi_{2}(R_{1}(\mathbf{g})),$

and

$\left|R_{1}(\mathbf{g})\cap\dots\cap
R_{n}(\mathbf{g})\right|=\left|\pi_{1}(R_{n}(\mathbf{g}))\times\pi_{2}(R_{1}(\mathbf{g}))\right|\lesssim\kappa^{-n\nu_{1}}\left|R_{1}(\mathbf{g})\right|.$

We say that a rectangle $T\in\\{\tilde{R}_{l}\\}$ is a _descendant_ of a
rectangle $R\in\\{\tilde{R}_{l}\\}$, and we write $T\succ R$, if $R\cap
T\neq\emptyset$ and both
$\operatorname{\ell}_{1}(T)>\operatorname{\ell}_{1}(R)$ and
$\operatorname{\ell}_{2}(T)<\operatorname{\ell}_{2}(R)$, and we say that $T$
is a _child_ of $R$ if $T\succeq R$ and if $S\in\\{\tilde{R}_{l}\\}$ and
$T\succeq S\succeq R$ then $S=T$ or $S=R$. We may define _ancestors_ and
_parents_ similarly, with the relations reversed.

Fix a rectangle $R\in\\{\tilde{R}_{l}\\}$, and consider $\mathbf{g}\in R$ that
lies in at least $n$ distinct rectangles of $\\{\tilde{R}_{l}\\}$. Then
$\mathbf{g}$ lies in at least $\lceil n/2\rceil$ distinct rectangles $S$ such
that $S\succeq R$, or $\mathbf{g}$ lies in at least $\lceil n/2\rceil$
distinct rectangles $S$ such that $S\preceq R$. Thus

$\displaystyle\left|\left\\{\mathbf{g}\in
R\colon\sum\nolimits_{l}\chi_{\tilde{R}_{l}}(\mathbf{g})\geq
n\right\\}\right|$ $\displaystyle\qquad\leq\left|\left\\{\mathbf{g}\in
R\colon\sum\nolimits_{l}^{\succeq}\chi_{\tilde{R}_{l}}(\mathbf{g})\geq\frac{n}{2}\right\\}\right|+\left|\left\\{\mathbf{g}\in
R\colon\sum\nolimits_{l}^{\preceq}\chi_{\tilde{R}_{l}}(\mathbf{g})\geq\frac{n}{2}\right\\}\right|,$

where $\sum^{\succeq}_{l}$ indicates that we sum over $l$ such that
$\tilde{R}_{l}\succeq R$, and $\sum^{\preceq}_{l}$ is defined analogously. We
estimate the measure of one of these two sets: the other may be estimated
similarly.

Consider all $S\in\\{\tilde{R}_{l}\\}$ such that $S\succeq R$. Inside this
collection of rectangles, we may identify the children $S_{1}$, …, $S_{p}$ of
$R$, which are pairwise disjoint, the children of the children of $R$, which
are again pairwise disjoint, and so on. Observe that

$R\cap(S_{1}\cup\dots\cup
S_{p})=\pi_{1}(R)\times(\pi_{2}(S_{1})\cup\dots\cup\pi_{2}(S_{p}))\subseteq\pi_{1}(R)\times\pi_{2}(R),$

and the near disjointness condition (3.1) implies that

$\left|\pi_{2}(S_{1})\cup\dots\cup\pi_{2}(S_{p})\right|\leq\frac{1}{2}\left|\pi_{2}(R)\right|.$

Then the measure of the set of all $\mathbf{g}\in R$ that also belong to
another rectangle $S\succ R$ is at most $\left|R\right|/2$.

Likewise, similar inequalities hold for the children of the children of $R$,
which we may label as $T_{1}$, …, $T_{q}$, and so

$\left|\pi_{2}(T_{1})\cup\dots\cup\pi_{2}(T_{q})\right|\leq\frac{1}{2}\left|\pi_{2}(S_{1})\cup\dots\cup\pi_{2}(S_{p})\right|\leq\frac{1}{2^{2}}\left|\pi_{2}(R)\right|.$

and the measure of the set of all $\mathbf{g}\in R$ that also belong to two
more rectangles $S,T\succ R$ is at most $2^{-2}\left|R\right|$. Continuing
inductively, the measure of the set of all $\mathbf{g}\in R$ that lie in at
least $m$ distinct rectangles $S\succ R$ is at most $2^{-m}\left|R\right|$.

Combining this estimate with an almost identical estimate for the measures of
sets defined using ancestors and parents, we conclude that the measure of the
set of all $\mathbf{g}\in R$ that belong to at least $n$ rectangles of
$\\{\tilde{R}_{l}\\}$ is at most

$2\times 2^{-\lceil(n-1)/2\rceil}\left|R\right|\lesssim
2^{-n/2}\left|R\right|.$

Then the measure of all $\mathbf{g}\in G$ that belong to at least $n$
rectangles of $\\{\tilde{R}_{l}\\}$ is at most a multiple of

$\sum_{l}2^{-n/2}\bigl{|}\tilde{R}_{l}\bigr{|}=2^{-n/2}\sum_{l}\bigl{|}\tilde{R}_{l}\bigr{|}\lesssim
2^{-n/2}|E|,$

which is what we needed to prove. ∎

Now we can prove an endpoint estimate for the strong maximal function on
$\mathbf{G}$.

######  Theorem 3.3.

The strong maximal function satisfies the following:

$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon|\mathcal{M}_{s}f(\mathbf{g})|>\lambda\big{\\}}\big{|}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}.$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$.

###### Proof.

We recall a fundamental estimate on the strong maximal function and
pseudodyadic strong maximal functions associated with adjacent pseudodyadic
systems, the prototype of which (in the setting of the product
$\mathbb{R}\times\mathbb{R}$) was proved in [20, Theorem 6.1]. More
explicitly, we define the pseudodyadic strong maximal function as follows:

$\displaystyle\mathcal{M}_{s}^{{\tau}_{1},{\tau}_{2}}(f)(\mathbf{g}):=\sup\left\\{\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\left|f(\mathbf{h})\right|\,\mathrm{d}\mathbf{h}\colon
R\ni\mathbf{g},R\in\mathscr{Q}^{{\tau}_{1}}\times\mathscr{Q}^{{\tau}_{2}}\right\\},$

where ${\tau}_{i}=1,2,\dots,\mathrm{T}_{i}$. Then

(3.4)
$\displaystyle\mathcal{M}_{s}(f)(\mathbf{g})\lesssim\sum_{{\tau}_{1}=1}^{\mathrm{T}_{1}}\sum_{{\tau}_{2}=1}^{\mathrm{T}_{2}}\mathcal{M}_{s}^{{\tau}_{1},{\tau}_{2}}(f)(\mathbf{g})\qquad\forall\mathbf{g}\in\mathbf{G}.$

This inequality means that instead of considering general products of balls,
it suffices to consider rectangles, and Theorem 3.2 may be applied.

In light of (3.4), sublinearity and Proposition 2.5, it suffices to show that

(3.5)
$\big{|}\big{\\{}g\in\mathbf{G}\colon|\mathcal{M}_{s}^{{\tau}_{1},{\tau}_{2}}f(\mathbf{g})|>1\big{\\}}\big{|}\lesssim
F_{\Phi}(f)\qquad\forall
f\in\mathsf{L}^{2}\cap\mathsf{L\,log\,L}(\mathbf{G}),$

when ${\tau}_{i}=1,2,\dots,\mathrm{T}_{i}$; we may assume that $f$ takes
nonnegative real values.

Let
$E=\big{\\{}\mathbf{g}\in\mathbf{G}\colon|\mathcal{M}_{s}^{{\tau}_{1},{\tau}_{2}}f(\mathbf{g})|>1\big{\\}}$.
Since $f\in\mathsf{L}^{2}(\mathbf{G})$ and
$\mathcal{M}_{s}^{{\tau}_{1},{\tau}_{2}}$ is $\mathsf{L}^{2}$ bounded, $|E|$
is finite. For every $\mathbf{g}\in E$ there exists
$R_{\mathbf{g}}\in\mathscr{R}^{{\tau}_{1},{\tau}_{2}}(\mathbf{G})$ satisfying

(3.6)
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R_{\mathbf{g}}}f(\mathbf{h})\,\mathrm{d}\mathbf{h}>1.$

Then $E=\bigcup_{\mathbf{g}\in E}R_{\mathbf{g}}$ by definition.

By Theorem 3.2, there is a sequence of rectangles
$\\{\tilde{R}_{l}\\}\subseteq\\{R_{\mathbf{g}}\\}_{\mathbf{g}\in E}$ such that

(3.7) $|E|=\Bigl{|}\bigcup_{\mathbf{g}\in
E}R_{\mathbf{g}}\Bigr{|}\lesssim\Bigl{|}\bigcup_{l}\tilde{R}_{l}\Bigr{|}\qquad\text{and}\qquad\Bigl{\|}\sum_{l}\chi_{\tilde{R}_{l}}\Bigr{\|}_{\mathrm{e}^{\mathsf{L}(\mathbf{G})}}\lesssim\Bigl{|}\bigcup_{\mathbf{g}\in
E}R_{\mathbf{g}}\Bigr{|}.$

Write $\tilde{E}$ for $\bigcup_{l}\tilde{R}_{l}$ and $\tilde{E}_{n}$ for
$\left\\{\mathbf{g}\in\mathbf{G}:\sum_{l}\chi_{\tilde{R}_{l}}(\mathbf{g})=n\right\\}$.
From (3.6), (2.7), (2.10) and (2.11), we deduce that $|E|\lesssim|\tilde{E}|$
and, for all $\lambda\in[1,\infty)$,

$\displaystyle|\tilde{E}|$
$\displaystyle=\Bigl{|}\bigcup_{l}\tilde{R}_{l}\Bigr{|}\leq\sum_{l}|\tilde{R}_{l}|<\sum_{l}\int_{\tilde{R}_{l}}f(\mathbf{g})\,\mathrm{d}\mathbf{g}$
$\displaystyle=\int_{\tilde{E}}\sum_{l}\chi_{\tilde{R}_{l}}(\mathbf{g})f(\mathbf{g})\,\mathrm{d}\mathbf{g}$
$\displaystyle\leq\int_{\tilde{E}}\Phi(\lambda^{2}f(\mathbf{g}))\,\mathrm{d}\mathbf{g}+\int_{\tilde{E}}\Psi(\sum_{l}\chi_{\tilde{R}_{l}}(\mathbf{g})/\lambda^{2})\,\mathrm{d}\mathbf{g}$
$\displaystyle\leq\Phi(\lambda^{2})\int_{\mathbf{G}}\Phi(f(\mathbf{g}))\,\mathrm{d}\mathbf{g}+\frac{1}{\lambda}\int_{\tilde{E}}\Psi(\sum_{l}\chi_{\tilde{R}_{l}}(\mathbf{g})/\lambda)\,\mathrm{d}\mathbf{g}.$

Since $|\tilde{E}_{n}|\leq C_{1}2^{-n/2}|\tilde{E}|$, where $C_{1}$ is a
geometric constant,

$\displaystyle|\tilde{E}|$
$\displaystyle\leq\Phi(\lambda^{2})F_{\Phi}(f)+\frac{1}{\lambda}\sum_{n=1}^{\infty}\mathrm{e}^{n/\lambda}\bigl{|}\tilde{E}_{n}\bigr{|}$
$\displaystyle\leq\Phi(\lambda^{2})F_{\Phi}(f)+\frac{C_{1}}{\lambda}\sum_{n=1}^{\infty}\mathrm{e}^{n/\lambda}2^{-n/2}\bigl{|}\tilde{E}\bigr{|}.$

We fix $\lambda$ that is large enough that the series converges and the right
hand term is less than $\bigl{|}\tilde{E}\bigr{|}/2$; then
$\bigl{|}\tilde{E}\bigr{|}\leq 2\Phi(\lambda^{2})F_{\Phi}(f)$, as required. ∎

######  Corollary 3.4.

The maximal operator $\mathcal{M}_{\zeta,\eta}$ satisfies the endpoint
estimate:

$\bigl{|}\bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\left|\mathcal{M}_{\zeta,\eta}(f)(\mathbf{g})\right|>\lambda\bigr{\\}}\bigr{|}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$.

###### Proof.

It follows from (2.5) with a little care that
$\mathcal{M}_{\zeta,\eta}(f)\lesssim\mathcal{M}_{s}f$ pointwise. ∎

The following lemma is another well known consequence of the boundedness of
the strong maximal operator. Recall that $\mathscr{M}(U)$ denotes the
collection of all maximal rectangles contained in $U$.

######  Lemma 3.5.

Let $U$ be an open subset of $\mathbf{G}$ of finite measure and
$\beta\in\mathbb{R}_{+}$. Given $R$ in $\mathscr{M}(U)$, let $\beta R$ be the
set $\mathbf{z}(R)\delta_{\beta}(\mathbf{z}(R)^{-1}R)$. Then

$\biggl{|}\bigcup_{R\in\mathscr{M}(U)}\beta
R\biggr{|}\lesssim(\beta+1)^{\nu_{1}+\nu_{2}}\left|U\right|.$

###### Proof.

Recall that $B(z(Q),c\operatorname{\ell}(Q)/3)\subseteq Q\subseteq
B(z(Q),2C\operatorname{\ell}(Q))$ for all $Q\in\mathscr{Q}(G)$ and
$P(\mathbf{z},\operatorname{\boldsymbol{\ell}})$ is the product
$B_{1}(z_{1},\operatorname{\ell}_{1})\times
B_{2}(z_{2},\operatorname{\ell}_{2})$. It follows that, if $R\in\mathscr{R}$
and $\mathbf{g}\in\beta R$, then

$P(\mathbf{z}(R),c\operatorname{\boldsymbol{\ell}}(R)/3)\subseteq R\subseteq
P(\mathbf{g},2C(\beta+1)\operatorname{\boldsymbol{\ell}}(R)),$

and we deduce, much as in the proof of Lemma 3.1, that

$\beta
R\subseteq\biggl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{R})(\mathbf{g})\geq
C(\mathbf{G})(\beta+1)^{-\nu_{1}-\nu_{2}}\biggr{\\}}.$

Hence

$\biggl{|}\bigcup_{R\in\mathscr{M}(U)}\beta
R\biggr{|}\leq\biggl{|}\biggl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{R})(\mathbf{g})\geq
C(\mathbf{G})(\beta+1)^{-\nu_{1}-\nu_{2}}\biggr{\\}}\biggr{|},$

and the hyperweak boundedness of $\mathcal{M}_{s}$ implies that

$\biggl{|}\bigcup_{R\in\mathscr{M}(U)}\beta
R\biggr{|}\lesssim(\beta+1)^{\nu_{1}+\nu_{2}}F_{\Phi}(\chi_{U})\lesssim(\beta+1)^{\nu_{1}+\nu_{2}}\left|U\right|,$

as claimed. ∎

##  4\. The atomic decomposition

In this section, we first prove a hyperweak boundedness estimate for the Lusin
area function $\mathcal{S}_{\not{\nabla}p,1}(f)$, and then use this estimate
to decompose $\mathsf{L\,log\,L}(\mathbf{G})$ functions.

Recall that $p^{[i]}_{t_{i}}$ and $q^{[i]}_{t_{i}}$ denote the convolution
kernels of the operators $e^{-t_{i}\sqrt{\mathcal{L}_{i}}}$ and
$t_{i}\partial_{t_{i}}e^{-t_{i}\sqrt{\mathcal{L}_{i}}}$; then
$q^{[i]}_{t_{i}}=t_{i}\partial_{t_{i}}p^{[i]}_{t_{i}}$. We write
$p_{\mathbf{t}}:=p^{[1]}_{t_{1}}\otimes p^{[2]}_{t_{2}}$ and
${{q}}_{\mathbf{t}}:=q^{[1]}_{t_{1}}\otimes q^{[2]}_{t_{2}}$.

###  4.1. Hyperweak boundedness and the area function
$\mathcal{S}_{\not{\nabla}p,1}(f)$

The first step is to apply Corollary 3.4.

######  Proposition 4.1.

The nontangential maximal operator $\mathcal{M}_{p,\eta}$ is hyperweakly
bounded. That is,

$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon|\mathcal{M}_{p,\eta}f(\mathbf{g})|>\lambda\big{\\}}\big{|}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$.

This weak type estimate for the Poisson maximal operator implies a similar
estimate for the Lusin area function $\mathcal{S}_{\not{\nabla}p,1}$, which is
the key to establishing the atomic decomposition for
$\mathsf{L\,log\,L}(\mathbf{G})$ functions. More explicitly, we denote the
tensor
$\not{\nabla}^{[1]}p_{1}^{[1]}(g_{1})\otimes\not{\nabla}^{[2]}p_{1}^{[2]}(g_{2})$
by $\not{\nabla}p(\mathbf{g})$ and define

$\mathcal{S}_{\not{\nabla}p,\eta}f(\mathbf{g}):=\biggl{(}\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma(\mathbf{g})}\left|(f\ast(\not{\nabla}p)_{\mathbf{t}})(\mathbf{h})\right|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2},$

where

(4.1)
$\int\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma(\mathbf{g})}f(\mathbf{h},\mathbf{t})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}:=\int_{\mathbf{T}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{P(\mathbf{g},\mathbf{t})}f(\mathbf{h},\mathbf{t})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}.$

######  Theorem 4.2.

The following estimate holds:

$\left|\left\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\not{\nabla}p,1}(f)(\mathbf{g})>\lambda\right\\}\right|\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$.

###### Proof.

A recent result of Fan, Yan and the first and third authors [6] shows that for
$f$ such that
$\left\|\mathcal{M}_{p,\eta}f\right\|_{\mathsf{L}^{1}(\mathbf{G})}<\infty$,

$\displaystyle\left\|\mathcal{S}_{\not{\nabla}p,1}f\right\|_{\mathsf{L}^{1}(\mathbf{G})}\lesssim\left\|\mathcal{M}_{p,\eta}f\right\|_{\mathsf{L}^{1}(\mathbf{G})}.$

More precisely, define the sublevel set
$L_{\eta}(\lambda):=\left\\{\mathbf{g}\in{\mathbf{G}}\colon\mathcal{M}_{p,\eta}(f)(\mathbf{g})\leq\lambda\right\\}$.
Then it is shown that, when $\eta$ is sufficiently large,

(4.2)
$\displaystyle\bigl{|}\bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\not{\nabla}p,1}(f)(\mathbf{g})>\lambda\bigr{\\}}\bigr{|}$
$\displaystyle\lesssim\bigl{|}L_{\eta}(\lambda)^{c}\bigr{|}+\frac{1}{\lambda^{2}}\int_{L_{\eta}(\lambda)}\mathcal{M}_{p,\eta}^{2}(f)(\mathbf{g})\,\mathrm{d}\mathbf{g}$

for all $\lambda\in\mathbb{R}_{+}$. Now
$\bigl{|}L_{\eta}(\lambda)^{c}\bigr{|}\lesssim F_{\Phi}(f/\lambda)$ by
Proposition 4.1, so the layer-cake formula and (2.10) show that

$\displaystyle\bigl{|}\bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\not{\nabla}p,1}(f)(\mathbf{g})>\lambda\bigr{\\}}\bigr{|}$
$\displaystyle\lesssim\bigl{|}L_{\eta}(\lambda)^{c}\bigr{|}+\frac{1}{\lambda^{2}}\int_{L_{\eta}(\lambda)}\mathcal{M}_{p,\eta}^{2}(f)(\mathbf{g})\,\mathrm{d}\mathbf{g}$
$\displaystyle=\bigl{|}L_{\eta}(\lambda)^{c}\bigr{|}+\frac{1}{\lambda^{2}}\int_{0}^{\lambda}2\mu\Big{|}L_{\eta}(\mu)^{c}\Big{|}\,\mathrm{d}\mu$
$\displaystyle\lesssim
F_{\Phi}(f/\lambda)+\frac{1}{\lambda^{2}}\int_{0}^{\lambda}F_{\Phi}(f/\mu)\mu\,\mathrm{d}\mu$
$\displaystyle\lesssim
F_{\Phi}(f/\lambda)+\frac{1}{\lambda}\int_{0}^{\lambda}\log(\mathrm{e}+\lambda/\mu)F_{\Phi}(f/\lambda)\,\mathrm{d}\mu$
$\displaystyle\lesssim F_{\Phi}(f/\lambda).$

This implies that the right-hand side of (4.2) is finite. By repeating the
argument of [6], we see that (4.2) also holds for
$f\in\mathsf{L\,log\,L}(\mathbf{G})$, and we conclude that

$\displaystyle\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon|\mathcal{S}_{\not{\nabla}p,1}(f)(\mathbf{g})|>\lambda\big{\\}}\big{|}$
$\displaystyle\lesssim\big{|}L_{\eta}(\lambda)^{c}\big{|}+\frac{1}{\lambda^{2}}\int_{L_{\eta}(\lambda)}\mathcal{M}_{p,\eta}^{2}(f)(\mathbf{g})\,\mathrm{d}\mathbf{g}$
$\displaystyle\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$, as required. ∎

Recall that ${{q}}_{\mathbf{t}}$ denotes
$t_{1}\partial_{t_{1}}p_{t_{1}}^{[1]}\otimes
t_{2}\partial_{t_{2}}p^{[2]}_{t_{2}}$ and

$\mathcal{S}_{q,1}(f)(\mathbf{g}):=\biggl{(}\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma(\mathbf{g})}\left|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})\right|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2}.$

######  Corollary 4.3.

The area function $\mathcal{S}_{q,1}(f)$ satisfies the estimate:

$\bigl{|}\bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{q,1}(f)(\mathbf{g})>\lambda\bigr{\\}}\bigr{|}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$.

###### Proof.

This holds as $f*{{q}}_{\mathbf{t}}$ is one of the components of the tensor
$f*(\not{\nabla}p)_{\mathbf{t}}$. ∎

###  4.2. The atomic decomposition for $\mathsf{L\,log\,L}$

Recall that the $q^{[i]}_{1}$ satisfy the standard smoothness, decay and
cancellation conditions (2.1) and (2.2), and by [11] there exist compactly
supported smooth functions $\varphi^{[i]}$ on $G_{i}$ with integral $0$ such
that

$\displaystyle f$
$\displaystyle=\int_{\mathbf{T}}f\ast{{q}}_{\mathbf{t}}\ast\varphi_{\mathbf{t}}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$

for all $f\in\mathsf{L}^{2}(\mathbf{G})$, where
$\varphi_{\mathbf{t}}=\varphi_{t_{1}}\otimes\varphi_{t_{2}}$. This is the
Calderón reproducing formula. We suppose that
$\operatorname{supp}(\varphi^{[i]})\subseteq B_{i}(o,1)$, by rescaling if
necessary.

######  Theorem 4.4.

Let $f\in\mathsf{L}^{2}(\mathbf{G})\cap\mathsf{L\,log\,L}(\mathbf{G})$. Then
we may write

$f=\sum_{k\in\mathbb{Z}}a_{k},$

where the sum converges unconditionally in $L^{2}(\mathbf{G})$ and the _atoms_
$a_{k}$ satisfy:

  1. (1)

$a_{k}$ vanishes outside a subset $U^{\dagger}_{k}$ of $\mathbf{G}$ such that
$|U^{\dagger}_{k}|\lesssim F_{\Phi}(2^{-k}{f})$;

  2. (2)

$\|a_{k}\|_{\mathsf{L}^{2}(\mathbf{G})}^{2}\lesssim F_{\Phi}(2^{k}f)$;

  3. (3)

each $a_{k}$ can be further decomposed:
$a_{k}=\sum_{R\in\mathscr{M}(U^{*}_{k})}a_{k,R}$, where the sum converges
unconditionally in $\mathsf{L}^{2}(\mathbf{G})$, and

    1. (a)

$\operatorname{supp}a_{k,R}\subseteq\beta R$, for a suitable
$\beta\in\mathbb{R}_{+}$;

    2. (b)

$\int_{G_{1}}a_{k,R}(g_{1},g_{2})\,\mathrm{d}g_{1}=0$ for all $g_{2}\in
G_{2}$;

    3. (c)

$\int_{G_{2}}a_{k,R}(g_{1},g_{2})\,\mathrm{d}g_{2}=0$ for all $g_{1}\in
G_{1}$;

    4. (d)

$\sum_{R\in\mathscr{M}(U^{*}_{k})}\left\|a_{k,R}\right\|^{2}_{\mathsf{L}^{2}(\mathbf{G})}\lesssim
2^{2k}F_{\Phi}(2^{-k}f)$.

###### Proof.

From Corollary 4.3, $\mathcal{S}_{q,1}$ is hyperweakly bounded. Assume that
$f\in\mathsf{L}^{2}(\mathbf{G})\cap\mathsf{L\,log\,L}(\mathbf{G})$. Then
$\mathcal{S}_{q,1}(f)\in\mathsf{L}^{2}(\mathbf{G})$. Now for all
$k\in\mathbb{Z}$, we set

(4.3) $\displaystyle U_{k}$
$\displaystyle:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{q,1}(f)(\mathbf{g})>2^{k}\\};$
(4.4) $\displaystyle U^{*}_{k}$
$\displaystyle:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{U_{k}})(\mathbf{g})>\alpha\\};$
(4.5) $\displaystyle U^{\dagger}_{k}$
$\displaystyle:=\bigcup\nolimits_{R\in\mathscr{M}(U^{*}_{k})}\beta R;$ (4.6)
$\displaystyle\mathscr{B}_{k}$
$\displaystyle:=\\{R\in\mathscr{R}(\mathbf{G})\colon\left|R\cap
U_{k}\right|\geq\left|R\right|/2,\left|R\cap
U_{k+1}\right|<\left|R\right|/2\\};$

here $\alpha$ and $\beta$ are positive numbers that will be specified during
the proof. By definition $U_{k}\supseteq U_{k+1}$, so the sequence
$\left|R\cap U_{k}\right|/\left|R\right|$ is decreasing and the sets
$\mathscr{B}_{k}$ are pairwise disjoint. If $\mathbf{g}\in
R\in\mathscr{B}_{k}$, then $|R\cap U_{k}|/\left|R\right|>1/2$ so
$\mathbf{g}\in U^{*}_{k}$ by Lemma 3.1 provided that $\alpha<C(\mathbf{G})/2$;
coupled with (4.6), this shows that

(4.7) $\left|R\cap(U^{*}_{k}\setminus U_{k+1})\right|=\left|R\setminus(R\cap
U_{k+1})\right|\geq\frac{1}{2}\left|R\right|.$

For each rectangle $R\in\mathscr{B}_{k}$, we define the tent $R_{+}$ over $R$
to be the set

(4.8)
$\Bigl{\\{}(\mathbf{g},\mathbf{t})\in\mathbf{G}\times\mathbf{T}\colon\mathbf{g}\in
R,4C\operatorname{\ell}_{1}(R)<t_{1}\leq
8C\operatorname{\ell}_{1}(R),4C\operatorname{\ell}_{2}(R)<t_{2}\leq
8C\operatorname{\ell}_{2}(R)\Bigr{\\}}.$

This definition implies that, if $R\in\mathscr{B}_{k}$ and
$(\mathbf{h},\mathbf{t})\in R_{+}$, then

(4.9) $R\subseteq B_{1}(h_{1},4C\operatorname{\ell}_{1}(R))\times
B_{2}(h_{2},4C\operatorname{\ell}_{2}(R))\subseteq B_{1}(h_{1},t_{1})\times
B_{2}(h_{2},t_{2})=P(\mathbf{h},\mathbf{t}).$

For future use, we note that, for any measurable set $V$ in $\mathbf{G}$,

(4.10) $\displaystyle\chi_{V}*\chi_{P(\mathbf{o},\mathbf{t})}(\mathbf{h})$
$\displaystyle=\int_{\mathbf{G}}\chi_{V}(\mathbf{g})\chi_{P(\mathbf{o},\mathbf{t})}(\mathbf{g}^{-1}\mathbf{h})\,\mathrm{d}\mathbf{g}=\int_{\mathbf{G}}\chi_{V}(\mathbf{g})\chi_{P(\mathbf{h},\mathbf{t})}(\mathbf{g})\,\mathrm{d}\mathbf{g}$
$\displaystyle=\left|P(\mathbf{h},\mathbf{t})\cap V\right|.$

Since $f\in\mathsf{L}^{2}(\mathbf{G})$, the reproducing formula implies that

$\displaystyle f(\mathbf{g})$
$\displaystyle=\int_{\mathbf{T}}f\ast{{q}}_{\mathbf{t}}\ast\varphi_{\mathbf{t}}(\mathbf{g})\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle=\int_{\mathbf{T}}\int_{\mathbf{G}}f\ast{{q}}_{\mathbf{t}}(\mathbf{h})\varphi_{\mathbf{t}}(\mathbf{h}^{-1}\mathbf{g})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}f\ast{{q}}_{\mathbf{t}}(\mathbf{h})\varphi_{\mathbf{t}}(\mathbf{h}^{-1}\mathbf{g})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle=\sum_{k\in\mathbb{Z}}a_{k}(\mathbf{g}),$

say. Next, for all $R\in\mathscr{B}_{k}$, choose
$\tilde{R}\in\mathscr{M}(U^{*}_{k})$ such that $R\subseteq\tilde{R}$. Then for
all $S\in\mathscr{M}(U^{*}_{k})$, set
$a_{k,S}=\sum_{R\in\mathscr{B}_{k}\colon\tilde{R}=S}b_{k,R}$, where

$b_{k,R}(\mathbf{g})=\iint_{R_{+}}f\ast{{q}}_{\mathbf{t}}(\mathbf{h})\varphi_{\mathbf{t}}(\mathbf{h}^{-1}\mathbf{g})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}.$

By construction,

$a_{k}=\sum_{S\in\mathscr{M}(U^{*}_{k})}a_{k,S}.$

Now

$\operatorname{supp}(\varphi_{\mathbf{t}})\subseteq B(o,t_{1})\times
B(o,t_{2})\subseteq B_{1}(o,8C\operatorname{\ell}_{1}(R))\times
B_{2}(o,8C\operatorname{\ell}_{2}(R)),$

and hence

$\displaystyle\operatorname{supp}(b_{k,R})$ $\displaystyle\subseteq
B_{1}(z_{1}(R),10C\operatorname{\ell}_{1}(R))\times
B_{2}(z_{2}(R),10C\operatorname{\ell}_{2}(R))$
$\displaystyle=B_{1}(z_{1}(R),\beta c\operatorname{\ell}_{1}(R)/3)\times
B_{2}(z_{2}(R),\beta c\operatorname{\ell}_{2}(R)/3)$
$\displaystyle\subseteq\beta R,$

where $\beta=30C/c$. Thus $a_{k}$ vanishes on $(U^{\dagger}_{k})^{c}$. By
Lemmas 3.5 and 3.1 and Theorem 4.4,

(4.11) $|U^{\dagger}_{k}|\lesssim|U^{*}_{k}|\lesssim|U_{k}|\lesssim
F_{\Phi}(2^{-k}f).$

Next, we claim that for all $k\in\mathbb{Z}$,

(4.12) $\displaystyle\|a_{k}\|_{\mathsf{L}^{2}(\mathbf{G})}^{2}\lesssim
2^{2k}F_{\Phi}(2^{-k}f).$

To see this, take $h\in\mathsf{L}^{2}(\mathbf{G})$ such that
$\|h\|_{\mathsf{L}^{2}(\mathbf{G})}=1$; then, writing $\check{\varphi}$ for
the reflected version of $\varphi$, that is,
$\check{\varphi}(\mathbf{g})=\varphi(\mathbf{g}^{-1})$, we see that

$\displaystyle\big{|}\langle a_{k},h\rangle\big{|}$
$\displaystyle=\bigg{|}\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}f\ast{{q}}_{\mathbf{t}}(\mathbf{g})h\ast\check{\varphi}_{\mathbf{t}}(\mathbf{g})\,\mathrm{d}\mathbf{g}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\bigg{|}$
$\displaystyle\leq\biggl{(}\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{g})|^{2}\,\mathrm{d}\mathbf{g}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2}\biggl{(}\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}|h\ast\check{\varphi}_{\mathbf{t}}(\mathbf{g})|^{2}\,\mathrm{d}\mathbf{g}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2}$
$\displaystyle\lesssim\biggl{(}\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{g})|^{2}\,\mathrm{d}\mathbf{g}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2}.$

By definition, (4.7), (4.8), (4.9), a convolution identity and (4.11),

$\displaystyle\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle\qquad\leq
2\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}\frac{\bigl{|}R\cap(U^{*}_{k}\setminus
U_{k+1})\bigr{|}}{\left|R\right|}\left|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})\right|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle\qquad\lesssim\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}\frac{\bigl{|}P(\mathbf{h},\mathbf{t})\cap(U^{*}_{k}\setminus
U_{k+1})\bigr{|}}{|P(\mathbf{o},\mathbf{t})|}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle\qquad=\iint_{\mathbf{G}\times\mathbf{T}}\frac{\bigl{|}P(\mathbf{h},\mathbf{t})\cap(U^{*}_{k}\setminus
U_{k+1})\bigr{|}}{|P(\mathbf{o},\mathbf{t})|}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\,\mathrm{d}\mathbf{h}$
$\displaystyle\qquad=\iint_{\mathbf{G}\times\mathbf{T}}\chi_{U^{*}_{k}\setminus
U_{k+1}}*\frac{\chi_{P(\mathbf{o},\mathbf{t})}}{|P(\mathbf{o},\mathbf{t})|}(\mathbf{h})|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\,\mathrm{d}\mathbf{h}$
$\displaystyle\qquad=\int_{\mathbf{G}}\chi_{U^{*}_{k}\setminus
U_{k+1}}(\mathbf{h})\int_{\mathbf{T}}\left|f\ast{{q}}_{\mathbf{t}}\right|^{2}*\frac{\chi_{P(\mathbf{o},\mathbf{t})}}{|P(\mathbf{o},\mathbf{t})|}(\mathbf{h})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle\qquad=\int_{U^{*}_{k}\backslash
U_{k+1}}|\mathcal{S}_{q,1}(f)(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}$
$\displaystyle\qquad\leq 2^{2k}|U^{*}_{k}|$ $\displaystyle\qquad\lesssim
2^{2k}F_{\Phi}(2^{-k}f).$

This shows that $|\langle a_{k},h\rangle|^{2}\lesssim
2^{2k}F_{\Phi}(2^{-k}f)$, whence
$\|a_{k}\|_{\mathsf{L}^{2}(\mathbf{G})}^{2}\lesssim 2^{2k}F_{\Phi}(2^{-k}f)$,
and (4.12) holds.

We now claim that

(4.13)
$\displaystyle\sum_{S\in\mathscr{M}(U^{*}_{k})}\|a_{k,S}\|^{2}_{\mathsf{L}^{2}(\mathbf{G})}$
$\displaystyle\lesssim 2^{2k}F_{\Phi}(2^{-k}f).$

Arguing as above, we see that, if $h\in\mathsf{L}^{2}(\mathbf{G})$ and
$\|h\|_{\mathsf{L}^{2}(\mathbf{G})}=1$, then

$\displaystyle\big{|}\langle a_{k,S},h\rangle\big{|}$
$\displaystyle=\bigg{|}\sum_{R\in\mathscr{B}_{k}(S)}\iint_{R_{+}}f\ast{{q}}_{\mathbf{t}}(\mathbf{h})\
h\ast\check{\varphi}_{\mathbf{t}}(\mathbf{h})\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\bigg{|}$
$\displaystyle\leq\biggl{(}\sum_{R\in\mathscr{B}_{k}(S)}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2}$
$\displaystyle\hskip
113.81102pt\times\biggl{(}\sum_{R\in\mathscr{B}_{k}(S)}\iint_{R_{+}}|h\ast\check{\varphi}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2}$
$\displaystyle\lesssim\biggl{(}\sum_{R\in\mathscr{B}_{k}(S)}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2},$

where $\mathscr{B}_{k}(S):=\\{R\in\mathscr{B}_{k}\colon\tilde{R}=S\\}$. Hence

$\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}\leq\biggl{(}\sum_{R\in\mathscr{B}_{k}(S)}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}\biggr{)}^{1/2},$

which implies that

$\sum_{R\in\mathscr{M}(U^{*}_{k})}\left\|a_{k,R}\right\|^{2}_{\mathsf{L}^{2}(\mathbf{G})}\leq\sum_{R\in\mathscr{B}_{k}}\iint_{R_{+}}|f\ast{{q}}_{\mathbf{t}}(\mathbf{h})|^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}.$

Then, much as argued to prove (4.12), we deduce (4.13). The unconditional
convergence of the sum follows as in [2]. ∎

##  5\. Proof of Theorem 1.1

In this section, we use the atomic decomposition to prove Theorem 1.1. Recall
that, if $f\in\mathsf{L}^{2}(\mathbf{G})\cap\mathsf{L\,log\,L}(\mathbf{G})$,
then we may write

$f=\sum_{k\in\mathbb{Z}}a_{k}\qquad\text{and}\qquad
a_{k}=\sum_{R\in\mathscr{M}(U^{*}_{k})}a_{k,R},$

where $U^{*}_{k}$ is as in Theorem 4.4, and $a_{k}$ and $a_{k.R}$ satisfy
support and cancellation conditions. In Theorem 4.4, we defined

$U_{k}:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{q,1}(f)(\mathbf{g})>2^{k}\\};$

and

$U^{*}_{k}:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{U_{k}})(\mathbf{g})>\alpha\\},$

where $\alpha<C(\mathbf{G})/2$; now we also define

(5.1)
$U^{**}_{k}:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{U^{*}_{k}})(\mathbf{g})>\alpha\\}.$

and

(5.2)
$U^{***}_{k}:=\\{\mathbf{g}\in\mathbf{G}\colon\mathcal{M}_{s}(\chi_{U^{**}_{k}})(\mathbf{g})>\alpha\\}.$

###  5.1. Proof of Theorem 1.1 for area functions $\mathcal{S}_{\psi,\eta}(f)$

Recall from Section 4.2 that
$\mathsf{L}^{2}(\mathbf{G})\cap\mathsf{L\,log\,L}(\mathbf{G})$ is dense in
$\mathsf{L\,log\,L}(\mathbf{G})$. Now we take a general $\psi$ satisfying the
decay, smoothness and cancellation conditions (2.1) and (2.2), and prove that

(5.3)
$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\psi,\eta}(f)(\mathbf{g})>\lambda\big{\\}}\big{|}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$. By sublinearity and density, it
suffices to prove that

(5.4)
$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\psi,\eta}(f)(\mathbf{g})>1\big{\\}}\big{|}\lesssim
F_{\Phi}(f)$

for all $f\in\mathsf{L}^{2}(\mathbf{G})\cap\mathsf{L\,log\,L}(\mathbf{G})$. We
may and shall suppose that $\eta>1$.

From Theorem 4.4, we may write

$f=\sum_{k}a_{k},$

where the $a_{k}$ satisfy the conditions of the lemma.

Observe that

$\displaystyle\Bigl{\|}\sum_{k\leq
0}a_{k}\Bigr{\|}_{\mathsf{L}^{2}(\mathbf{G})}$ $\displaystyle\leq\sum_{k\leq
0}\left\|a_{k}\right\|_{\mathsf{L}^{2}(\mathbf{G})}\lesssim\sum_{k\leq
0}2^{k}F_{\Phi}(2^{-k}f)^{1/2}\lesssim F_{\Phi}(f)^{1/2}.$

Thus, by the $\mathsf{L}^{2}(\mathbf{G})$ boundedness of
$\mathcal{S}_{\psi,\eta}$,

(5.5)
$\bigg{|}\bigg{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\psi,\eta}\biggl{(}\sum_{k\leq
0}a_{k}\biggr{)}(\mathbf{g})>1\bigg{\\}}\bigg{|}\lesssim\Bigl{\|}\sum_{k\leq
0}a_{k}\Bigr{\|}_{\mathsf{L}^{2}(\mathbf{G})}^{2}\lesssim F_{\Phi}(f).$

To handle $\mathcal{S}_{\psi,\eta}(\sum_{k\geq 1}a_{k})$, we use the fine
structure of the atoms $a_{k}$: each $a_{k}$ is supported in
$U_{k}^{\dagger}$, where $|U_{k}^{\dagger}|\lesssim 2^{-k}F_{\Phi}(f)$, and
$a_{k}=\sum_{R\in\mathscr{M}(U^{*}_{k})}a_{k,R}$.

For all $R={Q_{1}}\times{Q_{2}}\in\mathscr{M}(U^{*}_{k})$, let
${\tilde{Q}_{1}}$ be the biggest pseudodyadic cube containing ${Q_{1}}$ such
that ${\tilde{Q}_{1}}\times{Q_{2}}\subset U^{**}_{k}$, where $U^{**}_{k}$ is
defined in (5.1). Next, let ${\tilde{Q}_{2}}$ be the biggest pseudodyadic cube
containing ${Q_{2}}$ such that ${\tilde{Q}_{1}}\times{\tilde{Q}_{2}}\subseteq
U^{***}_{k}$, where $U^{***}_{k}$ is defined in (5.2). Finally, let
$R^{\dagger}$ be $100\beta_{k}({\tilde{Q}_{1}}\times{\tilde{Q}_{2}})$, where
$\beta_{k}=2^{k/(2\nu_{1}+2\nu_{2})}$. By Lemmas 3.5 and 3.1 and Theorem 4.4,

(5.6)
$\displaystyle\biggl{|}\bigcup_{R\in\mathscr{M}(U^{*}_{k})}R^{\dagger}\biggr{|}$
$\displaystyle\lesssim 2^{k/2}|U_{k}^{\dagger}|\lesssim
2^{k/2}|U_{k}^{***}|\lesssim 2^{k/2}|U_{k}^{**}|$ $\displaystyle\lesssim
2^{k/2}|U_{k}^{*}|\lesssim 2^{k/2}|U_{k}|\lesssim 2^{k/2}F_{\Phi}(2^{-k}f).$

We claim that there exists $\delta\in\mathbb{R}_{+}$ such that

(5.7)
$\sum_{R\in\mathscr{M}(U^{*}_{k})}\int_{(R^{\dagger})^{c}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}\lesssim
2^{-\delta k}F_{\Phi}(f)\qquad\forall k\in\mathbb{N}.$

Assume (5.7) for the moment, and let $E^{\dagger}:=\bigcup_{k\geq
1}\bigcup_{R\in\mathscr{M}(U^{*}_{k})}R^{\dagger}$. Then on the one hand,

$\displaystyle|E^{\dagger}|=\Big{|}\bigcup_{k\geq
1}\bigcup_{R\in\mathscr{M}(U^{*}_{k})}R^{\dagger}\Big{|}$
$\displaystyle\lesssim\sum_{k\geq 1}2^{k/2}F_{\Phi}(2^{-k}f)\lesssim
F_{\Phi}(f)$

by (5.6). On the other hand, summing (5.7) over positive $k$ shows that

$\displaystyle\int_{(E^{\dagger})^{c}}\Big{|}\mathcal{S}_{\psi,\eta}\Big{(}\sum_{k\geq
1}a_{k}\Big{)}(\mathbf{g})\Big{|}\,\mathrm{d}\mathbf{g}\lesssim F_{\Phi}(f).$

By Chebychev’s inequality,

$\displaystyle\Big{|}\Bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\psi,\eta}\Big{(}\sum_{k\geq
1}a_{k}(\mathbf{g})\Big{)}>1\Bigr{\\}}\Big{|}$
$\displaystyle\leq|E^{\dagger}|+\int_{(E^{\dagger})^{c}}\Big{|}\mathcal{S}_{\psi,\eta}\Bigl{(}\sum_{k\geq
1}a_{k}\Bigr{)}(\mathbf{g})\Big{|}\,\mathrm{d}\mathbf{g}\lesssim F_{\Phi}(f).$

Together with (5.5), this implies (5.3).

It remains to prove (5.7). Now

$\displaystyle\int_{(R^{\dagger})^{c}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|\,\mathrm{d}g_{2}\,\mathrm{d}g_{1}$
$\displaystyle\leq\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\int_{100{Q_{2}}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|\,\mathrm{d}g_{2}\,\mathrm{d}g_{1}$
$\displaystyle\qquad+\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\int_{(100{Q_{2}})^{c}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|\,\mathrm{d}g_{2}\,\mathrm{d}g_{1}$
$\displaystyle\qquad+\int_{(100\beta_{k}{\tilde{Q}_{2}})^{c}}\int_{100{Q_{1}}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$
$\displaystyle\qquad+\int_{(100\beta_{k}{\tilde{Q}_{2}})^{c}}\int_{(100{Q_{1}})^{c}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$
$\displaystyle=\mathrm{I}_{1}(R)+\mathrm{I}_{2}(R)+\mathrm{I}_{3}(R)+\mathrm{I}_{4}(R),$

say, where $\beta_{k}=2^{k/(2\nu_{1}+2\nu_{2})}\eta$. It suffices to control
$\mathrm{I}_{1}(R)$ and $\mathrm{I}_{2}(R)$, as the other two terms are
similar.

By Hölder’s inequality and the $\mathsf{L}^{2}(G_{2})$-boundedness of
$\mathcal{S}_{\psi^{[2]},\eta}$,

$\displaystyle\mathrm{I}_{1}$
$\displaystyle\lesssim|{Q_{2}}|^{1/2}\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\biggl{(}\int_{100{Q_{2}}}\bigl{(}\mathcal{S}_{\psi^{[1]},\eta}(a_{k,R})(g_{1},g_{2})\bigr{)}^{2}\,\mathrm{d}g_{2}\biggr{)}^{1/2}\,\mathrm{d}g_{1}.$

We use the cancellation and support restrictions on $a_{k,R}$ in the first
variable, (2.1), homogeneity, the geometry ($h_{1},z({Q_{1}})\in{Q_{1}}$ and
$g_{1}\in(100\beta_{k}{\tilde{Q}_{1}})^{c}$) and Hölder’s inequality to deduce
that

$\displaystyle|a_{k,R}\ast_{1}\psi_{t}^{[1]}(g_{1},g_{2})|$
$\displaystyle\qquad=\biggl{|}\int_{G_{1}}a_{k,R}(h_{1},g_{2})t^{-\nu_{1}}\bigl{(}\psi^{[1]}(\delta_{1/t}(h_{1}^{-1}g_{1}))-\psi^{[1]}(\delta_{1/t}(z({Q_{1}})^{-1}g_{1}))\bigr{)}\,\mathrm{d}{h}_{1}\biggr{|}$
$\displaystyle\qquad\lesssim\int_{G_{1}}\left|a_{k,R}(h_{1},g_{2})\right|\frac{t^{-\nu_{1}}\bigl{(}\rho(\delta_{1/t}(z({Q_{1}})^{-1}g_{1})\delta_{1/t}(h_{1}^{-1}g_{1})^{-1})\bigr{)}^{\varepsilon}}{\bigl{(}1+\rho(\delta_{1/t}(h_{1}^{-1}g_{1}))+\rho(\delta_{1/t}(z({Q_{1}})^{-1}g_{1}))\bigr{)}^{\nu_{1}+2\varepsilon}}\,\mathrm{d}{h}_{1}$
$\displaystyle\qquad=\int_{G_{1}}\left|a_{k,R}(h_{1},g_{2})\right|\frac{t^{\varepsilon}\bigl{(}\rho(z({Q_{1}})^{-1}h_{1})\bigr{)}^{\varepsilon}}{\bigl{(}t+\rho(h_{1}^{-1}g_{1})+\rho(z({Q_{1}})^{-1}g_{1})\bigr{)}^{\nu_{1}+2\varepsilon}}\,\mathrm{d}{h}_{1}$
$\displaystyle\qquad\lesssim\int_{G_{1}}\left|a_{k,R}(h_{1},g_{2})\right|\frac{t^{\varepsilon}\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\bigl{(}t+\rho(z({Q_{1}})^{-1}g_{1})\bigr{)}^{\nu_{1}+2\varepsilon}}\,\mathrm{d}{h}_{1}$
$\displaystyle\qquad\eqsim\frac{t^{\varepsilon}\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\bigl{(}t+\rho(z({Q_{1}})^{-1}g_{1})\bigr{)}^{\nu_{1}+2\varepsilon}}\left\|a_{k,R}(\cdot,g_{2})\right\|_{\mathsf{L}^{1}(G_{1})},$

where $z({Q_{1}})$ is the center of ${Q_{1}}$. Hence

$\displaystyle\bigl{(}\mathcal{S}_{\psi^{[1]},\eta}(a_{k,R})(g_{1},g_{2})\bigr{)}^{2}$
$\displaystyle=\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma_{1}^{\eta}(g_{1})}\bigl{|}a_{k,R}\ast_{1}\psi_{t}^{[1]}(h_{1},g_{2})\bigr{|}^{2}\,\mathrm{d}h_{1},\frac{\,\mathrm{d}t}{t}$
$\displaystyle\lesssim\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma_{1}^{\eta}(g_{1})}\frac{t^{2\varepsilon}\operatorname{\ell}_{1}({Q_{1}})^{2\varepsilon}\left\|a_{k,R}(\cdot,g_{2})\right\|_{\mathsf{L}^{1}(G_{1})}^{2}}{\bigl{(}t+\rho_{1}(z({Q_{1}})^{-1}h_{1})\bigr{)}^{2\nu_{1}+4\varepsilon}}\,\mathrm{d}h_{1}\,\frac{\,\mathrm{d}t}{t}$
$\displaystyle\eqsim\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma_{1}^{\eta}(g_{1})}\frac{t^{2\varepsilon}\operatorname{\ell}_{1}({Q_{1}})^{2\varepsilon}\left\|a_{k,R}(\cdot,g_{2})\right\|_{\mathsf{L}^{1}(G_{1})}^{2}}{\bigl{(}t+\rho_{1}(z({Q_{1}})^{-1}g_{1})\bigr{)}^{2\nu_{1}+4\varepsilon}}\,\mathrm{d}h_{1}\,\frac{\,\mathrm{d}t}{t}$
$\displaystyle=\frac{\operatorname{\ell}_{1}({Q_{1}})^{2\varepsilon}\left\|a_{k,R}(\cdot,g_{2})\right\|_{\mathsf{L}^{1}(G_{1})}^{2}}{\bigl{(}\rho_{1}(z({Q_{1}})^{-1}g_{1})\bigr{)}^{2\nu_{1}+2\varepsilon}}\int_{\mathbb{R}_{+}}\frac{t^{2\varepsilon}}{\bigl{(}t+1\bigr{)}^{2\nu_{1}+4\varepsilon}}\,\frac{\,\mathrm{d}t}{t}$
$\displaystyle\eqsim\frac{\operatorname{\ell}({Q_{1}})^{2\varepsilon}}{\bigl{(}\rho_{1}(z({Q_{1}})^{-1}g_{1})\bigr{)}^{2\nu_{1}+2\varepsilon}}\left\|a_{k,R}(\cdot,g_{2})\right\|^{2}_{\mathsf{L}^{1}(G_{1})}$
$\displaystyle\lesssim\frac{\operatorname{\ell}({Q_{1}})^{2\varepsilon}\left|{Q_{1}}\right|}{\bigl{(}\rho_{1}(z({Q_{1}})^{-1}g_{1})\bigr{)}^{2\nu_{1}+2\varepsilon}}\left\|a_{k,R}(\cdot,g_{2})\right\|^{2}_{\mathsf{L}^{2}(G_{1})}$

(recall that
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int$
denotes the “average integral”). This estimate implies that

$\displaystyle\mathrm{I}_{1}(R)$
$\displaystyle=\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\int_{100{Q_{2}}}\left|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})\right|\,\mathrm{d}g_{2}\,\mathrm{d}g_{1}$
$\displaystyle\leq\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}|100{Q_{2}}|^{1/2}\left(\int_{100{Q_{2}}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|^{2}\,\mathrm{d}g_{2}\right)^{1/2}\,\mathrm{d}g_{1}$
$\displaystyle\leq\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}|100{Q_{2}}|^{1/2}\left(\int_{G_{2}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|^{2}\,\mathrm{d}g_{2}\right)^{1/2}\,\mathrm{d}g_{1}$
$\displaystyle\lesssim\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}|{Q_{2}}|^{1/2}\frac{\operatorname{\ell}({Q_{1}})^{\varepsilon}|{Q_{1}}|^{1/2}}{\rho_{1}(z({Q_{1}})^{-1}g_{1})^{\nu_{1}+\varepsilon}}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}\,\mathrm{d}g_{1}$
$\displaystyle=\operatorname{\ell}({Q_{1}})^{\varepsilon}\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\frac{1}{\rho_{1}(z({Q_{1}})^{-1}g_{1})^{\nu_{1}+\varepsilon}}\,\mathrm{d}g_{1}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}$
$\displaystyle\eqsim\beta_{k}^{-\varepsilon}\gamma_{1}(R)^{-\varepsilon}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}.$

To estimate $\mathrm{I}_{2}(R)$, we argue as follows. For $g_{1}\notin
100\beta_{k}{\tilde{Q}_{1}}$ and $g_{2}\notin 100{Q_{2}}$,

$\displaystyle\bigl{(}\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})\bigr{)}^{2}$
$\displaystyle=\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma^{\eta}(\mathbf{g})}\bigl{|}a_{k,R}\ast\psi_{\mathbf{t}}(\mathbf{g})\bigr{|}^{2}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}},$

and because of the cancellation of $a_{k,R}$ and properties of $\psi^{[1]}$
and $\psi^{[2]}$,

$\displaystyle\bigl{|}a_{k,R}\ast\psi_{\mathbf{t}}(\mathbf{g})\bigr{|}$
$\displaystyle\qquad=\biggl{|}\int_{\mathbf{G}}a_{k,R}(\mathbf{h})\psi_{\mathbf{t}}(\mathbf{h}^{-1}\mathbf{g})\,\mathrm{d}\mathbf{h}\biggr{|}$
$\displaystyle\qquad=\biggl{|}\int_{\mathbf{G}}a_{k,R}(h_{1},h_{2})\psi^{[1]}_{t_{1}}(h_{1}^{-1}g_{1})\psi^{[2]}_{t_{2}}(h_{2}^{-1}g_{2})\,\mathrm{d}h_{2}\,\mathrm{d}h_{1}\biggr{|}$
$\displaystyle\qquad=\biggl{|}\int_{\mathbf{G}}a_{k,R}(h_{1},h_{2})\bigl{(}\psi^{[1]}_{t_{1}}(h_{1}^{-1}g_{1})-\psi^{[1]}_{t_{1}}(z({Q_{1}})^{-1}g_{1})\bigr{)}$
$\displaystyle\qquad\qquad\qquad\qquad\times\bigl{(}\psi^{[2]}_{t_{2}}(h_{2}^{-1}g_{2})-\psi^{[2]}_{t_{2}}z({Q_{2}})^{-1}g_{2})\bigr{)}\,\mathrm{d}h_{2}\,\mathrm{d}h_{1}\biggr{|}$
$\displaystyle\qquad\leq\left\|a_{k,R}\right\|_{\mathsf{L}^{1}(\mathbf{G})}\sup\left\\{\left|\psi^{[1]}_{t_{1}}(h_{1}^{-1}g_{1})-\psi^{[1]}_{t_{1}}(z({Q_{1}})^{-1}g_{1})\right|\colon
h_{1}\in{Q_{1}},g_{1}\in(100b{Q_{1}})^{c}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\times\sup\left\\{\left|\psi^{[2]}_{t_{2}}(h_{2}^{-1}g_{2})-\psi^{[2]}_{t_{2}}(z({Q_{2}})^{-1}g_{2})\right|\colon
h_{2}\in{Q_{2}},g_{2}\in(100b{Q_{2}})^{c}\right\\}$
$\displaystyle\qquad\lesssim\left\|a_{k,R}\right\|_{\mathsf{L}^{1}(\mathbf{G})}\frac{t_{1}^{\varepsilon}\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\bigl{(}t_{1}+\rho_{1}(g_{1}^{-1}z({Q_{1}}))\bigr{)}^{\nu_{1}+2\varepsilon}}\frac{t_{2}^{\varepsilon}\operatorname{\ell}({Q_{2}})^{\varepsilon}}{\bigl{(}t_{2}+\rho_{2}(g_{2}^{-1}z({Q_{2}}))\bigr{)}^{\nu_{2}+2\varepsilon}}\,,$

by the same geometrical arguments as used to treat $\mathrm{I}_{1}(R)$,
applied in both variables. Hence
$\bigl{(}\mathcal{S}_{\psi,\eta}(a_{k,R})(\mathbf{g})\bigr{)}^{2}$ is
dominated by a multiple of
$\left\|a_{k,R}\right\|_{\mathsf{L}^{1}(\mathbf{G})}$ by

$\displaystyle\int\\!\\!\\!\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Gamma^{\eta}(\mathbf{g})}\frac{t_{1}^{2\varepsilon}\operatorname{\ell}({Q_{1}})^{2\varepsilon}}{\bigl{(}t_{1}+\rho_{1}(h_{1}^{-1}z({Q_{1}}))\bigr{)}^{2\nu_{1}+2\varepsilon}}\frac{t_{2}^{2\varepsilon}\operatorname{\ell}({Q_{2}})^{2\varepsilon}}{\bigl{(}t_{2}+\rho_{2}(h_{2}^{-1}z({Q_{2}}))\bigr{)}^{2\nu_{2}+2\varepsilon}}\,\mathrm{d}\mathbf{h}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle\eqsim\int_{\mathbf{T}}\frac{t_{1}^{2\varepsilon}\operatorname{\ell}({Q_{1}})^{2\varepsilon}}{\bigl{(}t_{1}+\rho_{1}(g_{1}^{-1}z({Q_{1}}))\bigr{)}^{2\nu_{1}+4\varepsilon}}\frac{t_{2}^{2\varepsilon}\operatorname{\ell}({Q_{2}})^{2\varepsilon}}{\bigl{(}t_{2}+\rho_{2}(g_{2}^{-1}z({Q_{2}}))\bigr{)}^{2\nu_{2}+4\varepsilon}}\,\frac{\mathrm{d}\mathbf{t}}{\mathbf{t}}$
$\displaystyle\eqsim\frac{\operatorname{\ell}({Q_{1}})^{2\varepsilon}}{\bigl{(}\rho_{1}(g_{1}^{-1}z({Q_{1}}))\bigr{)}^{2\nu_{1}+2\varepsilon}}\frac{\operatorname{\ell}({Q_{2}})^{2\varepsilon}}{\bigl{(}\rho_{2}(g_{2}^{-1}z({Q_{2}}))\bigr{)}^{2\nu_{2}+2\varepsilon}}\,.$

Our estimation of $\mathrm{I}_{2}(R)$ concludes with the observation that

$\displaystyle\mathrm{I}_{2}(R)$
$\displaystyle=\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\int_{(100{Q_{2}})^{c}}|\mathcal{S}_{\psi,\eta}(a_{k,R})(g_{1},g_{2})|\,\mathrm{d}g_{2}\,\mathrm{d}g_{1}$
$\displaystyle\lesssim\left\|a_{k,R}\right\|_{\mathsf{L}^{1}(\mathbf{G})}\int_{(100\beta_{k}{\tilde{Q}_{1}})^{c}}\int_{(100{Q_{2}})^{c}}\frac{\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\bigl{(}\rho_{1}(g_{1}^{-1}z({Q_{1}}))\bigr{)}^{\nu_{1}+\varepsilon}}\frac{\operatorname{\ell}({Q_{2}})^{\varepsilon}}{\bigl{(}\rho_{2}(g_{2}^{-1}z({Q_{2}}))\bigr{)}^{\nu_{2}+\varepsilon}}\,\mathrm{d}g_{2}\,\mathrm{d}g_{1}$
$\displaystyle\lesssim\frac{\operatorname{\ell}({Q_{1}})^{\varepsilon}}{(100\beta_{k}\operatorname{\ell}({\tilde{Q}_{1}}))^{\varepsilon}}\frac{\operatorname{\ell}({Q_{2}})^{\varepsilon}}{(100\operatorname{\ell}({Q_{2}}))^{\varepsilon}}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}$
$\displaystyle\eqsim\beta_{k}^{-\varepsilon}\gamma_{1}(R)^{-\varepsilon}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}.$

Then, by combining the estimates of $\mathrm{I}_{1}(R)$ and
$\mathrm{I}_{2}(R)$, and then using Hölder’s inequality and Lemma 2.4, we see
that for all $k\geq 1$,

$\displaystyle\sum_{R\in\mathscr{M}(U^{*}_{k})}\mathrm{I}_{1}(R)+\mathrm{I}_{2}(R)$
$\displaystyle\lesssim\beta_{k}^{-\varepsilon}\sum_{R\in\mathscr{M}(U^{*}_{k})}\gamma(R)^{-\varepsilon}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}$
$\displaystyle\leq\beta_{k}^{-\varepsilon}\biggl{(}\sum_{R\in\mathscr{M}(U^{*}_{k})}\gamma_{1}(R)^{-2\varepsilon}\left|R\right|\biggr{)}^{1/2}\biggl{(}\sum_{R\in\mathscr{M}(U^{*}_{k})}\left\|a_{k,R}\right\|^{2}_{\mathsf{L}^{2}(\mathbf{G})}\biggr{)}^{1/2}$
$\displaystyle\lesssim\beta_{k}^{-\varepsilon}|U_{k}|^{1/2}\Bigl{(}2^{2k}F_{\Phi}(2^{-k}f)\Bigr{)}^{1/2}$
$\displaystyle\lesssim\beta_{k}^{-\varepsilon}2^{k}F_{\Phi}(2^{-k}f)$
$\displaystyle\lesssim 2^{-\varepsilon k/(2\nu_{1}+2\nu_{2})}F_{\Phi}(f).$

Similar estimates hold for $\mathrm{I}_{3}(R)$ and $\mathrm{I}_{4}(R)$, but
with $\gamma_{2}$ in place of $\gamma_{1}$, and hence (5.7) holds, and the
proof is complete.

###  5.2. Proof of Theorem 1.1 for square functions $\mathcal{S}_{\psi,0}(f)$

Arguing much as in Section 5.1, we may also check that

(5.8)
$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{S}_{\psi,\eta}(f)(\mathbf{g})>\lambda\big{\\}}\big{|}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$. We leave the details to the
reader.

###  5.3. Hyperweak boundedness of Riesz transformations

We write $\mathcal{R}$ for a double Riesz transformation
$\mathcal{R}^{[1]}_{j_{1}}\otimes\mathcal{R}^{[2]}_{j_{2}}$.

######  Theorem 5.1.

The double Riesz transformations satisfy the endpoint estimate:

(5.9)
$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{R}(f)(\mathbf{g})>\lambda\big{\\}}\lesssim
F_{\Phi}(f/\lambda)\qquad\forall\lambda\in\mathbb{R}_{+}$

for all $f\in\mathsf{L\,log\,L}(\mathbf{G})$.

###### Proof.

By density and sublinearity, we need only prove that, for all
$f\in\mathsf{L}^{2}(\mathbf{G})\cap\mathsf{L\,log\,L}(\mathbf{G})$,

(5.10)
$\big{|}\big{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{R}(f)(\mathbf{g})>1\big{\\}}\big{|}\lesssim
F_{\Phi}(f).$

From Theorem 4.4, we may write

$f=\sum_{k}a_{k}$

where the $a_{k}$ satisfy the conditions specified there. By the
$\mathsf{L}^{2}(\mathbf{G})$ boundedness of $\mathcal{R}$,

(5.11)
$\Bigl{|}\Bigl{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{R}\biggl{(}\sum_{k\leq
0}a_{k}\biggr{)}(\mathbf{g})>1\Bigr{\\}}\Bigr{|}\lesssim\Bigl{\|}\sum_{k\leq
1}a_{k}\Bigr{\|}_{\mathsf{L}^{2}(\mathbf{G})}^{2}\lesssim F_{\Phi}(f).$

To handle $\mathcal{R}(\sum_{k\geq 1}a_{k})$, we consider the $a_{k}$ in more
detail. The atom $a_{k}$ is supported in $U^{\dagger}_{k}$, where
$|U^{\dagger}_{k}|\lesssim F_{\Phi}(2^{-k}f)$, and we may write
$a_{k}=\sum_{R\in\mathscr{M}(U^{*}_{k})}a_{k,R}$.

Again, for all $R={Q_{1}}\times{Q_{2}}\in\mathscr{M}(U^{*}_{k})$, let
${\tilde{Q}_{1}}$ be the biggest pseudodyadic cube containing ${Q_{1}}$ such
that ${\tilde{Q}_{1}}\times{Q_{2}}\subset U^{**}_{k}$, where $U^{**}_{k}$ is
defined in (5.1). Next, let ${\tilde{Q}_{2}}$ be the biggest pseudodyadic cube
containing ${Q_{2}}$ such that ${\tilde{Q}_{1}}\times{\tilde{Q}_{2}}\subseteq
U^{***}_{k}$, where $U^{***}_{k}$ is defined in (5.2). Now let $R^{\dagger}$
be $100\beta_{k}({\tilde{Q}_{1}}\times{\tilde{Q}_{2}})$, where
$\beta_{k}=2^{k/(2\nu_{1}+2\nu_{2})}$. By Lemmas 3.5 and 3.1 and Theorem 4.4,

$\displaystyle\biggl{|}\bigcup_{R\in\mathscr{M}(U^{*}_{k})}R^{\dagger}\biggr{|}$
$\displaystyle\lesssim 2^{k/2}|U_{k}^{***}|\lesssim
2^{k/2}|U_{k}^{**}|\lesssim 2^{k/2}|U_{k}^{*}|\lesssim 2^{k/2}|U_{k}|\lesssim
2^{k/2}F_{\Phi}(2^{-k}f).$

As in the proof of Theorem 1.1, to prove that

(5.12)
$\bigg{|}\bigg{\\{}\mathbf{g}\in\mathbf{G}\colon\mathcal{R}\biggl{(}\sum_{k\geq
1}a_{k}\biggr{)}(\mathbf{g})>1\bigg{\\}}\bigg{|}\lesssim F_{\Phi}(f),$

it suffices to show that, for some $\delta\in\mathbb{R}_{+}$,

(5.13)
$\sum_{R\in\mathscr{M}(U^{*}_{k})}\int_{(R^{\dagger})^{c}}|\mathcal{R}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}\lesssim
2^{-\delta k}F_{\Phi}(f)$

for every $k\geq 1$. If (5.13) holds, then (5.11) and (5.12) imply (5.10). We
now prove (5.13).

$\displaystyle\int_{(R^{\dagger})^{c}}|\mathcal{R}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}$
$\displaystyle\leq\iint_{g_{1}\notin
100\beta_{k}{\tilde{Q}_{1}}}|\mathcal{R}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}+\iint_{g_{2}\notin
100\beta_{k}{\tilde{Q}_{2}}}|\mathcal{R}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}$
$\displaystyle=:\mathrm{I}_{1}+\mathrm{I}_{2},$

say, where $\beta_{k}=2^{{k/(2\nu_{1}+2\nu_{2})}}$. Since the estimates of
$\mathrm{I}_{1}$ and $\mathrm{I}_{2}$ are symmetric, we only estimate
$\mathrm{I}_{1}$. Note that

$\displaystyle\mathrm{I}_{1}$ $\displaystyle=\int_{g_{1}\notin
100\beta_{k}{\tilde{Q}_{1}}}\int_{100{Q_{2}}}|\mathcal{R}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}+\int_{g_{1}\notin
100\beta_{k}{\tilde{Q}_{1}}}\int_{(100{Q_{2}})^{c}}|\mathcal{R}(a_{k,R})(\mathbf{g})|\,\mathrm{d}\mathbf{g}$
$\displaystyle=:\mathrm{I}_{11}+\mathrm{I}_{12},$

say. By Hölder’s inequality and the $\mathsf{L}^{2}(G_{2})$-boundedness of
$\mathcal{R}^{[2]}_{j_{2}}$,

$\displaystyle\mathrm{I}_{11}$
$\displaystyle\lesssim|{\tilde{Q}_{2}}|^{1/2}\int_{g_{1}\notin
100\beta_{k}{\tilde{Q}_{1}}}\biggl{(}\int_{Q_{2}}\bigl{(}\mathcal{R}^{[1]}_{j_{1}}(a_{k,R})(g_{1},g_{2})\bigr{)}^{2}\,\mathrm{d}g_{2}\biggr{)}^{1/2}\,\mathrm{d}g_{1}.$

By the cancellation condition on $a_{k,R}(\cdot,g_{2})$ and the smoothness of
$\mathcal{R}^{[1]}_{j_{1}}$,

$\displaystyle\left|\mathcal{R}^{[1]}_{j_{1}}(a_{k,R})(\mathbf{g})\right|$
$\displaystyle=\left|\int_{{G_{1}}}\bigl{(}\mathcal{R}^{[1]}_{j_{1}}(g_{1},h_{1})-\mathcal{R}^{[1]}_{j_{1}}(g_{1},z({Q_{1}}))\bigr{)}a_{k,R}(h_{1},g_{2})\,\mathrm{d}h_{1}\right|$
$\displaystyle\lesssim\frac{\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\rho_{1}(g_{1},z({Q_{1}}))^{\nu_{1}+\varepsilon}}\int_{{G_{1}}}a_{k,R}(h_{1},g_{2})\,\mathrm{d}h_{1}$
$\displaystyle\lesssim\frac{\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\rho_{1}(g_{1},z({Q_{1}}))^{\nu_{1}+\varepsilon}}|{Q_{1}}|^{1/2}\biggl{(}\int_{{G_{1}}}|a_{k,R}(h_{1},g_{2})|^{2}\,\mathrm{d}h_{1}\biggr{)}^{1/2},$

where $z({Q_{1}})$ is the center of ${Q_{1}}$. Hence

$\displaystyle\mathrm{I}_{11}\lesssim\beta_{k}^{-\varepsilon}\gamma_{1}(R)^{-\varepsilon}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}.$

To estimate $\mathrm{I}_{12}$, we use the cancellation of $a_{k,R}$ to write
$\mathcal{R}(a_{k,R})(\mathbf{g})$ as

$\displaystyle\int_{\beta_{k}R}\bigl{(}\mathcal{R}^{[1]}_{j_{1}}(g_{1},h_{1})-\mathcal{R}^{[1]}_{j_{1}}(g_{1},z({Q_{1}}))\bigr{)}\bigl{(}\mathcal{R}^{[2]}_{j_{2}}(g_{2},h_{2})-\mathcal{R}^{[2]}_{j_{2}}(g_{2},z({Q_{2}}))\bigr{)}a_{k,R}(h_{1},h_{2})\,\mathrm{d}h_{1}\,\mathrm{d}h_{2},$

where $z({Q_{2}})$ is the center of ${Q_{2}}$. Using the smoothness of
$\mathcal{R}^{[1]}_{j_{1}}$ and $\mathcal{R}^{[2]}_{j_{2}}$, we deduce that

$\displaystyle\mathrm{I}_{12}$ $\displaystyle\lesssim\int_{g_{1}\notin
100b{\tilde{Q}_{1}}}\int_{(100{Q_{2}})^{c}}\frac{\operatorname{\ell}({Q_{1}})^{\varepsilon}}{\rho_{1}(g_{1},z({Q_{1}}))^{\nu_{1}+\varepsilon}}\frac{\operatorname{\ell}({Q_{2}})^{\varepsilon}}{\rho_{2}(g_{2},z({Q_{2}}))^{\nu_{2}+\varepsilon}}\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}$
$\displaystyle\lesssim
b^{-\varepsilon}\gamma_{1}(R)^{-\varepsilon}\left|R\right|^{1/2}\left\|a_{k,R}\right\|_{\mathsf{L}^{2}(\mathbf{G})}.$

Then, by continuing as in Section 5.1, we establish (5.13) and complete the
proof. ∎

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