$\tilde{a}_{S}=\Delta_{(1)}^{M}\Delta_{(2)}^{N}\tilde{b}_{S}$ and
$\operatorname{supp}\tilde{b}_{S}\subseteq S^{*}$. Further, for any sign
function $\sigma:\mathfrak{M}(\Omega_{1})\to\\{\pm 1\\}$ we may define a sign
function $\tilde{\sigma}:\mathfrak{M}_{1}\to\\{\pm 1\\}$ so that
$\tilde{\sigma}_{R}=\sigma_{R^{\uparrow}}$, and then

$\sum_{S\in\mathfrak{M}(\Omega_{1})}\sigma_{S}\left(\sum_{R:R^{\uparrow}=S}a_{R}\ast_{(1)}\varphi_{r}\right)=\sum_{R\in\mathfrak{M}_{1}}\tilde{\sigma}_{R}a_{R}\ast_{(1)}\varphi_{r},$

whence

$\displaystyle\left\|\sum_{S\in\mathfrak{M}(\Omega_{1})}\sigma_{S}\tilde{a}_{S}\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq\left\|\sum_{R\in\mathfrak{M}_{1}}\tilde{\sigma}_{R}a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq\eta_{1}(r)\left|\Omega\right|^{-1/2}.$

It follows that $a_{1}$ is $(|\Omega_{1}|/|\Omega|)^{1/2}$ multiplied by an
atom associated to $\Omega_{1}$.

We deal with $a_{2}$ by grouping the rectangles in $\mathfrak{M}_{2}$
according to their widths, that is, $q$. By definition, if
$R\in\mathfrak{M}_{2}$, then $q\leq r\leq(q^{2}+h)^{1/2}$. Further, since
$a_{R}\ast_{(1)}\varphi_{r}=\Delta_{(2)}^{N}b_{R}\ast_{(1)}\Delta_{(1)}^{M}\varphi_{r}$,
we have

$\displaystyle\left\|a_{R}\ast_{(1)}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq\left\|\Delta_{(2)}^{N}b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left\|\Delta_{(1)}^{M}\varphi_{r}\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim_{\varphi,N}\frac{q^{2M}}{r^{2M}}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})},$
(3.4)

by Lemma 2.23, and $\operatorname{supp}(b_{R}\ast_{(1)}\varphi_{r})\subseteq
R^{*}\cdot B^{(1)}(o,r)\subseteq g\cdot B^{(1)}(2\kappa r)\cdot
B^{(2)}(\kappa^{2}h)$. For $j\in\mathbb{Z}$ such that $(2\nu+1)^{j}\leq r$, we
write $\mathfrak{M}_{2,j}$ for the family of all $R\in\mathfrak{M}_{2}$ for
which $q=(2\nu+1)^{j}$. The rectangles in $\mathfrak{M}_{2,j}$ are pairwise
disjoint because they are maximal. This disjointness implies that the number
of overlapping supports $R^{*}\cdot B^{(1)}(o,r)$ where
$R\in\mathfrak{M}_{2,j}$ that contain a given point $g\in\mathbb{H}^{\nu}$,
which is the number of these rectangles $R$ that meet $B^{(1)}(g,r)$, is
bounded by a constant multiple of $(r/q)^{2\nu}$, and so for any sign sequence
$\sigma:\mathfrak{M}_{2,j}\to\\{\pm 1\\}$,

$\displaystyle\left\|\sum_{R\in\mathfrak{M}_{2,j}}\sigma_{R}a_{R}\ast_{(1)}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\lesssim\left(\frac{r}{q}\right)^{\nu}\left(\sum_{R\in\mathfrak{M}_{2,j}}\left\|a_{R}\ast_{(1)}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\right)^{1/2}$
(3.5)
$\displaystyle\lesssim_{\varphi,N}\left(\frac{q}{r}\right)^{2M-\nu}\left(\sum_{R\in\mathfrak{M}_{2,j}}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\right)^{1/2}$
$\displaystyle\eqsim\left(\frac{q}{r}\right)^{2M-\nu}\left\|\sum_{R\in\mathfrak{M}_{2,j}}\tilde{\sigma}_{R}a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\leq\left(\frac{q}{r}\right)^{2M-\nu}\eta_{2}(r)\left|\Omega\right|^{-1/2},$

by (3.4), the almost disjointness of the supports of the $a_{R}$ in
consideration, and the fact that $a$ is an atom.

Finally, for $R\in\mathfrak{M}_{2,j}$, we choose $R^{+}$ to be the smallest
adapted rectangle such that $R\subseteq R^{+}$ and $R^{*}\cdot
B^{(1)}(o,r)\subseteq(R^{+})^{*}$. For this, it is sufficient that
$\operatorname{width}(R^{+})\geq r/(\kappa-1)$, and so $|R^{+}|\lesssim
r^{2\nu}h\lesssim(r/q)^{2\nu}|R|$. Define
$\Omega_{2,j}=\bigcup_{R\in\mathfrak{M}_{2,j}}R^{+}$; then

$|\Omega_{2,j}|\leq\sum_{R\in\mathfrak{M}_{2,j}}\left|R^{+}\right|\lesssim\left(\frac{r}{q}\right)^{2\nu}\sum_{R\in\mathfrak{M}_{2,j}}\left|R\right|\leq\left(\frac{r}{q}\right)^{2\nu}\left|\Omega\right|.$
(3.6)

We write $a_{2,j}=\sum_{S\in\mathfrak{R}(\Omega_{2,j})}\tilde{a}_{S}$, where
$\tilde{a}_{S}=\sum_{R\in\mathfrak{M}_{2,j}:R^{+}=S}a_{R}$. By Lemma 3.3,
(3.5) and (3.6), for any sign sequence
$\sigma:\mathfrak{R}(\Omega_{2,j})\to\\{\pm 1\\}$,

$\left\|\sum_{S\in\mathfrak{R}(\Omega_{2,j})}\sigma_{S}\tilde{a}_{S}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim_{\varphi,N}\left(\frac{q}{r}\right)^{2M-2\nu}\eta_{2}(r)\left|\Omega_{2,j}\right|^{-1/2},$

and so $a_{2,j}$ is a multiple of $(q/r)^{2M-2\nu}\eta_{2}(r)$ times an atom
associated with $\Omega_{2,j}$; since $q=(2\nu+1)^{j}$ and
$\sum_{j}((2\nu+1)^{j}/r)^{2M-2\nu}$ is bounded when we sum over those
integers $j$ for which $(2\nu+1)^{j}\leq r$, we have controlled $a_{2}$.

In the third case, where $(q^{2}+h)^{1/2}\leq r$, we observe that
$\operatorname{supp}(b\ast_{(1)}\varphi_{r})\subseteq g\cdot B^{(1)}(3r)$, and

$\displaystyle\left\|a\ast_{(1)}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle=\left\|\Delta_{(1)}^{M}\Delta_{(2)}^{N}b\ast_{(1)}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}=\left\|b\ast_{(1)}\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
(3.7)
$\displaystyle\leq\left\|b\right\|_{L^{2}(\mathbb{H}^{\nu})}\left\|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r}\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim
q^{2M}(q^{2}+h)^{2N}\left\|a\right\|_{L^{2}(\mathbb{H}^{\nu})}r^{-2M-4N}$
$\displaystyle=\frac{q^{2M}(q^{2}+h)^{2N}}{r^{2M+4N}}\left\|a\right\|_{L^{2}(\mathbb{H}^{\nu})},$

again by Lemma 2.23. A very similar argument to that used for $a_{2}$ deals
with $a_{3}$; a double sum is needed to manage the widths and the heights of
the rectangles and (3.7) replaces (3.4).

Finally, it is evident that $a\ast_{(1)}\varphi_{r}\to a$ in
$L^{2}(\mathbb{H}^{\nu})$ as $r\to 0+$. To show that
$a\ast_{(1)}\varphi_{r}\to a$ in $H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$
as $r\to 0+$, we consider

$\sum_{R\in\mathfrak{M}_{1}}(a_{R}-a_{R}\ast_{(1)}\varphi_{r})+\sum_{R\in\mathfrak{M}_{2}}a_{R}-\sum_{R\in\mathfrak{M}_{2}}a_{R}\ast_{(1)}\varphi_{r}+\sum_{R\in\mathfrak{M}_{3}}a_{R}-\sum_{R\in\mathfrak{M}_{3}}a_{R}\ast_{(1)}\varphi_{r}.$

The second and the fourth sum tend to $0$ as $r\to 0+$ because of the
unconditional convergence in (A2); the third and fifth sums tend to $0$ as
$r\to 0+$ because $\eta_{2}(r)\to 0$ and $\eta_{3}(r)\to 0$ as $r\to 0+$,
again because of the unconditional convergence. We may show that the first sum
tends to $0$ as $r\to 0+$ by considering the supports of $b_{R}$ and
$b_{R}\ast_{(1)}\varphi_{r}$ to be the rectangle $R^{+}$ and observing that
$\left\|a_{R}-a_{R}\ast_{(1)}\varphi_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}\to
0$ as $r\to 0+$, and summing these using unconditional convergence. We leave
the details of this verification to the reader. ∎

###  3.3 Boundedness on particles and atoms

######  Proposition 3.10.

Suppose that $\mathcal{A}$ is a linear operator, or a nonnegative sublinear
operator, that satisfies a strong type $(2,2)$ bound
$\left\|\mathcal{A}f\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq
C_{1}\left\|f\right\|_{L^{2}(\mathbb{H}^{\nu})}$ or a weak-type $(2,2)$ bound

$\displaystyle\left|{\left\\{g\in\mathbb{H}^{\nu}:{\left|\mathcal{A}f(g)\right|}>\lambda\right\\}}\right|\leq\frac{C_{2}}{\lambda^{2}}\left\|f\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\qquad\forall\lambda\in\mathbb{R}^{+}$

for all $f\in L^{2}(\mathbb{H}^{\nu})$. Let $M,N\in\mathbb{Z}^{+}$.

(a) Suppose also that there exist $C,\delta_{1},\delta_{2}\in\mathbb{R}^{+}$
such that

$\int_{(S^{*})^{c}}\left|\mathcal{A}a_{R}(g)\right|\,dg\leq
C\rho_{\delta_{1},\delta_{2}}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
(3.8)

for all $S\in\mathfrak{R}$ such that $R\subseteq S\in\mathfrak{R}$ and all
particles $a_{R}\in L^{2}(\mathbb{H}^{\nu})$ such that
$a_{R}=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}$ where $b_{R}\in
L^{2}(\mathbb{H}^{\nu})$ and $\operatorname{supp}b_{R}\subseteq R^{*}$ and

$\displaystyle\rho_{\delta_{1},\delta_{2}}(R,S):=\left(\frac{\operatorname{height}(R)}{\operatorname{height}(S)}\right)^{\delta_{1}}+\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{\delta_{2}}.$
(3.9)

Then there is a constant $C_{0}$ such that

$\displaystyle\left\|\mathcal{A}a\right\|_{L^{1}(\mathbb{H}^{\nu})}\leq C_{0}$
(3.10)

for all $(1,2,M,N)$-atoms $a$.

(b) Suppose also that (3.10) holds. Then $\mathcal{A}$ maps
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu})$ into
$L^{1}(\mathbb{H}^{\nu})$, and

$\left\|\mathcal{A}f\right\|_{L^{1}(\mathbb{H}^{\nu})}\leq
C_{0}\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu}).$
(3.11)

Hence $\mathcal{A}$ may be extended uniquely by continuity to a bounded
operator from $H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to
$L^{1}(\mathbb{H}^{\nu})$ that satisfies the same inequality for all $f\in
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$.

###### Proof.

Take a $(1,2,M,N,\kappa)$-atom $a$ associated to an open set $\Omega$ of
finite measure, and write $a\sim\sum_{R\in\mathfrak{M}(\Omega)}a_{R}$ as in
Definition 3.1. Define the set $\Upsilon$ by

$\Upsilon=\\{g\in\mathbb{H}^{\nu}:\mathcal{M}_{s}\mathbbm{1}_{\Omega}\geq
1/100^{D}\\}.$

If $R$ is any maximal adapted subrectangle of $\Omega$, and $g\in S^{*}=100R$,
then
${1}/{100^{D}}\leq\mathcal{M}_{s}\mathbbm{1}_{R}(g)\leq\mathcal{M}_{s}\mathbbm{1}_{\Omega}(g)$,
so that $S^{*}\subseteq\Upsilon$. As the flag maximal function is $L^{2}$
bounded, Chebyshev’s inequality shows that

$\left|\Upsilon\right|\lesssim\frac{\left\|\mathbbm{1}_{\Omega}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}}{100^{2D}}\lesssim|\Omega|\,,$

where the implicit constants are independent of $\Omega$. Now

$\left\|\mathcal{A}(a)\right\|_{L^{1}(\mathbb{H}^{\nu})}=\int_{\Upsilon}|\mathcal{A}(a)(g)|\,dg+\int_{\Upsilon^{c}}|\mathcal{A}(a)(g)|\,dg.$

From the estimate
$\left\|a\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq|\Omega|^{-1/2}$, the $L^{2}$
boundedness of $\mathcal{A}$, and Hölder’s inequality

$\displaystyle\int_{\Upsilon}|\mathcal{A}(a)(g)|\,dg$
$\displaystyle\leq|\Upsilon|^{1/2}\left\|\mathcal{A}(a)\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim|\Omega|^{1/2}\left\|a\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq
1,$

where the implicit constant is geometric. Hence, to prove (3.10), it suffices
to prove that there exists a constant $C$ such that
$\int_{\Upsilon^{c}}|\mathcal{A}(a)(g)|\,dg\leq C$. We write $a$ as
$\sum_{R\in\mathfrak{M}(\Omega)}a_{R}$, and then
$\mathcal{A}a=\sum_{R\in\mathfrak{M}(\Omega)}\mathcal{A}a_{R}$, where the sum
converges in $L^{2}(\mathbb{H}^{\nu})$ by $L^{2}$-boundedness, so

$\displaystyle\int_{\Upsilon^{c}}|\mathcal{A}(a)(g)|\,dg$
$\displaystyle\leq\sum_{R\in\mathfrak{M}(\Omega)}\int_{\Upsilon^{c}}|\mathcal{A}(a_{R})(g)|\,dg\leq\sum_{R\in\mathfrak{M}(\Omega)}\int_{(R^{\dagger*})^{c}}\left|\mathcal{A}(a_{R})(g)\right|\,dg.$

By (3.8), the Cauchy–Schwarz inequality and Journé’s lemma (Lemma 2.13), it
follows that

$\displaystyle\left\|\mathcal{A}(a)\right\|_{L^{1}(\mathbb{H}^{\nu})}$
$\displaystyle\lesssim\left(\sum_{R\in\mathfrak{M}(\Omega)}\rho_{\delta_{1},\delta_{2}}(R,S)^{2}|R|\right)^{1/2}\left(\sum_{R\in\mathfrak{M}(\Omega)}\left\|a_{R}\right\|^{2}_{L^{2}(\mathbb{H}^{\nu})}\right)^{1/2}\lesssim|\Omega|^{1/2}|\Omega|^{-1/2}=1,$

as required.

To prove (b), it will suffice to prove (3.11). Take $f\in
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu})$ and
$\varepsilon\in\mathbb{R}^{+}$. Then we may write
$f\sim\sum_{j}\lambda_{j}a_{j}$, where the $a_{j}$ are atoms and
$\sum_{j}|\lambda_{j}|<\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}+\varepsilon$.
Take a smooth nonnegative-real-valued radial function $\varphi$ with support
in $B^{(1)}(o,1)$ such that $\int_{\mathbb{H}^{\nu}}\varphi_{r}(g)\,dg=1$, and
set $\varphi_{r}(\cdot)=r^{-D}\varphi(\delta_{1/r}\cdot)$, as in Lemma 3.9.

Now $f=\lim_{r\to 0+}f\ast_{(1)}\varphi_{r}$ in $L^{2}(\mathbb{H}^{\nu})$ and
so $\mathcal{A}f=\lim_{r\to 0+}\mathcal{A}(f\ast_{(1)}\varphi_{r})$ in
$L^{2}(\mathbb{H}^{\nu})$. Further, the sum $\sum_{j}\lambda_{j}a_{j}$
converges in $L^{1}(\mathbb{H}^{\nu})$, and so the sum
$\sum_{j}\lambda_{j}a_{j}\ast_{(1)}\varphi_{r}$ converges in
$L^{2}(\mathbb{H}^{\nu})$, and

$\displaystyle\left|\mathcal{A}f\right|$ $\displaystyle=\left|\lim_{r\to
0+}\lim_{J\to\infty}\mathcal{A}\left(\sum_{j=1}^{J}\lambda_{j}a_{j}\ast_{(1)}\varphi_{r}\right)\right|$
$\displaystyle\leq\lim_{r\to
0+}\lim_{J\to\infty}\sum_{j=1}^{J}\left|\lambda_{j}\mathcal{A}(a_{j}\ast_{(1)}\varphi_{r})\right|$
$\displaystyle=\lim_{r\to
0+}\sum_{j=1}^{\infty}\left|\lambda_{j}\right|\left|\mathcal{A}(a_{j}\ast_{(1)}\varphi_{r})\right|.$

By Lemma 3.9, $a_{j}\ast_{(1)}\varphi_{r}\sim\sum_{k}c_{j,k,r}a_{j,k,r}$ where
the sum converges in $L^{2}(\mathbb{H}^{\nu})$; further,
$\sum_{k}\left|c_{j,k,r}\right|\leq C$ for all $r\in\mathbb{R}^{+}$; and
$\lim_{r\to 0+}\sum_{j,k\in\mathbb{Z}}\left|c_{j,k,r}\right|=1$. Then
$\left|\mathcal{A}(a_{j}\ast_{(1)}\varphi_{r})\right|\leq\sum_{k}\left|c_{j,k,r}\right|\left|\mathcal{A}a_{j,k,r}\right|$,
so

$\displaystyle\left|\mathcal{A}f\right|$ $\displaystyle\leq\lim_{r\to
0+}\sum_{j=1}^{\infty}\left|\lambda_{j}\right|\sum_{k}\left|c_{j,k,r}\right|\left|\mathcal{A}a_{j,k,r}\right|\leq\lim_{r\to
0+}\sum_{j,k\in\mathbb{Z}}\left|\lambda_{j}\right|\left|c_{j,k,r}\right|\left|\mathcal{A}a_{j,k,r}\right|$

whence

$\left\|\mathcal{A}f\right\|_{L^{1}(\mathbb{H}^{\nu})}\leq\lim_{r\to
0+}C_{0}\sum_{j,k\in\mathbb{Z}}\left|\lambda_{j}\right|\left|c_{j,k,r}\right|\leq
C_{0}(\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}+\varepsilon)$

by dominated convergence. Since $\varepsilon$ is arbitrary, (3.11) holds. ∎

###  3.4 The inclusion $H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$

In this section, we consider the Lusin–Littlewood–Paley operator
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}$, and show that it is
bounded on the atomic Hardy space. We first recall the definition of this
operator. We suppose that the integers $M$ and $N$ satisfy $M>D/4$ and $N\geq
1$, and consider a pair $\boldsymbol{\tilde{\varphi}}$ of Poisson-bounded
functions, as in Definition 2.20. We define
$\varphi^{(1)}=\Delta_{(1)}\tilde{\varphi}^{(1)}$ and
$\varphi^{(2)}=\Delta_{(2)}\tilde{\varphi}^{(2)}$, and take
$\varphi^{(1)}_{r}$ to be $r^{-D}\varphi^{(1)}(\delta_{1/r}\cdot)$ and
$\varphi^{(2)}_{s}$ to be $s^{-1}\varphi^{(2)}(\cdot/s)$. Further, we define
$\chi^{(1)}_{r}$ to be the $L^{1}(\mathbb{H}^{\nu})$-normalised characteristic
function of the ball $B^{(1)}(o,r)$ and $\chi^{(2)}_{s}$ to be the
$L^{1}(\mathbb{R})$-normalised characteristic function of the ball
$B^{(2)}(0,s)$.

######  Definition.

Suppose that $\varphi^{(1)}$, $\varphi^{(2)}$, $\chi^{(1)}$ and $\chi^{(2)}$
are as above. For $f\in L^{1}(\mathbb{H}^{\nu})$, we define the
Lusin–Littlewood–Paley area function
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)$ associated to
$\varphi^{(1)}$ and $\varphi^{(2)}$ by

$\displaystyle\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)(g):=\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}\bigr{|}^{2}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}$

for all $g\in\mathbb{H}^{\nu}$, and we define the Hardy space
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$, usually
abbreviated to $H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$, to be the set of
all $f\in L^{1}(\mathbb{H}^{\nu})$ for which
$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$,
with norm

$\left\|f\right\|_{H^{1}_{F,\mathrm{area},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})}:=\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$

######  Theorem 3.11.

Suppose that $M>D/4$, $N\geq 1$ and $\kappa>1$, and that the sublinear
operator $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}$ and the space
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$ are as
defined above. Then
$H^{1}_{F,\mathrm{atom},M,N,\kappa}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area},\boldsymbol{\varphi}}$, and

$\left\|f\right\|_{H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})}\lesssim_{M,N,\boldsymbol{\varphi}}\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu}).$

###### Proof.

The proof involves two steps: first we suppose that $\varphi^{(1)}$ and
$\varphi^{(2)}$ are supported in the unit balls in $\mathbb{H}^{\nu}$ and in
$\mathbb{R}$, and show that there exists
$\delta_{1},\delta_{2}\in\mathbb{R}^{+}$ such that, for all particles $a_{R}$
for which $b_{R}$ is supported in $R^{*}$,

$\displaystyle\int_{(S^{*})^{c}}|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(a_{R})(g)|\,dg\lesssim_{\delta_{1},\delta_{2},\boldsymbol{\varphi}}\rho_{\delta_{1},\delta_{2}}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})},$

where $S$ is any adapted rectangle that contains $R$; this step is carried out
in Proposition 3.12. Proposition 3.10 then proves the theorem for such
$\varphi^{(1)}$ and $\varphi^{(2)}$. The second step is to show that the
result holds for more general functions $\varphi^{(1)}$ and $\varphi^{(2)}$;
this step is carried out in Proposition 3.14. ∎

For technical reasons, and to prove more general results, we consider a more
general operator.

######  Definition.

Suppose that $\varphi^{(1)}$ and $\varphi^{(2)}$ are as described above, that
$\chi^{(1)}$ and $\chi^{(2)}$ are the $L^{1}$-normalised characteristic
functions of the unit balls in $\mathbb{H}^{\nu}$ and $\mathbb{R}$, and that
$\beta,\gamma\in[0,1]$. For $f\in L^{1}(\mathbb{H}^{\nu})$, we define the
generalised area function
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)$

$\displaystyle\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)(g):=\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}\bigr{|}^{2}\ast_{(1)}\chi^{(1)}_{\beta
r}\ast_{(2)}\chi^{(2)}_{\gamma
s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}$ (3.12)

for all $g\in\mathbb{H}^{\nu}$, where we interpret $\chi^{(1)}_{0}$ and
$\chi^{(2)}_{0}$ as Dirac delta distributions on $\mathbb{H}^{\nu}$ and
$\mathbb{R}$.

######  Proposition 3.12.

Suppose that $M>D/4$ and $N\geq 1$; that $\tilde{\varphi}^{(1)}\in
C^{2M+2}(\mathbb{H}^{\nu})$ is radial and supported in $B^{(1)}(o,1)$ and
$\tilde{\varphi}^{(2)}\in C^{2N+2}(\mathbb{R})$ is even and supported in
$B^{(2)}(0,1)$; and that $\beta,\gamma\in[0,1]$. Then

$\displaystyle\int_{(S^{*})^{c}}\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(a_{R})(g)\,dg\leq
C(M,N,\nu,\boldsymbol{\varphi})\rho_{{3/2},2M-{D/2}}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
(3.13)

for all particles $a_{R}$ such that $\operatorname{supp}(b_{R})\subseteq
R^{*}$ and all $S\in\mathfrak{R}$ such that $R\subseteq S$ with
$\rho_{{3/2},2M-{D/2}}(R,S)$ defined as in (3.9). Here,

$C(M,N,\nu,\boldsymbol{\varphi})\lesssim_{M,N,\nu}\left\|\tilde{\varphi}^{(1)}\right\|_{C^{2M+2}(\mathbb{H}^{\nu})}\left\|\tilde{\varphi}^{(2)}\right\|_{C^{2M+2}(\mathbb{R})}.$

###### Proof.

We begin with a summary of the argument: given a particle $a_{R}$ such that
$a_{R}=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}$, where
$\operatorname{supp}(b_{R})\subseteq R^{*}$, we first translate and dilate
$a_{R}$ so that $R^{*}\subseteq T(o,1,h)$, where $h\geq 1$, and
$S^{*}=T(o,r^{*},h^{*})$; next we simplify what has to be proved by splitting
the sublinear operator
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}$ into five
pieces, which we treat separately; for several of these, we decompose
$(S^{*})^{c}$ into annular regions and bound the $L^{1}$ norm of the part of
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(a_{R})$ under
consideration in each of the regions using the Cauchy–Schwarz inequality. We
now provide the details. In this proof, we write $\varphi_{r,s}$ for
$\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}$ and $\chi_{r,s}$ for
$\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}$.

The estimate (3.13) is invariant under translations and normalised dilations,
so there is no loss of generality in supposing that $R\subseteq
R^{*}=T(o,1,h),$ where $h\geq 1$; then $|R|\eqsim|R^{*}|\eqsim h$. Moreover,
we may suppose that $S^{*}=T(o,r^{*},h^{*}),$ where $r^{*}\geq 1$ and
$h^{*}\geq h$. We observe that

$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(a_{R})\eqsim\sum_{j=1}^{5}\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{j}(a_{R}),$

where

$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{j}(a_{R})(g)=\biggl{(}\iint_{T^{j}}\bigl{|}a_{R}\ast_{(1)}\varphi_{r,s}\bigr{|}^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2},$

and the regions $T^{j}$, which partition $\mathbb{R}^{+}\times\mathbb{R}^{+}$
into five, are defined by

$\displaystyle T^{1}$ $\displaystyle=\\{(r,s):0<r\leq 1,0<s\leq h\\},\quad
T^{2}=\\{(r,s):0<r\leq 1,h<s<\infty\\}$ $\displaystyle T^{3}$
$\displaystyle=\\{(r,s):1<r\leq h^{1/2},r^{2}<s\leq h\\},\quad
T^{4}=\\{(r,s):h^{1/2}<r<\infty,0<s\leq r^{2}\\}$ $\displaystyle T^{5}$
$\displaystyle=\\{(r,s):1<r<\infty,\max\\{r^{2},h\\}<s<\infty\\}.$

The key to our estimation of these terms is the observation that

$\displaystyle\operatorname{supp}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}\subseteq T(o,1,h)\cdot T(o,r,s)\cdot T(o,\beta r,\gamma
s)\subseteq T(o,2r+1,2s+h).$ (3.14)

First, (3.14) implies that the support of
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}^{1}(a_{R})$ is so small
that it does not meet $(S^{*})^{c}$ and so no estimation is needed;
nevertheless, we note that

$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{1}(a_{R})\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$

by the Cauchy–Schwarz inequality and the $L^{2}$-boundedness of the
Lusin–Littlewood–Paley area operator.

Second, to treat
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{2}(a_{R})$,
that is, to estimate

$\int_{(S^{*})^{c}}\biggl{(}\iint_{[h,\infty)\times(0,1]}\bigl{|}a_{R}\ast_{(1)}\varphi_{r,s}\bigr{|}^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg,$

we define $V_{k}=T(o,3,3\times 2^{k}h)$ when $k\in\mathbb{N}$; then
$\left|V_{k}\right|\eqsim 2^{k}h$, and
$\operatorname{supp}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}\subseteq V_{k}$ when $s\leq 2^{k}h$, from (3.14). Moreover,

$\displaystyle a_{R}\ast_{(1)}\varphi_{r,s}$
$\displaystyle=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}=\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\Delta_{(2)}^{N}(\varphi^{(2)}_{s})$
$\displaystyle=s^{-2N}\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}(\Delta_{(2)}^{N}\varphi^{(2)})_{s}.$

From these considerations,

$\displaystyle\int_{\mathbb{H}^{\nu}\setminus
V_{0}}\biggl{(}\iint_{[h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad=\sum_{k\in\mathbb{Z}^{+}}\int_{V_{k}\setminus
V_{{k-1}}}\biggl{(}\iint_{[h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad=\sum_{k\in\mathbb{Z}^{+}}\int_{V_{k}\setminus
V_{{k-1}}}\biggl{(}\iint_{[2^{k-1}h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\leq\sum_{k\in\mathbb{Z}^{+}}|{V_{k}\setminus
V_{{k-1}}}|^{1/2}\left(\int_{{V_{k}\setminus
V_{{k-1}}}}\iint_{[2^{k-1}h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}\left(\iiint_{\mathbb{H}^{\nu}\times[2^{k-1}h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad=\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}\left(\iint_{[2^{k-1}h,\infty)\times(0,1]}\left\|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}$
$\displaystyle\qquad=\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}\left(\iint_{[2^{k-1}h,\infty)\times(0,1]}s^{-4N}\left\|\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}(\Delta_{(2)}^{N}\varphi^{(2)})_{s}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{1}(\mathbb{R})}\left(\iint_{[2^{k-1}h,\infty)\times(0,1]}s^{-4N}\left\|\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}$
$\displaystyle\qquad=\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{1}(\mathbb{R})}\left(\int_{[2^{k-1}h,\infty)}s^{-4N}\,\frac{ds}{s}\right)^{1/2}\left(\int_{(0,1]}\left\|\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{dr}{r}\right)^{1/2}$
$\displaystyle\qquad\lesssim_{\boldsymbol{\varphi},N}\sum_{k\in\mathbb{Z}^{+}}\frac{|V_{k}|^{1/2}}{2^{2Nk}h^{2N}}\left\|\Delta_{(1)}^{M}b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim_{N}\sum_{k\in\mathbb{Z}^{+}}\frac{2^{k/2}h^{1/2}}{2^{2kN}}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}$
$\displaystyle\qquad\eqsim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2};$

we need $N\geq 1$ to ensure convergence of the sum. Since
$S^{*}\cap\operatorname{supp}\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{2}(a_{R})\subseteq
V_{k}$ when $k\geq k^{*}:=\lceil\log_{2}(((r^{*})^{2}+h^{*}-9)/3h)\rceil$, it
follows that

$\displaystyle\int_{(S^{*})^{c}}\biggl{(}\iint_{[h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\leq\sum_{k=k^{*}-1}^{\infty}\int_{V_{k}\setminus
V_{k-1}}\biggl{(}\iint_{[h,\infty)\times(0,1]}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\lesssim_{\boldsymbol{\varphi},N}\sum_{k=k^{*}-1}^{\infty}\frac{2^{k/2}h^{1/2}}{2^{2kN}}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}$
$\displaystyle\qquad\lesssim_{N}\left(\frac{h}{(r^{*})^{2}+h^{*}-9}\right)^{2N-1/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\eqsim_{N}\left(\frac{\operatorname{height}(R)}{\operatorname{height}(S)}\right)^{2N-1/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Third, we deal with
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{3}(a_{R})$,
that is, with

$\biggl{(}\int_{1}^{h^{1/2}}\int_{0}^{h}\bigl{|}a_{R}\ast_{(1)}\varphi_{r,s}\bigr{|}^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g)\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}.$

We define $U_{j}$ to be the set

$\\{(z,t)\in\mathbb{H}^{\nu}:|z|\leq 3\times 2^{j},|t|\leq 12h\\}$

when $j\in\mathbb{N}$; then $\left|U_{j}\right|\lesssim 2^{2\nu j}h$ for all
$j$. By (3.14),
$\operatorname{supp}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}\subseteq U_{j}$ if $1\leq r<2^{j}$ and $0<s\leq h$. Further,

$\displaystyle a_{R}\ast_{(1)}\varphi_{r,s}$
$\displaystyle=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}=\Delta_{(2)}^{N}b_{R}\ast_{(1)}\Delta_{(1)}^{M}(\varphi^{(1)}_{r})\ast_{(2)}\varphi^{(2)}_{s}$
$\displaystyle=r^{-2M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(2)}\varphi^{(2)}_{s}.$

Matters are now arranged such that

$\displaystyle\int_{\mathbb{H}^{\nu}\setminus
U_{0}}\biggl{(}\int_{1}^{h^{1/2}}\int_{0}^{h}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad=\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}\int_{U_{j}\setminus
U_{j-1}}\biggl{(}\int_{2^{j-1}}^{h^{1/2}}\int_{0}^{h}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\leq\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}|U_{j}\setminus
U_{j-1}|^{1/2}\left(\int_{U_{j}\setminus
U_{j-1}}\int_{2^{j-1}}^{h^{1/2}}\int_{0}^{h}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{\mathbb{H}^{\nu}}\int_{2^{j-1}}^{\infty}\int_{0}^{h}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad=\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{2^{j-1}}^{\infty}\int_{0}^{h}\left\|\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(2)}\varphi^{(2)}_{s}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{ds}{s}\,\frac{dr}{r^{4M+1}}\right)^{1/2}$
$\displaystyle\qquad\lesssim_{\boldsymbol{\varphi}}\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{2^{j-1}}^{\infty}\left\|\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{dr}{r^{4M+1}}\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{2^{j-1}}^{\infty}\left\|\Delta_{(2)}^{N}b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\left\|(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\right\|_{L^{1}(\mathbb{H}^{\nu})}^{2}\,\frac{dr}{r^{4M+1}}\right)^{1/2}$
$\displaystyle\qquad\lesssim_{M}\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left\|\Delta_{(2)}^{N}b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left\|\Delta_{(1)}^{M}\varphi^{(1)}\right\|_{L^{1}(\mathbb{H}^{\nu})}2^{-2jM}$
$\displaystyle\qquad\lesssim_{\boldsymbol{\varphi}}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\sum_{j\in\mathbb{Z}^{+}}2^{jn}|R|^{1/2}2^{-2jM}$
$\displaystyle\qquad\simeq_{M}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})};$

we need $M>\nu/2$ to ensure convergence of the sum. Since
$S^{*}\cap\operatorname{supp}\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{3}(a_{R})\subseteq
U_{j}$ when $j\geq j^{*}:=\lfloor\log_{2}(r^{*}/3)\rfloor$, it follows that

$\displaystyle\int_{(S^{*})^{c}}\biggl{(}\int_{1}^{h^{1/2}}\int_{0}^{h}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\leq\sum_{j=j^{*}}^{\infty}|U_{j}|^{1/2}\left(\int_{\mathbb{H}^{\nu}}\int_{2^{j-1}}^{\infty}\int_{0}^{h}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad\simeq_{M,\boldsymbol{\varphi}}\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{2M-\nu}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Fourth, to treat
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{4}(a_{R})$,
that is, to estimate

$\int_{(S^{*})^{c}}\biggl{(}\int_{1}^{\infty}\int_{0}^{r^{2}}\bigl{|}a_{R}\ast_{(1)}\varphi_{r,s}\bigr{|}^{2}\ast_{(1)}\chi_{r,s}(g)\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg,$

we define $U_{j}=T(o,2^{j+1}+1,2^{2j+1}+h)$ when $j\in\mathbb{N}$; then
$\left|U_{j}\right|\lesssim 2^{Dj}+2^{2\nu j}h$.

By (3.14), if $r<2^{j}$ and $s\leq r^{2}$, then
$\operatorname{supp}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}\subseteq U_{j}$. Further, by hypothesis,
$\varphi^{(2)}=\Delta_{(2)}\tilde{\varphi}^{(2)}$, whence

$\displaystyle a_{R}\ast_{(1)}\varphi_{r,s}$
$\displaystyle=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}(\Delta_{(2)}\tilde{\varphi}^{(2)})_{s}$
$\displaystyle=s^{2}\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\Delta_{(2)}(\tilde{\varphi}^{(2)}_{s})$
$\displaystyle=s^{2}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\Delta_{(1)}^{M}\Delta_{(2)}(\varphi^{(1)}_{r})\ast_{(2)}\tilde{\varphi}^{(2)}_{s}$
$\displaystyle=r^{-2M-4}s^{2}\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\Delta_{(2)}\varphi^{(1)})_{r}\ast_{(2)}\tilde{\varphi}^{(2)}_{s}$

so that
$\left\|a_{R}\ast_{(1)}\varphi_{r,s}\right\|_{L^{2}(\mathbb{H}^{\nu})}$ is
bounded by

$\displaystyle
r^{-2M-4}s^{2}\left\|\Delta_{(2)}^{N}b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left\|\Delta_{(1)}^{M}\varphi^{(1)}\right\|_{L^{1}(\mathbb{H}^{\nu})}\left\|\tilde{\varphi}^{(2)}\right\|_{L^{1}(\mathbb{R})}\lesssim_{\boldsymbol{\varphi},M}r^{-2M-4}s^{2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Matters are now arranged such that

$\displaystyle\int_{\mathbb{H}^{\nu}\setminus
U_{0}}\biggl{(}\int_{1}^{\infty}\int_{0}^{r^{2}}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad=\sum_{j\in\mathbb{Z}^{+}}\int_{U_{j}\setminus
U_{j-1}}\biggl{(}\int_{2^{j-1}}^{\infty}\int_{0}^{r^{2}}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}\setminus
U_{j-1}|^{1/2}\left(\int_{U_{j}\setminus
U_{j-1}}\int_{2^{j-1}}^{\infty}\int_{0}^{r^{2}}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{\mathbb{H}^{\nu}}\int_{2^{j-1}}^{\infty}\int_{0}^{r^{2}}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad=\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{2^{j-1}}^{\infty}\int_{0}^{r^{2}}\left\|a_{R}\ast_{(1)}\varphi_{r,s}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{ds}{s}\,\frac{dr}{r}\right)^{1/2}$
$\displaystyle\qquad\lesssim_{\boldsymbol{\varphi}}\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}\left(\int_{2^{j-1}}^{\infty}\int_{0}^{r^{2}}r^{-8-4M}s^{4}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\,\frac{ds}{s}\,\frac{dr}{r}\right)^{1/2}$
$\displaystyle\qquad\simeq_{M}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\sum_{j\in\mathbb{Z}^{+}}(2^{Dj}+2^{2\nu
j}h)^{1/2}2^{-2Mj}$
$\displaystyle\qquad\eqsim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})};$

we need $M>D/4$ to ensure that the sum converges. Since
$S^{*}\cap\operatorname{supp}\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{4}(a_{R})\subseteq
U_{j}$ when $j\geq j^{*}$, where $j^{*}$ is the smallest integer $j$ such that
$r^{*}\leq 2^{j+1}+1$ and $(r^{*})^{2}+h^{*}\leq(2^{j+1}+1)^{2}+2^{2j+1}+h$,
it follows that

$\displaystyle\int_{(S^{*})^{c}}\biggl{(}\int_{1}^{\infty}\int_{0}^{r^{2}}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}(g^{\prime})\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\,dg^{\prime}\lesssim\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{2M-D/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Fifth, to estimate the term
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{5}(a_{R})$,
that is,

$\displaystyle\biggl{(}\int_{1}^{\infty}\int_{\max\\{r^{2},h\\}}^{\infty}\bigl{|}a_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}\bigr{|}^{2}\ast_{(1)}\chi^{(1)}_{\beta
r}\ast_{(2)}\chi^{(2)}_{\gamma
s}(g)\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2},$

we use a simple pointwise estimate that follows from (3.14), the normalisation
of $\chi_{\beta r,\gamma s}$, the commutativity of convolution on radial
functions and distributions, and the Cauchy–Schwarz inequality, namely,

$\displaystyle\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}$
$\displaystyle\leq\left\|\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,2r+1,2s+h)}$ (3.15)
$\displaystyle\leq\left\|\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,2r+1,2s+h)}$
$\displaystyle=\left\|b_{R}\ast_{(1)}\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}^{2}\mathbbm{1}_{T(o,2r+1,2s+h)}$
$\displaystyle\leq\left\|b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\left\|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\mathbbm{1}_{T(o,2r+1,2s+h)}.$

Further, we see that

$\displaystyle\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}(z,t)$
$\displaystyle=\Delta_{(1)}^{M}(\varphi^{(1)}_{r})\ast_{(1)}\Delta_{(2)}^{N}(\varphi^{(2)}_{s})(z,t)=r^{-2M}s^{-2N}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(1)}(\Delta_{(2)}^{N}\varphi^{(2)})_{s}(z,t)$
$\displaystyle=r^{-2M}s^{-2N}\int_{\mathbb{R}}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}(z,t^{\prime})(\Delta_{(2)}^{N}\varphi^{(2)})_{s}(t-t^{\prime})\,dt^{\prime};$

the integrand vanishes off an interval of length $O(r^{2})$ because
$(\varphi^{(1)})_{r}$ is supported in $B^{(1)}(o,r)$, so

$\displaystyle\left|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}(z,t)\right|$
$\displaystyle\lesssim
r^{2-2M}s^{-2N}\left\|(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\left\|(\Delta_{(2)}^{N}\varphi^{(2)})_{s}\right\|_{L^{\infty}(\mathbb{R})}$
$\displaystyle=\left\|\Delta_{(1)}^{M}\varphi^{(1)}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}r^{2-2M-D}s^{-2N-1};$

since moreover $\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}$ is supported in
$T(o,r,s)$, we deduce that

$\left\|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim_{\boldsymbol{\varphi},M,N}r^{2-2M-D}s^{-2N-1}\left|T(o,r,s)\right|^{1/2}\lesssim
r^{-2M-\nu}s^{-2N-1/2}.$ (3.16)

Moreover, from Corollary 3.2,
$\left\|b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim
h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$. Combining this with the
estimates (3.15) and (3.16), we deduce that

$\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta r,\gamma
s}\lesssim
r^{-4M-2\nu}s^{-4N-1}h^{4N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}\mathbbm{1}_{T(o,2r+1,2s+h)}.$
(3.17)

When $r\geq 1$ and $s\geq\max\\{r^{2},h\\}$, and $(z,t)\in T(o,2r+1,2s+h)$,

$|z|\leq 2r+1\leq 3r\qquad\text{and}\qquad|t|\leq 2s+h+(2r+1)^{2}\leq 12s.$
(3.18)

It follows that

$\displaystyle\biggl{(}\int_{1}^{\infty}\int_{\max\\{r^{2},h\\}}^{\infty}\bigl{|}a_{R}\ast_{(1)}\varphi_{r,s}\bigr{|}^{2}\ast_{(1)}\chi_{\beta
r,\gamma s}(z,t)\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}$
$\displaystyle\qquad\leq\left(\int_{1}^{\infty}\int_{\max\\{r^{2},h\\}}^{\infty}\left\|\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta
r,\gamma
s}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,2r+1,2s+h)}(z,t)\,\frac{ds}{s}\,\frac{dr}{r}\right)^{1/2}$
$\displaystyle\qquad\lesssim_{\varphi,M,N}h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left(\int_{\max\\{1,|z|/3\\}}^{\infty}\int_{\max\\{r^{2},h,|t|/12\\}}^{\infty}\frac{1}{r^{4M+2\nu}}\frac{1}{s^{4N+1}}\,\frac{ds}{s}\,\frac{dr}{r}\right)^{1/2}$
$\displaystyle\qquad=h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left(\frac{1}{4N+1}\int_{\max\\{1,|z|/3\\}}^{\infty}\frac{1}{\max\\{r^{2},h,|t|/12\\}^{4N+1}}\frac{1}{r^{4M+2\nu}}\,\frac{dr}{r}\right)^{1/2}$
$\displaystyle\qquad\lesssim_{M,N}h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\frac{1}{\max\\{1,|z|/3\\}^{2M+\nu}\max\\{1,|z|/3,h,|t|/12\\}^{2N+1/2}}$
$\displaystyle\qquad\lesssim_{M,N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\frac{h^{2N}}{\max\\{1,|z|/3\\}^{2M+\nu}\max\\{1,|z|/3,h,|t|/12\\}^{{2N+1/2}}}$

for all $(z,t)\in\mathbb{H}^{\nu}$. It follows immediately that
$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{5}(a_{R})\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
provided that $M>\nu/2$ and $N\geq 1$. Further,

$\displaystyle\int_{(S^{*})^{c}}\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}^{5}(a_{R})(g)\,dg$
$\displaystyle\qquad\lesssim_{M,N,\varphi}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\int_{\mathbb{C}^{\nu}}\frac{1}{\max\\{1,|z|/3\\}^{2M+\nu}}\,dz\int_{|t|\geq
h^{*}}\frac{h^{2N}}{(|t|/12)^{{2N+1/2}}}\,dt$
$\displaystyle\qquad\qquad+\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\int_{|z|\geq
r^{*}}\frac{1}{\max\\{1,|z|/3\\}^{2M+\nu}}\,dz\int_{\mathbb{R}}\frac{h^{2N}}{\max\\{h,|t|/12\\}^{{2N+1/2}}}\,dt$
$\displaystyle\qquad\lesssim_{M,N}\left(\frac{\operatorname{height}(R)}{\operatorname{height}(S)}\right)^{2N-1/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}+\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{2M-\nu}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

The quotients that precede $|R|^{1/2}$ in our estimates are dominated by
$\rho_{2N-1/2,2M-{D/2}}(R,S)$, as defined in (3.9). Further, examination of
the argument shows that the implicit constants depend on
$\boldsymbol{\varphi}$ through products such as
$\left\|\varphi^{(1)}\right\|_{L^{1}(\mathbb{H}^{\nu})}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}$,
all of which may be controlled by the product
$\left\|\tilde{\varphi}^{(1)}\right\|_{C^{2M+2}(\mathbb{H}^{\nu})}\left\|\tilde{\varphi}^{(2)}\right\|_{C^{2N+2}(\mathbb{R})}$
because of the support restriction. ∎

######  Corollary 3.13.

Let $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}$ be as in
Proposition 3.12. Then
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)\in
L^{1}(\mathbb{H}^{\nu})$, and

$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},1,1}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}$

for all $f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap
L^{2}(\mathbb{H}^{\nu})$. Hence
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}$ extends
uniquely by density and continuity to a mapping from
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to
$H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$.

###### Proof.

This follows from Proposition 3.10. ∎

We now consider what happens when the functions $\varphi^{(1)}$ and
$\varphi^{(2)}$ in the definition of the Lusin–Littlewood–Paley area function
do not have compact support. As in the statement of Theorem 3.11, we consider
a Poisson-bounded pair of functions $\tilde{\varphi}^{(1)}$ on
$\mathbb{H}^{\nu}$ and $\tilde{\varphi}^{(2)}$ on $\mathbb{R}$, that is,
functions such that

$\displaystyle\left|\mathcal{D}\tilde{\varphi}^{(1,r)}(g)\right|$
$\displaystyle\lesssim(\left\|g\right\|+1)^{-D-1-m}\qquad\forall
g\in\mathbb{H}^{\nu}$

for all differential operators $\mathcal{D}$ that are products of $m$ vector
fields, each chosen from $X_{1},\dots,X_{2\nu}$, and

$\displaystyle\left|\mathcal{T}^{n}\tilde{\varphi}^{(2,s)}(t)\right|$
$\displaystyle\lesssim(|t|+1)^{-2-n}\qquad\forall t\in\mathbb{R}.$

We define $\varphi^{(1)}=\Delta_{(1)}\tilde{\varphi}^{(1)}$ and
$\varphi^{(2)}=\Delta_{(2)}\tilde{\varphi}^{(2)}$. As before, we write
$\varphi_{r,s}=\varphi^{(1)}_{r}\ast_{(1)}\varphi^{(2)}_{s}$, where
$\varphi^{(1)}_{r}$ and $\varphi^{(2)}_{s}$ are $L^{1}$-normalised dilates,
and define $\chi_{r,s}$ similarly.

######  Proposition 3.14.

Suppose that the functions $\varphi^{(1)}$ and $\varphi^{(2)}$ are as defined
above and $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)$
is defined by (3.12). Then
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},1,1}(f)\in
L^{1}(\mathbb{H}^{\nu})$, and

$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},1,1}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu}).$

###### Proof.

It is easy to construct smooth functions $\eta^{(k)}$ on $\mathbb{R}$, where
$k\in\mathbb{N}$, such that $\sum_{k\in\mathbb{N}}\eta^{(k)}=1$,
$\eta^{(0)}(t)=0$ unless $|t|\leq 1$, $\eta^{(1)}(t)=0$ unless $1/2\leq|t|\leq
2$, and $\eta^{(k)}=\eta^{(1)}(\cdot/2^{k-1})$ when $k\geq 2$. Take an even
function $\varphi^{(2)}$ on $\mathbb{R}$. Then we may write

$\varphi^{(2)}=\sum_{k\in\mathbb{N}}\varphi^{(2)}\eta^{(k)}=\sum_{k\in\mathbb{N}}(\varphi^{(2,k)})_{2^{k}},$

where $\varphi^{(2,k)}$ is the normalised dilate
$(\varphi^{(2)}\eta^{(k)})_{2^{-k}}$, which is supported in $B^{(2)}(0,1)$.
Then

$\displaystyle\sum_{k\in\mathbb{N}}\left\|\varphi^{(2,k)}\right\|_{C^{2N+2}(\mathbb{R})}$
$\displaystyle=\left\|(\varphi^{(2)}\eta^{(0)})\right\|_{C^{2N+2}(\mathbb{R})}+\sum_{k\in\mathbb{Z}^{+}}2^{2Nk}\left\|(\varphi^{(2)}\eta^{(k)})\right\|_{C^{2N+2}(\mathbb{R})}<\infty,$

since we may write each derivative $\mathcal{T}^{m}(\varphi^{(2)}\eta^{(k)})$
as a finite weighted sum of terms
$\mathcal{T}^{n}\varphi^{(2)}\mathcal{T}^{m-n}\eta^{(k)}$, and

$\left|\mathcal{T}^{n}\varphi^{(2)}(t)\mathcal{T}^{m-n}\eta^{(k)}(t)\right|\lesssim(1+|t|)^{-2-n}2^{-(m-n)k}\mathbbm{1}_{\operatorname{supp}(\eta^{(k)})}(t)\lesssim
2^{-2k-mk}$

for all $t\in\mathbb{R}$. Similarly, given a Poisson-bounded function
$\varphi^{(1)}$, we may write
$\varphi^{(1)}=\sum_{j\in\mathbb{N}}(\varphi^{(1,j)})_{2^{j}},$ where
$\varphi^{(1,j)}$ is supported in $B^{(1)}(o,1)$ and
$\sum_{j\in\mathbb{N}}\left\|\varphi^{(1,j)}\right\|_{C^{2M+2}(\mathbb{H}^{\nu})}<\infty.$
To do this, one uses cut-off functions that depend on a smooth norm on
$\mathbb{H}^{\nu}$, such as the Korányi norm, and homogeneous dilations.

Suppose that $\beta,\gamma\geq 1$. By a change of variables and the previous
proposition,

$\displaystyle\int_{\mathbb{H}^{\nu}}\left(\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{\beta
r,\gamma
s}\right|^{2}\ast_{(1)}\chi_{r,s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}\,dg$
(3.19)
$\displaystyle\qquad=\int_{\mathbb{H}^{\nu}}\left(\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}\right|^{2}\ast_{(1)}\chi_{\beta^{-1}r,\gamma^{-1}s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}\,dg$
$\displaystyle\qquad\lesssim_{M,N}\left\|\varphi^{(1)}\right\|_{C^{2M+2}(\mathbb{H}^{\nu})}\left\|\varphi^{(2)}\right\|_{C^{2N+2}(\mathbb{R})}\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}.$

We conclude by taking functions $\varphi^{(1)}$ and $\varphi^{(2)}$ as in the
statement of the proposition, decomposing them as sums
$\sum_{j\in\mathbb{N}}(\varphi^{(1,j)})_{2^{j}}$ and
$\sum_{k\in\mathbb{N}}(\varphi^{(2,k)})_{2^{k}}$, and applying the estimate
(3.19):

$\displaystyle\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},1,1}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad=\int_{\mathbb{H}^{\nu}}\left(\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|\sum_{j,k\in\mathbb{N}}f\ast_{(1)}(\varphi^{(1,j)})_{2^{j}r}\ast_{(2)}(\varphi^{(2,k)})_{2^{k}s}\right|^{2}\ast_{(1)}\chi_{r,s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}\,dg$
$\displaystyle\qquad\leq\sum_{j,k\in\mathbb{N}}\int_{\mathbb{H}^{\nu}}\left(\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}(\varphi^{(1,j)})_{2^{j}r}\ast_{(2)}(\varphi^{(2,k)})_{2^{k}s}\right|^{2}\ast_{(1)}\chi_{r,s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}\,dg$
$\displaystyle\qquad\lesssim_{M,N}\sum_{j,k\in\mathbb{N}}\left\|\varphi^{(1,j)}\right\|_{C^{2M+2}(\mathbb{H}^{\nu})}\left\|\varphi^{(2,k)}\right\|_{C^{2N+2}(\mathbb{R})}\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})},$

as required. ∎

For the next corollary, recall that the letters $h$ and $p$ are used for the
heat and Poisson kernels.

######  Corollary 3.15.

Suppose that $\varphi^{(1)}$ and $\varphi^{(2)}$ are either
$\Delta_{(1)}h^{(1)}_{1}$ and $\Delta_{(2)}h^{(2)}_{1}$, or
$\Delta_{(1)}p^{(1)}_{1}$ and $\Delta_{(2)}p^{(2)}_{1}$ on $\mathbb{H}^{\nu}$
and on $\mathbb{R}$, and that
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)$ is defined by (4.2).
Then $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\in
L^{1}(\mathbb{H}^{\nu})$, and

$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}\leq\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu}).$

Hence $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}$ extends
continuously to a unique mapping from
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to
$H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$.

###### Proof.

This result follows from the proposition above and our estimates for the heat
and Poisson kernels in Section 2.5. ∎

###  3.5 The inclusion $H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})$

In this section, we consider the continuous square function Hardy space. We
begin by recalling the definition.

######  Definition.

Suppose that $\varphi^{(1)}$ and $\varphi^{(2)}$ are as at the beginning of
Section 3.4. For $f\in L^{1}(\mathbb{H}^{\nu})$, we define the continuous
Littlewood–Paley square function
$\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)$ associated to
$\varphi^{(1)}$ and $\varphi^{(2)}$ by

$\displaystyle\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)(g):=\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g)\bigr{|}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}$

for all $g\in\mathbb{H}^{\nu}$. We define the continuous square function Hardy
space $H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$, often
abbreviated to $H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})$, to be the set of
all $f\in L^{1}(\mathbb{H}^{\nu})$ for which
$\left\|\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$,
with norms

$\displaystyle\left\|f\right\|_{H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})}$
$\displaystyle:=\left\|\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$

######  Theorem 3.16.

Suppose that the pair $\boldsymbol{\varphi}$ of radial functions
$\varphi^{(1)}$ and $\varphi^{(2)}$ on $\mathbb{H}^{\nu}$ and $\mathbb{R}$ is
Poisson-bounded, as in Definition 2.20. Then the operator
$\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}$ is bounded from
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to $L^{1}(\mathbb{H}^{\nu})$.

###### Proof.

This result follows from Proposition 3.12, with the parameters $\beta$ and
$\gamma$ there taken to be $0$, combined with an almost identical version of
the proof of Proposition 3.14. ∎

###  3.6 The inclusion $H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$

In this section, we consider the discrete square function Hardy space, whose
definition we now recall.

######  Definition.

Suppose that $\varphi^{(1)}$ and $\varphi^{(2)}$ are as at the beginning of
Section 3.4. For $f\in L^{1}(\mathbb{H}^{\nu})$, we define the discrete
Littlewood–Paley square function
$\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)$ associated to
$\varphi^{(1)}$ and $\varphi^{(2)}$ by

$\displaystyle\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)(g):=\biggl{(}\sum_{(m,n)\in\mathbb{Z}\times\mathbb{Z}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{2^{m}}\ast_{(2)}\varphi^{(2)}_{2^{n}}(g)\bigr{|}^{2}\biggr{)}^{1/2}$

for all $g\in\mathbb{H}^{\nu}$. We define the discrete square function Hardy
space $H^{1}_{F,\mathrm{dis},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$, often
abbreviated to $H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$, to be the set of
all $f\in L^{1}(\mathbb{H}^{\nu})$ for which
$\left\|\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$,
with norm

$\displaystyle\left\|f\right\|_{H^{1}_{F,\mathrm{dis},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})}$
$\displaystyle:=\left\|\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$

By replacing integrals by sums in the arguments of the previous section, we
may show that the discrete operator
$\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}$ is bounded from
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to $L^{1}(\mathbb{H}^{\nu})$.

###  3.7 The inclusion $H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})$

We recall the key points of Definition 2.20. A family $F(\mathbb{H}^{\nu})$ of
pairs of functions $(\varphi^{(1)},\varphi^{(2)})$, where
$\varphi^{(1)}:\mathbb{H}^{\nu}\to\mathbb{C}$ and
$\varphi^{(2)}:\mathbb{R}\to\mathbb{C}$, is said to be _Poisson-bounded_ if
there are constants $C_{m}$ and $C_{n}$ for all $m\in\mathbb{N}$ and all
$n\in\mathbb{N}$ such that

$\displaystyle\left|\mathcal{D}\varphi^{(1)}(g)\right|$
$\displaystyle\leq\frac{C_{m}}{(1+\left\|g\right\|^{2})^{2m+D/2+1/2}}\qquad\forall
g\in\mathbb{H}^{\nu}$

for all differential operators $\mathcal{D}$ that are products of $m$ vector
fields, each chosen from $\mathcal{X}_{1},\dots,\mathcal{X}_{2\nu}$, and

$\displaystyle\left|\mathcal{T}^{e}\varphi^{(2)}(t)\right|$
$\displaystyle\leq\frac{C_{n}}{(1+\left\|t\right\|^{2})^{n+1}}\qquad\forall
t\in\mathbb{R}$

for all pairs $(\varphi^{(1)},\varphi^{(2)})\in F(\mathbb{H}^{\nu})$. As
usual, we write $\varphi_{r,s}$ for the convolution
$\varphi^{(1)}_{r}\ast_{(1)}\varphi^{(2)}_{s}$, where $\varphi^{(1)}_{r}$ and
$\varphi^{(2)}_{s}$ denote the normalised dilates of $\varphi^{(1)}$ and
$\varphi^{(2)}$.

Given such a Poisson-bounded family, the _grand maximal function
$\mathcal{M}_{F,\mathrm{gmax}}(f)$ of $f\in L^{1}(\mathbb{H}^{\nu})$_ is
defined by

$\displaystyle\mathcal{M}_{F,\mathrm{gmax}}(f)(g)=\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r,s\in\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(g)\right|\qquad\forall
g\in\mathbb{H}^{\nu}.$

######  Definition 3.17.

The space $H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})$ is defined to be the set
of all $f\in L^{1}(\mathbb{H}^{\nu})$ such that
$\mathcal{M}_{F,\mathrm{gmax}}(f)\in L^{1}(\mathbb{H}^{\nu})$, with norm

$\left\|f\right\|_{H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})}:=\left\|\mathcal{M}_{F,\mathrm{gmax}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$
(3.20)

We do not show the dependence of this space on the choice of family
$F(\mathbb{H}^{\nu})$, though evidently this is important.

######  Theorem 3.18.

Suppose that $M>D/4$, $N\geq 1$ and $\kappa>1$. Then
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})$, and

$\left\|f\right\|_{H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})}\lesssim_{M,N}\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu}).$

###### Proof.

The proof involves two steps: first we suppose that $\varphi^{(1)}$ and
$\varphi^{(2)}$ have compact support and show that there exists
$\delta\in\mathbb{R}^{+}$ such that, for all particles $a_{R}$ for which
$b_{R}$ is supported in $R^{*}$,

$\displaystyle\int_{(S^{*})^{c}}|\mathcal{M}_{F,\mathrm{gmax}}(a_{R})(g)|\,dg\lesssim\rho_{\delta_{1},\delta_{2}}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})},$

where $S$ is any adapted rectangle that contains $R$; this step is carried out
in Proposition 3.12. Proposition 3.10 then proves the theorem for such
$\varphi^{(1)}$ and $\varphi^{(2)}$. The second step is to show that the
result holds for more general functions $\varphi^{(1)}$ and $\varphi^{(2)}$;
this step is carried out in Proposition 3.14. ∎

######  Proposition 3.19.

Suppose that $M>D/4$ and $N\geq 1$, that $\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})$ and that $\varphi^{(1)}$ and $\varphi^{(2)}$ are
supported in the balls $B^{(1)}(o,1)$ and $B^{(2)}(0,1)$. Then

$\displaystyle\int_{(S^{*})^{c}}\mathcal{M}_{F,\mathrm{gmax}}(a_{R})(g)\,dg\lesssim_{M,N}\rho_{{3/2},2M-D/2}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
(3.21)

for all particles $a_{R}$ such that $\operatorname{supp}(b_{R})\subseteq
R^{*}$ and all $S\in\mathfrak{R}$ such that $R\subseteq S$. Here,
$\rho_{{3/2},2M-{D/2}}(R,S)$, as defined in (3.9).

###### Proof.

We begin with a summary of the argument: given a particle $a_{R}$ such that
$a_{R}=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}$, where
$\operatorname{supp}(b_{R})\subseteq R^{*}$, we first translate and dilate
$a_{R}$ so that $R^{*}\subseteq T(o,1,h)$, where $h\geq 1$, and
$S^{*}=T(o,r^{*},h^{*})$; next we simplify what has to be proved by splitting
the sublinear operator $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}$
into five pieces, which we treat separately; for several of these, we
decompose $(S^{*})^{c}$ into annular regions and bound the $L^{1}$ norm of the
part of $\mathcal{M}_{F,\mathrm{gmax}}(a_{R})$ under consideration in each of
the regions using the Cauchy–Schwarz inequality. We now provide the details.
Once again, we write $\varphi_{r,s}$ for
$\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}$ and $\chi_{r,s}$ for
$\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}$.

The estimate (3.13) is invariant under translations and normalised dilations,
so there is no loss of generality in supposing that $R\subseteq
R^{*}=T(o,1,h);$ then $|R|\eqsim|R^{*}|\eqsim h$. Moreover, we may suppose
that $S^{*}=T(o,r^{*},h^{*}),$ where $r^{*}\geq 1$ and $h^{*}\geq h$. We
observe that

$\mathcal{M}_{F,\mathrm{gmax}}(a_{R})\eqsim\max_{j=1,\dots,5}\mathcal{M}_{F,\mathrm{gmax}}^{j}(a_{R}),$

where

$\mathcal{M}_{F,\mathrm{gmax}}^{j}(a_{R})(g)=\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{(r,s)\in
T^{j}}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|,$

and the regions $T^{j}$, which partition $\mathbb{R}^{+}\times\mathbb{R}^{+}$
into five, are defined by

$\displaystyle T^{1}$ $\displaystyle=\\{(r,s):0<r\leq 1,0<s\leq h\\},\quad
T^{2}=\\{(r,s):0<r\leq 1,h<s<\infty\\}$ $\displaystyle T^{3}$
$\displaystyle=\\{(r,s):1<r\leq h^{1/2},r^{2}<s\leq h\\},\quad
T^{4}=\\{(r,s):h^{1/2}<r<\infty,0<s\leq r^{2}\\}$ $\displaystyle T^{5}$
$\displaystyle=\\{(r,s):1<r<\infty,\max\\{r^{2},h\\}<s<\infty\\}.$

The key to our estimation of these terms is the observation that

$\displaystyle\operatorname{supp}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|\subseteq
T(o,1,h)\cdot T(o,r,s)=T(o,r+1,s+h).$ (3.22)

First, the support of $\mathcal{M}_{F,\mathrm{gmax}}^{1}(a_{R})$ is so small
that it does not meet $(S^{*})^{c}$ and so no estimation is needed;
nevertheless, we note that
$\left\|\mathcal{M}_{F,\mathrm{gmax}}^{1}(a_{R})\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
by the Cauchy–Schwarz inequality and the $L^{2}$-boundedness of
$\mathcal{M}_{F,\mathrm{gmax}}^{1}$, which follows from its control by the
flag maximal function $\mathcal{M}_{F}$ (as in Definition 2.7).

Second, to treat $\mathcal{M}_{F,\mathrm{gmax}}^{2}(a_{R})$, that is, to
estimate

$\displaystyle\int_{(S^{*})^{c}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(0,1]}\sup_{s\in[h,\infty)}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg,$
(3.23)

we define $V_{k}=T(o,2,2\times 2^{k}h)$ when $k\in\mathbb{N}$; then
$\left|V_{k}\right|\eqsim 2^{k}h$, and
$\operatorname{supp}\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|\subseteq V_{k}$
when $s\leq 2^{k}h$, from (3.22). Moreover,

$\displaystyle a_{R}\ast_{(1)}\varphi_{r,s}$
$\displaystyle=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}=\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\Delta_{(2)}^{N}(\varphi^{(2)}_{s})$
$\displaystyle=s^{-2N}\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}(\Delta_{(2)}^{N}\varphi^{(2)})_{s}.$

From these considerations,

$\displaystyle\int_{\mathbb{H}^{\nu}\setminus
V_{0}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(0,1]}\sup_{s\in[h,\infty)}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad=\sum_{k\in\mathbb{Z}^{+}}\int_{V_{k}\setminus
V_{{k-1}}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(0,1]}\sup_{s\in[2^{k-1}h,\infty)}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad\leq\sum_{k\in\mathbb{Z}^{+}}|{V_{k}\setminus
V_{{k-1}}}|^{1/2}\left(\int_{{V_{k}\setminus
V_{{k-1}}}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(0,1]}\sup_{s\in[2^{k-1}h,\infty)}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|^{2}\,dg^{\prime}\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}(2^{k-1}h)^{-2N}$
$\displaystyle\qquad\qquad\qquad\left(\int_{\mathbb{H}^{\nu}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(0,1]}\sup_{s\in[2^{k-1}h,\infty)}\left|\Delta_{(1)}^{M}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}(\Delta_{(2)}^{N}\varphi^{(2)})_{s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}(2^{k-1}h)^{-2N}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\left(\left\|\varphi^{(1)}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}\right)$
$\displaystyle\qquad\qquad\qquad\left(\int_{\mathbb{H}^{\nu}}\sup_{r\in\mathbb{R}^{+}}\sup_{s\in\mathbb{R}^{+}}\left|\Delta_{(1)}^{M}b_{R}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\lesssim_{F}\sum_{k\in\mathbb{Z}^{+}}|V_{k}|^{1/2}(2^{k-1}h)^{-2N}\left\|\mathcal{M}_{s}(\Delta_{(1)}^{M}b_{R}f)\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim_{N}h^{1/2}h^{-2N}\left\|\Delta_{(1)}^{M}b_{R}f\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2};$

we need $N\geq 1$ to ensure convergence of the sum. Since
$S^{*}\cap\operatorname{supp}\mathcal{M}_{F,\mathrm{gmax}}^{2}(a_{R})\subseteq
V_{k}$ when $k\geq k^{*}:=\lceil\log_{2}(((r^{*})^{2}+h^{*}-9)/3h)\rceil$, it
follows that the left-hand side of (3.23) is bounded by

$\displaystyle\sum_{k=k^{*}-1}^{\infty}\int_{V_{k}\setminus
V_{{k-1}}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(0,1]}\sup_{s\in[2^{k-1}h,\infty)}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad\lesssim_{F}\sum_{k=k^{*}-1}^{\infty}|V_{k}|^{1/2}(2^{k-1}h)^{-2N}\left\|\mathcal{M}_{s}(\Delta_{(1)}^{M}b_{R}f)\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim_{N}\left(\frac{h}{(r^{*})^{2}+h^{*}-9}\right)^{2N-1/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\eqsim_{N}\left(\frac{\operatorname{height}(R)}{\operatorname{height}(S)}\right)^{2N-1/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Third, we deal with $\mathcal{M}_{F,\mathrm{gmax}}^{3}(a_{R})$, that is, with

$\int_{(S^{*})^{c}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,h^{1/2}]}\sup_{s\in(0,h]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg.$

We define $U_{j}$ as $U_{j}:=\\{(z,t)\in\mathbb{H}^{\nu}:|z|\leq
2^{j+1},|t|\leq 8h\\}$ when $j\in\mathbb{N}$; then $\left|U_{j}\right|\lesssim
2^{2\nu j}h$ for all $j$. By (3.22),
$\operatorname{supp}(a_{R}\ast_{(1)}\varphi_{r,s})\subseteq U_{j}$ if $1\leq
r<2^{j}$ and $0<s\leq h$. Further,

$\displaystyle a_{R}\ast_{(1)}\varphi_{r,s}$
$\displaystyle=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}=\Delta_{(2)}^{N}b_{R}\ast_{(1)}\Delta_{(1)}^{M}(\varphi^{(1)}_{r})\ast_{(2)}\varphi^{(2)}_{s}$
$\displaystyle=r^{-2M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(2)}\varphi^{(2)}_{s}.$

Matters are now arranged such that

$\displaystyle\int_{\mathbb{H}^{\nu}\setminus
U_{0}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,h^{1/2}]}\sup_{s\in(0,h]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad=\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}\int_{U_{j}\setminus
U_{j-1}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(2^{j-1},h^{1/2}]}\sup_{s\in(0,h]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad\leq\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}|U_{j}\setminus
U_{j-1}|^{1/2}\left(\int_{U_{j}\setminus U_{j-1}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(2^{j-1},h^{1/2}]}\sup_{s\in(0,h]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}|U_{j}|^{1/2}2^{-2M(j-1)}$
$\displaystyle\qquad\qquad\qquad\left(\int_{U_{j}\setminus
U_{j-1}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(2^{j-1},h^{1/2}]}\sup_{s\in(0,h]}\left|\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(2)}\varphi^{(2)}_{s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}|U_{j}|^{1/2}2^{-2M(j-1)}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\left(\left\|\Delta_{(1)}^{M}\varphi^{(1)}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\left\|\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}\right)$
$\displaystyle\qquad\qquad\qquad\left(\int_{\mathbb{H}^{\nu}}\sup_{r\in\mathbb{R}^{+}}\sup_{s\in\mathbb{R}^{+}}\left|\Delta_{(2)}^{N}b_{R}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\lesssim_{F}\sum_{j=1}^{\lfloor\log_{2}(h)\rfloor+1}|U_{j}|^{1/2}2^{-2M(j-1)}\left\|\mathcal{M}_{s}(\Delta_{(2)}^{N}b_{R})\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim_{M}|h|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\simeq|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})};$

we need $M>\nu/2$ to ensure convergence of the sum. Since
$S^{*}\cap\operatorname{supp}\mathcal{M}_{F,\mathrm{gmax}}^{3}(a_{R})\subseteq
U_{j}$ when $j\geq j^{*}:=\lfloor\log_{2}(r^{*}/2)\rfloor$, it follows that

$\displaystyle\int_{(S^{*})^{c}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,h^{1/2}]}\sup_{s\in(0,h]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\lesssim_{F}\sum_{j=j^{*}}^{\infty}|U_{j}|^{1/2}2^{-2M(j-1)}\left\|\mathcal{M}_{s}(\Delta_{(2)}^{N}b_{R})\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\simeq_{M}\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{2M-\nu}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Fourth, to treat $\mathcal{M}_{F,\mathrm{gmax}}^{4}(a_{R})$, that is, to
estimate

$\int_{(S^{*})^{c}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,\infty]}\sup_{s\in(0,r^{2}]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg,$

we define $U_{j}=T(o,2^{j}+1,2^{2j}+h)$ when $j\in\mathbb{N}$; then
$\left|U_{j}\right|\lesssim 2^{Dj}+2^{2\nu j}h$.

By (3.22), if $r<2^{j}$ and $s\leq r^{2}$, then
$\operatorname{supp}(a_{R}\ast_{(1)}\varphi_{r,s})\subseteq U_{j}$. Further,
as before,

$\displaystyle a_{R}\ast_{(1)}\varphi_{r,s}$
$\displaystyle=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}=r^{-2M}\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(2)}\varphi^{(2)}_{s}.$

Matters are now arranged such that

$\displaystyle\int_{\mathbb{H}^{\nu}\setminus
U_{0}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,\infty]}\sup_{s\in(0,r^{2}]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad=\sum_{j\in\mathbb{Z}^{+}}\int_{U_{j}\setminus
U_{j-1}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(2^{j-1},\infty]}\sup_{s\in(0,r^{2}]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}\setminus
U_{j-1}|^{1/2}\left(\int_{U_{j}\setminus U_{j-1}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(2^{j-1},\infty]}\sup_{s\in(0,r^{2}]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}2^{-2M(j-1)}$
$\displaystyle\qquad\qquad\qquad\left(\int_{U_{j}\setminus
U_{j-1}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(2^{j-1},\infty]}\sup_{s\in(0,r^{2}]}\left|\Delta_{(2)}^{N}b_{R}\ast_{(1)}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(2)}\varphi^{(2)}_{s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\leq\sum_{j\in\mathbb{Z}^{+}}|U_{j}|^{1/2}2^{-2M(j-1)}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\left(\left\|\Delta_{(1)}^{M}\varphi^{(1)}\right\|_{L^{\infty})\mathbb{H}^{\nu})}\left\|\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}\right)$
$\displaystyle\qquad\qquad\qquad\left(\int_{\mathbb{H}^{\nu}}\sup_{r\in\mathbb{R}^{+}}\sup_{s\in\mathbb{R}^{+}}\left|\Delta_{(2)}^{N}b_{R}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\right|^{2}\,dg\right)^{1/2}$
$\displaystyle\qquad\lesssim_{F}\sum_{j=1}^{\infty}|U_{j}|^{1/2}2^{-2M(j-1)}\left\|\mathcal{M}_{s}(\Delta_{(2)}^{N}b_{R})\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\lesssim_{M}|h|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\simeq|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})};$

we need $M>D/4$ to ensure that the sum converges. Since
$S^{*}\cap\operatorname{supp}\mathcal{M}_{F,\mathrm{gmax}}^{4}(a_{R})\subseteq
U_{j}$ when $j\geq j^{*}$, where $j^{*}$ is the smallest integer $j$ such that
$r^{*}\leq 2^{j+1}+1$ and $(r^{*})^{2}+h^{*}\leq(2^{j+1}+1)^{2}+2^{2j+1}+h$,
it follows that

$\displaystyle\int_{(S^{*})^{c}}\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,\infty]}\sup_{s\in(0,r^{2}]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|\,dg\lesssim\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{2M-D/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Fifth, to estimate the term $\mathcal{M}_{F,\mathrm{gmax}}^{5}(a_{R})$, that
is,

$\displaystyle\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,\infty]}\sup_{s\in(\max\\{r^{2},h\\},\infty]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(g)\right|,$

we use the pointwise estimate

$\displaystyle\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|$
$\displaystyle\leq\left\|a_{R}\ast_{(1)}\varphi_{r,s}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,r+1,s+h)}=\left\|b_{R}\ast_{(1)}\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,r+1,s+h)}$
(3.24)
$\displaystyle\leq\left\|b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left\|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}\right\|_{L^{2}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,r+1,s+h)}.$

Arguing very much as in the proof of (3.17), we see that

$\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|\lesssim
C(M,N,\boldsymbol{\varphi})r^{-2M-\nu}s^{-2N-1/2}h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,r+1,s+h)}$

where

$C(M,N,\boldsymbol{\varphi})=\left\|\Delta_{(1)}^{M}\varphi^{(1)}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}.$

Further, we see that

$\displaystyle\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}(z,t)$
$\displaystyle=\Delta_{(1)}^{M}(\varphi^{(1)}_{r})\ast_{(1)}\Delta_{(2)}^{N}(\varphi^{(2)}_{s})(z,t)=r^{-2M}s^{-2N}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\ast_{(1)}(\Delta_{(2)}^{N}\varphi^{(2)})_{s}(z,t)$
$\displaystyle=r^{-2M}s^{-2N}\int_{\mathbb{R}}(\Delta_{(1)}^{M}\varphi^{(1)})_{r}(z,t^{\prime})(\Delta_{(2)}^{N}\varphi^{(2)})_{s}(t-t^{\prime})\,dt^{\prime};$

the integrand vanishes off an interval of length $O(r^{2})$ because
$(\varphi^{(1)})_{r}$ is supported in $B^{(1)}(o,r)$, so

$\displaystyle\left|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}(z,t)\right|$
$\displaystyle\lesssim
r^{2-2M}s^{-2N}\left\|(\Delta_{(1)}^{M}\varphi^{(1)})_{r}\right\|_{L^{\infty}(\mathbb{H}^{\nu})}\left\|(\Delta_{(2)}^{N}\varphi^{(2)})_{s}\right\|_{L^{\infty}(\mathbb{R})}\lesssim_{\varphi,M,N}r^{2-2M-D}s^{-2N-1};$

since moreover $\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}$ is supported in
$T(o,r,s)$, we deduce that

$\left\|\Delta_{(1)}^{M}\Delta_{(2)}^{N}\varphi_{r,s}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim_{\varphi,M,N}r^{2-2M-D}s^{-2N-1}\left|T(o,r,s)\right|^{1/2}\lesssim
r^{-2M-\nu}s^{-2N-1/2}.$ (3.25)

Moreover, from Corollary 3.2,
$\left\|b_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim
h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$. Combining this with the
estimates in (3.24) and (3.25), we deduce that

$\left|a_{R}\ast_{(1)}\varphi_{r,s}\right|\lesssim
r^{-2M-\nu}s^{-2N-1/2}h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,r+1,s+h)}.$
(3.26)

When $r\geq 1$ and $s\geq\max\\{r^{2},h\\}$, and $(z,t)\in T(o,r+1,s+h)$,

$|z|\leq r+1\leq 2r\qquad\text{and}\qquad|t|\leq s+h+(r+1)^{2}\leq 6s.$ (3.27)

Consequently,

$\displaystyle\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r\in(1,\infty]}\sup_{s\in(\max\\{r^{2},h\\},\infty]}\left|a_{R}\ast_{(1)}\varphi_{r,s}(z,t)\right|$
$\displaystyle\qquad\lesssim_{F}\sup_{r\in(1,\infty]}\sup_{s\in(\max\\{r^{2},h\\},\infty]}r^{-2M-\nu}s^{-2N-1/2}h^{2N}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\mathbbm{1}_{T(o,r+1,s+h)}(z,t)$
$\displaystyle\qquad\lesssim_{M,N}\frac{1}{\max\\{1,|z|/2\\}^{2M+\nu}}\frac{h^{2N}}{\max\\{h,|t|/g\\}^{2M+\nu}}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$

It follows immediately that
$\left\|\mathcal{M}_{F,\mathrm{gmax}}^{5}(a_{R})\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
as long as $M>\nu/2$ and $N\geq 1$. Further,

$\displaystyle\int_{(S^{*})^{c}}\mathcal{M}_{F,\mathrm{gmax}}^{5}(a_{R})(g)\,dg$
$\displaystyle\qquad\lesssim_{M,N,\varphi}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\int_{\mathbb{C}^{\nu}}\frac{1}{\max\\{1,|z|/2\\}^{2M+\nu}}\,dz\int_{|t|\geq
h^{*}}\frac{h^{2N}}{(|t|/6)^{{2N+1/2}}}\,dt$
$\displaystyle\qquad\qquad+\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\int_{|z|\geq
r^{*}}\frac{1}{\max\\{1,|z|/2\\}^{2M+\nu}}\,dz\int_{\mathbb{R}}\frac{h^{2N}}{\max\\{h,|t|/6\\}^{{2N+1/2}}}\,dt$
$\displaystyle\qquad\lesssim_{M,N}\left(\frac{\operatorname{height}(R)}{\operatorname{height}(S)}\right)^{2N-1/2}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}+\left(\frac{\operatorname{width}(R)}{\operatorname{width}(S)}\right)^{2M-\nu}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

The quotients that precede $|R|^{1/2}$ in our estimates are dominated by
$\rho_{2N-1/2,2M-{D/2}}(R,S)$, as defined in (3.9).

Further, examination of the argument shows that $\boldsymbol{\varphi}$ comes
into the constants in the form of products such as
$\left\|\varphi^{(1)}\right\|_{L^{1}(\mathbb{H}^{\nu})}\left\|\Delta_{(2)}^{N}\varphi^{(2)}\right\|_{L^{\infty}(\mathbb{R})}$,
which are all dominated by
$\left\|\varphi^{(1)}\right\|_{C^{2M}(\mathbb{H}^{\nu})}\cdot\left\|\varphi^{(2)}\right\|_{C^{2N}(\mathbb{R})}$
because of the support restriction. ∎

We now consider what happens when the functions $\varphi^{(1)}$ and
$\varphi^{(2)}$ in the definition of the grand maximal function are not
supported in the unit balls of $\mathbb{H}^{\nu}$ and $\mathbb{R}$.

######  Proposition 3.20.

Suppose that the functions $\varphi^{(1)}$ and $\varphi^{(2)}$ are Poisson-
bounded, as in Definition 2.20, and that $\mathcal{M}_{F,\mathrm{gmax}}(f)$ is
defined by (3.17). Then $\mathcal{M}_{F,\mathrm{gmax}}(f)\in
L^{1}(\mathbb{H}^{\nu})$, and

$\left\|\mathcal{M}_{F,\mathrm{gmax}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu}).$

###### Proof.

The proof of this proposition is essentially the same as the proof of
Proposition 3.14. ∎

We recall that $p^{(1)}_{r}$ and $p^{(2)}_{s}$ denote the Poisson kernels on
$\mathbb{H}^{\nu}$ and on $\mathbb{R}$ (here $r,s\in\mathbb{R}^{+}$), and, for
$f\in L^{1}(\mathbb{H}^{\nu})$, the Poisson integral $u$ of $f$ is given by

$u(g,r,s):=f\ast_{(1)}p^{(1)}_{r}\ast_{(2)}p^{(2)}_{s}(g)\qquad\forall
g\in\mathbb{H}^{\nu}.$

The nontangential maximal function of $f\in L^{1}(\mathbb{H}^{\nu})$ is
defined by

$\displaystyle
u^{*}(g)=\sup_{(g^{\prime},r,s)\in\Gamma_{\beta}(g)}|u(g^{\prime},r,s)|,$

and the radial maximal function of $f\in L^{1}(\mathbb{H}^{\nu})$ is defined
by

$\displaystyle u^{+}(g)=\sup_{r,s\in\mathbb{R}^{+}}|u(g^{\prime},r,s)|.$

######  Corollary 3.21.

Suppose that $f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$. The the radial
maximal function $u^{+}$ and the nontangential maximal function $U^{*}$ are in
$L^{1}(\mathbb{H}^{\nu})$, and

$\left\|u^{+}\right\|_{L^{1}(\mathbb{H}^{\nu})}\leq\left\|u^{*}\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\qquad\forall
f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu}).$

###### Proof.

It is obvious that the grand maximal function dominates the radial and
nontangential maximal functions. ∎

###  3.8 Simple singular integral operators on the atomic Hardy space

In this section, we consider two types of singular integral operator acting on
the atomic Hardy space. We denote by $\mathcal{H}$ the Hilbert transform in
the central variable, that is,

$\mathcal{H}f(z,t^{\prime})=\int_{\mathbb{R}}f(z,t^{\prime}-t)\,\frac{dt}{t}\qquad\forall(z,t^{\prime})\in\mathbb{H}^{\nu}.$

We shall be interested mainly in the case where $f\in
L^{2}(\mathbb{H}^{\nu})$, in which case one may interpret $\mathcal{H}$ as a
convolution operator acting on the functions $f(z,\cdot)$ for almost all
$z\in\mathbb{C}^{\nu}$. The second type of operator was discussed in Section
2.6; we write $\mathcal{K}$ for the operator of right convolution with a
smooth singular integral kernel that is homogeneous of degree $-D$.

######  Theorem 3.22.

Suppose that $\mathcal{H}$ and $\mathcal{K}$ are as discussed above. Then
$\mathcal{K}\mathcal{H}$ and $\mathcal{K}$ are bounded operators from
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to $L^{1}(\mathbb{H}^{\nu})$.

###### Proof.

We are going to apply Proposition 3.10, and need to estimate

$\int_{(S^{*})^{c}}\left|\mathcal{K}a_{R}(g)\right|\,dg,$

where $a_{R}$ is a particle associated to an adapted rectangle $R$ and $S$ is
an adapted rectangle that contains $R$. We also need to estimate a similar
integral with $\mathcal{K}\mathcal{H}a_{R}$ in place of $\mathcal{K}a_{R}$. We
begin by considering the action of $\mathcal{H}$, and reducing the examination
of the second integral to the examination of the first.

Suppose that $\tilde{a}\in L^{2}(\mathbb{R})$ and
$\tilde{a}=\Delta_{(2)}\tilde{b}$, where $\tilde{b}\in L^{2}(\mathbb{R})$ and
is supported in $[-h,h]$. Using a standard partition of unity argument, we may
write $\mathcal{H}\tilde{a}$ as a sum $\sum_{n\in\mathbb{Z}}a\ast_{(2)}k_{n}$,
where $k_{n}$ is an odd kernel, which vanishes outside
$[-2^{n+1}h,-2^{n-1}h]\cup[2^{n-1}h,2^{n+1}h]$ and
$k_{n}(\cdot)=k_{0}(2^{n}\cdot)$ for all $n$. The kernel
$\sum_{n=-\infty}^{0}k_{n}$ is a truncated version of the Hilbert kernel, so
the associated convolution operator is bounded on $L^{2}(\mathbb{R})$; the
bound is independent of $h$. Hence for $\tilde{a}$ and $\tilde{b}$ as above,
we may view $\mathcal{H}\tilde{a}$ as the sum of one function
$\tilde{a}\ast_{(2)}(\sum_{n=-\infty}^{0}k_{n})$, which has controlled
$L^{2}(\mathbb{R})$-norm and is supported in $[-3h,3h]$, and functions
$\tilde{a}\ast_{(2)}k_{n}$, where $n>0$, which are supported in
$[-(2^{n+1}+1)h,(2^{n+1}+1)h]$. Since

$\tilde{a}\ast_{(2)}k_{n}=\tilde{b}\ast_{(2)}\Delta_{(2)}k_{n}=\Delta_{(2)}(\tilde{b}\ast_{(2)}k_{n}),$

and $\left\|\Delta_{(2)}k_{n}\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim
2^{-2n}h^{-2}$, it follows that

$\displaystyle\left|\operatorname{supp}(\tilde{a}\ast_{(1)}k_{n})\right|^{1/2}\left\|\tilde{a}\ast_{(1)}k_{n}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\lesssim
2^{n}h^{1/2}\left\|\Delta_{(2)}k_{n}\right\|\left\|\tilde{b}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim
2^{-n}h^{1/2}\left\|\tilde{a}\right\|_{L^{2}(\mathbb{H}^{\nu})}$ (3.28)

by Lemma 2.23. In short, $\tilde{a}\ast_{(1)}k_{n}$ is a multiple of a
particle with support in $[-2^{n+2}h,2^{n+2}h]$.

Suppose now that $a_{R}$ is a normalised particle supported in an enlargement
$R^{*}$ of an adapted rectangle of base $q$ and height $h$. We may write

$\mathcal{H}a_{R}=a_{R}\ast_{(2)}\left(\sum_{n=-\infty}^{0}k_{n}\right)+\sum_{n\in\mathbb{Z}^{+}}a_{R}\ast_{(2)}k_{n}=:a_{R,0}+\sum_{n\in\mathbb{Z}^{+}}a_{R,n},$

which is a weighted sum of normalised particles supported in tubes
$R^{*}\cdot[-2^{n+1}h,2^{n+1}h]$. The left-hand summand $a_{R,0}$ has similar
$L^{2}(\mathbb{H}^{\nu})$-norm and support size to the particle $a_{R}$, and
poses no problems. The other terms $a_{R,n}$ are particles with smaller
$L^{2}(\mathbb{H}^{\nu})$-norms and larger support sizes, and some thought is
required to deal with these. To do this, we write $R_{n}$ for the smallest
adapted rectangle that contains
$R\cdot[-2^{n+1}\kappa^{-1}h,2^{n+1}\kappa^{-1}h]$; it has the same base as
$R$, but a larger height, and $R^{*}\cdot[-2^{n+1}h,2^{n+1}h]\subseteq
R_{n}^{*}$. Then the summands $a_{R,n}$ may be associated with the adapted
rectangles $R_{n}$, and from (3.28),

$|R_{n}|^{1/2}\left\|a_{R,n}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim
2^{-n}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

Suppose now that we can show that there exists
$\delta_{1},\delta_{2}\in\mathbb{R}^{+}$ such that

$\int_{(S^{*})^{c}}\left|\mathcal{K}a_{R}(g)\right|\,dg\lesssim_{M,N,\delta}\rho_{\delta_{1},\delta_{2}}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
(3.29)

for all particles $a_{R}$ associated to $R\in\mathfrak{R}$ and all
$S\in\mathfrak{R}$ with $\rho_{\delta_{1},\delta_{2}}(R,S)$, as defined in
(3.9). It will then follow that $\mathcal{K}$ is bounded from
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ to $L^{1}(\mathbb{H}^{\nu})$ by
Proposition 3.10. Further, from above,
$\mathcal{H}a_{R}=\sum_{n\in\mathbb{N}}a_{R,n}$, whence

$\displaystyle\int_{(S^{*})^{c}}\left|\mathcal{K}\mathcal{H}a_{R}(g)\right|\,dg$
$\displaystyle=\int_{(S^{*})^{c}}\left|\mathcal{K}\sum_{n\in\mathbb{N}}a_{R,n}(g)\right|\,dg$
$\displaystyle\leq\sum_{n\in\mathbb{N}}\int_{(S^{*})^{c}}\left|\mathcal{K}a_{R,n}(g)\right|\,dg$
$\displaystyle\leq\sum_{n\in\mathbb{N}}\rho_{\delta_{1},\delta_{2}}(R_{n},S)|R_{n}|^{1/2}\left\|a_{R,n}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\lesssim\sum_{n\in\mathbb{N}}\rho\rho_{\delta_{1},\delta_{2}}(R_{n},S)2^{-n}|R|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\lesssim\rho_{\delta_{1},\delta_{2}}(R,S)\left|R\right|^{1/2}\left\|a_{R}\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

It remains to prove (3.29). To do this, we use a partition of unity to
decompose the convolution kernel $k$ of the operator $\mathcal{K}$ as a sum
$\sum_{j\in\mathbb{N}}k_{j}$, where $\operatorname{supp}(k_{0})\subseteq
B^{(1)}(o,1)$ and $\operatorname{supp}(k_{j})\subseteq
B^{(1)}(o,2^{j-2})\setminus B^{(1)}(o,2^{j})$ when $j\geq 1$; we may do this
in such a way that convolution with $k_{0}$ is bounded on
$L^{2}(\mathbb{H}^{\nu})$, and the $L^{1}(\mathbb{H}^{\nu})$-norms of the
derivatives of the kernels $k_{j}$ vanish rapidly as $j$ increases. When we
consider the convolutions $a_{R}\ast_{(1)}k_{j}$, we can pass a derivative of
$a_{R}$ onto $k_{j}$ to obtain a convergent sum. ∎

We note that the argument of the proof above may easily be improved to show
that $\mathcal{K}$ and $\mathcal{K}\mathcal{H}$ send particles to
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$-functions. However, it appears to
be nontrivial to then show that these operators also send atoms to
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$-functions. Nevertheless, this
conclusion follows from our later characterisation of the Hardy space using
Riesz transforms.

##  4 The Lusin–Littlewood–Paley area function

In this section examine the Hardy space defined via area functions. We first
examine the definition and properties of the space, and then show that if
$f\in H^{1}_{F,\mathrm{area},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$, and
$\varphi^{(1)}$ and $\varphi^{(2)}$ satisfy appropriate spectral conditions,
then $f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$.

###  4.1 Properties of the area function Hardy space

We show that the space is independent of some of the paramenters used in its
definition, and prove that the area operator is (when suitably normalised) an
isometry on $L^{2}(\mathbb{H}^{\nu})$.

We begin by introducing three cones that we shall discuss.

######  Definition 4.1.

Suppose that $g\in\mathbb{H}^{\nu}$ and $\beta,\gamma\in\mathbb{R}^{+}$. The
cones $\Gamma_{\beta,\gamma}(g)$, $\Gamma^{(1)}_{\beta}(g)$, and
$\Gamma^{(2)}_{\beta}(g)$ are defined as follows:

$\displaystyle\Gamma_{\beta,\gamma}(g)$
$\displaystyle=\\{(g^{\prime},r,s)\in\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}:g^{\prime}\in
T(g,\beta r,\gamma s)\\},$ $\displaystyle\Gamma^{(1)}_{\beta}(g)$
$\displaystyle=\\{(g^{\prime},r)\in\mathbb{H}^{\nu}\times\mathbb{R}^{+}:g^{\prime}\in
g\cdot B^{(1)}(o,\beta r)\\},$
$\displaystyle\Gamma^{(2)}_{\gamma}(g^{\prime\prime})$
$\displaystyle=\\{(g^{\prime\prime\prime},s)\in\mathbb{R}\times\mathbb{R}^{+}:g^{\prime\prime\prime}\in
g^{\prime\prime}\cdot B^{(2)}(0,\gamma s)\\}.$

The cone $\Gamma_{\beta}$ defined earlier (as in (1.1)) corresponds to the
cone $\Gamma_{\beta,\beta^{2}}$ above.

Next, we recall from Section 2.3 that $S_{\Delta_{(1)}}(\mathbb{H}^{\nu})$ is
the subspace of $S(\mathbb{H}^{\nu})$ of convolution kernels of operators
$\Phi^{(1)}(\Delta_{(1)})$ where $\Phi^{(1)}\in S(\mathbb{R})$, and
$S_{\Delta_{(2)}}(\mathbb{R})$ is the subspace of $S(\mathbb{R})$ of
convolution kernels of operators $\Phi^{(2)}(\Delta_{(2)})$ where
$\Phi^{(2)}\in S(\mathbb{R})$. Spectral theory provides an abundance of such
functions $\tilde{\varphi}^{(1)}$ and $\tilde{\varphi}^{(2)}$, and in
particular, we may also suppose that $\tilde{\varphi}^{(1)}$ and hence
$\varphi^{(1)}$, and $\tilde{\varphi}^{(2)}$ and hence $\varphi^{(2)}$ are
compactly supported. Moreover, by multiplying $\varphi^{(1)}$ and
$\varphi^{(2)}$ by appropriate scalars, we may suppose that

$\int_{\mathbb{R}^{+}}|\Phi^{(1)}(ru)|^{2}\,\frac{dr}{r}=\int_{\mathbb{R}^{+}}|\Phi^{(2)}(sv)|^{2}\,\frac{ds}{s}=1\qquad\forall
u,v\in\mathbb{R}^{+}.$ (4.1)

Finally, the normalised characteristic functions
$|B^{(1)}(o,r)|^{-1}\mathbbm{1}_{B^{(1)}(o,r)}$ and
$|B^{(2)}(0,s)|^{-1}\mathbbm{1}_{B^{(2)}(0,s)}$ are denoted by
$\chi^{(1)}_{r}$ and $\chi^{(2)}_{s}$. We indicate the pair of functions
$\varphi^{(1)}$ and $\varphi^{(2)}$ by the symbol $\boldsymbol{\varphi}$.

######  Definition 4.2.

Suppose that $\varphi^{(1)}_{r}$, $\varphi^{(2)}_{s}$, $\chi^{(1)}_{r}$ and
$\chi^{(2)}_{s}$ are as above, and that $\beta,\gamma\in\mathbb{R}^{+}$. For
$f\in L^{1}(\mathbb{H}^{\nu})$, we define the Lusin–Littlewood–Paley area
function $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)$
associated to $\varphi^{(1)}$ and $\varphi^{(2)}$ by

$\displaystyle\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)(g):=\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}\bigr{|}^{2}\ast_{(1)}\chi^{(1)}_{\beta
r}\ast_{(2)}\chi^{(2)}_{\gamma
s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}$ (4.2)

for all $g\in\mathbb{H}^{\nu}$, and we define the Hardy space
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(\mathbb{H}^{\nu})$,
usually abbreviated to $H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$, to be the
set of all $f\in L^{1}(\mathbb{H}^{\nu})$ for which
$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$,
with norm

$\left\|f\right\|_{H^{1}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(\mathbb{H}^{\nu})}:=\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$
(4.3)

Important special cases of the Lusin–Littlewood–Paley area function include
when $\varphi^{(1)}$ and $\varphi^{(2)}$ are the convolution kernels of the
heat semigroup or the Poisson semigroup on $\mathbb{H}^{\nu}$ and
$\mathbb{R}$.

There are several equivalent definitions of the area integral, as illustrated
by the following lemma.

######  Lemma 4.3.

The following integrals are equivalent:

$\displaystyle\mathrm{I}_{\beta,\gamma}$
$\displaystyle:=\int_{\mathbb{H}^{\nu}}\left(\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}(F(\cdot,r,s)\ast_{(1)}\chi^{(1)}_{\beta
r}\ast_{(2)}\chi^{(2)}_{\gamma
s})(g)\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}\,dg;$
$\displaystyle\mathrm{J}_{\beta,\gamma}$
$\displaystyle:=\int_{\mathbb{H}^{\nu}}\left(\iiint_{\Gamma_{\beta,\gamma}(g)}\frac{F(g^{\prime},r,s)}{\left|T(o,\beta
r,\gamma
s)\right|}\,dg^{\prime}\,\frac{dr}{r}\,\frac{ds}{s}\right)^{1/2}\,dg,$

for all functions
$F:\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}\to[0,+\infty)$ and
all $\beta,\gamma\in\mathbb{R}^{+}$. The implicit constants are geometric.

###### Proof.

We claim that
$\mathrm{I}_{\beta,\gamma}\eqsim\mathrm{I}_{\beta^{\prime},\gamma^{\prime}}$
for all $\beta$, $\gamma$, $\beta^{\prime}$ and $\gamma^{\prime}$, and then
that
$\mathrm{I}_{\beta,\gamma/2}\lesssim\mathrm{J}_{\beta,\gamma}\lesssim\mathrm{I}_{\beta,\gamma}$,
which implies the stated result.

First,

$\displaystyle\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}(F(\cdot,r,s)\ast_{(1)}\chi^{(1)}_{\beta
r}\ast_{(2)}\chi^{(2)}_{\gamma s})(g)\,\frac{dr}{r}\,\frac{ds}{s}$ (4.4)
$\displaystyle\qquad=\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}F(gg_{1},r,s)(\chi^{(1)}_{\beta
r}*_{(2)}\chi^{(2)}_{\gamma
s})(g_{1}^{-1})\,dg_{1}\,\frac{dr}{r}\,\frac{ds}{s}$
$\displaystyle\qquad=\frac{C}{\beta^{D}\gamma}\iiiint_{\mathbb{H}^{\nu}\times\mathbb{R}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}F(gg_{1}g_{2},r,s)\mathbbm{1}_{B^{(1)}(o,\beta
r)}(g_{1})\mathbbm{1}_{B^{(2)}(0,\gamma
s)}(g_{2})\,dg_{1}\,dg_{2}\frac{dr}{r^{D+1}}\,\frac{ds}{s^{2}}\,,$

where $C$ depends on $\nu$. The last expression may be rewritten as either

$\displaystyle\frac{C}{\beta^{D}\gamma}\iint_{\Gamma^{(1)}_{\beta}(o)}\left(\iint_{\Gamma^{(2)}_{\gamma}(0)}F(gg_{1}g_{2},r,s)\,\frac{ds}{s^{2}}\,dg_{2}\right)\,\frac{dr}{r^{D+1}}\,dg_{1}$
$\displaystyle\frac{C}{\beta^{D}\gamma}\iint_{\Gamma^{(2)}_{\gamma}(0)}\left(\iint_{\Gamma^{(1)}_{\beta}(o)}F(gg_{1}g_{2},r,s)\,\frac{dr}{r^{D+1}}\,dg_{1}\right)\,\frac{ds}{s^{2}}\,dg_{2}.$

Hence $\mathrm{I}_{\beta,\gamma}$ is equivalent to either

$\displaystyle\frac{C}{\beta^{D}\gamma}\int_{\mathbb{H}^{\nu}}\left(\iint_{\Gamma^{(1)}_{\beta}(o)}\left(\iint_{\Gamma^{(2)}_{\gamma}(0)}F(gg_{1}g_{2},r,s)\,\frac{ds}{s^{2}}\,dg_{2}\right)\,\frac{dr}{r^{D+1}}\,dg_{1}\right)^{1/2}\,dg$
$\displaystyle\frac{C}{\beta^{D}\gamma}\int_{\mathbb{H}^{\nu}}\left(\iint_{\Gamma^{(2)}_{\gamma}(0)}\left(\iint_{\Gamma^{(1)}_{\beta}(o)}F(gg_{1}g_{2},r,s)\,\frac{dr}{r^{D+1}}\,dg_{1}\right)\,\frac{ds}{s^{2}}\,dg_{2}\right)^{1/2}\,dg.$

It may be argued, as in the classical case (see [59, pp. 125–126]) that
changing the parameter $\beta$ changes the first of these two expressions to
an equivalent expression, and that changing the parameter $\gamma$ changes the
second to an equivalent expression (in this case, we need to write $g$ as
$(z,t)$ and integrate with respect to first $t$ and then $z$). We conclude
that changing $\beta$ and $\gamma$ changes $\mathrm{I}_{\beta,\gamma}$ to an
equivalent integral.

Finally, from Lemma 2.8,

$\mathrm{I}_{\beta,\gamma/2}\lesssim\mathrm{J}_{\beta,\gamma}\lesssim\mathrm{I}_{\beta,\gamma},$
(4.5)

and the lemma follows. ∎

######  Corollary 4.4.

The space
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(\mathbb{H}^{\nu})$
coincides with the space of functions on $\mathbb{H}^{\nu}$ for which

$\int_{\mathbb{H}^{\nu}}\biggl{(}\iiint_{\Gamma_{\beta}(g)}\frac{1}{\left|T(o,\beta
r,\gamma
s)\right|}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g^{\prime})\bigr{|}^{2}\,dg^{\prime}\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg$

and the integral above is equivalent to the
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(\mathbb{H}^{\nu})$
norm.

Since the Hardy space
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi},\beta,\gamma}(\mathbb{H}^{\nu})$
does not depend on the parameters $\beta$ and $\gamma$, henceforth we take
these to be $1$ unless explicitly stated otherwise, and write
$H^{1}_{F,\mathrm{area},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$ or just
$H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$ for the space.

######  Lemma 4.5.

There exists a positive geometric constant $C_{P}$ such that, if $\Omega$ is a
subset of $\mathbb{H}^{\nu}$ of finite measure, and
$W=\bigcup_{g\in\Omega}\Gamma(g)$, then
$\mathbbm{1}_{\Omega}\ast_{(1)}p_{r,s}(g^{\prime})\geq
C_{P}\qquad\forall(g^{\prime},r,s)\in W.$

###### Proof.

By Lemma 2.21, for all $\beta,\gamma\in\mathbb{R}^{+}$,
$\mathbbm{1}_{\Omega}\ast_{(1)}p_{r,s}(g^{\prime})\gtrsim\mathbbm{1}_{\Omega}\ast_{(1)}\chi_{\alpha
r,\beta s}(g^{\prime}),\
\forall(g^{\prime},r,s)\in\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}.$
The proof is complete. ∎

Finally, we show that the Lusin–Littlewood–Paley area operator is bounded on
$L^{2}(\mathbb{H}^{\nu})$.

######  Proposition 4.6.

Suppose that $\varphi^{(1)}$ and $\varphi^{(2)}$ are as described before
Definition 4.2. For all $f\in L^{2}(\mathbb{H}^{\nu})$,
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\in
L^{2}(\mathbb{H}^{\nu})$, and
$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\right\|_{L^{2}(\mathbb{H}^{\nu})}=\left\|f\right\|_{L^{2}(\mathbb{H}^{\nu})}.$

###### Proof.

Since the functions $\chi^{(1)}_{r}$ and $\chi^{(2)}_{s}$ are normalised
characteristic functions, for all $f\in L^{2}(\mathbb{H}^{\nu})$,

$\displaystyle\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\right\|_{L^{2}(\mathbb{H}^{\nu})}^{2}$
$\displaystyle:=\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}\right|^{2}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\,dg$
$\displaystyle=\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g)\right|^{2}\,\frac{dr}{r}\,\frac{ds}{s}\,dg$
$\displaystyle=\left\|f\right\|_{L^{2}(\mathbb{H}^{\nu})},$

as required. ∎

###  4.2 The inclusion $H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$

Here we show that every $f\in H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$ has
an atomic decomposition, and control the decomposition appropriately, provided
$\varphi^{(1)}$ and $\varphi^{(2)}$ satisfy certain spectral conditions. To be
more specific, we will show the following result.

######  Theorem 4.7.

Suppose that $M,N\geq 1$ and that $\psi^{(1)}$ and $\psi^{(2)}$ satisfy
….111What conditions? There exists a constant $C$, depending on $M$, $N$,
$\nu$, and $\boldsymbol{\varphi}$, such that for all $f\in
H^{1}_{F,\mathrm{area},\boldsymbol{\psi}}(\mathbb{H}^{\nu})\cap
L^{2}(\mathbb{H}^{\nu})$, there exist $(1,2,M,N,3)$ atoms $a_{j}$ and numbers
$\lambda_{j}$, for all $j\in\mathbb{N}$, such that
$f\sim\sum\lambda_{j}a_{j}$, where the sum converges in
$L^{2}(\mathbb{H}^{\nu})$, and

$\left\|f\right\|_{H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})}\leq\sum_{j=0}^{\infty}|\lambda_{j}|\leq
C\left\|f\right\|_{H^{1}_{F,\mathrm{area},\boldsymbol{\psi}}(\mathbb{H}^{\nu})}.$

Hence, by continuity and density,
$H^{1}_{F,\mathrm{area},\boldsymbol{\psi}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$.

###### Proof.

We denote by $\mathfrak{R}$ the collection of all adapted rectangles, as in
Section 2.2. We take the enlargement parameter $\kappa$ (defined at the start
of Section 3) to be $3$, and then (2.13) implies that

$\displaystyle R\cdot B^{(1)}(o,q)\cdot B^{(2)}(0,h)$ $\displaystyle\subset
g\cdot\bar{B}^{(1)}(o,q/2)\cdot\bar{B}^{(2)}(0,(q^{2}+4h)/8)\cdot
B^{(1)}(o,q)\cdot B^{(2)}(0,h)$ $\displaystyle=g\cdot B^{(1)}(o,3q/2)\cdot
B^{(2)}(0,(q^{2}+12h)/8)$ $\displaystyle\subseteq B^{(1)}(o,\kappa q/2)\cdot
B^{(2)}(0,\kappa^{2}(q^{2}+4h)/8)$ $\displaystyle=R^{*},$

when $R\in\mathfrak{R}$, $\operatorname{width}(R)=q$ and
$\operatorname{height}(R)=h$. By Lemma 2.9, for all open sets $\Omega$ and
$R\in\mathfrak{R}(\Omega)$

$R^{*}\subseteq\left\\{g\in\mathbb{H}^{\nu}:\mathcal{M}_{F}(\mathbbm{1}_{\Omega})(g)>\frac{1}{2^{2\nu}\kappa^{D}(5\nu+2)}\right\\}.$

For each tile $R$ in $\mathfrak{R}$, the tent $T(R)$ over $R$ is defined by

$T(R):=\left\\{(g,r,s)\in\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}:g\in
R,r\in(q/2,q],s\in(0,q^{2}]\right\\},$

where $q=\operatorname{width}(R)$, and for each adapted rectangle $R$ in
$\mathfrak{R}$ that is not a tile, the tent $T(R)$ over $R$ is defined by

$T(R):=\left\\{(g,r,s)\in\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}:g\in
R,r\in(q/2,q],s\in(h/2,h]\right\\},$

where $q=\operatorname{width}(R)$ and $h=\operatorname{height}(R)$. It is
evident that $\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}$
decomposes as a disjoint union:

$\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}=\bigsqcup_{\begin{subarray}{c}\ell\in\mathbb{Z}\\\
R\in\mathfrak{B}_{\ell}\end{subarray}}T(R);$

indeed, for
$(g,r,s)\in\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}$, the
coordinates $r$ and $s$ determine the dimensions and the coordinate $g$
determines the location of an adapted rectangle $R$ such that $g\in R$.

Take $f\in H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})\cap
L^{2}(\mathbb{H}^{\nu})$. For each $\ell\in\mathbb{Z}$, we define

$\displaystyle\Omega_{\ell}$
$\displaystyle:=\left\\{g\in\mathbb{H}^{\nu}:\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)(g)>2^{\ell}\right\\},$
$\displaystyle\mathfrak{B}_{\ell}$
$\displaystyle:=\left\\{R\in\mathfrak{R}:|R^{*}\cap\Omega_{\ell}|>\frac{1}{3\kappa^{D}}|R^{*}|,|R^{*}\cap\Omega_{\ell+1}|\leq\frac{1}{3\kappa^{D}}|R^{*}|\right\\}$
$\displaystyle\widetilde{\Omega}_{\ell}$
$\displaystyle:=\left\\{g\in\mathbb{H}^{\nu}:\mathcal{M}_{F}(\mathbbm{1}_{\Omega_{\ell}})(g)>\frac{1}{2^{2\nu}\kappa^{D}(5\nu+2)}\right\\}.$

The second of these exotic-looking constants is chosen so that
$R^{*}\subseteq\widetilde{\Omega}_{\ell}$; we leave it to the reader to check
that the first ensures that, if $R\in\mathfrak{R}$ and $(g,r,s)\in T(R)$, then

$\frac{1}{3\kappa^{D}}|R^{*}|\leq\frac{1}{2}|T(g,r,s)|.$ (4.6)

We take smooth even radial functions $\tilde{\psi}^{(1)}$ on
$\mathbb{H}^{\nu}$ and $\tilde{\psi}^{(2)}$ on $\mathbb{R}$ such that
$\operatorname{supp}\tilde{\psi}^{(1)}\subseteq B^{(1)}(o,1)$ and
$\operatorname{supp}\tilde{\psi}^{(2)}\subseteq B^{(2)}(0,1)$, and we set
$\psi^{(1)}=\Delta_{(1)}^{M}\tilde{\psi}^{(1)}$ and
$\psi^{(2)}=\Delta_{(2)}^{N}\tilde{\psi}^{(2)}$; by rescaling $\varphi^{(1)}$
and $\varphi^{(2)}$ if necessary, and renormalising, we may and shall assume
that

$\int_{\mathbb{R}^{+}}f\ast_{(1)}\varphi_{r}^{(1)}\ast_{(1)}\psi_{r}^{(1)}\,\frac{dr}{r}=f\qquad\text{and}\qquad\int_{\mathbb{R}^{+}}f\ast_{(2)}\varphi_{s}^{(2)}\ast_{(2)}\psi_{s}^{(2)}\,\frac{dr}{r}=f$

for all $f\in L^{2}(\mathbb{H}^{\nu})$; here, as usual, $\varphi^{(1)}_{r}$
denotes the normalised dilate of $\varphi^{(1)}$, and so on. As before, we
abbreviate $\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}$ to $\varphi_{r,s}$,
and define $\tilde{\psi}_{r,s}$ and $\psi_{r,s}$ analogously. It follows from
spectral theory, as in Corollary 2.16 and Remark 2.17, that for all $f\in
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})\cap L^{2}(\mathbb{H}^{\nu})$,

$\displaystyle f$
$\displaystyle=\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}f\ast_{(1)}\varphi_{r,s}\ast_{(1)}\psi_{r,s}\,\frac{ds}{s}\,\frac{dr}{r}=\int_{\mathbb{H}^{\nu}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}f\ast_{(1)}\varphi_{r,s}(g)\,{}_{g}\psi_{r,s}\,\frac{ds}{s}\,\frac{dr}{r}\,dg$
$\displaystyle=\sum_{\ell\in\mathbb{Z}}\sum_{R\in\mathcal{B}_{\ell}}\iiint_{T(R)}f\ast_{(1)}\varphi_{r,s}(g)\,{}_{g}\psi_{r,s}\,\frac{dr}{r}\,\frac{ds}{s}\,dg$
$\displaystyle=\sum_{\ell\in\mathbb{Z}}\lambda_{\ell}\sum_{R\in\mathfrak{B}_{\ell}}a_{\ell,R},$

say, where

$\displaystyle
a_{\ell,R}:=\frac{1}{\lambda_{\ell}}\iiint_{T(R)}f\ast_{(1)}\varphi_{r,s}(g)\,{}_{g}\psi_{r,s}\,dg\,\frac{dr}{r}\,\frac{ds}{s}$
$\displaystyle\lambda_{\ell}:=\bigg{\|}\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(\cdot)\right|^{2}\mathbbm{1}_{T(R)}(\cdot)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\bigg{\|}_{L^{2}(\mathbb{H}^{\nu})}|\widetilde{\Omega}_{\ell}|^{1/2}.$

(If $\lambda_{\ell}=0$, then all $a_{\ell,R}$ are taken to be $0$, as the
integral involved in the definition of $a_{\ell,R}$ vanishes. At the end of
this proof we show that each $\lambda_{\ell}$ is finite, and so these
expressions make sense.)

Fix $\ell\in\mathbb{Z}$; we claim that
$\sum_{R\in\mathfrak{B}_{\ell}}a_{\ell,R}$ is a geometric multiple of a flag
atom. In light of Lemma 3.3, it suffices to prove that there exist functions
$b_{\ell,R}$ in $\operatorname{Dom}(\Delta_{(1)}^{M}\Delta_{(2)}^{N})$ in
$L^{2}(\mathbb{H}^{\nu})$, for all $R\in\mathfrak{B}_{\ell}$, such that

  1. (A1)

$a_{\ell,R}={\Delta_{(1)}}^{M}{\Delta_{(2)}}^{N}b_{R}$ and
$\operatorname{supp}b_{R}\subseteq R^{*}$; and

  2. (A2)

for all sign sequences $\sigma:\mathfrak{B}_{\ell}\to\\{\pm 1\\}$, the sum
$\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\,a_{\ell,R}$ converges in
$L^{2}(\mathbb{H}^{\nu})$, to $a_{\sigma}$ say, and
$\left\|a_{\sigma}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim\left|\Omega\right|^{-1/2}.$

Suppose that $\operatorname{width}(R)=q$ and $\operatorname{height}(R)=h$. It
is evident that for each $\ell\in\mathbb{Z}$ and $R\in\mathfrak{B}_{\ell}$,

$a_{\ell,R}:=\Delta_{(1)}^{M}\Delta_{(2)}^{N}b_{\ell,R},$

where

$\displaystyle
b_{\ell,R}:=\frac{1}{\lambda_{\ell}}\iiint_{T(R)}f\ast_{(1)}\varphi_{r,s}(g)\,{}_{g}\tilde{\psi}_{r,s}\,dg\,\frac{dr}{r}\,\frac{ds}{s}\,.$
(4.7)

By construction, when $(g,r,s)\in T(R)$,
$\operatorname{supp}{}_{g}\tilde{\psi}_{r,s}\subseteq g\cdot T(o,r,s)\subseteq
R\cdot T(o,q,h).$ Hence

$\displaystyle\operatorname{supp}b_{\ell,R}\subseteq R^{*}.$ (4.8)

Next we take a sign sequence $\sigma:\mathfrak{B}_{\ell}\to\\{\pm 1\\}$ and
estimate the $L^{2}(\mathbb{H}^{\nu})$-norm of
$\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\,a_{\ell,R}$. For all smooth
compactly supported $h$ on $\mathbb{H}^{\nu}$ such that
$\left\|h\right\|_{L^{2}(\mathbb{H}^{\nu})}\leq 1$,

$\displaystyle\Bigl{|}\int_{\mathbb{H}^{\nu}}\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\,a_{\ell,R}(g^{\prime})h(g^{\prime})\,dg^{\prime}\Bigr{|}$
$\displaystyle\qquad=\Biggl{|}\frac{1}{\lambda_{\ell}}\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\iiint_{T(R)}\int_{\mathbb{H}^{\nu}}f\ast_{(1)}\varphi_{r,s}(g){}_{g}\psi_{r,s}(g^{\prime})h(g^{\prime})\,dg^{\prime}\,\frac{ds}{s}\frac{dr}{r}\,dg\Biggr{|}$
$\displaystyle\qquad=\Biggl{|}\frac{1}{\lambda_{\ell}}\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\iiint_{T(R)}h\ast_{(1)}\psi_{r,s}(g)f\ast_{(1)}\varphi_{r,s}(g)\,\frac{ds}{s}\frac{dr}{r}\,dg\Biggr{|}$
$\displaystyle\qquad=\Biggl{|}\frac{1}{\lambda_{\ell}}\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}h\ast_{(1)}\psi_{r,s}(g)f\ast_{(1)}\varphi_{r,s}(g)\mathbbm{1}_{T(R)}(g,r,s)\,\frac{ds}{s}\frac{dr}{r}\,dg\Biggr{|}$
$\displaystyle\qquad\leq\frac{1}{\lambda_{\ell}}\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}|h\ast_{(1)}\psi_{r,s}(g)|^{2}\mathbbm{1}_{T(R)}(g,r,s)\,\frac{ds}{s}\frac{dr}{r}\,dg\biggr{)}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(g)\right|^{2}\mathbbm{1}_{T(R)}(g,r,s)\,\frac{ds}{s}\frac{dr}{r}\,dg\biggr{)}^{1/2}$
$\displaystyle\qquad\leq\frac{1}{\lambda_{\ell}}\biggl{(}\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}|h\ast_{(1)}\psi_{r,s}(g)|^{2}\,dg\biggr{)}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(g)\right|^{2}\mathbbm{1}_{R}(g)\,\mathbbm{1}_{T_{d}(R)}(m,n)\,dg\biggr{)}^{1/2}$
$\displaystyle\qquad\lesssim\frac{1}{\lambda_{\ell}}\left\|h\right\|_{L^{2}(\mathbb{H}^{\nu})}\left\|\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(\cdot)\right|^{2}\mathbbm{1}_{T(R)}(\cdot,r,s)\,\frac{ds}{s}\,\frac{dr}{r}\biggr{)}^{1/2}\right\|_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\leq|\widetilde{\Omega}_{\ell}|^{-{1}/{2}},$

by the Cauchy–Schwarz inequality and the definition of $\lambda_{\ell}$, and
so
$\left\|\sum_{R\in\mathfrak{B}_{\ell}}\sigma_{\ell,R}\,a_{\ell,R}\right\|_{L^{2}(\mathbb{H}^{\nu})}\lesssim|\widetilde{\Omega}_{\ell}|^{-1/2}$.
Hence $a$ is a multiple of an atom, and the multiple depends only on $M$, $N$,
$\nu$, $\boldsymbol{\varphi}$ and $\boldsymbol{\psi}$.

Finally, we verify the convergence of the series
$\sum_{\ell}|\lambda_{\ell}|$. To do this, we first fix $\ell\in\mathbb{Z}$,
and observe that if $R\in\mathfrak{R}$ and $(g^{\prime},r,s)\in T(R)$, then,
from (2.13), $T(g^{\prime},r,s)\subseteq
R^{*}\subseteq\widetilde{\Omega}_{\ell}$, whence

$\left|\widetilde{\Omega}_{\ell}\cap
T(g^{\prime},r,s)\right|=\left|T(g^{\prime},r,s)\right|=\left|T(o,r,s)\right|,$

while, by definition of $\mathfrak{B}_{\ell}$ and (4.6),

$\left|\Omega_{\ell+1}\cap
T(g^{\prime},r,s)\right|\leq\left|R^{*}\cap\Omega_{\ell+1}\right|\leq\frac{1}{3\kappa^{D}}\left|R^{*}\right|\leq\frac{1}{2}\left|T(o,r,s)\right|.$

It follows that

$\frac{\left|(\widetilde{\Omega}_{\ell}\setminus\Omega_{\ell+1})\cap
T(g^{\prime},r,s)\right|}{\left|T(g^{\prime},r,s)\right|}\geq\frac{1}{2}\,.$

Now by the definitions of $\mathfrak{B}_{\ell}$, the Lusin–Littlewood–Paley
area function, and $\Omega_{\ell}$,

$\displaystyle\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{T(R)}\left|f\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\,\frac{ds}{s}\,\frac{dr}{r}\,dg^{\prime}$
$\displaystyle\qquad\leq
2\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{T(R)}\left|f\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\frac{\left|(\widetilde{\Omega}_{\ell}\setminus\Omega_{\ell+1})\cap
T(g^{\prime},r,s)\right|}{\left|T(g^{\prime},r,s)\right|}\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}$
$\displaystyle\qquad\leq
2\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\frac{\left|(\widetilde{\Omega}_{\ell}\setminus\Omega_{\ell+1})\cap
T(g^{\prime},r,s)\right|}{\left|T(g^{\prime},r,s)\right|}\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}$
$\displaystyle\qquad=2\int_{\widetilde{\Omega}_{\ell}\backslash\Omega_{\ell+1}}\iiint_{\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\frac{\mathbbm{1}_{T(o,r,s)}(g^{\prime}\cdot
g^{-1})}{\left|T(o,r,s)\right|}\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}\,dg$
$\displaystyle\qquad=2\int_{\widetilde{\Omega}_{\ell}\backslash\Omega_{\ell+1}}\iiint_{\Gamma(g)}\left|f\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\frac{1}{\left|T(o,r,s)\right|}\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}\,dg$
$\displaystyle\qquad=2\int_{\widetilde{\Omega}_{\ell}\backslash\Omega_{\ell+1}}\left|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)(g)\right|^{2}\,dg$
$\displaystyle\qquad\leq 2^{2\ell+3}|\widetilde{\Omega}_{\ell}|.$

Hence

$\displaystyle\sum_{\ell\in\mathbb{Z}}|\lambda_{\ell}|$
$\displaystyle\leq\sum_{\ell\in\mathbb{Z}}\left\|\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}\psi_{r,s}(\cdot)\right|^{2}\mathbbm{1}_{T(R)}(\cdot,r,s)\,\frac{dr}{r}\frac{ds}{s}\biggr{)}^{1/2}\right\|_{L^{2}(\mathbb{H}^{\nu})}\left|\widetilde{\Omega}_{\ell}\right|^{1/2}$
$\displaystyle=\sum_{\ell\in\mathbb{Z}}\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{T(R)}\left|f\ast_{(1)}\psi_{r,s}(g^{\prime})\right|^{2}\,dg^{\prime}\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}|\widetilde{\Omega}_{\ell}|^{1/2}$
$\displaystyle\leq\sum_{\ell\in\mathbb{Z}}2^{\ell+3/2}|\widetilde{\Omega}_{\ell}|\lesssim\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\psi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}$
$\displaystyle=\left\|f\right\|_{H^{1}_{F,\mathrm{area},\boldsymbol{\psi}}(\mathbb{H}^{\nu})}.$

This completes the proof of Theorem 4.7. ∎

We remark that the atomic decomposition
$\sum_{\ell\in\mathbb{Z}}\lambda_{\ell}a_{\ell}$ in the proof above converges
to $f$ in the $L^{2}(\mathbb{H}^{\nu})$ norm. Indeed, to show this, we only
need to show that
$\|\sum_{|\ell|>L}\lambda_{\ell}a_{\ell}\|_{L^{2}(\mathbb{H}^{\nu})}\rightarrow
0$ as $L$ tends to infinity. To see this, first note that

$\displaystyle\Bigl{|}\big{\langle}\sum_{|\ell|>L}\lambda_{\ell}a_{\ell},h\big{\rangle}\Bigr{|}$
$\displaystyle\qquad\leq\biggl{|}\sum_{R\in\mathfrak{B}_{\ell}}\iiint_{T(R)}h\ast_{(1)}\psi_{r,s}(g^{\prime})\sum_{|\ell|>L}\lambda_{\ell}a_{\ell}\ast_{(1)}\varphi_{r,s}(g^{\prime})\,dg^{\prime}\,\frac{dr}{r}\,\frac{ds}{s}\biggr{|}$
$\displaystyle\qquad\leq\int_{\mathbb{H}^{\nu}}\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|h\ast_{(1)}\psi_{r,s}(g^{\prime})\right|^{2}\mathbbm{1}_{T(R)}(g^{\prime},r,s)\,\frac{dr}{r}\frac{ds}{s}\biggr{)}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|\sum_{|\ell|>L}\lambda_{\ell}a_{\ell}\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\mathbbm{1}_{T(R)}(g^{\prime},r,s)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg^{\prime}$
$\displaystyle\qquad\leq
C\left\|h\right\|_{L^{2}(\mathbb{H}^{\nu})}\bigg{\|}\biggl{(}\sum_{R\in\mathfrak{B}_{\ell}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|\sum_{|\ell|>L}\lambda_{\ell}a_{\ell}\ast_{(1)}\varphi_{r,s}(g^{\prime})\right|^{2}\mathbbm{1}_{T(R)}(g^{\prime},r,s)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\bigg{\|}_{L^{2}(\mathbb{H}^{\nu})}$
$\displaystyle\qquad\to 0$

when $L$ tends to $\infty$, as the area integral operator is bounded on
$L^{2}(\mathbb{H}^{\nu})$. Since

$\bigg{\\{}\sum_{|\ell|>L}\lambda_{\ell}a_{\ell}\bigg{\\}}_{L^{2}(\mathbb{H}^{\nu})}=\sup_{h:\left\|h\right\|_{L^{2}(\mathbb{H}^{\nu})=1}}\Bigl{|}\big{\langle}\sum_{|\ell|>L}\lambda_{\ell}a_{\ell},h\big{\rangle}\Bigr{|},$

the sum $\sum_{\ell\in\mathbb{Z}}\lambda_{\ell}a_{\ell}$ converges to $f$ in
the $L^{2}(\mathbb{H}^{\nu})$ norm.

###  4.3 Remarks on a “discrete area function”

We could replace the integrals over $r$ and $s$ in the definition of the area
function $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}$ (Definition 4.2
by sums, taking $r$ and $s$ to be $2^{j}$ and $2^{k}$, or more generally
$\mathrm{e}^{\alpha j}$ and $\mathrm{e}^{\beta k}$.

The proofs that $\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\in
L^{1}(\mathbb{H}^{\nu})$ if $f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$
in Section 3.4 and that $f\in H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$ if
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\varphi}}(f)\in
L^{1}(\mathbb{H}^{\nu})$ in Section 4.2 go through with minor modifications,
and show that the atomic Hardy space may also be characterised by a “discrete
area function”. We spare the reader the details!

##  5 The square function Hardy space

We recall the definitions of the continuous and discrete Littlewood–Paley
square functions. We suppose that the integers $M$ and $N$ satisfy $M>D/4$ and
$N\geq 1$, and consider a pair $\boldsymbol{\tilde{\varphi}}$ of Poisson-
bounded functions, as in Definition 2.20. We define
$\varphi^{(1)}=\Delta_{(1)}\tilde{\varphi}^{(1)}$ and
$\varphi^{(2)}=\Delta_{(2)}\tilde{\varphi}^{(2)}$, and take
$\varphi^{(1)}_{r}$ to be $r^{-D}\varphi^{(1)}(\delta_{1/r}\cdot)$ and
$\varphi^{(2)}_{s}$ to be $s^{-1}\varphi^{(2)}(\cdot/s)$.

######  Definition 5.1.

Suppose that $\varphi^{(1)}$ and $\varphi^{(2)}$ are as above. For $f\in
L^{1}(\mathbb{H}^{\nu})$, we define the continuous and discrete
Littlewood–Paley square functions
$\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)$ and
$\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)$ associated to
$\varphi^{(1)}$ and $\varphi^{(2)}$ by

$\displaystyle\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)(g):=\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g)\bigr{|}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}$
$\displaystyle\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)(g):=\biggl{(}\sum_{(m,n)\in\mathbb{Z}\times\mathbb{Z}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{2^{m}}\ast_{(2)}\varphi^{(2)}_{2^{n}}(g)\bigr{|}^{2}\biggr{)}^{1/2}$

for all $g\in\mathbb{H}^{\nu}$. We define the square function Hardy spaces
$H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$ and
$H^{1}_{F,\mathrm{dis},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$, often
abbreviated to $H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})$ and
$H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$, to be the set of all $f\in
L^{1}(\mathbb{H}^{\nu})$ for which
$\left\|\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$
or
$\left\|\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$,
with norms

$\left\|f\right\|_{H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})}:=\left\|\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})},\quad\left\|f\right\|_{H^{1}_{F,\mathrm{dis},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})}:=\left\|\mathcal{S}_{F,\mathrm{dis},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$

We showed in Section 3 that
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})$ and
$H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$. Now we show that
$H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$; coupled with the result of the
last section that $H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$, this completes the proof of the
equivalence of these spaces. A key technical result on the way to showing that
$H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$ is that
$H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})=H^{1}_{F,\mathrm{cts},\boldsymbol{\psi}}(\mathbb{H}^{\nu})$
for suitable pairs of functions $\boldsymbol{\varphi}$ and
$\boldsymbol{\psi}$.

Very similar arguments apply to the discrete square function space
$H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$, but we just summarise the changes
that need to be made rather than go through all the details.

Of course, for spaces such as
$H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$ to be
meaningful, we need some additional conditions on the pair of functions
$\boldsymbol{\varphi}$ involved in the definition of the square function
spaces. For instance if $\varphi^{(1)}=0$, then the spaces
$H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})$ and
$H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$ are just $L^{1}(\mathbb{H}^{\nu})$
with a trivial norm.

###  5.1 The inclusion $H^{1}_{F,\mathrm{cts}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$

A simple trick now enables us to pass from a square function to an area
function.

######  Lemma 5.2.

Suppose that $(\varphi^{(1)},\varphi^{(2)})$ and $(\psi^{(1)},\psi^{(2)})$ are
as in Definition 5.1. Then

$\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\psi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}\lesssim\left\|\mathcal{S}_{F,\mathrm{cts},\boldsymbol{\varphi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}\qquad\forall
f\in L^{1}(\mathbb{H}^{\nu}).$ (5.1)

###### Proof.

To begin with, we note that $(\varphi^{(1)},\varphi^{(2)})$ and
$(\psi^{(1)},\psi^{(2)})$ given as in Definition 5.1 satisfy the properties as
in Definition def:ctssqfnphi. From the definition of
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{\psi}}(f)$, we have

$\displaystyle\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\psi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}=\int_{\mathbb{H}^{\nu}}\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}\bigr{|}^{2}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg$
$\displaystyle=\int_{\mathbb{H}^{\nu}}\bigg{\\{}\sum_{j,k\in\mathbb{Z}}\sum_{R\in\mathfrak{R}(N,j,k)}\int_{2^{-j-N}}^{2^{-j+1-N}}\int_{2^{-k-N}}^{2^{-k+1-N}}$
$\displaystyle\qquad\qquad\int_{\mathbb{H}^{\nu}}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(\tilde{g}^{-1}\cdot
g)\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(\tilde{g})\bigr{|}^{2}\,d\tilde{g}\,\frac{dr}{r}\,\frac{ds}{s}\chi_{R}(g)\bigg{\\}}^{1/2}\,dg$
$\displaystyle\leq\int_{\mathbb{H}^{\nu}}\bigg{\\{}\sum_{j,k\in\mathbb{Z}}\sum_{R\in\mathfrak{R}(N,j,k)}\int_{2^{-j-N}}^{2^{-j+1-N}}\int_{2^{-k-N}}^{2^{-k+1-N}}$
$\displaystyle\qquad\qquad\int_{\mathbb{H}^{\nu}}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(\tilde{g}^{-1}\cdot
g)\sup_{g_{R}\in
CR}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g_{R})\bigr{|}^{2}\,d\tilde{g}\,\frac{dr}{r}\,\frac{ds}{s}\chi_{R}(g)\bigg{\\}}^{1/2}\,dg$
$\displaystyle\lesssim\int_{\mathbb{H}^{\nu}}\bigg{\\{}\sum_{j,k\in\mathbb{Z}}\sum_{R\in\mathfrak{R}(N,j,k)}\int_{2^{-j-N}}^{2^{-j+1-N}}\int_{2^{-k-N}}^{2^{-k+1-N}}\sup_{g_{R}\in
CR}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g_{R})\bigr{|}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\chi_{R}(g)\bigg{\\}}^{1/2}\,dg,$

where the first inequality follows from the fact that for
$R\in\mathfrak{R}(N,j,k)$, $g\in R$,
$\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(\tilde{g}^{-1}\cdot g)\neq 0$ and for
$r\in(2^{-j-N},2^{-j+1-N})$, $s\in(2^{-k-N},2^{-k+1-N})$, we see that there
exists an absolute constant $C$ such that $\tilde{g}\in CR$; and second
inequality follows from the fact that

$\int_{\mathbb{H}^{\nu}}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(\tilde{g}^{-1}\cdot
g)d\tilde{g}\lesssim 1.$

Then, by using the flag Plancherel–Pólya inequality, we deduce that

$\displaystyle\left\|\mathcal{S}_{F,\mathrm{area},\boldsymbol{\psi}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}$
$\displaystyle\lesssim\int_{\mathbb{H}^{\nu}}\bigg{\\{}\sum_{j,k\in\mathbb{Z}}\sum_{R\in\mathfrak{R}(N,j,k)}\int_{2^{-j-N}}^{2^{-j+1-N}}\int_{2^{-k-N}}^{2^{-k+1-N}}\inf_{g_{R}\in
R}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g_{R})\bigr{|}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\chi_{R}(g)\bigg{\\}}^{1/2}\,dg$
$\displaystyle\leq\int_{\mathbb{H}^{\nu}}\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}\varphi^{(1)}_{r}\ast_{(2)}\varphi^{(2)}_{s}(g)\bigr{|}^{2}\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}\,dg,$

which proves (5.1). ∎

###  5.2 The inclusion $H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})$

The arguments of the previous sections may be extendeded to show that the
Hardy spaces $H^{1}_{F,\mathrm{cts},\boldsymbol{\varphi}}(\mathbb{H}^{\nu})$
and $H^{1}_{F,\mathrm{cts},\boldsymbol{\psi}}(\mathbb{H}^{\nu})$ are
equivalent, under appropriate conditions on the pairs of functions
$\boldsymbol{\varphi}$ and $\boldsymbol{\psi}$, which are very similar to the
conditions on the pairs of functions in Lemma 2.25.

The argument of the previous section then implies that if $f\in
H^{1}_{F,\mathrm{dis}}(\mathbb{H}^{\nu})$, then $f$ also belongs to the
“discrete area function” Hardy space, which we discussed (without defining
precisely) in Section 4.3, and explained why this space coincides with the
atomic Hardy space.

It follows that the discrete square function Hardy space coincides with the
atomic Hardy space.

##  6 Maximal function flag Hardy space

In this section, we characterise the flag Hardy space by maximal functions. We
begin by recalling three definitions.

######  Definition 6.1.

Fix $\beta>0$. The nontangential maximal function of $f\in
L^{1}(\mathbb{H}^{\nu})$ is defined by

$\displaystyle
u^{*}(g)=u^{*}(f)(g)=\sup_{(g^{\prime},r,s)\in\Gamma_{\beta}(g)}|u(g^{\prime},r,s)|.$

The space $H^{1}_{F,\mathrm{nontan}}(\mathbb{H}^{\nu})$ is defined to be the
linear space of all $f\in L^{1}(\mathbb{H}^{\nu})$ such that
$\left\|u^{*}\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$, with norm

$\left\|f\right\|_{H^{1}_{F,\mathrm{nontan}}(\mathbb{H}^{\nu})}:=\left\|u^{*}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$
(6.1)

######  Definition 6.2.

The radial maximal function of $f\in L^{1}(\mathbb{H}^{\nu})$ is defined by

$\displaystyle u^{+}(g)=u^{+}(f)(g)=\sup_{r,s\in\mathbb{R}^{+}}|u(g,r,s)|.$

The space $H^{1}_{F,\mathrm{radial}}(\mathbb{H}^{\nu})$ is defined to be the
set of all $f\in L^{1}(\mathbb{H}^{\nu})$ such that
$\left\|u^{+}\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$, with norm

$\left\|f\right\|_{H^{1}_{F,\mathrm{radial}}(\mathbb{H}^{\nu})}:=\left\|u^{+}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$
(6.2)

######  Definition 6.3.

Let $F(\mathbb{H}^{\nu})$ be the collection of all pairs
$\boldsymbol{\varphi}$ of functions $\varphi^{(1)}$ on $\mathbb{H}^{\nu}$ and
$\varphi^{(2)}$ on $\mathbb{R}$ such that

$\displaystyle\left|\Delta_{(1)}^{M}\varphi^{(1)}(g)\right|$
$\displaystyle\leq\frac{1}{(1+\left\|g\right\|^{2})^{M+2N+D/2+1/2}}\qquad\forall
g\in\mathbb{H}^{\nu}$
$\displaystyle\left|\Delta_{(2)}^{N}\varphi^{(2)}(t)\right|$
$\displaystyle\leq\frac{1}{(1+\left\|t\right\|^{2})^{M+1}}\qquad\forall
t\in\mathbb{R}.$

The _grand maximal function $\mathcal{M}_{F,\mathrm{gmax}}(f)$ of $f\in
L^{1}(\mathbb{H}^{\nu})$_ is defined by

$\displaystyle\mathcal{M}_{F,\mathrm{gmax}}(f)(g)=\sup_{\boldsymbol{\varphi}\in
F(\mathbb{H}^{\nu})}\sup_{r,s\in\mathbb{R}^{+}}\left|f\ast_{(1)}\varphi_{r,s}(g)\right|\qquad\forall
g\in\mathbb{H}^{\nu}.$

The space $H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})$ is defined to be the set
of all $f\in L^{1}(\mathbb{H}^{\nu})$ such that
$\left\|\mathcal{M}_{F,\mathrm{gmax}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}<\infty$,
with norm

$\left\|f\right\|_{H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})}:=\left\|\mathcal{M}_{F,\mathrm{gmax}}(f)\right\|_{L^{1}(\mathbb{H}^{\nu})}.$
(6.3)

In this definition, we write $\varphi_{r,s}$ for the convolution
$\varphi^{(1)}_{r}\ast_{(1)}\varphi^{(2)}_{s}$, where $\varphi^{(1)}_{r}$ and
$\varphi^{(2)}_{s}$ denote the normalised dilates of $\varphi^{(1)}$ and
$\varphi^{(2)}$.

In particular, the pairs of heat kernels $(h^{(1)}_{1},h^{(2)}_{1})$ and pairs
of Poisson kernels $(p^{(1)}_{1},p^{(2)}_{1})$ and all their translates by
elements of bounded subsets of $\mathbb{H}^{\nu}$ and of $\mathbb{R}$ belong
to $F(\mathbb{H}^{\nu})$, at least up to multiples. It follows that the grand
maximal function dominates the nontangential and the radial maximal functions.
Further, we may dominate $\mathcal{M}_{F,\mathrm{gmax}}$ by the flag maximal
operator (see Section 2.5), so $\mathcal{M}_{F,\mathrm{gmax}}$ is
$L^{p}$-bounded for all $p\in(1,\infty]$.

We have seen in Sections 4 and 3 that
$H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{atom}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})$. The remarks above show that
$H^{1}_{F,\mathrm{gmax}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{nontan}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{radial}}(\mathbb{H}^{\nu})$. We complete the identification
of these five spaces by showing that
$H^{1}_{F,\mathrm{nontan}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$ and
$H^{1}_{F,\mathrm{radial}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{nontan}}(\mathbb{H}^{\nu})$. Here, all inclusions are
continuous, and there are also corresponding norm inequalities.

###  6.1 The inclusion $H^{1}_{F,\mathrm{nontan}}(\mathbb{H}^{\nu})\subseteq
H^{1}_{F,\mathrm{area}}(\mathbb{H}^{\nu})$

Let $p^{(1)}_{r}$ denote the Poisson kernel associated to $\Delta_{(1)}$ on
$\mathbb{H}^{\nu}$. It is easy to see that, if $f\in L^{p}(\mathbb{H}^{\nu})$,
then $u$, given by $u(r,g):=f\ast_{(1)}p_{r}(g)$ for all
$(r,g)\in\mathbb{R}^{+}\times\mathbb{H}^{\nu}$, is harmonic, in the sense that
$\Delta^{(1)}u=0$, where $\Delta^{(1)}=\Delta_{(1)}-\partial_{r}^{2}$ (recall
that the operator $\Delta_{(1)}$ was normalised to be positive). Moreover, if
$u$ is a harmonic function on $\mathbb{R}^{+}\times\mathbb{H}^{\nu}$, that is,
$\Delta^{(1)}u=0$, then

$\displaystyle\Delta^{(1)}(u^{2})(r,g)=2\left|\nabla^{(1)}u(r,g)\right|^{2}$
(6.4)

in $\mathbb{R}^{+}\times\mathbb{H}^{\nu}$, where

$\displaystyle\nabla^{(1)}=(\nabla_{(1)},\partial_{r})=(X_{1},\ldots,X_{\nu},Y_{1},\ldots,Y_{\nu},\partial_{r}).$
(6.5)

Write $p^{(2)}_{s}$ for the usual Poisson kernel on $\mathbb{R}$, and
$\nabla^{(2)}$ for the full gradient on $\mathbb{R}\times\mathbb{R}+$, that
is,

$\nabla^{(2)}=\bigl{(}\partial_{x},\partial_{s}\bigr{)}\qquad\forall(x,s)\in\mathbb{R}\times\mathbb{R}^{+},$

and we define $\Delta^{(2)}:=\partial_{x}^{2}-\partial_{s}^{2}$. As usual, we
shall write $p_{r,s}$ for $p^{(1)}_{r}\ast_{(2)}p^{(2)}_{s}$.

######  Definition 6.4.

For $f\in L^{1}(\mathbb{H}^{\nu})$, the Lusin–Littlewood–Paley area integral
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{p}}(f)$ of $f$ associated to the
Poisson kernel is defined by

$\displaystyle\mathcal{S}_{F,\mathrm{area},\boldsymbol{p}}(f)(g):=\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\bigl{|}f\ast_{(1)}r\nabla^{(1)}p^{(1)}_{r}\ast_{(2)}s\nabla^{(2)}p^{(2)}_{s}\bigr{|}^{2}\ast_{(1)}\chi^{(1)}_{r}\ast_{(2)}\chi^{(2)}_{s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}.$
(6.6)

where $\chi_{r}^{(1)}$ and $\chi_{s}^{(2)}$ are the normalised indicator
functions of the unit balls in $\mathbb{H}^{\nu}$ and $\mathbb{R}$, as in
Section 2.1.

We may write the area integral
$\mathcal{S}_{F,\mathrm{area},\boldsymbol{p}}(f)$ more briefly as

$\biggl{(}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}(\not{\nabla}p)_{r,s}\right|^{2}\ast_{(1)}\chi_{r,s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\biggr{)}^{1/2}$

where $\not{\nabla}p_{r,s}$ denotes the tensor-valued function
$(\nabla^{(1)}p^{(1)})_{r}\ast_{(2)}(\nabla^{(2)}p^{(2)})_{s}$ and
$\chi_{r,s}=\chi_{r}^{(1)}\ast_{(2)}\chi_{s}^{(2)}$. Now

$\displaystyle\int_{A(\alpha)}\mathcal{S}_{F,\mathrm{area},\boldsymbol{p}}(f)^{2}(g)\,dg$
$\displaystyle=\int_{A(\alpha)}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}(\not{\nabla}p)_{r,s}\right|^{2}\ast_{(1)}\chi_{r,s}(g)\,\frac{dr}{r}\,\frac{ds}{s}\,dg$
(6.7)
$\displaystyle=\int_{A(\alpha)}\int_{\mathbb{H}^{\nu}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\left|f\ast_{(1)}(\not{\nabla}p)_{r,s}(gg^{\prime})\right|^{2}\chi_{r,s}(g^{\prime-1})\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}\,dg$
$\displaystyle\leq\int_{\mathbb{H}^{\nu}}\int_{\mathbb{H}^{\nu}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\mathrm{H}(gg^{\prime},r,s)\left|f\ast_{(1)}(\not{\nabla}p)_{r,s}(gg^{\prime})\right|^{2}\chi_{r,s}(g^{\prime-1})\,\frac{dr}{r}\,\frac{ds}{s}\,dg^{\prime}\,dg$
$\displaystyle=\int_{\mathbb{H}^{\nu}}\int_{\mathbb{H}^{\nu}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\mathrm{H}(gg^{\prime},r,s)\left|f\ast_{(1)}(\not{\nabla}p)_{r,s}(gg^{\prime})\right|^{2}\chi_{r,s}(g^{\prime-1})\,\frac{dr}{r}\,\frac{ds}{s}\,dg\,dg^{\prime}$
$\displaystyle=\int_{\mathbb{H}^{\nu}}\iint_{\mathbb{R}^{+}\times\mathbb{R}^{+}}\mathrm{H}(g,r,s)\left|f\ast_{(1)}(\not{\nabla}p)_{r,s}(g)\right|^{2}\,\frac{dr}{r}\,\frac{ds}{s}\,dg,$

provided that
$\mathrm{H}:\mathbb{H}^{\nu}\times\mathbb{R}^{+}\times\mathbb{R}^{+}\to[0,\infty)$
is chosen such that $\mathrm{H}(gg^{\prime},r,s)=1$ whenever $g\in A(\alpha)$
and $g^{\prime}\in\operatorname{supp}(\chi_{r,s})$.

######  Theorem 6.5.