# Marcinkiewicz multipliers associated with the Kohn Laplacian on the Shilov
boundary
of the product domain in $\mathbb{C}^{2n}$

Peng Chen  Department of Mathematics, Sun Yat-Sen University, Guangzhou, P. R.
China<EMAIL_ADDRESS>,  Michael G. Cowling  School of Mathematics and
Statistics, University of New South Wales, Sydney 2052, Australia
<EMAIL_ADDRESS>,  Guorong Hu  College of Mathematics and Information
Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China
<EMAIL_ADDRESS>and  Ji Li  Department of Mathematics, Macquarie University,
Sydney, NSW 2109, Australia<EMAIL_ADDRESS>
###### Abstract.

Let $M^{(k)}$, $k=1,2,\ldots,n$, be the boundary of an unbounded polynomial
domain $\Omega^{(k)}$ of finite type in $\mathbb{C}^{2}$, and let
$\Box_{b}^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study
multivariable spectral multipliers $m(\Box_{b}^{(1)},\ldots,\Box_{b}^{(n)})$
acting on the Shilov boundary $\widetilde{M}=M^{(1)}\times\cdots\times
M^{(n)}$ of the product domain $\Omega^{(1)}\times\cdots\times\Omega^{(n)}$.
We show that if a function $F(\lambda_{1},\ldots,\lambda_{n})$ satisfies a
Marcinkiewicz-type differential condition, then the spectral multiplier
operator $m(\Box_{b}^{(1)},\ldots,\Box_{b}^{(n)})$ is a product
Calderón–Zygmund operator of Journé type.

###### Key words and phrases:

Nonisotropic smoothing operator of order zero, product singular integral of
Journé type, Marcinkiewicz multiplier, Kohn Laplacian

###### 2010 Mathematics Subject Classification:

32A50, 32A55, 32T15, 32W30, 32W10

##  1\. Introduction and statement of main result

The Shilov boundary $\widetilde{M}=M^{(1)}\times\cdots\times M^{(n)}$ of a
product domain $\Omega^{(1)}\times\cdots\times\Omega^{(n)}$ in
$\mathbb{C}^{2n}$ is an important model in several complex variables that
extends the standard case of the bidisc [12]. Here each $M^{(k)}$ is the
boundary of an unbounded polynomial domain
$\Omega^{(k)}:=\\{(z,w)\in\mathbb{C}^{2}:\operatorname{Im}(w)>P^{(k)}(z)\\}$,
where $P^{(k)}$ is a real, subharmonic, nonharmonic polynomial of degree
$m_{k}$ (see [11]). In [12], Nagel and Stein built the theory of product
singular integrals and Littlewood–Paley theory on $\widetilde{M}$, based on
the solution to the initial value problem and the regularity properties of the
heat operator for the Kohn Laplacian $\Box_{b}^{(k)}$ on each $M^{(k)}$. This
work leads immediately to optimal estimates for the behaviour of the solution
of the Kohn–Laplace equation on the decoupled boundary in $\mathbb{C}^{n+1}$
[13]. It is worth pointing out that the heat kernels of the Kohn Laplacian
$\Box_{b}$ on the boundary $M$ (here we drop the superscript (k)) do not
satisfy standard Gaussian upper bounds; indeed, Nagel and Müller (unpublished)
showed that the kernel of $e^{-s\Box_{b}}$ satisfies the pointwise upper bound

(1.1)
$\left|K_{e^{-s\Box_{b}}}(x,y)\right|\leq\frac{C}{V(x,\rho(x,y))}\exp\left(-c{\rho(x,y)^{2}/s}\right)$

for some positive constants $C$ and $c$ (see also [15]). Here $\rho$ is the
control metric on $M$ and $V(x,\delta)$ is the volume of the metric ball
$B(x,\delta):=\\{y\in M:\rho(x,y)<\delta\\}$. The measure $V$ is doubling,
with upper dimension $Q$, and reverse doubling, with lower dimension $4$. See
Section 2 for the details.

One of the important types of singular integrals on the Shilov boundary
$\widetilde{M}$ is the family of Marcinkiewicz multipliers associated with the
Kohn Laplacians, that is, the operators
$m(\Box_{b}^{(1)},\ldots,\Box_{b}^{(n)})$. The aim of this paper is to
investigate the behaviour of these multipliers. We show that if the function
$m(\lambda_{1},\ldots,\lambda_{n})$ satisfies a suitable smoothness condition,
defined using Sobolev norms, then $m(\Box_{b}^{(1)},\ldots,\Box_{b}^{(n)})$ is
a product Calderón–Zygmund operator of Journé type.

To ease the burden of notational complexity, in what follows we consider only
the two parameter case, that is, $n=2$.

Marcinkiewicz multipliers in the Euclidean setting provide one of the most
important examples of product Calderón–Zygmund operators. It is well-known
that such multipliers are bounded on $L^{p}$ for $1<p<\infty$. To the best of
our knowledge, the earliest such result is due to Gundy and Stein [4] in 1979;
they investigated the operator $m(\Delta_{1},\Delta_{2})$ with
$m:{\mathbb{R}^{2}}\to{\mathbb{C}}$ a bounded function and $\Delta_{i}$ the
standard Laplace operator on the Euclidean space $\mathbb{R}^{n_{i}}$, for
$i=1$, 2. They proved that, if $\alpha_{0},\beta_{0}\in\mathbb{N}$ are
sufficiently large, then $m(\Delta_{1},\Delta_{2})$, initially defined on
$L^{2}({\mathbb{R}}^{n_{1}}\times\mathbb{R}^{n_{2}})$, extends to a bounded
operator on $L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})$ for all
$p\in(1,\infty)$ provided the function $m$ satisfies

(1.2)
$\left|\partial_{\lambda_{2}}^{\beta}\partial_{\lambda_{1}}^{\alpha}m(\lambda_{1},\lambda_{2})\right|\lesssim\lambda_{1}^{-\alpha}\lambda_{2}^{-\beta}$

for all $\alpha\leq\alpha_{0}$ and $\beta\leq\beta_{0}$. Their proof uses the
pointwise majorization of $m(\Delta_{1},\Delta_{2})$ by the Littlewood–Paley
product $g$ and $g_{\lambda}$ functions. Another approach, due to L.K. Chen
[1], uses $(H^{1},L^{1})$ estimates for $L^{2}$-bounded linear operators,
where $H^{1}$ is the product Hardy space introduced by Gundy and Stein and
further studied by Chang and Fefferman. Chen proved that it suffices to take
$\alpha_{0}$ and $\beta_{0}$ equal to when $\lfloor{n_{1}}/{2}\rfloor+1$ and
$\lfloor{n_{2}}/{2}\rfloor+1$.

Recently, this Marcinkiewicz multiplier was investigated in the abstract
setting of spaces of homogeneous type with the Marcinkiewicz multiplier
$m(L_{1},L_{2})$ associated with nonnegative self-adjoint second-order
operators whose heat kernels satisfy Gaussian upper bounds [2]. Suppose that
$X_{1}\times X_{2}$ is the Cartesian product of measure spaces $X_{1}$ and
$X_{2}$ and that $L_{1}$ and $L_{2}$ are nonnegative self-adjoint operators
acting on the spaces $L^{2}(X_{1})$ and $L^{2}(X_{2})$. Let $E_{L_{i}}$ be the
projection valued measure associated to $L_{i}$. There is a unique spectral
decomposition $E$ such that for all Borel subsets $A\subset\mathbb{R}^{2}$,
$E(A)$ is a projection on $L^{2}({X_{1}\times X_{2}})$ and $E(A_{1}\times
A_{2})=E_{L_{1}}(A_{1})\otimes E_{L_{2}}(A_{2})$ for all Borel subsets $A_{1}$
and $A_{2}$ of $\mathbb{R}$. Hence for a bounded Borel function
$m:\mathbb{R}^{2}\to{\mathbb{C}}$, one may define the spectral multiplier
operator $m(L_{1},L_{2})$ acting on the space $L^{2}({X_{1}\times X_{2}})$ by
the formula

(1.3)
$m(L_{1},L_{2}):=\iint_{\mathbb{R}\times\mathbb{R}}m(\lambda_{1},\lambda_{2})\,dE(\lambda_{1},\lambda_{2}).$

Clearly $m(L_{1},L_{2})$ is bounded on $L^{2}({X_{1}\times X_{2}})$. If the
function $F$ satisfies a suitable mixed Sobolev condition with regularity
$s_{1}>n_{1}/2$ and $s_{2}>n_{2}/2$, then $m(L_{1},L_{2})$ is bounded on
$L^{p}({X_{1}\times X_{2}})$ for $1<p<\infty$ ([2]). Here $n_{i}$ is the upper
dimension of $X_{i}$. The proof uses a ($H^{1}$, $L^{1}$) estimate, where the
$H^{1}$ is the product Hardy space associated with $L_{1}$ and $L_{2}$
introduced in [3].

In the single-parameter compact manifold $M$, Street [16] proved that the
spectral multiplier $m(\Box_{b})$ is an anisotropic smooth (NIS) operator of
order zero if $m:[0,\infty)\to\mathbb{C}$ satisfies the Mihlin–Hörmander
condition $|(\lambda\partial_{\lambda})^{a}m(\lambda)|\leq C_{a}$ for all
$\lambda\in(0,\infty)$ and all $a\in\mathbb{N}_{0}$, then $m(\Box_{b})$ is
bounded on $L^{p}(M)$ for all $1<p<\infty$. Moreover, he pointed out that if
one is only concerned with $L^{p}$ boundedness, then a sufficient condition is

$\sup_{t>0}\|\eta(\cdot)m(t\cdot)\|_{W^{\sigma,2}(\mathbb{R})}<\infty$

where $\sigma>{Q+1/2}$; here $\eta$ is a nontrivial cut-off function on
$(0,\infty)$ and $W^{\sigma,2}(\mathbb{R})$ is the usual fractional Sobolev
space on $\mathbb{R}$. The expected result for $m(\Box_{b})$ in the unbounded
polynomial domain is also discussed in [16, Section 8].

In light of [15], it is natural to expect that the Marcinkiewicz multiplier
$m(\Box_{b}^{(1)},\Box_{b}^{(2)})$ is a product NIS operator of order zero
when $m$ satisfies the decay condition for infinitely many partial
derivatives. Here we provide a more general characterisation. We aim to prove
that when $m$ satisfies a suitable Sobolev estimate,
$m(\Box_{b}^{(1)},\Box_{b}^{(2)})$ is a product Calderón–Zygmund operator of
Journé type [6]. In this metric space setting, Journé-type product
Calderón–Zygmund operators, and the corresponding product Hardy and BMO spaces
and product T1 theorem were studied in [5].

To state our main result, we write $W^{\sigma,2}(\mathbb{R})$ for the usual
fractional Sobolev space on $\mathbb{R}$, and
$W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})$ for the product
Sobolev space whose norm is given by

$\|f\|_{W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}=\|(1+|\xi_{1}|^{2})^{\sigma_{1}/2}(1+|\xi_{2}|^{2})^{\sigma_{2}/2}\widehat{f}(\xi_{1},\xi_{2})\|_{L^{2}(\mathbb{R}\times\mathbb{R})},$

and $\eta$ for a $C_{0}^{\infty}(\mathbb{R})$-function supported in $[1/2,2]$
such that $\eta=1$ in a neighborhood of $1$.

######  Theorem 1.1.

Let $Q_{1}$ and $Q_{2}$ be the upper dimensions of the spaces $M^{(1)}$ and
$M^{(2)}$. Suppose that $\sigma_{1}>(Q_{1}+3)/{2}$ and
$\sigma_{2}>(Q_{2}+3)/{2}$, and that
$m:[0,\infty)\times[0,\infty)\to\mathbb{C}$ satisfies

(1.4)
$\displaystyle\sup_{t_{1},t_{2}>0}\|\eta(\lambda_{1})\eta(\lambda_{2})m(t_{1}\lambda_{1},t_{2}\lambda_{2})\|_{W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}<\infty,$
(1.5)
$\displaystyle\sup_{t_{1}>0}\|\eta(\lambda_{1})m(t_{1}\lambda_{1},0)\|_{W^{\sigma_{1},2}(\mathbb{R})}<\infty$

and

(1.6)
$\displaystyle\sup_{t_{2}>0}\|\eta(\lambda_{2})m(0,t_{2}\lambda_{2})\|_{W^{\sigma_{2},2}(\mathbb{R})}<\infty.$

Then the Marcinkiewicz multiplier $m(\Box_{b}^{(1)},\Box_{b}^{(2)})$, defined
on $L^{2}(\widetilde{M})$ by spectral theory, is a product Calderón–Zygmund
operator on $\widetilde{M}$ as defined in [5], and hence is bounded on
$L^{p}(\widetilde{M})$ whenever $1<p<\infty$.

We recall that the upper bound (1.1) for the heat kernel of the Kohn Laplacian
$\Box_{b}^{(i)}$ is singular on the diagonal, that is, $\\{(x,y)\in
M^{(i)}\times M^{(i)}:x=y\\}$. Thus, our setting is quite different from the
setting where the semigroup has Gaussian upper bounds.

######  Remark 1.2.

Suppose that $m:[0,\infty)\times[0,\infty)\to\mathbb{C}$ satisfies

$\left|\partial_{\lambda_{1}}^{k_{1}}\partial_{\lambda_{2}}^{k_{2}}m(\lambda_{1},\lambda_{2})\right|\leq
C_{k_{1},k_{2}}\lambda_{1}^{-k_{1}}\lambda_{2}^{-k_{2}}$

for all $k_{1},k_{2}\in\mathbb{N}_{0}$ with
$k_{1}\leq\lfloor\sigma_{1}\rfloor+1$ and
$k_{2}\leq\lfloor\sigma_{2}\rfloor+1$, and all $\lambda_{1},\lambda_{2}>0$.
Then $m$ satisfies conditions (1.4) to (1.6).

######  Remark 1.3.

In view of [15], it should be possible to use product Littlewood–Paley theory
and domination by the strong maximal function to obtain the $L^{p}$
boundedness of the multiplier $m(\Box_{b}^{(1)},\Box_{b}^{(2)})$ under the
conditions $\sigma_{1}>{(Q_{1}+1)}/{2}$ and $\sigma_{2}>{(Q_{2}+1)}/{2}$.
However, it is an open question whether one can get rid of the ${1/2}$, since
tools such as the Plancherel theorem or restriction estimates are not
available.

The paper is organised as follows. In Section 2 we recall necessary notions on
polynomial domains in $\mathbb{C}^{2}$ and the Shilov boundary in
$\mathbb{C}^{2n}$, as well as the product singular integrals of Journé type.
In Section 3, we establish pointwise kernel estimate for the multipliers when
the function $m$ is smooth enough and has compact support. In the last section
we prove the main result.

Throughout this paper, $\mathbb{N}$ denotes the set of positive integers,
while $\mathbb{N}_{0}$ denotes the set of nonnegative integers, that is,
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. We will use the symbols $c$ and $C$ to
denote positive constants, whose values may vary at every occurrence. If
$f\leq Cg$ for some constant $C$ which is independent of the main variables
involved, then we write $f\lesssim g$. If $f\lesssim g\lesssim f$, we write
$f\sim g$.

##  2\. Preliminaries

In this section we recall some basic notions about polynomial domains in
$\mathbb{C}^{2}$ and Shilov boundaries in $\mathbb{C}^{2n}$, as well as
product Calderón–Zygmund operators on $M^{(1)}\times M^{(2)}$.

We start by discussing the single factor case. Let
$\Omega:=\\{(z,w)\in\mathbb{C}^{2}:\operatorname{Im}(w)>P(z)\\}$, where $P$ is
a real, subharmonic, nonharmonic polynomial of finite degree $\nu$, as in
[11]. Then $M:=\partial\Omega$ can be identified with
$\mathbb{C}\times\mathbb{R}=\\{(z,t):\ z\in\mathbb{C},t\in\mathbb{R}\\}$. The
basic $(0,1)$ Levi vector field is then
$\bar{Z}={\partial/{\partial\bar{z}}}-i{({\partial
P}/{\partial\bar{z}})}{\partial/{\partial t}}$, and we write
$\bar{Z}=X_{1}+iX_{2}$. The real vector fields $\\{X_{1},X_{2}\\}$ and their
commutators of order at most $m$ span the tangent space to $M$ at each point
on $M$. A curve $\gamma:[0,1]\to M$ is said to be _horizontal_ if
$\gamma^{\prime}(t)\in\mathrm{span}\\{X_{1},X_{2}\\}$ at each point, and we
define the length of such a horizontal curve by

$\mathrm{Len}(\gamma)=\int_{0}^{1}|\gamma^{\prime}(t)|\,dt,$

where $|\gamma^{\prime}(t)|^{2}=|c_{1}(t)|^{2}+|c_{2}(t)|^{2}$ when
$\gamma^{\prime}(t)=c_{1}(t)X_{1}+c_{2}(t)X_{2}$. We denote by $\rho$ the
control (or Carnot–Carathéodory) distance in $M$ generated by $X_{1}$ and
$X_{2}$, defined by

$\displaystyle\rho(x,y):=\inf\bigg{\\{}\mathrm{Len}(\gamma):\gamma\text{
horizontal},\gamma(0)=x,\gamma(1)=y\biggr{\\}}.$

We put $B(x,r):=\\{y\in M:\rho(x,y)<r\\}$. Let $\mu$ be the Lebesgue measure
on $M$ and define $V(x,r):=\mu(B(x,r))$. Then

(2.1)
$V(x,\delta)=\biggl{(}\sum_{k=2}^{m}\Lambda_{k}(x)\delta^{k}\biggr{)}\delta^{2},$

where the $\Lambda_{k}(x)$ are continuous, non-negative functions on $M$ (see
[10, Section 4.1]). Moreover, if $Q=m+2$, then for all $x\in M,\lambda\geq 1$
and $\delta>0$,

(2.2) $\displaystyle V(x,\lambda\delta)$ $\displaystyle\leq
C\lambda^{Q}V(x,\delta)$ (2.3) $\displaystyle V(x,\lambda\delta)$
$\displaystyle\geq C\lambda^{4}V(x,\delta)$

for all $x\in M$, $\lambda\geq 1$ and $\delta>0$,

Denote by $X_{j}^{\ast}$ the formal adjoint of $X_{j}$, that is,

$\langle X_{j}^{\ast}\varphi,\psi\rangle=\langle\varphi,X_{j}\psi\rangle,$

where $\langle\varphi,\psi\rangle=\int_{M}\varphi(x)\psi(x)\,dx$ for all
$\varphi,\psi\in C^{\infty}(M)$. In general, $X_{j}^{\ast}=-X_{j}+a_{j}$,
where $a_{j}\in C^{\infty}(M)$. The Kohn Laplacian $\Box_{b}$ on $M$ is
formally defined by

$\displaystyle\Box_{b}:=X_{1}^{\ast}X_{1}+X_{2}^{\ast}X_{2}.$

Set the heat operator $T_{t}:=e^{-t\Box_{b}}$ for $t>0$. As recalled in (1.1),
the kernel of $T_{t}$ satisfies the non-Gaussian upper bounds in terms of the
control distance $\rho$ on $M$.

Since $\Box_{b}$ is a self-adjoint operator, it admits a spectral
decomposition $E(\lambda)$; in particular, $E(0)=\pi$, where $\pi$ is the
Szegő projection on $M$. Hence, for any bounded Borel measurable function
$m:[0,\infty)\to\mathbb{C}$, we may define

$\displaystyle m(\Box_{b})=\int_{[0,\infty)}m(\lambda)\,dE(\lambda)$

and, with an abuse of notation, we define

$\displaystyle
m(\widehat{\square}_{b})=\int_{(0,\infty)}m(\lambda)\,dE(\lambda),$

so that

$\displaystyle
m(\widehat{\square}_{b})=(1-\pi)m(\Box_{b})=m(\Box_{b})-m(0)\pi.$

To describe higher-order derivatives on $M$ we need more notation. For each
$j\in\mathbb{N}$, we set

$\displaystyle\mathcal{I}_{j}:=\\{1,2\\}^{j}=\\{(\alpha_{1},\cdots,\alpha_{j}):\alpha_{1},\cdots,\alpha_{j}\in\\{1,2\\}\\}.$

For $\alpha=(\alpha_{1},\cdots,\alpha_{j})\in\mathcal{I}_{j}$, we define
$|\alpha|=j$ and call $|\alpha|$ the length of the multi-index $\alpha$; we
also set

$X_{\alpha}:=X_{\alpha_{1}}\cdots X_{\alpha_{j}}.$

For instance, if $\alpha=(1,2,2,1)$, then $|\alpha|=4$ and
$X_{\alpha}=X_{1}X_{2}X_{2}X_{1}$. By convention, let
$\mathcal{I}_{0}=\\{(0)\\}$, and for $\alpha\in\mathcal{I}_{0}$ (that is,
$\alpha=(0)$), we define $|\alpha|=0$ and $X_{\alpha}=I$ (the identity
operator). We also put

$\displaystyle\mathcal{I}=\bigcup_{j=0}^{\infty}\mathcal{I}_{j},$

and write $f\in L^{p}_{k}(M)$ if $X_{\alpha}f\in L^{p}(M)$ whenever
$|\alpha|\leq k$.

Anisotropic smoothing operators (NIS operators for short) were introduced in
[9]. Here we use the definition from [7, Definition 5.10].

######  Definition 2.1.

Let $T:C_{0}^{\infty}(M)\to C^{\infty}(M)$ be a linear operator, with Schwartz
kernel $K_{T}(x,y)$. We say that $T$ is an NIS operator, smoothing of order
$r$, if $K_{T}$ is $C^{\infty}$ away from the diagonal of $M\times M$ and the
following conditions are satisfied:

  1. (i)

For all $s\geq 0$, there exist parameters $a(s)<\infty$ and $b<\infty$ such
that if $\zeta,\zeta^{\prime}\in C_{0}^{\infty}(M)$ and $\zeta^{\prime}\equiv
1$ on the support of $\zeta$, then there exists $C=C(s,\zeta,\zeta^{\prime})$
such that

$\displaystyle\|\zeta Tf\|_{L^{2}_{s}(M)}\leq
C\left(\|\zeta^{\prime}f\|_{L^{2}_{a(s)}(M)}+\|f\|_{L_{b}^{2}(M)}\right)\qquad\forall
f\in C_{0}^{\infty}(M).$

  2. (ii)

There exist constants $C_{\alpha,\beta}$ such that for $x\neq y$,

$\displaystyle\left|(X_{\alpha})_{x}(X_{\beta})_{y}K_{T}(x,y)\right|\leq
C_{\alpha,\beta}\frac{\rho(x,y)^{r-|\alpha|-|\beta|}}{V(x,\rho(x,y))}.$

  3. (iii)

For each integer $\ell\geq 0$, there is an integer $N=N(\ell)\geq 0$ and a
constant $C=C(\ell)$ such that if $\phi\in C_{0}^{\infty}(B(x,\delta))$,
then111What does $D^{\beta}\phi$ mean?

$\displaystyle\sum_{|\alpha|=\ell}\left|X_{\alpha}T(\phi)(x)\right|\leq
C\delta^{r-\ell}\sup_{y\in M}\sum_{|\beta|\leq
N}\delta^{|\beta|}|X_{\beta}\phi(y)|.$

  4. (iv)

The above conditions also hold for the adjoint operator $T^{\ast}$.

It is shown in [9, 7] that $\pi$ is an NIS operator of order zero and all NIS
operators of order zero are bounded on $L^{p}(M)$.

Nagel and Stein [12] introduced a class of singular integrals on $M$ which
includes NIS operators of order zero and proved that they are still bounded on
$L^{p}(M)$ for $1<p<\infty$. The main difference between their singular
integral operators and NIS operators of order zero is that the former are not
assumed to satisfy the condition (i) of Definition 2.1. In the same paper,
products of such singular integral operators are also studied.

For NIS operators and the singular integrals operators of Nagel and Stein
[12], condition (iii) is called a cancellation condition; it guarantees
boundedness on $L^{2}(M)$. For spectral multipliers, the $L^{2}$-boundedness
holds from the definition via spectral theory.

We will need the definition of single-parameter Calderón–Zygmund operators on
$M$. A continuous function $K(x,y)$ defined on $M\times
M\backslash\\{(x,y):x=y\\}$ is called a _Calderón–Zygmund kernel_ if there
exists a constant $C>0$ and a regularity exponent $\varepsilon\in(0,1]$ such
that

(i) $\displaystyle|K(x,y)|$ $\displaystyle\leq CV(x,y)^{-1};$ (ii)
$\displaystyle|K(x,y)-K(x,y^{\prime})|$ $\displaystyle\leq
C\biggl{(}\frac{\rho(y,y^{\prime})}{\rho(x,y)}\biggr{)}^{\varepsilon}V(x,y)^{-1}\quad\text{if}\quad\rho(y,y^{\prime})\leq\frac{\rho(x,y)}{2};$
(iii) $\displaystyle|K(x,y)-K(x^{\prime},y)|$ $\displaystyle\leq
C\biggl{(}\frac{\rho(x,x^{\prime})}{\rho(x,y)}\biggr{)}^{\varepsilon}V(x,y)^{-1}\quad\text{if}\quad\rho(x,x^{\prime})\leq\frac{\rho(x,y)}{2}.$

The smallest such constant $C$ is denoted by $|K|_{CZ}$. We say that an
operator $T$ is a _singular integral operator_ associated with a
Calderón–Zygmund kernel $K$ if the operator $T$ is a continuous linear
operator from $C^{\infty}_{0}(M)$ into its dual such that

$\langle Tf,g\rangle=\iint g(x)K(x,y)f(y)\,d\mu(y)\,d\mu(x)$

for all functions $f,g\in C^{\infty}_{0}(M)$ with disjoint supports; $T$ is
said to be a _Calderón–Zygmund operator_ if it extends to be a bounded
operator on $L^{2}(M)$. If $T$ is a Calderón–Zygmund operator associated with
a kernel $K$, its operator norm is defined by

$\|T\|_{CZ}=\|T\|_{L^{2}\to L^{2}}+|K|_{CZ}.$

We now turn to the product case, that is, the case of two factors. Let
$M^{(k)}$ be the boundary of an unbounded model polynomial domain
$\Omega^{(k)}$, where $k=1,2$. We will use superscripts to indicate operators
or quantities on $M^{(k)}$. For example, the Kohn Laplacian, control distance,
measure, volume of a ball and higher-order derivatives will be denoted by
$\Box_{b}^{(k)},\rho^{(k)}(\cdot,\cdot)$, $d\mu^{(k)}$, $V^{(k)}(x,r)$ and
$X_{\alpha}^{(k)}$ respectively.

Now we recall the definition of _product singular integral operators_ on
$\widetilde{M}=M^{(1)}\times M^{(2)}$. A linear operator
$T:C^{\infty}_{0}({\widetilde{M}})\mapsto[C^{\infty}_{0}({\widetilde{M}})]^{\prime}$
is said to be a singular integral operator if there exist Calderón–Zygmund-
operator-valued operators $K_{1}$ on $M^{(2)}$ and $K_{2}$ on $M^{(1)}$ with
the following properties:

(i) $\displaystyle\langle g\otimes k,Tf\otimes h\rangle=\iint g(x_{1})\langle
k,K_{1}(x_{1},y_{1})h\rangle f(y_{1})\,d\mu^{(1)}(x_{1})\,d\mu^{(1)}(y_{1})$
(ii) $\displaystyle\langle k\otimes g,Th\otimes f\rangle=\iint g(x_{2})\langle
k,K_{2}(x_{2},y_{2})h\rangle f(y_{2})\,d\mu^{(2)}(x_{2})\,d\mu^{(2)}(y_{2})$
(iii) $\displaystyle\hskip 65.44142pt\|K_{k}(x_{k},y_{k})\|_{CZ}\leq
CV^{(k)}(x_{k},y_{k})^{-1};$ (iv)
$\displaystyle\|K_{k}(x_{k},y_{k})-K_{k}(x_{k},y^{\prime}_{k})\|_{CZ}\leq
C\biggl{(}\frac{\rho^{(k)}(y_{k},y^{\prime}_{k})}{\rho^{(k)}(x_{k},y_{k})}\biggr{)}^{\varepsilon_{k}}V^{(k)}(x_{k},y_{k})^{-1}$
(v) $\displaystyle\|K_{k}(x_{k},y_{k})-K_{k}(y^{\prime}_{k},y_{k})\|_{CZ}\leq
C\biggl{(}\frac{\rho^{(k)}(x_{k},y^{\prime}_{k})}{\rho^{(k)}(x_{k},y_{k})}\biggr{)}^{\varepsilon_{k}}V^{(k)}(x_{k},y_{k})^{-1}$

A product singular integral operator $T$ is said to be a _product
Calderón–Zygmund operator_ if $T$ is bounded on $L^{2}(\widetilde{M})$.

######  Proposition 2.2 ([5]).

Suppose that $T$ is a product Calderón–Zygmund operator. Then $T$ extends to a
bounded operator on $L^{p}(\widetilde{M})$ when $1<p<\infty$.

For convenience in our application, we also provide the following version,
where the difference is replaced by derivative in the regularity condition of
the kernel. This is a special case of the above proposition.

######  Proposition 2.3.

Suppose that $T$ is bounded on $L^{2}(\widetilde{M})$, where
$\widetilde{M}=M^{(1)}\times M^{(2)}$. Suppose further that the distribution
kernel $K(x_{1},x_{2},y_{1},y_{2})$ of $T$ is $C^{\infty}$ away from the
“cross” $\\{(x_{1},x_{2},y_{1},y_{2}):(x_{1}-y_{1})(x_{2}-y_{2})=0\\}$ and
satisfies the following additional properties:
(i) $\langle T(\varphi_{1}\otimes\varphi_{2}),\psi_{1}\otimes\psi_{2}\rangle$
is given by

$\displaystyle\iiiint
K(x_{1},x_{2},y_{1},y_{2})\varphi_{1}(y_{1})\varphi_{2}(y_{2})\psi_{1}(x_{1})\psi_{2}(x_{2})$
$\displaystyle\hskip
130.88284pt\,d\mu^{(1)}(x_{1})\,d\mu^{(2)}(x_{2})\,d\mu^{(1)}(y_{1})\,d\mu^{(2)}(y_{2})$

whenever $\varphi_{1},\psi_{1}\in C_{0}^{\infty}(M^{(1)})$ have disjoint
supports and $\varphi_{2},\psi_{2}\in C_{0}^{\infty}(M^{(2)})$ have disjoint
supports.
(ii) For all functions $f_{2},g_{2}\in L^{2}(M^{(2)})$, there exists a
constant $C>0$ independent of $f_{2},g_{2}$ such that for all
$\alpha,\beta\in\\{0,1\\}$,

$\displaystyle\left|\int_{M^{(2)}}\int_{M^{(2)}}(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K(x_{1},x_{2},y_{1},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{(2)}(x_{2})\,d\mu^{(2)}(y_{2})\right|$
$\displaystyle\hskip
145.10922pt\leq\frac{C\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},y_{1})}\|f_{2}\|_{2}\|g_{2}\|_{2}.$

(iii) The kernel $K(x_{1},x_{2},y_{1},y_{2})$ satisfies the estimate

$\displaystyle|(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(Y^{(2)}_{\beta^{\prime}})_{y_{2}}K(x_{1},x_{2},y_{1},y_{2})|$
$\displaystyle\hskip
113.81102pt\lesssim\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}\rho^{(2)}(x_{2},y_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(1)}(x_{1},y_{1})V^{(2)}(x_{2},y_{2})}.$

(iv) Conditions (2) and (3) hold when the indices $1$ and $2$ are
interchanged, that is, if the roles of $M^{(1)}$ and $M^{(2)}$ are
interchanged.
(v) Conditions (2) to (4) hold for the three transposes of $T$, that is, the
operators that arise by interchanging $x_{1}$ and $y_{1}$, or interchanging
$x_{2}$ and $y_{2}$, or interchanging both pairs of variables.
Then $T$ extends to a bounded operator on $L^{p}(\widetilde{M})$ when
$1<p<\infty$.

######  Remark 2.4.

If $T_{1}$ and $T_{2}$ are (single-factor) singular integral operators on
$M^{(1)}$ and $M^{(2)}$, then $T=T_{1}\otimes T_{2}$ satisfies the above
assumptions. Here $T^{\varphi_{2},x_{2}}$ is equal to $T_{1}$ multiplied by
the factor $T_{2}(\varphi_{2})(x_{2})$.

##  3\. Functional calculus

In this section, our aim is to prove two pointwise kernel estimates for the
multipliers where the multiplier function is smooth enough and with compact
support.

We begin with two lemmas which state some useful elementary estimates for
functions. For one-parameter version of these estimates, see Street [15, 16].
We write $\mathcal{F}$ to indicate the Fourier transformation: subscripts $1$
and $2$ indicate Fourier transforms with respect to the first or second
variables only.

######  Lemma 3.1.

Suppose $m:[0,\infty)\times[0,\infty)\to\mathbb{C}$ is a function with
$\operatorname{supp}m\subset[1/4,4]\times[1/4,4]$, and let $\phi$ be a smooth
function on $\mathbb{R}$ such that

$\displaystyle\phi(\xi)=\begin{cases}0&\text{if $|\xi|\leq 1/4$}\\\ 1&\text{if
$|\xi|\geq 1/2$}.\end{cases}$

Set $\psi(\lambda_{1},\lambda_{2}):=m(\lambda_{1}^{2},\lambda_{2}^{2})$.
(i) Define
$F_{r_{1},r_{2}}(\lambda_{1},\lambda_{2}):=\psi(r_{1}\lambda_{1},r_{2}\lambda_{2})$.
Then, for any $\varepsilon>0$ and any $N_{1},N_{2}\in\mathbb{N}$, there exists
a constant $C_{\varepsilon,N_{1},N_{2}}$ such that

$\displaystyle|F_{r_{1},r_{2}}(\lambda_{1},\lambda_{2})|\leq
C_{\varepsilon,N_{1},N_{2}}\|m\|_{W^{(\varepsilon+1/2,\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}(1+r_{1}\lambda_{1})^{-N_{1}}(1+r_{2}\lambda_{2})^{-N_{2}}$

for all $r_{1},r_{2},\lambda_{1},\lambda_{2}>0$.
(ii) Define $F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ by the formula

$\displaystyle\mathcal{F}_{1}F_{s_{1},r_{1},r_{2}}(\xi_{1},\lambda_{2})=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}_{1}{\psi}\left(\frac{\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right).$

Then, for any $\varepsilon>0$, any $N_{1},N_{2}\in\mathbb{N}$ and any
$\sigma_{1}\in\mathbb{R}$, there exists a constant
$C_{\varepsilon,\sigma_{1},N_{1},N_{2}}$ such that

$\displaystyle|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})|$
$\displaystyle\leq
C_{\varepsilon,\sigma_{1},N_{1},N_{2}}\|m\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\quad\times\left[(1+s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\right](1+r_{2}\lambda_{2})^{-N_{2}}$

for all $r_{1},r_{2},s_{1},\lambda_{1},\lambda_{2}>0$.
(iii) Define $F_{s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ by the formula

$\displaystyle\mathcal{F}_{2}{F_{s_{2},r_{1},r_{2}}}(\lambda_{1},\xi_{2})=\phi\left(\frac{\xi_{2}}{s_{2}}\right)\frac{1}{r_{2}}\mathcal{F}_{2}{\psi}\left(r_{1}\lambda_{1},\frac{\xi_{2}}{r_{2}}\right).$

Then, for any $\varepsilon>0$, any $N_{1},N_{2}\in\mathbb{N}$, and any
$\sigma_{2}\in\mathbb{R}$, there exists a constant
$C_{\varepsilon,\sigma_{2},N_{1},N_{2}}$ such that

$\displaystyle|F_{s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2})|$
$\displaystyle\leq
C_{\varepsilon,\sigma_{2},N_{1},N_{2}}\|m\|_{W^{(\varepsilon+1/2,\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\qquad\times(1+r_{1}\lambda_{1})^{-N_{1}}\left[(1+s_{2}\lambda_{2})^{-N_{2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{(N_{2}+\varepsilon+{1/2})-\sigma_{2}}\right]$

for all $r_{1},r_{2},s_{2},\lambda_{1},\lambda_{2}>0$.
(iv) Define $F_{s_{1},s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ by the
formula

$\displaystyle\mathcal{F}{F_{s_{1},s_{2},r_{1},r_{2}}}(\xi_{1},\xi_{2})=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\phi\left(\frac{\xi_{2}}{s_{2}}\right)\frac{1}{r_{1}r_{2}}\mathcal{F}{\psi}\left(\frac{\xi_{1}}{r_{1}},\frac{\xi_{2}}{r_{2}}\right).$

Then, for any $\varepsilon>0$, any $N_{1},N_{2}\in\mathbb{N}$, and any
$\sigma_{1},\sigma_{2}\in\mathbb{R}$, there exists a constant
$C_{\varepsilon,\sigma_{1},\sigma_{2},N_{1},N_{2}}$ such that

$\displaystyle|F_{\sigma_{1},\sigma_{2},s_{1},s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2})|$
$\displaystyle\leq
C_{\varepsilon,\sigma_{1},\sigma_{2},N_{1},N_{2}}\|m\|_{W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\quad\times\left[(1+s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\right]$
$\displaystyle\quad\times\left[(1+s_{2}\lambda_{2})^{-N_{2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{(N_{2}+\varepsilon+{1/2})-\sigma_{2}}\right]$

for all $r_{1},r_{2},s_{1},s_{2},\lambda_{1},\lambda_{2}>0$.

###### Proof.

We first prove (i). Using the fact that
$W^{(\varepsilon+1/2,\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})\subset
L^{\infty}(\mathbb{R}\times\mathbb{R})\cap C(\mathbb{R}\times\mathbb{R})$ and
$\operatorname{supp}\psi\subset[1/2,2]\times[1/2,2]$, we see that

$\displaystyle|F_{r_{1},r_{2}}(\lambda_{1},\lambda_{2})|$ $\displaystyle\leq
3^{N_{1}}3^{N_{2}}\|\psi\|_{L^{\infty}(\mathbb{R}\times\mathbb{R})}(1+|\lambda_{1}|)^{-N_{1}}(1+|\lambda_{2}|)^{-N_{2}}$
$\displaystyle\leq
C_{\varepsilon,N_{1},N_{2}}\|\psi\|_{W^{(\varepsilon+1/2,\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}(1+|\lambda_{1}|)^{-N_{1}}(1+|\lambda_{2}|)^{-N_{2}}$
$\displaystyle\leq
C_{\varepsilon,N_{1},N_{2}}\|m\|_{W^{(\varepsilon+1/2,\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}(1+|\lambda_{1}|)^{-N_{1}}(1+|\lambda_{2}|)^{-N_{2}}.$

Next we prove (ii). Fix a function $\eta\in C_{0}^{\infty}(\mathbb{R})$ such
that $\eta(\lambda_{2})=1$ for all $\lambda_{2}\in[1/2,2]$. We consider four
cases.

### Case 1.

$s_{1}\lambda_{1}\leq 1$ and $s_{1}\leq r_{1}$. By the Cauchy–Schwarz
inequality,

$\displaystyle\left|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})\right|$
$\displaystyle=\left|\int_{\mathbb{R}}\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}_{1}{\psi}\left(\frac{\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{i\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle=\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\lesssim\frac{s_{1}}{r_{1}}\int_{|\xi_{1}|\geq{1/4}}\left|\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|d\xi_{1}.$

Now $\operatorname{supp}\psi\subset[1/2,2]\times[1/2,2]$, so
$\operatorname{supp}\mathcal{F}_{1}{\psi}\bigl{(}\frac{s_{1}\xi_{1}}{r_{1}},\cdot\bigr{)}\subseteq[1/2,2]$
for all $s_{1},r_{1}>0$ and $\xi_{1}\in\mathbb{R}$. Hence
$\mathcal{F}_{1}{\psi}\bigl{(}\frac{s_{1}\xi_{1}}{r_{1}},\lambda_{2}r_{2}\bigr{)}=\eta(\lambda_{2}r_{2})\mathcal{F}_{1}{\psi}\bigl{(}\frac{s_{1}\xi_{1}}{r_{1}},\lambda_{2}r_{2}\bigr{)}$,
and it follows that

$\displaystyle\left|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})\right|$
$\displaystyle\lesssim\frac{s_{1}}{r_{1}}|\eta(r_{2}\lambda_{2})|\int_{|\xi_{1}|\geq{1/4}}\left|\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}^{2}\lambda_{2}\right)\right|d\xi_{1}$
$\displaystyle\leq\frac{s_{1}}{r_{1}}|\eta(r_{2}\lambda_{2})|\int_{|\xi_{1}|\geq{1/4}}\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{\sigma_{1}-(\varepsilon+{1/2})}\left|\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}^{2}\lambda_{2}\right)\right|d\xi_{1}$
$\displaystyle\leq|\eta(r_{2}\lambda_{2})|\left(\int_{|\xi_{1}|\geq{1/4}}\left|\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{\sigma_{1}}\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}^{2}\lambda_{2}\right)\right|^{2}d\left(\frac{s_{1}\xi_{1}}{r_{1}}\right)\right)^{1/2}$
$\displaystyle\hskip
113.81102pt\times\left(\int_{|\xi_{1}|\geq{1/4}}\left|1+\frac{s\xi_{1}}{r_{1}}\right|^{-(1+2\varepsilon)}d\left(\frac{s_{1}\xi_{1}}{r_{1}}\right)\right)^{1/2}$
$\displaystyle\lesssim|\eta(r_{2}\lambda_{2})|\left(\int_{\mathbb{R}}\left|(1+|\xi|)^{\sigma_{1}}\mathcal{F}_{1}{\psi}\left(\xi_{1},r_{2}^{2}\lambda_{2}\right)\right|^{2}d\xi\right)^{1/2}.$

Using the continuous inclusion $W^{\varepsilon+1/2,2}(\mathbb{R})\subset
L^{\infty}(\mathbb{R})$, we see that

$\displaystyle\left|\mathcal{F}_{1}{\psi}\left(\xi_{1},r_{2}^{2}\lambda_{2}\right)\right|^{2}\lesssim\|\mathcal{F}_{1}{\psi}(\xi_{1},\cdot)\|_{L^{\infty}(\mathbb{R})}^{2}\lesssim\int_{\mathbb{R}}\big{|}(1+|\xi_{2}|)^{\varepsilon+{1/2}}\mathcal{F}{\psi}(\xi_{1},\xi_{2})\big{|}^{2}d\xi_{2}.$

Hence by Fubini’s theorem,

$\displaystyle|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})|$
$\displaystyle\qquad\lesssim|\eta(r_{2}\lambda_{2})|\left[\int_{\mathbb{R}\times\mathbb{R}}\left|\left(1+\left|\xi_{1}\right|\right)^{\sigma_{1}}(1+|\xi_{2}|)^{\varepsilon+{1/2}}\mathcal{F}{\psi}(\xi_{1},\xi_{2})\right|^{2}d\xi_{1}d\xi_{2}\right]^{1/2}$
$\displaystyle\qquad\lesssim(1+r_{2}^{2}\lambda_{2})^{-N_{2}}\|\psi\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\qquad\lesssim\|m\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}\left[(1+s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\right](1+r_{2}\lambda_{2})^{-N_{2}},$

where for the last inequality we used the estimate $s_{1}\lambda_{1}\sim
1+s_{1}\lambda_{1}$ and $1+{s_{1}}/{r_{1}}\sim 1$.

### Case 2.

$s_{1}\lambda_{1}\geq 1$ and ${s_{1}}\leq{r_{1}}$. In this case, by elementary
calculus,

$\displaystyle\left|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})\right|$
$\displaystyle=\left|\int_{\mathbb{R}}\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}_{1}{\psi}\left(\frac{\xi_{1}}{r_{1}},r_{2}^{2}\lambda_{2}\right)e^{i\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle=\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle=(s_{1}\lambda_{1})^{-N_{1}}\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\partial_{\xi_{1}}^{N_{1}}[e^{is_{1}\lambda_{1}\xi_{1}}]d\xi_{1}\right|$
$\displaystyle=(s_{1}\lambda_{1})^{-N_{1}}\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\partial_{\xi_{1}}^{N_{1}}\left[\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right]e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}}$
$\displaystyle\quad\quad\times\left|\int_{|\xi_{1}|\geq{1/4}}\left[\partial_{\xi_{1}}^{N_{1}}\phi(\xi_{1})\right]\left[(\partial_{\xi_{1}}^{k_{2}}\mathcal{F}_{1}{\psi})\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right]e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle=(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\hskip
56.9055pt\times\left|\int_{|\xi_{1}|\geq{1/4}}\left[\partial_{\xi_{1}}^{k_{1}}\phi(\xi_{1})\right]\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|,$

where $\varphi_{k_{2}}$ is the function on $\mathbb{R}\times\mathbb{R}$ given
by
$\varphi_{k_{2}}(\lambda_{1},\lambda_{2})=(-i\lambda_{1})^{k_{2}}\psi(\lambda_{1},\lambda_{2})$.
Much as in Case 1,
$\mathcal{F}_{1}{\varphi_{k_{2}}}({s_{1}\xi_{1}}/{r_{1}},r_{2}\lambda_{2})=\eta(\lambda_{2}r_{2})\mathcal{F}_{1}{\varphi_{k_{2}}}({s_{1}\xi_{1}}/{r_{1}},r_{2}\lambda_{2})$.
It follows that

$\displaystyle\left|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})\right|$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}}|\eta(r_{2}\lambda_{2})|\int_{|\xi_{1}|\geq{1/4}}\left|\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|d\xi_{1}$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\hskip
56.9055pt\times\int_{|\xi_{1}|\geq{1/4}}\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{\sigma_{1}-(\varepsilon+{1/2})}\left|\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|d\xi_{1}$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{k_{2}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\hskip
56.9055pt\times\left[\int_{|\xi_{1}|\geq{1/4}}\left|\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{\sigma_{1}}\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|^{2}d\left(\frac{s_{1}\xi_{1}}{r_{1}}\right)\right]^{1/2}$
$\displaystyle\qquad\hskip
56.9055pt\times\left[\int_{\mathbb{R}}\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{-1-2\varepsilon}d\left(\frac{s_{1}\xi_{1}}{r_{1}}\right)\right]^{1/2}$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{k_{2}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\hskip
56.9055pt\times\left[\int_{\mathbb{R}}\left|\left(1+\left|\xi_{1}\right|\right)^{\sigma_{1}}\mathcal{F}_{1}{\varphi_{k_{2}}}(\xi_{1},r_{2}\lambda_{2})\right|^{2}d\xi_{1}\right]^{1/2}$
$\displaystyle\qquad\lesssim(s_{1}\lambda_{1})^{-N_{1}}(1+r_{2}\lambda_{2})^{-N_{2}}\|\varphi_{k_{2}}\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\qquad\lesssim(s_{1}\lambda_{1})^{-N_{1}}(1+r_{2}\lambda_{2})^{-N_{2}}\|\psi\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\qquad\lesssim\|m\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}\left[(1+s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\right](1+r_{2}\lambda_{2})^{-N_{2}}$

where for the last inequality we used the estimate $1+s_{1}\lambda_{1}\sim 1$
and $1+{s_{1}}/{r_{1}}\sim 1$.

### Case 3.

$s_{1}\lambda_{1}\leq 1$ and $s_{1}\geq r_{1}$. This case is similar to Case
2, and we skip the details.

### Case 4.

$s_{1}\lambda_{1}\geq 1$ and ${s_{1}}\geq{r_{1}}$. Taking $\varphi_{k_{2}}$ as
in Case 2, we see that

$\displaystyle\left|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})\right|$
$\displaystyle\qquad=\left|\int_{\mathbb{R}}\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}_{1}{\psi}\left(\frac{\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{i\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\qquad=\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\qquad=(s_{1}\lambda_{1})^{-N_{1}}\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\partial_{\xi_{1}}^{N_{1}}[e^{is_{1}\lambda_{1}\xi_{1}}]d\xi_{1}\right|$
$\displaystyle\qquad=(s_{1}\lambda_{1})^{-N_{1}}\frac{s_{1}}{r_{1}}\left|\int_{|\xi_{1}|\geq{1/4}}\partial_{\xi_{1}}^{N_{1}}\left[\phi(\xi_{1})\mathcal{F}_{1}{\psi}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right]e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}}$
$\displaystyle\qquad\qquad\times\left|\int_{|\xi_{1}|\geq{1/4}}\left[\partial_{\xi_{1}}^{N_{1}}\phi(\xi_{1})\right]\left[(\partial_{\xi_{1}}^{k_{2}}\mathcal{F}_{1}{\psi})\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right]e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\qquad=(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\qquad\times\left|\int_{|\xi_{1}|\geq{1/4}}\left[\partial_{\xi_{1}}^{k_{1}}\phi(\xi_{1})\right]\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)e^{is_{1}\lambda_{1}\xi_{1}}d\xi_{1}\right|$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s}{r_{1}}\right)^{1+k_{2}}|\eta(r_{2}\lambda_{2})|\int_{|\xi_{1}|\geq{1/4}}\left|\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|d\xi_{1}$
$\displaystyle\qquad\lesssim(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{1+k_{2}-\sigma_{1}+\varepsilon+{1/2}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\qquad\times\int_{|\xi_{1}|\geq{1/4}}\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{\sigma_{1}-(\varepsilon+{1/2})}\left|\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|d\xi_{1}$
$\displaystyle\qquad\leq(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{k_{2}+\varepsilon+{1/2}-\sigma_{1}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\qquad\times\left[\int_{|\xi_{1}|\geq{1/4}}\left|\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{\sigma_{1}}\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\frac{s_{1}\xi_{1}}{r_{1}},r_{2}\lambda_{2}\right)\right|^{2}d\left(\frac{s_{1}\xi_{1}}{r_{1}}\right)\right]^{1/2}$
$\displaystyle\qquad\qquad\times\left[\int_{\mathbb{R}}\left(1+\left|\frac{s_{1}\xi_{1}}{r_{1}}\right|\right)^{-1-2\varepsilon}d\left(\frac{s_{1}\xi_{1}}{r_{1}}\right)\right]^{1/2}$
$\displaystyle\qquad\lesssim(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{k_{2}+\varepsilon+{1/2}-\sigma_{1}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\qquad\qquad\times\left[\int_{\mathbb{R}}\left|\left(1+\left|\xi_{1}\right|\right)^{\sigma_{1}}\mathcal{F}_{1}{\varphi_{k_{2}}}(\xi_{1},r_{2}\lambda_{2})\right|^{2}d\xi_{1}\right]^{1/2}.$

Since $W^{\varepsilon+1/2,2}(\mathbb{R})\subset L^{\infty}(\mathbb{R})$,

$\displaystyle\left|\mathcal{F}_{1}{\varphi_{k_{2}}}\left(\xi_{1},r_{2}\lambda_{2}\right)\right|^{2}\lesssim\|\mathcal{F}_{1}{\varphi_{k_{2}}}(\xi_{1},\cdot)\|_{L^{\infty}(\mathbb{R})}^{2}\lesssim\int_{\mathbb{R}}\big{|}(1+|\xi_{2}|)^{\varepsilon+{1/2}}\mathcal{F}{\varphi_{k_{2}}}(\xi_{1},\xi_{2})\big{|}^{2}d\xi_{2}.$

Therefore, it follows from Fubini’s theorem that

$\displaystyle|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})|$
$\displaystyle\qquad\lesssim(s_{1}\lambda_{1})^{-N_{1}}\sum_{k_{1}+k_{2}=N_{1}}\left(\frac{s_{1}}{r_{1}}\right)^{k_{2}+\varepsilon+{1/2}-\sigma_{1}}|\eta(r_{2}\lambda_{2})|$
$\displaystyle\hskip
113.81102pt\times\left[\int_{\mathbb{R}\times\mathbb{R}}\left|\left(1+\left|\xi_{1}\right|\right)^{\sigma_{1}}(1+|\xi_{2}|)^{\varepsilon+{1/2}}\mathcal{F}{\varphi_{k_{2}}}(\xi_{1},\xi_{2})\right|^{2}d\xi_{1}d\xi_{2}\right]^{1/2}$
$\displaystyle\qquad\lesssim(s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}(1+r_{2}\lambda_{2})^{-N_{2}}\|\varphi_{k_{2}}\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\qquad\lesssim\|m\|_{W^{(\sigma_{1},\varepsilon+1/2),2}(\mathbb{R}\times\mathbb{R})}\left[(1+s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\right](1+r_{2}\lambda_{2})^{-N_{2}}.$

The proof of (iii) is parallel to that of (ii), and we omit the details.

To prove (iv), we consider sixteen cases depending on whether each of
$\lambda_{1}s_{1},s_{1}/r_{1},\lambda_{2}s_{2}$ and $s_{2}/r_{2}$ is small or
large. Each of these cases can be handled by using a similar argument to that
in the proof of (ii) and we omit the details here. ∎

If $m(\cdot,\cdot)$ has compact support with respect to the first variable
only, then the following result holds; its proof is similar to that of Lemma
3.1 (ii) and so will be omitted.

######  Lemma 3.2.

Suppose $m:[0,\infty)\times[0,\infty)\to\mathbb{C}$ is a function with
$\operatorname{supp}m\subset[1/4,4]\times[0,\infty)$. Let $\phi$ be a smooth
cut-off function on $\mathbb{R}$ such that

$\displaystyle\phi(\xi)=\begin{cases}0&\text{if $|\xi|\leq 1/4$}\\\ 1&\text{if
$|\xi|\geq 1/2$}.\end{cases}$

Set $\psi(\lambda_{1},\lambda_{2}):=m(\lambda_{1}^{2},\lambda_{2}^{2})$ and
define $F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ by the formula

$\displaystyle\mathcal{F}_{1}F_{s_{1},r_{1}}(\xi_{1},\lambda_{2})=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}_{1}{\psi}\left(\frac{\xi_{1}}{r_{1}},\lambda_{2}\right),$

where $\mathcal{F}_{1}$ is the Fourier transform with respect to the first
variable. Then, for any $\varepsilon>0$, any $N_{1},N_{2}\in\mathbb{N}$ and
any $\sigma_{1}\in\mathbb{R}$, there exists a constant
$C_{\varepsilon,\sigma_{1},N_{1},N_{2}}$ such that

$\displaystyle|F_{s_{1},r_{1}}(\lambda_{1},\lambda_{2})|$ $\displaystyle\leq
C_{\varepsilon,\sigma_{1},N_{1},N_{2}}\|m\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\left[(1+\lambda_{1}s_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\right]$

for all $r_{1},s_{1},\lambda_{1},\lambda_{2}>0$. Here
$\|m\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}=\sup_{\lambda_{2}\in\mathbb{R}}\|m(\cdot,\lambda_{2})\|_{W^{\sigma_{1},2}(\mathbb{R})}$.

The following result concerning finite speed propagation was proved by Melrose
[8].

######  Lemma 3.3.

For $k=1,2$, there exists a constant $\kappa_{k}$ such that

$\displaystyle\operatorname{supp}\left(K_{\cos(t\Box_{b}^{(k)})}\right)\subset\\{(x,y)\in
M^{(k)}\times M^{(k)}:\rho^{(k)}(x,y)\leq\kappa_{k}t\\}.$

This result has the following bi-parameter analogue.

######  Proposition 3.4.

Let $F$ be a function on $\mathbb{R}\times\mathbb{R}$ such that
$F(-\lambda_{1},-\lambda_{2})=F(\lambda_{1},\lambda_{2})$.
(i) If
$\operatorname{supp}\mathcal{F}{F}\subset[-r_{1},r_{1}]\times[-r_{2},r_{2}]$,
then

$\displaystyle\operatorname{supp}K_{F(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}\subset\\{(x_{1},x_{2},y_{1},y_{2}):\rho^{(1)}(x_{1},y_{1})\leq\kappa_{1}r_{1}\mbox{
and }\rho^{(2)}(x_{2},y_{2})\leq\kappa_{2}r_{2}\\}.$

(ii) If
$\operatorname{supp}\mathcal{F}_{1}{F}(\cdot,\lambda_{2})\subset[-r_{1},r_{1}]$
for all $\lambda_{2}\in\mathbb{R}$, then

$\displaystyle\operatorname{supp}K_{F(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}\subset\\{(x_{1},x_{2},y_{1},y_{2}):\rho^{(1)}(x_{1},y_{1})\leq\kappa_{1}r_{1}\\}.$

(iii) If
$\operatorname{supp}\mathcal{F}_{2}{F}(\lambda_{1},\cdot)\subset[-r_{2},r_{2}]$
for all $\lambda_{1}\in\mathbb{R}$, then

$\displaystyle\operatorname{supp}K_{F(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}\subset\\{(x_{1},x_{2},y_{1},y_{2}):\rho^{(2)}(x_{2},y_{2})\leq\kappa_{2}r_{2}\\}.$

###### Proof.

We prove only (ii) since other assertions can be proved similarly. Since $F$
is even, by the Fourier inversion formula,

$\displaystyle
F(\lambda_{1},\lambda_{2})=\int_{\mathbb{R}}\mathcal{F}_{1}{F}(\xi_{1},\lambda_{2})e^{i\lambda_{1}\xi_{1}}d\xi_{1}=\int_{-r_{1}}^{r_{1}}\mathcal{F}_{1}{F}(\xi_{1},\lambda_{2})\cos(\lambda_{1}\xi_{1})d\xi_{1}.$

Hence

$\displaystyle
F(\Box_{b}^{(1)},\Box_{b}^{(2)})=\int_{-r_{1}}^{r_{1}}\mathcal{F}_{1}{F}(\xi_{1},\Box_{b}^{(2)})\cos(\xi_{1}\Box_{b}^{(1)})d\xi_{1}.$

It follows that

$\displaystyle
K_{F(\Box_{b}^{(1)},\Box_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})=\int_{-r_{1}}^{r_{1}}K_{\mathcal{F}_{1}{F}(\xi_{1},\Box_{b}^{(2)})}(x_{2},y_{2})K_{\cos(\xi_{1}\Box_{b}^{1})}(x_{1},y_{1})d\xi_{1}.$

By Lemma 3.3,

$\displaystyle\operatorname{supp}K_{\cos(\xi_{1}\Box_{b}^{(1)})}\subset\\{(x_{1},y_{1}):\rho^{(1)}(x_{1},y_{1})\leq\kappa_{1}r_{1}\\}$

for all $\xi_{1}\in[-r_{1},r_{1}]$, which yields the assertion (ii). ∎

We will need the following result. See [15, Proposition 4.6].

######  Lemma 3.5.

Let $k\in\\{1,2\\}$. For every multi-index $\alpha\in\mathcal{I}$ and every
$\sigma>Q_{k}+2|\alpha|$, there exists a constant $C_{k,\alpha,\sigma}$ such
that for all $t>0$,

(3.1)
$\displaystyle\big{\|}(X_{\alpha}^{(k)})_{x}K_{(I+t\widehat{\square}_{b}^{(k)})^{-\sigma/4}}(x,\cdot)\big{\|}_{L^{2}(M^{(k)})}\leq
C_{k,\alpha,m}\frac{\sqrt{t}^{-|\alpha|}}{V^{(k)}(x,\sqrt{t})^{1/2}}.$

######  Remark 3.6.

Inequality (3.1) can be rewritten as

$\displaystyle\big{\|}(X_{\alpha}^{(k)})_{x}K_{(I+t\sqrt{\widehat{\square}_{b}^{(k)}})^{-\sigma/2}}(x,\cdot)\big{\|}_{L^{2}(M^{(k)})}\leq
C_{k,\alpha,m}\frac{t^{-|\alpha|}}{V^{(k)}(x,t)^{1/2}}.$

Indeed, since

$\displaystyle\left(I+t\sqrt{\widehat{\square}_{b}^{(k)}}\right)^{-\sigma/2}=\phi(\Box_{b}^{(k)})\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4},$

where
$\phi(\lambda)=(1+t\sqrt{\lambda})^{-\sigma/2}(1+t^{2}\lambda)^{\sigma/4}$,

$\displaystyle\big{\|}(X_{\alpha}^{(k)})_{x}K_{\bigl{(}I+t\sqrt{\widehat{\square}_{b}^{(k)}}\bigr{)}^{-\sigma/2}}(x,\cdot)\big{\|}_{L^{2}(M^{(k)})}$
$\displaystyle=\big{\|}(X_{\alpha}^{(k)})_{x}K_{\bigl{(}I+t\sqrt{\widehat{\square}_{b}^{(k)}}\bigr{)}^{-\sigma/2}}(\cdot,x)\big{\|}_{L^{2}(M^{(k)})}$
$\displaystyle=\big{\|}(X_{\alpha}^{(k)})_{x}K_{\phi(\widehat{\square}_{b}^{(k)})\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4}}(\cdot,x)\big{\|}_{L^{2}(M^{(k)})}$
$\displaystyle=\left\|(X_{\alpha}^{(k)})_{x}\int
K_{\phi(\Box_{b}^{(k)})}(\cdot,z)K_{\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4}}(z,x)\,dz\right\|_{L^{2}(M^{(k)})}$
$\displaystyle=\left\|\int
K_{\phi(\Box_{b}^{(k)})}(\cdot,z)\left[(X_{\alpha}^{(k)})_{x}K_{\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4}}(z,x)\right]\,dz\right\|_{L^{2}(M^{(k)})}$
$\displaystyle=\left\|\phi(\Box_{b}^{(k)})\left[(X_{\alpha}^{(k)})_{x}K_{\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4}}(\cdot,x)\right]\right\|_{L^{2}(M^{(k)})}$
$\displaystyle\lesssim\left\|(X_{\alpha}^{(k)})_{x}\left[K_{\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4}}(\cdot,x)\right]\right\|_{L^{2}(M^{(k)})}$
$\displaystyle=\left\|(X_{\alpha}^{(k)})_{x}\left[K_{\left(I+t^{2}\widehat{\square}_{b}^{(k)}\right)^{-\sigma/4}}(x,\cdot)\right]\right\|_{L^{2}(M^{(k)})}$
$\displaystyle\leq C_{k,\alpha,m}\frac{t^{-|\alpha|}}{V^{(k)}(x,t)^{1/2}},$

where we used the fact that
$K_{\phi(\Box_{b}^{(k)})}(x,y)=K_{\phi(\Box_{b}^{(k)})}(y,x)\overline{\phantom{m}}$.

Our next two propositions establish pointwise kernel estimates for the
multipliers, when the multiplier function is smooth enough and has compact
support. The first proposition is for one group of variables and the second is
for both groups of variables.

######  Proposition 3.7.

Suppose that $m:[0,\infty)\times[0,\infty)\to\mathbb{C}$ is supported in
$[1/4,4]\times[0,\infty)$ and
$\|m\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}<\infty$ for
some $\sigma_{1}\in\mathbb{R}$. Then, for any $\varepsilon>0$, any
$\alpha,\beta\in\mathcal{I}$, any $N_{1}\in\mathbb{N}$ such that
$N_{1}>(|\alpha|\vee|\beta|)+{Q_{1}}/{2}$, there exists a constant
$C_{\varepsilon,\alpha,\beta,N_{1}}$ such that

$\displaystyle\left|\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{m(r_{1}^{2}\Box_{b}^{(1)},\Box_{b}^{(2)})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})\right|$
$\displaystyle\qquad\leq
C_{\varepsilon,\alpha,\beta,\sigma_{1},N_{1}}\|m\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|f_{2}\|_{2}\|g_{2}\|_{2}$
$\displaystyle\qquad\qquad\times\left(1+\frac{\rho^{(1)}(x_{1},y_{1})}{r_{1}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}\frac{(r_{1}\vee\rho^{(1)}(x_{1},y_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+r_{1})}$

for all $x_{1},y_{1}\in M^{(1)}$.

###### Proof.

Fix arbitrary $x_{1},y_{1}\in M^{(1)}$. We prove the assertion by considering
four cases.

### Case 1.

$\rho^{(1)}({x}_{1},{y}_{1})\geq r_{1}$. Let
$\psi(\lambda_{1},\lambda_{2}):=m(\lambda_{1}^{2},\lambda_{2}^{2})$,
$\psi_{r_{1}}(\lambda_{1},\lambda_{2}):=\psi(r_{1}\lambda_{1},\lambda_{2})$
and define $F_{s_{1},r_{1}}(\lambda_{1},\lambda_{2})$ by the formula

$\displaystyle\mathcal{F}{F_{s_{1},r_{1}}}(\xi_{1},\xi_{2})$
$\displaystyle=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}{\psi}\left(\frac{\xi_{1}}{r_{1}},\xi_{2}\right)=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\mathcal{F}{\psi_{r_{1}}}(\xi_{1},\xi_{2}),$

where $\phi$ is as in Lemma 3.2. Let
$J_{s_{1},s_{2},r_{1}}(\lambda_{1},\lambda_{2})$ be a complex-valued function
such that

$\displaystyle
J_{s_{1},r_{1}}(\lambda_{1},\lambda_{2})^{2}=F_{s_{1},r_{1}}(\lambda_{1},\lambda_{2})$

(any choice of the square root will do). Set
$s_{1}={\rho^{(1)}({x}_{1},{y}_{1})}/{(4\kappa_{1})}$. Since

$\displaystyle\operatorname{supp}\bigl{(}\mathcal{F}{F_{s_{1},r_{1}}}(\cdot,\cdot)-\mathcal{F}{\psi_{r_{1}}}(\cdot,\cdot)\bigr{)}\subset[0,2s_{1}]\times[0,\infty)=\left[0,\frac{\rho^{(1)}({x}_{1},{y}_{1})}{2\kappa_{1}}\right]\times[0,\infty),$

it follows from the finite speed propagation (Proposition 3.4) that

$\displaystyle\operatorname{supp}\left(K_{F_{s_{1},s_{2},r_{1}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(\cdot,\cdot,\cdot,\cdot)-K_{\psi_{r_{1}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(\cdot,\cdot,\cdot,\cdot)\right)$
$\displaystyle\qquad\subset\left\\{(u_{1},v_{1},x_{2},y_{2}):\rho^{(1)}(u_{1},v_{1})\leq\frac{\rho^{(1)}({x}_{1},{y}_{1})}{2}\right\\}.$

Hence we can write

$\displaystyle\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{m(r_{1}^{2}\Box_{b}^{(1)},\Box_{b}^{(2)})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad=\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{\psi_{r_{1}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad=\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{F_{s_{1},r_{1}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad=\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{F_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad\qquad+\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{\pi^{(1)}\otimes
F_{s_{1},r_{1}}(0,\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad:=I_{1}+I_{2}.$

Since $\pi^{(1)}$ is an NIS operator,

$\displaystyle|(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{\pi^{(1)}}(x_{1},y_{1})|$
$\displaystyle\leq
C_{\alpha,\beta}\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1}))}$
$\displaystyle\sim\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+r_{1})}.$

Therefore, from Lemma 3.2,

$\displaystyle|I_{2}|$ $\displaystyle\lesssim
C_{\alpha,\beta}\frac{\rho^{(1)}({x}_{1},{y}_{1})^{-|\alpha|-|\beta|}}{V^{(1)}({x}_{1},\rho^{(1)}({x}_{1},{y}_{1})+r_{1})}\|F_{s_{1},s_{2},r_{1}}(0,\sqrt{\Box_{b}^{(2)}})\|_{2\to
2}\|f_{2}\|_{2}\|g_{2}\|_{2}$ $\displaystyle\leq
C_{\varepsilon,\alpha,\beta,\sigma_{1},N_{1}}\|m\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|f_{2}\|_{2}\|g_{2}\|_{2}$
$\displaystyle\quad\times\left(1+\frac{\rho^{(1)}({x}_{1},{y}_{1})}{r_{1}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}\frac{(r_{1}\vee\rho^{(1)}({x}_{1},{y}_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}({x}_{1},\rho^{(1)}({x}_{1},{y}_{1})+r_{1})}.$

Next we estimate term $I_{1}$. Using the fact that

$\displaystyle
F_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})=J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})\circ
J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})$

and the Cauchy–Schwarz inequality, we see that

$\displaystyle|I_{1}|$
$\displaystyle=\left|\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{F_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})\right|$
$\displaystyle=\left|\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}\iint
K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})\right.$
$\displaystyle\qquad\times\left.K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)})}}(z_{1},y_{1},z_{2},y_{2})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})\right|$
$\displaystyle\leq\left\|\int(X_{\alpha}^{(1)})_{x_{1}}K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},\cdot,x_{2},\cdot)f_{2}(x_{2})\,d\mu^{2}(x_{2})\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$
$\displaystyle\quad\times\left\|\int(X_{\beta}^{(1)})_{y_{1}}K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(\cdot,y_{1},\cdot,y_{2})g_{2}(y_{2})\,d\mu^{2}(y_{2})\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$ $\displaystyle=:I_{11}\times I_{12}.$

We only estimate term $I_{11}$, as $I_{12}$ is similar. Set
$V^{(1)}_{s_{1}}(x_{1})=V^{(1)}({x}_{1},s_{1})^{1/2}$. We write

$\displaystyle V^{(1)}({x}_{1},s_{1})^{1/2}I_{11}$
$\displaystyle=\sup_{\|h\|_{2}=1}\biggl{|}V^{(1)}_{s_{1}}(x_{1})\iiint(X_{\alpha}^{(1)})_{x_{1}}K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})f_{2}(x_{2})\,d\mu^{2}(x_{2})$
$\displaystyle\hskip 142.26378pt\times
h(z_{1},z_{2})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})\biggr{|}$
$\displaystyle\leq\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iiiint
V^{(1)}_{s_{1}}(u_{1})(X_{\alpha}^{(1)})_{u_{1}}K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(u_{1},z_{1},x_{2},z_{2})f_{2}(x_{2})\,d\mu^{2}(x_{2})$
$\displaystyle\hskip 142.26378pt\times
h(z_{1},z_{2})g_{1}(u_{1})\,d\mu^{1}(u_{1})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})\biggr{|}$
$\displaystyle=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iiiint
K_{J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(u_{1},z_{1},x_{2},z_{2})f_{2}(x_{2})\,d\mu(x_{2})h(z_{1},z_{2})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})$
$\displaystyle\hskip
142.26378pt\times(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\,d\mu^{1}(u_{1})\biggr{|}$
$\displaystyle=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
J_{s_{1},r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})h(u_{1},x_{2})f_{2}(x_{2})(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})$
$\displaystyle\hskip 142.26378pt\times
g_{1}(u_{1})\,d\mu^{1}(u_{1})\,d\mu^{2}(x_{2})\biggr{|}$
$\displaystyle=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
G(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}\otimes
Ih(u_{1},x_{2})$ $\displaystyle\hskip 142.26378pt\times
f_{2}(x_{2})(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\,d\mu^{1}(u_{1})\,d\mu^{2}(x_{2})\biggr{|}$
$\displaystyle=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
G(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})h(u_{1},x_{2})$
$\displaystyle\hskip
56.9055pt\times\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}\otimes
I\left[f_{2}(x_{2})(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\right]\,d\mu^{1}(u_{1})\,d\mu^{2}(x_{2})\biggr{|}$

where

$\displaystyle
G(\lambda_{1},\lambda_{2})=(1+s_{1}\lambda_{1})^{N_{1}/2}J_{s_{1},r_{1}}(\lambda_{1},\lambda_{2}),$

and for the third step we used the estimate

$\displaystyle\zeta(x_{1})\leq\|\zeta\|_{\infty}=\sup_{\|\theta\|_{1}=1}\int_{M^{(1)}}\zeta(u_{1})\theta(u_{1})\,d\mu^{1}(u_{1})\qquad\forall
x_{1}\in M^{(1)}.$

By Lemma 3.2,

$\displaystyle|G(\lambda_{1},\lambda_{2})|$
$\displaystyle=|(1+s_{1}\lambda_{1})^{N_{1}/2}||J_{s_{1},r_{1}}(\lambda_{1},\lambda_{2})|$
$\displaystyle\lesssim|(1+s_{1}\lambda_{1})^{N_{1}/2}||F_{s_{1},r_{1}}(\lambda_{1},\lambda_{2})|^{1/2}$
$\displaystyle\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}}++{\varepsilon}+1/2)/2-{\sigma_{1}/2}}\|m\|^{1/2}_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}.$

From this and the Cauchy–Schwarz inequality, it follows that

$\displaystyle V^{(1)}({x}_{1},s_{1})^{1/2}I_{11}$ $\displaystyle\leq
C\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\left\|G(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})h\right\|_{2}$
$\displaystyle\hskip
142.26378pt\left\|\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\right\|_{2}\|f_{2}\|_{2}$
$\displaystyle\leq
C\sup_{\|g_{1}\|_{1}=1}\|G\|_{\infty}\left\|\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\right\|_{2}\|f_{2}\|_{2}$
$\displaystyle\leq
C\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\|m\|^{1/2}_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|f_{2}\|_{2}$
$\displaystyle\hskip
142.26378pt\times\sup_{\|g_{1}\|_{1}=1}\left\|\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\right\|_{2}.$

Hence, from Lemma 3.5 and Remark 3.6,

$\displaystyle\left\|\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})\right\|_{2}$
$\displaystyle\qquad\leq\sup_{\|f_{1}\|_{2}=1}\left|\int\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{u_{1}}V^{(1)}_{s_{1}}(u_{1})g_{1}(u_{1})f_{1}(u_{1})\,d\mu^{1}(u_{1})\right|$
$\displaystyle\qquad=\sup_{\|f_{1}\|_{2}=1}\left|\int
g_{1}(u_{1})V^{(1)}_{s_{1}}(u_{1})(X_{\alpha}^{(1)})_{u_{1}}\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}f_{1}(u_{1})\,d\mu^{1}(u_{1})\right|$
$\displaystyle\qquad\leq\sup_{\|f_{1}\|_{2}=1}\|g_{1}\|_{1}\sup_{u_{1}}V^{(1)}_{s_{1}}(u_{1})\left|(X_{\alpha}^{(1)})_{u_{1}}\int
K_{\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}}(u_{1},y_{1})f_{1}(y_{1})\,d\mu^{1}(y_{1})\right|$
$\displaystyle\qquad\leq
C\sup_{\|f_{1}\|_{2}=1}\|g_{1}\|_{1}\sup_{u_{1}}V^{(1)}_{s_{1}}(u_{1})\left\|(X_{\alpha}^{(1)})_{u_{1}}K_{\left(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}}(u_{1},\cdot)\,d\mu^{1}(u_{1})\right\|_{2}\|f_{1}\|_{2}$
$\displaystyle\qquad\leq C\|g_{1}\|_{1}s_{1}^{-|\alpha|}.$

Combining the two estimates above, we deduce that

$\displaystyle I_{11}\leq
C\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\|m\|^{1/2}_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|f_{2}\|_{2}s_{1}^{-|\alpha|}V^{(1)}({x}_{1},s_{1})^{-1/2}.$

Similarly,

$\displaystyle I_{12}\leq
C\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\|m\|^{1/2}_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|g_{2}\|_{2}s_{1}^{-|\beta|}V^{(1)}({y}_{1},s_{1})^{-1/2}.$

Consequently,

$\displaystyle|I_{1}|$ $\displaystyle\lesssim
C\|m\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|f_{2}\|_{2}\|g_{2}\|_{2}\left(1+\frac{\rho^{(1)}(x_{1},y_{1})}{r_{1}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}\frac{(r_{1}\vee\rho^{(1)}(x_{1},y_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+r_{1})}.$

### Case 2.

$\rho^{(1)}({x}_{1},{y}_{1})\leq r_{1}$. Let
$\psi(\lambda_{1},\lambda_{2}):=m(\lambda_{1}^{2},\lambda_{2}^{2})$,
$\psi_{r_{1}}(\lambda_{1},\lambda_{2}):=\psi(r_{1}\lambda_{1},\lambda_{2})$
and let $J_{r_{1}}(\lambda_{1},\lambda_{2})$ be any complex-valued function
such that

$\displaystyle
J_{r_{1}}(\lambda_{1},\lambda_{2})^{2}=\psi(r_{1}\lambda_{1},\lambda_{2})$

(any branch will do). Since
$J^{2}_{r_{1}}(0,\lambda_{2})=\psi(0,\lambda_{2})=m(0,\lambda_{2}^{2})=0$, it
follows that

$\displaystyle
J_{r_{1}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})=J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}}).$

Hence we can write

$\displaystyle\iint(X_{\alpha}^{1})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{m(r_{1}^{2}\Box_{b}^{(1)},\Box_{b}^{(2)})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad=\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{\psi_{r_{1}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad=\iint(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{J^{2}_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})$
$\displaystyle\qquad=:B,$

say. From the Cauchy–Schwarz inequality, it follows that

$\displaystyle B$
$\displaystyle=\left|\iint(X_{\alpha}^{(1)})_{x}(X_{\beta}^{(1)})_{y_{1}}K_{J^{2}_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})\right|$
$\displaystyle=\left|\iint(X_{\alpha}^{(1)})_{x}(X_{\beta}^{(1)})_{y_{1}}\int_{M^{(1)}\times
M^{(2)}}K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})\right.$
$\displaystyle\hskip
85.35826pt\times\left.K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)})}}(z_{1},y_{1},z_{2},y_{2})\,dz_{1}\,dz_{2}f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{2}(x_{2})\,d\mu^{2}(y_{2})\right|$
$\displaystyle\leq\left\|\int(X_{\alpha}^{(1)})_{x_{1}}K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})f_{2}(x_{2})\,d\mu^{2}(x_{2})\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$ $\displaystyle\hskip
85.35826pt\times\left\|\int(X_{\beta}^{(1)})_{y_{1}}K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(z_{1},y_{1},z_{2},y_{2})g_{2}(y_{2})\,d\mu^{2}(y_{2})\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$ $\displaystyle=:B_{1}\times B_{2}.$

Set $V^{(1)}_{r_{1}}(x_{1})=V^{(1)}({x}_{1},r_{1})^{1/2}$. We write

$\displaystyle V^{(1)}({x}_{1},r_{1})^{1/2}I_{11}$
$\displaystyle\qquad=\sup_{\|h\|_{2}=1}\biggl{|}V^{(1)}_{r_{1}}(x_{1})\iint(X_{\alpha}^{(1)})_{x_{1}}K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})$
$\displaystyle\qquad\hskip
199.16928ptf_{2}(x_{2})\,dx_{2}h(z_{1},z_{2})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})\biggr{|}$
$\displaystyle\qquad\leq\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
V^{(1)}_{r_{1}}(x_{1})(X_{\alpha}^{(1)})_{x_{1}}K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})f_{2}(x_{2})\,d\mu^{2}(x_{2})$
$\displaystyle\qquad\hskip
199.16928pth(z_{1},z_{2})g_{1}(x_{1})\,d\mu^{1}(x_{1})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})\biggr{|}$
$\displaystyle\qquad=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iiint
K_{J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})}(x_{1},z_{1},x_{2},z_{2})f_{2}(x_{2})\,d\mu^{2}(x_{2})h(z_{1},z_{2})\,d\mu^{1}(z_{1})$
$\displaystyle\qquad\hskip
199.16928pt\,d\mu^{2}(z_{2})(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\,d\mu^{1}(x_{1})\biggr{|}$
$\displaystyle\qquad=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
J_{r_{1}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{{\Box}_{b}^{(2)}})h(x_{1},x_{2})f_{2}(x_{2})(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})$
$\displaystyle\qquad\hskip
199.16928ptg_{1}(x_{1})\,d\mu^{1}(x_{1})\,d\mu^{2}(x_{2})\biggr{|}$
$\displaystyle\qquad=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
G(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}\otimes
Ih(x_{1},x_{2})f_{2}(x_{2})$ $\displaystyle\qquad\hskip
199.16928pt(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\,d\mu^{1}(x_{1})\,d\mu^{2}(x_{2})\biggr{|}$
$\displaystyle\qquad=\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\biggl{|}\iint
G(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})h(x_{1},x_{2})$
$\displaystyle\qquad\hskip
113.81102pt\times\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}\otimes
I\left[f_{2}(x_{2})(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\right]$
$\displaystyle\qquad\hskip
284.52756pt\times\,d\mu^{1}(x_{1})\,d\mu^{2}(x_{2})\biggr{|},$

where

$\displaystyle
G(\lambda_{1},\lambda_{2})=(1+r_{1}\lambda_{1})^{N_{1}/2}J_{r_{1}}(\lambda_{1},\lambda_{2}).$

Since the supports of $m$ and $\psi$ are compact,

$\displaystyle|G(\lambda_{1},\lambda_{2})|\lesssim|(1+r_{1}\lambda_{1})^{N_{1}/2}||\psi(r_{1}\lambda_{1},\lambda_{2})|^{1/2}\lesssim
C\|m\|^{1/2}_{\infty}.$

From this and the Cauchy–Schwarz inequality, it follows that

$\displaystyle V^{(1)}({x}_{1},r_{1})^{1/2}B_{1}$ $\displaystyle\qquad\leq
C\sup_{\|h\|_{2}=1,\|g_{1}\|_{1}=1}\left\|G(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})h\right\|_{2}$
$\displaystyle\qquad\hskip
113.81102pt\left\|\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\right\|_{2}\|f_{2}\|_{2}$
$\displaystyle\qquad\leq
C\sup_{\|g_{1}\|_{1}=1}\|G\|_{\infty}\left\|\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\right\|_{2}\|f_{2}\|_{2}$
$\displaystyle\qquad\leq
C\|m\|^{1/2}_{\infty}\|f_{2}\|_{2}\sup_{\|g_{1}\|_{1}=1}\left\|\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\right\|_{2}.$

Together with Lemma 3.5 and Remark 3.6, this yields

$\displaystyle\left\|\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})\right\|_{2}$
$\displaystyle\qquad\leq\sup_{\|f_{1}\|_{2}=1}\left|\int\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}(X_{\alpha}^{(1)})_{x_{1}}V^{(1)}_{r_{1}}(x_{1})g_{1}(x_{1})f_{1}(x_{1})\,d\mu^{1}(x_{1})\right|$
$\displaystyle\qquad=\sup_{\|f_{1}\|_{2}=1}\left|\int
g_{1}(x_{1})V^{(1)}_{r_{1}}(x_{1})(X_{\alpha}^{(1)})_{x_{1}}\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}f_{1}(x_{1})\,d\mu^{1}(x_{1})\right|$
$\displaystyle\qquad\leq\sup_{\|f_{1}\|_{2}=1}\|g_{1}\|_{1}\sup_{x_{1}}V^{(1)}_{r_{1}}(x_{1})\left|(X_{\alpha}^{(1)})_{x_{1}}\int
K_{\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}}(x_{1},y_{1})f_{1}(y_{1})\,d\mu^{1}(y_{1})\right|$
$\displaystyle\qquad\leq
C\sup_{\|f_{1}\|_{2}=1}\|g_{1}\|_{1}\sup_{x_{1}}V^{(1)}_{r_{1}}(x_{1})\left\|X_{\alpha}^{(1)})_{x_{1}}K_{\left(I+r_{1}\sqrt{\widehat{\square}_{b}^{(1)}}\right)^{-N_{1}/2}}(x_{1},\cdot)\right\|_{2}\|f_{1}\|_{2}$
$\displaystyle\qquad\leq C\|g_{1}\|_{1}r_{1}^{-|\alpha|}.$

Combining the two estimates above shows that

$\displaystyle B_{1}\leq
C\|m\|^{1/2}_{\infty}\|f_{2}\|_{2}r_{1}^{-|\alpha|}V^{(1)}({x}_{1},r_{1})^{-1/2}.$

Similarly,

$\displaystyle B_{2}\leq
C\|m\|^{1/2}_{\infty}\|g_{2}\|_{2}r_{1}^{-|\beta|}V^{(1)}({y}_{1},r_{1})^{-1/2}.$

Therefore

$\displaystyle|B|$ $\displaystyle\lesssim
C\|m\|_{\infty}\|f_{2}\|_{2}\|g_{2}\|_{2}\frac{(r_{1}\vee\rho^{(1)}(x_{1},y_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+r_{1})},$

as required. ∎

######  Proposition 3.8.

Suppose that $\sigma_{1},\sigma_{2}\in\mathbb{R}$ and
$m:[0,\infty)\times[0,\infty)\to\mathbb{C}$ has support in
$[1/4,4]\times[1/4,4]$ and
$\|m\|_{W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}<\infty$.
Then, for any $\varepsilon>0$, any
$\alpha,\beta,\alpha^{\prime},\beta^{\prime}\in\mathcal{I}$, and any
$N_{1},N_{2}\in\mathbb{N}$ with $N_{1}>\frac{Q_{1}}{2}+(|\alpha|\vee|\beta|)$
and $N_{2}>\frac{Q_{2}}{2}+(|\alpha^{\prime}|\vee|\beta^{\prime}|)$, there
exists a constant
$C=C(\varepsilon,\alpha,\beta,\alpha^{\prime},\beta^{\prime},\sigma_{1},\sigma_{2},N_{1},N_{2})$
such that

$\displaystyle\left|(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{m(r_{1}^{2}\Box_{b}^{(1)},r_{2}^{2}\Box_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})\right|$
$\displaystyle\qquad\leq
C\|m\|_{W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}\left(1+\frac{\rho^{(1)}(x_{1},y_{1})}{r_{1}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}\frac{(r_{1}\vee\rho^{(1)}(x_{1},y_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+r_{1})}$
$\displaystyle\hskip 142.26378pt\
\times\left(1+\frac{\rho^{(2)}(x_{2},y_{2})}{r_{2}}\right)^{N_{2}+\varepsilon+{1/2}-\sigma_{2}}\frac{(r_{2}\vee\rho^{(2)}(x_{2},y_{2}))^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(x_{2},\rho^{(2)}(x_{2},y_{2})+r_{2})}.$

###### Proof.

Fix $\bar{x}_{1},\bar{x}_{2}\in M^{(1)}$ and $\bar{y}_{1},\bar{y}_{2}\in
M_{2}$. We wish to bound

$\displaystyle(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{m(r_{1}^{2}\Box_{b}^{(1)},r_{2}^{2}\Box_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(x_{1},x_{2},y_{1},y_{2})=(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}.$

We divide the argument into several cases:

### Case 1.

$\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})\geq r_{1}$ and
$\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})\geq r_{2}$. Let
$\psi(\lambda_{1},\lambda_{2}):=m(\lambda_{1}^{2},\lambda_{2}^{2})$,
$\psi_{r_{1},r_{2}}(\lambda_{1},\lambda_{2}):=\psi(r_{1}\lambda_{1},r_{2}\lambda_{2})$
and let $F_{s_{1},s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ be given via
the relation

$\displaystyle\mathcal{F}{F_{s_{1},s_{2},r_{1},r_{2}}}(\xi_{1},\xi_{2})=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\phi\left(\frac{\xi_{2}}{s_{2}}\right)\frac{1}{r_{1}r_{2}}\mathcal{F}{\psi}\left(\frac{\xi_{1}}{r_{1}},\frac{\xi_{2}}{r_{2}}\right)=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\phi\left(\frac{\xi_{2}}{s_{2}}\right)\mathcal{F}{\psi_{r_{1},r_{2}}}(\xi_{1},\xi_{2}),$

where $\phi$ is as in Lemma 3.1. Let
$J_{s_{1},s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ be a complex-valued
function such that

$\displaystyle
J_{s_{1},s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2})^{2}=F_{s_{1},s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2}).$

Set $s_{1}=\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{4\kappa_{1}}$ and
$s_{2}=\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})}{4\kappa_{2}}$. Since

$\displaystyle\operatorname{supp}\mathcal{F}{F_{s_{1},s_{2},r_{1},r_{2}}}(\cdot,\cdot)-\operatorname{supp}\mathcal{F}{\psi_{r_{1},r_{2}}}(\cdot,\cdot)$
$\displaystyle\subset[0,2s_{1}]\times[0,2s_{2}]$
$\displaystyle=\left[0,\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{2\kappa_{1}}\right]\times\left[0,\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})}{2\kappa_{2}}\right],$

it follows from the finite speed propagation (Proposition 3.4) that

$\displaystyle\operatorname{supp}\left(K_{F_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(\cdot,\cdot,\cdot,\cdot)-K_{\psi_{r_{1},r_{2}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(\cdot,\cdot,\cdot,\cdot)\right)$
$\displaystyle\qquad\subset\left\\{(x_{1},x_{2},y_{1},y_{2}):\rho^{(1)}(x,y)\leq\frac{\rho^{(1)}(\bar{x}_{1},\bar{x}_{2})}{2}\mbox{
and
}\rho^{(2)}(x_{2},y_{2})\leq\frac{\rho^{(2)}(\bar{x}_{2},\bar{x}_{2})}{2}\right\\}.$

Hence we can write

$\displaystyle(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{m(r_{1}^{2}\Box_{b}^{(1)},r_{2}^{2}\Box_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad=(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{\psi_{r_{1},r_{2}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad=(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad=(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad\qquad+(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{\pi^{(1)}\otimes
F_{s_{1},s_{2},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},y_{1},x_{2},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad\qquad+(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},0)\otimes\pi^{(2)}}(x_{1},y_{1},x_{2},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad\qquad+F_{s_{1},s_{2},r_{1},r_{2}}(0,0)(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{\pi^{(1)}\otimes\pi^{(2)}}(x_{1},y_{1},x_{2},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad:=I_{1}+I_{2}+I_{3}+I_{4}.$

We first estimate $I_{1}$. Using the fact that

$\displaystyle
F_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})=J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})\circ
J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})$

and the Cauchy–Schwarz inequality, we see that

$\displaystyle|I_{1}|$
$\displaystyle=\left|(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}\right|$
$\displaystyle=\left|(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}\int_{M^{(1)}\times
M^{(2)}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},x_{2},z_{1},z_{2})\right.$
$\displaystyle\hskip
113.81102pt\times\left.K_{J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)})}}(z_{1},z_{2},y_{1},y_{2})\,d\mu^{1}(z_{1})\,d\mu^{2}(z_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}\right|$
$\displaystyle\leq\left\|(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},x_{2},\cdot,\cdot)\Bigm{|}_{(x_{1},x_{2})=(\bar{x}_{1},\bar{x}_{2})}\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$
$\displaystyle\qquad\times\left\|(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(\cdot,\cdot,y_{1},y_{2})\Bigm{|}_{(y_{1},y_{2})=(\bar{y}_{1},\bar{y}_{2})}\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$ $\displaystyle=:I_{11}\times I_{22}.$

We write

$\displaystyle(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},x_{2},\cdot,\cdot)$
$\displaystyle=G(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}})(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{\left(I+s_{1}\sqrt{\Box_{b}^{(1)}}\right)^{-N_{1}/2}\otimes\left(I+s_{2}\sqrt{\Box_{b}^{(1)}}\right)^{-N_{2}/2}}(x_{1},x_{2},\cdot,\cdot),$

where

$\displaystyle
G(\lambda_{1},\lambda_{2})=(1+s_{1}\lambda_{1})^{N_{1}/2}(1+s_{2}\lambda_{2})^{N_{2}/2}J_{s_{1},s_{2},r_{1},r_{2}}(\lambda_{1},\lambda_{2}).$

By Lemma 3.1 (iv),

$\displaystyle|G(\lambda_{1},\lambda_{2})|\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}.$

From this, Lemma 3.5 and Remark 3.6, it follows that

$\displaystyle I_{11}$
$\displaystyle\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}\left\|k\right\|_{L^{2}(M^{(1)}\times
M^{(2)})}$
$\displaystyle\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}\frac{s_{1}^{-|\alpha|}}{V^{(1)}(\bar{x}_{1},s_{1})^{1/2}}\frac{s_{2}^{-|\alpha^{\prime}|}}{V^{(2)}(\bar{x}_{2},s_{2})^{1/2}},$

where

$k(\cdot,\cdot)=(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{\left(I+s_{1}\sqrt{\Box_{b}^{(1)}}\right)^{-N_{1}/2}\otimes\left(I+s_{2}\sqrt{\Box_{b}^{(1)}}\right)^{-N_{2}/2}}(x_{1},x_{2},\cdot,\cdot)\Bigm{|}_{(x_{1},x_{2})=(\bar{x}_{1},\bar{x}_{2})};$

here we used the assumptions that $N_{1}>Q_{1}+2(|\alpha|\vee|\beta|)$ and
$N_{2}>Q_{2}+2(|\alpha^{\prime}|\vee|\beta^{\prime}|)$. Similarly,

$\displaystyle
I_{12}\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}\frac{s_{1}^{-|\beta|}}{V^{(1)}(\bar{y}_{1},s_{1})^{1/2}}\frac{s_{2}^{-|\beta^{\prime}|}}{V^{(2)}(\bar{y}_{2},s_{2})^{1/2}}.$

Therefore,

$\displaystyle|I_{1}|=I_{11}\times I_{12}$
$\displaystyle\lesssim\left(1+\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{r_{1}}\right)^{({N_{1}}+\varepsilon+{1/2})-\sigma_{1}}\left(1+\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})}{r_{2}}\right)^{({N_{2}}+\varepsilon+{1/2})-{\sigma_{2}}}$
$\displaystyle\quad\times\frac{s_{1}^{-|\alpha|}}{V^{(1)}(\bar{x}_{1},s_{1})^{1/2}}\frac{s_{2}^{-|\alpha^{\prime}|}}{V^{(2)}(\bar{x}_{2},s_{2})^{1/2}}\frac{s_{1}^{-|\beta|}}{V^{(1)}(\bar{y}_{1},s_{1})^{1/2}}\frac{s_{2}^{-|\beta^{\prime}|}}{V^{(2)}(\bar{y}_{2},s_{2})^{1/2}},$

Next we estimate $I_{2}$. By Lemma 3.1 (iv),

$\displaystyle|J_{s_{1},s_{2},r_{1},r_{2}}(0,\lambda_{2})|\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}(1+\lambda_{2}s_{2})^{-{N_{2}/2}}.$

Hence

$\displaystyle\left|(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},s_{2},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{2},y_{2})\Bigm{|}_{(\bar{x}_{2},\bar{y}_{2})}\right|$
$\displaystyle\qquad=\left|(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}\int_{M^{(2)}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{2},z_{2})K_{J_{s_{1},s_{2},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}})}(z_{2},y_{2})\,dz_{2}\Bigm{|}_{(\bar{x}_{2},\bar{y}_{2})}\right|$
$\displaystyle\qquad\leq\left\|(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{2},\cdot)\Bigm{|}_{x_{2}=\bar{x}_{2}}\right\|_{L^{2}(M^{(2)})}$
$\displaystyle\hskip
113.81102pt\left\|(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{J_{s_{1},s_{2},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}})}(\cdot,y_{2})\Bigm{|}_{y_{2}=\bar{y}_{2}}\right\|_{L^{2}(M^{(2)})}$
$\displaystyle\qquad\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}$
$\displaystyle\hskip
113.81102pt\left\|(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{(I+s_{2}\sqrt{\widehat{\square}_{b}^{(2)}})^{-N_{2}/2}}(x_{2},\cdot)\Bigm{|}_{x_{2}=\bar{x}_{2}}\right\|_{L^{2}(M^{(2)})}$
$\displaystyle\qquad\qquad\times\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}$
$\displaystyle\hskip
113.81102pt\left\|(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{(I+s_{2}\sqrt{\widehat{\square}_{b}^{(2)}})^{-N_{2}/2}}(\cdot,y_{2})\Bigm{|}_{y_{2}=\bar{y}_{2}}\right\|_{L^{2}(M^{(2)})}$
$\displaystyle\qquad\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\left(1+\frac{s_{2}}{r_{2}}\right)^{({N_{2}/2}+{1/4}+{\varepsilon/2})-{\sigma_{2}/2}}\frac{s_{2}^{-|\alpha|}}{V^{(2)}(\bar{x}_{2},s_{2})^{1/2}}\frac{s_{2}^{-|\beta|}}{V^{(2)}(\bar{y}_{2},s_{2})^{1/2}}$
$\displaystyle\qquad\sim\left(1+\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{r_{1}}\right)^{({N_{1}}+\varepsilon+{1/2})-{\sigma_{1}}}\left(1+\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})}{r_{2}}\right)^{({N_{2}}+\varepsilon+{1/2})-{\sigma_{2}}}$
$\displaystyle\hskip
113.81102pt\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(\bar{x}_{2},\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})+r_{2})},$

where we used the assumption that
$N_{2}>Q_{2}+2(|\alpha^{\prime}|\vee|\beta^{\prime}|)$.

Since $\pi^{(1)}$ is an NIS operator,

$\displaystyle\left|(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{\pi^{(1)}}(x_{1},y_{1})\Bigg{|}_{(\bar{x}_{1},\bar{y}_{1}))}\right|$
$\displaystyle\leq
C_{\alpha,\beta}\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(\bar{x}_{1},\rho^{(1)}(\bar{x}_{1},\bar{y}_{1}))}$
$\displaystyle\sim\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(\bar{x}_{1},\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})+r_{1})}.$

Therefore,

$\displaystyle|I_{2}|$
$\displaystyle\lesssim\left(1+\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{r_{1}}\right)^{({N_{1}}+\varepsilon+{1/2})-{\sigma_{1}}}\left(1+\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})}{r_{2}}\right)^{({N_{2}}+\varepsilon+{1/2})-{\sigma_{2}}}$
$\displaystyle\qquad\times\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(\bar{x}_{1},\rho(\bar{x}_{1},\bar{y}_{1})+r_{1})}\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(\bar{x}_{2},\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})+r_{2})}.$

The estimate of $I_{3}$ is similar to that of $I_{2}$.

Finally we estimate $I_{4}$. By Lemma 3.2 (iv),

$\displaystyle|F_{s_{1},s_{2},r_{1},r_{2}}(0,0)|\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\left(1+\frac{s_{2}}{r_{2}}\right)^{(N_{2}+\varepsilon+{1/2})-\sigma_{2}},$

and

$\displaystyle|I_{4}|$
$\displaystyle\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}\left(1+\frac{s_{2}}{r_{2}}\right)^{(N_{2}+\varepsilon+{1/2})-\sigma_{2}}$
$\displaystyle\hskip
142.26378pt\times\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1}))}\frac{\rho^{(2)}(x_{2},y_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(x_{2},\rho^{(2)}(x_{2},y_{2}))}$
$\displaystyle\sim\left(1+\frac{\rho^{(1)}(x_{1},y_{1})}{r_{1}}\right)^{({N_{1}}+\varepsilon+{1/2})-{\sigma_{1}}}\left(1+\frac{\rho^{(2)}(x_{2},y_{2})}{r_{2}}\right)^{({N_{2}}+\varepsilon+{1/2})-{\sigma_{2}}}$
$\displaystyle\hskip
142.26378pt\times\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+r_{1})}\frac{\rho^{(2)}(x_{2},y_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(x_{2},\rho^{(2)}(x_{2},y_{2})+r_{2})}.$

### Case 2.

$\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})\geq r_{1}$ and
$\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})\leq r_{2}$. Let
$\psi(\lambda_{1},\lambda_{2}):=m(\lambda_{1}^{2},\lambda_{2}^{2})$ and define
$F_{s}(\lambda_{1},\lambda_{2})$ by the formula

$\displaystyle\mathcal{F}_{1}{F_{s_{1},r_{1},r_{2}}}(\xi_{1},\lambda_{2})=\phi\left(\frac{\xi_{1}}{s_{1}}\right)\frac{1}{r_{1}}\mathcal{F}_{1}{\psi}\left(\frac{\xi_{1}}{r_{1}},r_{2}\lambda\right).$

Let $J_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})$ be a complex-valued
function such that

$\displaystyle
J_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})^{2}=F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2}).$

Set $s_{1}={\rho^{(1)}(x_{1},y_{1})/(4\kappa_{1})}$. Then by the finite speed
propagation (as in Case 1),

$\displaystyle(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{m(r_{1}^{2}\Box_{b}^{(1)},r_{2}^{2}\Box_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad=(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},r_{1},r_{2}}(\sqrt{\Box_{b}^{(1)}},\sqrt{\Box_{b}^{(2)}}})(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad=(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{F_{s_{1},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\sqrt{\widehat{\square}_{b}^{(2)}})}(x_{1},x_{2},y_{1},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad\qquad+(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{\pi^{1}\otimes
F_{s_{1},r_{1},r_{2}}(0,\sqrt{\widehat{\square}_{b}^{(2)}}})(x_{1},y_{1},x_{2},y_{2})\Bigm{|}_{(\bar{x}_{1},\bar{x}_{2},\bar{y}_{1},\bar{y}_{2})}$
$\displaystyle\qquad=:I_{1}+I_{2}.$

We first estimate $I_{1}$. By Lemma 3.1 (ii),

$\displaystyle|F_{s_{1},r_{1},r_{2}}(\lambda_{1},\lambda_{2})|\lesssim(1+s_{1}\lambda_{1})^{-N_{1}}\left(1+\frac{s_{1}}{r_{1}}\right)^{(N_{1}+\varepsilon+{1/2})-\sigma_{1}}(1+r_{2}\lambda_{2})^{-N_{2}}.$

Hence

$\displaystyle\left|I_{1}\right|$
$\displaystyle=\biggl{|}(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}$
$\displaystyle\qquad\int_{M_{1}\times
M_{2}}K_{J_{s_{1},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\widehat{\square}_{b}^{(2)})}(x_{1},x_{2},z_{1},z_{2})K_{J_{s_{1},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\widehat{\square}_{b}^{(2)})}(z_{1},z_{2},y_{1},y_{2})\,dz_{1}\,dz_{2}\biggr{|}$
$\displaystyle\leq\left\|(X_{\alpha}^{(1)})_{x_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{J_{s_{1},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\widehat{\square}_{b}^{(2)})}(x_{1},x_{2},\cdot,\cdot)\Bigm{|}_{(x_{1},x_{2})=(\bar{x}_{1},\bar{x}_{2})}\right\|_{L^{2}(M_{1}\times
M_{2})}$
$\displaystyle\qquad\times\left\|(X_{\beta}^{(1)})_{y_{1}}(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{J_{s_{1},r_{1},r_{2}}(\sqrt{\widehat{\square}_{b}^{(1)}},\widehat{\square}_{b}^{(2)})}(\cdot,\cdot,y_{1},y_{2})\Bigm{|}_{(y_{1},y_{2})=(\bar{y}_{1},\bar{y}_{2})}\right\|_{L^{2}(M_{1}\times
M_{2})}$
$\displaystyle\lesssim\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}\left(1+\frac{s_{1}}{r_{1}}\right)^{({N_{1}/2}+{1/4}+{\varepsilon/2})-{\sigma_{1}/2}}$
$\displaystyle\qquad\times\left\|(X_{\alpha}^{(1)})_{x_{1}}K_{(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}})^{-N_{1}/2}}(x_{1},\cdot)\Bigm{|}_{x_{1}=\bar{x}_{1}}\right\|_{L^{2}(M^{(1)})}$
$\displaystyle\qquad\times\left\|(X_{\alpha^{\prime}}^{(2)})_{x_{2}}K_{(I+r_{2}\sqrt{\widehat{\square}_{b}^{(2)}})^{-N_{2}/2}}(x_{2},\cdot)\Bigm{|}_{x_{2}=\bar{x}_{2}}\right\|_{L^{2}(M^{(2)})}$
$\displaystyle\qquad\times\left\|(X_{\beta}^{(1)})_{y_{1}}K_{(I+s_{1}\sqrt{\widehat{\square}_{b}^{(1)}})^{-N_{1}/2}}(\cdot,y_{1})\Bigm{|}_{y_{1}=\bar{y}_{1}}\right\|_{L^{2}(M^{(1)})}$
$\displaystyle\qquad\times\left\|(X_{\beta^{\prime}}^{(2)})_{y_{2}}K_{(I+r_{2}\sqrt{\widehat{\square}_{b}^{(2)}})^{-N_{2}/2}}(\cdot,y_{2})\Bigm{|}_{y_{2}=\bar{y}_{2}}\right\|_{L^{2}(M^{(2)})}$
$\displaystyle\lesssim\left(1+\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{r_{1}}\right)^{({N_{1}}+\varepsilon+{1/2})-{\sigma_{1}}}\frac{s_{1}^{-|\alpha|-|\beta|}}{V^{(1)}(\bar{x}_{1},s_{1})}\frac{r_{2}^{-|\alpha|-|\beta|}}{V^{(2)}(\bar{x}_{2},r_{2})}$
$\displaystyle\sim\left(1+\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})}{r_{1}}\right)^{({N_{1}}+\varepsilon+{1/2})-{\sigma_{1}}}\left(1+\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})}{r_{2}}\right)^{({N_{2}}+\varepsilon+{1/2})-{\sigma_{2}}}$
$\displaystyle\qquad\times\frac{\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(\bar{x}_{1},\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})+r_{1})}\frac{\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(\bar{x}_{2},\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})+r_{2})}$

since $\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})\leq r_{2}$.

### Case 3.

$\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})\leq r_{1}$ and
$\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})\geq r_{2}$. This case is very similar to
Case 2, except that we use Lemma 3.1 (iii) instead of Lemma 3.1 (ii).

### Case 4.

$\rho^{(1)}(\bar{x}_{1},\bar{y}_{1})\leq r_{1}$ and
$\rho^{(2)}(\bar{x}_{2},\bar{y}_{2})\leq r_{2}$. This case can be easily
handled by using Lemma 3.1 (i). We omit the details. ∎

##  4\. Proof of Theorem 1.1

Our aim is to show that $m(\Box_{b}^{(1)},\Box_{b}^{(2)})$ is bounded on
$L^{p}(M^{(1)}\times M^{(2)})$. To do this, we will show that
$m(\Box_{b}^{(1)},\Box_{b}^{(2)})$ is a product singular integral operator
satisfying all the conditions in Proposition 2.3.

First, we write

(4.1)
$\begin{split}m(\Box_{b}^{(1)},\Box_{b}^{(2)})&=[m(0,0)\pi^{(2)}\otimes\pi^{(2)}]+\big{[}\pi^{(1)}\otimes
m^{2}(\widehat{\square}_{b}^{(2)})\big{]}\\\
&\qquad+\big{[}m^{1}(\widehat{\square}_{b}^{(1)})\otimes\pi^{(2)}\big{]}+\big{[}m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})\big{]},\end{split}$

where

$\displaystyle m^{1}(\lambda_{1}):=m(\lambda_{1},0)\quad\mbox{and}\quad
m^{2}(\lambda_{2}):=m(0,\lambda_{2}).$

Since $\pi^{(k)}$ is an NIS operator on $M^{(k)}$, it is bounded on
$L^{2}(M^{(k)})$ for $1<p<\infty$ and satisfies (i) and (ii) of Definition 2.1
for $k=1,2$. Hence $\pi^{(1)}\otimes\pi^{(2)}$ satisfies all the conditions in
Proposition 2.3. Moreover, since $m$ satisfies (1.5) and (1.6), it follows
from Street’s result on one-parameter spectral multipliers associated with
Kohn Laplacians that $m^{1}(\widehat{\square}_{b}^{(1)})$ and
$m^{2}(\widehat{\square}_{b}^{(2)})$ satisfy (i) and (ii) of Definition 2.1.
This, together with the fact that $\pi^{(1)}$ and $\pi^{(2)}$ are bounded on
$L^{2}$, implies that both $\pi^{(1)}\otimes
m^{2}(\widehat{\square}_{b}^{(2)})$ and
$m^{1}(\widehat{\square}_{b}^{(1)})\otimes\pi^{(2)}$ satisfy all the
conditions in Proposition 2.3. Thus, it remains to show that
$m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})$ satisfies the
conditions in Proposition 2.3.

Let $\omega$ be a nonnegative smooth function on $[0,\infty)$ supported in
$[0,2]$ such that $\omega(\lambda)=1$ for $\lambda\in[0,1]$. Let
$\theta(\lambda)=\omega(\lambda)-\omega(2\lambda)$ for all
$\lambda\in[0,\infty)$. Then $\operatorname{supp}\theta\subset[1/2,2]$ and

$\displaystyle\sum_{j=-\infty}^{+\infty}\theta(2^{-j}\lambda)$
$\displaystyle=\sum_{j=-\infty}^{+\infty}\big{[}\omega(2^{-j}\lambda)-\omega(2^{-(j-1)}\lambda)\big{]}$
$\displaystyle=\sum_{j=-\infty}^{-1}\big{[}\omega(2^{-j}\lambda)-\omega(2^{-(j-1)}\lambda)\big{]}+\sum_{j=0}^{+\infty}\big{[}\omega(2^{-j}\lambda)-\omega(2^{-(j-1)}\lambda)\big{]}$
$\displaystyle=\lim_{j\to-\infty}\big{[}\omega(2\lambda)-\omega(2^{-j}\lambda)\big{]}+\lim_{j\to+\infty}\big{[}\omega(2^{-j}\lambda)-\omega(2\lambda)\big{]}$
$\displaystyle=-\lim_{j\to-\infty}\omega(2^{-j}\lambda)+\lim_{j\to+\infty}\omega(2^{-j}\lambda)$
$\displaystyle=0+1=1.$

Set
$m_{j_{1},j_{2}}({\Box}_{b}^{(1)},{\Box}_{b}^{(2)})=\theta(2^{-j_{1}}{\Box}_{b}^{(1)})\theta(2^{-j_{2}}{\Box}_{b}^{(2)})m({\Box}_{b}^{(1)},{\Box}_{b}^{(2)})$.
Now $m_{j_{1},j_{2}}(0,0)=0$, and it follows that
$m_{j_{1},j_{2}}({\Box}_{b}^{(1)},{\Box}_{b}^{(2)})=m_{j_{1},j_{2}}(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})$.
This implies that

$m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})=\sum_{j_{1},j_{2}}m_{j_{1},j_{2}}({\Box}_{b}^{(1)},{\Box}_{b}^{(2)}).$

Similarly, define

$m_{j_{1}}({\Box}_{b}^{(1)},\widehat{\square}_{b}^{(2)})=\theta(2^{-j_{1}}{\Box}_{b}^{(1)})m({\Box}_{b}^{(1)},\widehat{\square}_{b}^{(2)})$

and

$m_{j_{2}}({\Box}_{b}^{(1)},\widehat{\square}_{b}^{(2)})=\theta(2^{-j_{2}}{\Box}_{b}^{(2)})m(\widehat{\square}_{b}^{(1)},{\Box}_{b}^{(2)}).$

Then

$m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})=\sum_{j_{1}}m_{j_{1}}({\Box}_{b}^{(1)},\widehat{\square}_{b}^{(2)})$

and

$m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})=\sum_{j_{2}}m_{j_{2}}(\widehat{\square}_{b}^{(1)},{\Box}_{b}^{(2)}).$

Next we need the following auxiliary lemma.

######  Lemma 4.1.

Let $\delta=\rho(x,y)$ and $N>a\geq 0$.

(4.2)
$\sum_{j=-\infty}^{\infty}\left(1+\frac{\delta}{2^{-j}}\right)^{-N}\frac{2^{ja}}{V(x,\delta+2^{-j})}\leq
C\frac{\delta^{-a}}{V(x,\delta)}.$

###### Proof.

The volume of the ball satisfies the reverse doubling condition (2.3) and thus

$\displaystyle\sum_{2^{-j}\geq\delta}\left(1+\frac{\delta}{2^{-j}}\right)^{-N}\frac{2^{ja}}{V(x,\delta+2^{-j})}$
$\displaystyle\qquad\leq
C_{N}\sum_{2^{-j}\geq\delta}\left(\frac{\delta}{2^{-j}}\right)^{4}\frac{2^{ja}}{V(x,\delta)}$
$\displaystyle\qquad\leq C_{N}\frac{\delta^{-a}}{V(x,\delta)}.$

On the other hand, since $N>a$,

$\displaystyle\sum_{2^{-j}<\delta}\left(1+\frac{\delta}{2^{-j}}\right)^{-N}\frac{2^{ja}}{V(x,\delta+2^{-j})}$
$\displaystyle\qquad\leq
C_{N}\sum_{2^{-j}<\delta}\left(\frac{\delta}{2^{-j}}\right)^{-N}\frac{2^{ja}}{V(x,\delta)}$
$\displaystyle\qquad\leq C_{N}\frac{\delta^{-a}}{V(x,\delta)}.$

The proof of Lemma 4.1 is complete. ∎

###### Proof of Theorem 1.1.

We only need to check that
$K_{m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})}$ satisfies the
conditions of Proposition 2.3. Define $m_{j_{1},j_{2}},m_{j_{1}}$ and
$m_{j_{2}}$ as above. Then from Proposition 3.8 and Lemma 4.1, it follows that

$\displaystyle\left|(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(Y^{(2)}_{\beta^{\prime}})_{y_{2}}K_{m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})\right|$
$\displaystyle\qquad\lesssim\sum_{j_{1},j_{2}}\left|(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}(X_{\alpha^{\prime}}^{(2)})_{x_{2}}(Y^{(2)}_{\beta^{\prime}})_{y_{2}}m_{j_{1},j_{2}}({\Box}_{b}^{(1)},{\Box}_{b}^{(2)})(x_{1},x_{2},y_{1},y_{2})\right|$
$\displaystyle\qquad\leq\sum_{j_{1},j_{2}}\|\theta(\cdot_{1})\theta(\cdot_{2})m(2^{j_{1}}\cdot_{1},2^{j_{2}}\cdot_{2})\|_{W^{(\sigma_{1},\sigma_{2}),2}(\mathbb{R}\times\mathbb{R})}$
$\displaystyle\qquad\qquad\times\left(1+\frac{\rho^{(1)}(x_{1},y_{1})}{2^{-j_{1}}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}\frac{(2^{-j_{1}}\vee\rho^{(1)}(x_{1},y_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+2^{-j_{1}})}$
$\displaystyle\qquad\qquad\times\left(1+\frac{\rho^{(2)}(x_{2},y_{2})}{2^{-j_{2}}}\right)^{N_{2}+\varepsilon+{1/2}-\sigma_{2}}\frac{(2^{-j_{2}}\vee\rho^{(2)}(x_{2},y_{2}))^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(2)}(x_{2},\rho^{(2)}(x_{2},y_{2})+2^{-j_{2}})}$
$\displaystyle\qquad\leq
C\sup_{t_{1},t_{2}>0}\|\eta(\lambda_{1})\eta(\lambda_{2})m(t_{1}\lambda_{1},t_{2}\lambda_{2})\|_{W^{(\sigma_{1},\sigma_{2}),2}}\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}\rho^{(2)}(x_{2},y_{2})^{-|\alpha^{\prime}|-|\beta^{\prime}|}}{V^{(1)}(x_{1},y_{1})V^{(2)}(x_{2},y_{2})}.$

Similarly, from Proposition 3.7 and Lemma 4.1 it follows that

$\displaystyle\left|\int_{M^{(2)}}\int_{M^{(2)}}(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{m(\widehat{\square}_{b}^{(1)},\widehat{\square}_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})f_{2}(x_{2})g_{2}(y_{2})\,d\mu^{(2)}(x_{2})\,d\mu^{(2)}(y_{2})\right|$
$\displaystyle\qquad\leq\sum_{j_{1}}\biggl{|}\int_{M^{(2)}}\int_{M^{(2)}}(X_{\alpha}^{(1)})_{x_{1}}(X_{\beta}^{(1)})_{y_{1}}K_{m_{j_{1}}({\Box}_{b}^{(1)},\widehat{\square}_{b}^{(2)})}(x_{1},x_{2},y_{1},y_{2})$
$\displaystyle\hskip
113.81102ptf_{2}(x_{2})g_{2}(y_{2})\,d\mu^{(2)}(x_{2})\,d\mu^{(2)}(y_{2})\biggr{|}$
$\displaystyle\qquad\leq
C\sum_{j_{1}}\|\theta(\cdot_{1})m(2^{j_{1}}\cdot_{1},\cdot_{2})\|_{W^{\sigma_{1},2}L^{\infty}(\mathbb{R}\times\mathbb{R})}\|f_{2}\|_{2}\|g_{2}\|_{2}$
$\displaystyle\hskip
113.81102pt\times\left(1+\frac{\rho^{(1)}(x_{1},y_{1})}{2^{-j_{1}}}\right)^{N_{1}+\varepsilon+{1/2}-\sigma_{1}}\frac{(2^{-j_{1}}\vee\rho^{(1)}(x_{1},y_{1}))^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},\rho^{(1)}(x_{1},y_{1})+2^{-j_{1}})}$
$\displaystyle\qquad\leq
C\sup_{t_{1},t_{2}>0}\|\eta(\lambda_{1})\eta(\lambda_{2})m(t_{1}\lambda_{1},t_{2}\lambda_{2})\|_{W^{(\sigma_{1},\sigma_{2}),2}}\|f_{2}\|_{2}\|g_{2}\|_{2}\frac{\rho^{(1)}(x_{1},y_{1})^{-|\alpha|-|\beta|}}{V^{(1)}(x_{1},y_{1})}.$

Note that this estimate still holds if the indices $1$ and $2$ are
interchanged, that is, if the roles of $M^{(1)}$ and $M^{(2)}$ are changed.
Moreover, since $m(\Box_{b}^{(1)},\Box_{b}^{(2)})$ is a self-adjoint operator,
its transpose also satisfies these conditions. Hence,
$m(\Box_{b}^{(1)},\Box_{b}^{(2)})$, defined spectrally on
$L^{2}(\widetilde{M})$, is a product Calderón–Zygmund operator on
$\widetilde{M}$ as studied in [5]. ∎

## Acknowledgements

Cowling is supported by the Australia Research Council, through grant
DP170103025. Hu is supported by the National Natural Science Foundation of
China (Grant No. 11901256) and the Natural Science Foundation of Jiangxi
Province (Grant No. 20192BAB211001). Li is supported by the Australia Research
Council, through grant DP170101060.

## References

  * [1] L.K. Chen, The multiplier operators on the product spaces, Illinois J. Math. 38 (1994), 420–433.
  * [2] P. Chen, X. Duong, J. Li, L.A. Ward and L. Yan, Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type, J. Fourier Anal. Appl. 23 (2017), 21–64.
  * [3] P. Chen, X. Duong, J. Li, L.A. Ward and L. Yan, Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type, Math. Z. 282 (2016), 1033–1065.
  * [4] R. Gundy and E.M. Stein, $H^{p}$ theory for the poly-disc, Proc. Nat. Acad. Sci. 76 (1979), 1026–1029.
  * [5] Y.S. Han, J. Li and C.C. Lin, Criterions of the $L^{2}$ boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) XVI (2016), 845–907.
  * [6] J.-L. Journé, Calderón–Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), 55–91.
  * [7] K.D Koenig, On maximal Sobolev and Hölder estimates for the tangential Cauchy–Riemann operator and boundary Laplacian, Amer. J. Math. 124 (2002), 129–197.
  * [8] R. Melrose, Propagation for the wave group of a positive subelliptic second-order differential operator. Hyperbolic equations and related topics (Katata/Kyoto, 1984), pp. 181–192. Academic Press, Boston, (1986).
  * [9] A. Nagel, J.-P. Rosay, E.M. Stein and S. Wainger, Estimates for the Bergman and Szegö kernels in $\mathbb{C}^{2}$, Ann. of Math. (2) 129 (1989), 113–149.
  * [10] A. Nagel and E. M. Stein, Differentiable control metrics and scaled bump functions, J. Differential Geom. 57 (2001), 465–492.
  * [11] A. Nagel and E. M. Stein, The $\Box_{b}$-heat equation on pseudoconvex manifolds of finite type in $\mathbb{C}^{2}$, Math. Z. 238 (2001), 37–88.
  * [12] A. Nagel and E. M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoamericana 20 (2004), 531–561.
  * [13] A. Nagel and E. M. Stein, The $\bar{\partial}_{b}$-complex on decoupled boundaries in $\mathbb{C}^{n}$, Ann. Math. 164 (2006), 649–713.
  * [14] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), 103–147.
  * [15] B. Street, The $\Box_{b}$-heat equation and multipliers via the wave equation, Math. Z. 263 (2009), 861–886.
  * [16] B. Street, Multi-paramter singular integrals. Annals of Mathematics Studies, 189. Princeton University Press, Princeton, NJ, 2014.