# On a class of pseudodifferential operators on the product of compact Lie
groups

Serena Federico  Serena Federico: Department of Mathematics: Analysis Logic
and Discrete Mathematics Ghent University Krijgslaan 281, Ghent, B 9000
Belgium E-mail address<EMAIL_ADDRESS>and  Alberto Parmeggiani
Alberto Parmeggiani: Department of Mathematics University of Bologna Piazza di
Porta S. Donato 5, 40126 Bologna Italy E-mail address
<EMAIL_ADDRESS>
###### Abstract.

In this paper a bisingular pseudodifferential calculus, along the lines of the
one introduced by L. Rodino in [Rodino_ASNSP_1975], is developed in the global
setting of a product of compact Lie groups. The approach follows that
introduced by M. Ruzhansky and V. Turunen [MR-VT-Book_2010] (see also V.
Fischer [Fischer-JFA_2015]), in that it exploits the harmonic analysis of the
groups involved.

Serena Federico has received funding from the European Unions Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 838661 and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Bisingular symbols on $G=G_{1}\times G_{2}$
  4. 4 Kernel estimates
  5. 5 Calculus of bisingular pseudodifferential operators
  6. A

##  1\. Introduction

In this paper we shall introduce a class of global pseudodifferential
operators on the product of compact Lie groups and develope the corresponding
global symbolic calculus in the spirit of the one introduced by Ruzhansky and
Turunen in [MR-VT-Book_2010] and of that introduced subsequently by Fischer in
[Fischer-JFA_2015].

On the product of two manifolds the class we consider here was first studied
by Rodino in [Rodino_ASNSP_1975], where, in particular, the author used the
classical theory of pseudodifferential operators developed by Hörmander in
[Ho-FIO1] to construct an algebra of pseudodifferential operators containing
the so-called bisingular operators.

The interest of our approach lies in the fact that it is global and based on
the group structure and on its related representation theory.

As the classes $S^{m_{1},m_{2}}(\Omega_{1}\times\Omega_{2})$ in
[Rodino_ASNSP_1975] are not in general contained in any of the Hörmander
classes $S^{m}(\Omega_{1}\times\Omega_{2}\times\mathbb{R}^{n_{1}+n_{2}})$,
similarily in our case the classes $S^{m_{1},m_{2}}(G_{1}\times
G_{2}\times\widehat{G}_{1}\times\widehat{G}_{2})$ are not in general contained
in any class $S^{m}(G)$, with $G=G_{1}\times G_{2}$, defined by Ruzhansky and
Turunen in [MR-VT-Book_2010].

Natural examples of bisingular pseudodifferential operators in our setting, as
in the general compact manifold setting, are tensor products of the form
$A_{1}\otimes A_{2}$, where $A_{i}$ for $i=1,2$ is a pseudodifferential
operator with symbol in the class $S^{m_{i}}(G_{i})$ introduced in [MR-VT-JW-
JFAA_2014], that is, $A_{i}\in L^{m}(G_{i}):=\mathrm{Op}(S^{m_{i}}(G_{i}))$,
with $G_{i}$ being a compact Lie group.

The study of these operators goes back to 1971, when Pilidi in [Pilidi-
DAN1971] reduced the boundary value problem for functions of two complex
variables in bicylinders to the analysis of a bisingular equation on the two
distinguished boundaries. In [Pilidi-FA1971] the same author also developed a
product calculus to deal with these objects and considered the corresponding
index problem. Afterwards, a priori estimates and Fredholm properties for
bisingular operators were studied by Rabinovic̆ in [RAB-1973], while in 1975
Rodino in [Rodino_ASNSP_1975] introduced the so-called calculus of bisingular
pseudodifferential operators. Other related questions, such as residues and
index problems, have been recently considered by Nicola and Rodino in [MR-
NR-2006], while microlocal properties have been studied by Borsero and Schulz
in [BO-SH-2014].

Let us also recall that a global version (i.e. in the Shubin setting of
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$) of the calculus in
[Rodino_ASNSP_1975] was developed by Battisti, Gramchev, Pilipovič and Rodino
in [BA-GR-RO-PIL-2013], and that other calculi of product type were developed
by Dudučava in [DUD-I-1979] and [DUDI-II-1979], and more recently by Melrose
and Rochon in [ME-ROC-2006].

Note that a natural and immediate generalization of bisingular operators are
the multisingular ones, whose prototype are tensor products of the form
$\bigotimes_{\begin{subarray}{c}i=1\end{subarray}}^{N}A_{i}$, with $A_{i}\in
L^{m_{i}}(G_{i})$. We will not pursue this topic here, but with suitable
arrangements in the arguments used below one can define a multisingular
pseudodifferential calculus on the direct product of finitely many compact Lie
groups and define the corresponding multisingular pseudodifferential
operators.

We want to remark that due to the intrinsic product stucture of the bisingular
calculus, the suitable version of the celebrated Gårding inequality for
elliptic operators is not available for bielliptic operators (see for instance
[RAB-1973], where such inequality is attained only under very specific
assumptions). Hence, it seems that for the class of bisingular operators, that
serves as a model for degenerate elliptic operators, a more natural inequality
to consider is the sharp Gårding inequality. We will analyze the problem of
the validity of this inequality in a future paper that will be part II of the
present work.

We finally conclude this introduction by giving the plan of the paper.

In Section 2 we shall recall some basic definitions on compact Lie groups,
such as the notions of Fourier tranform, difference operators and Taylor
expansion, as well as the standard quantization formula.

In Section 3 we introduce the class of bisingular symbols and define the
corresponding pseudodifferential operators.

Section 4 will be devoted to the derivation of some fundamental kernel
estimates needed to prove some asymptotic properties that are the object of
Section 5.

Finally in Section 5 we develop the calculus, that is, we prove asymptotic
formulas for the composition and for the adjoint of bisingular operators, and
prove, after introducing ellipticity in the bisingular setting, the existence
of parametrices for bielliptic operators.

##  2\. Preliminaries

In the sequel $G$ will be a compact Lie group, $\widehat{G}$ its unitary dual,
that is the set of all equivalence classes of unitary representations of $G$,
and $\mathrm{Rep}(G)$ the set of all the irreducible unitary representations
of $G$. Since $G$ is compact, any given $\xi\in\mathrm{Rep}(G)$ is finite
dimensional and we shall denote by $\mathcal{H}_{\xi}$ the associated
representation space, and by $\mathcal{U}(\mathcal{H}_{\xi})$ the
corresponding space of unitary operators on $\mathcal{H}_{\xi}$.

The Fourier and inverse Fourier transforms on $G$ are given in terms of the
representations of the group as follows.

Given a function $f\in C^{\infty}(G)$, and $\xi\in\mathrm{Rep}(G)$, the
(matrix-valued) global Fourier transform of $f$ at $\xi$ is defined by

$\widehat{f}(\xi)=\int_{G}f(x)\xi^{*}(x)dx,$

where $\xi^{*}(x):=\,^{t}\overline{\xi(x)}$ stands for the adjoint
representation of $\xi$, while $dx$ denotes the Haar measure on the group.
Notice that, given
$\xi:\mathcal{H}_{\xi}\rightarrow\mathcal{U}(\mathcal{H}_{\xi})$, and
$d_{\xi}:=\mathrm{dim}(\xi):=\mathrm{dim}(\mathcal{H}_{\xi})$, then
$\widehat{f}(\xi)\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$. Correspondingly, the
inverse Fourier transform is given by

$f(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}(\xi(x)\widehat{f}(\xi)),$

where $\mathrm{Tr}(A)$ denotes the trace of the matrix $A$.

Related to the previous formulas one has the following Parseval identity

$\|f\|^{2}_{L^{2}(G)}=\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\widehat{f}(\xi)\|^{2}_{HS}=:\|\widehat{f}\|_{\ell^{2}(\widehat{G})}^{2},$

where
$\|\widehat{f}(\xi)\|_{HS}:=\left(\mathrm{Tr}(\widehat{f}(\xi)\widehat{f}(\xi)^{*})\right)^{1/2}$
is the Hilbert-Schmidt norm .

In order to deal with (matrix-valued) functions on $\widehat{G}$ we will need
to make use of the so called difference operators that we next define
following [MR-VT-JW-JFAA_2014].

######  Definition 2.1.

We say that $Q_{\xi}$ is a difference operator of order $k$ on
$\mathcal{F}(\mathcal{D}^{\prime}(G))$ (the image of the group Fourier
transform of distributions on $G$) if

$Q_{\xi}\widehat{f}(\xi)=\widehat{q_{Q}f}(x),$

for a function $q_{Q}\in C^{\infty}(G)$ vanishing of order $k$ at the identity
element $e$ of $G$, that is, $q_{Q}$ is such that $q_{Q}(e)=P_{x}q_{Q}(e)=0$
for all left-invariant differential operators $P_{x}\in\mathrm{Diff^{k-1}(G)}$
of order $k-1$.

We shall denote by $\mathrm{diff}^{k}(\widehat{G})$ the set of all difference
operators of order $k$ on $\widehat{G}$.

######  Definition 2.2.

A collection of $n_{\triangle}\geq n=\mathrm{dim}(G)$ difference operators
$\triangle_{1},...,\triangle_{n_{\triangle}}$ in
$\mathrm{diff}^{1}(\widehat{G})$ is called admissible if the corresponding
functions $q_{1},...,q_{n_{\triangle}}\in C^{\infty}(G)$ are such that
$q_{1}(e)=\ldots=q_{n_{\triangle}}(e)=0$, and $dq_{j}(e)\neq 0$ for all
$j=1,...,{n_{\triangle}},$ with
$\mathrm{rank}(dq_{1}(e),...,dq_{n_{\triangle}}(e))=n$. Finally, a collection
of difference operators is called strongly admissible if $\bigcap_{j}\\{x\in
G;q_{j}(x)=0\\}=\\{e\\}$.

Given a fixed family of functions $Q=\\{q_{j}\\}_{j=1,...,{n_{\triangle}}}$,
we shall denote by

  * •

$\triangle_{Q}$ the associated admissible collection of difference operators;

  * •

$q^{\alpha}:=q_{1}^{\alpha_{1}}\ldots
q_{n_{\triangle}}^{\alpha_{n_{\triangle}}}$;

  * •

$\triangle_{Q,j}=\triangle_{q_{j}}$ and
$\triangle_{Q}^{\alpha}:=\triangle_{Q,1}^{\alpha_{1}}\cdots\triangle_{Q,n_{\triangle}}^{\alpha_{n_{\triangle}}}$
the corresponding element in $\mathrm{diff}^{|\alpha|}(\widehat{G})$.

Additionally, once the collection of difference operators is fixed, namely the
corresponding family of functions $Q$ is fixed, one can find a family of
differential operators in $\mathrm{Diff}^{|\alpha|}(G)$, denoted by
$\partial_{x}^{(\alpha)}$, such that the following Taylor’s formula holds

$f(x)=\sum_{|\alpha|<N}\frac{1}{\alpha!}q(x)^{\alpha}\partial_{x}^{(\alpha)}f(e)+\mathcal{O}(h(x)^{N}),\quad
h(x)\rightarrow 0,$

for all $f\in C^{\infty}(G)$, where $h(x)$ is the geodesic distance from $x$
to $e_{G}$. The differential operators $\partial_{x}^{(\alpha)}$ can be
replaced by
$\partial^{\alpha}_{x}:=\partial_{x_{1}}^{\alpha_{1}}\ldots\partial_{x_{n}}^{\alpha_{n}}$,
with $\partial_{x_{j}}$, $j=1,\ldots,n$, being a collection of left-invariant
first order differental operators corresponding to some linearly independent
left-invariant vector fields on $G$ ($\partial_{x_{j}}$ are not the Euclidean
directional derivatives here).

######  Remark 2.3.

Note that we are assuming the Lie algebra $\mathfrak{g}$ to be the space of
left-invariant vector fields. In particular, we shall use the notation
$\partial_{x_{j}}$ and $\tilde{\partial}_{x_{j}}$ for the left and right
invariant vector fields, respectively. Once we fix an orthonormal basis of
left-invariant vector fields for $\mathfrak{g}$, then any element of
$\mathrm{Diff^{k}(G)}$ (the space of smooth vector fields on $G$) can be
written as a linear combination in terms of the elements of the basis. Note
also that a similar property holds for right-invariant vector fields.

By Lemma 4.4 in [MR-VT-JW-JFAA_2014], the family of functions
$\\{q_{ij}=\xi_{ij}-\delta_{ij}\\}_{[\xi]\in\widehat{G},1\leq i,j\leq
d_{\xi}}$ always induces a strongly admissible collection of difference
operators, therefore we choose the latter as the fixed admissible collection
for the rest of the paper.

In the context of the difference operators defined above, the following notion
of Leibniz formula is adopted (see [Fischer-JFA_2015])

######  Definition 2.4.

A collection of $\triangle=\triangle_{Q}$ difference operators satisfies the
Leibniz-like property if, for any Fourier transforms $\widehat{f_{1}}$ and
$\widehat{f_{2}}$ (with $f_{1},f_{2}\in\mathcal{D}^{\prime}(G)$),

$\triangle_{Q,j}(\widehat{f_{1}}\widehat{f_{2}})=\triangle_{Q,j}(\widehat{f_{1}})\widehat{f_{2}}+\widehat{f_{1}}\triangle_{Q,j}(\widehat{f_{2}})+\sum_{1\leq
l,k\leq
n_{\triangle}}c_{l,k}^{(j)}\triangle_{Q,l}(\widehat{f_{1}})\triangle_{Q,k}(\widehat{f_{2}}),$

for some coefficients $c_{l,k}^{(j)}\in\mathbb{C}$ depending only on $l,k,j$
and $\triangle$.

If $\triangle$ is a collection satisfying the Leibniz-like formula, then,
recursively, for any given $\alpha\in\mathbb{N}^{n_{\triangle}}_{0}$, one has

$\triangle_{Q}^{\alpha}(\widehat{f_{1}}\widehat{f_{2}})=\sum_{|\alpha|\leq|\alpha_{1}|+|\alpha_{2}|\leq
2|\alpha|}c_{\alpha_{1},\alpha_{2}}^{\alpha}(\triangle_{Q}^{\alpha_{1}}\widehat{f_{1}})(\triangle_{Q}^{\alpha_{2}}\widehat{f_{2}}),$
(2.1)

for some coefficients $c_{\alpha_{1},\alpha_{2}}^{\alpha}\in\mathbb{C}$
depending on $\alpha_{1},\alpha_{2},\alpha$ and $\triangle$, with
$c^{\alpha}_{\alpha,0}=c^{\alpha}_{0,\alpha,}=1$.

######  Remark 2.5.

We remark that there always exists a strongly admissible collection of
difference operators as in Definition 2.2 for which the Leibniz property above
holds (see Corollary 5.13 in [Fischer-JFA_2015]). In particular this is always
the case for the strongly admissible collection $\triangle_{Q}$ with
$Q=\\{q_{ij}=\xi_{ij}-\delta_{ij},1\leq i,j\leq d_{\xi},\xi\in\widehat{G}\\}$.

In order to introduce the precise difference operators we are going to exploit
in our setting, it is more convenient for us to use the definition of
difference operators given in [Fischer-JFA_2015], which, in turn, gives rise
to the same difference operators defined above by means of the admissible
collection $\\{q_{ij}=\xi_{ij}-\delta_{ij}\\}_{[\xi]\in\widehat{G},1\leq
i,j\leq d_{\xi}}$.

######  Definition 2.6.

For each $\tau,\xi\in\mathrm{Rep}(G)$ we define the linear mapping
$\triangle_{\tau}\widehat{f}(\xi)$ on
$\mathcal{H}_{\tau}\otimes\mathcal{H}_{\xi}$ by

$\triangle_{\tau}\widehat{f}(\xi)=\widehat{f}(\tau\otimes\xi)-\widehat{f}(I_{d_{\tau}}\otimes\xi).$
(2.2)

The restriction of $\triangle_{\tau}\widehat{f}(\xi)$ to any occurrence of
$\rho\in\widehat{G}$ in the decomposition into irreducibles of
$\tau\otimes\xi$ defines the same mapping on $\mathcal{H}_{\rho}$, while the
restriction to any $\rho\in\widehat{G}$ not appearing in the decomposition of
$\tau\otimes\xi$ is fixed to be zero. With these conventions the operation
$\triangle_{\tau}$ is called difference operator associated with
$\tau\in\mathrm{Rep}(G)$.

######  Definition 2.7.

Let $G=G_{1}\times G_{2}$ a compact Lie group such that $G_{i}$ is compact for
all $i=1,2$, and $n_{i}:=\mathrm{dim}(G_{i})$. Let $e=(e_{1},e_{2})$ be the
neutral element of $G$. A collection of $n_{P}:=n_{\triangle_{P}}\geq n_{1}$
of difference operators
$\triangle_{p_{1}},\ldots,\triangle_{p_{n_{P}}}\in\mathrm{diff}^{1}(\widehat{G})$
is called admissible relative to $G_{1}$ if the corresponding functions
$p_{1},\ldots,p_{n_{P}}\in C^{\infty}(G)$ are such that
$p_{1}(e)=\ldots=p_{n_{P}}(e)=0$, and $dp_{j}(e)\neq 0$ for all
$j=1,...,n_{P}$, with $\mathrm{rank}(dp_{1}(e),\ldots,dp_{n_{P}}(e))=n_{1}$.

The collection is called strongly admissible relative to $G_{1}$ if
$\bigcap_{j=1}^{n_{\triangle_{P}}}\\{x\in G;p_{j}(x)=0\\}=\\{e_{1}\\}\times
G_{2}.$ Admissible and strongly admissible collections relative to $G_{2}$ are
defined similarily by reversing the role of $G_{1}$ and $G_{2}$.

We then consider the family of functions

$R=\\{r_{ij}^{(\tau)};1\leq i,j\leq
d_{\tau},\tau\in\widehat{G}\\}=\\{(I_{d_{\tau_{1}}}\otimes\tau_{2}-I_{d_{\tau}})_{ij};1\leq
i,j\leq d_{\tau},\tau\in\widehat{G}\\}$ (2.3)

and

$P=\\{p_{ij}^{(\tau)};1\leq i,j\leq
d_{\tau},\tau\in\widehat{G}\\}=\\{(\tau_{1}\otimes
I_{d_{\tau_{2}}}-I_{d_{\tau}})_{ij};1\leq i,j\leq
d_{\tau},\tau\in\widehat{G}\\},$ (2.4)

so that both $\triangle_{R}$ and $\triangle_{P}$ are strongly admissible
collections relative to $G_{1}\simeq G_{1}\times\\{e_{2}\\}\subset G$ and
$G_{2}\simeq\\{e_{1}\\}\times G_{2}\subset G$ respectively (see Definition
2.7). After ordering the (huge but finite) families $P$ and $R$ above, that
is, writing

$P=\\{p_{k},k=1,...,n_{{P}}\\},\quad R=\\{r_{k},k=1,...,n_{{R}}\\},$

where each $p_{k},r_{k}$ are functions of the form
$p_{ij}^{(\tau)},r_{ij}^{(\tau)}$ respectively, for some $\tau\in\widehat{G}$
and some $i,j\in\\{0,...,d_{\tau}\\}$, we may define

$\triangle^{\alpha,\beta}:=\triangle_{P}^{\alpha}\triangle_{R}^{\beta}=\triangle^{\alpha_{1}}_{p_{1}}\cdots\triangle^{\alpha_{n_{P}}}_{p_{n_{P}}}\,\triangle^{\beta_{1}}_{r_{1}}\cdots\triangle^{\beta_{n_{R}}}_{r_{n_{R}}}.$
(2.5)

Note that the function $p_{j}$, for all $j=1,...,n_{{P}},$ is independent of
$x_{2}\in G_{2}$, and, similarily, the function $r_{j}$, for all
$j=1,...,n_{{R}}$, is independent of $x_{1}$.

These difference operators, namely of the form (2.5), will be the ones used
for the rest of the paper. Note that $\triangle_{P}^{\alpha}$ and
$\triangle_{Q}^{\beta}$ may be tought of as ”partial difference operators” in
the “directions” of $\widehat{G}_{1}$ and $\widehat{G}_{2}$ respectively.

By Remark 2.5 and formula (2.4) applied iteratively, we have the following
Leibniz-like formula for the difference operators we are considering.

######  Proposition 2.8.

Let $G=G_{1}\times G_{2}$ , with $G_{1}$ and $G_{2}$ compact Lie groups. Then,
for any $\alpha\in\mathbb{N}_{0}^{n_{\triangle_{P}}}$,
$\beta\in\mathbb{N}_{0}^{n_{\triangle_{R}}}$, and for all Fourier transforms
$\widehat{f_{1}},\widehat{f_{2}}$ (with
$f_{1},f_{2}\in\mathcal{D}^{\prime}(G)$), we have

$\triangle^{\alpha,\beta}(\widehat{f_{1}}\widehat{f_{2}})=\sum_{|\alpha|\leq|\alpha_{1}|+|\alpha_{2}|\leq
2|\alpha|}\sum_{|\beta|\leq|\beta_{1}|+|\beta_{2}|\leq
2|\beta|}c^{\alpha}_{\alpha_{1},\alpha_{2}}c^{\beta}_{\beta_{1},\beta_{2}}(\triangle^{\alpha_{1},\beta_{1}}\widehat{f_{1}})(\triangle^{\alpha_{2},\beta_{2}}\widehat{f_{2}}),$
(2.6)

for some coefficients
$c^{\alpha}_{\alpha_{1},\alpha_{2}},c^{\beta}_{\beta_{1},\beta_{2}}\in\mathbb{C}$
such that
$c^{\beta}_{\beta,0}=c^{\beta}_{0,\beta}=c^{\alpha}_{\alpha,0}=c^{\alpha}_{0,\alpha}=1$.

###### Proof.

The proof immediately follows by application of formula (2.1) twice, that is,
for $\triangle^{\alpha}_{P}$ and $\triangle_{Q}^{\beta}$ respectively. ∎

Observe now that, since the families of functions in (2.3) and (2.4) defining
admissible collections of difference operators on $\widehat{G}$ relative to
$G_{1}$ and $G_{2}$ are fixed, on denoting by $x=(x_{1},x_{2})$ an element of
$G=G_{1}\times G_{2}$, with $\mathrm{dim}(G_{1})=n_{1}$ and
$\mathrm{dim}(G_{2})=n_{2}$, we can find a family of differential operators

$\partial^{\alpha,\beta}_{x}:=\partial_{x_{1}}^{\alpha}\partial^{\beta}_{x_{2}}$

such that the following form of Taylor’s formula holds (see, for instance,
[Fischer-JFA_2015])

$f(x)=\sum_{|\alpha|<N}\sum_{|\beta|<N}\frac{1}{\alpha!\beta!}q^{\alpha,\beta}(x^{-1})\partial_{x}^{\alpha,\beta}f(e)+\sum_{\begin{subarray}{c}|\alpha+\beta|=2N,\\\
|\alpha|\geq N\vee|\beta|\geq
N\end{subarray}}\frac{1}{\alpha!\beta!}q^{\alpha,\beta}(x^{-1})f_{\alpha,\beta}(x),$

where

$q^{\alpha,\beta}(x):=r(x)^{\alpha}p(x)^{\beta}=r_{1}(x)^{\alpha_{1}}\ldots
r_{n_{R}}(x)^{\alpha_{n_{R}}}\,p_{1}(x)^{\beta_{1}}\ldots
p_{n_{P}}(x)^{\beta_{n_{P}}}.$

Recall that, in particular, we will have
${\triangle_{R,j}}\widehat{f}(\xi):=\widehat{r_{j}f}(\xi)$ and
${\triangle_{P,k}}=\widehat{p_{k}f}(\xi)$. Moreover, the differential
operators $\partial^{\alpha,\beta}_{x}$ are chosen so as to satisfy
$\partial_{x_{1}}^{\alpha}(p(x)^{\alpha})=\partial_{x_{2}}^{\beta}(r(x)^{\beta})=1$
for all $\alpha,\beta$ such that $|\alpha|=|\beta|=1$. In particular, since
$P$ and $R$ are strongly admissible collections relative to $G_{1}$ and
$G_{2}$ respectively, we have that there are $n_{1}$ and $n_{2}$ elements in
$P$ and $R$ respectively, say $(p_{1},\ldots,p_{n_{1}})$ and
$(r_{1},\ldots,r_{n_{2}})$, such that
$(\partial_{x_{1,1}},\ldots,\partial_{x_{1,n_{1}}},\partial_{x_{2,1}},\ldots,\partial_{x_{2,n_{2}}})$
can be identified with
$(dp_{1}(e),\ldots,dp_{n_{1}}(e),dr_{1}(e),\ldots,dr_{n_{2}}(e))$ (where
$df(e)$ denotes the differential computed at $e$) by duality, and we fix the
former as the basis of the Lie algebra $\mathfrak{g}$. We stress that the
choice of $q^{\alpha,\beta}(x^{-1})$ istead of $q^{\alpha,\beta}(x)$ is
technical (see [MR-VT-Book_2010]). Note finally that the formula above can be
derived by application of Taylor’s formula twice, that is, first with respect
to the variable $x_{1}$ by using the functions
$q^{\alpha,0}(x)=q^{\alpha,0}(x_{1})$, and then by expanding again with
respect to $x_{2}$ and using $q^{0,\beta}(x)=q^{0,\beta}(x_{2})$.

##  3\. Bisingular symbols on $G=G_{1}\times G_{2}$

In this section we define what we shall call class of bisingular symbols,
since, as pointed out by L. Rodino in [Rodino_ASNSP_1975], it contains symbols
of operators of bisingular type (see [Pilidi-DAN1971],[Pilidi-FA1971] and
[Simonenko-FAP-1971]).

Notation. In what follows we call $x=(x_{1},x_{2})$ an element of
$G=G_{1}\times G_{2}$ and $\xi:=\xi_{1}\otimes\xi_{2}$ an element of
$\widehat{G}$, where $\xi_{j}\in\widehat{G}_{j}$. By using the definitions
above, and fixing the families $R$ and $P$, we define
$\triangle_{1}^{\alpha}:=\triangle_{P}^{\alpha}$,
$\triangle_{2}^{\beta}:=\triangle_{Q}^{\beta}$, and
$\partial_{1}^{\alpha}:=\partial^{\alpha}_{x_{1}}=\partial^{\alpha_{1}}_{x_{11}}\ldots\partial^{\alpha_{n_{1}}}_{x_{1n_{1}}}$,
$\partial_{2}^{\beta}:=\partial_{x_{2}}^{\beta}=\partial^{\beta_{1}}_{x_{21}}\ldots\partial^{\beta_{n_{2}}}_{x_{2n_{2}}}$
as above (where, as previously mentioned, $\partial_{x_{j}}$ are not the
Euclidean directional derivatives). We shall finally put
$\partial^{\alpha,\beta}:=\partial^{\alpha}_{x_{1}}\partial^{\beta}_{x_{2}}$
and analogously for the difference operators $\triangle^{\alpha,\beta}$.

We recall that, given a continuous linear operator $A$ from $C^{\infty}(G)$ to
$\mathcal{D}^{\prime}(G)$, its matrix-valued symbol
$\sigma_{A}(x,\xi)\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$ (as introduced in
[MR-VT-Book_2010]) is given by

$\sigma_{A}(x,\xi)=\xi^{*}(x)(A\xi)(x),$ (3.1)

and that

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}(\xi(x)\sigma_{A}(x,\xi)\widehat{f}(\xi)),\quad
f\in C^{\infty}(G),$

holds in the sense of distributions and the sum is independent of the choice
of the rapresentative $\xi$ of the class $[\xi]$.

######  Definition 3.1.

Let $G=G_{1}\times G_{2}$ be a compact Lie group and define
$n_{i}:=\mathrm{dim}(G_{i})$. We call class of bisingular symbols of order
$(m_{1},m_{2})\in\mathbb{R}^{2}$ the set $S^{m_{1},m_{2}}(G\times\widehat{G})$
of all
$a:G\times\widehat{G}\longrightarrow\bigcup_{[\xi]\in\widehat{G}}\mathbb{C}^{d_{\xi}\times
d_{\xi}}$ that are smooth in $x\in G$ and such that, for all multiindices
$\alpha_{1}\in\mathbb{N}_{0}^{n_{1}},\alpha_{2}\in\mathbb{N}_{0}^{n_{2}},\beta_{1}\in\mathbb{N}_{0}^{\triangle_{P}},\beta_{2}\in\mathbb{N}_{0}^{\triangle_{R}}$,

$\|\partial^{\alpha_{1}}_{x_{1}}\partial^{\alpha_{2}}_{x_{2}}\triangle_{1}^{\beta_{1}}\triangle_{2}^{\beta_{2}}a(x_{1},x_{2},\xi_{1},\xi_{2})\|_{op}\leq
C_{\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}}\langle\xi_{1}\rangle^{m_{1}-|\beta_{1}|}\langle\xi_{2}\rangle^{m_{2}-|\beta_{2}|},$

where

$\|a\|_{op}:=\sup\\{|a(x,\xi)v|_{\ell^{2}};v\in\mathbb{C}^{d_{\xi}},|v|_{\ell^{2}}\leq
1\\}.$

Additionally we shall denote by
$S^{-\infty,-\infty}(G\times\widehat{G}):=\bigcap_{(m_{1},m_{2})\in\mathbb{Z}^{2}}S^{m_{1},m_{2}}(G\times\widehat{G})$
the class of smoothing elements.

It is important to bear in mind that
$\widehat{G}=\widehat{G}_{1}\times\widehat{G}_{2}$.

Due to the equivalence of $\|a\|_{\mathscr{L}(\mathcal{H}_{\xi})}$ and
$\|a\|_{op}$, we will freely use both notations below.

Let us remark that, as in the standard case, the space
$S^{m_{1},m_{2}}(G\times\widehat{G})$ is a Fréchet space equipped with the
seminorms

$\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),(b_{1},b_{2})}}:=\max_{\begin{subarray}{c}|\alpha_{1}|\leq
a_{1},|\alpha_{2}|\leq a_{2}\\\ |\beta_{1}|\leq b_{1},|\beta_{2}|\leq
b_{2}\end{subarray}}\sup_{(x,\xi)\in
G\times\widehat{G}}\langle\xi_{1}\rangle^{-m_{1}+|\alpha_{1}|}\langle\xi_{2}\rangle^{-m_{2}+|\alpha_{2}|}\|\triangle^{\alpha_{1},\alpha_{2}}\partial_{x_{1}}^{\beta_{1},\beta_{2}}\sigma(x,\xi)\|_{\mathscr{L}(\mathcal{H}_{\xi})},$

with $a_{1},a_{2},b_{1},b_{2}\in\mathbb{N}_{0}$.

To each matrix-valued symbol $a\in
S^{m_{1},m_{2}}(G\times\widehat{G}_{1}\times\widehat{G}_{2})$ one can
associate an operator $\mathrm{Op}(a)$ by means of the following quantization
formula

$\mathrm{Op}(a)\varphi(x):=\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}(\xi(x)a(x,\xi)\widehat{\varphi}(\xi))$
$=\sum_{[\xi_{1}]\in\widehat{G}_{1}}\sum_{[\xi_{2}]\in\widehat{G}_{2}}d_{\xi_{1}}d_{\xi_{2}}\mathrm{Tr}((\xi_{1}\otimes\xi_{2})(x)a(x,\xi_{1},\xi_{2})\widehat{\varphi}(\xi_{1}\otimes\xi_{2})),$
(3.2)

and we shall denote by $L^{m_{1},m_{2}}(G)$ the class of operators of the
previous form, that is, those obtained by quantizing symbols in
$S^{m_{1},m_{2}}(G\times\widehat{G})$ as in (3.2). These operators will be
called binsingular operators of order $(m_{1},m_{2})$ on $G=G_{1}\times
G_{2}$.

Moreover, with any $a\in S^{m_{1},m_{2}}(G\times\widehat{G})$, we associate
the maps

$G_{1}\times\widehat{G}_{1}\ni(x_{1},\xi_{1})\longmapsto
a(x_{1},x_{2},\xi_{1},D_{2})\in L^{m_{2}}(G_{2}),$
$G_{2}\times\widehat{G}_{2}\ni(x_{2},\xi_{2})\longmapsto
a(x_{1},x_{2},D_{1},\xi_{2})\in L^{m_{1}}(G_{1}),$

where $L^{m_{1}}(G_{1})$ and $L^{m_{2}}(G)$ are classes of operators on
$G_{1}$ and $G_{2}$ respectively obtained by means of the quantization
formulas

$a(x_{1},x_{2},\xi_{1},D_{2})\varphi(x_{2})=\sum_{[\xi_{2}]\in\widehat{G}_{2}}d_{\xi_{2}}\mathrm{Tr}\left((I_{d_{\xi_{1}}}\otimes\xi_{2}(x_{2}))a(x_{1},x_{2},\xi_{1},\xi_{2})\times(I_{d_{\xi_{1}}}\otimes\widehat{\varphi}(\xi_{2}))\right)$

and

$a(x_{1},x_{2},D_{1},\xi_{2})\varphi(x_{1})=\sum_{[\xi_{1}]\in\widehat{G}_{1}}d_{\xi_{1}}\mathrm{Tr}\left((\xi_{1}(x_{1})\otimes
I_{d_{\xi_{2}}})a(x_{1},x_{2},\xi_{1},\xi_{2})\times(\widehat{\varphi}(\xi_{1})\otimes
I_{d_{\xi_{2}}})\right).$

It is important to stress that the symbol $a\in
S^{m_{1},m_{2}}(G\times\widehat{G})$ is uniquely determined by one of these
maps.

Troughout the paper we will often write $a(x,\xi)$ in place of
$a(x_{1},x_{2},\xi_{1},\xi_{2})$, where
$\xi=\xi_{1}\otimes\xi_{2}\in\widehat{G}$, and
$S^{m_{1},m_{2}}(G\times\widehat{G}_{1}\times\widehat{G}_{2})$ in place of
$S^{m_{1},m_{2}}(G\times\widehat{G})$.

######  Remark 3.2.

Notice that, in general, there is no $m\in\mathbb{R}$ such that
$S^{m_{1},m_{2}}(G\times\widehat{G})\subset S^{m}(G\times\widehat{G})$.
However we always have that $S^{m_{1},m_{2}}(G\times\widehat{G})\subset
S^{m}_{0,0}(G\times\widehat{G})$ for some $m\in\mathbb{R}$.

Given a continuous linear operator
$A:\mathcal{D}(G)\rightarrow\mathcal{D}^{\prime}(G)$, (where
$\mathcal{D}(G):=C^{\infty}(G)$), its right-convolution kernel
$R_{A}\in\mathcal{D}^{\prime}(G\times G)$ is defined by

$A\varphi(x)=\int_{G}\varphi(y)R_{A}(x,y^{-1}x)dy=(R_{A}(x,\cdot)\ast\varphi)(x).$
(3.3)

Therefore, given $A\in L^{m_{1},m_{2}}(G)$ with simbol $\sigma_{A}\in
S^{m_{1},m_{2}}(G\times\widehat{G})$, one has

$\sigma_{A}(x,\xi):=(\mathcal{F}_{y\rightarrow\xi}\,R_{A})(x,\xi),$

where

$R_{A}(x,y):=\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}\left(\xi(y)a(x,\xi)\right),$

with $y=(y_{1},y_{2})\in G=G_{1}\times G_{2}$ and $\xi\in\widehat{G}$ of the
form $\xi=\xi_{1}\otimes\xi_{2}$, with
$(\xi_{1},\xi_{2})\in\widehat{G}_{1}\times\widehat{G}_{2}$.

For any fixed $(x_{1},\xi_{1})\in G_{1}\times\widehat{G}_{1}$ and
$(x_{2},\xi_{2})\in G_{2}\times\widehat{G}_{2}$, we can write, respectively,
the operators $a(x_{1},x_{2},\xi_{1},D_{2})$ and
$a(x_{1},x_{2},D_{1},\xi_{2})$ defined above in terms of their
(right-)convolution kernels, that is,

$a(x_{1},x_{2},\xi_{1},D_{2})\varphi(x_{2})=(R^{2}_{a}(x_{1},x_{2},\xi_{1},\cdot)\ast_{G_{2}}\varphi)(x_{2})$
$a(x_{1},x_{2},D_{1},\xi_{2})\varphi(x_{1})=(R^{1}_{a}(x_{1},x_{2},\cdot,\xi_{2})\ast_{G_{1}}\varphi)(x_{1})$

where

$R_{a}^{2}(x_{1},x_{2},\xi_{1},y_{2}):=\sum_{[\xi_{2}]\in\widehat{G}_{2}}d_{\xi_{2}}\mathrm{Tr}\left((I_{\xi_{1}}\otimes\xi_{2}(y_{2}))a(x_{1},x_{2},\xi_{1},\xi_{2})\right)$
(3.4)

and

$R_{a}^{1}(x_{1},x_{2},y_{1},\xi_{2}):=\sum_{[\xi_{1}]\in\widehat{G}_{1}}d_{\xi_{1}}\mathrm{Tr}\left((\xi_{1}(x_{1})\otimes
I_{\xi_{2}})a(x_{1},x_{2},\xi_{1},\xi_{2})\right).$ (3.5)

Due to the orthogonality property of irriducible representations we have that

$a(x_{1},x_{2},\xi_{1},\xi_{2})=\int_{G_{1}}R_{a}^{1}(x_{1},x_{2},y_{1},\xi_{2})(\xi_{1}(y_{1})^{*}\otimes
I_{\xi_{2}})dy_{1}$

and

$a(x_{1},x_{2},\xi_{1},\xi_{2})=\int_{G_{2}}R_{a}^{2}(x_{1},x_{2},\xi_{1},y_{2})(I_{\xi_{1}}\otimes\xi_{2}(y_{2})^{*})dy_{2}.$

######  Definition 3.3.

Given $a\in S^{m_{1},m_{2}}(G\times\widehat{G})$ and $b\in
S^{m^{\prime}_{1},m^{\prime}_{2}}(G\times\widehat{G})$, we shall denote by
$(a\circ_{\xi_{1}}b)(x_{1},x_{2},\xi_{1},\xi_{2})$ and
$(a\circ_{\xi_{2}}b)(x_{1},x_{2},\xi_{1},\xi_{2})$ the symbols in
$S^{m_{1}+m^{\prime}_{1},m_{2}+m^{\prime}_{2}}(G\times\widehat{G})$
corresponding to the operators

$(a\circ_{\xi_{1}}b)(x_{1},x_{2},D_{1},\xi_{2})\varphi(x_{1})=a(x_{1},x_{2},D_{1},\xi_{2})b(x_{1},x_{2},D_{1},\xi_{2})\varphi(x_{1}),\quad\forall\varphi\in
C^{\infty}(G_{1}),$

and

$(a\circ_{\xi_{2}}b)(x_{1},x_{2},\xi_{1},D_{2})\psi(x_{2})=a(x_{1},x_{2},\xi_{1},D_{2})b(x_{1},x_{2},\xi_{1},D_{2})\psi(x_{2}),\quad\forall\psi\in
C^{\infty}(G_{2}).$

By considering the right-convolution kernels it is not difficult to show that

$(a\circ_{\xi_{1}}b)(x_{1},x_{2},\xi_{1},\xi_{2})\sim\sum_{|\alpha_{1}|\geq
0}(\triangle^{\alpha_{1},0}a(x,\xi))\,\partial^{\alpha_{1},0}b(x,\xi)$

and

$(a\circ_{\xi_{2}}b)(x_{1},x_{2},\xi_{1},\xi_{2})\sim\sum_{|\alpha_{2}|\geq
0}(\triangle^{0,\alpha_{2}}a(x,\xi))\,\partial^{0,\alpha_{2}}b(x,\xi),$

where for all $N>0$ we have

$r_{N}^{1}(x,\xi):=(a\circ_{\xi_{1}}b)(x_{1},x_{2},\xi_{1},\xi_{2})-\sum_{|\alpha_{1}|<N}(\triangle^{\alpha_{1},0}a(x,\xi))\,\partial^{\alpha_{1},0}b(x,\xi)=\sum_{|\alpha_{1}|=N}(\triangle^{\alpha_{1},0}a(x,\xi))b_{\alpha_{1}}(x,\xi)$

$r_{N}^{2}(x,\xi):=(a\circ_{\xi_{2}}b)(x_{1},x_{2},\xi_{1},\xi_{2})-\sum_{|\alpha_{2}|\geq
0}(\triangle^{0,\alpha_{2}}a(x,\xi))\,\partial^{0,\alpha_{2}}b(x,\xi)=\sum_{|\alpha_{2}|=N}(\triangle^{0,\alpha_{2}}a(x,\xi))b_{\alpha_{2}}(x,\xi),$

for suitable $b_{\alpha_{1}},b_{\alpha_{2}}$ having the same properties as
$b$, that is, $b_{\alpha_{1}},b_{\alpha_{2}}\in
S^{m^{\prime}_{1},m^{\prime}_{2}}(G\times\widehat{G})$.

Let $a\in S^{m_{1},m_{2}}(G\times\widehat{G})$ and denote by
$\mathrm{Op}(a_{(x_{2},\xi_{2})})(x_{1},D_{1}):=a(x_{1},x_{2},D_{1},\xi_{2})$
the operator defined above and belonging to $L^{m_{1}}(G_{1})$ for all
$(x_{2},\xi_{2})\in G_{2}\times\widehat{G}_{2}$. Then it is possible to define
the adjoint of $\mathrm{Op}(a_{(x_{2},\xi_{2})})$ (as an operator on $G_{1}$),
denoted by
$\mathrm{Op}(a_{(x_{2},\xi_{2})})(x_{1},D_{1})^{*_{1}}:=a(x_{1},x_{2},D_{1},\xi_{2})^{*_{1}}$,
as the operator satisfying

$(\mathrm{Op}(a_{(x_{2},\xi_{2})})u,v)_{L^{2}(G_{1})}=(u,\mathrm{Op}(a_{(x_{2},\xi_{2})})^{*_{1}}v)_{L^{2}(G_{1})},\quad
u,v\in\mathcal{D}(G_{1}),$ (3.6)

where $(\cdot,\cdot)_{L^{2}(G_{1})}$ stands for the scalar product on
$L^{2}(G_{1})$.

In a similar way, on denoting by

$\mathrm{Op}(a_{(x_{1},\xi_{1})})(x_{2},D_{2}):=a(x_{1},x_{2},\xi_{1},D_{2})$

the operator belonging to $L^{m_{2}}(G_{2})$ for all $(x_{1},\xi_{1})\in
G_{1}\times\widehat{G}_{1}$, one can define the adjoint operator
$\mathrm{Op}(a_{(x_{1},\xi_{1})})(x_{2},D_{2})^{*_{2}}:=a(x_{1},x_{2},\xi_{1},D_{2})^{*_{2}}$
as the one satisfying

$(\mathrm{Op}(a_{(x_{1},\xi_{1})})u,v)_{L^{2}(G_{2})}=(u,\mathrm{Op}(a_{(x_{1},\xi_{1})})^{*_{2}}v)_{L^{2}(G_{2})},\quad
u,v\in\mathcal{D}(G_{2}),$ (3.7)

with $(\cdot,\cdot)_{L^{2}(G_{2})}$ denoting the scalar product on
$L^{2}(G_{2})$.

Sobolev spaces $H^{s_{1},s_{2}}(G)$. We shall now define what we shall call
bisingular Sobolev spaces which are the ones to be naturally used in this
setting. We will also see that bisingular operators exhibit continuity
properties when acting on these spaces.

Let us consider the operator $L$ on $G=G_{1}\times G_{2}$, defined as

$L:=(I_{1}+L_{G_{1}})\otimes(I_{2}+L_{G_{2}}),$

where $L_{G_{i}}$ and $I_{i}$ denote the positive Laplace operator and the
identity operator on $G_{i}$ respectively.

The operator $L$ will be called bilaplacian, since, as expected, it will play
the role of the Laplacian in this setting.

By formula (3.1) we have that the symbol of the bilaplacian is given by

$\sigma_{L}(\xi)=\sigma_{L}(\xi_{1}\otimes\xi_{2})=\langle\xi_{1}\rangle^{2}\langle\xi_{2}\rangle^{2}I_{d_{\xi}},$

where $\langle\xi_{i}\rangle:=(1+\lambda^{2}_{\xi_{1}})^{1/2}$, with
$\lambda^{2}_{\xi_{i}}>0$ being the eigenvalue of $L_{G_{i}}$ relative to the
representation $\xi_{i}\in\widehat{G}_{i}$, and
$I_{d_{\xi_{i}}}\in\mathbb{C}^{d_{\xi_{i}}\times d_{\xi_{i}}}$ is the identity
matrix.

######  Definition 3.4 (Bisingular Sobolev space of order $(s_{1},s_{2})$).

We shall call bisingular Sobolev space of order $(s_{1},s_{2})$ the space

$H^{s_{1},s_{2}}(G):=\\{f\in\mathcal{D}^{\prime}(G);\langle\xi_{1}\rangle^{s_{1}}\langle\xi_{2}\rangle^{s_{2}}\widehat{f}(\xi)\in\ell^{2}(\widehat{G})\\},$

equipped with the norm

$\|f\|_{s_{1},s_{2}}:=\left(\sum_{[\xi]\in\widehat{G}}d_{\xi}\langle\xi_{1}\rangle^{2s_{1}}\langle\xi_{2}\rangle^{2s_{2}}\mathrm{Tr}(\widehat{f}(\xi)^{*}\widehat{f}(\xi))\right)^{1/2}$
$=\|\langle\xi_{1}\rangle^{s_{1}}\langle\xi_{2}\rangle^{s_{2}}\widehat{f}\|_{\ell^{2}(\widehat{G})}=:\|\widehat{f}\|_{h^{s_{1},s_{2}}(\widehat{G})},$

where

$h^{s_{1},s_{2}}(\widehat{G}):=\\{\widehat{f}\in\mathcal{F}(\mathcal{D}^{\prime}(G));\langle\xi_{1}\rangle^{s_{1}}\langle\xi_{2}\rangle^{s_{2}}\widehat{f}\in\ell^{2}(\widehat{G})\\},$

where $F\in\ell^{2}(\widehat{G})$ if and only if
$\sum_{[\xi]\in\widehat{G}}d_{\xi}\|F(\xi)\|_{HS}^{2}<\infty.$

One may check that the spaces $h^{s_{1},s_{2}}(\widehat{G})$ are indeed
complete with respect to the scalar product

$(f,g)_{s_{1},s_{2}}:=\sum_{[\xi]\in\widehat{G}}d_{\xi}\langle\xi_{1}\rangle^{2s_{1}}\langle\xi_{2}\rangle^{2s_{2}}\mathrm{Tr}(\widehat{g}(\xi)^{*}\widehat{f}(\xi)).$

Therefore, the Sobolev spaces $H^{s_{1},s_{2}}(G)$ are also complete.

##  4\. Kernel estimates

This section is devoted to the proof of some estimates for the (right-
convolution) kernels of bisingular pseudodifferential operators on compact Lie
groups. These estimates will be employed in the next section to develop the
global calculus of bisingular operators.

Before proving the estimates, we will first give some properties representing
the suitable bisingular generalization of certain results holding in the
standard (global) compact case.

Notation. Recall that
$\langle\xi_{j}\rangle^{s}:=(1+\lambda_{\xi_{j}})^{s/2}$, $j=1,2$.
Additionally, we assume $\triangle_{1},\triangle_{2}$ to be the the admissible
collections of difference operators previously defined. Note that we shall
often use the notation $S^{m_{1},m_{2}}(G)$ for
$S^{m_{1},m_{2}}(G\times\widehat{G})$.

######  Proposition 4.1.

Let $\triangle^{\alpha,\beta}:=\triangle_{1}^{\alpha}\triangle_{2}^{\beta}$,
then, for any $m_{1},m_{2}\in\mathbb{R}$ and multi-indeces
$\alpha\in\mathbb{N}^{n_{P}}$, $\beta\in\mathbb{N}^{n_{R}}$, there exists
$d\in\mathbb{N}_{0}$ and $C>0$ such that, for all $f_{1},f_{2}\in
C^{d}([0,+\infty))$, $\xi=\xi_{1}\otimes\xi_{2}\in\widehat{G}$, and
$t_{1},t_{2}\in(0,1)$, we have

$\|\triangle^{\alpha,\beta}f_{1}(t_{1}\lambda_{\xi_{1}})f_{2}(t_{2}\lambda_{\xi_{2}})\|_{\mathscr{L}(\mathcal{H}_{\xi})}\leq
Ct_{1}^{m_{1}/2}\langle\xi_{1}\rangle^{m_{1}-|\alpha|}\underset{\ell_{1}=0,...,d}{\sup_{\lambda_{\xi_{1}}\geq
0}}|\partial^{\ell_{1}}_{\lambda_{\xi_{1}}}f_{1}(\lambda_{\xi_{1}})|$
$\times\,\,t_{2}^{m_{2}/2}\langle\xi_{2}\rangle^{m_{2}-|\beta|}\underset{\ell_{2}=0,...,d}{\sup_{\lambda_{\xi_{2}}\geq
0}}|\partial^{\ell_{2}}_{\lambda_{\xi_{2}}}f_{2}(\lambda_{\xi_{2}})|,$

in the sense that if the supremum on the right-hand side is finite, then the
left-hand side is also finite and the inequality holds.

###### Sketch of proof.

Due to the form of $\triangle^{\alpha,\beta}$ we have

$\|\triangle^{\alpha,\beta}f_{1}(t_{1}\lambda_{\xi_{1}})f_{2}(t_{2}\lambda_{\xi_{2}})\|_{\mathscr{L}(\mathcal{H}_{\xi})}=\|\triangle^{\alpha}_{1}f_{1}(t_{1}\lambda_{\xi_{1}})\|_{\mathscr{L}(\mathcal{H}_{\xi_{1}})}\|\triangle^{\beta}_{2}f_{2}(t_{2}\lambda_{\xi_{2}})\|_{\mathscr{L}(\mathcal{H}_{\xi_{2}})}.$

Therefore, by Proposition 6.1 in [Fischer-JFA_2015] applied separately to each
term on the right-hand side of the previous identity, the result follows. ∎

######  Lemma 4.2.

Let $k\in\mathcal{D}^{\prime}(G)$, with $G=G_{1}\times G_{2}$ and
$n_{i}=\mathrm{dim}(G_{i})$. Then, if $s_{1}>n_{1}/2$ and $s_{2}>n_{2}/2$,

$\|k\|_{L^{2}(G)}\lesssim\sup_{\xi\in\widehat{G}}\,\,\langle\xi_{1}\rangle^{s_{1}/2}\langle\xi_{2}\rangle^{s_{2}/2}\,\,\|\widehat{k}\|_{\mathscr{L}(\mathcal{H}_{\xi})}.$

Hence $k\in L^{2}(G)$ when there exist $s_{1}>n_{1}/2$ and $s_{2}>n_{2}/2$
such that the right-hand side is finite.

###### Proof.

Let $B_{s_{1},s_{2}}(x,y)=B_{s_{1}}(x_{1},y_{1})\otimes
B_{s_{2}}(x_{2},y_{2})=B_{s_{1}}(y_{1})\otimes B_{s_{2}}(y_{2})$ as in Lemma
A.3 ($B_{s_{1},s_{2}}(x,y)$ is independent of $x$). Then, for $s_{1},s_{2}>0$,
we can write

$k(y)=((I_{1}+L_{G_{1}})^{s_{1}/2}\otimes(I_{2}+L_{G_{2}})^{s_{2}/2})(k\ast(B_{s_{1}}\otimes
B_{s_{2}}))(y),$

which gives, in particular, that

$\widehat{k}(\xi)=\langle\xi_{1}\rangle^{s_{1}/2}\langle\xi_{2}\rangle^{s_{2}/2}\widehat{B_{s_{1}}\otimes
B_{s_{2}}}(\xi)\,\widehat{k}(\xi).$

Therefore, for $s_{1}>n_{1}/2$ and $s_{2}>n_{2}/2$, we get

$\|k\|^{2}_{L^{2}(G)}\underset{\text{Plancherel}}{=}\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\widehat{k}(\xi)\|_{HS}^{2}$
$\leq\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\widehat{B_{s_{1}}\otimes
B_{s_{2}}}(\xi)\|^{2}_{HS}\|\langle\xi_{1}\rangle^{s_{1}/2}\langle\xi_{2}\rangle^{s_{2}/2}\widehat{k}(\xi)\|^{2}_{\mathscr{L}(\mathcal{H}_{\xi})}$
$\leq\|B_{s_{1}}\otimes
B_{s_{2}}\|^{2}_{L^{2}(G)}\sup_{[\xi]\in\widehat{G}}\langle\xi_{1}\rangle^{s_{1}}\langle\xi_{2}\rangle^{s_{2}}\|\widehat{k}(\xi)\|^{2}_{\mathscr{L}(\mathcal{H}_{\xi})}$
$\underset{\text{Lemma
\ref{lemma.embedding}}}{\lesssim}\sup_{[\xi]\in\widehat{G}}\langle\xi_{1}\rangle^{s_{1}}\langle\xi_{2}\rangle^{s_{2}}\|\widehat{k}(\xi)\|^{2}_{\mathscr{L}(\mathcal{H}_{\xi})},$

which concludes the proof. ∎

######  Lemma 4.3.

Let $\sigma\in S^{m_{1},m_{2}}(G)$ with (right-convolution) kernel
$k_{x}(\cdot):=k(x,\cdot)$. Then the following properties hold:

  1. 1.

The kernel associated with
$\partial^{\gamma_{1},\gamma_{2}}\triangle^{\alpha_{1},\alpha_{2}}\sigma\in
S^{m_{1}-|\alpha_{1}|,m_{2}-|\alpha_{2}|}(G)$, for any
$\alpha_{i}\in\mathbb{N}_{0}^{n_{\triangle_{i}}}$, and
$\gamma_{1}\in\mathbb{N}_{0}^{n_{1}},\gamma_{2}\in\mathbb{N}_{0}^{n_{2}}$, is
given by
$q^{\alpha_{1},\alpha_{2}}\partial^{\gamma_{1},\gamma_{2}}_{x_{1},x_{2}}k_{x}$;

  2. 2.

If $\sigma_{1},\sigma_{2}$ are two bisingular symbols with kernels $k^{1}_{x}$
and $k^{2}_{x}$, respectively, then the kernel of the product
$\sigma_{1}\sigma_{2}$ is given by $k_{x}^{1}\ast k_{x}^{2}$.

###### Proof.

The proof of Lemma 4.3 follows immediately by the form and the properties of
bisingular symbols. ∎

As a consequence of Lemma 4.3 we get Corollary 4.4 below giving a first key
estimate for the kernels of bisingular pseudodifferential operators.

######  Corollary 4.4.

If $\sigma\in S^{m_{1},m_{2}}(G)$, then, for any
$\gamma:=(\gamma_{1},\gamma_{2})\in\mathbb{N}_{0}^{n_{1}}\times\mathbb{N}_{0}^{n_{2}}$
and
$\theta=(\theta_{1},\theta_{2})\in\mathbb{N}_{0}^{n_{1}}\times\mathbb{N}_{0}^{n_{2}}$
such that, for all $i=1,2$, $\gamma_{i}+m_{i}+n_{i}<|\alpha_{i}|$, the
function
$\partial^{\gamma}_{x}\partial^{\theta}_{z}(q^{\alpha_{1},\alpha_{2}}(z)k_{x}(z))$
is continuous on $G$ and bounded as follows:

$|\partial^{\gamma}_{x}\partial^{\theta}_{z}(q^{\alpha_{1},\alpha_{2}}(z)k_{x}(z))|\leq
C\sup_{[\xi]\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}},|\alpha_{1}|,|\alpha_{2}|,\gamma},$

where $\|\cdot\|_{S^{m_{1},m_{2}},|\alpha_{1}|,|\alpha_{2}|,\gamma}$ is the
suitable seminorm. The constant $C$ above depends on the parameters
$m_{i},\triangle,\gamma_{i},\vartheta_{i}$ for all $i=1,2$.

###### Proof.

The proof follows from the proof of Corollary 6.5 in [Fischer-JFA_2015]
together with Lemma 4.3 and Lemma A.3. ∎

Corollary 4.4 immediately gives the proposition below.

######  Proposition 4.5.

If $\sigma\in S^{m_{1},m_{2}}(G)$ then the associated kernel $(x,y)\mapsto
k_{x}(y)$ is smooth on $G\times(G\setminus S)$, with $S=\\{x\in
G;x_{1}=e_{1}\\}\cup\\{x\in G;x_{2}=e_{2}\\}$. If $\sigma\in
S^{-\infty,-\infty}(G)$ is smoothing then the associated kernel $(x,y)\mapsto
k_{x}(y)$ is smooth on $G\times G$. The converse is also true, namely, if
$(x,y)\mapsto k_{x}(y)$ is smooth on $G\times G$ then the associated symbol is
smoothing, that is, it belongs to $S^{-\infty,-\infty}(G)$.

In order to show some estimates for the kernels, we will need to work inside
dyadic pieces where the eigenvalues (i.e. the frequencies in this setting) of
$L_{G_{1}}$ and $L_{G_{2}}$ are localized. In that perspective the following
lemma will be crucial to understand how the localized symbol and the
corresponding kernel behave.

######  Lemma 4.6.

Let $\chi\in C_{0}^{\infty}(\mathbb{R})$ be a given function with values in
$[0,1]$ and $\chi\equiv 1$ in a neighborhood of $0$. Let $\sigma\in
S^{m_{1},m_{2}}(G)$ and let $k_{x}$ be the associated kernel. For each
$\ell_{1},\ell_{2}\in\mathbb{N}$ we define

$\sigma_{\ell_{1},\ell_{2}}(x,\xi):=\sigma(x,\xi)\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}).$

Then $\sigma_{\ell_{1},\ell_{2}}\in S^{-\infty,-\infty}(G)$ and, for any
$\gamma=(\gamma_{1},\gamma_{2})\in\mathbb{N}_{0}^{n_{1}}\times\mathbb{N}_{0}^{n_{2}}$,

$\|\sigma_{\ell_{1},\ell_{2}}\|_{S^{m_{1},m_{2}},\gamma}\leq
C(G,m_{1},m_{2},\gamma)\|\sigma\|_{S^{m_{1},m_{2}},\gamma}.$

Additionally, the kernel $k_{x,\ell_{1},\ell_{2}}(y)$ associated with
$\sigma_{\ell_{1},\ell_{2}}$ is smooth on $G\times G$, and, for all
$\beta\in\mathbb{N}_{0}^{n_{1}+n_{2}}$,
$\partial^{\beta}k_{x,\ell_{1},\ell_{2}}\rightarrow\partial^{\beta}k_{x}$ in
$\mathcal{D}^{\prime}(G)$ uniformly in $x\in G$ as
$\ell_{1},\ell_{2}\rightarrow\infty$.

###### Proof.

The proof follows the proof of Lemma 6.6 in [Fischer-JFA_2015] with suitable
modifications, namely by using the function
$\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}})$ as
a cutoff function in the proof (note that $(1-\chi(\ell^{-1}\lambda_{\pi}))$
in [Fischer-JFA_2015] is replaced by
$(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))$
here), and by replacing the standard Sobolev spaces $H^{s}(G)$ by the Sobolev
spaces $H^{s_{1},s_{2}}(G_{1}\times G_{2})$. For the sake of completness we
shall give the proof of the secod part of the lemma, that is the convergence
of the kernels, where a few arrangements are needed.

Let $s_{1}=\lceil\frac{n_{1}}{2}\rceil$ and
$s_{2}=\lceil\frac{n_{2}}{2}\rceil$, where $\lceil\cdot\rceil$ stands for the
upper integer part. By using the bisingular Sobolev spaces we get

$\|\partial^{\beta}(k_{x,\ell_{1},\ell_{2}}-k_{x})\|_{H^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}=\|\partial^{\beta}(\sigma_{\ell_{1},\ell_{2}}-\sigma)\|_{h^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}$
$=\|(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))\partial^{\beta}\sigma\|_{h^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}$
$\lesssim\|\langle\xi_{1}\rangle^{-m_{1}-1}\langle\xi_{2}\rangle^{-m_{2}-1}(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))\partial^{\beta}\sigma\|_{h^{-s_{1},-s_{2}}}$
$\lesssim\|\langle\xi_{1}\rangle^{-m_{1}-1}\langle\xi_{2}\rangle^{-m_{2}-1}(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))\partial^{\beta}\sigma\|_{L^{\infty}(\widehat{G})}.$

Due to the hypothesis on $\chi$, for some $\varepsilon_{1},\varepsilon_{2}>0$,
with $0<\varepsilon_{1}<\varepsilon_{2}$, we have that $\chi\equiv 1$ on
$[0,\varepsilon_{1}]$ and $\chi\equiv 0$ on $[\varepsilon_{2},+\infty)$.
Therefore we get that
$(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))\not\equiv
0$ in the following three cases

  1. (1)

$\lambda_{\xi_{1}}>\varepsilon_{1}\ell_{1},\lambda_{\xi_{2}}>\varepsilon_{1}\ell_{2}$,

  2. (2)

$\lambda_{\xi_{1}}>\varepsilon_{1}\ell_{1},\lambda_{\xi_{2}}\leq\varepsilon_{1}\ell_{2}$,

  3. (3)

$\lambda_{\xi_{1}}\leq\varepsilon_{1}\ell_{1},\lambda_{\xi_{2}}>\varepsilon_{1}\ell_{2}$.

Let us start with the proof of the convergence in case $(1)$. The inequalities
above lead to

$\|\partial^{\beta}(k_{x,\ell_{1},\ell_{2}}-k_{x})\|_{H^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}$
$\leq\underset{\lambda_{\xi_{2}}>\varepsilon_{\chi}\ell_{2}}{\max_{\lambda_{\xi_{1}}>\varepsilon_{\chi}\ell_{1}}}\|(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))\partial^{\beta}\sigma\|_{h^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}$
$\leq(1+\varepsilon_{1}\ell_{1})^{-1}(1+\varepsilon_{1}\ell_{2})^{-1}\|\langle\xi_{1}\rangle^{-m_{1}}\langle\xi_{2}\rangle^{-m_{2}}\partial^{\beta}\sigma\|_{L^{\infty}(\widehat{G})}$
$\lesssim(1+\varepsilon_{1}\ell_{1})^{-1}(1+\varepsilon_{1}\ell_{2})^{-1}\|\sigma\|_{S^{m_{1},m_{2}},\beta},$

which gives, in particular, that

$\max_{x\in
G}\|\partial^{\beta}(k_{x,\ell_{1},\ell_{2}}-k_{x})\|_{H^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}\lesssim(1+\varepsilon_{1}\ell_{1})^{-1}(1+\varepsilon_{1}\ell_{2})^{-1}\|\sigma\|_{S^{m_{1},m_{2}},\beta}.$

This finally yields the convergence
$\partial^{\beta}k_{x,\ell_{1},\ell_{2}}\overset{\mathscr{D}^{\prime}}{\rightarrow}\partial^{\beta}k_{x}$
uniformly in $x\in G$ as $\ell_{1},\ell_{2}\rightarrow\infty$.

For cases $(2)$ and $(3)$ the proof is the same (by reversing the roles of the
parameters) and it is similar to the one in the case $(1)$. For completeness
we show the steps in case $(2)$, that is, when
$\lambda_{\xi_{1}}>\varepsilon_{1}\ell_{1}$ and
$\lambda_{\xi_{2}}\leq\varepsilon_{1}\ell_{2}$. Under these hypotheses we have

$\|\partial^{\beta}(k_{x,\ell_{1},\ell_{2}}-k_{x})\|_{H^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}$
$\leq\underset{\lambda_{\xi_{2}}\leq\varepsilon_{1}\ell_{2}}{\max_{\lambda_{\xi_{1}}>\varepsilon_{1}\ell_{1}}}\|(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}})\chi(\ell_{2}^{-1}\lambda_{\xi_{2}}))\partial^{\beta}\sigma\|_{h^{-s_{1}-m_{1}-1,-s_{2}-m_{2}-1}}$
$\leq\underset{\lambda_{\xi_{2}}\leq\varepsilon_{1}\ell_{2}}{\max_{\lambda_{\xi_{1}}>\varepsilon_{1}\ell_{1}}}\|\langle\xi_{1}\rangle^{-m_{1}-1}\langle\xi_{2}\rangle^{-m_{2}-1}(1-\chi(\ell_{1}^{-1}\lambda_{\xi_{1}}))\partial^{\beta}\sigma\|_{h^{-s_{1},-s_{2}}}$
$\leq(1+\varepsilon_{1}\ell_{1})^{-1}\|\langle\xi_{1}\rangle^{-m_{1}}\langle\xi_{2}\rangle^{-m_{2}}\partial^{\beta}\sigma\|_{L^{\infty}(\widehat{G})}$
$\lesssim(1+\varepsilon_{1}\ell_{1})^{-1}\|\sigma\|_{S^{m_{1},m_{2}},\beta},$

yielding, as before, the convergence in $\mathcal{D}^{\prime}$ uniformly in
$x$, which completes the proof. ∎

######  Lemma 4.7.

Let $\sigma\in S^{m_{1},m_{2}}(G)$, and $\eta\in C_{0}^{\infty}(\mathbb{R})$.
For any $t_{1},t_{2}\in(0,1)$ we define
$\sigma_{t_{1},t_{2}}(x,\xi):=\sigma(x,\xi)\eta(t_{1}\lambda_{\xi_{1}})\eta(t_{2}\lambda_{\xi_{2}})$.
Then, for any $m_{1}^{\prime},m_{2}^{\prime}\in\mathbb{R}$ we have

$\|\sigma_{t_{1},t_{2}}\|_{S^{m^{\prime}_{1},m^{\prime}_{2}},\gamma}\leq
Ct_{1}^{\frac{m_{1}-m^{\prime}_{1}}{2}}t_{2}^{\frac{m_{2}-m^{\prime}_{2}}{2}}\|\sigma\|_{S^{m_{1},m_{2}},\gamma},$

where $C=C(m_{1},m_{2},m^{\prime}_{1},m^{\prime}_{2},\gamma,\eta)$ is
independent of $\sigma$, $t_{1}$ and $t_{2}$.

For the proof of Lemma 4.7 see [Fischer-JFA_2015] (Lemma 6.8).

We are now ready to prove the main result of this section concerning some
estimates for the (right-convolution) kernel of bisingular pseudodifferential
operators. Let us remark that these estimates are the suitable generalization
to our setting of those holding in the standard (non-bisingular) case (see
[Fischer-JFA_2015]). Note that below we shall denote by $|y|:=d_{G}(y,e_{G})$,
where $d_{G}(\cdot,\cdot)$ is the geodesic distance (and analogously for
$|y_{j}|,$ $j=1,2$). Additionally, for any given $x=(x_{1},x_{2})\in G$, for a
neighborhood of $x$ we shall mean a Cartesian products of the form
$U_{1}\times U_{2}$, with $U_{i}$ being a geodesic neigborhood of $x_{i}$ for
$i=1,2$.

######  Theorem 4.8.

Let $\sigma\in S^{m_{1},m_{2}}(G)$ and $(x,y)\mapsto k_{x}(y)\in
C^{\infty}(G\times(G\setminus S))$ be its associated kernel. Then, for
$n_{i}=\mathrm{dim}(G_{i}),i=1,2$, the following estimates hold

  * •

If $n_{i}+m_{i}>0$ for $i=1,2$, then there exists $C>0$ and $a,b\in\mathbb{N}$
(independent of $\sigma$) such that for all $y\not\in S$

$|k_{x}(y)|\leq
C\sup_{\xi\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}}_{a,b}}|y_{1}|^{-n_{1}-m_{1}}|y_{2}|^{-n_{2}-m_{2}}.$

  * •

If $n_{i}+m_{i}=0$ for $i=1,2$, then there exists $C>0$ and $a,b\in\mathbb{N}$
(independent of $\sigma$) such that for all $y\not\in S$

$|k_{x}(y)|\leq
C\sup_{\xi\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}}_{a,b}}|\ln|y_{1}|||\ln|y_{2}||.$

  * •

If $n_{i}+m_{i}<0$ for $i=1,2$, then $k_{x}$ is continuous on $G$ and for all
$y\not\in S$

$|k_{x}(y)|\leq
C\sup_{\xi\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}}_{0,0}}.$

  * •

If $n_{i}+m_{i}>0$ and $n_{j}+m_{j}=0$ for $i,j\in\\{1,2\\},i\neq j$, then
$C>0$ and $a,b\in\mathbb{N}$ (independent of $\sigma$) such that for all
$y\not\in S$

$|k_{x}(y)|\leq
C\sup_{\xi\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}}_{a,b}}|y_{i}|^{-n_{i}-m_{i}}|\ln|y_{j}||.$

  * •

If $n_{i}+m_{i}<0$ and $n_{j}+m_{j}=0$ for $i,j\in\\{1,2\\},i\neq j$, then
$C>0$ and $\gamma_{j}\in\mathbb{N}^{2}$ (independent of $\sigma$) being either
of the form $\gamma_{j}=(a_{j},0)$ or of the form $\gamma_{j}=(0,a_{j})$, such
that for all $y\not\in S$

$|k_{x}(y)|\leq
C\sup_{\xi\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}}_{\gamma_{j},0}}|\ln|y_{j}||.$

  * •

If $n_{i}+m_{i}>0$ and $n_{j}+m_{j}<0$ for $i,j\in\\{1,2\\},i\neq j$, then
$C>0$ and $\gamma_{i}\in\mathbb{N}^{2}$ (independent of $\sigma$, of the same
form as above) such that for all $y\not\in S$

$|k_{x}(y)|\leq
C\sup_{\xi\in\widehat{G}}\|\sigma(x,\xi)\|_{S^{m_{1},m_{2}}_{\gamma_{i},0}}|y_{i}|^{-n_{i}-m_{i}}.$

###### Proof.

We shall separately analyze the above cases. Let us remark that throughout the
proof we shall use the notation $\mathcal{L}_{1}:=I_{1}+L_{G_{1}}$ and
$\mathcal{L}_{2}:=I_{2}+L_{G_{2}}$, where $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ are tought of as operators on $G_{1}$ and $G_{2}$
respectively, while $L:=\mathcal{L}_{1}\otimes\mathcal{L}_{2}$ is defined on
$G=G_{1}\times G_{2}$.

Case $n_{i}+m_{i}>0$. The estimate in this case trivially follows from
Corollary 4.4.

Toolkit. Let $\eta_{0},\eta_{1}\in C_{0}^{\infty}(\mathbb{R})$ be supported in
$[-1,1]$ and $[1/2,2]$ respectively, taking values in $[0,1]$, and such that

$\forall\lambda>0\quad\sum_{\ell=0}^{\infty}\eta_{\ell}(\lambda)=1,\quad\text{where}\,\,\eta_{\ell}(\lambda):=\eta_{1}(2^{-(\ell-1)}\lambda),\ell\geq
1.$

Now, for each $\ell_{1},\ell_{2}\in\mathbb{N}_{0}$, we define
$\sigma_{\ell_{1},\ell_{2}}(x,\xi):=\sigma(x,\xi)\eta_{\ell_{1}}(\lambda_{\xi_{1}})\eta_{\ell_{2}}(\lambda_{\xi_{2}})$
(with $\lambda_{\xi_{1}}$,$\lambda_{\xi_{2}}$, recall, being the eigenvalues
of $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, respectively), and denote by
$k_{x,\ell_{1},\ell_{2}}$ the corresponding kernel. Notice that, since
$\eta_{\ell_{1}}(\lambda_{\xi_{1}})\eta_{\ell_{2}}(\lambda_{\xi_{2}})$ is
smoothing, then $\sigma_{\ell_{1},\ell_{2}}$ is smoothing too. Moreover, also
the mapping $(x,y)\mapsto
k_{x,\ell_{1},\ell_{2}}(y)=k_{x}\ast\eta_{\ell_{1}}(\mathcal{L}_{1})\eta_{\ell_{2}}(\mathcal{L}_{2})\delta_{e_{1}}\otimes\delta_{e_{2}}$
is smooth, as $(x,y)\mapsto k_{x}(y)$ is smooth on $G\times G\setminus S$ and
$\eta_{\ell_{1}}(\mathcal{L}_{1})\eta_{\ell_{2}}(\mathcal{L}_{2})\delta_{e_{1}}\otimes\delta_{e_{2}}$
is smooth on $G$.

Observe now that one has the following convergence in
$C^{\infty}(G\times(G\setminus S))$

$k_{x}(y)=\lim_{N_{1},N_{2}\rightarrow\infty}\sum_{\ell_{1}=0}^{N_{1}}\sum_{\ell_{2}=0}^{N_{2}}k_{x,\ell_{1},\ell_{2}}(y)=\Big{(}k_{x}\ast\sum_{\ell_{1}=0}^{N_{1}}\sum_{\ell_{2}=0}^{N_{2}}\eta_{\ell_{1}}(\mathcal{L}_{1})\eta_{\ell_{2}}(\mathcal{L}_{2})\delta_{e_{1}}\otimes\delta_{e_{2}}\Big{)}(y),$

and that the following bound holds for $y\not\in S$

$|k_{x}(y)|\leq\sum_{\ell_{1},\ell_{2}}|k_{x,\ell_{1},\ell_{2}}(y)|.$

With this in mind we have, by Corollary 4.4 and Lemma 4.6, that for any given
$\alpha_{i}\in N_{0}^{n_{\triangle_{i}}}$, with $i=1,2$, and for any given
$m^{\prime}_{i}\in\mathbb{R}$, $i=1,2$, such that
$m^{\prime}_{i}+n_{i}<|\alpha_{i}|$,

$\sup_{x\in
G}|q^{\alpha_{1},\alpha_{2}}(z)k_{x,\ell_{1},\ell_{2}}|\lesssim\sup_{[\xi]\in\widehat{G}}\|\sigma_{\ell_{1},\ell_{2}}(x,\xi)\|_{S^{m^{\prime}_{1},m^{\prime}_{2}}_{(|\alpha_{1}|,|\alpha_{2}|),0}}$
$\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(|\alpha_{1}|,|\alpha_{2}|),0}}2^{-(\ell_{1}-1)\frac{m^{\prime}_{1}-m_{1}}{2}}2^{-(\ell_{2}-1)\frac{m^{\prime}_{2}-m_{2}}{2}}.$
(4.1)

Note that, for all $z\in G$ and for all $a_{1},a_{2}\in 2\mathbb{N}_{0}$, we
have

$|z_{1}|^{a_{1}}|z_{2}|^{a_{2}}\lesssim\sum_{|\alpha_{1}|=a_{1},|\alpha_{2}|=a_{2}}|q^{\alpha_{1},\alpha_{2}}(z)|.$

The previous estimate is of course meaningful in a neghborhhod $U=U_{1}\times
U_{2}$ where $U_{1}$ and $U_{2}$ are geodesic neigborhoods of $e_{1}$ and
$e_{2}$ respectively, in which, in the following, we will be working. Note
that outside that neighborhood the estimates in the statement are
straightforward, because of the smoothness of the kernel. Therefore, for all
$a_{1},a_{2}\in 2\mathbb{N}_{0}$ and $m^{\prime}_{1},m^{\prime}_{2}$ such that
$m^{\prime}_{i}+n_{i}<a_{i}$, $i=1,2$, (4.1) implies

$|z_{1}|^{a_{1}}|z_{2}|^{a_{2}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}2^{\ell_{1}\frac{m_{1}-m^{\prime}_{1}}{2}}2^{\ell_{2}\frac{m_{2}-m^{\prime}_{2}}{2}}.$
(4.2)

Since we want to study the behaviour of $k_{x}(y)$ close to the set $S$, we
will be considering each of the following situations

  1. 1.

$|z_{1}|<1$ and $|z_{2}|<1$;

  2. 2.

$|z_{1}|<1$ and $|z_{2}|\geq 1$ (resp. $|z_{1}|\geq 1$ and $|z_{2}|<1$).

Case $n_{i}+m_{i}>0$ for all $i=1,2$. When $|z_{1}|<1$ and $|z_{2}|<1$, we can
chose $\ell_{0_{i}}\in\mathbb{N}_{0}$ such that

$2^{-\ell_{0_{i}}}\leq|z_{i}|\leq 2^{-\ell_{0_{i}}+1},\quad i=1,2.$

In order to derive the desired estimate we write

$\sum_{\ell_{1}=0}^{N_{1}}\sum_{\ell_{2}=0}^{N_{2}}=\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
\ell_{2}\leq\ell_{0_{2}}\end{subarray}}+\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}+\sum_{\begin{subarray}{c}N_{1}\geq\ell_{1}>\ell_{0_{1}}\\\
\ell_{2}\leq\ell_{0_{2}}\end{subarray}}+\sum_{\begin{subarray}{c}N_{1}\geq\ell_{1}>\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}},$

and study the behaviour of $k_{x,\ell_{1},\ell_{2}}$ in the cases

  * 1.

$\ell_{i}\leq\ell_{0_{i}}$ for $i=1,2$,

  * 2.

$\ell_{i}>\ell_{0_{i}}$ for $i=1,2$,

  * 3.

$\ell_{1}\leq\ell_{0_{1}}$ and $\ell_{2}>\ell_{0_{2}}$ (resp.
$\ell_{2}\leq\ell_{0_{2}}$ and $\ell_{1}>\ell_{0_{1}}$),

separately.

For $\ell_{i}\leq\ell_{0_{i}}$, for $i=1,2$, from (4.2) we get

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
\ell_{2}\leq\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-a_{1}}2^{\ell_{0_{1}}\frac{m_{1}-m^{\prime}_{1}}{2}}|z_{2}|^{-a_{2}}2^{\ell_{0_{2}}\frac{m_{2}-m^{\prime}_{2}}{2}}.$

We then choose $a_{i}\in 2\mathbb{N}_{0}$ and $m^{\prime}_{i}\in\mathbb{R}$,
for $i=1,2$, such that

$m_{i}+n_{i}>a_{i}\geq
m_{i}+n_{i}-2\quad\text{and}\quad\frac{m_{i}-m_{i}^{\prime}}{2}=m_{i}+n_{i}-a_{i}>0,$
(4.3)

which yields

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
\ell_{2}\leq\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-a_{1}-\frac{m_{1}-m_{1}^{\prime}}{2}}|z_{2}|^{-a_{2}-\frac{m_{2}-m_{2}^{\prime}}{2}}$
$\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-m_{1}-n_{1}}|z_{2}|^{-m_{2}-n_{2}}.$

For $\ell_{i}>\ell_{0_{i}}$ ($\ell_{i}\leq N_{i}$) we make a different choice
for $a_{i}$ and $m^{\prime}_{i}$ in (4.3) that we call
$a_{i}^{\prime},m^{\prime\prime}_{i}$ in order to keep the notation
$a_{i},m_{i}^{\prime}$ for the choices we made in the previous case
$\ell_{i}\leq\ell_{0_{i}}$. We now choose $a_{i}^{\prime}=a_{i}+2$ and
$m^{\prime\prime}_{i}$ satisfying

$\frac{m_{i}-m_{i}^{\prime\prime}}{2}=m_{i}+n_{i}-a^{\prime}_{i},\quad i=1,2.$

Since $m_{i}<m^{\prime\prime}_{i}$ now, we have that

$\sum_{\begin{subarray}{c}N_{1}\geq\ell_{1}>\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-a^{\prime}_{1}}2^{\ell_{0_{1}}\frac{m_{1}-m_{1}^{\prime\prime}}{2}}|z_{2}|^{-a^{\prime}_{2}}2^{\ell_{0_{2}}\frac{m_{2}-m_{2}^{\prime\prime}}{2}}$
$\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-m_{1}-n_{1}}|z_{2}|^{-m_{2}-n_{2}}.$

For $\ell_{1}\leq\ell_{0_{1}}$ and $\ell_{2}>\ell_{0_{2}}$ (resp.
$\ell_{2}\leq\ell_{0_{2}}$ and $\ell_{1}>\ell_{0_{1}}$) we make a different
choice of $a_{i}$ and $m^{\prime}_{i}$ that we call
$a_{i}^{\prime\prime},m^{\prime\prime\prime}_{i}$ in order to keep the
previous notation for the other cases. By choosing
$a_{1}^{\prime\prime}=a_{1},m_{1}^{\prime\prime\prime}=m_{1}^{\prime}$,
$a_{2}^{\prime\prime}=a_{2}^{\prime}$ and
$m_{2}^{\prime\prime\prime}=m_{2}^{\prime\prime}$ we get, once again from
(4.2), that

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-a^{\prime\prime}_{1}}2^{\ell_{0_{1}}\frac{m_{1}-m_{1}^{\prime\prime\prime}}{2}}|z_{2}|^{-a^{\prime\prime}_{2}}2^{\ell_{0_{2}}\frac{m_{2}-m_{2}^{\prime\prime\prime}}{2}}$
$\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-m_{1}-n_{1}}|z_{2}|^{-m_{2}-n_{2}}.$

The estimate in the case when $\ell_{2}\leq\ell_{0_{2}}$ and
$\ell_{1}>\ell_{0_{1}}$ follows similarly by exchanging the role of $\ell_{1}$
an $\ell_{2}$. Collecting the (four) estimates together we get the desired
result (keeping the biggest seminorm) in the case when $|z_{1}|<1$ and
$|z_{2}|<1$.

In the case when $|z_{1}|<1$ and $|z_{2}|\geq 1$, we can choose $\ell_{0_{1}}$
as before, and, once again, split the analysis into the cases
$\ell_{1}\leq\ell_{0_{1}}$ and $\ell_{1}>\ell_{0_{1}}$. Note that we do not
split the sum in $\ell_{2}$ in this case, so we will make a single choice for
$a_{2}$ and $m_{2}^{\prime}$. By choosing $a_{1}$, $m^{\prime}_{1}$,
$a^{\prime}_{1}$, $m^{\prime\prime}_{1}$ as before, and $a_{2}=n_{2}+m_{2}+3$,
$m_{2}^{\prime}=m_{2}+2$ (so that $m^{\prime}_{2}+n_{2}<a_{2}$), we will get
the result in this case (again the result is given in terms of the biggest
seminorm).

Finally the case $|z_{1}|\geq 1$ and $|z_{2}|<1$ is proved as the last one by
reversing the role of $z_{1}$ and $z_{2}$.

Collecting all the estimates above we obtain the result in terms of the
biggest seminorm.

Case: $m_{i}+n_{i}=0$ for $i=1,2$. We consider again all the cases
$|z_{i}|<1$, $|z_{j}|\geq 1$, $i,j=1,2$ ($i\neq j$). When $|z_{1}|<1$ and
$|z_{2}|<1$ we fix $\ell_{0_{i}}$ as before and consider the cases 1, 2 and 3
(and the respective case of the last one) as above. Then, for
$\ell_{i}\leq\ell_{0_{i}}$, for $i=1,2$, from (4.2) with $a_{i}=0$,
$m^{\prime}_{i}=m_{i}$, for all $i=1,2$, we get

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
\ell_{2}\leq\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{a_{1},a_{2},0}}\,\,\ell_{0_{1}}\ell_{0_{2}}\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|\ln|z_{1}||\,\,|\ln|z_{2}||.$

When $\ell_{i}>\ell_{0_{i}}$ for $i=1,2$ we choose $a^{\prime}_{i}=2$ and
$m^{\prime\prime}_{i}=m_{i}-4$ for all $i=1,2,$ and get (from (4.2) with
$a_{i}^{\prime},m_{i}^{\prime\prime}$)

$\sum_{\begin{subarray}{c}N_{1}\geq\ell_{1}>\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}.$

When $\ell_{1}\leq\ell_{0_{1}}$ and $\ell_{2}>\ell_{0_{2}}$
($\ell_{2}\leq\ell_{0_{2}}$ and $\ell_{1}>\ell_{0_{1}}$), by choosing
$a_{1}^{\prime\prime}=a_{1}$, $m_{1}^{\prime\prime\prime}=m_{1}^{\prime}$ and
$a_{2}^{\prime\prime}=a_{2}^{\prime}$,
$m_{2}^{\prime\prime\prime}=m_{2}^{\prime\prime}$, we obtain

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|\ln|z_{1}||.$

Collecting the estimates together the result when $|z_{1}|<1$ and $|z_{2}|<1$
follows.

When $|z_{1}|<1$ and $|z_{2}|\geq 1$ we fix again $\ell_{0_{1}}$ as before.
Recall that now we do not split the sum in $\ell_{2}$ and that we will make a
single choice for $a_{2}$ and $m_{2}^{\prime}$ in (4.2). Then, using estimate
(4.2) with $a_{1}$ and $m_{1}^{\prime}$ (when $\ell_{1}\leq\ell_{0_{1}}$), and
$a^{\prime}_{1}$ and $m^{\prime\prime}_{1}$ (when $\ell_{1}>\ell_{0_{1}}$) as
in the previous case, the result follows by choosing $a_{2}=n_{2}+m_{2}+3=3$
and $m_{2}^{\prime}=m_{2}+2$ (where $m_{2}+n_{2}<a_{2}$ is still satisfied).

The case $|z_{1}|\geq 1$ and $|z_{2}|<1$ is treated as the previous one
reversing the roles of $z_{1}$ and $z_{2}$.

Finally, collecting all the cases above, we get the result in terms of the
biggest seminorm.

Case $n_{i}+m_{i}>0$, $n_{j}+m_{j}=0$ for $i,j\in\\{1,2\\},i\neq j$. To fix
ideas suppose $n_{1}+m_{1}>0$ and $n_{2}+m_{2}=0$ since the other case is
treated analogously. We then combine the strategies used in the cases
$n_{i}+m_{i}>0$ for all $i=1,2$ and $n_{i}+m_{i}=0$ for all $i=1,2$.

When $|z_{1}|<1$ and $|z_{2}|<1$ we fix again $\ell_{0_{i}}$ such that
$|z_{i}|\sim 2^{-\ell_{0_{i}}}$, $i=1,2$. Then, for
$\ell_{i}\leq\ell_{0_{i}}$, we choose $a_{i}\in 2\mathbb{N}_{0}$ and
$m_{i}\in\mathbb{R}$, for all $i=1,2$, such that

$m_{1}+n_{1}>a_{1}\geq
m_{1}+n_{1}-2\quad\text{and}\quad\frac{m_{1}-m_{1}^{\prime}}{2}=m_{1}+n_{1}-a_{1}>0,$
$a_{2}=0,\quad m^{\prime}_{2}=m_{2}.$

so that, from (4.2), we obtain

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
\ell_{2}\leq\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-m_{1}-n_{1}}|\ln|z_{2}||.$

For $\ell_{i}>\ell_{0_{i}}$, for all $i=1,2$, we apply (4.2) with
$a^{\prime}_{1}=a_{1}+2$, $m^{\prime\prime}_{1}$ satisfying the same
conditions as $m^{\prime}_{1}$ with $a_{1}^{\prime}$ in place of $a_{1}$
(where, recall, $a_{1},m^{\prime}_{1}$ are the parameters used for
$\ell_{i}\leq\ell_{0_{i}}$), $a^{\prime}_{2}=2$ and
$m^{\prime\prime}_{2}=m_{2}-4$. We then have

$\sum_{\begin{subarray}{c}N_{1}\geq\ell_{1}>\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-n_{1}-m_{1}}.$

For $\ell_{1}\leq\ell_{0_{1}}$ and $\ell_{2}>\ell_{0_{2}}$
($\ell_{2}\leq\ell_{0_{2}}$ and $\ell_{1}>\ell_{0_{1}}$) we repeat the
strategy used before, that is, we choose $a_{1}^{\prime\prime}=a_{1}$,
$m_{1}^{\prime\prime\prime}=m_{1}^{\prime}$,
$a_{2}^{\prime\prime}=a_{2}^{\prime}$, and
$m_{2}^{\prime\prime\prime}=m_{2}^{\prime\prime}$ in (4.2) and get

$\sum_{\begin{subarray}{c}\ell_{1}\leq\ell_{0_{1}}\\\
N_{2}\geq\ell_{2}>\ell_{0_{2}}\end{subarray}}|k_{x,\ell_{1},\ell_{2}}(z)|\lesssim\|\sigma\|_{S^{m_{1},m_{2}}_{(a_{1},a_{2}),0}}|z_{1}|^{-n_{1}-m_{1}}.$

Hence, collecting all the estimates we get the result when $|z_{1}|<1$ and
$|z_{2}|<1$.

When $|z_{1}|<1$ and $|z_{2}|\geq 1$ the proof follows by considering again
only the two cases $\ell_{1}\leq\ell_{0_{1}}$ and $\ell_{1}>\ell_{0_{1}}$
(here we do not split the sum in $\ell_{2}$, so $0\leq\ell_{2}\leq N_{2}$).
Using the same choices as before for
$a_{1},a^{\prime}_{1},m^{\prime}_{1},m^{\prime\prime}_{1}$, and choosing
$a_{2}=3$ and $m^{\prime}_{2}=m_{2}+2$ (so that $m_{2}^{\prime}+n_{2}<a_{2}$)
in (4.2), where, recall, $a_{1},m_{1}^{\prime}$ are the parameters used when
$\ell_{1}\leq\ell_{0_{1}}$, while $a^{\prime}_{1},m^{\prime\prime}_{1}$ are
those used for $\ell_{1}>\ell_{0_{1}}$ (we make a single choice for $a_{2}$
and $m^{\prime}_{2}$ here), then the desired estimates hold when $|z_{1}|<1$
and $|z_{2}|\geq 1$.

When $|z_{1}|\geq 1$ and $|z_{2}|<1$ the result is proved by reversing the
roles of $z_{1}$ and $z_{2}$ in the last case.

Cases $n_{i}+m_{i}<0$ and $n_{j}+m_{j}=0$; $n_{i}+m_{i}>0$ and $n_{j}+m_{j}<0$
($i\neq j$).

These cases can be treated as the last one, that is, by combing the strategies
used for the other cases in the different regions $|z_{i}|<1,|z_{j}|\geq 1$,
$i,j=1,2$ ($i\neq j$). The proof is left to the reader.

∎

##  5\. Calculus of bisingular pseudodifferential operators

In what follows we will use the previous properties to prove a composition
formula for bisingular operators.

######  Theorem 5.1 (Composition formula).

Let $\sigma_{A}\in S^{m_{1},m_{2}}(G\times\widehat{G})$ and $\sigma_{B}\in
S^{m^{\prime}_{1},m^{\prime}_{2}}(G\times\widehat{G})$, and
$A:=\mathrm{Op}(a)$ and $B=\mathrm{Op}(b)$ the corresponding
pseudodifferential operators. Then the symbol $\sigma_{AB}$ of $AB$ is,
asymptotically,

$a\\#b(x,\xi):=\sigma_{AB}(x,\xi)\sim\sum_{j\geq
0}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(x,\xi),$ (5.1)

where

$c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}\in
S^{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(G\times\widehat{G}),$
$c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(x,\xi)=d^{\prime}_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}+d^{\prime\prime}_{m_{1}+m^{\prime}_{1}-j-1,m_{2}+m^{\prime}_{2}-j}$
$+d^{\prime\prime\prime}_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j-1},$
(5.2)
$d^{\prime}_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}=\sum_{|\alpha_{1}|=|\alpha_{2}|=j}\frac{1}{\alpha_{1}!\alpha_{2}!}(\triangle^{\alpha_{1},\alpha_{2}}\sigma_{A}(x,\xi))\partial^{\alpha_{1},\alpha_{2}}\sigma_{B}(x,\xi),$
$d^{\prime\prime}_{m_{1}+m^{\prime}_{1}-j-1,m_{2}+m^{\prime}_{2}-j}=\sum_{|\alpha_{2}|=j}\frac{1}{\alpha_{2}!}\left(\triangle^{0,\alpha_{2}}\sigma_{A}\circ_{\xi_{1}}\partial^{0,\alpha_{2}}\sigma_{B}-\sum_{|\alpha_{1}|\leq
j}\frac{1}{\alpha_{1}!}(\triangle^{\alpha_{1},\alpha_{2}}\sigma_{A}(x,\xi))\partial^{\alpha_{1},\alpha_{2}}\sigma_{B}(x,\xi)\right),$

and

$d^{\prime\prime\prime}_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j-1}=\sum_{|\alpha_{1}|=j}\frac{1}{\alpha_{1}!}\left(\triangle^{\alpha_{1},0}\sigma_{A}\circ_{\xi_{2}}\partial^{\alpha_{1},0}\sigma_{B}-\sum_{|\alpha_{2}|\leq
j}\frac{1}{\alpha_{2}!}(\triangle^{\alpha_{1},\alpha_{2}}\sigma_{A}(x,\xi))\partial^{\alpha_{1},\alpha_{2}}\sigma_{B}(x,\xi)\right)$

are such that they belong to
$S^{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(G\times\widehat{G})$. In
particular the asymptotic formula (5.1) means that, for any given $N>0$,

$r_{N}=\sigma_{AB}-\sum_{j<N}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}\in
S^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N}(G\times\widehat{G}).$

###### Proof.

Let $A$ and $B$ the operators above, then, by (3.3) we have

$ABf(x)=\int_{G}(Bf)(xz)R_{A}(x,z^{-1})dz$
$=\int_{G}f(xy^{-1})\left(\int_{G}R_{B}(xz,yz)R_{A}(x,z^{-1})dz\right)dy$
$\underset{y\rightarrow
y^{-1}x}{=}\int_{G}f(y)\left(\int_{G}R_{B}(xz,y^{-1}xz)R_{A}(x,z^{-1})dz\right)dy$
$=\int_{G}f(y)R_{AB}(x,y^{-1}x)dy,$

where

$R_{AB}(x,y):=\int_{G}R_{B}(xz,yz)R_{A}(x,z^{-1})dz.$

Since $\sigma_{AB}(x,\xi)=\widehat{R_{A,B}}(x,\xi)$ we have

$\sigma_{AB}(x,\xi)=\int_{G}\int_{G}R_{A}(x,z^{-1})R_{B}(xz,yz)\xi^{*}(y)dzdy$
$=\int_{G}\int_{G}R_{A}(x,z^{-1})\xi^{*}(z^{-1})R_{B}(xz,yz)\xi^{*}(yz)dzdy.$
(5.3)

Then we write
$R_{B}(xz,yz)=R_{B}(x_{1}z_{1},x_{2}z_{2},y_{1}z_{1},y_{2}z_{2})$ and take the
Taylor expansion of $R_{B}$ with respect to the first variable at
$z_{1}=e_{1}$, that is,

$R_{B}(xz,yz)=\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}q^{\alpha_{1},0}(z_{1}^{-1},x_{2}z_{2})\partial^{\alpha_{1},0}R_{B}(x_{1},x_{2}z_{2},yz)$
$+\sum_{|\alpha_{1}|=N}\frac{1}{\alpha_{1}!}q^{\alpha_{1},0}(z_{1}^{-1},x_{2}z_{2})(R_{B})_{\alpha_{1}}(x_{1}z_{1},x_{2}z_{2},yz),$

where $q^{\alpha_{1},0}(x)=r^{\alpha_{1}}(x_{1})$ is constant with tespect to
$x_{2}$. Now, taking into account that $q^{\alpha_{1},0}(x_{1},x_{2})$ does
not depend on the choice of the second variable and that
$q^{0,\alpha_{2}}(x_{1},x_{2})$ does not depend on the choice of the first
variable, we expand the previous quantity with respect to the second variable
at $z_{2}=e_{2}$ and have

$R_{B}(xz,yz)=\sum_{|\alpha_{2}|<N}\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(z_{1}^{-1},z_{2}^{-1})\partial^{\alpha_{1},\alpha_{2}}R_{B}(x_{1},x_{2},yz)$
$+\sum_{|\alpha_{2}|=N}\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(z_{1}^{-1},z_{2}^{-1})(\partial^{\alpha_{1},0}R_{B})_{\alpha_{2}}(x_{1},x_{2},yz),$
$+\sum_{|\alpha_{2}|<N}\sum_{|\alpha_{1}|=N}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(z_{1}^{-1},z_{2}^{-1})\partial^{0,\alpha_{2}}(R_{B})_{\alpha_{1}}(x_{1},x_{2},yz)$
$+\sum_{|\alpha_{2}|=N}\sum_{|\alpha_{1}|=N}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(z_{1}^{-1},z_{2}^{-1})(R_{B})_{\alpha_{1},\alpha_{2}}(x_{1}z_{1},x_{2}z_{2},yz).$

Therefore we have

$\displaystyle\sigma_{AB}(x,\xi)$ $\displaystyle=$
$\displaystyle\sum_{|\alpha_{1}|<N,|\alpha_{2}|<N}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G\times
G}\xi^{*}(z^{-1})q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(yz)\partial^{\alpha_{1},\alpha_{2}}R_{B}(x,yz)dzdy$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}\int_{G\times
G}\Bigg{(}q^{\alpha_{1},0}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(z^{-1})\partial^{\alpha_{1},0}R_{B}(x,yz)\xi^{*}(yz)$
$\displaystyle-$
$\displaystyle\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(z^{-1})\partial^{\alpha_{1},\alpha_{2}}R_{B}(x,yz)\xi^{*}(yz)\Bigg{)}dzdy$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}\int_{G\times
G}\Bigg{(}q^{0,\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(z^{-1})\partial^{0,\alpha_{2}}R_{B}(x,yz)\xi^{*}(yz)$
$\displaystyle-$
$\displaystyle\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(z^{-1})\partial^{\alpha_{1},\alpha_{2}}R_{B}(x,yz)\xi^{*}(yz)\Bigg{)}dzdy$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{1}|=N,|\alpha_{2}|=N}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G\times
G}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(z^{-1})(R_{B})_{\alpha_{1},\alpha_{2}}(xz,yz)\xi^{*}(yz)dzdy,$

and, rearranging the terms, get

$\sigma_{AB}=\sum_{|\alpha_{1}|=|\alpha_{2}|<N}\frac{1}{\alpha_{1}!\alpha_{2}!}(\triangle^{\alpha_{1},\alpha_{2}}\sigma_{A}(x,\xi))\partial^{\alpha_{1},\alpha_{2}}\sigma_{B}(x,\xi)$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}\left((\triangle^{\alpha_{1},0}\sigma_{A}\circ_{\xi_{2}}\partial^{\alpha_{1},0}\sigma_{B})(x,\xi)-\sum_{|\alpha_{2}|\leq|\alpha_{1}|}\frac{1}{\alpha_{2}!}(\triangle^{\alpha_{1},\alpha_{2}}\sigma_{A}(x,\xi))\partial^{\alpha_{1},\alpha_{2}}\sigma_{B}(x,\xi)\right)$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}\left((\triangle^{0,\alpha_{2}}\sigma_{A}\circ_{\xi_{1}}\partial^{0,\alpha_{2}}\sigma_{B})(x,\xi)-\sum_{|\alpha_{1}|\leq|\alpha_{2}|}\frac{1}{\alpha_{1}!}(\triangle^{\alpha_{1},\alpha_{2}}\sigma_{A}(x,\xi))\partial^{\alpha_{1},\alpha_{2}}\sigma_{B}(x,\xi)\right)$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{1}|=N,|\alpha_{2}|=N}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G\times
G}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\xi^{*}(z^{-1})(R_{B})_{\alpha_{1},\alpha_{2}}(xz,yz)\xi^{*}(yz)dzdy$
$\displaystyle=$
$\displaystyle\sum_{j<N}\left(d^{\prime}_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}+d^{\prime\prime\prime}_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j-1}+d^{\prime\prime}_{m_{1}+m^{\prime}_{1}-j-1,m_{2}+m^{\prime}_{2}-j}\right)+r_{N}.$

In order to complete the proof we only need to show that $r_{N}\in
S^{m_{1}+m_{1}^{\prime}-N,m_{2}+m_{2}^{\prime}-N}(G\times\widehat{G})$ for all
$N\in\mathbb{N}_{0}$, that is, we have to check that

$\sup_{x\in
G}\|\partial^{\gamma_{1},\gamma_{2}}\Delta^{\beta_{1},\beta_{2}}r_{N}(x,\xi)\|_{\mathscr{L}(\mathcal{H}_{\xi})}\lesssim\langle\xi_{2}\rangle^{m_{1}+m^{\prime}_{1}-|\beta_{1}|-N}\langle\xi_{2}\rangle^{m_{2}+m^{\prime}_{2}-|\beta_{2}|-N},$
(5.4)

for all $\gamma_{1},\gamma_{2},\beta_{1},\beta_{2}$. For simplicity we
consider the case $\alpha_{1}=\alpha_{2}=\beta_{1}=\beta_{2}=0$ since the
general case follows similarly. We then write
$\xi^{*}(z)=\langle\xi_{1}\rangle^{-s_{1}}\langle\xi_{2}\rangle^{-s_{2}}(I_{1}+L_{G_{1}})_{z_{1}}^{s_{1}}\otimes(I_{2}+L_{G_{2}})_{z_{2}}^{s_{2}}\,\xi^{*}(z)$,
with integers $s_{1},s_{2}\geq 1$, and have, after integrating by parts and
using the fact that $(R_{B})_{\alpha_{1},\alpha_{2}}(x,y)$ is the kernel of a
symbol in $S^{m^{\prime}_{1},m^{\prime}_{2}}(G\times\widehat{G})$ ,

$\displaystyle r_{N}(x,\xi)$ $\displaystyle=$
$\displaystyle\langle\xi_{1}\rangle^{-s_{1}}\langle\xi_{2}\rangle^{-s_{2}}\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|=N\\\
|\gamma_{1}|+|\gamma_{2}|=2s_{1}\\\
|\tau_{1}|+|\tau_{2}|=2s_{2}\end{subarray}}c_{\gamma_{1},\gamma_{2},\tau_{1},\tau_{2}}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G}\left(\partial_{z}^{\gamma_{1},\tau_{1}}(q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1}))\right)\xi^{*}(z^{-1})$
$\displaystyle\times$
$\displaystyle\partial_{z}^{\gamma_{2},\tau_{2}}\widehat{({R}_{B})}_{\alpha_{1},\alpha_{2}}(xz,\xi)dz$
$\displaystyle=$
$\displaystyle\langle\xi_{1}\rangle^{-s_{1}}\langle\xi_{2}\rangle^{-s_{2}}\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|=N\\\
|\gamma_{1}|+|\gamma_{2}|=2s_{1}\\\
|\tau_{1}|+|\tau_{2}|=2s_{2}\end{subarray}}c_{\gamma_{1},\gamma_{2},\tau_{1},\tau_{2}}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G}\left(\tilde{\partial}_{z^{-1}}^{\gamma_{1},\tau_{1}}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})\right)\xi^{*}(z^{-1})$
$\displaystyle\times$
$\displaystyle\partial_{z_{1}=xz}^{\gamma_{2},\tau_{2}}(\widehat{R}_{B})_{\alpha_{1},\alpha_{2}}(z_{1},\xi)dz,$

where in the second equality we applied the relation beetween left-invariant
and right-invariant vector fields given by
$\partial^{\alpha,\beta}\\{\phi(\cdot^{-1})\\}(x)=(-1)^{|\alpha|+|\beta|}(\tilde{\partial}^{\alpha,\beta}\phi)(x^{-1})$
($\tilde{\partial}$ denoting the right invariant vector field in our
notation), and used the left invariance of $\partial^{\gamma_{2},\tau_{2}}$.

The previous computations, in particular, give

$\displaystyle\|r_{N}(x,\xi)\|_{\mathscr{L}(\mathcal{H}_{\xi})}$
$\displaystyle\leq$ $\displaystyle
C_{s_{1},s_{2}}\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|=N\\\
|\gamma_{1}|+|\gamma_{2}|=2s_{1}\\\
|\tau_{1}|+|\tau_{2}|=2s_{2}\end{subarray}}\langle\xi_{1}\rangle^{-s_{1}}\langle\xi_{2}\rangle^{-s_{2}}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G}|\tilde{\partial}_{z^{-1}}^{\gamma_{1},\tau_{1}}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})|dz$
$\displaystyle\times$ $\displaystyle\sup_{z_{1}\in
G}\|\partial^{\gamma_{2},\tau_{2}}_{z_{1}}(\widehat{R}_{B})_{\alpha_{1},\alpha_{2}}(z_{1},\xi)\|_{\mathscr{L}(\mathcal{H}_{\xi})}$
$\displaystyle\leq$ $\displaystyle
C_{s_{1},s_{2}}\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|=N\\\
|\gamma_{1}|+|\gamma_{2}|=2s_{1}\\\
|\tau_{1}|+|\tau_{2}|=2s_{2}\end{subarray}}\langle\xi_{1}\rangle^{m^{\prime}_{1}-s_{1}}\langle\xi_{2}\rangle^{m^{\prime}_{2}-s_{2}}\frac{1}{\alpha_{1}!\alpha_{2}!}\int_{G}|\tilde{\partial}_{z^{-1}}^{\gamma_{1},\tau_{1}}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})|dz$
$\displaystyle\times$
$\displaystyle\|(\widehat{R}_{B})_{\alpha_{1},\alpha_{2}}\|_{S^{m^{\prime}_{1},m^{\prime}_{2}}_{(2s_{1},2s_{2})}}.$

We now assume that $N$ is sufficiently large, namely
$N>N_{0}:=\max\\{m_{1},m_{2}\\}$, and choose $s_{1}=N-m_{1}$ and
$s_{2}=N-m_{2}$. In this case, by using Proposition 4.8, we obtain

$\int_{G}|\tilde{\partial}_{z^{-1}}^{\gamma_{1},\tau_{1}}q^{\alpha_{1},\alpha_{2}}(z^{-1})R_{A}(x,z^{-1})|dz\lesssim\|\tilde{\partial}^{\gamma_{1},\tau_{1}}\Delta^{\alpha_{1},\alpha_{2}}\sigma_{A}\|_{S^{m_{1}-N,m_{2}-N}}\leq\|\sigma_{A}\|_{S^{m_{1}-N,m_{2}-N}_{(N,N),(2(N-m_{1}),2(N-m_{2}))}},$

and, consequently,

$\|r_{N}(x,\xi)\|_{\mathscr{L}(\mathcal{H}_{\xi})\lesssim}\langle\xi_{1}\rangle^{m_{1}+m^{\prime}_{1}-N}\langle\xi_{2}\rangle^{m_{2}-m^{\prime}_{2}-N},\quad\forall
N>N_{0},$

which proves (5.4) for every $N>N_{0}$ when
$\gamma_{1}=\gamma_{2}=\beta_{1}=\beta_{2}=0$. By using similar arguments
together with the Leibniz formula one proves (5.4) in the general form
(possibly with a different $N_{0}$), which, in particular, gives that
$r_{N}\in S^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N}$ for every
$N>N_{0}$.

We are now left with proving that $r_{N}\in
S^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N}$ for every $N\leq N_{0}$.
Observe that

$\displaystyle r_{N}(x,\xi)$ $\displaystyle=$
$\displaystyle\sigma_{AB}(x,\xi)-\sum_{j<N}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(x,\xi)$
$\displaystyle=$
$\displaystyle\sigma_{AB}(x,\xi)-\sum_{j<N_{0}+1}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(x,\xi)+\sum_{N\leq
j<N_{0}+1}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(x,\xi)$
$\displaystyle=$ $\displaystyle r_{N_{0}+1}(x,\xi)+\sum_{N\leq
j<N_{0}+1}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}(x,\xi),$

therefore, since

$r_{N_{0}+1}\in
S^{m_{1}+m^{\prime}_{1}-N_{0}-1,m_{2}+m^{\prime}_{2}-N_{0}-1},$ $\sum_{N\leq
j<N_{0}+1}c_{m_{1}+m^{\prime}_{1}-j,m_{2}+m^{\prime}_{2}-j}\in
S^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N},$

and

$S^{m_{1}+m^{\prime}_{1}-N_{0}-1,m_{2}+m^{\prime}_{2}-N_{0}-1}\subset
S^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N},$

we finally get that $r_{N}\in
S^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N}$ for every $N\leq N_{0}$.
This concludes the proof.

∎

######  Theorem 5.2.

Let $\sigma\in S^{m_{1},m_{2}}(G\times\widehat{G})$, then the symbol of the
operator $\mathrm{Op}(\sigma)^{*}$, denoted by $\sigma^{*}$, is asymptotically
given by

$\sigma^{*}(x,\xi)\sim\sum_{j\geq 0}c_{m_{1}-j,m_{2}-j}(x,\xi),$ (5.5)

where $c_{m_{1}-j,m_{2}-j}\in S^{m_{1}-j,m_{2}-j}(G\times\widehat{G})$ and

$c_{m_{1}-j,m_{2}-j}(x,\xi)=d^{\prime}_{m_{1}-j,m_{2}-j}+d^{\prime\prime}_{m_{1}-j-1,m_{2}-j}+d^{\prime\prime\prime}_{m_{1}-j,m_{2}-j-1},$

with, using the notations in (3.6) and (3.7) for $\sigma^{*_{1}}(x,\xi)$ and
$\sigma^{*_{2}}(x,\xi)$,

$\displaystyle d^{\prime}_{m_{1}-j,m_{2}-j}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|=|\alpha_{2}|=j\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{1}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*},$
$\displaystyle d^{\prime\prime}_{m_{1}-j-1,m_{2}-j}$ $\displaystyle=$
$\displaystyle\sum_{|\alpha_{1}|=j}\frac{1}{\alpha_{1}!}\Big{(}\Delta^{\alpha_{1},0}\partial^{0,\alpha_{1}}\sigma^{*_{1}}(x,\xi)-\sum_{|\alpha_{2}|\leq|\alpha_{1}|}\frac{1}{\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{2}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}\Big{)},$
$\displaystyle d^{\prime\prime\prime}_{m_{1}-j,m_{2}-j-1}$ $\displaystyle=$
$\displaystyle\sum_{|\alpha_{2}|<j}\frac{1}{\alpha_{2}!}\Big{(}\Delta^{0,\alpha_{2}}\partial^{0,\alpha_{2}}\sigma^{*_{2}}(x,\xi)-\sum_{|\alpha_{1}|\leq|\alpha_{2}|}\frac{1}{\alpha_{1}!}\Delta^{\alpha_{1},\alpha_{2}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}\Big{)},$

belonging to $S^{m_{1}-j,n_{2}-j}(G\times\widehat{G})$. In particular the
asymptotic formula (5.5) means that, for any $N>0$,

$r_{N}=\sigma^{*}-\sum_{j<N}c_{m_{1}-j,m_{2}-j}\in
S^{m_{1}-N,m_{2}-N}(G\times\widehat{G}).$

###### Proof.

The strategy here is similar to the one used for the asymptotic composition
formula. Notice that, since the kernel of $\sigma^{*}(x,D)$ satisfies
$k_{\sigma^{*}}(x,v)=\overline{k_{\sigma}(xv^{-1},v^{-1})}$, by taking the
Fourier transform in the second variable we have

$\sigma^{*}(x,\xi)=\int_{G}\overline{k_{\sigma}(xv^{-1},v^{-1})}\,\,\xi_{1}^{*}(v_{1})\otimes\xi^{*}_{2}(v_{2})dv.$

We now expand
$\overline{k_{\sigma}(xv^{-1},v^{-1})}=\overline{k_{\sigma}(x_{1}v_{1}^{-1},x_{2}v_{2}^{-1},v^{-1})}$
in the first variable at $v_{1}=e_{1}$, and, afterwards, in the second
variable at $v_{2}=e_{2}$, and get

$\displaystyle\overline{k_{\sigma}(xv^{-1},v^{-1})}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|<N,|\alpha_{2}|<N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{1}}(v)\partial^{\alpha_{1},\alpha_{2}}\overline{k_{\sigma}(x,v^{-1})}$
$\displaystyle+$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|<N,|\alpha_{2}|=N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(v)\overline{(\partial^{\alpha_{1},0}k_{\sigma})_{\alpha_{2}}(x_{1},x_{2}v_{2}^{-1},v^{-1})}$
$\displaystyle+$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|<N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(v)\overline{(\partial^{0,\alpha_{2}}(k_{\sigma})_{\alpha_{1}})(x_{1}v_{1}^{-1},x_{2},v^{-1})}$
$\displaystyle+$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|=N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(v)\overline{(k_{\sigma})_{\alpha_{1},\alpha_{2}}(x_{1}v_{1}^{-1},x_{2}v_{2}^{-1},v^{-1})}.$
$\displaystyle=$ $\displaystyle I+II+III+IV.$

Now observe that for $II$ we have

$\displaystyle II$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|<N,|\alpha_{2}|=N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(v)\overline{(\partial^{\alpha_{1},0}k_{\sigma})_{\alpha_{2}}(x_{1},x_{2}v_{2}^{-1},v^{-1})}$
$\displaystyle=$
$\displaystyle\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}\Big{(}q^{\alpha_{1}}(v_{1})\overline{\partial^{\alpha_{1},0}k_{\sigma}(x_{1},x_{2}v_{2}^{-1},v^{-1})}-\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}q^{\alpha_{1},\alpha_{2}}(v)\overline{\partial^{\alpha_{1},\alpha_{2}}k_{\sigma}(x_{1},x_{2},v^{-1})}\Big{)},$

which shows that $II$ (by the calculus introduced in [MR-VT-Book_2010]) is the
kernel of the pseudodifferential operator with symbol

$\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}\Big{(}\Delta^{\alpha_{1},0}\partial^{\alpha_{1},0}\sigma^{*_{2}}(x,\xi)-\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{2}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}\Big{)}.$

For the term $III$ with similar arguments one concludes that $III$ is the
kernel of

$\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}\Big{(}\Delta^{0,\alpha_{2}}\partial^{0,\alpha_{2}}\sigma^{*_{1}}(x,\xi)-\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{2}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}\Big{)}.$

For the term $I$ it is immediate to see that it is the kernel of the operator
whose symbol is given by

$\sum_{\begin{subarray}{c}|\alpha_{1}|<N,|\alpha_{2}|<N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{1}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}.$

Therefore, putting together the properties above and rearranging the terms,
one gets

$\displaystyle\sigma^{*}(x,\xi)$ $\displaystyle\sim$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|=|\alpha_{2}|<N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{1}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{1}|<N}\frac{1}{\alpha_{1}!}\Big{(}\Delta^{\alpha_{1},0}\partial^{\alpha_{1},0}\sigma^{*_{1}}(x,\xi)-\sum_{|\alpha_{2}|\leq|\alpha_{1}|}\frac{1}{\alpha_{2}!}\Delta^{\alpha_{1},\alpha_{2}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}\Big{)}$
$\displaystyle+$
$\displaystyle\sum_{|\alpha_{2}|<N}\frac{1}{\alpha_{2}!}\Big{(}\Delta^{0,\alpha_{2}}\partial^{0,\alpha_{2}}\sigma^{*_{2}}(x,\xi)-\sum_{|\alpha_{1}|\leq|\alpha_{2}|}\frac{1}{\alpha_{1}!}\Delta^{\alpha_{1},\alpha_{2}}\partial^{\alpha_{1},\alpha_{2}}\sigma(x,\xi)^{*}\Big{)}$
$\displaystyle+$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha_{1}|=N,|\alpha_{2}|=N\end{subarray}}\frac{1}{\alpha_{1}!\alpha_{2}!}\int
q_{\alpha_{1},\alpha_{2}}(v)\overline{(k_{\sigma})_{\alpha_{1},\alpha_{2}}(x_{1}v_{1}^{-1},x_{2}v_{2}^{-1},v^{-1})}(\xi_{1}^{*}(v_{1})\otimes\xi^{*}_{2}(v_{2}))dv$
$\displaystyle=$
$\displaystyle\sum_{j<N}\left(d^{\prime}_{m_{1}-j,m_{2}-j}+d^{\prime\prime}_{m_{1}-j-1,m_{2}-j}+d^{\prime\prime\prime}_{m_{1}-j,m_{2}-j-1}\right)+r_{N}.$

In order to complete the proof it remains to show that $r_{N}\in
S^{m_{1}-N,m_{2}-N}(G\times\widehat{G})$ which follows by arguments similar to
those used in Theorem 5.1. This concludes the proof. ∎

######  Theorem 5.3 (Asymptotic expansion).

Let $\sigma_{j}$ be a sequence of symbols in
$S^{m^{\prime}_{j},m^{\prime\prime}_{j}}(G\times\widehat{G})$ with
$m^{\prime}_{j},m^{\prime\prime}_{j}$ decreasing to $-\infty$. Then there
exists $\sigma\in
S^{m^{\prime}_{0},m^{\prime\prime}_{0}}(G\times\widehat{G})$, unique modulo
$S^{-\infty,-\infty}$, such that

$\sigma-\sum_{j=0}^{M}\sigma_{j}\in
S^{m^{\prime}_{M+1},m^{\prime\prime}_{M+1}}(G\times\widehat{G}),\quad\forall
M\in\mathbb{N}.$ (5.7)

###### Proof.

Let $\psi\in C^{\infty}(\mathbb{R};[0,1])$ be such that $\psi\equiv 0$ on
$(-\infty,1/2)$ and $\psi\equiv 1$ on $(1,\infty)$. Then, by Proposition 2.8
and Proposition 4.1, we have that, for any given
$\tilde{m}_{1},\tilde{m}_{2}\in\mathbb{R}$,

$\|\Delta^{\alpha,\beta}\partial^{\gamma_{1},\gamma_{2}}\sigma_{j}(x,\xi)\psi(t_{1}\lambda_{\xi_{1}})\psi(t_{2}\lambda_{\xi_{2}})\|_{\mathscr{L}(\mathcal{H}_{\xi})}\hskip
142.26378pt$ $\displaystyle\lesssim$
$\displaystyle\sum_{\begin{subarray}{c}|\alpha|\leq|\alpha_{1}|+|\alpha_{2}|\leq
2|\alpha|\\\ |\beta|\leq|\beta_{1}|+|\beta_{2}|\leq
2|\beta|\end{subarray}}\|\Delta^{\alpha_{1},\beta_{1}}\partial^{\gamma_{1},\gamma_{2}}\sigma_{j}(x,\xi)\Delta^{\alpha_{2},\beta_{2}}\psi(t_{1}\lambda_{\xi_{1}})\psi(t_{2}\lambda_{\xi_{2}})\|_{\mathscr{L}(\mathcal{H}_{\xi})}$
$\displaystyle\lesssim$
$\displaystyle\|\sigma_{j}\|_{S^{m^{\prime}_{j},m^{\prime\prime}_{j}}_{(2\alpha,2\beta),\gamma}}\sum_{\begin{subarray}{c}|\alpha|\leq|\alpha_{1}|+|\alpha_{2}|\leq
2|\alpha|\\\ |\beta|\leq|\beta_{1}|+|\beta_{2}|\leq
2|\beta|\end{subarray}}\langle\xi_{1}\rangle^{m^{\prime}_{j}-|\alpha_{1}|}\langle\xi_{2}\rangle^{m^{\prime\prime}_{j}-|\alpha_{2}|}t_{1}^{\tilde{m}_{1}/2}\langle\xi_{1}\rangle^{\tilde{m}_{1}-|\alpha_{2}|}t_{2}^{\tilde{m}_{2}/2}\langle\xi_{2}\rangle^{\tilde{m}_{2}-|\beta_{2}|}.$

We then choose $\tilde{m}_{1}=m^{\prime}_{0}-m^{\prime}_{j}$ and
$\tilde{m}_{2}=m^{\prime\prime}_{0}-m^{\prime\prime}_{j}$ and get

$\|\Delta^{\alpha,\beta}\partial^{\gamma_{1},\gamma_{2}}\sigma_{j}(x,\xi)\psi(t_{1}\lambda_{\xi_{1}})\psi(t_{2}\lambda_{\xi_{2}})\|_{\mathscr{L}(\mathcal{H}_{\xi})}\lesssim\|\sigma_{j}\|_{S^{m^{\prime}_{j},m^{\prime\prime}_{j}}_{(|2\alpha|,|2\beta|),(|\gamma_{1}|,|\gamma_{2}|)}}t_{1}^{\frac{m^{\prime}_{0}-m^{\prime}_{j}}{2}}t_{2}^{\frac{m^{\prime\prime}_{0}-m^{\prime\prime}_{j}}{2}}\langle\xi_{1}\rangle^{m^{\prime}_{0}-|\alpha|}\langle\xi_{2}\rangle^{m^{\prime\prime}_{0}-|\beta|},$

which, in particular, gives that for any given
$a=(a_{1},a_{2})\in\mathbb{N}_{0}\times\mathbb{N}_{0}$ and
$b=(b_{1},b_{2})\in\mathbb{N}_{0}\times\mathbb{N}_{0}$,

$\|\sigma_{j}(x,\xi)\psi(t_{1}\lambda_{\xi_{1}})\psi(t_{2}\lambda_{\xi_{2}})\|_{S^{m^{\prime}_{0},m^{\prime\prime}_{0}}_{a,b}}\leq
C_{a,b,m^{\prime}_{0},m^{\prime\prime}_{0},\sigma_{j}}t_{1}^{\frac{m^{\prime}_{0}-m^{\prime}_{j}}{2}}t_{2}^{\frac{m^{\prime\prime}_{0}-m^{\prime\prime}_{j}}{2}}.$

We now choose a decreasing sequence $t_{j}$ such that

$t_{j}\in(0,2^{-j})\quad\text{and}\quad
C_{(j,j),(j,j),m^{\prime}_{0},m^{\prime\prime}_{0},\sigma_{j}}t_{j}^{\frac{m^{\prime}_{0}-m^{\prime}_{j}}{2}}t_{j}^{\frac{m^{\prime\prime}_{0}-m^{\prime\prime}_{j}}{2}}\leq
2^{-j},$

and define
$\tilde{\sigma}_{j}(x,\xi):=\sigma_{j}(x,\xi)\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})$.
By using the properties above we get, for all $\ell\in\mathbb{N}_{0}$,

$\sum_{j=0}^{\infty}\|\tilde{\sigma}_{j}\|_{S^{m^{\prime}_{0},m^{\prime\prime}_{0}}_{(\ell,\ell),(\ell,\ell)}}\leq\sum_{j=0}^{\ell}\|\tilde{\sigma}_{j}\|_{S^{m^{\prime}_{0},m^{\prime\prime}_{0}}_{(\ell,\ell),(\ell,\ell)}}+\sum_{j=\ell+1}^{\infty}2^{-j}<\infty,$

which implies that
$\sigma=\sum_{j=0}^{\infty}\sigma_{j}(x,\xi)\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})\in
S^{m^{\prime}_{0},m^{\prime\prime}_{0}}(G\times\widehat{G})$, and,
consequently, by taking the sum for $j\geq M$, also that
$\sum_{j=M}^{\infty}\sigma_{j}(x,\xi)\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})\in
S^{m^{\prime}_{M},m^{\prime\prime}_{M}}(G\times\widehat{G})$ for all
$M\in\mathbb{N}$. We then have that

$\displaystyle\sigma-\sum_{j=0}^{M-1}\sigma_{j}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\sigma_{j}(x,\xi)\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})-\sum_{j=0}^{M-1}\Big{(}1-\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})+\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})\Big{)}\sigma_{j}$
$\displaystyle=$
$\displaystyle-\sum_{j=0}^{M-1}\Big{(}1-\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})\Big{)}\sigma_{j}+\sum_{j=M}^{\infty}\tilde{\sigma}_{j}$

belongs to $S^{m^{\prime}_{M},m^{\prime\prime}_{M}}(G\times\widehat{G})$,
since, by Proposition 4.1,
$1-\psi(t_{j}\lambda_{\xi_{1}})\psi(t_{j}\lambda_{\xi_{2}})$ is smoothing. In
order to conclude the proof, we just have to show that $\sigma$ is unique up
to smoothing operators. This last property easily follows by observing that,
if $\tau$ is another symbol with the same asymptotic expansion as $\sigma$,
then, for any given $M\in\mathbb{N}$,

$\sigma-\tau=\Big{(}\sigma-\sum_{j=1}^{M-1}\sigma_{j}\Big{)}-\Big{(}\tau-\sum_{j=1}^{M-1}\sigma_{j}\Big{)}\in
S^{m^{\prime}_{M},m^{\prime\prime}_{M}}(G\times\widehat{G}),$

which, finally, shows that $\sigma=\tau$ modulo
$S^{-\infty,-\infty}(G\times\widehat{G})$ and proves the result. ∎

We will now introduce the definition of bielliptic operators and derive, for
these objects, the existence of biparametrices.

######  Definition 5.4.

Let $a\in S^{m_{1},m_{2}}(G\times\widehat{G})$ and $A=\mathrm{Op}(a)\in
L^{m_{1},m_{2}}(G)$. We say that $A$ is bielliptic if

  * (i)

$a(x,\xi)$ is invertible for all but finitely many $[\xi]\in\widehat{G}$ and,
for such $\xi$, its inverse $a(x,\xi)^{-1}$ satisfies

$\|a(x,\xi)^{-1}\|_{\mathscr{L}(H_{\xi})}\leq\langle\xi_{1}\rangle^{-m_{1}}\langle\xi_{2}\rangle^{-m_{2}};$

  * (ii)

$a(x_{1},x_{2},D_{1},\xi_{2})$ is exactly invertible as an operator in
$L^{m_{1}}(G_{1})$ for all $(x_{2},\xi_{2})\in G_{2}\times\widehat{G}_{2}$
with inverse in $L^{-m_{1}}(G_{1})$, and, in particular,

$(a\circ_{1}a^{-1})(x_{1},x_{2},D_{1},\xi_{2})=\mathrm{Id}_{\mathscr{D}^{\prime}(G_{1})};$

  * (iii)

$a(x_{1},x_{2},\xi_{1},D_{2})$ is exactly invertible as an operator in
$L^{m_{2}}(G_{2})$ for all $(x_{2},\xi_{2})\in G_{2}\times\widehat{G}_{2}$with
inverse in $L^{-m_{2}}(G_{2})$, and, in particular,

$(a\circ_{2}a^{-1})(x_{1},x_{2},\xi_{1},D_{2})=\mathrm{Id}_{\mathscr{D}^{\prime}(G_{2})}.$

######  Theorem 5.5.

Let $A\in L^{m_{1},m_{2}}(G)$ be bielliptic. Then there exists $B\in
L^{-m_{1},-m_{2}}(G)$ such that

$AB=I+K_{1},$ $BA=I+K_{2},$

where $I:=\mathrm{Id}_{\mathcal{D}^{\prime}(G)}$ is the identity map and
$K_{1},K_{2}$ are smoothing bisingular operators.

###### Proof.

We start with the proof of the first assertion, namely, the existence of $B$
such that $AB=I+K_{1},$ with $K_{1}$ smoothing.

First observe that, by definition of biellipticity, one has that $a^{-1}\in
S^{-m_{1},-m_{2}}(G\times\widehat{G})$. Then, by taking
$b_{0}(x,\xi)=a(x,\xi)^{-1}$ and by using the asymptotic composition formula
together with $(ii)$ and $(iii)$ of Definition 5.4, we have that
$a\\#b_{0}=\boldsymbol{1}-r_{1}$, with $r_{1}\in
S^{-1,-1}(G\times\widehat{G})$ and
$\boldsymbol{1}(\xi)=I_{\mathbb{C}^{d_{\xi}}}$. We now define
$b_{j}:=b_{0}\\#r_{j}$, with $r_{j}:=r_{1}\\#r_{j-1}\in
S^{-j,-j}(G\times\widehat{G})$ for $j\geq 2$, and have
$a\\#b_{j}=(\boldsymbol{1}-r_{1})\\#r_{j}$. Then, for $b\sim\sum_{j\geq
0}b_{j}$, we obtain, for any $k\in\mathbb{N}$,

$a\\#\sum_{j<k}b_{j}=(\boldsymbol{1}-r_{1})\\#\Big{(}\boldsymbol{1}+\sum_{0<j<k}r_{j}\Big{)}$
$=\boldsymbol{1}+\sum_{0<j<k}r_{j}-r_{1}-r_{1}+r_{1}\\#\sum_{0<j<k}r_{j}=\boldsymbol{1}-r_{k},$

where, recall, $r_{k}\in S^{-k,-k}(G\times\widehat{G})$. This, finally, gives
that

$a\\#b-\boldsymbol{1}\in S^{-\infty,\infty}(G\times\widehat{G}),$

which proves the first assertion.

In order to prove the existence of a left parametrix $B$, that is such that
$BA=I+K_{2}$, with $K_{2}$ smoothing, one proceeds as before, namely, one
takes $b_{0}=a^{-1}$ and defines $b_{0}\\#a-\boldsymbol{1}=-s_{1}\in
S^{-1,-1}(G\times\widehat{G})$ and $s_{j}:=s_{j-1}\\#s_{1}$ for all $j\geq 2$.
Then, taking $b_{j}:=s_{j}\\#b_{0}$, the result follows for $b\sim\sum_{j\geq
0}b_{j}$. This concludes the proof. ∎

## Appendix A

######  Lemma A.1.

Let $G=G_{1}\times G_{2}$ be a compact Lie group, with $G_{i}$, $i=1,2,$ being
a compact Lie group of dimension $n_{i}=\mathrm{dim}(G_{i})$, and let also
$q\in\mathcal{D}(G)$ and $a_{1},a_{2}\in\mathbb{N}$. Then the fallowing
properties are equivalent

  1. 1.

For all
$(\alpha_{1},\alpha_{2})\in\mathbb{N}^{n_{1}}\times\mathbb{N}^{n_{2}}$, with
$|\alpha_{i}|<a_{i}$, then $\partial^{\alpha_{1},\alpha_{2}}q(e_{G})=0$, that
is, $q$ vanishes of order $(a_{1}-1,a_{2}-1)$ at $e_{G}$.

  2. 2.

For any given differential operator
$D^{k_{1},k_{2}}:=D_{1}^{k_{1}}D_{2}^{k_{2}}\in\mathrm{Diff}^{k_{1}+k_{2}}(G)$,
$D_{j}^{k_{j}}\in\mathrm{Diff}^{k_{j}}(G_{j})$, such that $k_{i}<a_{i}$, we
have $D^{k_{1},k_{2}}q(e_{G})=0$.

  3. 3.

There exists a constant $C_{q}$ such that, for all $x\in G$, we have $q(x)\leq
C_{q}|x_{1}|^{a_{1}}|x_{2}|^{a_{2}}$.

Lemma A.1, whose proof is left to the reader, gives a notion of vanishing
order of a function suitable in our setting, where, in particular, the
vanishing order with respect to each variable is considered. For the standard
(non adapted to the bisingular case) notion of vanishing orderer of a function
see Lemma A.1 in [Fischer-JFA_2015].

######  Proposition A.2.

Let $m_{1},m_{2}\in\mathbb{R}$ and $a_{1},a_{2}\in\mathbb{N}$. For any given
function $q\in\mathcal{D}(G)$ vanishing of order $(a_{1}-1,a_{2}-1)$ at
$e_{G}$, there exists $d_{1},d_{2}\in\mathbb{N}_{0}$ such that, for all $f\in
C^{d_{1}}([0,+\infty);C^{d_{2}}[0,+\infty))$ satisfying

$\|f\|_{\mathcal{M}_{m_{1},m_{2},d_{1},d_{2}}}:=\sup_{\lambda_{1},\lambda_{2}\geq
0,\ell_{1}=0,...,d_{1},\ell_{2}=0,...,d_{2}}(1+\lambda_{1})^{-m_{1}+\ell_{1}}(1+\lambda_{2})^{-m_{2}+\ell_{2}}|\partial^{\ell_{1}}_{\lambda_{1}}\partial^{\ell_{2}}_{\lambda_{2}}f(\lambda_{1},\lambda_{2})|<\infty,$

we have

$\|\triangle_{q}{f(t_{1}\lambda_{\xi_{1}},t_{2}\lambda_{\xi_{2}})}\|_{\mathcal{L}(\mathcal{H}_{\pi})}\leq
Ct_{1}^{m_{1}/2}t_{2}^{m_{2}/2}(1+\lambda_{\xi_{1}})^{\frac{m_{1}-a_{1}}{2}}(1+\lambda_{\xi_{2}})^{\frac{m_{2}-a_{2}}{2}},\quad\forall\pi\in\widehat{G},\quad
t_{1},t_{2}\in(0,1).$

The constant $C$ may be chosen as
$C^{\prime}\|f\|_{\mathcal{M}_{m_{1},m_{2},d_{1},d_{2}}}$, with
$C^{\prime}=C^{\prime}(m_{1},m_{2},q,a_{1},a_{2})$ also depending on the group
$G$ but not on $f,t_{1},t_{2}$ and $\xi=\xi_{1}\otimes\xi_{2}$.

The proof of the proposition is done following that of [Fischer-JFA_2015] and
is also left to the reader.

######  Lemma A.3.

Let $G=G_{1}\times G_{2}$ be such that $\mathrm{dim}(G_{1})=n_{1}$. If
$s_{1}>n_{1}/2,s_{2}>n_{2}/2$, then the kernel $\mathcal{B}_{s_{1},s_{2}}$ of
the operator $(I_{1}+L_{G_{1}})^{-s_{1}/2}\otimes(I_{2}+L_{G_{2}})^{-s_{2}/2}$
is square integrable and the continuous inclusion $H^{s_{1},s_{2}}(G)\subset
C(G)$ holds.

###### Sketch of the proof of Lemma A.3.

Notice that

$B_{s_{2},s_{2}}(x,y)=B_{s_{1}}(x_{1},y_{1})\otimes B_{s_{2}}(x_{2},y_{2})$

where $B_{s_{j}}(x_{j},y_{j})$, defined on $G_{j}\times G_{j}$, is the kernel
of the operator $(I_{j}+L_{G_{j}})^{-s_{j}}$, $j=1,2$. Then (see Lemma A.5 in
[Fischer-JFA_2015]), we have

$B_{s_{j}}=\frac{1}{\Gamma(s_{j}/2)}\int_{t_{j}=0}^{\infty}t_{j}^{s_{j}/2-1}e^{-t_{j}}p^{(j)}_{t_{j}}dt_{j},$

where

$p^{(j)}_{t_{j}}:=e^{-t_{j}\Delta_{j}}\delta_{e_{G_{j}}},\quad t_{j}>0,$

and $\Gamma$ is the gamma function. Since (see Lemma A.5 in [Fischer-
JFA_2015]) for $s_{j}>n_{j}/2$

$\|B_{s_{j}}\|_{L^{2}(G_{j})}<\infty,\quad j=1,2,$

we have

$\|B_{s}\|_{L^{2}(G)}=\|B_{s_{1}}\|_{L^{2}(G_{1})}\|B_{s_{2}}\|_{L^{2}(G_{2})}<\infty.$

Finally the Sobolev embedding will follow from the fact that one can write $f$
as

$f=\\{\left((I_{1}+L_{G_{1}})^{-s_{1}/2}\otimes(I_{2}+L_{G_{2}})^{-s_{2}/2}\right)f\\}\ast
B_{s},$

for all $f\in H^{s_{1},s_{2}}(G)$ with $s_{1}>n_{1}/2$ and $s_{2}>n_{2}/2$. ∎

## References