# Toolbox of Para-differential Calculus on Compact Lie Groups

Chengyang Shao

###### Abstract.

This paper provides a toolbox of para-differential calculus on compact Lie
groups. The toolbox is based on representation theory of compact Lie groups
and contains exact formulas of symbolic calculus. Para-differential operators
are constructed in a global, coordinate-free manner, giving lower order terms
in symbolic calculus a clear form. The toolbox helps to understand non-local,
nonlinear differential operators defined on certain manifolds with high
symmetry.

##  1\. Introduction

In this paper, we will develop a toolbox of harmonic analysis necessary for
the study of motion of a water droplet under zero gravity. This is the
starting point of a project proposed by the author in [Sha22] to understand
nonlinear oscillation of water droplets. The paper is split from Section 1-4
of [Sha23a]. The application of this toolbox to the Cauchy problem of
oscillating water droplet can be found in [Sha23a].

###  1.1. Spectral Localization and Dispersive PDEs on Compact Manifolds

The study of nonlinear dispersive partial differential equations and harmonic
analysis are closely linked to each other and mutually reinforcing. If
discussed on a compact manifold, the geometry of the underlying space shall
also come on the scene. We will be exploring an aspect of this topic in this
paper. In order to extract the key ideas out of several important examples, we
start our discussion by reviewing some previous results to illustrate the
intersectionality of harmonic analysis on compact manifolds and nonlinear
dispersive PDEs.

In the pioneering work [BGT04], Burq, Gérad and Tzvetkov obtained Strichartz
estimates for the unitary group $e^{it\Delta_{g}}$ on a compact Riemannian
manifold $(M,g)$, and discussed its application to the Cauchy problem of semi-
linear Schrödinger equation on compact Riemannian manifold. The discussion
relies on “$L^{p}$ decoupling of Laplace eigenfunctions”, discovered by Sogge
[Sog88]. As a result, they showed that on a compact Riemann surface $(M,g)$,
the Cauchy problem of the cubic Schrödinger equation

$i\partial_{t}u+\Delta_{g}u=|u|^{2}u,\quad u(0)\in H^{s}(M)$

with $s>1/2$ is locally well-posed in $C^{0}_{t}H^{s}(M)$. They further showed
in [BGT05] that the regime of local well-posedness can be extended to $s>1/4$
if $(M,g)$ is a Zoll surface (in particular, if it is the standard 2-sphere).
A crucial harmonic analysis result needed by this is a bilinear eigenfunction
estimate on Zoll surfaces, an inequality bounding the $L^{2}$ norm of the
product of two eigenfunctions.

Hani continued the study in [Han12], showing that the Cauchy problem of the
cubic Schrödinger equation on $(M,g)$ is globally well-posed for $s>2/3$,
regardless of the geometry. The proof is a modification of the “I-method” due
to Colliander-Keel-Staffilani-Takaoka-Tao (see [CKS+02]). When transplanting
the I-method from $\mathbb{R}^{n}$ or $\mathbb{T}^{n}$ to a general manifold,
an immediate issue appears: one replaces Fourier analysis by spectral
decomposition using Laplace eigenfunctions, but unlike Fourier modes, the
product of two Laplace eigenfunctions is, in general, not an eigenfunction
anymore. In order to overcome this difficulty, Hani established _spectral
licalization property_ as a substitute. That is, the product of two
eigenfunctions with eigenvalues $\lambda$ and $\mu$ respectively should
sharply concentrate around the range
$\sqrt{-\Delta_{g}}\in\big{[}|\sqrt{\lambda}-\sqrt{\mu}|,\sqrt{\lambda}+\sqrt{\mu}\big{]}$.
This turns out to be an important ingredient for the proof.

Not surprisingly, the spectral localization property could be improved if
certain geometric conditions are posed for the underlying manifold. Such
improvement should then lead to stronger results for nonlinear dispersive
PDEs. In fact, this is the case for compact Lie groups and homogeneous spaces.
Highest weight theory for representations of a compact Lie group implies the
following: the product of two Laplace eigenfunctions on a compact Lie group is
a _finite_ linear combination of Laplace eigenfunctions. In [BP11] and [BCP15]
(see also [BBP10]), precise form of this algebraic result plays a fundamental
role for the search of periodic or quasi-periodic solutions of the semi-linear
Schrödinger equation

$i\partial_{t}u+\Delta u+\mu u=f(x,|u|^{2})u$

on a compact Lie group or homogeneous space. Another example is [DS04], where
the underlying manifold is the standard sphere $\mathbb{S}^{d}$. The
aforementioned spectral localization on Lie groups then becomes an algebraic
property of spherical harmonic functions: the product of spherical harmonics
of degree $p$ and $q$ is a linear combination of spherical harmonics of degree
between $|p-q|$ and $p+q$. Using this fact, the authors casted a normal form
reduction for semi-linear Klein-Gordon equations on $\mathbb{S}^{d}$.

Furthermore, the spectral localization property plays a fundamental role in
the theory of pseudo-differential operators on compact Lie groups.
Construction of pseudo-differential calculus on compact Lie groups depends
very much on representation theory of compact Lie groups. Being the non-
commutative analogy of Fourier theory on $\mathbb{R}^{n}$ or $\mathbb{T}^{n}$,
spectral localization shows how matrix elements of different representations
(“modes of different frequencies”) interact. It thus occupies a similar part
as the additive formula $e^{i\xi_{1}\cdot x}e^{i\xi_{2}\cdot
x}=e^{i(\xi_{1}+\xi_{2})\cdot x}$ does in commutative Fourier analysis and
symbolic calculus.

Global symbolic calculus on compact Lie groups has been proposed by Ruzhansky-
Turunen-Wirth [RT09]-[RTW14], while the ideas seem to date back to Taylor
[Tay86]. However, as pointed out by Fischer [Fis15], the pseudo-differential
calculus in [RT09]-[RTW14] may lack rigorous justification. Fischer’s work
[Fis15] provides a mathematically rigorous and complete construction in which
the symbolic calculus formulas are proved. In particular, Fischer proved that
pseudo-differential operators of type $(1,0)$ defined via global symbols
indeed coincides with the Hörmander class of $(1,0)$ pseudo-differential
operators defined via local coordinates. To the author’s knowledge, the first
result on _linear_ dispersive PDEs on a compact Lie group with essential usage
of global symbolic calculus is Bambusi-Langella [BL22]. The paper states a
general frame work concerning “abstract pseudo-differential operators” and
obtained a growth estimate of the solution to _linear_ Schrödinger equation
with possibly time-dependent potential. Taking into account the pseudo-
differential calculus constructed by [Fis15], linear Schrödinger equations on
a compact Lie group thus realizes this abstract framework.

###  1.2. Para-differential Calculus on Manifolds

When coming to quasi-linear dispersive PDEs, however, some more technicalities
beyond spectral localization will be necessarily required. Within our scope,
we shall focus on one of them, namely para-differential calculus.

Para-differential calculus originates from a systematic application of
Littlewood-Paley decomposition to nonlinear PDEs. It first appeared in the
form of _para-product estimate_ , leading to inequalities of the form
$\|fg\|_{H^{s}}\lesssim|f|_{L^{\infty}}\|g\|_{H^{s}}+\|f\|_{H^{s}}|g|_{L^{\infty}}$
or composition estimates of the form $\|F(u)\|_{H^{s}}\lesssim\|u\|_{H^{s}}$.
Such inequalities were crucial for local well-posedness of semi-linear
dispersive PDEs; see for example, Saut-Temam [ST76] Linares-Ponce [LP14] for
discussion of KdV equation, or Kato [Kat95] and Staffilani [Sta95] for
discussion of semi-linear Schrödinger equation. General estimates may also be
found in [Tay00].

The defining character of such methodology was extracted as _isolating the
part with lowest regularity_. It has been extended to a class of operators
called _para-differential operators_ ; see Bony [Bon81] and Meyer
[Mey80]-[Mey81]). Modelled on symbols $a(x,\xi)$ that are of limited
regularity with respect to the spatial variable $x$, a symbolic calculus is
still available for para-differential operators. Para-differential calculus
thus becomes a powerful tool when usual pseudo-differential calculus is of
limited usage due to low regularity.

Para-differential calculus has been successfully used in the study of quasi-
linear PDEs (and also micro-local properties for fully nonlinear ones, see
[Bon81] or Hörmander [Hör97]; but we do not discuss that aspect). Taking fluid
dynamics equations as an example, the gravity-capillary water waves equation
was re-written into a para-differential form by Alazard-Burq-Zuilly [ABZ11],
and a local well-posedness result in the regime $s>3.5$ was deduced. A
pleasant feature of this para-differential form is that the spectral
properties of the dispersive relation for the problem are explicit. Not only
is it an improvement of previous local well-posedness results (see for example
[Wu97]-[Wu99], [Lan05], [CS07], [SZ08], [MZ09]), it also reduces the water
waves equation into a form convenient for the study of long time behaviour.
See the review [IP18] for detailed discussion, or [BD18] [DIPP17] [IP19] as
examples of applying this para-differential form.

Being such a powerful tool, it is natural to ask whether para-differential
calculus admits generalization to curved manifolds instead of $\mathbb{R}^{n}$
or $\mathbb{T}^{n}$. Let us briefly summarize previous attempts towards it.

In [KR06], Klainerman and Rodnianski extended the Littlewood-Paley theory to
compact surfaces. They defined Littlewood-Paley decomposition of a tensor
field via spectral cut-off of the tensorial Laplacian operator, studied the
interaction of different frequencies, and deduced Bernstein type inequalities.
This served as a toolbox in proving the $L^{2}$ bounded curvature conjecture
for the Einstein equation. See [KRS15] for the details. The argument actually
does not require para-differential calculus, but only an invariant version of
Littlewood-Paley decomposition on compact surfaces. Not surprisingly, these
properties can be reproduced and improved for compact Lie groups (see
[Fis15]).

In [BGdP21], Bonthonneau, Guillarmou and de Poyferré developed a para-
differential calculus for any compact manifold via local coordinate charts.
This approach is a natural generalization of the Hörmander calculus on a
curved manifold $M$: a $(1,1)$ symbol $a(x,\xi)$ is defined on the cotangent
bundle of $M$, as a function that becomes a $(1,1)$ symbol under any local
chart; para-differential operators corresponding to rough symbols are also
defined by coordinate charts. A key feature is that symbolic calculus, just as
the usual Hörmander calculus does, only preserves information at the order of
principle symbol. The authors used this para-differential calculus to study
microlocal regularity for an Anosov flow.

On the other hand, a global definition of para-differential operators on a
compact manifold is introduced in Delort [Del15]. Observing the commutator
characterization of usual pseudo-differential operators (see e.g. Beals
[Bea77]), Delort defined para-differential operators on a compact manifold via
commutators with vector fields and Laplacian spectral projections. The most
important result from this para-differential calculus is that, the commutator
of para-differential operators of order $m$ and $m^{\prime}$ is an operator of
$m+m^{\prime}-1$. With these preparations, Delort successfully obtained a
normal form reduction for quasi-linear Hamiltonian Klein-Gordon equations on
the standard sphere $\mathbb{S}^{d}$, and thus obtained extended lifespan
estimate beyond the energy estimate regime.

Both constructions in [BGdP21] and [Del15] share the common feature that
formulas of symbolic calculus for para-differential operators involve only the
_principal symbol_ , just as the usual Hörmander calculus on curved manifolds.
Consequently, if one applies either one to a quasi-linear dispersive PDE, much
of the spectral information related to the dispersive relation becomes
implicit, although not lost. Besides, lower order information apart from the
principal symbol is blurred. These disadvantages might cause extra difficulty
in studying the long-time behavior of the solution.

###  1.3. Organization of the Paper

Our goal is to develop an analytic toolbox for PDEs on compact Lie groups and
homogeneous spaces. In particular, the toolbox should enable us to reduce the
spherical capillary water waves system in [Sha22] to a para-differential form
convenient for the study of its long-time behaviour. Observing the advantages
of incorporating geometric information, our goal will be accomplished by
employing the notion of symbols in [Fis15] as the narrative formalism for
para-differential calculus.

Thus we transplant the sequence of ideas for para-differential calculus on
$\mathbb{R}^{n}$ to a compact Lie group $\mathbb{G}$ (see for example Chapter
8-10 of [Hör97]). The Fourier modes $e^{i\xi\cdot x}$ are replaced by
irreducible unitary representations $\xi$, so that a symbol $a(x,\xi)$ is a
field acting on the representation space of $\xi$. Due to unbounded degree of
degeneracy of Laplace eigenfunctions (see the Weyl type lemma 2.4), it is
necessary to consider symbols as _matrices_ acting on representation spaces
instead of _scalar-valued functions_.

We start by defining symbols and symbol classes on $\mathbb{G}$ in Section 2,
then Littlewood-Paley decomposition in Section 3. Littlewood-Paley
characterization for Sobolev and Zygmund space will be given here. The Stein
theorem 3.3 for pseudo-differential operators of type $(1,1)$ will be proved
in Section 3. The spectral condition is introduced in Section 3, and Theorem
3.8 establishes the arbitrary $H^{s}$ boundedness for operators satisfying a
spectral condition. Para-product estimates and Bony’s para-linearization
theorem are deduced as corollaries. In Section 4, para-differential operators
are defined corresponding to rough symbols $a(x,\xi)$ with $C^{r}_{*}$
regularity in $x$. Formulas for symbolic calculus, i.e. composition and
adjoint formulas are then proved for para-differential operators in Theorem
4.1-4.5. That the commutator of para-differential operators of order $m$ and
$m^{\prime}$ is of $m+m^{\prime}-1$ follows as a direct consequence.

None of the steps listed above are trivial. Let us discuss the major
difficulties. As mentioned before, unlike Fourier modes on commutative groups,
the product of two Laplace eigenfunctions on a non-commutative Lie group can
_never_ be a Laplace eigenfunction. Furthermore, the dual of $\mathbb{G}$ does
not form a group anymore, so the definition of “differentiation” for a symbol
$a(x,\xi)$ with respect to $\xi$ requires extra effort; see Section 2 for the
details. The spectral localization property for eigenfunctions on $\mathbb{G}$
will serve as the substitute for $e^{i\xi_{1}\cdot x}e^{i\xi_{2}\cdot
x}=e^{i(\xi_{1}+\xi_{2})\cdot x}$. In order to prove arbitrary $H^{s}$
boundedness for para-differential operators, we actually need a _lower
spectral bound_ of the product. Corollary 2.3.1, being stronger than Hani’s
general result [Han12] in this special geometry, never appeared in previous
literature on representation theory to the author’s knowledge. It explicitly
describes such bounds. The role it plays in establishing arbitrary $H^{s}$
boundedness for para-differential operators is clearly indicated in the proof
of Theorem 3.8.

Additionally, functions of symbols thus cannot be manipulated so easily as in
symbolic calculus on $\mathbb{R}^{n}$. We introduce the notion of _quasi-
homogeneous symbols_ in Definition 4.5. They are essentially functions of
classical differential symbols, hence still enjoys, to some extent, the good
properties of the latter. It seems that this class of quasi-homogeneous
symbols is broad enough to cover operators arising from PDEs that we are
concerned with.

We emphasize that the algebraic structure of $\mathbb{G}$ is fundamental
throughout our para-differential calculus. It enables us to write down the
precise form of composition and adjoint operators. To this extent, this paper,
together with the up-coming one (which will be the second split from
[Sha23a]), provide a novel formalism for non-local, _quasi-linear_ dispersive
partial differential equations defined on certain manifolds with high
symmetry. It can be expected that the formalism may be applied to other quasi-
linear dispersive problems besides the spherical capillary waves equation.

##  2\. Preliminary: Representation of Compact Lie Groups

This section is devoted to preliminaries of representation theory of compact
Lie groups. Notation will also be fixed in this section. Results from
representation theory can be found in any standard textbook on representation
theory, for example [Pro07]. Concise descriptions can also be found in [BP11].

###  2.1. Representation and Spectral Properties

To simplify our discussion, throughout this paper, we fix $\mathbb{G}$ to be a
simply connected, compact, semisimple Lie group with dimension $n$ and rank
$\varrho$, i.e. the dimension of any maximal torus in $\mathbb{G}$. By
standard classification results of compact connected Lie groups, this is the
essential part for our discussion. It is known from standard theory of Cartan
subalgebra that all maximal tori in $\mathbb{G}$ are conjugate to each other,
so the rank $\varrho$ is uniquely determined. The group $\mathbb{G}$ is in
fact a direct product of groups of the following types: the special unitary
groups $\mathrm{SU}(n)$, the compact symplectic groups
$\mathrm{Sp}(n)=\mathrm{Sp}(2n;\mathbb{C})\cap\mathrm{U}(2n)$, the spin groups
$\mathrm{Spin}(n)$, and the exceptional groups
$G_{2},F_{4},E_{6},E_{7},E_{8}$.

We write $e$ for the identity element of $\mathbb{G}$. Let $\mathfrak{g}$ be
the Lie algebra of $\mathbb{G}$, identified with the space of left-invariant
vector fields on $\mathbb{G}$. If a basis $X_{1},\cdots,X_{n}$ for
$\mathfrak{g}$ is given, then for any multi-index
$\alpha\in\mathbb{N}_{0}^{n}$, write

(2.1) $X^{\alpha}=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$

for the left-invariant differential operator composed by these basis vectors,
in the “normal” ordering. When $\alpha$ runs through all multi-indices, the
operators $\\{X^{\alpha}\\}$ span the universal enveloping algebra of
$\mathfrak{g}$, which is the content of the Poincaré–Birkhoff–Witt theorem;
however, this fact is not needed within our scope.

Since $\mathbb{G}$ is semi-simple and compact, the Killing form

$\operatorname{Tr}\big{(}\mathrm{Ad}(X)\circ\mathrm{Ad}(Y)\big{)},\quad
X,Y\in\mathfrak{g}$

on $\mathbb{G}$ is non-degenerate and negative definite, where $\mathrm{Ad}$
is the adjoint representation. A Riemann metric on $\mathbb{G}$ can thus be
introduced as the opposite of the Killing form. The Laplace operator
corresponding to this metric, usually called _Casimir element_ in
representation theory, is given by

$\Delta=\sum_{i=1}^{n}X_{i}^{2},$

where $\\{X_{i}\\}$ is any orthonormal basis of $\mathfrak{g}$; the operator
is independent from the choice of basis.

Let $\widehat{\mathbb{G}}$ be the dual of $\mathbb{G}$, i.e. the set of
equivalence classes of irreducible unitary representations of $\mathbb{G}$.
For simplicity, we do not distinguish between an equivalence class
$\xi\in\widehat{\mathbb{G}}$ and a certain unitary representation that
realizes it. The ambient space of $\xi\in\widehat{\mathbb{G}}$ is an Hermite
space $\mathcal{H}_{[}\xi]$, with (complex) dimension $d_{\xi}$. If we equip
the space $\mathcal{H}_{[}\xi]$ with an orthonormal basis containing $d_{\xi}$
elements, then $\xi:\mathbb{G}\to\mathrm{U}(\mathcal{H}_{[}\xi])$ can be
equivalently viewed as a unitary-matrix-valued function
$(\xi_{jk})_{j,k=1}^{d_{\xi}}$.

We shall equip the Lie group $\mathbb{G}$ with the normalized Haar measure.
For simplicity, we will denote the integration with respect to this measure by
$dx$. The _Fourier transform_ of $f\in\mathcal{D}^{\prime}(\mathbb{G})$ is
defined by

$\widehat{f}(\xi):=\int_{\mathbb{G}}f(x)\xi^{*}(x)dx\in\mathrm{End}(\mathcal{H}_{[}\xi]).$

Conversely, for every field $a(\xi)$ on $\widehat{\mathbb{G}}$ such that
$a(\xi)\in\mathrm{End}(\mathcal{H}_{[}\xi])$ for all
$\xi\in\widehat{\mathbb{G}}$, the _Fourier inversion_ of $a$ is defined by

$\check{a}(x)=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\operatorname{Tr}\big{(}a(\xi)\cdot\xi(x)\big{)},$

as long as the right-hand-side converges at least in the sense of
distribution.

Two different norms for $a(\xi)\in\mathrm{End}(\mathcal{H}_{[}\xi])$ will be
used in this paper, one being the operator norm

$\|a(\xi)\|:=\sup_{v\in\mathcal{H}_{[}\xi],v\neq 0}\frac{|a(\xi)v|}{|v|},$

one being the Hilbert-Schmidt norm

$\llbracket a(\xi)\rrbracket:=\sqrt{\operatorname{Tr}(a(\xi)\cdot
a^{*}(\xi))}.$

To simplify notation, when there is no risk of confusion, we shall omit the
dependence of these norms on the representation $\xi$.

We fix the convolution on $\mathbb{G}$ to be _right_ convolution:

$(f*g)(x)=\int_{\mathbb{G}}f(y)g(y^{-1}x)dy.$

The only property not inherited from the usual convolution on Euclidean spaces
is commutativity. Every other property, including the convolution-product
duality, Young’s inequality, is still valid. The only modification is that

$\widehat{f*g}(\xi)=(\widehat{g}\cdot\widehat{f})(\xi),$

which in general does not coincide with $(\widehat{f}\cdot\widehat{g})(\xi)$
since the Fourier transform is now a matrix.

The Peter-Weyl theorem is fundamental for harmonic analysis on Lie groups:

######  Theorem 2.1 (Peter-Weyl).

Let $\mathcal{M}_{\xi}$ be the subspace in $L^{2}(\mathbb{G})$ spanned by
matrix entries of the representation $\xi\in\widehat{\mathbb{G}}$. Then
$\mathcal{M}_{\xi}$ is a bi-invariant subspace of $L^{2}(\mathbb{G})$ of
dimension $d_{\xi}^{2}$, and there is a Hilbert space decomposition

$L^{2}(\mathbb{G})=\bigoplus_{\xi\in\widehat{\mathbb{G}}}\mathcal{M}_{\xi}.$

If $f\in L^{2}(\mathbb{G})$, the Fourier inversion

$(\widehat{f})^{\vee}(x)=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\operatorname{Tr}\left(\widehat{f}(\xi)\cdot\xi(x)\right)$

converges to $f$ in the $L^{2}$ norm, and there holds the Plancherel identity

$\|f\|_{L^{2}(\mathbb{G})}^{2}=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\big{\llbracket}\widehat{f}(\xi)\big{\rrbracket}^{2}.$

In the following, standard constructions and results in highest weight theory
will be directly cited. They are already collected in Berti-Procesi [BP11].

Let $\mathfrak{t}\subset\mathfrak{g}$ be the Lie algebra of a maximal torus of
$\mathbb{G}$, and write $\mathfrak{t}^{*}$ for its dual. Let
$\\{\bm{\alpha}_{i}\\}_{i=1}^{\varrho}\subset\mathfrak{t}^{*}$ be the set of
_positive simple roots_ of $\mathfrak{g}$ with respect to $\mathfrak{t}$. They
form a basis for $\mathfrak{t}^{*}$. The _fundamental weights_
$\bm{\varpi}_{1},\cdots,\bm{\varpi}_{\varrho}\in\mathfrak{t}^{*}$ admit
several equivalent definitions, one of them being the unique set of vectors
$\\{\bm{\varpi}_{i}\\}_{i=1}^{\varrho}$ satisfying

$(\bm{\varpi}_{i},\bm{\alpha}_{j})=\frac{1}{2}\delta_{ij}|\bm{\alpha}_{j}|^{2},\quad
i,j=1,\cdots,\varrho.$

Here the inner product on the dual is inherited from the Killing form on
$\mathfrak{g}$.

######  Theorem 2.2 (Highest Weight Theorem).

Irreducible unitary representations of $\mathbb{G}$ are in 1-1 correspondence
with the discrete cone

$\mathscr{L}^{+}:=\left\\{\sum_{i=1}^{\varrho}n_{i}\bm{\varpi}_{i},\,n_{i}\in\mathbb{N}_{0}\right\\}.$

The correspondence assigns an irreducible unitary representation to its
highest weight vector.

In particular, the trivial representation corresponds to the point
$0\in\mathscr{L}^{+}$. From now on, we shall denote the assignment between
representations $\xi\in\widehat{\mathbb{G}}$ and highest weights
$\bm{j}\in\mathscr{L}^{+}$ by $\bm{j}=\bm{J}(\xi)$, or $\xi=\Xi(\bm{j})$. The
highest weight theorem implies a complete description of the spectrum of the
Laplace operator $\Delta$ of $\mathbb{G}$.

######  Theorem 2.3.

Each space $\mathcal{M}_{\xi}$ is an eigenspace of the Laplace operator
$\Delta$, with eigenvalue

$-|\bm{J}(\xi)+\bm{\varpi}|^{2}+|\bm{\varpi}|^{2},$

where $\bm{\varpi}=\sum_{i=1}^{\varrho}\bm{\varpi}_{i}$.

We write $\lambda_{\xi}=|\bm{J}(\xi)+\bm{\varpi}|^{2}-|\bm{\varpi}|^{2}$, so
that $\\{\lambda_{\xi}\\}$ is the spectrum of the positive self-adjoint
operator $-\Delta$. Imitating commutative harmonic analysis, we set the size
functions $|\xi|:=\sqrt{|\lambda_{\xi}|}$,
$\langle{[}\rangle\xi]:=(1+\lambda_{\xi})^{1/2}$. The operator $|\nabla|$,
which is the square-root of $-\Delta$ defined via spectral analysis, will be
frequently used. The values of $\langle{[}\rangle\xi]$ are exactly the
eigenvalues of $(1-\Delta)^{1/2}$. We will also use the following notation:

(2.2)
$\mathfrak{S}[\gamma_{1},\gamma_{2}]:=\left\\{\xi\in\widehat{\mathbb{G}}:\gamma_{1}\leq|\xi|\leq\gamma_{2}\right\\}.$

For $s\in\mathbb{R}$, the Sobolev space $H^{s}(\mathbb{G})$ is defined to be
the subspace of $f\in\mathcal{D}^{\prime}(\mathbb{G})$ such that
$\|(1-\Delta)^{s/2}f\|_{L^{2}(\mathbb{G})}<\infty$. By the Peter-Weyl theorem
and the above characterization of Laplacian, the Sobolev norm is computed by

$\|f\|_{H^{s}(\mathbb{G})}^{2}=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\langle{[}\rangle\xi]^{2s}\big{\llbracket}\widehat{f}(\xi)\big{\rrbracket}^{2}.$

To proceed further, we need to introduce a notion of partial ordering on
$\mathscr{L}^{+}$. Define the cone

$\mathscr{R}^{+}:=\left\\{\sum_{i=1}^{\varrho}n_{i}\bm{\alpha}_{i},\,n_{i}\in\mathbb{N}_{0}\right\\}.$

We then introduce the _dominance order_ on $\mathscr{L}^{+}$ as follows: for
$\bm{j},\bm{k}\in\mathscr{L}^{+}$, we set $\bm{j}\prec\bm{k}$ if
$\bm{j}\neq\bm{k}$ and $\bm{k}-\bm{j}\in\mathscr{R}^{+}$, and set
$\bm{j}\preceq\bm{k}$ to include the possibility $\bm{j}=\bm{k}$. The relation
$\prec$ defines a partial ordering on $\mathscr{L}^{+}$. With the aid of this
root system, we obtain the following characterization of product of Laplace
eigenfunctions (see, for example, [Pro07], Proposition 3 of page 345):

######  Proposition 2.1.

Let $u_{\bm{j}}\in\mathcal{M}_{\Xi(\bm{j})}$,
$u_{\bm{k}}\in\mathcal{M}_{\Xi(\bm{k})}$ be Laplace eigenfunctions
corresponding to highest weights $\bm{j},\bm{k}\in\mathscr{L}^{+}$
respectively. Then the product $u_{\bm{j}}u_{\bm{k}}$ is in the space

$\bigoplus_{\bm{l}\in\mathscr{L}^{+}:\,\bm{l}\preceq\bm{j}+\bm{k}}\mathcal{M}_{\Xi(\bm{l})}.$

We deduce from this characterization an important spectral localization
property for products that will play a fundamental role in dyadic analysis. It
is compatible with Hani’s spectral localization [Han12]: for a general compact
Riemannian manifold, the product of eigenfunctions with eigenvalue
$\mu,\lambda$ respectively should sharply concentrate in
$\big{[}|\sqrt{\lambda}-\sqrt{\mu}|,\sqrt{\lambda}+\sqrt{\mu}\big{]}$ on the
spectral side.

######  Corollary 2.3.1.

There is a constant $0<c\leq 1$, depending only on the algebraic structure of
$\mathbb{G}$, with the following property. If $u_{1}\in\mathcal{M}_{\xi_{1}}$,
$u_{2}\in\mathcal{M}_{\xi_{2}}$, then the product
$u_{1}u_{2}\in\bigoplus_{\xi}\mathcal{M}_{\xi}$, where the range of sum is for

$\xi\in\mathfrak{S}\Big{[}c\big{|}|\xi_{1}|-|\xi_{2}|\big{|},|\xi_{1}|+|\xi_{2}|\Big{]}.$

###### Proof.

With a little abuse of notation, given a highest weight vector
$\bm{j}\in\mathscr{L}^{+}$, we denote by $\bar{\bm{j}}$ the highest weight
vector of $\Xi(\bm{j})^{*}$, the dual representation, which is also
irreducible. It is known from representation theory that the highest weight of
$\Xi(\bm{j})^{*}$ is just $-W_{0}(\bm{j})$, where $W_{0}$ is the element in
the Weyl group with longest length. Since the Weyl group is orthogonal, there
holds $|\bm{j}|=|\bar{\bm{j}}|$. By self-adjointness of $\Delta$, matrix
entries of $\Xi(\bar{\bm{j}})$ correspond to the same eigenvalue of $\Delta$
as those of $\Xi(\bm{j})$ do.

Now suppose there are highest weights
$\bm{j},\bm{k},\bm{l}\in\mathscr{L}^{+}$, and let $u_{\bm{j}}$, $u_{\bm{k}}$,
$u_{\bm{l}}$ be linear combinations of matrix entries of $\Xi(\bm{j})$,
$\Xi(\bm{k})$, $\Xi(\bm{l})$, respectively. From the above we know that e.g.
$\bar{u}_{\bm{j}}\in\mathcal{M}_{\Xi(\bar{\bm{j}})}$. By Proposition 2.1 and
Schur orthogonality, the integral

$\int_{\mathbb{G}}u_{\bm{j}}u_{\bm{k}}\bar{u}_{\bm{l}}dx$

does not vanish only if the following relations are simultaneously satisfied:

$\bm{l}\preceq\bm{j}+\bm{k},\quad\bm{j}\preceq\bar{\bm{k}}+\bm{l},\quad\bar{\bm{k}}\preceq\bm{j}+\bar{\bm{l}}.$

Note that this is only a necessary condition. Denoting the pre-dual vector of
$\bm{\alpha}_{i}$ by $\tilde{\bm{\alpha}}_{i}$ (i.e.
$\tilde{\bm{\alpha}}_{i}\in\mathfrak{t}$, and
$\bm{\alpha}_{i}(\tilde{\bm{\alpha}}_{i^{\prime}})=\delta_{ii^{\prime}}$),
these relations are equivalent to

$\left.\begin{aligned}
\bm{l}(\tilde{\bm{\alpha}}_{i})&\leq\bm{j}(\tilde{\bm{\alpha}}_{i})+\bm{k}(\tilde{\bm{\alpha}}_{i})\\\
\bm{j}(\tilde{\bm{\alpha}}_{i})&\leq\bar{\bm{k}}(\tilde{\bm{\alpha}}_{i})+\bm{l}(\tilde{\bm{\alpha}}_{i})\\\
\bar{\bm{k}}(\tilde{\bm{\alpha}}_{i})&\leq\bm{j}(\tilde{\bm{\alpha}}_{i})+\bar{\bm{l}}(\tilde{\bm{\alpha}}_{i})\\\
\end{aligned}\right\\}\quad i=1,\cdots,\varrho.$

Due to monotonicity of the eigenvalues with respect to the dominance order,
the first inequality implies

$|\bm{l}+\bm{\varpi}|^{2}\leq|\bm{j}+\bm{k}+\bm{\varpi}|^{2}$

so we conclude that $|\bm{l}|\leq|\bm{j}|+|\bm{k}|$. Since
$\\{\bm{\alpha}_{i}\\}_{i=1}^{\varrho}$ is a basis for $\mathfrak{t}^{*}$, it
follows that $\\{\tilde{\bm{\alpha}}_{i}\\}_{i=1}^{\varrho}$ is a basis for
$\mathfrak{t}$, so the second and third inequalities, together with
$|\bm{l}|=|\bar{\bm{l}}|$, imply

$|\bm{j}-\bar{\bm{k}}|\lesssim|\bm{l}|.$

Using $|\bm{k}|=|\bar{\bm{k}}|$ once again, we obtain
$\big{|}|\bm{j}|-|\bm{k}|\big{|}\lesssim|\bm{l}|$. Thus the integral does not
vanish only if

$\big{|}|\bm{j}|-|\bm{k}|\big{|}\lesssim|\bm{l}|\leq|\bm{j}|+|\bm{k}|.$

Using the formula for eigenvalues in Theorem 2.3, with
$\bm{j}=\bm{J}(\xi_{1})$, $\bm{k}=\bm{J}(\xi_{2})$, $\bm{l}=\bm{J}(\xi)$, this
is implies that the spectral projection of $u_{\bm{j}}u_{\bm{k}}$ to
$\mathcal{M}_{\xi}$ is nontrivial only if

$\big{|}|\xi_{1}|-|\xi_{2}|\big{|}\lesssim|\xi|\leq|\xi_{1}|+|\xi_{2}|.$

This is exactly what to prove. ∎

The dual representation of $\xi\in\widehat{\mathbb{G}}$ is, in general, not
equivalent to $\xi$, although they have the same dimension. This is why
$\bar{\bm{j}}$ should be studied first. But the rank one case can be treated
trivially: the special unitary group $\mathrm{SU}(2)$ is the only simply
connected compact Lie group of rank 1. Irreducible unitary representations of
$\mathrm{SU}(2)$ are uniquely labeled by their dimensions. For the Lie algebra
$\mathfrak{su}(2)$ and the subalgebra corresponding to the subgroup
$\mathrm{diag}(e^{i\psi},e^{-i\psi})$, the dominance order is equivalent to
the usual ordering on natural numbers. Corollary 2.3.1 then becomes a well-
known fact: the product of spherical harmonics of degree $p$ and $q$ is a
linear combination of spherical harmonics of degree between $|p-q|$ and $p+q$,
with coefficients given by the _Clebsch–Gordan coefficients_ (see e.g.
[Vil78]). This fact has been used by [DS04] as the substitute of spectral
localization properties on spheres. On the other hand, the constant $c$ can be
strictly less than 1. For example, the root system of $\mathrm{SU}(3)$ forms a
hexagonal lattice, and $c=1/2$ in that case.

In estimating the magnitude of certain symbols, we will also be employing the
following asymptotic formula of Weyl type. It is the discrete version of
volume growth estimate for Euclidean balls:

######  Lemma 2.4.

As $t\to\infty$,

$\sum_{\xi\in\mathfrak{S}[0,t]}d_{\xi}^{2}\simeq t^{n}.$

###### Proof.

By Schur orthogonality, recalling $|\xi|=\sqrt{\lambda_{\xi}}$, we find that

$\sum_{\xi\in\mathfrak{S}[0,t]}d_{\xi}^{2}=\sum_{|\xi|\leq
t}\dim\mathcal{M}_{\xi}=\sum_{\lambda\in\sigma(-\Delta):\lambda\leq t^{2}}1,$

i.e. the number of eigenvalues of $-\Delta$ with magnitude less than $t^{2}$
(counting multiplicity). Using the Weyl law, we find that the ratio between

$\sum_{|\xi|\leq
t}d_{\xi}^{2}\quad\text{and}\quad\frac{\omega_{n}}{(2\pi)^{n}}\mathrm{Vol}(\mathbb{G})t^{n}$

has limit 1 as $t\to\infty$. Here $\omega_{n}$ is the volume of Euclidean
$n$-unit ball. ∎

###  2.2. Symbol and Symbolic Calculus

A global notion of symbol can be defined on Lie groups due to their high
symmetry. An explicit symbolic calculus was formally constructed by a series
of works of Ruzhansky, Turunen, Wirth. Fischer’s work [Fis15] provided a
complete study of the object. We will closely follow [Fis15].

We fix our compact Lie group $\mathbb{G}$ as in the previous subsection. The
starting point will be a Taylor’s formula on $\mathbb{G}$.

######  Proposition 2.2 (Taylor expansion).

Let $q=(q_{1},\cdots,q_{M})$ be an $M$-tuple of smooth functions on
$\mathbb{G}$, all vanishing at $e\in\mathbb{G}$, such that $(\nabla
q_{i})_{i=1}^{M}$ has rank $n$. For a multi-index
$\alpha\in\mathbb{N}_{0}^{m}$, set $q^{\alpha}:=q_{1}^{\alpha_{1}}\cdots
q_{M}^{\alpha_{M}}$. Corresponding to each multi-index $\alpha$, there is a
left-invariant differential operator $X_{q}^{(\alpha)}$ of order $|\alpha|$ on
$\mathbb{G}$, such that the following Taylor’s formula holds for every smooth
function $f$ on $\mathbb{G}$ and every $N\in\mathbb{N}_{0}$:

$f(xy)=\sum_{|\alpha|<N}q^{\alpha}(y^{-1})X_{q}^{(\alpha)}f(x)+R_{N}(f;x,y).$

Here the remainder $R_{N}(f;x,y)$ depends linearly on derivatives of $f$ of
order $\leq N$, is smooth in $x,y$, and satisfies
$R_{N}(f;x,y)=O\big{(}|f|_{C^{N}}\mathrm{dist}(y,e)^{N}\big{)}$.

###### Sketch of proof.

Suppose without loss of generality that the first $n$ functions are of maximal
rank at $e$. By the implicit function theorem, the mapping
$x\to(q_{1}(x),\cdots,q_{M}(x))\in\mathbb{R}^{M}$ is a smooth embedding in
some neighbourhood of $e\in\mathbb{G}$. Consider first $x=e$. Fix a basis
$X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, such that $X_{i}q_{j}(e)=\delta_{ij}$
for $i,j\leq n$. For multi-indices $\alpha\in\mathbb{N}_{0}^{M}$, write
$X^{\alpha}=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$ (the “normal” order);
if the first $n$ components of $\alpha$ all vanish, then just set
$X^{\alpha}=0$. Choose constants $c_{\alpha}$ inductively so that

$\left(\sum_{\alpha}c_{\alpha}X^{\alpha}\right)q^{\beta}(e)=\delta_{\alpha\beta},\quad\beta_{i}\neq
0\text{ only for }1\leq i\leq n.$

Then with $X^{(\alpha)}_{q}:=\sum_{\alpha}c_{\alpha}X^{\alpha}$, the formula
holds for $x=e$ as an implication of the usual Taylor’s formula. In order to
pass to general $x\in\mathbb{G}$, just use the left translation operator
$L_{x}$ to act on both sides, and notice that $X^{(\alpha)}_{q}$ are left-
invariant differential operators. ∎

######  Remark 2.1.

If $f$ takes value in a finite dimensional normed space $V$, the Taylor’s
formula given above is still valid. If we define the $C^{N}$ norm of $f$ by

$|f|_{C^{N};V}:=\sup_{v^{*}\in V^{*}:v^{*}\neq 0}\frac{\big{|}\langle
f,v^{*}\rangle\big{|}_{C^{N}}}{|v^{*}|},$

then the estimate
$|R_{N}(f;x,y)|_{V}=O\big{(}|f|_{C^{N};V}\mathrm{dist}(y,e)^{N}\big{)}$ is
valid, with the implicit constant independent from the dimension of $V$.

Note that the differential operators $X_{q}^{(\alpha)}$ are defined
independently for every multi-index, so the equality
$X_{q}^{(\alpha+\beta)}=X_{q}^{(\alpha)}X_{q}^{(\beta)}$ is, in general, not
valid.

A _symbol_ $a$ on $\mathbb{G}$ is simply a field $a$ defined on
$\mathbb{G}\times\widehat{\mathbb{G}}$, such that $a(x,\xi)$ is a distribution
of value in $\mathrm{End}(\mathcal{H}_{[}\xi])$ for each
$\xi\in\widehat{\mathbb{G}}$. If a basis for $\mathcal{H}_{[}\xi]$ is chosen,
the value $a(x,\xi)$ can be simply understood as a matrix function of size
$d_{\xi}\times d_{\xi}$. We define the _quantization_ of a symbol $a$ formally
by

(2.3)
$\mathbf{Op}(a)f(x):=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\operatorname{Tr}\left(a(x,\xi)\cdot\widehat{f}(\xi)\cdot\xi(x)\right).$

Conversely, if $A:C^{\infty}(\mathbb{G})\to\mathcal{D}^{\prime}(\mathbb{G})$
is a continuous operator, then it is the quantization of the symbol

(2.4) $\sigma[A](x,\xi):=\xi^{*}(x)\cdot(A\xi)(x).$

Here $A\xi$ is understood as entry-wise action. In this case the series (2.3)
then converges in $\mathcal{D}^{\prime}(\mathbb{G})$. The _associated right
convolution kernel_ for $a(x,\xi)$ is defined by

(2.5)
$\mathcal{K}(x,y):=\left(a(x,\cdot)\right)^{\vee}(y)=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\operatorname{Tr}\left(a(x,\xi)\cdot\xi(y)\right),$

where the Fourier inversion is taken with respect to $\xi$. Formally, the
action $\mathbf{Op}(a)f$ can be written as a convolution:

(2.6)
$f(\cdot)*\mathcal{K}(x,\cdot)=\int_{\mathbb{G}}f(y)\mathcal{K}(x,y^{-1}x)dy.$

An intrinsic notion of difference operators acting on symbols was introduced
by Fischer [Fis15], generalizing the differential operator with respect to
$\xi$ in harmonic analysis on $\mathbb{R}^{n}$. For any (continuous) unitary
representation $(\tau,\mathcal{H}_{\tau})$, Maschke’s theorem ensures that
$\tau=\oplus_{j}\xi_{j}$ for finitely many $\xi_{j}\in\widehat{\mathbb{G}}$. A
symbol $a(x,\xi)$ can be naturally extended to any $\tau$ by
$a(x,\tau)=\oplus_{j}a(x,\xi_{j})$, up to equivalence of representations. The
definition of difference operators is then given by

######  Definition 2.1.

Given any extended symbol $a(x,\cdot)$ and representation $\tau$, the
difference operator $\mathbf{D}_{\tau}$ gives rise to a new extended symbol in
the following manner:

$\mathbf{D}_{\tau}a(x,\pi):=a(x,\tau\otimes\pi)-a(x,\mathrm{Id}_{\mathcal{H}_{[}}\tau]\otimes\pi).$

For a tuple of representations $\bm{\tau}=(\tau_{1},\cdots,\tau_{p})$, write
$\mathbf{D}^{\bm{\tau}}=\mathbf{D}_{\tau_{1}}\cdots\mathbf{D}_{\tau_{p}}$, and
$\mathbf{D}^{\bm{\tau}}a(x,\xi)$ is then an endomorphism of
$\mathcal{H}_{\tau_{1}}\otimes\cdots\otimes\mathcal{H}_{\tau_{p}}\otimes\mathcal{H}_{[}\xi]$.

Corresponding to the functions $q=\\{q_{i}\\}_{i=1}^{M}$ as in Proposition
2.2, Ruzhansky and Turunen introduced the so-called _admissible difference
operators_ , also generalizing the differential operator with respect to $\xi$
in harmonic analysis on $\mathbb{R}^{n}$, but in a more computable way
compared to the intrinsic definition. We employ the following definitions from
[RT09], [RT13] and [RTW14]:

######  Definition 2.2.

An $M$-tuple of smooth functions $q=(q_{i})_{i=1}^{M}$ on $\mathbb{G}$ is said
to be RT-admissible if they all vanishing at $e\in\mathbb{G}$ and $(\nabla
q_{i})_{i=1}^{M}$ has rank $n$. If in addition the only common zero of
$(q_{i})_{i=1}^{M}$ is the identity element, then the $M$-tuple is said to be
strongly RT-admissible.

######  Definition 2.3.

Given a function $q\in C^{\infty}(\mathbb{G})$, the corresponding RT-
difference operator $\mathbf{D}_{q}$ acts on the Fourier transform of a
$f\in\mathcal{D}^{\prime}(\mathbb{G})$ by

$\Big{(}\mathbf{D}_{q}\widehat{f}\Big{)}(\xi):=\widehat{qf}(\xi).$

The corresponding collection of RT-difference operators corresponding to an
$M$-tuple $q=(q_{i})_{i=1}^{M}$ is the set of difference operators

$\mathbf{D}_{q}^{\alpha}:=\mathbf{D}_{q^{\alpha}}=\mathbf{D}_{q_{1}}^{\alpha_{1}}\cdots\mathbf{D}_{q_{M}}^{\alpha_{M}}.$

If the tuple $q$ is RT-admissible (strongly RT-admissible), the corresponding
collection of RT difference operators is said to be RT-admissible (strongly
RT-admissible).

We write $\mathbf{D}_{q,\xi}$ for the action of a difference operator on the
$\xi$ variable. Formally, we have

(2.7)
$(\mathbf{D}_{q,\xi}a)(x,\xi)=\int_{\mathbb{G}}q(y)\left(\sum_{\eta\in\widehat{\mathbb{G}}}d_{\eta}\operatorname{Tr}\left(a(x,\eta)\cdot\eta(y)\right)\right)\xi^{*}(y)dy,$

so $\mathbf{D}_{q,\xi}$ commutes with any differential operator acting on $x$.
To compare with $\mathbb{R}^{n}$, we simply notice that given a symbol
$a(x,\xi)$ on $\mathbb{R}^{n}$, the Fourier inversion of
$\partial_{\xi}a(x,\xi)$ with respect to $\xi$ is $iya^{\vee}(x,y)$, i.e.
multiplication by a polynomial. The functions $q$ on $\mathbb{G}$ then play
the role of monomials on $\mathbb{R}^{n}$.

With the aid of difference operators, [Fis15] introduced symbol classes of
interest.

######  Definition 2.4.

Let $m\in\mathbb{R}$, $0\leq\delta\leq\rho\leq 1$. Fix a basis
$X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, and define $X^{\alpha}$ as in
Proposition 2.1. The symbol class $\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G})$
is the set of all symbols $a(x,\xi)$, such that $a(x,\xi)$ is smooth in
$x\in\mathbb{G}$, and for any tuple of representation
$\bm{\tau}=(\tau_{1},\cdots,\tau_{p})$ and any left-invariant differential
operator $X^{\alpha}$, there is a constant $C_{\alpha\bm{\tau}}$ such that

$\big{\|}X^{\alpha}\mathbf{D}^{\bm{\tau}}a(x,\xi)\big{\|}\leq
C_{\alpha\bm{\tau}}\langle{[}\rangle\xi]^{m-\rho p+\delta|\alpha|}.$

Here the norm is taken to be the operator norm of
$\mathcal{H}_{\tau_{1}}\otimes\cdots\otimes\mathcal{H}_{\tau_{p}}\otimes\mathcal{H}_{[}\xi]$.

######  Definition 2.5.

Let $m\in\mathbb{R}$, $0\leq\delta\leq\rho\leq 1$. Fix a basis
$X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, and define $X^{\alpha}$ as in
Proposition 2.1. Let $q=(q_{1},\cdots,q_{M})$ be an RT-admissible $M$-tuple.
The symbol class $\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G};\mathbf{D}_{q})$ is
the set of all symbols $a(x,\xi)$, such that $a(x,\xi)$ is smooth in
$x\in\mathbb{G}$, and

$\big{\|}X^{\alpha}_{x}\mathbf{D}_{q,\xi}^{\beta}a(x,\xi)\big{\|}\leq
C_{\alpha}\langle{[}\rangle\xi]^{m+\delta|\alpha|-\rho|\beta|}.$

We can also introduce the norms

(2.8) $\mathbf{M}_{k,\rho;l,\delta;q}^{m}(a):=\sup_{|\alpha|\leq
k}\sup_{|\beta|\leq
l}\sup_{x,\xi}\langle{[}\rangle\xi]^{\rho|\beta|-m-\delta|\alpha|}\left\|X^{\alpha}_{x}\mathbf{D}_{q;\xi}^{\beta}a(x,\xi)\right\|.$

In particular, for $\rho=\delta=1$, we write $\mathbf{M}_{k,l;q}^{m}(a)$ for
simplicity.

In [Fis15], Fischer proved that if $q=(q_{i})_{i=1}^{M}$ is a strongly RT-
admissible tuple, then the symbol class in Definition 2.5 does not depend on
the choice of $q$, and in fact gives rise to the usual Hörmander class of
pseudo-differential operators.

######  Theorem 2.5 (Fischer [Fis15], Theorem 5.9. and Corollary 8.13.).

(1) Suppose $0\leq\delta\leq\rho\leq 1$. If $q=(q_{i})_{i=1}^{M}$ is a
strongly RT-admissible tuple, then a symbol
$a\in\mathscr{S}^{m}_{\rho,\delta}$ if and only if all the norms
$\mathbf{M}_{r,\rho;l,\delta;q}^{m}(a)$ are finite.

(2) Moreover, if $\rho>\delta$ and $\rho\geq 1-\delta$, then the operator
class $\mathbf{Op}\mathscr{S}^{m}_{\rho,\delta}$ coincides with the Hörmander
class $\Psi^{m}_{\rho,\delta}$ of $(\rho,\delta)$-pseudo-differential
operators defined via local charts.

Let us briefly sketch Fischer’s proof of (1) in Theorem 2.5. The following
lemma is a key ingredient, and is worthy of a specific mention:

######  Lemma 2.6 ([Fis15], Lemma 5.10).

Suppose $q,q^{\prime}\in\mathcal{D}(\mathbb{G})$, such that $q^{\prime}/q$
extends to a smooth function on $\mathbb{G}$. If a symbol $a=a(\xi)$ is such
that $\|\mathbf{D}_{q}a(\xi)\|$ does not grow faster than a power of
$\langle{[}\rangle\xi]$, then

$\|\mathbf{D}_{q^{\prime}}a(\xi)\|\lesssim_{q,q^{\prime}}\|\mathbf{D}_{q}a(\xi)\|.$

Using this lemma, Fischer showed that for any two strongly RT-admissible tuple
$q,q^{\prime}$, the collection of norms
$\mathbf{M}_{r,\rho;l,\delta;q}^{m}(a)$ and
$\mathbf{M}_{r,\rho;l,\delta;q^{\prime}}^{m}(a)$ are equivalent to each other.

A convenient choice of strongly RT-admissible tuple is necessary for
calculations. Fischer found that matrix entries of the fundamental
representations produce a strongly RT-admissible tuple in the following way:
for each unitary representation $\tau$ of $\mathbb{G}$, set

$q_{jk}^{(\tau)}(x)=\tau_{jk}(x)-\delta_{jk},$

i.e. the matrix entries of $\tau-\mathrm{Id}_{\mathcal{H}_{[}}\tau]$. Fischer
showed in Lemma 5.11. of [Fis15] that, as $\tau$ exhausts all of the
fundamental representations $\Xi(\bm{\varpi}_{i}),i=1,\cdots,\varrho$, the
collection $q=\\{q_{jk}^{(\tau)}\\}$ is strongly RT-admissible. The advantage
of this choice is that it is essentially intrinsically defined for
$\mathbb{G}$, and exploits the algebraic structure of $\mathbb{G}$. From now
on, we will just defined the _fundamental tuple of $\mathbb{G}$_ as

(2.9) $Q:=\bigcup_{\tau:\text{ fundamental
representation}}\left\\{\tau_{ij}-\delta_{ij}:i,j=1,\cdots,d_{\tau}\right\\}.$

Thus, as Fischer pointed out, the symbol class
$\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G};\mathbf{D}_{Q})$ in Definition 2.5,
with $Q$ being the fundamental tuple (2.9), in fact coincides with the
intrinsic definition of symbols 2.4. This is because for each representation
$\tau$, the matrix entries of $\mathbf{D}_{\tau}a$ are exactly given by
$\mathbf{D}_{\tau_{jk}-\delta_{jk}}a$, so as $\tau$ exhausts all of the
fundamental representations, the corresponding difference operators should
give rise to the class $\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G})$. It is
legitimate to write $\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G})$ and omit the
dependence on strongly RT-admissible difference operators. This is how (1) of
Theorem 2.5 was proved.

A particular property of the fundamental tuple $Q$ deserves a specific
mention. For harmonic analysis on $\mathbb{R}^{n}$, differentiation in
frequency usually simplifies the computation drastically. For analysis on
compact group, the dual $\widehat{\mathbb{G}}$ is discrete, whence no natural
notion of “continuous differentiation” is available. But the RT difference
operators can be constructed in such a manner that they possess _Leibniz type
property_ : given a tuple $q=(q_{i})_{i=1}^{M}$, the corresponding RT
difference operators are said to possess Leibniz type property, if there are
constants $c_{j,k}^{i}\in\mathbb{C}$ such that for symbols $a,b$,

(2.10) $\mathbf{D}_{q_{i}}(ab)=\mathbf{D}_{q_{i}}a\cdot
b+a\cdot\mathbf{D}_{q_{i}}b+\sum_{j,k=1}^{M}c_{j,k}^{i}\mathbf{D}_{q_{j}}a\cdot\mathbf{D}_{q_{k}}b.$

Inductively, this implies

$\mathbf{D}_{q}^{\alpha}(ab)=\sum_{\begin{subarray}{c}0\leq|\beta|,|\gamma|\leq|\alpha|\\\
|\alpha|\leq|\beta|+|\gamma|\leq
2|\alpha|\end{subarray}}c_{\beta\gamma}^{\alpha}\mathbf{D}_{q}^{\beta}a\cdot\mathbf{D}_{q}^{\gamma}b.$

Condition (2.10) is equivalent to

$q_{i}(xy)=q_{i}(x)+q_{i}(y)+\sum_{j,k=1}c_{j,k}^{i}q_{j}(x)q_{k}(y),$

so the matrix entries of $\tau-\mathrm{Id}_{\mathcal{H}_{[}}\tau]$, where
$\tau$ exhausts all fundamental representations of $\mathbb{G}$, is a good
choice, because due to the equality

$\tau(xy)=\tau(x)\tau(y),$

they give rise to strongly admissible RT difference operators that satisfies
Leibniz type property: for $q_{ij}(x)=\tau_{ij}(x)-\delta_{ij}$,

(2.11)
$q_{ij}(xy)=q_{ij}(x)+q_{ij}(y)+\sum_{k=1}^{d_{\tau}}q_{ik}(x)q_{kj}(y).$

The advantage of the symbol class $\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G})$
is that a symbolic calculus is available. The proof of the following results
is quite parallel as in Euclidean spaces (not without technicality in
verifying the necessary kernel properties):

######  Theorem 2.7 ([Fis15], Corollary 7.9.).

Suppose $0\leq\delta<\rho\leq 1$. If $a\in\mathscr{S}^{m}_{\rho,\delta}$,
$b\in\mathscr{S}^{m^{\prime}}_{\rho,\delta}$, then the composition
$\mathbf{Op}(a)\circ\mathbf{Op}(b)\in\mathbf{Op}\mathscr{S}^{m+m^{\prime}}_{\rho,\delta}$,
and in fact if $q$ is any strongly RT-admissible tuple and $X^{(\alpha)}_{q}$
is as in Proposition 2.2, then the symbol $\sigma$ of
$\mathbf{Op}(a)\circ\mathbf{Op}(b)$ satisfies

$\sigma(x,\xi)-\sum_{|\alpha|<N}\left(\mathbf{D}_{q,\xi}^{\alpha}a\cdot
X_{q,x}^{(\alpha)}b\right)(x,\xi)\in\mathscr{S}^{m+m^{\prime}-(\rho-\delta)N}_{\rho,\delta}.$

One can thus write

$\sigma(x,\xi)\sim\sum_{\alpha}\left(\mathbf{D}_{q,\xi}^{\alpha}a\cdot
X_{q,x}^{(\alpha)}b\right)(x,\xi).$

######  Theorem 2.8 ([Fis15], Corollary 7.6.).

Suppose $0\leq\delta<\rho\leq 1$. If $a\in\mathscr{S}^{m}_{\rho,\delta}$, then
the adjoint $\mathbf{Op}(a)^{*}\in\mathbf{Op}\mathscr{S}^{m}_{\rho,\delta}$,
and in fact if $q$ is any strongly RT-admissible tuple and $X^{(\alpha)}_{q}$
is as in Proposition 2.2, then the symbol $a^{\bullet;q}$ of
$\mathbf{Op}(a)^{*}$ satisfies

$a^{\bullet;q}(x,\xi)-\sum_{|\alpha|<N}\left(\mathbf{D}_{q,\xi}^{\alpha}X_{q,x}^{(\alpha)}a^{*}\right)(x,\xi)\in\mathscr{S}^{m+m^{\prime}-(\rho-\delta)N}_{\rho,\delta}.$

One can thus write

$a^{\bullet;q}(x,\xi)\sim\sum_{\alpha}\left(\mathbf{D}_{q,\xi}^{\alpha}X_{q,x}^{(\alpha)}a^{*}\right)(x,\xi).$

Fischer also proved that for $0\leq\delta\leq\rho\leq 1$ with $\delta<1$, if
$a\in\mathscr{S}^{m}_{\rho,\delta}$, then $\mathbf{Op}(a)$ maps $H^{s+m}$
continuously to $H^{s}$ for any $s\in\mathbb{R}$. The proof is still parallel
to that on Euclidean spaces.

Finally, to prove (2) in Theorem 2.5, i.e.
$\mathbf{Op}\mathscr{S}^{m}_{\rho,\delta}(\mathbb{G})=\Psi^{m}_{\rho,\delta}(\mathbb{G})$,
the Hörmander class of operators for $\rho>\delta$ and $\rho\geq 1-\delta$,
Fischer showed that the commutators of $\mathbf{Op}(a)$ with vector fields and
$C^{\infty}$-multiplications satisfy Beals’ characterization of pseudo-
differential operators as stated in [Bea77]. In particular, the operator class
$\mathbf{Op}\mathscr{S}^{m}_{1,0}(\mathbb{G})$ coincides with the Hörmander
class $\Psi^{m}(\mathbb{G})$ defined via local charts. Thus, it is possible to
manipulate pseudo-differential operators on $\mathbb{G}$ using globally
defined symbolic calculus, and all the lower-order information lost in
operation with principal symbols can be retained.

###  2.3. Order of a Symbol

Unlike the case of $\mathbb{R}^{n}$, symbolic calculus on the non-commutative
Lie group $\mathbb{G}$ involves endomorphisms of the representation spaces,
hence suffers from non-commutativity. Thus, for example, properties of the
commutator111Note that this is not the commutator of pseudo-differential
operators. $(ab-ba)(x,\xi)=:[a,b](x,\xi)$ of two symbols $a(x,\xi)$ and
$b(x,\xi)$ is not as clear as on $\mathbb{R}^{n}$ (it simply vanishes for
symbols on $\mathbb{R}^{n}$). With the aid of Fischer’s theorem, however, we
are able to show that the commutator of symbols of order $m$ and $m^{\prime}$
respectively “is reduced by order 1”.

We introduce a formal definition of order as follows.

######  Definition 2.6.

Let $m\in\mathbb{R}$. We say that a symbol $a(x,\xi)$ on $\mathbb{G}$,
regardless of regularity in $x$, is of order $m$, if for some strongly RT-
admissible tuple $q$, there always holds

$\sup_{x\in\mathbb{G}}\big{\|}\mathbf{D}_{q}^{\beta}a(x,\xi)\big{\|}\lesssim\langle{[}\rangle\xi]^{m-|\beta|}.$

By Fischer’s Lemma 2.6, the class does not depend on the choice of strongly RT
-admissible tuple.

Thus the class of symbols of order $m$, in our convention, is the collection
of symbols that “possess best decays upon differentiation in $\xi$”. It
necessarily includes all the $\mathscr{S}^{m}_{1,\delta}(\mathbb{G})$ with
$0\leq\delta\leq 1$. The property that we need to prove is

######  Proposition 2.3.

Let $a,b$ be symbols on $\mathbb{G}$ of order $m$ and $m^{\prime}$ and type 1
respectively. Then the commutator $[a,b]$ is of order $m+m^{\prime}-1$. In
particular, if $\delta\in[0,1]$, $a\in\mathscr{S}^{m}_{1,\delta}(\mathbb{G})$
and $b\in\mathscr{S}^{m^{\prime}}_{1,\delta}(\mathbb{G})$, then
$[a,b]\in\mathscr{S}^{m+m^{\prime}-1}_{1,\delta}(\mathbb{G})$.

###### Proof.

It is quite alright to disregard the dependence on $x$ and consider Fourier
multipliers $a=a(\xi)$, $b=b(\xi)$ alone, as no differentiation in $x$ is
involved. By Fischer’s theorem 2.5, the operators $A=\mathbf{Op}(a)$ and
$B=\mathbf{Op}(b)$ are in fact in the Hörmander class of pseudo-differential
operators $\Psi^{m}_{1,0}$ and $\Psi^{m^{\prime}}_{1,0}$ respectively. The
Hörmander calculus on principal symbols (as functions on the cotangent bundle)
then asserts that $[A,B]\in\Psi^{m+m^{\prime}-1}_{1,0}$, with principal symbol
given by the Poisson bracket. But $[A,B]$ is nothing but
$[\mathbf{Op}(a),\mathbf{Op}(b)]=\mathbf{Op}\big{(}[a,b]\big{)}$ as $a,b$ are
assumed to be Fourier multipliers. Fischer’s theorem again implies that
$[a,b]\in\mathscr{S}^{m+m^{\prime}-1}_{1,0}$. To extend this to $x$-dependent
symbols $a(x,\xi)$ and $b(x,\xi)$, it suffices to note that, for example,
given a vector field $X$, the Leibniz rule
$X_{x}\big{(}[a,b]\big{)}=[X_{x}a,b]+[a,X_{x}b]$ holds. The operator norm of
$X_{x}\big{(}[a,b]\big{)}$ at a given representation $\xi$ can thus be
estiamted similarly. ∎

##  3\. (1,1) Pseudo-differential Operators on Compact Lie Group

###  3.1. Littlewood-Paley Decomposition

We start with a continuous version of Littlewood-Paley decomposition. Such
construction has already been explored by Klainerman and Rodnianski in [KR06]
for a general compact manifold. Here we aim to expoit the Lie group structures
to deduce more. Fix an even function $\phi\in C_{0}^{\infty}(\mathbb{R})$,
such that $\phi(\lambda)=1$ for $|\lambda|\leq 1/2$, and $\phi(\lambda)=0$ for
$|\lambda|\geq 1$. Setting $\psi(\lambda)=-\lambda\phi^{\prime}(\lambda)$, we
obtain a continuous partition of unity

$1=\phi(\lambda)+\int_{1}^{\infty}\psi\Big{(}\frac{\lambda}{t}\Big{)}\frac{dt}{t}.$

The _continuous Littlewood-Paley decomposition_ of a distribution
$f\in\mathcal{D}^{\prime}(\mathbb{G})$ will then be defined by

(3.1)
$f=\phi\big{(}|\nabla|\big{)}f+\int_{1}^{\infty}\psi_{t}\big{(}|\nabla|\big{)}f\frac{dt}{t},\quad\psi_{t}(\cdot)=\psi\left(\frac{\cdot}{t}\right)$

We also write the _partial sum operator_ as

$\phi_{T}(|\nabla|)f:=\phi\left(\frac{|\nabla|}{T}\right)f=\phi\big{(}|\nabla|\big{)}f+\int_{1}^{T}\psi_{t}\big{(}|\nabla|\big{)}f\frac{dt}{t}.$

This is the convention employed by Hörmander [Hör97], Chapter 9. It is easy to
deduce the Bernstein inequality:

######  Proposition 3.1.

If $f\in\mathcal{D}(\mathbb{G})$ has Fourier support contained in
$\mathfrak{S}[0,t]$, then for any $s\in\mathbb{R}$,

$\|f\|_{H^{s}}\lesssim t^{s}\|f\|_{L^{2}}.$

For each $t\geq 1$, the integrand $\psi(|\nabla|/t)f$ is a smooth function on
$\mathbb{G}$, whose Fourier support is contained in $\mathfrak{S}[t/2,t]$,
where the definition is given in (2.2). In particular, the mode of
$\psi(|\nabla|/t)f$ corresponding to zero eigenvalue is always zero. The
integral (3.1) converges at least in the sense of distribution, and just as in
the case of Littlewood-Paley decomposition on Euclidean space, the speed of
convergence reflects the regularity property of $f$. Sometimes the _discrete
Littlewood-Paley decomposition_ is also employed: setting
$\vartheta(\lambda)=\phi_{1}(\lambda)-\phi_{1}(\lambda)$ and
$\vartheta_{j}(\lambda)=\vartheta(\lambda/2^{j})$, then

(3.2) $f=\phi\big{(}|\nabla|\big{)}f+\sum_{j\geq
0}\vartheta_{j}\big{(}|\nabla|\big{)}f.$

The continuous and discrete Littlewood-Paley decomposition are parallel to
each other, with summation in place of integration or vice versa, but in
different scenarios one version can make the computation simpler than the
other.

Since the integrand in (3.1) is the convolution of $f$ with kernel

$\left(\psi_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]\right)^{\vee}(x)=\sum_{\xi\in\mathbb{G}}d_{\xi}\psi_{t}(|\xi|)\operatorname{Tr}\left(\xi(x)\right),$

it is thus important to understand a kernel with finite Fourier support and is
merely a scaling on each representation space. Notice that the order of
convolution does not really matter for this specific case. We thus state the
following fundamental lemma concerning kernels of spectral cut-off operators:

######  Lemma 3.1.

Let $h\in C_{0}^{\infty}(\mathbb{R})$. Let $P$ be any classical differential
operator with smooth coefficients of degree $N\in\mathbb{N}_{0}$. Then for
$t\geq 1$, the function

$\check{h}_{t}:=\left(h_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]\right)^{\vee}=\sum_{\xi\in\mathbb{G}}d_{\xi}h\left(\frac{|\xi|}{t}\right)\operatorname{Tr}\left(\xi\right)$

satisfies

$\|P\check{h}_{t}\|_{L^{1}}\lesssim_{N}t^{N}|h|_{L^{\infty}}.$

If $f\in C(\mathbb{G})$ is such that for some $r>0$,
$f(x)=O(\mathrm{dist}(x,e)^{r})$ when $x\to e$, then

$\|f\check{h}_{t}\|_{L^{1}}\lesssim|h|_{L^{\infty}}t^{-r}.$

With a little abuse of notation, we denote the Fourier inversion of the symbol
$h_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]$ as $\check{h}_{t}$. The proof
can be found in [Fis15]. It in fact only depends on the compact manifold
structure of $\mathbb{G}$, and follows from properties of heat kernel on a
compact manifold. As a corollary, the symbol
$h_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]$ is of class
$\mathscr{S}^{0}_{1,0}$ if $h$ is smooth and has compact support. We deduce a
corollary that will be used later.

######  Corollary 3.1.1.

Let $h\in C_{0}^{\infty}(\mathbb{R})$. If $f\in\mathcal{D}(\mathbb{G})$ is
such that $f(x)=O(\mathrm{dist}(x,e)^{N})$ for some $N\in\mathbb{N}_{0}$, then
$h_{t}*f$ remains in a bounded subset of $\mathcal{D}(\mathbb{G})$, for $t\geq
1$, and

$\big{|}(\check{h}_{t}*f)(x)\big{|}\lesssim_{N}|h|_{L^{\infty}}t^{-N}\sum_{k=0}^{N}t^{k}|f|_{C^{k}}\mathrm{dist}(x,e)^{k}.$

###### Proof.

For simplicity we prove for $N=1$; the general case follows similarly. The
order of convolution does not matter since
$h_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]$ is a scaling on each
$\mathcal{H}_{[}\xi]$. We use Taylor’s formula to write

$\displaystyle(\check{h}_{t}*f)(x)$
$\displaystyle=\int_{\mathbb{G}}\check{h}_{t}(y)f(y^{-1}x)dy$
$\displaystyle=\int_{\mathbb{G}}\check{h}_{t}(y)f(y)dy+\int_{\mathbb{G}}\check{h}_{t}(y)R_{1}(f;y^{-1},x)dy.$

The first term is controlled by $|h|_{L^{\infty}}|f|_{C^{0}}t^{-1}$ by Lemma
2.6, and the second is controlled by
$|h|_{L^{\infty}}\mathrm{dist}(x,e)|f|_{C^{1}}$, by Taylor’s formula. ∎

A more general multiplier theorem was deduced in [Fis15] from Lemma 3.1:

######  Theorem 3.2.

Let $h$ be a smooth function on $[0,\infty)$. Define the multiplier norm

$\|h\|_{m;k}=\sup_{\lambda\geq 0,l\leq
k}\left(1+|\lambda|\right)^{-m+l}\big{|}h^{(l)}(\lambda)\big{|}.$

If $q\in C^{\infty}(\mathbb{G})$ vanishes of order $N-1$ at $e$, where $N\geq
1$ is an integer, then with

$a_{t}(\xi)=h_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi],$

the symbol $a_{t}$ is of class $\mathscr{S}^{m}_{1,0}$, and the difference
norm is estimated by

$\|\mathbf{D}_{q}a_{t}(\xi)\|\lesssim_{q}\|h\|_{m;k}\langle{[}\rangle\xi]^{m-N}t^{-m}.$

The Littlewood-Paley characterization of Sobolev space is obtained
immediately:

######  Proposition 3.2.

Given $s\in\mathbb{R}$, a distribution $f$ belongs to $H^{s}(\mathbb{G})$ if
and only if for some non-vanishing $h\in C_{0}^{\infty}(0,\infty)$, there
holds

$\left\|\phi\big{(}|\nabla|\big{)}f\right\|_{L^{2}}^{2}+\int_{0}^{\infty}t^{2s-1}\left\|h_{t}\big{(}|\nabla|\big{)}f\right\|_{L^{2}_{x}}^{2}dt<\infty,$

where $h_{t}(\lambda)=h(\lambda/t)$. The lower bound of integral does not
cause singularity since $h=0$ near 0. Similarly, distribution $f$ belongs to
$H^{s}(\mathbb{G})$ if and only if for some non-zero $h\in
C_{0}^{\infty}(0,\infty)$, there holds

$\left\|\phi\big{(}|\nabla|\big{)}f\right\|_{L^{2}}^{2}+\sum_{j\geq
0}2^{2sj}\left\|h_{2^{j}}\big{(}|\nabla|\big{)}f\right\|_{L^{2}}^{2}.$

The square root of either of the above quadratic forms is equivalent to
$\|f\|_{H^{s}}$.

From Lemma 3.1, we also obtain a characterization of the Zygmund space
$C^{r}_{*}(\mathbb{G})$ with $r\geq 0$:

######  Proposition 3.3.

For $r\geq 0$, a distribution $f\in\mathcal{D}^{\prime}(\mathbb{G})$ is in the
Zygmund class $C^{r}_{*}(\mathbb{G})$ if and only if

$\sup|\phi(|\nabla|)f|+\sup_{t\geq
1}t^{r}\left|\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}<\infty.$

This quantity is equivalent to the Zygmund space norm defined via local
coordinate charts.

###### Sketch of proof.

The proof does not differ very much from the Euclidean case. Suppose first
$0<r<1$ is not an integer, and $f\in C^{r}$. Then

$\psi_{t}\big{(}|\nabla|\big{)}f(x)=\int_{\mathbb{G}}\check{\psi}_{t}(y^{-1}x)f(y)dy=\int_{\mathbb{G}}\check{\psi}_{t}(y^{-1}x)\left(f(y)-f(x)\right)dy.$

Note that the order of convolution does not matter, since on the Fourier side,
$\psi_{t}\big{(}|\nabla|\big{)}$ is just a scaling on each representation
space, and the last step is because $\check{\psi}_{t}$ always has mean zero.
Then

$|\psi_{t}\big{(}|\nabla|\big{)}f|_{L^{\infty}}\lesssim_{r}|f|_{C^{r}}\int_{\mathbb{G}}\big{|}\check{\psi}_{t}(y^{-1}x)\big{|}\cdot\mathrm{dist}(x,y)^{r}dy,$

and the right-hand-side is controlled by $|f|_{C^{r}}t^{-r}$ by Lemma 3.1. As
for the opposite direction, if we know that
$\big{|}\psi_{t}\big{(}|\nabla|\big{)}f\big{|}_{L^{\infty}}\lesssim t^{-r}$,
then write

$\displaystyle\int_{1}^{\infty}\left(\psi_{t}\big{(}|\nabla|\big{)}f(x)-\psi_{t}\big{(}|\nabla|\big{)}f(y)\right)\frac{dt}{t}=\int_{t\leq\mathrm{dist}(x,y)^{-1}}+\int_{t>\mathrm{dist}(x,y)^{-1}}.$

When $x$ is close to $y$, the first integral is estimated, using Lemma 3.1
again, by

$\sum_{|\beta|=1}\int_{1}^{\mathrm{dist}(x,y)^{-1}}\left|X^{\beta}\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}\cdot\mathrm{dist}(x,y)\frac{dt}{t}\,\lesssim_{r}\,\sup_{t\geq
1}t^{r}\left|\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}\cdot\mathrm{dist}(x,y)^{r}.$

The second integral is simply estimated by

$2\sup_{t\geq
1}t^{r}\left|\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}\int_{t>\mathrm{dist}(x,y)^{-1}}t^{-r-1}dt\,\simeq_{r}\,\sup_{t\geq
1}t^{r}\left|\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}\cdot\mathrm{dist}(x,y)^{r}.$

This proves the equivalence for $0<r<1$. As for the general case, we can
simply apply the classical Schauder theory and interpolation for Hölder
spaces; recall that interpolation of Hölder spaces results in the Zygmund
space when the intermediate index is an integer. ∎

With the aid of Lemma 2.6, we immediately obtain

######  Corollary 3.2.1.

Suppose $r\geq 0$ and $f\in C^{r}_{*}$. Fix a basis $X_{1},\cdots,X_{n}$ of
$\mathfrak{g}$, and define $X^{\alpha}$ as in Proposition 2.1. Then there
holds, for $t\geq 1$,

$\left|X^{\alpha}\phi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}\lesssim_{r,\alpha}\left\\{\begin{aligned}
&t^{(|\alpha|-r)_{+}}|f|_{C^{r}_{*}}&\quad|\alpha|\neq r\\\
&\log(1+t)|f|_{C^{r}_{*}}&\quad|\alpha|=r\end{aligned}\right.$

where $s_{+}=\max(s,0)$. Furthermore, for $r>0$,
$\left|f-\phi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}\lesssim
t^{-r}|f|_{C^{r}_{*}}$ when $t\geq 1$.

######  Remark 3.1.

Just as in the Euclidean case, we can _define_ Zygmund spaces with index $<0$
by Littlewood-Paley characterization: we say that $f\in C^{r}_{*}$ for $r<0$,
iff

$|f|_{C^{-r}_{*}}:=\sup|\phi(|\nabla|)f|+\sup_{t\geq
1}t^{r}\left|\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty}}<\infty.$

This gives a united definition of Zygmund spaces on $\mathbb{G}$. In
particular, $|\nabla|^{s}$ is continuous from $C^{r+s}_{*}$ to $C^{r}_{*}$ for
all $r\in\mathbb{R}$. Furthermore, for $r<0$, there holds a growth estimate

$\displaystyle\left|\phi_{t}\big{(}|\nabla|\big{)}f\right|\leq\sup|\phi(|\nabla|)f|+\int_{1}^{t}\left|\psi_{\tau}\big{(}|\nabla|\big{)}f\right|d\tau\lesssim
t^{-r}|f|_{C^{r}_{*}}.$

######  Remark 3.2.

These results easily generalize to vector-valued functions. If $r>0$, $V$ is a
finite dimensional normed space, $f\in C^{r}_{*}(\mathbb{G};V)$, then the norm
$|f|_{C^{r}_{*};V}$ is defined as

$\sup_{v^{*}\in V^{*},v^{*}\neq 0}\frac{|\langle
v^{*},f\rangle|_{C^{r}_{*}}}{|v^{*}|_{V^{*}}}.$

We have e.g.

$|f|_{C^{r}_{*};V}\simeq\sup|f|_{L^{\infty};V}+\sup_{x,y\in\mathbb{G}}\frac{|f(x)-f(y)|_{V}}{\mathrm{dist}(x,y)^{r}},\quad
0<r<1,$

where $|f|_{L^{\infty};V}=\sup|f|_{V}$, and the equivalence is _independent_
from $V$. Repeating the same argument as in the proof of Proposition 3.3, just
with $\langle v^{*},f\rangle$ in place of $f$, then taking supremum over
$v^{*}\in V^{*}$, we arrive at

$|f|_{C^{r}_{*};V}\simeq_{r}|\phi(|\nabla|)f|_{L^{\infty};V}+\sup_{t\geq
1}t^{r}\left|\psi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty};V},$
$\left|X^{\alpha}\phi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty};V}\lesssim_{r,\alpha}\left\\{\begin{aligned}
&t^{(|\alpha|-r)_{+}}|f|_{C^{r}_{*};V}&\quad|\alpha|\neq r\\\
&\log(1+t)|f|_{C^{r}_{*};V}&\quad|\alpha|=r\end{aligned}\right.$

and $\left|f-\phi_{t}\big{(}|\nabla|\big{)}f\right|_{L^{\infty};V}\lesssim
t^{-r}|f|_{C^{r}_{*};V}$ for $t\geq 1$. Results for Zygmund spaces of index
$\leq 0$ in Remark 3.1 can be generalized to vector-valued functions
similarly. All the implicit constants above _do not depend on_ $V$.

It is quite legitimate to refer the decomposition (3.1) or (3.2) as _geometric
Littlewood-Paley decomposition_. In [KR06], Klainerman and Rodnianski
commented that, unlike Euclidean case, the geometric Littlewood-Paley
decomposition on a general manifold does not exclude high-low interactions,
although this drawback does not affect applications within their scope. As
Berti-Procesi [BP11] pointed out, this is because that on a general compact
manifold, the product of Laplacian eigenfunctions does not enjoy good spectral
property (unlike the Fourier modes $e^{ix\xi}$), and this might cause
transmission of energy between arbitrary modes for nonlinear dispersive
systems. However, when working on a compact Lie group, this issue is resolved
by spectral localization properties, Proposition 2.1 and Corollary 2.3.1.

###  3.2. (1,1) Pseudo-differential Operator

Even in the Euclidean case, the symbol class
$\mathscr{S}^{m}_{1,1}(\mathbb{R}^{n})$ exhibits exotic properties compared to
smaller classes $\mathscr{S}^{m}_{1,\delta}(\mathbb{R}^{n})$ with $\delta<1$,
and “must remain forbidden fruit” as commented by Stein [SM93] (Subsection
1.2., Chapter 7). We thus cannot expect a satisfactory calculus for general
$\mathscr{S}^{m}_{1,1}(\mathbb{G})$ symbols. But in analogy to the Euclidean
case, a series of theorems and constructions still remain valid for
$\mathscr{S}^{m}_{1,1}(\mathbb{G})$.

######  Theorem 3.3 (Stein).

Suppose $a\in\mathscr{S}^{m}_{1,1}(\mathbb{G})$. Then for $s>0$,
$\mathbf{Op}(a)$ is a bounded linear operator from $H^{s+m}$ to $H^{s}$. The
effective version reads

$\|\mathbf{Op}(a)f\|_{H^{s}}\leq
C_{s;q}\mathbf{M}^{m}_{[s]+1,n+2;q}(a)\|f\|_{H^{s+m}},$

where $q$ is a strongly RT-admissible tuple, and the norm
$\mathbf{M}^{m}_{k,l;q}$ is defined in (2.8).

Before we proceed to the proof, let us first state several lemmas that will be
used later.

######  Lemma 3.4 (Taylor [Tay00], Lemma 4.2 of Chapter 2).

For $s<1$, there holds

$\sum_{j\in\mathbb{Z}}2^{2sj}\left|\sum_{k:k<j}2^{k-j}a_{k}\right|^{2}\leq
C_{s}\sum_{k\in\mathbb{Z}}2^{2sk}|a_{k}|^{2}.$

For $s>0$, there holds

$\sum_{j\in\mathbb{Z}}2^{2sj}\left|\sum_{k:k\geq j}a_{k}\right|^{2}\leq
C_{s}\sum_{k\in\mathbb{Z}}2^{2sk}|a_{k}|^{2}.$

As a corollary, we also have the following lemma:

######  Lemma 3.5.

Let $s>0$, $l\geq s+1$ be an integer. Fix a basis $X_{1},\cdots,X_{n}$ of
$\mathfrak{g}$, and define $X^{\alpha}$ as in Proposition 2.1. There is a
constant $C=C(s,l)$ with the following properties. If $\\{f_{k}\\}_{k\geq 0}$
is a sequence in $H^{l}(\mathbb{G})$, such that for any multi-index $\alpha$
with $|\alpha|\leq l$, there holds

$\|X^{\alpha}f_{k}\|_{L^{2}}\leq
2^{k(|\alpha|-s)}c_{k},\quad(c_{k})\in\ell^{2}(\mathbb{N}),$

then in fact $\sum_{k}f_{k}\in H^{s}(\mathbb{G})$, and

$\left\|\sum_{k\geq 0}f_{k}\right\|_{H^{s}}^{2}\leq C\sum_{k\geq 0}c_{k}^{2}.$

###### Proof.

We will use the discrete Littlewood-Paley decomposition (3.2) to conduct the
proof. If $j\leq k$, then by taking $\alpha=0$, obviously

$\|\vartheta_{j}\big{(}|\nabla|\big{)}f_{k}\|_{L^{2}}\leq\|f_{k}\|_{L^{2}}\leq
2^{-ks}c_{k}.$

If $j>k$, then by Bernstein’s inequality (Proposition 3.1),

$\displaystyle\|\vartheta_{j}\big{(}|\nabla|\big{)}f_{k}\|_{L^{2}}$
$\displaystyle\leq
C{2^{-lj}}\|\vartheta_{j}\big{(}|\nabla|\big{)}f_{k}\|_{H^{l}}$
$\displaystyle\simeq C2^{-lj}\sum_{|\alpha|=l}\|X^{\alpha}f_{k}\|_{L^{2}}$
$\displaystyle\leq C2^{-ks}2^{l(k-j)}c_{k}.$

Thus, for $f=\sum_{k}f_{k}$, we have

$2^{js}\|\vartheta_{j}\big{(}|\nabla|\big{)}f\|_{L^{2}}\leq
C\sum_{k:k<j}2^{(l-s)(k-j)}c_{k}+\sum_{k:k\geq j}2^{(j-k)s}c_{k}.$

Using Lemma 3.4, we find that
$\left\\{2^{js}\|\vartheta_{j}\big{(}|\nabla|\big{)}f\|_{L^{2}}\right\\}_{j\geq
0}$ is an $\ell^{2}$ sequence. Thus $f\in H^{s}$. ∎

###### Proof of Stein’s Theorem.

The proof here is very much like the one in [Mét08], Section 4.3. Writing
$a_{0}(x,\xi)=a(x,\xi)\phi(|\xi|)$, consider the dyadic decomposition

$a(x,\xi)=a_{0}(x,\xi)+\sum_{j=1}^{\infty}a_{j}(x,\xi),\quad\text{with}\quad
a_{j}(x,\xi)=a(x,\xi)\vartheta_{j}(|\xi|).$

Define $A_{j}=\mathbf{Op}a_{j}(x,\xi)$. By formula (2.3)-(2.6), it follows
that $A_{j}$ is represented as a convergent integral operator

$A_{j}u(x)=\int_{\mathbb{G}}u(y)\mathcal{K}_{j}(x,y^{-1}x)dy,$

where the kernel

$\mathcal{K}_{j}(x,y)=\left(a_{j}(x,\cdot)\right)^{\vee}(y)=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\operatorname{Tr}\left(a_{j}(x,\xi)\cdot\xi(y)\right).$

We fix $Q$ to be the fundamental tuple, as defined in (2.9). For any multi-
index $\beta$, by definition of RT difference operator,

$Q^{\beta}(y)\mathcal{K}_{j}(x,y)=\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\operatorname{Tr}\left((\mathbf{D}_{Q^{\beta},\xi}a_{j})(x,\xi)\cdot\xi(y)\right).$

Using the Cauchy-Schwartz inequality for trace inner product (note that if $A$
is an $n\times n$ matrix then $\llbracket A\rrbracket^{2}\leq n\|A\|^{2}$),
Weyl-type Lemma 2.4, the Leibniz type property (2.10) and Theorem 3.2, we
obtain

(3.3) $\displaystyle\left|Q^{\beta}(y)\mathcal{K}_{j}(x,y)\right|$
$\displaystyle\lesssim\sum_{\xi\in\mathfrak{S}[0,2^{j+1}]}d_{\xi}^{2}\cdot\mathbf{M}^{m}_{0,|\beta|;Q}(a)\langle{[}\rangle\xi]^{m-|\beta|}$
$\displaystyle\lesssim
2^{jn}\cdot\mathbf{M}^{m}_{0,|\beta|;Q}(a)2^{j(m-|\beta|)}.$

Summing over $\beta$ whose whose length is $\leq L$ for $L\in\mathbb{N}$,
using that the only common zero of the $Q_{i}$’s is the identity element, then
Fischer’s lemma 2.6 for any other strongly RT-admissible tuple
$q=(q_{i})_{i=1}^{M}$, we find

(3.4)
$\displaystyle\big{(}1+|2^{j}\mathrm{dist}(y,e)|^{2}\big{)}^{L}\cdot|\mathcal{K}_{j}(x,y)|^{2}$
$\displaystyle\lesssim\left[\mathbf{M}^{m}_{0,L;Q}(a)\right]^{2}2^{2jm+2jn}$
$\displaystyle\simeq_{q}\left[\mathbf{M}^{m}_{0,L;q}(a)\right]^{2}2^{2jm+2jn}.$

Hence we can estimate the weighted $L^{2}$ norm of the kernel:

$\displaystyle\int_{\mathbb{G}}\big{(}1+|2^{j}\mathrm{dist}(x,y)|^{2}$
$\displaystyle\big{)}^{[n/2]+1}|\mathcal{K}_{j}(x,y^{-1}x)|^{2}dy$
$\displaystyle\leq\int_{\mathbb{G}}\frac{\big{(}1+|2^{j}\mathrm{dist}(x,y)|^{2}\big{)}^{n+2}}{\big{(}1+|2^{j}\mathrm{dist}(x,y)|^{2}\big{)}^{n+1-[n/2]}}\cdot|\mathcal{K}_{j}(x,y^{-1}x)|^{2}dy$
$\displaystyle\lesssim\left[\mathbf{M}^{m}_{0,n+2;q}(a)\right]^{2}2^{2jm+jn}\cdot
2^{jn}\int_{\mathbb{G}}\frac{dy}{\big{(}1+|2^{j}\mathrm{dist}(x,y)|^{2}\big{)}^{n+1-[n/2]}}$
$\displaystyle\lesssim\left[\mathbf{M}^{m}_{0,n+2;q}(a)\right]^{2}2^{2jm+jn}.$

Here we estimate the last integral by splitting it into two parts
$\mathrm{dist}(x,y)<R$ and $\mathrm{dist}(x,y)\geq R$, where $R$ is the
injective radius of $\mathbb{G}$; the first part can then be estimated as an
Euclidean integral, while the second is just bounded by $2^{-nj}$, and they
both absorbs the factor $2^{jn}$.

We then proceed to $X^{\alpha}\circ A_{j}$, whose integral kernel will be
denoted as $\mathcal{K}_{j,\alpha}(x,y)$, which is just $\mathcal{K}_{j}(x,y)$
when $\alpha=0$. Using the quantization formulas (2.3)-(2.4), we find that,
with $\sigma_{X_{i}}(x,\xi)=\xi^{*}(x)\cdot(X_{i}\xi)(x)$ being the symbol of
$X_{i}$ (which is of class $S^{1}_{1,0}(\mathbb{G})$ by Fischer’s theorem),
the symbol of $X_{i}\circ A_{j}$ is

$(X_{i;x}a_{j})(x,\xi)+\sigma_{X_{i}}(x,\xi)\cdot
a_{j}(x,\xi)\in\mathscr{S}^{m+1}_{1,1}(\mathbb{G}).$

By induction on $|\alpha|$, we obtain

$\int_{\mathbb{G}}\big{(}1+|2^{j}\mathrm{dist}(x,y)|^{2}\big{)}^{[n/2]+1}|\mathcal{K}_{j,\alpha}(x,y^{-1}x)|^{2}dy\lesssim\left[\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)\right]^{2}2^{2j(m+|\alpha|)+jn}.$

Applying the Cauchy-Schwarz inequality to
$\int\mathcal{K}_{j,\alpha}(x,y^{-1}x)u(y)dy$, we find that for any $u\in
L^{2}$,

$\displaystyle\int_{\mathbb{G}}|(X^{\alpha}\circ A_{j})u(x)|^{2}dx$
$\displaystyle\lesssim\left[\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)\right]^{2}2^{2j(m+|\alpha|)+jn}\int_{\mathbb{G}}\int_{\mathbb{G}}\frac{|u(y)|^{2}}{\big{(}1+|2^{j}\mathrm{dist}(x,y)|^{2}\big{)}^{[n/2]+1}}dydx$
$\displaystyle=C\left[\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)\right]^{2}2^{2j(m+|\alpha|)}\|u\|_{L^{2}}^{2},$

or

(3.5) $\|X^{\alpha}\circ
A_{j}u\|_{L^{2}}\lesssim_{\alpha}\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)2^{j(m+|\alpha|)}\|u\|_{L^{2}}.$

Now substitute $u$ by any Littlewood-Paley building block
$\vartheta_{k}\big{(}|\nabla|\big{)}f$ of $f\in\mathcal{D}(\mathbb{G})$ in
(3.5). Since the support of $a_{j}(x,\xi)$ with respect to $\xi$ is contained
in $\mathfrak{S}[0,2^{j+1}]$, we have

$A_{j}f=\sum_{k:|k-j|\leq 3}A_{j}\vartheta_{k}\big{(}|\nabla|\big{)}f.$

Applying (3.5) with $u=\vartheta_{k}\big{(}|\nabla|\big{)}f$, we have

$\displaystyle\|X^{\alpha}A_{j}f\|_{L^{2}}$
$\displaystyle\lesssim_{\alpha}\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)2^{j(m+|\alpha|)}\sum_{k:|k-j|\leq
3}\big{\|}\vartheta_{k}\big{(}|\nabla|\big{)}f\big{\|}_{L^{2}}$
$\displaystyle\lesssim_{\alpha}\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)2^{j(|\alpha|-s)}\sum_{k:|k-j|\leq
3}2^{k(s+m)}\big{\|}\vartheta_{k}\big{(}|\nabla|\big{)}f\big{\|}_{L^{2}}.$

Since
$\sum_{k=1}^{\infty}2^{2k(s+m)}\big{\|}\vartheta_{k}\big{(}|\nabla|\big{)}f\big{\|}_{L^{2}}^{2}\simeq\|f\|_{H^{s+m}}^{2}$,
we apply Lemma 3.5 to obtain

$\|\mathbf{Op}(a)f\|_{H^{s}}=\left\|\sum_{j=0}^{\infty}A_{j}f\right\|_{H^{s}}\leq
C\mathbf{M}^{m}_{[s]+1,n+2;q}(a)\|f\|_{H^{s+m}}.$

∎

######  Remark 3.3.

We point out that the proof generalizes without much modification to Zygmund
spaces $C^{r}_{*}$ with $r>0$. Since this is not needed for our goal, we omit
the details.

We do not know whether the class $\mathbf{Op}\mathscr{S}^{m}_{1,1}$ given in
Definition 2.4-2.5 coincides with any locally defined Hörmander class of
operators or not. However, this is not a question of concern within our scope,
since all we need is a calculus for a suitable subclass of symbols and
operators, broad enough to cover operators arising from classical differential
operators.

###  3.3. Spectral Condition

The constructions and theorems so far rely mostly, if not completely, on the
compact manifold structure of $\mathbb{G}$. From now on, we will exploit more
algebraic properties of $\mathbb{G}$. In particular, the spectral localization
property, namely Corollary 2.3.1, will play an important role. Just as in the
Euclidean case, we introduce a subclass of $\mathscr{S}^{m}_{1,1}(\mathbb{G})$
that satisfies a spectral condition. The exposition in this subsection is
parallel to that of [Mét08], Chapter 4, or [Hör97], Chapter 9-10.

Before we pass to the definition of spectral condition, we need a lemma
concerning difference operators acting on symbols with finite Fourier support.
Being trivial for Euclidean or toric case, for non-commutative groups it
relies on spectral localization.

######  Lemma 3.6.

Suppose the symbol $a=a(\xi)$ has Fourier support contained in
$\mathfrak{S}[\gamma_{1},\gamma_{2}]$. Then with $Q$ being the fundamental
tuple defined in (2.9), there is a constant $C$ depending only on the
algebraic structure of $\mathbb{G}$, such that for any $Q_{i}$, the difference
$\mathbf{D}_{Q_{i}}a$ vanishes for $|\xi|\notin[\gamma_{1}-C,\gamma_{2}+C]$.

###### Proof.

We use formula (2.7):

$(\mathbf{D}_{Q_{i}}a)(\xi)=\int_{\mathbb{G}}Q_{i}(y)\left(\sum_{\eta\in\mathfrak{S}[\gamma_{1},\gamma_{2}]}d_{\eta}\operatorname{Tr}\left(a(\eta)\cdot\eta(y)\right)\right)\xi^{*}(y)dy.$

By the proof of Corollary 2.3.1, the product $Q_{i}(y)\eta(y)$ is the sum of
matrix entries of irreducible representations whose highest weights
$\bm{j}\in\mathscr{L}_{+}$ satisfy

$\bm{J}(\eta)-\bm{\varpi}_{i}\preceq\bm{j}\preceq\bm{J}(\eta)+\bm{\varpi}_{i}.$

Such $\bm{j}$’s must fall into a bounded parallelogram in $\mathfrak{t}^{*}$
that contains $\bm{J}(\eta)$, so there must hold

$|\bm{J}(\eta)|-C\leq|\bm{j}|\leq|\bm{J}(\eta)|+C,$

where $C\simeq\sum_{i=1}^{\varrho}|\bm{\varpi}_{i}|$. Hence the Fourier
support of

$Q_{i}(y)\left(\sum_{\eta\in\mathfrak{S}[\gamma_{1},\gamma_{2}]}d_{\eta}\operatorname{Tr}\left(a(\eta)\cdot\eta(y)\right)\right)$

is contained in $\mathfrak{S}\big{[}\gamma_{1}-C,\gamma_{2}+C\big{]}$, with
$C\simeq\sum_{i=1}^{\varrho}|\bm{\varpi}_{i}|$. ∎

We turn to the definition of spectral condition. Since no natural “addition”
is available on $\widehat{\mathbb{G}}$, we restrict ourselves to a narrower
definition of spectral condition:

######  Definition 3.1.

Fix $\delta>0$. The subclass
$\Sigma_{\delta}^{m}(\mathbb{G})\subset\mathscr{S}^{m}_{1,1}(\mathbb{G})$
consists of all symbols $a\in\mathscr{S}^{m}_{1,1}(\mathbb{G})$ such that the
partial Fourier transform of matrix entries of $a(x,\xi)$ satisfies

$\widehat{a_{ij}}(\eta,\xi):=\int_{\mathbb{G}}a_{ij}(x,\xi)\eta^{*}(x)dx=0\quad\mathrm{if}\quad|\eta|\geq\delta\langle{[}\rangle\xi],\quad
i,j=1,\cdots,d_{\xi},$

when $\langle{[}\rangle\xi]$ is large enough. This is called the spectral
condition with parameter $\delta$.

With a little abuse of notation, we may write

$\widehat{a}(\eta,\xi)=\big{[}\widehat{a_{ij}}(\eta,\xi)\big{]}_{i,j=1}^{d_{\xi}}\in\mathrm{End}(\mathcal{H}_{[}\eta]\otimes\mathcal{H}_{[}\xi]),$

and abbreviate the spectral condition as $\widehat{a}(\eta,\xi)=0$ for
$|\eta|\geq\delta\langle{[}\rangle\xi]$. Since the symbol $a$ is smooth in
$x$, the spectral condition is only important for high modes, i.e. when
$\langle{[}\rangle\xi]$ is large: if $a(x,\xi)$ vanishes for all but finitely
many $\xi\in\widehat{\mathbb{G}}$, then in fact $a\in\mathscr{S}^{-\infty}$,
so the “low frequency part of $a(x,\xi)$” does not cause any problem in
regularity. This is why the spectral condition is required only for
_sufficiently large $\langle{[}\rangle\xi]$._

We list some basic properties of $\Sigma^{m}_{\delta}(\mathbb{G})$ that will
be used repeatedly.

######  Lemma 3.7.

(1) If $a\in\Sigma^{m}_{\delta}(\mathbb{G})$,
$b\in\Sigma^{m^{\prime}}_{\delta^{\prime}}(\mathbb{G})$, then $a\cdot
b\in\Sigma^{m+m^{\prime}}_{\delta+\delta^{\prime}}(\mathbb{G})$.

(2) If $a\in\Sigma^{m}_{\delta}(\mathbb{G})$, $X\in\mathfrak{g}$ is a left-
invariant vector field, then the symbol of $X\circ\mathbf{Op}(a)$ is of order
$m+1$, and still satisfies a spectral condition with parameter $\delta$.

(3) Suppose $a\in\Sigma^{m}_{\delta}$. Let $Q$ be the fundamental tuple as
defined in (2.9). Then given any multi-index $\alpha$, there is a constant
$C_{\alpha}$ depending only on $\alpha$ and the algebraic structure of
$\mathbb{G}$, such that the symbol
$\mathbf{D}_{Q}^{\alpha}a\in\mathscr{S}^{m-|\alpha|}_{1,1}$, and satisfies the
spectral condition with parameter $C_{\alpha}\delta$.

###### Proof.

Part (1) is a direct consequence of the spectral localization property, namely
Corollary 2.3.1. For part (2), write $A=\mathbf{Op}(a)$. For a left-invariant
vector field $X\in\mathfrak{g}$, the symbol of $X\circ A$ reads

(3.6) $(X_{x}a)(x,\xi)+\sigma_{X}(x,\xi)\cdot
a(x,\xi)\in\mathscr{S}^{m+1}_{1,1},$

as seen in the proof of Stein’s theorem. By the right convolution kernel
formula (2.5)-(2.6), we find that the right convolution kernel
$\mathcal{K}(x,y)$ of $X$ is invariant under left translation with respect to
$x$, so in fact $\mathcal{K}(x,y)\equiv\mathcal{K}(e,y)$, and the symbol
$\sigma_{X}(x,\xi)$ is thus independent from $x$. As a result, it is
legitimate to write it as
$\sigma_{X}(\xi)\in\mathrm{End}(\mathcal{H}_{[}\xi])$, and refer it as a _non-
commutative Fourier multiplier_. Consequently, the Fourier transform of (3.6)
with respect to the $x$ variable evaluated at $\eta\in\widehat{\mathbb{G}}$ is

$\sigma_{X}(\eta)\cdot\widehat{a}(\eta,\xi)+\left(\sigma_{X}(\xi)\cdot
a\right)^{\wedge}(\eta,\xi),$

which still vanishes for $|\eta|\geq\delta\langle{[}\rangle\xi]$.

As for part (3), we just have to prove the claim for $|\alpha|=1$ as the rest
follows by induction. Theorem 2.5 ensures that
$\mathbf{D}_{Q}a\in\mathscr{S}^{m-1}_{1,1}$. Here with a little abuse of
notation we use $Q$ to denote any component of the fundamental tuple. To
verify the spectral condition, we employ formula (2.7):

$(\mathbf{D}_{Q}a)(x,\xi)=\int_{\mathbb{G}}Q(y)\left(\sum_{\zeta\in\widehat{\mathbb{G}}}d_{\zeta}\sum_{i,j=1}^{d_{\zeta}}a_{ij}(x,\zeta)\zeta_{ij}(y)\right)\xi^{*}(y)dy.$

Taking Fourier transform with respect to $x$, evaluating at
$\eta\in\widehat{\mathbb{G}}$ and using the spectral condition, we obtain

(3.7)
$\widehat{\mathbf{D}_{Q}a}(\eta,\xi)=\sum_{\zeta\in\widehat{\mathbb{G}}:\langle{[}\rangle\zeta]\geq\delta^{-1}|\eta|}d_{\zeta}\sum_{i,j=1}^{d_{\zeta}}\widehat{a_{ij}}(\eta,\zeta)\otimes\int_{\mathbb{G}}Q(y)\zeta_{ij}(y)\xi^{*}(y)dy\in\mathrm{End}(\mathcal{H}_{[}\eta]\otimes\mathcal{H}_{[}\xi]).$

From the proof of Lemma 3.6, we find that $Q(y)\xi^{*}(y)$ has Fourier support
contained in $\mathfrak{S}\big{[}|\xi|-C,|\xi|+C\big{]}$, where $C$ depends
only on the algebraic property of $\mathbb{G}$.

Now suppose $|\eta|\geq 2\delta\langle{[}\rangle\xi]$. Then for each summand
in (3.7), the representation $\zeta$ satisfies $|\zeta|\geq
2\langle{[}\rangle\xi]$. By Schur orthogonality, matrix entries of $\zeta$ is
thus $L^{2}$ orthogonal to matrix entries of $\bar{Q}(y)\xi(y)$. This shows
that for $|\eta|\geq 2\delta\langle{[}\rangle\xi]$, each summand in (3.7)
vanishes. ∎

We are now at the place to state the fundamental theorem for para-differential
calculus: (1,1) symbols satisfying a suitable spectral condition give rise to
operators bounded in all Sobolev spaces.

######  Theorem 3.8.

Let $\delta\in(0,1/2)$, $m\in\mathbb{R}$. Suppose
$a\in\Sigma_{\delta}^{m}(\mathbb{G})$. Then the operator $\mathbf{Op}(a)$ maps
$H^{s+m}(\mathbb{G})$ to $H^{s}(\mathbb{G})$ continuously for all
$s\in\mathbb{R}$. Quantitatively, with $s_{+}=0$ for $s\leq 0$ and $s_{+}=s$
for $s>0$, for any strongly RT-admissible tuple $q$,

$\|\mathbf{Op}(a)f\|_{H^{s}}\leq
C_{s,\delta;q}\mathbf{M}^{m}_{[s_{+}]+1,n+2;q}(a)\|f\|_{H^{s+m}}.$

Before we proceed to the proof, we still need a summation Lemma in Sobolev
spaces.

######  Lemma 3.9.

Let $s\in\mathbb{R}$, $l>s$ be a positive integer, and $\gamma>0$. Fix a basis
$X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, and define $X^{\alpha}$ as in
Proposition 2.1. There is a constant $C=C(s,l,\gamma)$ with the following
properties. If $\\{f_{k}\\}_{k\geq 0}$ is a sequence in $H^{l}(\mathbb{G})$,
such that for any multi-index $\alpha$ with $|\alpha|\leq l$, there holds

$\|X^{\alpha}f_{k}\|_{L^{2}}\leq
2^{k(|\alpha|-s)}c_{k},\quad(c_{k})\in\ell^{2}(\mathbb{N}),$

and furthermore $\widehat{f}_{k}$ is supported in
$\\{\xi:\langle{[}\rangle\xi]\geq\gamma 2^{k}\\}$, then $\sum_{k}f_{k}\in
H^{s}(\mathbb{G})$ and

$\left\|\sum_{k\geq 0}f_{k}\right\|_{H^{s}}^{2}\leq C\sum_{k\geq 0}c_{k}^{2}.$

###### Proof.

The proof is quite similar for that of Lemma 3.5. The only difference is as
follows: the spectral condition
$\mathrm{supp}\widehat{f}_{k}\subset\\{\xi:\langle{[}\rangle\xi]\geq\gamma
2^{k}\\}$ implies that for some integer $N\sim\log\gamma$, the Littlewood-
Paley building block $\vartheta_{j}\big{(}|\nabla|\big{)}f_{k}$ vanishes for
$j<k-N$. So for $f=\sum_{k}f_{k}$, we obtain similarly as in Lemma 3.5,

$2^{js}\|\vartheta_{j}\big{(}|\nabla|\big{)}f\|_{L^{2}}\leq
C\sum_{k:k<j}2^{(l-s)(k-j)}c_{k}+\sum_{k:j\leq k<j+N}2^{(j-k)s}c_{k}.$

The first sum gives rise to terms of an $\ell^{2}$ sequence again by Lemma
3.4, while the second obviously gives rise to terms of an $\ell^{2}$ sequence.
Thus again
$\\{2^{js}\big{\|}\vartheta_{j}\big{(}|\nabla|\big{)}f\big{\|}_{L^{2}}\\}_{j\geq
0}$ is an $\ell^{2}$ sequence. The rest of the proof is identical to that of
Lemma 3.5. ∎

###### Proof of Theorem 3.8.

The proof is still very much similar to that of Theorem 4.3.4. in [Mét08], and
proceeds basically parallelly as Stein’s theorem. We still decompose
$\mathbf{Op}(a)f$ into a dyadic sum $\sum_{j\geq 0}A_{j}f$, where $A_{j}$ is
the quantization of $a_{j}(x,\xi):=a(x,\xi)\vartheta_{j}(|\xi|)$. The estimate
of operator norms for $X^{\alpha}\circ A_{j}$ is identical to that in the
proof of Theorem 3.3. There are only some minor differences: since
$\widehat{a}(\eta,\xi)=0$ for $|\eta|>\delta\langle{[}\rangle\xi]$, by the
quantization formula (2.3)-(2.4) we have

$\widehat{A_{j}f}(\eta)=\int_{\mathbb{G}}\sum_{\xi\in\mathfrak{S}[2^{j-1},2^{j}]}d_{\xi}\operatorname{Tr}\left(\xi(x)a_{j}(x,\xi)\widehat{f}(\xi)\right)\eta^{*}(x)dx.$

By spectral localization property 2.3.1 and the spectral condition
$\widehat{a_{j}}(\eta,\xi)=0$ for $|\eta|\geq\delta 2^{j}$ (since $|\xi|\leq
2^{j}$), we find that the integrand
$\operatorname{Tr}\left(\xi(x)a_{j}(x,\xi)\widehat{f}(\xi)\right)$ has Fourier
support (with respect to $x$) contained in

$\left\\{\eta\in\widehat{\mathbb{G}}:|\eta|\geq
c\left(\frac{1}{2}-\delta\right)2^{j}\right\\},$

where $c\in(0,1)$ depends only on the algebraic structure of $\mathbb{G}$,
i.e. $\widehat{A_{j}f}(\eta)=0$ for $|\eta|\leq c(1/2-\delta)2^{j}$. By Lemma
3.7 and induction on $|\alpha|$, the symbol of $X^{\alpha}\circ a$ still
satisfies the spectral condition with parameter $\delta$. To summarize, for
any $f\in\mathcal{D}^{\prime}(\mathbb{G})$, the Fourier support of
$X^{\alpha}A_{j}f$ is within
$\\{\eta\in\widehat{\mathbb{G}}:|\eta|>c(1/2-\delta)2^{j}\\}$.

With exactly the same argument as in the proof of Theorem 3.3, the estimate

$\|X^{\alpha}A_{j}f\|\lesssim_{\alpha}\mathbf{M}^{m}_{|\alpha|,n+2;q}(a)2^{j(|\alpha|-s)}\sum_{k:|k-j|\leq
3}2^{k(s+m)}\big{\|}\vartheta_{k}\big{(}|\nabla|\big{)}f\big{\|}_{L^{2}}$

is valid. We can then apply Lemma 3.9 to conclude. ∎

######  Remark 3.4.

The proof for Theorem 3.3 and 3.8 proceed with exactly the same idea as in the
Euclidean case, since the product spectral localization property does not
differ very much from the Euclidean case under dyadic decomposition. We thus
expect that Hörmander’s characterization of the class
$\tilde{\Psi}^{m}_{1,1}(\mathbb{R}^{n})$, namely (1,1) pseudo-differential
operators that are continuous on all Sobolev spaces (see [Hör88] or Chapter 9
of [Hör97] for the details), admits a direct generalization to $\mathbb{G}$.
But we content ourselves with operators satisfying the spectral condition,
because this is what we actually need for para-differential calculus. We only
write $\tilde{\mathscr{S}}^{m}_{1,1}(\mathbb{G})$ for the subclass of $(1,1)$
symbols whose quantization are continuous on every Sobolev space, and do not
characterize it explicitly. The inclusion
$\Sigma_{<1/2}^{m}\subset\tilde{\mathscr{S}}^{m}_{1,1}$ is quite sufficient
for most of our applications.

###  3.4. Para-product and Para-linearization on $\mathbb{G}$

As an initial application, we are able to recover the para-product estimate
and Bony’s para-linearization theorem on $\mathbb{G}$, just as in [BGdP21],
but this time with a global notion of para-product. We point out that in
[KR06], the authors also constructed para-product in terms of spectral
calculus on a general compact surface. Since $\mathbb{G}$ obviously enjoys
more structure than a general manifold, we are able to recover almost
everything in the Euclidean setting.

Given two distributions $a$ and $u$, we define the _para-product_ $T_{a}u$ by

(3.8)
$T_{a}u:=\int_{1}^{\infty}\left[\phi_{2^{-10}t}\big{(}|\nabla|\big{)}a\right]\cdot\psi_{t}\big{(}|\nabla|\big{)}u\frac{dt}{t},$

where $\phi$ and $\psi$ are as in the continuous Littlewood-Paley
decomposition (3.1). Similarly as in the Euclidean case, the gap 10 is
inessential. By the spectral localization property i.e. Corollary 2.3.1, the
integrand has Fourier support contained in $\mathfrak{S}[ct,c^{-1}t]$ for some
constant $c\in(0,1)$, depending only on the algebraic structure of
$\mathbb{G}$. The symbol of $T_{a}$ is

$\sigma[T_{a}](x,\xi)=\int_{1}^{\infty}\left[\phi_{2^{-10}t}\big{(}|\nabla|\big{)}a(x)\right]\cdot\psi_{t}\big{(}|\xi|\big{)}\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi]\frac{dt}{t}.$

######  Proposition 3.4.

If $a\in L^{\infty}$, the symbol $\sigma[T_{a}]$ of the para-product operator
$T_{a}$ is of class $\Sigma^{0}_{2^{-10}}$.

###### Proof.

The spectral condition is obvious since $\psi_{t}(|\xi|)\neq 0$ only for
$|\xi|\leq t$. We turn to estimate the operator norm of
$X^{\alpha}_{x}\mathbf{D}_{q,\xi}^{\beta}\sigma[T_{a}](x,\xi)(x,\xi)$ on
$\mathcal{H}_{[}\xi]$, for a given basis $X_{1},\cdots,X_{n}$ of
$\mathfrak{g}$ and $q$ is a strongly RT-admissible tuple. But this is
immediate from Lemma 3.1 and Theorem 3.2, and quantatively
$\mathbf{M}^{0}_{k,l;q}\big{(}\sigma[T_{a}]\big{)}\lesssim_{k,l}\langle{[}\rangle\xi]^{k-l}|a|_{L^{\infty}}$.
∎

By Theorem 3.8, the para-product operator $T_{a}$ is thus continuous from
$H^{s}$ to itself for all $s\in\mathbb{R}$. This of course also admits a more
direct proof. We can simply insert the multiplier corresponding to a function
$h\in C_{0}^{\infty}(0,\infty)$, such that $h(\lambda)\equiv 1$ for
$\lambda\in[c,c^{-1}]$ and $h\geq 0$. Then the Fourier support of $T_{a}u$ is
away from the trivial representation $\xi\equiv 1$, and

$T_{a}u=\int_{0}^{\infty}h_{t}\big{(}|\nabla|\big{)}\Big{(}\phi_{2^{-10}t}\big{(}|\nabla|\big{)}a\cdot\psi_{t}\big{(}|\nabla|\big{)}u\Big{)}\frac{dt}{t}.$

Write
$v(t)=\phi_{2^{-10}t}\big{(}|\nabla|\big{)}a\cdot\psi_{t}\big{(}|\nabla|\big{)}u$
for the integrand. It has Fourier support contained in
$\mathfrak{S}[ct,c^{-1}t]$. By Bernstein’s inequality and Lemma 3.1, we find
that

$\displaystyle\|v(t)\|_{H^{s}}$ $\displaystyle\lesssim
t^{s}\left|\phi_{2^{-10}t}\big{(}|\nabla|\big{)}a\right|_{L^{\infty}}\big{\|}\psi_{t}\big{(}|\nabla|\big{)}u\big{\|}_{L^{2}}$
$\displaystyle\lesssim
t^{s}|a|_{L^{\infty}}\big{\|}\psi_{t}\big{(}|\nabla|\big{)}u\big{\|}_{L^{2}}.$

So by the Peter-Weyl theorem,

$\displaystyle\|T_{a}u\|_{H^{s}}^{2}$
$\displaystyle\simeq\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}|\xi|^{2s}\left\llbracket\int_{0}^{\infty}h_{t}(|\xi|)\cdot\widehat{v(t)}(\xi)\frac{dt}{t}\right\rrbracket^{2}$
$\displaystyle\leq\sum_{\xi\in\widehat{\mathbb{G}}}d_{\xi}\int_{0}^{\infty}\big{|}h_{t}(|\xi|)\big{|}^{2}\frac{dt}{t}\cdot\int_{0}^{\infty}|\xi|^{2s}\big{\llbracket}\widehat{v(t)}(\xi)\big{\rrbracket}^{2}\frac{dt}{t}$
$\displaystyle\lesssim\int_{0}^{\infty}\|v(t)\|_{H^{s}}^{2}\frac{dt}{t}\lesssim|a|_{L^{\infty}}^{2}\|u\|_{H^{s}}^{2},$

where we used Proposition 3.2 at the last step. Note that the integrals in the
above are not singular since the integrands all vanish near $t=0$.

Just as in the Euclidean case, we have the para-product decomposition on
$\mathbb{G}$:

######  Theorem 3.10.

If $s>0$, $a,u\in(L^{\infty}\cap H^{s})(\mathbb{G})$, then

$au=T_{a}u+T_{u}a+R(a,u),$

where the smoothing remainder

$\|R(a,u)\|_{H^{s}}\lesssim|a|_{L^{\infty}}\|u\|_{H^{s}}.$

###### Proof.

It suffices to investigate the remainder $R(a,u)$. Using the continuous
Littlewood-Paley decomposition (3.1),

$R(a,u)=\int_{\begin{subarray}{c}2^{-10}\leq t_{1}/t_{2}\leq 2^{10}\\\ 1\leq
t_{1},t_{2}\end{subarray}}\psi_{t_{1}}\big{(}|\nabla|\big{)}a\cdot\psi_{t_{2}}\big{(}|\nabla|\big{)}u\cdot\frac{dt_{1}dt_{2}}{t_{1}t_{2}}+\phi\big{(}|\nabla|\big{)}a\cdot\phi\big{(}|\nabla|\big{)}u.$

Only the integral is non-trivial. We make a change of variable
$(t_{1},t_{2})=t(\cos\varphi,\sin\varphi)$, so that it equals

$\int_{2^{-10}\leq\tan\varphi\leq
2^{10}}\left(\int_{1}^{\infty}\psi_{t\cos\varphi}\big{(}|\nabla|\big{)}a\cdot\psi_{t\sin\varphi}\big{(}|\nabla|\big{)}u\cdot\frac{dt}{t}\right)\sin\varphi
d\varphi.$

Then $R(a,u)=\mathbf{Op}(R[a])u$, where the symbol

$R[a](x,\xi)=\int_{2^{-10}\leq\tan\varphi\leq
2^{10}}\left(\int_{1}^{\infty}\psi_{t\cos\varphi}\big{(}|\nabla|\big{)}a\cdot\psi_{t\sin\varphi}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]\cdot\frac{dt}{t}\right)\sin\varphi
d\varphi.$

Just as in the proof of the previous proposition, we employ Lemma 3.1 and
Theorem 3.2 to conclude, for a strongly RT-admissible tuple $q$,

$\mathbf{M}^{0}_{k,l;q}(R[a])\lesssim_{k,l}\langle{[}\rangle\xi]^{k-l}|a|_{L^{\infty}}.$

Thus by Stein’s theorem, $\mathbf{Op}(R[a])$ maps $H^{s}$ to itself for $s>0$.
∎

######  Remark 3.5.

If in addition $a\in C^{r}_{*}(\mathbb{G})$, then from the spectral
characterization of Zygmund spaces, we find that in fact
$R[a]\in\mathscr{S}^{-r}_{1,1}$, so

$\|R(a,u)\|_{H^{s}}\lesssim|a|_{C^{r}_{*}}\|u\|_{H^{s-r}}.$

We also have a direct generalization of Bony’s para-linearization theorem,
proposed by Bony [Bon81]. Such a result is neither proved true or false for
general compact manifolds in [KR06].

######  Theorem 3.11 (Bony).

Suppose $F\in C^{\infty}(\mathbb{C};\mathbb{C})$ (understood as smooth mapping
on the plane instead of holomorphic function), $F(0)=0$. Let $r>0$, and
suppose $u\in C^{r}_{*}$. Write $u_{t}=\phi(D/t)u$, where $\phi$ is the
function in the Littlewood-Paley decomposition (3.1). Then with the symbol

$l_{u}(x,\xi):=\int_{1}^{\infty}F^{\prime}(u_{t}(x))\cdot\psi_{t}\big{(}|\xi|\big{)}\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi]\frac{dt}{t},$

there holds $F(u)=F(u_{1})+\mathbf{Op}(l_{u})u$, and we have the para-
linearization formula

$F(u)=F(u_{1})+T_{F^{\prime}(u)}u+\mathbf{Op}(R[u])u$

with the symbol $R[u](x,\xi)\in\mathscr{S}^{-r}_{1,1}$. Consequently, if $u\in
C^{r}_{*}\cap H^{s}$, then $F(u)-T_{F^{\prime}(u)}u\in H^{s+r}$. In
particular, for $s>n/2$, $F(u)-T_{F^{\prime}(u)}u\in H^{2s-n/2}$.

###### Proof.

The proof resembles that of Theorem 10.3.1. in [Hör97] very much. We first set
$u_{t}=\phi_{t}\big{(}|\nabla|\big{)}u$, then define a symbol

$l_{u}(x,\xi):=\int_{1}^{\infty}F^{\prime}(u_{t}(x))\cdot\psi_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]\frac{dt}{t}$

That $F(u)=F(u_{1})+l_{u}(x,D)u$ is obtained by a direct computation already
done when estimating the symbol of $T_{a}$, which also shows
$l_{u}(x,\xi)\in\mathscr{S}^{0}_{1,1}$. Working in local coordinate charts, we
find $F(u),F^{\prime}(u)\in C^{r}_{*}$.

Let us set $p(x,\xi)$ as the symbol of the para-product operator
$T_{F^{\prime}(u)}$. For any strongly RT-admissible tuple $q$, we estimate the
$\mathbf{M}_{k,l;q}^{-r}$ norm of

$R[u](x,\xi):=l_{u}(x,\xi)-p(x,\xi)=\int_{1}^{\infty}\Big{[}F^{\prime}(u_{t}(x))-\phi_{2^{-10}t}\big{(}|\nabla|\big{)}F^{\prime}(u(x))\Big{]}\psi_{t}(|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\xi]\frac{dt}{t}.$

In view of Lemma 3.1 and Corollary 2.5 once again, all we need to show is that
the magnitude of

(3.9)
$X^{\alpha}_{x}\Big{[}F^{\prime}(u_{t})-\phi_{2^{-10}t}\big{(}|\nabla|\big{)}F^{\prime}(u)\Big{]}$

has an upper bound $K_{\alpha}(|u|_{C^{r}_{*}})t^{|\alpha|-r}$, where
$K_{\alpha}$ is an increasing function.

For $\alpha=0$, we find the quantity (3.9) equals

$F^{\prime}(u_{t})-\phi_{2^{-10}t}\big{(}|\nabla|\big{)}F^{\prime}(u)=F^{\prime}(u_{t})-F^{\prime}(u)+F^{\prime}(u)-\phi_{2^{-10}t}\big{(}|\nabla|\big{)}F^{\prime}(u).$

By Corollary 3.2.1, we have $|u-u_{t}|\lesssim t^{-r}|u|_{C^{r}_{*}}$, and
since $F^{\prime}(u)\in C^{r}_{*}$, we find that (3.9) is controlled by
$t^{-r}$ for $\alpha=0$.

For $|\alpha|>r$, the second term in (3.9) is controlled by
$t^{|\alpha|-r}|u|_{C^{r}_{*}}$ with the aid of Corollary 3.2.1, since
$F^{\prime}(u)\in C^{r}_{*}$. We note that being $C^{r}_{*}$ is a local
property, so it is legitimate to work in any local coordinate chart $\\{x\\}$
and estimate the usual local partial derivatives of the ambient function.
Using the chain rule, we find that the first term in (3.9) is a
$C^{\infty}$-linear combination of terms of the form

$F^{(1+|\alpha|-|\beta|)}(u_{t})\left(\partial_{x}^{\beta_{1}}u_{t}\right)^{j_{1}}\cdots\left(\partial_{x}^{\beta_{k}}u_{t}\right)^{j_{k}},$

where $\beta=\beta_{1}+\cdots+\beta_{k}$, and
$j_{1}|\beta_{1}|+\cdots+j_{k}|\beta_{k}|=|\alpha|$. Using Corollary 3.2.1
again, we can bound this term by

$K(|u|_{C^{r}_{*}})\prod_{i:\beta_{i}>r}\left(|u|_{C^{r}_{*}}t\right)^{j_{i}(|\beta_{i}|-r)}\cdot\prod_{i:\beta_{i}\leq
r}\left(|u|_{C^{r}_{*}}\log(1+t)\right)^{j_{i}}=O(t^{|\alpha|-r}).$

As for $0<|\alpha|\leq r$, we can just use the classical interpolation
inequality for smooth functions with all derivatives bounded, in between 0 and
$r+1$:

$|D^{j}f|_{L^{\infty}}\lesssim_{j,k}|f|_{C^{k}}^{j/k}\cdot|f|_{L^{\infty}}^{1-j/k},\quad
0<j<k.$

This implies that (3.9) is controlled by $t^{|\alpha|-r}$, hence ensures
$R[u]\in\mathscr{S}^{-r}_{1,1}$. The claim of the theorem now follows from
Stein’s theorem. ∎

##  4\. Para-differential Operators on Compact Lie Group

###  4.1. Rough Symbols and Their Smoothing

Because of the spectral localization property on $\mathbb{G}$, the theory of
para-differential operators resembles that on $\mathbb{R}^{n}$ very much. The
details of the latter may be found in Chapter 9-10 in [Hör97]. For
completeness and convenience of future reference, we still write down the
details here.

######  Definition 4.1.

For $r\geq 0$ and $m\in\mathbb{R}$, define the symbol class
$\mathcal{A}_{r}^{m}(\mathbb{G})$ to be the collection of all symbols
$a(x,\xi)$ on $\mathbb{G}\times\widehat{\mathbb{G}}$, such that if
$q=(q_{i})_{i=1}^{M}$ is a strongly RT-admissible tuple, then

$\big{\|}\mathbf{D}_{q,\xi}^{\beta}a(x,\xi)\big{\|}_{r;x}\leq
C_{\beta}\langle{[}\rangle\xi]^{m-|\beta|},\quad\forall\xi\in\widehat{\mathbb{G}}.$

Here the norm $\|a(x,\xi)\|_{r;x}$ for
$a(\cdot,\xi):\mathbb{G}\to\mathrm{End}(\mathcal{H}_{[}\xi])$ is defined by

$\|a(x,\xi)\|_{r;x}:=\sup_{B^{*}\in\mathrm{End}(\mathcal{H}_{[}\xi])^{*}}\frac{\big{|}\langle
B^{*},a(x,\xi)\rangle\big{|}_{C^{r}_{*x}}}{\|B^{*}\|},$

as in Remark 3.2, that is we consider $\mathrm{End}(\mathcal{H}_{[}\xi])$
merely as a normed space instead of a normed algebra. Introduce the following
norm on $\mathcal{A}_{r}^{m}(\mathbb{G})$:

$\mathbf{W}^{m;r}_{l;q}(a):=\sup_{\xi\in\widehat{\mathbb{G}}}\sum_{|\beta|\leq
l}\langle{[}\rangle\xi]^{|\beta|-m}\big{\|}\mathbf{D}_{q,\xi}^{\beta}a(x,\xi)\big{\|}_{r;x}$

We note that Fischer’s Lemma 2.6 ensures that the definition of
$\mathcal{A}^{m}_{r}(\mathbb{G})$ does not depend on the choice of $q$, so we
can choose any set of functions $q$ that facilitates our computation. In
particular, the fundamental tuple $Q$ defined in (2.9) is a convenient choice.

######  Definition 4.2 (Admissible cut-off).

An admissible cut-off function $\chi$ with parameter $\delta\in(0,1/2)$ is a
smooth function on $\mathbb{R}\times\mathbb{R}$ such that

$\chi(\mu,\lambda)=\left\\{\begin{aligned}
1,&\quad|\mu|\leq\frac{\delta}{2}\langle{[}\rangle\lambda],\\\
0,&\quad|\mu|\geq\delta\langle{[}\rangle\lambda],\end{aligned}\right.$

and there also holds

$|\partial_{\mu}^{k}\partial_{\lambda}^{l}\chi(\mu,\lambda)|\lesssim_{k,l}\langle\lambda\rangle^{-k-l}.$

An obvious choice of admissible cut-off function is the one that we used to
construct para-product (3.8):

(4.1) $\chi(\mu,\lambda)=\int_{1}^{\infty}\phi_{\delta
t/2}(\mu)\psi_{t}(\lambda)\frac{dt}{t}.$

However, it will be shown shortly that there is a lot of flexibility in
choosing an admissible cut-off function.

######  Definition 4.3 (Para-differential Operator).

Let $\chi$ be an admissible cut-off function with some parameter $\delta$. For
$a\in\mathcal{A}_{r}^{m}(\mathbb{G})$, set
$a^{\chi}(x,\xi)=\chi(|\nabla_{x}|,|\xi|)a(x,\xi)$, the regularized symbol
corresponding to $a$. Define the para-differential operator $T_{a}^{\chi}$
corresponding to $a\in\mathcal{A}_{r}^{m}(\mathbb{G})$ as

$T_{a}^{\chi}u(x):=\mathbf{Op}(a^{\chi})u(x).$

We immediately deduce that $a^{\chi}$ is a (1,1) symbol satisfying the
spectral condition with parameter $\delta$ with improved growth estimate:

######  Proposition 4.1.

Suppose $r\geq 0$. Fix a basis $X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, and
define $X^{\alpha}$ as in Proposition 2.1. Suppose
$a\in\mathcal{A}_{r}^{m}(\mathbb{G})$ and $\chi$ is an admissible cut-off
function with parameter $\delta$. If $r=0$, then
$a^{\chi}\in\Sigma^{m+\varepsilon}_{\delta}(\mathbb{G})\subset\mathscr{S}^{m+\varepsilon}_{1,1}(\mathbb{G})$.
If $r>0$, then
$a^{\chi}\in\Sigma^{m}_{\delta}(\mathbb{G})\subset\mathscr{S}^{m}_{1,1}(\mathbb{G})$
for some $\delta>0$. In fact, for any strongly RT-admissible tuple $q$, we
have

$\big{\|}X_{x}^{\alpha}\mathbf{D}_{q,\xi}^{\beta}a^{\chi}(x,\xi)\big{\|}\lesssim_{\alpha,\beta;q}\left\\{\begin{aligned}
&\mathbf{W}^{m;r}_{|\beta|;q}(a)\cdot\langle{[}\rangle\xi]^{m+(|\alpha|-r)_{+}-|\beta|}&\quad|\alpha|\neq
r\\\
&\mathbf{W}^{m;r}_{|\beta|;q}(a)\cdot\langle{[}\rangle\xi]^{m-|\beta|}\log(1+\langle{[}\rangle\xi])&\quad|\alpha|=r\end{aligned}\right.,$

where $s_{+}=\max(s,0)$. The second inequality matters only when $r$ is a
positive integer.

###### Proof.

That $a^{\chi}$ satisfies the spectral condition with parameter $\delta$ is
immediate from the definition: the Fourier cut-off operator
$\chi(|\nabla_{x}|,|\xi|)$ annihilates all frequencies of size greater than
$\delta\langle{[}\rangle\xi]$. We turn to verify the estimates for the
$\mathscr{S}^{m}_{1,1}$ norms of $a^{\chi}$. By Fischer’s lemma i.e. Lemma
2.6, it suffices to prove the inequality for $q=Q$, the fundamental tuple
(2.9). The advantage is that the corresponding RT difference operators have
Leibniz type property, and all the $\mathbf{D}_{Q}^{\beta}a^{\chi}$’s also
satisfy spectral conditions (with parameters proportional to $\delta$) by
Lemma 3.7.

For $\beta=0$, the estimate for $\|X^{\alpha}_{x}a^{\chi}(x,\xi)\|$ follows
from the vector-valued version of Corollary 3.2.1 (see Remark 3.2), where the
normed space considered is $\mathcal{H}_{[}\xi]$. To obtain the estimate for
$\|X_{x}^{\alpha}\mathbf{D}_{Q,\xi}^{\beta}a^{\chi}(x,\xi)\|$ with $\beta\neq
0$, we just have to apply Theorem 3.2. ∎

We notice that the norm of $a^{\chi}(x,\xi)$ grows slower upon differentiation
in $x$ than generic $\mathscr{S}^{m}_{1,1}$ symbols. This property is called
by Hörmander as having _reduced order $m-r$_ for symbols on $\mathbb{R}^{n}$.

The next proposition shows that the para-differential operator
$T_{a}^{\chi}=\mathbf{Op}(a^{\chi})$ extracts the “highest order
differentiation” from $\mathbf{Op}(a)$. The para-product decomposition can be
seen as a special case of it.

######  Proposition 4.2.

Suppose $r>0$, $a\in\mathcal{A}_{r}^{m}(\mathbb{G})$. Fix a basis
$X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, and define $X^{\alpha}$ as in
Proposition 2.1. If $\chi$ is an admissible cut-off function, then
$a-a^{\chi}\in\mathcal{A}_{0}^{m-r}$, and in fact for any strongly RT-
admissible tuple $q$,

$\sup_{x\in\mathbb{G}}\big{\|}\mathbf{D}_{q,\xi}^{\beta}(a-a^{\chi})(x,\xi)\big{\|}\lesssim_{\beta,q}\mathbf{W}^{m;r}_{|\beta|;q}(a)\cdot\langle\xi\rangle^{m-r-|\beta|}.$

###### Proof.

We still just need to prove for $q=Q$, the fundamental tuple. By induction, it
suffices to prove for $|\beta|=1$. By the Leibniz type property (2.10),
setting $M_{i}$ to be the dimension of the $i$’th fundamental representation,

$\mathbf{D}_{Q_{i},\xi}(a-a^{\chi})(x,\xi)=\big{(}\mathbf{D}_{Q_{i},\xi}a-(\mathbf{D}_{Q_{i},\xi}a)^{\chi}\big{)}(x,\xi)-\sum_{j,k=1}^{M_{i}}c^{i}_{j,k}\mathbf{D}_{Q_{j},\xi}(\chi\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi])\big{(}|\nabla_{x}|,|\xi|\big{)}\mathbf{D}_{Q_{k}}a(x,\xi).$

By Lemma 3.7, $\mathbf{D}_{Q_{i}}a$ satisfies the spectral condition with
parameter proportional to $\delta$, so we may simply apply Corollary 3.2.1
(and Remark 3.2) to conclude that

$\sup_{x\in\mathbb{G}}\left\|\big{(}\mathbf{D}_{Q_{i},\xi}a-(\mathbf{D}_{Q_{i},\xi}a)^{\chi}\big{)}(x,\xi)\right\|\lesssim_{Q}\mathbf{W}^{m;r}_{1;Q}(a)\cdot\langle\xi\rangle^{m-r-1}.$

The sum $\sum_{j,k=1}^{M_{i}}$ needs greater care to handle. For a fixed
$\eta\in\widehat{\mathbb{G}}$, Theorem 3.2 ensures that the symbol
$\mathbf{D}_{Q_{j},\xi}(\chi\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi])\big{(}|\eta|,|\xi|\big{)}$
is in a bounded subset in $\mathscr{S}^{m-1}_{1,1}$, and by Lemma 3.6 it does
not vanish only if

$\frac{1}{\delta}|\eta|-C\leq|\xi|\leq\frac{2}{\delta}|\eta|+C,$

where $C$ depends on the algebraic structure of $\mathbb{G}$ only.
Consequently,
$\mathbf{D}_{Q_{j},\xi}(\chi\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi])\big{(}|\eta|,|\xi|\big{)}\neq
0$ only if $|\eta|/\langle{[}\rangle\xi]\simeq\delta$, so when the highest
weight of $\xi$ is large enough, the Fourier support of

$\mathbf{D}_{Q_{j},\xi}(\chi\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi])\big{(}|\nabla_{x}|,|\xi|\big{)}\mathbf{D}_{Q_{k}}a(x,\xi)$

with respect to $x$ is contained in
$\mathfrak{S}[C\delta\langle{[}\rangle\xi],C^{\prime}\delta\langle{[}\rangle\xi]]$,
with $C,C^{\prime}>0$ depending on the algebraic structure of $\mathbb{G}$
only. Corollary 3.2.1 and Remark 3.2 then implies that

$\sup_{x\in\mathbb{G}}\left\|\mathbf{D}_{Q_{j},\xi}(\chi\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi])\big{(}|\nabla_{x}|,|\xi|\big{)}\mathbf{D}_{Q_{k}}a(x,\xi)\right\|\lesssim_{Q}\mathbf{W}^{m;r}_{1;Q}(a)\cdot\langle\xi\rangle^{m-r-1}.$

∎

We next estimate the difference $a^{\chi_{1}}-a^{\chi_{2}}$ for different
choices of admissible cut-off functions. The result shows that the choice of
$\chi$ is in fact very flexible:

######  Proposition 4.3.

Suppose $r\geq 0$, $\chi_{1},\chi_{2}$ are two admissible cut-off functions
with parameters $0<\delta_{1}<\delta_{2}<1/2$ respectively. Fix a basis
$X_{1},\cdots,X_{n}$ of $\mathfrak{g}$, and define $X^{\alpha}$ as in
Proposition 2.1. Then for $a\in\mathcal{A}_{r}^{m}$, the difference
$a^{\chi_{1}}-a^{\chi_{2}}\in\Sigma^{m-r}_{\delta_{2}}$. In fact for any
strongly RT-admissible tuple $q$,

$\big{\|}X^{\alpha}_{x}\mathbf{D}_{q,\xi}^{\beta}(a^{\chi_{1}}-a^{\chi_{2}})(x,\xi)\big{\|}\lesssim_{\alpha,\beta,q}\mathbf{W}^{m;r}_{|\beta|;q}(a)\cdot\langle\xi\rangle^{m-r+|\alpha|-|\beta|}.$

Thus for any $s\in\mathbb{R}$, $T_{a}^{\chi_{1}}-T_{a}^{\chi_{2}}$ maps
$H^{s+m}$ continuously to $H^{s}$, and

$\|(T_{a}^{\chi_{1}}-T_{a}^{\chi_{2}})u\|_{H^{s}}\lesssim_{s,q}\mathbf{W}^{m;r}_{n+2;q}(a)\|u\|_{H^{s+m-r}}.$

###### Proof.

This follows immediately from an observation and Corollary 3.2.1 (also Remark
3.2): the Fourier support of $\chi_{1}(\eta,\xi)-\chi_{2}(\eta,\xi)$ with
respect to $x$ is in the set
$\mathfrak{S}[\delta_{1}\langle{[}\rangle\xi],\delta_{2}\langle{[}\rangle\xi]]$.
∎

Thus, given a rough symbol $a\in\mathcal{A}^{m}_{r}$ with $r>0$, it is
legitimate to abbreviate the dependence on admissible cut-off function, and
define _the_ para-differential operator $T_{a}$ modulo
$\mathbf{Op}\big{(}\Sigma^{m-r}_{<1/2}\big{)}$.

###  4.2. Symbolic Calculus of Para-differential Operators

In this subsection, we state the core results in our toolbox: the composition
and adjoint formula for para-differential operators. The spirit of proof does
not differ much from that in Métivier’s book [Mét08], but technically we have
to be extra careful dealing with the operator norm of symbols. The spectral
localization property i.e. Corollary 2.3.1 and results obtained in Subsection
4.1. will be used repeatedly.

######  Definition 4.4.

Let $r>0$ be a real number. Let $q$ be a RT-admissible tuple, and
$X_{q}^{(\alpha)}$ be the left-invariant differential operators as in
Proposition 2.2. If $a,b$ are any symbol of at least $C^{r}_{*}$ regularity in
$x$, define

$(a\\#_{r;q}b)(x,\xi):=\sum_{\alpha:|\alpha|\leq
r}\mathbf{D}_{q,\xi}^{\alpha}a(x,\xi)\cdot X_{q,x}^{(\alpha)}b(x,\xi).$

We now state the first main theorem of this section.

######  Theorem 4.1 (Composition of Para-differential Operators, I).

Suppose $r>0$, $m,m^{\prime}$ are real numbers. Let $q$ be a RT-admissible
tuple, whose components are linear combinations of the fundamental tuple $Q$
of $\mathbb{G}$, and let $X_{q}^{(\alpha)}$ be the left-invariant differential
operators as in Proposition 2.2. Given $a\in\mathscr{S}^{m}_{1,1}$,
$b\in\mathcal{A}_{r}^{m^{\prime}}$, for an admissible cut-off function $\chi$
with parameter $\delta$ sufficiently small (depending on the magnitude of $r$
and the algebraic structure of $\mathbb{G}$ only),

$\mathbf{Op}(a)\circ\mathbf{Op}(b^{\chi})-\mathbf{Op}(a\\#_{r;q}b^{\chi})\in\mathbf{Op}\mathscr{S}^{m+m^{\prime}-r}_{1,1}(\mathbb{G}).$

More precisely, the operator norm of
$\mathbf{Op}(a)\circ\mathbf{Op}(b^{\chi})-\mathbf{Op}(a\\#_{r;q}b^{\chi})$ for
$H^{s+m+m^{\prime}-r}\to H^{s}$ is bounded by

$C_{s}\mathbf{M}^{m}_{l,n+2;q}(a)\mathbf{W}^{m^{\prime};r}_{l;q}(b),$

where the integer $l$ does not depend on $a,b$.

###### Proof.

We observe a simple fact: for symbols $\sigma_{1},\sigma_{2}$ of type (1,1),
the symbol of $\mathbf{Op}(\sigma_{1})\circ\mathbf{Op}(\sigma_{2})$ is

(4.2)
$\int_{\mathbb{G}}\mathcal{K}_{\sigma_{1}}(x,y)\xi^{*}(y)\sigma_{2}(xy^{-1},\xi)dy,$

where $\mathcal{K}_{\sigma_{1}}$ is the convolution kernel of
$\mathbf{Op}(\sigma_{1})$, defined by (2.5)-(2.6).

Let us now examine the symbol $\sigma$ of
$\mathbf{Op}(a)\circ\mathbf{Op}(b^{\chi})$. We start from $b^{\chi}(x,\xi)$:
since $b^{\chi}(x,\xi)$ satisfies the spectral condition with parameter
$\delta$, we can choose another admissible cut-off function
$\chi_{1}(\mu,\lambda)$ with parameter $2\delta$, and find

$\widehat{b^{\chi}}(\eta,\xi)=\chi_{1}(|\eta|,|\xi|)\widehat{b^{\chi}}(\eta,\xi.$

Converting back to the physical space, this implies

$b^{\chi}(x,\xi)=\int_{\mathbb{G}}\check{\chi}_{1}(z^{-1}x,|\xi|)b^{\chi}(z)dz.$

Here with a little abuse of notation, we write $\check{\chi}_{1}(y,|\xi|)$ for
the Fourier inversion of
$\chi_{1}(|\eta|,|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\eta]$ with respect to
$\eta$. Note that $\chi_{1}(|\eta|,|\xi|)\mathrm{Id}_{\mathcal{H}_{[}}\eta]$
is a scaling on each representation space $\mathcal{H}_{[}\eta]$, so the order
of convolution is irrelevant. By formula (4.2),

$\displaystyle\sigma(x,\xi)$
$\displaystyle=\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)\left(\int_{\mathbb{G}}\check{\chi}_{1}(z^{-1}xy^{-1},|\xi|)b^{\chi}(z,\xi)dz\right)dy$
$\displaystyle=\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)\left(\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)b^{\chi}(xz^{-1},\xi)dz\right)dy.$

Using Proposition 2.2 with $N$ being the smallest integer $>r$, we compute the
inner integral as

$\displaystyle\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)b^{\chi}(xz^{-1},\xi)dz$
$\displaystyle=\sum_{|\alpha|\leq
r}\left(\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)q^{\alpha}(z)dz\right)X^{(\alpha)}_{q,x}b^{\chi}(x,\xi)$
$\displaystyle\quad+\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)R_{N}\big{(}b^{\chi}(\cdot,\xi);x,z^{-1}\big{)}dz$

The integrals in the sum $\sum_{|\alpha|\leq r}$ are of convolution type. By
Proposition 2.1, the Fourier support of $q^{\alpha}$ for $|\alpha|\leq r$ is
within $\mathfrak{S}[0,C]$, where $C$ depends on $r$ and the algebraic
structure of $\mathbb{G}$ only. Thus, for $|\xi|\geq 2C/\delta$, we have

$\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)q^{\alpha}(z)dz=q^{\alpha}(y),$

and consequently,

(4.3) $\displaystyle\sigma(x,\xi)$ $\displaystyle=\sum_{|\alpha|\leq
r}\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)q^{\alpha}(y)X^{(\alpha)}_{q,x}b^{\chi}(x,\xi)dy$
$\displaystyle\quad+\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)\left(\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)R_{N}\big{(}b^{\chi}(\cdot,\xi);x,z^{-1}\big{)}dz\right)dy$
$\displaystyle=:\sum_{|\alpha|\leq r}\mathbf{D}_{q,\xi}^{\alpha}a(x,\xi)\cdot
X_{q,x}^{(\alpha)}b^{\chi}(x,\xi)+\varsigma_{N}(x,\xi).$

The sum in the right-hand-side of (4.3) is just $a\\#_{r;q}b^{\chi}$, which is
in $\mathscr{S}^{m+m^{\prime}}_{1,1}$. There is then only one assertion to be
verified: the term $\varsigma_{N}(x,\xi)$ is of class
$\mathscr{S}^{m+m^{\prime}-r}_{1,1}$, from which it follows that the symbol
$\sigma$ really is of class $\mathscr{S}^{m+m^{\prime}}_{1,1}$. Note that the
lower frequency terms give smoothing symbols in $\mathscr{S}^{-\infty}$, so we
just have to verify the assertion for $|\xi|\geq 2C/\delta$.

Let us first show that
$\varsigma_{N}(x,\xi)\lesssim\langle{[}\rangle\xi]^{m+m^{\prime}-r}$. We can
change the order of integration as follows:

(4.4)
$\displaystyle\varsigma_{N}(x,\xi)=\int_{\mathbb{G}}\left(\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)\check{\chi}_{1}(zy^{-1},|\xi|)dy\right)R_{N}\big{(}b^{\chi}(\cdot,\xi);x,z^{-1}\big{)}dz.$

Just as in the proof of Stein’s theorem, we decompose
$\mathcal{K}_{a}=\sum_{j=0}^{\infty}\mathcal{K}_{a_{j}}$, where
$\mathcal{K}_{a_{j}}$ is the convolution kernel of
$a_{j}(x,\xi):=a(x,\xi)\vartheta_{j}\big{(}|\xi|\big{)}$, with $\vartheta$ as
in (3.2). For the inner integral with respect to $y$, each summand

$P_{j}(x,z):=\int_{\mathbb{G}}\mathcal{K}_{a_{j}}(x,y)\xi^{*}(y)\check{\chi}_{1}(zy^{-1},|\xi|)dy$

is of convolution type. By spectral property i.e. Corollary 2.3.1, the Fourier
support of $\mathcal{K}_{a_{j}}(x,y)\xi^{*}(y)$ with respect to $y$ is in the
set

$\mathfrak{S}\Big{[}c\big{|}2^{j}-|\xi|\big{|},2^{j}+|\xi|\Big{]}.$

Hence $\widehat{P_{j}}(x,\eta)$ (the Fourier transform is taken with respect
to $z$) does not vanish only for

$c\big{|}2^{j}-|\xi|\big{|}\leq 2\delta\langle{[}\rangle\xi].$

If we choose $\delta\ll c$, this implies that $2^{j}$ must be comparable with
$\langle{[}\rangle\xi]$. As a result,

$\varsigma_{N}(x,\xi)=\sum_{j:2^{j}\simeq\langle{[}\rangle\xi]}\int_{\mathbb{G}}\mathcal{K}_{a_{j}}(x,y)\xi^{*}(y)\left(\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)R_{N}\big{(}b^{\chi}(\cdot,\xi);x,z^{-1}\big{)}dz\right)dy.$

Just as inequalitites (3.3)-(3.4) in the proof of Stein’s theorem, the
estimate

(4.5)
$\big{(}1+|2^{j}\mathrm{dist}(y,e)|^{2}\big{)}^{L/2}\cdot|\mathcal{K}_{a_{j}}(x,y)|\lesssim\mathbf{M}^{m}_{0,L;Q}(a)\cdot
2^{jm+jn}$

is valid for $L\in\mathbb{N}$. The remainder in Taylor’s formula can be
estimated using the remark following Proposition 4.1: since $N>r$, we have, by
the improved growth estimate (Proposition 4.1),

$\displaystyle\big{\|}R_{N}\big{(}b^{\chi}(\cdot,\xi);x,z^{-1}\big{)}\big{\|}$
$\displaystyle\lesssim\|b^{\chi}(x,\xi)\|_{C^{N}_{x};\mathrm{End}(\mathcal{H}_{[}\xi])}\cdot\mathrm{dist}(z,e)^{N}$
$\displaystyle\lesssim\mathbf{W}^{m;r}_{0;Q}(b)\cdot\langle{[}\rangle\xi]^{m^{\prime}+N-r}\mathrm{dist}(z,e)^{N}.$

Here the norm is the operator norm on $\mathrm{End}(\mathcal{H}_{[}\xi])$, and
the implicit constants do not depend on $\xi$. By Corollary 3.1.1, noting the
integrand is smooth and equals $O(\mathrm{dist}(z,e)^{N})$, we thus estimate

(4.6)
$\displaystyle\int_{\mathbb{G}}\check{\chi}_{1}(zy^{-1},|\xi|)R_{N}\big{(}b^{\chi}(\cdot,\xi);x,z^{-1}\big{)}dz=\sum_{k=0}^{N}\upsilon_{k}(x,y,\xi),$

where each $\upsilon_{k}(x,y,\xi)$ is smooth in $x$, and satisfies

$\|\upsilon_{k}(x,y,\xi)\|\lesssim\mathbf{W}^{m;r}_{0;Q}(b)\cdot\langle{[}\rangle\xi]^{m^{\prime}+k-r}\mathrm{dist}(y,e)^{k}.$

Combining every inequality (4.5)-(4.6), using Lemma 2.6, we estimate the
operator norm of $\varsigma_{N}(x,\xi)$ as (noting that $\|\xi^{*}(y)\|\equiv
1$)

$\displaystyle\|\varsigma_{N}(x,\xi)\|$
$\displaystyle\lesssim\sum_{j:2^{j}\simeq\langle{[}\rangle\xi]}\sum_{k=0}^{N}\left\|\int_{\mathbb{G}}\mathcal{K}_{a_{j}}(x,y)\xi^{*}(y)\upsilon_{k}(x,y,\xi)dy\right\|$
$\displaystyle\lesssim\sum_{k=0}^{N}\mathbf{M}^{m}_{0,n+1;Q}(a)\cdot\mathbf{W}^{m;r}_{0;Q}(b)\cdot\langle{[}\rangle\xi]^{m^{\prime}+k-r}\sum_{j:2^{j}\simeq\langle{[}\rangle\xi]}2^{jm+jn}\int_{\mathbb{G}}\frac{\min\left(\mathrm{dist}(y,e)^{k},1\right)dy}{\big{(}1+|2^{j}\mathrm{dist}(y,e)|^{2}\big{)}^{(n+1)/2}}$
$\displaystyle\lesssim\mathbf{M}^{m}_{0,n+1;Q}(a)\cdot\mathbf{W}^{m;r}_{0;Q}(b)\cdot\langle{[}\rangle\xi]^{m+m^{\prime}-r}.$

This shows that the second sum in (4.3) is controlled by
$\langle{[}\rangle\xi]^{m+m^{\prime}-r}$.

Finally, we use induction on the order of differentiation in (4.2). Noticing
that
$\mathbf{D}_{Q^{\beta},\xi}\big{(}\xi^{*}(y)\big{)}=Q^{\beta}(y)\xi^{*}(y)$,
we have, for example, given any $X\in\mathfrak{g}$,

(4.7)
$X_{x}\sigma(x,\xi)=\int_{\mathbb{G}}X_{x}\mathcal{K}_{a}(x,y)\xi^{*}(y)b^{\chi}(xy^{-1},\xi)dy+\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)X_{x}\big{(}b^{\chi}(xy^{-1},\xi)\big{)}dy$

and by Leibniz type property of the fundamental tuple $Q$,

(4.8) $\displaystyle\mathbf{D}_{Q_{i},\xi}\sigma(x,\xi)$
$\displaystyle=\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)Q_{i}(y)\xi^{*}(y)b^{\chi}(xy^{-1},\xi)dy+\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)\xi^{*}(y)(\mathbf{D}_{Q_{i},\xi}b^{\chi})(xy^{-1},\xi)dy$
$\displaystyle\quad+\sum_{j,k}c_{j,k}^{i}\int_{\mathbb{G}}\mathcal{K}_{a}(x,y)Q_{j}(y)\xi^{*}(y)(\mathbf{D}_{Q_{k},\xi}b^{\chi})(xy^{-1},\xi)dy.$

Each of the above integrals can be estimated in exactly the same method, with
$m$ replaced by $m+1$ in (4.7) and by $m-1$ in (4.8); we just have to observe
that $X_{x}\mathcal{K}_{a}(x,y)$ is the convolution kernel of $X_{x}a(x,\xi)$,
and $\mathcal{K}_{a}(x,y)Q_{i}(y)$ is the convolution kernel of
$\mathbf{D}_{Q_{i},\xi}a(x,\xi)$. ∎

Now if $\mathbf{Op}(a)$ is in fact a para-differential operator corresponding
to some rough symbol of class $\mathcal{A}^{m}_{r}$, we can actually
strengthen Theorem 4.1, and deduce the composition formula of para-
differential operators:

######  Theorem 4.2 (Composition of Para-differential Operators, II).

Suppose $r>0$, $m,m^{\prime}$ are real numbers. Let $q$ be a RT-admissible
tuple, whose components are are linear combinations of the fundamental tuple
$Q$ of $\mathbb{G}$, and let $X_{q}^{(\alpha)}$ be the left-invariant
differential operators as in Proposition 2.2. Given $a\in\mathcal{A}_{r}^{m}$,
$b\in\mathcal{A}_{r}^{m^{\prime}}$, it follows that

$T_{a}\circ
T_{b}-T_{a\\#_{r,q}b}\in\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-r}.$

More precisely, the operator norm of $T_{a}\circ T_{b}-T_{a\\#_{r,q}b}$ for
$H^{s+m+m^{\prime}-r}\to H^{s}$ is bounded by

$C_{s}\mathbf{W}^{m;r}_{l;q}(a)\mathbf{W}^{m^{\prime};r}_{l;q}(b),$

where the integer $l$ does not depend on $a,b$.

Before we prove Theorem 4.2, we state two auxiliary results.

######  Lemma 4.3.

Suppose $r>0$, $m,m^{\prime}$ are real numbers, and $a\in\mathcal{A}_{r}^{m}$,
$b\in\mathcal{A}_{r}^{m^{\prime}}$. There are constants $\delta_{0}\in(0,1/2)$
and $C>0$ depending only on the algebraic structure of $\mathbb{G}$ with the
following property: if $\chi$ is an admissible cut-off function with parameter
$\delta<\delta_{0}$, the symbol of $T_{a}^{\chi}\circ T_{b}^{\chi}$ is of
class $\Sigma^{m+m^{\prime}-r}_{C\delta}$.

######  Lemma 4.4.

There is a constant $C>1$, depending on the algebraic structure of
$\mathbb{G}$ only, with the following property. Let $A$ be a finite
dimensional normed algebra. Suppose $0\leq r_{1}\leq r_{2}$, with $r_{2}>0$.
Suppose $u\in C^{r_{1}}_{*}$, $v\in C^{r_{2}}_{*}$ are $A$-valued functions.
Suppose $\vartheta\in C_{0}^{\infty}(\mathbb{R})$ is such that
$\vartheta(\lambda)=1$ for $|\lambda|\leq 1$ and vanishes for $|\lambda|\geq
2$. Then for $t\geq 1$,

$\left\|\vartheta\left(\frac{|\nabla|}{Ct}\right)(uv)-\vartheta\left(\frac{|\nabla|}{t}\right)u\cdot\vartheta\left(\frac{|\nabla|}{t}\right)v\right\|_{L^{\infty};A}\lesssim_{r_{1},r_{2}}t^{-r_{1}}\|u\|_{C^{r_{1}}_{*};A}\|v\|_{C^{r_{2}}_{*};A}.$

The implicit constants do not depend on $A$.

###### Proof of Lemma 4.3.

By formula (4.2), the symbol $\sigma$ of $T_{a}^{\chi}\circ T_{b}^{\chi}$ is

$\displaystyle\sigma(x,\xi)$
$\displaystyle=\int_{\mathbb{G}}\mathcal{K}_{a^{\chi}}(x,y)\xi^{*}(y)b^{\chi}(xy^{-1},\xi)dy$
$\displaystyle=\sum_{\zeta\in\widehat{\mathbb{G}}}d_{\zeta}\int_{\mathbb{G}}\operatorname{Tr}\Big{[}a^{\chi}(x,\zeta)\zeta(y)\Big{]}\xi^{*}(y)b^{\chi}(xy^{-1},\xi)dy.$

Let us fix $\xi\in\widehat{\mathbb{G}}$. By the spectral localization property
of products (Corollary 2.3.1), the Fourier support of $\zeta\otimes\xi^{*}$ is
in

$\mathfrak{S}\Big{[}c\big{|}|\zeta|-|\xi|\big{|},|\zeta|+|\xi|\Big{]},$

where $0<c<1$ depends on the algebraic structure of $\mathbb{G}$ only. Since
the integral is of convolution type, we find that the Fourier support of
$\int_{\mathbb{G}}(\zeta\otimes\xi)^{*}(y)b^{\chi}(xy^{-1},\xi)dy$ with
respect to $x$ is contained in the set

$\mathfrak{S}\Big{[}c\big{|}|\zeta|-|\xi|\big{|},|\zeta|+|\xi|\Big{]}\cap\mathfrak{S}[0,\delta\langle{[}\rangle\xi]],$

so the integral does not vanish only for $|\zeta|\leq C\langle{[}\rangle\xi]$,
where $C$ depends on the algebraic structure of $\mathbb{G}$ only. Now
applying the spectral condition satisfied by $a^{\chi}(x,\zeta)$, using the
spectral localization property of products again, we obtain that the Fourier
support of $\sigma(x,\xi)$ is contained in the set
$\mathfrak{S}[0,C\delta\langle{[}\rangle\xi]]$. That
$\sigma\in\mathscr{S}^{m+m^{\prime}}_{1,1}$ for sufficiently small $\delta$ is
a direct consequence of Theorem 4.1. ∎

The proof of Lemma 4.4 is identical to that of Proposition 8.6.9. of [Hör97],
with the application of Corollary 2.3.1, Corollary 3.2.1 and its vector-valued
version as described in Remark 3.2.

###### Proof of Theorem 4.2.

We notice a simple fact: if $q$ is an RT-admissible tuple and
$X_{q}^{(\alpha)}$ are the corresponding left-invariant differential operators
as in Proposition 2.2, then for $a\in\mathcal{A}^{m}_{r}$,
$b\in\mathcal{A}^{m^{\prime}}_{r}$, it follows that $a\\#_{r;q}b$ is an
element in $\bigcup_{j\leq r}\mathcal{A}_{r-j}^{m+m^{\prime}-j}$ since by the
product rule in Zygmund spaces,

(4.9) $\sum_{|\alpha|=j}\mathbf{D}_{q,\xi}^{\alpha}a(x,\xi)\cdot
X_{q,x}^{(\alpha)}b(x,\xi)\in\mathcal{A}_{r-j}^{m+m^{\prime}-j}.$

First of all, we need to justify the notation that disregards the admissible
cut-off function. In fact, Lemma 3.7, Proposition 4.3 and formula (4.9)
ensures that $T_{a\\#_{r;q}b}$ is defined modulo
$\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-r}$, provided that the parameter of
the admissible cut-off function is sufficiently small (depending on the
algebraic structure of $\mathbb{G}$ only). We also compute, for admissible
cut-off functions $\chi_{1},\chi_{2}$ with parameter $\delta_{1}<\delta_{2}$
respectively,

$\mathbf{Op}(a^{\chi_{1}})\circ\mathbf{Op}(b^{\chi_{1}})-\mathbf{Op}(a^{\chi_{2}})\circ\mathbf{Op}(b^{\chi_{2}})=\mathbf{Op}(a^{\chi_{1}}-a^{\chi_{2}})\circ\mathbf{Op}(b^{\chi_{1}})+\mathbf{Op}(a^{\chi_{2}})\circ\mathbf{Op}(b^{\chi_{1}}-b^{\chi_{2}}).$

By Proposition 4.3, $a^{\chi_{1}}-a^{\chi_{2}}\in\Sigma_{\delta_{2}}^{m-r}$,
$b^{\chi_{1}}-b^{\chi_{2}}\in\Sigma_{\delta_{2}}^{m^{\prime}-r}$. Thus if
$\delta_{1},\delta_{2}$ are sufficiently small (depending on the algebraic
structure of $\mathbb{G}$ only), Lemma 4.3 then implies that the right-hand-
side is in
$\mathbf{Op}\Sigma_{C\delta_{2}}^{m+m^{\prime}-r}\subset\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-r}$.
Thus $T_{a}\circ T_{b}$ is defined modulo
$\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-r}$. Applying Lemma 3.7 and Theorem
4.1 once again, we find that if $\chi$ is an admissible cut-off function with
sufficiently small parameter $\delta$ (depending on the algebraic structure of
$\mathbb{G}$ only), then

$T_{a}^{\chi}\circ
T_{b}^{\chi}-\mathbf{Op}\big{(}a^{\chi}\\#_{r;q}b^{\chi}\big{)}\in\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-r}.$

From now on, we fix $\chi$ to be the function in (4.1).

It remains to show that $a^{\chi}\\#_{r;q}b^{\chi}$ equals
$(a\\#_{r;q}b)^{\chi_{1}}$ modulo $\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-r}$
for some suitable admissible cut-off function $\chi_{1}$. For simplicity we
write $a^{(\alpha)}=\mathbf{D}_{q,\xi}^{\alpha}a$,
$b_{(\alpha)}=X_{q,x}^{(\alpha)}b$. We just need to show that for
$|\alpha|\leq r$, each summand of
$(a^{\chi}\\#_{r}b^{\chi})-(a\\#_{r;q}b)^{\chi_{1}}$, being

$(a^{\chi})^{(\alpha)}\cdot(b^{\chi})_{(\alpha)}-\left(a^{(\alpha)}\cdot
b_{(\alpha)}\right)^{\chi_{1}},$

is in $\Sigma^{m+m^{\prime}-r}_{<1/2}$. The only technical complexity comes
from $(a^{\chi})^{(\alpha)}$. In fact, from the proof of Proposition 4.2, we
know that

$\mathbf{D}_{Q_{i},\xi}a^{\chi}(x,\xi)=(\mathbf{D}_{Q_{i},\xi}a)^{\chi}(x,\xi)+\sum_{j,k=1}^{M_{i}}c^{i}_{j,k}\mathbf{D}_{Q_{j},\xi}(\chi\cdot\mathrm{Id}_{\mathcal{H}_{[}}\xi])\big{(}|\nabla_{x}|,|\xi|\big{)}\mathbf{D}_{Q_{k}}a(x,\xi),$

and belongs to $\Sigma_{C\delta}^{m-1}$ for some $C$ depending on the
algebraic structure of $\mathbb{G}$ only. Thus, if $\delta$ is sufficiently
small, we obtain by induction that

$(a^{\chi})^{(\alpha)}\cdot(b^{\chi})_{(\alpha)}=(a^{(\alpha)})^{\chi}\cdot(b_{(\alpha)})^{\chi}\,\mod\,\Sigma_{<1/2}^{m+m^{\prime}-r},$

since $|\alpha|\leq r$.

The claim is then reduced to checking

(4.10)
$(a^{(\alpha)})^{\chi}\cdot(b_{(\alpha)})^{\chi}-\left(a^{(\alpha)}\cdot
b_{(\alpha)}\right)^{\chi_{1}}\in\Sigma^{m+m^{\prime}-r}_{<1/2}.$

To verify this, we just need the technical Lemma 4.4. Recall that we chose
$\chi$ as in (3.1). We regard $\xi$ as fixed, then set
$\chi_{1}(|\eta|,|\xi|)=\chi(|\eta|/C,|\xi|)$ where $C$ is as in Lemma 4.4,
and $\vartheta(|\eta|)=\chi_{1}(|\eta|,|\xi|)$. Since
$a^{(\alpha)}\in\mathcal{A}_{r}^{m-|\alpha|}$,
$b_{(\alpha)}\in\mathcal{A}_{r-|\alpha|}^{m}$, Lemma 4.4 gives

$\displaystyle\left\|(a^{(\alpha)})^{\chi}\cdot(b_{(\alpha)})^{\chi}-\left(a^{(\alpha)}\cdot
b_{(\alpha)}\right)^{\chi_{1}}\right\|$
$\displaystyle\lesssim\langle\xi\rangle^{-(r-|\alpha|)}\cdot\mathbf{W}^{m-|\alpha|;r}_{0;Q}(a)\langle\xi\rangle^{m-|\alpha|}\cdot\mathbf{W}^{m^{\prime};r-|\alpha|}_{0;Q}(b)\langle\xi\rangle^{m^{\prime}}$
$\displaystyle\lesssim\mathbf{W}^{m-|\alpha|;r}_{0;Q}(a)\cdot\mathbf{W}^{m^{\prime};r-|\alpha|}_{0;Q}(b)\cdot\langle\xi\rangle^{m+m^{\prime}-r}.$

Here the norm to be estimated is the operator norm on
$\mathrm{End}(\mathcal{H}_{[}\xi])$, and the implicit constants do not depend
on $\xi$. Derivatives of (4.10) can be estimated similarly. By Lemma 3.7,
(4.10) still satisfies a spectral condition. This completes the proof. ∎

As a corollary, we obtain the commutator property of para-differential
operators: the commutator of two para-differential operators of order $m$ and
$m^{\prime}$ gives rise to a para-differential operator of order
$m+m^{\prime}-1$.

######  Corollary 4.4.1.

Suppose $r>0$, $m,m^{\prime}$ are real numbers. Given
$a\in\mathcal{A}_{r}^{m}$, $b\in\mathcal{A}_{r}^{m^{\prime}}$, the commutator
$[T_{a},T_{b}]$ is in the class $\mathbf{Op}\Sigma_{<1/2}^{m+m^{\prime}-1}$.
More precisely, the operator norm of $[T_{a},T_{b}]$ for
$H^{s+m+m^{\prime}-1}\to H^{s}$ is bounded by

$C_{s}\mathbf{W}^{m;r}_{l;q}(a)\mathbf{W}^{m^{\prime};r}_{l;q}(b),$

where the integer $l$ does not depend on $a,b$.

###### Proof.

Imitating the proof of Proposition 2.3, we find that the commutator symbol
$[a,b](x,\xi)$ is of class $\mathcal{A}^{m+m^{\prime}-1}_{r}$; we just need to
repeat the proof of Proposition 2.3 and, in addition, estimate the $C^{r}_{*}$
norms with respect to $x$.

On the other hand,

$\displaystyle(a\\#_{r;q}b)(x,\xi)-(b\\#_{r;q}a)(x,\xi)$
$\displaystyle=[a,b](x,\xi)$
$\displaystyle\quad+\sum_{\alpha:1\leq|\alpha|\leq
r}\left(\mathbf{D}_{q,\xi}^{\alpha}a(x,\xi)\cdot
X_{q,x}^{(\alpha)}b(x,\xi)-\mathbf{D}_{q,\xi}^{\alpha}b(x,\xi)\cdot
X_{q,x}^{(\alpha)}a(x,\xi)\right).$

The sum is easily seen to give rise to symbols of order $m+m^{\prime}-1$.
Theorem 4.2 then shows that

$[T_{a},T_{b}]\in\mathbf{Op}\Sigma^{m+m^{\prime}-1}_{<1/2}.$

∎

We can compare this with [Del15]: in order to cast the normal form reduction
for quasi-linear Hamiltonian Klein-Gordon equation, such commutator property
is necessary. The definition of para-differential operator within out
formalism is of course different from that in [Del15]. While it is not known
whether these two definitions coincide, this does not affect our goal, as a
calculus is already available.

We also state the adjoint formula for para-differential operators and omit the
proof.

######  Theorem 4.5 (Adjoint of Para-differential Operator).

Suppose $r>0$, $m$ is a real number. Let $q$ be a RT-admissible tuple, whose