#  Drift diffusion equations with fractional diffusion on compact Lie groups

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Julio Delgado  Julio
Delgado: Departmento de Matemáticas Universidad del Valle Cali-Colombia E-mail
address<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael
Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University, Belgium and School of Mathematical Sciences Queen Mary
University of London United Kingdom E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this work we investigate the well-posedness for difussion equations
associated to subelliptic pseudo-differential operators on compact Lie groups.
The diffusion by strongly elliptic operators is considered as a special case
and in particular the fractional diffusion with respect to the Laplacian. The
general case is studied within the Hörmander classes associated to a sub-
Riemannian structure on the group (encoded by a Hörmander system of vector
fields). Applications to diffusion equations for fractional sub-Laplacians,
fractional powers of more general subelliptic operators, and the corresponding
quasi-geostrophic model with drift $D$ are investigated. Examples on
$\textnormal{SU}(2)$ for diffusion problems with fractional diffusion are
analysed.

###### Key words and phrases:

Diffusion equations, quasi-geostrophic model, diffusion with drift, pseudo-
differential operators

###### 2020 Mathematics Subject Classification:

Primary 22E30, 35S10; Secondary 35R11, 58J40.

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). J. Delgado is
also supported by Vice. Inv. Universidad del Valle Grant CI-71281, MathAmSud
and Minciencias-Colombia under the project MATHAMSUD 21-MATH-03. Michael
Ruzhansky is also supported by EPSRC grant EP/R003025/2.

###### Contents

  1. 1 Introduction
  2. 2 Fourier analysis on Lie groups and subelliptic Hörmander classes
    1. 2.1 Pseudo-differential operators via localisations
    2. 2.2 The positive sub-Laplacian and pseudo-differential operators via global symbols
    3. 2.3 Subelliptic Hörmander classes on compact Lie groups
  3. 3 Analysis of diffusion problems on compact Lie groups
    1. 3.1 Preliminary results
    2. 3.2 Case I: Diffusion problems for strongly elliptic operators with positive order
    3. 3.3 Case II: Diffusion problems for operators with positive symbols on the phase space
    4. 3.4 Consequences: Well posedness for diffusion equations in elliptic Hörmander classes
      1. 3.4.1 The strongly elliptic case
      2. 3.4.2 The case of non-negative symbols
    5. 3.5 Analysis of diffusion equation for fractional sub-Laplacians
    6. 3.6 Analysis of diffusion equations for fractional Laplacians with drift
    7. 3.7 Examples: diffusion problems with drift for the Laplacian on $\textnormal{SU}(2)$

##  1\. Introduction

In this manuscript we establish the well-posedness of the Cauchy problem

$(\textnormal{IVP}):\begin{cases}\frac{\partial v}{\partial t}-K(t)v=f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G),\\\ v(0)=u_{0},&\text{ }\end{cases}$
(1.1)

on a compact Lie group $G,$ where the the time-dependent family of pseudo-
differential operators $K(t)$ includes subelliptic operators, fractional
Laplacians and sub-Laplacians with or without drifts (extra perturbations like
smooth vector fields). The motivation for the consideration of drifts came
from quasi-geostrophic theory, see [6, 26] and references therein for details.

The investigation of fractional diffusion equations has been intensively
active in the last decades, in particular motivated by the diverse
applications e.g fractional dynamics [27], [28], fluid mechanics [29], [10]
and mathematical finance [11]. Moreover, the paper [6] by Caffarelli and
Vasseur has considered the analysis of drift diffusion equations with
fractional diffusion inspired by the applications of such models in quasi-
geostrophic theory and control theory, see Constantin [10] and Duvaut and
Lions [16], respectively. In the Euclidean setting, analysis of the fractional
Laplacian in terms of the pseudo-differential operators theory and singular
integrals can be found e.g. in G. Grubb [23] (and the references therein). On
compact manifolds, in particular, compact Lie groups a theory of initial value
problems for pseudo-differential operators can be found in the book of M.
Taylor [38] and in Chazarain and Piriou [7].

On the other hand, the research on the fractional Laplacian and the fractional
diffusion on different group structures have been of recent interest, for
instance the discrete fractional diffusion is considered in [9]. The
fractional Laplacian on the torus has been studied in [32] and several aspects
of the harmonic analysis of the fractional Laplacian on lattices has been
investigated in [8]. Other recent works on the fractional Laplacian and its
different generalisations have been considered in [1], [12], [13] [14], [37],
[15], [17], [20], [23], [30], [31] and the references therein. In particular,
the counterpart of the results proved here for compact Lie groups have been
analysed on graded Lie groups in [4] in the case of hypoelliptic problems for
Rockland operators. A nice exposition on the fractional Sobolev spaces and the
fractional Laplacian can be found in [37]. On the other hand, drift diffusion
equations with fractional diffusion have intensely attracted the interest in
the last 10 years starting with the works of Caffarelli and Vasseur in [6].

Now, we are going to explain in more detail, the family of initial value
problems (1.1) covered by our approach. For a compact Lie group $G,$ the set
$\widehat{G}$ denotes its unitary dual. Motivated by the applications of the
previous paragraph and by the recent techniques in the analysis of
differential problems on compact Lie groups [3, 33], built on the harmonic
analysis of operators on the phase space $G\times\widehat{G},$ in this paper
we analyse the existence and uniqueness for the Cauchy problem (1.1)
associated to a time-dependent continuous family of pseudo-differential
operators $K(t)$ in the Hörmander classes
$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ (introduced in [3]
and that include e.g. the Hörmander sub-Laplacian
$\mathcal{L}=-\sum_{j=1}^{k}X_{j}^{2}$ and its functional calculus, its
fractional powers, etc.) with positive order $m>0$. The functional calculus of
the pseudo-differential operators in [3, 35] allows us to consider the case of
the fractional diffusion and of the fractional diffusion with drift. Our
analysis includes the case where

  * •

$K(t)$ is strongly elliptic with respect to $\mathcal{L},$ of arbitrary order
$m>0.$ Examples are fractional operators of the form
$K(t)=a(x,t)\mathcal{L}^{\frac{m}{2}},$ $a(x,t)(1+\mathcal{L})^{\frac{m}{2}},$
where $a\in C([0,T],C^{\infty}(G))$ is a real-valued function such that
$a(x,t)\neq 0,$ for all $(x,t)\in G\times[0,T].$ Our analysis also covers the
case where $K(t)=a(x,t)\mathcal{L}^{\frac{m}{2}}_{G}$ with
$\mathcal{L}_{G}=-\sum_{j=1}^{k}X_{j}^{2}$ being the Laplace-Beltrami operator
on $G.$

  * •

$K(t)=a(x,t)\mathcal{L}_{G}^{\frac{m}{2}}+D,$ where $0\leq m\leq 1,$ and the
drift $D=\sum_{i=1}^{n}a_{i}X_{i}\in\textnormal{Diff}^{1}(G)$ is a left-
invariant differential operator of first order such that the matrix-valued
symbol of

$-K(t)=-(a(x,t)\mathcal{L}_{G}^{\frac{m}{2}}+D)$

is positive on every representation space. Moreover, the positivity condition
is removed when $m>1,$ and $a(x,t)\neq 0$ for all $(x,t)\in G\times[0,T].$ In
both cases we say that the diffusion problem (3.1) has drift $D$ following the
terminology e.g. in [6].

In many of the cases, the study of the Cauchy problem

$Lu=f,\,\partial_{t}^{j}u(x,0)=g_{j+1},\,0\leq j\leq m-1,$ (1.2)

where $L=\partial_{t}^{m}-\sum_{j=0}^{m-1}a_{m-j}(x,t,D)\partial_{t}^{j},$
with any $a_{m-j}(x,t,D)$ being a differential operator of order $m-j$ on a
closed manifold $M$ (or on $\mathbb{R}^{n}$) can be reduced to an equivalent
first order system of the form

$\begin{cases}\frac{\partial u}{\partial t}-K(t)u=f,&\text{
}u\in\mathscr{D}^{\prime}((0,T)\times M),\\\ u(0)=u_{0}.&\text{ }\end{cases}$
(1.3)

Indeed, if $\Delta_{M}$ denotes the positive Laplacian on $M,$ and
$\Gamma=(1+\Delta_{M})^{\frac{1}{2}}$ is the corresponding Bessel potential,
setting

$u_{j}:=\partial_{t}^{j-1}\Gamma^{m-j}u,\,\,1\leq j\leq m,$ (1.4)

then (1.2) is equivalent to the system

$\partial_{t}\begin{pmatrix}u_{1}&\\\ u_{2}&\\\ \vdots&\\\ u_{m-1}&\\\
u_{m}\end{pmatrix}=\begin{pmatrix}0&\Gamma&0&\cdots&0\\\
0&0&\Gamma&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\ddots&\Gamma\\\ b_{1}(x,t,D)&b_{2}(x,t,D)&\cdots&\cdots&b_{m}(x,t,D)\\\
\end{pmatrix}\begin{pmatrix}0&\\\ 0&\\\ \vdots&\\\ 0&\\\ \,f\end{pmatrix},$
(1.5)

where the sequence $b_{m}(x,t,D)=a_{m-j+1}(x,t,D)\Gamma^{j-m}$ of pseudo-
differential operators are of order one, with the functions $u_{j}$ having
initial conditions $u_{j}|_{t=0}=\Gamma^{m-j}g_{j}$ at $t=0.$ Changing the
notation and denoting the function vector

$(u_{1},\cdots,u_{m})^{t}$

by $u$ and the matrix in (1.5) by $K(t)$ we have the equivalence of (1.5),
(1.3) and the initial higher order initial value problem (1.2). For a complete
analysis of this equivalence in the setting of parabolic and hyperbolic
problems we refer the reader to the book of M. Taylor [38, Chapters IV-V].

The matrix $K(t)$ in (1.3) has as entries pseudo-differential operators or
order one and it illustrates the relevance of the analysis of the initial
value problem (1.1), even in the case of a compact Lie group $M=G$. Our choice
of the setting of compact Lie groups is motivated by exploiting the group
Fourier transform, and the global notion of a matrix-valued symbol that allows
the consideration of a variety of pseudo-differential operators $K(t),$ [33],
allowing a coordinates-free approach. Of particular interest for this work, is
the case when $K(t)$ becomes (a parametrised in $t\in[0,T]$) family of
subelliptic pseudo-differential operators [3].

The main results of this work are Theorem 3.6 in Subsection 3.3 and Theorem
3.8 in Subsection 3.3 where we prove that

  * •

The Cauchy problem 1.1 is well posedness for any continuous family of strongly
elliptic pseudo-differential operators $K(t)$ in the subelliptic Hörmander
classes in the Hörmander classes
$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ with the solution
operator

$(v_{0},f)\in H^{s,\mathcal{L}}(G)\times
L^{2}([0,T],H^{s,\mathcal{L}}(G))\mapsto u\in
L^{\infty}([0,T],H^{s,\mathcal{L}}(G)),$ (1.6)

being a continuous mapping for any $s\in\mathbb{R}.$ The continuity of the
operator is proved with the corresponding energy estimates and the Sobolev
spaces $H^{s,\mathcal{L}}(G)$ are then adapted to the regularity scale of the
sub-Laplacian
$\|w\|_{H^{s,\mathcal{L}}(G)}:=\|(1+\mathcal{L})^{\frac{s}{2}}\|_{L^{2}(G)}.$

  * •

The strongly elliptic condition in the continuity of (1.6) can be relaxed
assuming the positivity of the matrix-valued symbol of the operator $-K(t),$
when the order $m$ is assumed to be in the interval
$[0,\rho/\kappa-(2-1/\kappa)\delta].$ The upper bound of this interval, or the
maximum order of tolerance for the order, is related with the sharp-Gårding
inequality for the classes
$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ (see [5]).

In the case where the Laplacian takes the place of the sub-Laplacian, we
recover the Hörmander classes $\Psi^{m}_{\rho,\delta}(G\times\widehat{G})$ in
[33] (and then, the Hörmander classes defined by localisations [25]) we allow
the order $0\leq m\leq\rho-\delta,$ and even the criterion in Theorem 3.8 for
a type of operators of the form
$K(t)=a(t,x)(\mathcal{L}_{G})^{\frac{1}{2}}+D,$ $D\in
TG^{\mathbb{C}}=TG\otimes iTG,$ (see Corollary 3.15 for details) provides new
results beyond the known criteria given e.g. in Taylor [38, Chapters IV-V]. In
that case, the well posedness for the (quasi-geostrophic) differential problem

$v_{t}=a(t,x)(\mathcal{L}_{G})^{\frac{1}{2}}v+\sum_{j=1}^{n}a_{j}(t,x)iX_{j}u,\,v_{0}=u,$

is analysed, taking care of the drift $D=\sum_{j=1}^{n}a_{j}(t,x)iX_{j}\in
TG^{\mathbb{C}}=TG\otimes iTG.$

A concrete example in this direction is given in Subsection 3.7 in the case of
$\textnormal{SU}(2),$ showing the viability of the positivity criterion of the
symbol of the operator $-K(t)$ in Theorem 3.8.

  * •

Our criteria for the strongly elliptic case where
$K(t)\in\Psi^{m}_{\rho,\delta}(G\times\widehat{G})$ recovers the well
posedness result for hyperbolic problems in Taylor [38] in the case of the
fractional Laplacian on $G$.

This paper is organised as follows.

  * •

In Section 2 we provide the preliminaries about the Fourier analysis on
compact Lie groups and the related topics about the subelliptic Hörmander
classes $\Psi^{m}_{\rho,\delta}(G\times\widehat{G}).$

  * •

Finally, Section 3 is dedicated to the proof of our main results and to
presenting several consequences, finalising the section with the investigation
of fractional diffusion models (with drift) on $G=\textnormal{SU}(2).$

##  2\. Fourier analysis on Lie groups and subelliptic Hörmander classes

###  2.1. Pseudo-differential operators via localisations

Pseudo-differential operators on compact manifolds, and consequently on
compact Lie groups, can be defined by using local coordinate charts (see
Hörmander [25] and also M. Taylor [38] for a good introductory background on
the subject). Although here we will use the matrix-symbols introduced in [33],
as we will see later there is a global description of the Hörmander classes in
[25] in terms of such a global notion of matrix-symbols.

Let us briefly introduce these classes starting with the definition in the
Euclidean setting. Let $U$ be an open subset of $\mathbb{R}^{n}.$ We say that
the ``symbol" $a\in C^{\infty}(U\times\mathbb{R}^{n},\mathbb{C})$ belongs to
the Hörmander class of order $m$ and of $(\rho,\delta)$-type,
$S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n}),$ $0\leqslant\rho,\delta\leqslant
1,$ if for every compact subset $K\subset U$ and for all
$\alpha,\beta\in\mathbb{N}_{0}^{n}$, the symbol inequalities

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leqslant
C_{\alpha,\beta,K}(1+|\xi|)^{m-\rho|\alpha|+\delta|\beta|},$

hold true uniformly in $x\in K$ for all $\xi\in\mathbb{R}^{n}.$ Then, a
continuous linear operator $A:C^{\infty}_{0}(U)\rightarrow C^{\infty}(U)$ is a
pseudo-differential operator of order $m$ of $(\rho,\delta)$-type, if there
exists a symbol $a\in S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n})$ such that

$Af(x)=\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
ix\cdot\xi}a(x,\xi)(\mathscr{F}_{\mathbb{R}^{n}}{f})(\xi)d\xi,$

for all $f\in C^{\infty}_{0}(U),$ where

$(\mathscr{F}_{\mathbb{R}^{n}}{f})(\xi):=\smallint\limits_{U}e^{-i2\pi
x\cdot\xi}f(x)dx$

is the Euclidean Fourier transform of $f$ at $\xi\in\mathbb{R}^{n}.$

Once the definition of Hörmander classes on open subsets of $\mathbb{R}^{n}$
is established, it can be extended to smooth manifolds as follows. Given a
$C^{\infty}$-manifold $M,$ a linear continuous operator
$A:C^{\infty}_{0}(M)\rightarrow C^{\infty}(M)$ is a pseudo-differential
operator of order $m$ of $(\rho,\delta)$-type, with $\rho\geqslant 1-\delta,$
and $0\leq\delta<\rho\leq 1,$ if for every local coordinate patch
$\omega:M_{\omega}\subset M\rightarrow U_{\omega}\subset\mathbb{R}^{n},$ and
for every $\phi,\psi\in C^{\infty}_{0}(U_{\omega}),$ the operator

$Tu:=\psi(\omega^{-1})^{*}A\omega^{*}(\phi u),\,\,u\in
C^{\infty}(U_{\omega}),$

is a standard pseudo-differential operator with symbol $a_{T}\in
S^{m}_{\rho,\delta}(U_{\omega}\times\mathbb{R}^{n}).$ In this case we write
$A\in\Psi^{m}_{\rho,\delta}(M,\textnormal{loc}).$

###  2.2. The positive sub-Laplacian and pseudo-differential operators via
global symbols

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}\simeq
T_{e_{G}}G$, where $e_{G}$ is the neutral element of $G$, and let

$X=\\{X_{1},\cdots,X_{k}\\}\subset\mathfrak{g}$

be a system of $C^{\infty}$-vector fields. For all
$I=(i_{1},\cdots,i_{\omega})\in\\{1,2,\cdots,k\\}^{\omega}$ of length
$\omega\geqslant 1$, we denote by

$X_{I}:=[X_{i_{1}},[X_{i_{2}},\cdots[X_{i_{\omega-1}},X_{i_{\omega}}]\cdots]]$

a commutator of length $\omega$, where $X_{I}:=X_{i}$ when $\omega=1$ and
$I=(i)$. The system $X$ is said to satisfy Hörmander's condition of step (or
order) $\kappa$ if $\mathfrak{g}=\mathrm{span}\\{X_{I}:|I|\leq\kappa\\}$, that
is, in other words, the vector fields $X_{j}$, $j=1,\ldots,k$, together with
their commutator up to length $\kappa$, generate the whole Lie algebra
$\mathfrak{g}$.

Note that we are assuming that there is no subsystem
$Y=\\{Y_{1},\cdots,Y_{\ell}\\}\subset X$, $\ell<k$, of smooth vector fields
such that $\mathfrak{g}=\mathrm{span}\\{Y_{I}:|I|\leq\kappa\\}$. In this case
we call $X$ a system of Hörmander's vector fields.

Given a system $X=\\{X_{1},\cdots,X_{k}\\}$ of Hörmander's vector fields, then
the operator defined as

$\mathcal{L}\equiv\mathcal{L}_{X}:=-(X_{1}^{2}+\cdots+X_{k}^{2}),$

is a hypoelliptic operator by Hörmander theorem on sums of the squares of
vector fields (see Hörmander [24]). In particular the operator $\mathcal{L}$
is also subelliptic, and it is called the subelliptic Laplacian associated
with the system $X$, or simply sub-Laplacian. It is clear from the definition
that one can define different sub-Laplacians by using different systems of
Hörmander's vector fields (and that satisfy Hörmander condition of different
step).

As already mentioned in the introduction, we will make use of the quantization
process developed by the third author and V. Turunen in [33]. We briefly
recall below how this global quantization is defined.

Let $A$ be a continuous linear operator from $C^{\infty}(G)$ into
$\mathscr{D}^{\prime}(G),$ and let $\widehat{G}$ be the algebraic unitary dual
of $G.$ Then, there exists a function

$a:G\times\widehat{G}\rightarrow\cup_{\ell\in\mathbb{N}}\mathbb{C}^{\ell\times\ell},$
(2.1)

that we call the symbol of $A,$ such that
$a(x,\xi):=a(x,[\xi])\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$ for every
equivalence class $[\xi]\in\widehat{G},$ where
$\xi:G\rightarrow\textnormal{Hom}(H_{\xi}),$
$H_{\xi}\cong\mathbb{C}^{d_{\xi}},$ and such that

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}{\textnormal{Tr}}[\xi(x)a(x,\xi)\widehat{f}(\xi)],\,\,\forall
f\in C^{\infty}(G).$ (2.2)

Note that we have denoted by

$\widehat{f}(\xi)\equiv(\mathscr{F}f)(\xi):=\smallint\limits_{G}f(x)\xi(x)^{*}dx\in\mathbb{C}^{d_{\xi}\times
d_{\xi}},\,\,\,[\xi]\in\widehat{G},$

the Fourier transform of $f$ at $\xi\cong(\xi_{ij})_{i,j=1}^{d_{\xi}},$ where
the matrix representation of $\xi$ is induced by an orthonormal basis of the
representation space $H_{\xi}.$ The function $a$ in (2.1) satisfying (2.2) is
unique, and satisfies the identity

$a(x,\xi)=\xi(x)^{*}(A\xi)(x),\,\,A\xi:=(A\xi_{ij})_{i,j=1}^{d_{\xi}},\,\,\,[\xi]\in\widehat{G}.$

Note that the previous identity is well defined. Indeed, it is well known that
the functions $\xi_{ij}$, which are of $C^{\infty}$-class, are the
eigenfunctions of the positive Laplace operator $\mathcal{L}_{G}$, that is
$\mathcal{L}_{G}\xi_{ij}=\lambda_{[\xi]}\xi_{ij}$ for some non-negative real
number $\lambda_{[\xi]}\geq 0$ depending only of the equivalence class $[\xi]$
and not on the representation $\xi$.

In general, we refer to the function $a$ as the (global or full) symbol of the
operator $A,$ and we will use the notation $A=\textnormal{Op}(a)$ to indicate
that $a:=\sigma_{A}$ is the symbol associated with the operator $A.$

In order to classify symbols in the Hörmander classes, in [33] the authors
defined the notion of difference operators, which endows $\widehat{G}$ with a
difference structure. Following [36], a difference operator
$Q_{\xi}:\mathscr{D}^{\prime}(\widehat{G})\rightarrow\mathscr{D}^{\prime}(\widehat{G})$
of order $k$ is defined as

$Q_{\xi}\widehat{f}(\xi)=\widehat{qf}(\xi),\,[\xi]\in\widehat{G},$ (2.3)

for some function $q$ vanishing of order $k$ at the neutral element $e=e_{G}.$
We will denote by $\textnormal{diff}^{k}(\widehat{G})$ the class of all
difference operators of order $k.$ For a fixed smooth function $q,$ the
associated difference operator will be denoted by $\Delta_{q}\equiv Q_{\xi}.$
A system of difference operators (see e.g. [36])

$\Delta_{\xi}^{\alpha}:=\Delta_{q_{(1)}}^{\alpha_{1}}\cdots\Delta_{q_{(i)}}^{\alpha_{i}},\,\,\alpha=(\alpha_{j})_{1\leqslant
j\leqslant i},$

with $i\geq n$, is called admissible if

$\textnormal{rank}\\{\nabla q_{(j)}(e):1\leqslant j\leqslant
i\\}=\textnormal{dim}(G),\textnormal{ and
}\Delta_{q_{(j)}}\in\textnormal{diff}^{1}(\widehat{G}).$

An admissible collection is said to be strongly admissible if, additionally,

$\bigcap_{j=1}^{i}\\{x\in G:q_{(j)}(x)=0\\}=\\{e_{G}\\}.$

######  Remark 2.1.

Matrix components of unitary representations induce difference operators.
Indeed, if $\xi_{1},\xi_{2},\cdots,\xi_{k},$ are fixed irreducible and unitary
representation of $G$, which not necessarily belong to the same equivalence
class, then each coefficient of the matrix

$\xi_{\ell}(g)-I_{d_{\xi_{\ell}}}=[\xi_{\ell}(g)_{ij}-\delta_{ij}]_{i,j=1}^{d_{\xi_{\ell}}},\,\quad
g\in G,\,\,1\leq\ell\leq k,$ (2.4)

that is each function $q^{\ell}_{ij}(g):=\xi_{\ell}(g)_{ij}-\delta_{ij}$,
$g\in G,$ defines a difference operator

$\mathbb{D}_{\xi_{\ell},i,j}:=\mathscr{F}(\xi_{\ell}(g)_{ij}-\delta_{ij})\mathscr{F}^{-1}.$
(2.5)

We can fix $k\geq\mathrm{dim}(G)$ of these representations in such a way that
the corresponding family of difference operators is admissible, that is,

$\textnormal{rank}\\{\nabla q^{\ell}_{i,j}(e):1\leqslant\ell\leqslant
k\\}=\textnormal{dim}(G).$

To define higher order difference operators of this kind, let us fix a unitary
irreducible representation $\xi_{\ell}$. Since the representation is fixed we
omit the index $\ell$ of the representations $\xi_{\ell}$ in the notation that
will follow. Then, for any given multi-index
$\alpha\in\mathbb{N}_{0}^{d_{\xi_{\ell}}^{2}}$, with
$|\alpha|=\sum_{i,j=1}^{d_{\xi_{\ell}}}\alpha_{i,j}$, we write

$\mathbb{D}^{\alpha}:=\mathbb{D}_{1,1}^{\alpha_{11}}\cdots\mathbb{D}^{\alpha_{d_{\xi_{\ell}},d_{\xi_{\ell}}}}_{d_{\xi_{\ell}}d_{\xi_{\ell}}}$

for a difference operator of order $|\alpha|$.

The difference operators endow the unitary dual $\widehat{G}$ with a
difference structure. For difference operators of the previous form, the
following finite Leibniz-like formula holds true (see [34] for details). Note
that below we are still assuming that the representation $\xi_{\ell}$ is
fixed.

######  Proposition 2.2 (Leibniz rule for difference operators).

Let $G$ be a compact Lie group and let $\mathbb{D}^{\alpha},$
$\alpha\in\mathbb{N}^{d_{\xi_{\ell}}}_{0},$ be the family of difference
operators defined in (2.5). Then, the following Leibniz rule

$\displaystyle(\mathbb{D}^{\alpha}(a_{1}a_{2}))(x_{0},\xi)=\sum_{|\gamma|,|\varepsilon|\leqslant|\alpha|\leqslant|\gamma|+|\varepsilon|}C_{\varepsilon,\gamma}(\mathbb{D}^{\gamma}a_{1})(x_{0},\xi)(\mathbb{D}^{\varepsilon}a_{2})(x_{0},\xi),\quad
x_{0}\in G,$

holds for all $a_{1},a_{2}\in
C^{\infty}(G)\times\mathscr{S}^{\prime}(\widehat{G})$, where the summation is
taken over all $\varepsilon,\gamma$ such that
$|\varepsilon|,|\delta|\leq|\alpha|\leq|\gamma|+|\varepsilon|$.

Note that for a different kind of difference operators, namely for those given
by compositions of difference operators of higher order associated with
different representations, a Leibniz-like formula still holds true by
iteration. For more details about difference operators and Leibniz-like
formulas for admissible collections see also Corollary 5.13 in [19].

We are now going to introduce the global Hörmander classes of symbols defined
in [33]. First let us recall that every left-invariant vector field
$Y\in\mathfrak{g}$ can be identified with the first order differential
operator $\partial_{Y}:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ given
by

$\partial_{Y}f(x)=(Y_{x}f)(x)=\frac{d}{dt}f(x\exp(tY))|_{t=0}.$

If $\\{X_{1},\cdots,X_{n}\\}$ is a basis of the Lie algebra $\mathfrak{g},$
then we will use the standard multi-index notation

$\partial_{X}^{\alpha}=X_{x}^{\alpha}=\partial_{X_{1}}^{\alpha_{1}}\cdots\partial_{X_{n}}^{\alpha_{n}},$

for a canonical left-invariant differential operator of order $|\alpha|.$

By using this property, together with the following notation for the so-called
elliptic weight

$\langle\xi\rangle:=(1+\lambda_{[\xi]})^{1/2},\,\,[\xi]\in\widehat{G},$

we can finally give the definition of global symbol classes.

######  Definition 2.3.

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ Let

$\sigma:G\times\widehat{G}\rightarrow\bigcup_{[\xi]\in\widehat{G}}\mathbb{C}^{d_{\xi}\times
d_{\xi}},$

be a matrix-valued function such that for any $[\xi]\in\widehat{G},$
$\sigma(\cdot,[\xi])$ is of $C^{\infty}$-class, and such that, for any given
$x\in G$ there is a distribution $k_{x}\in\mathscr{D}^{\prime}(G),$ smooth in
$x,$ satisfying $\sigma(x,\xi)=\widehat{k}_{x}(\xi),$ $[\xi]\in\widehat{G}$.
We say that $\sigma\in\mathscr{S}^{m}_{\rho,\delta}(G)$ if the following
symbol inequalities

$\|\partial_{X}^{\beta}\Delta_{\xi}^{\gamma}\sigma_{A}(x,\xi)\|_{\textnormal{op}}\leqslant
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\gamma|+\delta|\beta|},$ (2.6)

are satisfied for all $\beta$ and $\gamma$ multi-indices and for all
$(x,[\xi])\in G\times\widehat{G}.$ For
$\sigma_{A}\in\mathscr{S}^{m}_{\rho,\delta}(G)$ we will write
$A\in\Psi^{m}_{\rho,\delta}(G)\equiv\textnormal{Op}(\mathscr{S}^{m}_{\rho,\delta}(G)).$

The global Hörmander classes on compact Lie groups can be used to describe the
Hörmander classes defined by local coordinate systems. We present the
corresponding statement as follows.

######  Theorem 2.4 (Equivalence of classes, [33, 34]).

Let $A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a continuous linear
operator and let $0\leq\delta<\rho\leq 1,$ with $\rho\geq 1-\delta.$ Then,
$A\in\Psi^{m}_{\rho,\delta}(G,\textnormal{loc}),$ if and only if
$\sigma_{A}\in\mathscr{S}^{m}_{\rho,\delta}(G).$ Consequently,

$\textnormal{Op}(\mathscr{S}^{m}_{\rho,\delta}(G))=\Psi^{m}_{\rho,\delta}(G,\textnormal{loc}),\,\,\,0\leqslant\delta<\rho\leqslant
1,\,\rho\geqslant 1-\delta.$ (2.7)

###  2.3. Subelliptic Hörmander classes on compact Lie groups

In order to define the subelliptic Hörmander calculus, we will use a suitable
basis of the Lie algebra arising from Taylor expansions. We explain the choice
of this basis by means of the following lemma (see Lemma 7.4 in [19]).

######  Lemma 2.5.

Let $G$ be a compact Lie group of dimension $n.$ Let
$\mathfrak{D}=\\{\Delta_{q_{(j)}}\\}_{1\leqslant j\leqslant n}$ be a strongly
admissible admissible collection of difference operators, that is

$\textnormal{rank}\\{\nabla q_{(j)}(e):1\leqslant j\leqslant
n\\}=n,\,\,\,\bigcap_{j=1}^{n}\\{x\in G:q_{(j)}(x)=0\\}=\\{e_{G}\\}.$

Then there exists a basis
$X_{\mathfrak{D}}=\\{X_{1,\mathfrak{D}},\cdots,X_{n,\mathfrak{D}}\\}$ of
$\mathfrak{g}$ such that

$X_{j,\mathfrak{D}}q_{(k)}(\cdot^{-1})(e_{G})=\delta_{jk}.$

Moreover, by using the multi-index notation

$\partial_{X}^{(\beta)}=\partial_{X_{1,\mathfrak{D}}}^{\beta_{1}}\cdots\partial_{X_{n,\mathfrak{D}}}^{\beta_{n}},$

for any $\beta\in\mathbb{N}_{0}^{n},$ where

$\partial_{X_{i,\mathfrak{D}}}f(x)=\frac{d}{dt}f(x\exp(tX_{i,\mathfrak{D}}))|_{t=0},\,\,f\in
C^{\infty}(G),$

and denoting by

$R_{x,N}^{f}(y)=f(xy)-\sum_{|\alpha|<N}q_{(1)}^{\alpha_{1}}(y^{-1})\cdots
q_{(n)}^{\alpha_{n}}(y^{-1})\partial_{X}^{(\alpha)}f(x)$

the Taylor remainder, we have that

$|R_{x,N}^{f}(y)|\leqslant C|y|^{N}\max_{|\alpha|\leqslant
N}\|\partial_{X}^{(\alpha)}f\|_{L^{\infty}(G)},$

where the constant $C>0$ is dependent on $N,$ $G$ and $\mathfrak{D}$ (but not
on $f\in C^{\infty}(G)).$ In addition we have that
$\partial_{X}^{(\beta)}|_{x_{1}=x}R_{x_{1},N}^{f}=R_{x,N}^{\partial_{X}^{(\beta)}f}$,
and

$|\partial_{X}^{(\beta)}|_{y_{1}=y}R_{x,N}^{f}(y_{1})|\leqslant
C|y|^{N-|\beta|}\max_{|\alpha|\leqslant
N-|\beta|}\|\partial_{X}^{(\alpha+\beta)}f\|_{L^{\infty}(G)},$

provided that $|\beta|\leqslant N.$

Using the notation above, and denoting by
$\Delta_{\xi}^{\alpha}:=\Delta_{q_{(1)}}^{\alpha_{1}}\cdots\Delta_{q_{(n)}}^{\alpha_{n}},$
we can introduce the subelliptic Hörmander class of symbols of order
$m\in\mathbb{R}$ of type $(\rho,\delta)$.

######  Definition 2.6 (Subelliptic Hörmander classes).

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ Let
us consider a sub-Laplacian $\mathcal{L}=-(X_{1}^{2}+\cdots+X_{k}^{2})$ on
$G,$ where the system of vector fields $X=\\{X_{i}\\}_{i=1}^{k}$ satisfies the
Hörmander condition of step $\kappa$. We say that
$\sigma\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ if

$p_{\alpha,\beta,\rho,\delta,m,\textnormal{left}}(\sigma)^{\prime}:=\sup_{(x,[\xi])\in
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta|\beta|-m)}\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi)\|_{\textnormal{op}}<\infty,$
(2.8)
$p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(\sigma)^{\prime}:=\sup_{(x,[\xi])\in
G\times\widehat{G}}\|(\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi))\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta|\beta|-m)}\|_{\textnormal{op}}<\infty,$
(2.9)

holds true for all $\alpha,\beta\in\mathbb{N}^{n}_{0}$.

By following the usual nomenclature, we define:

$\textnormal{Op}({S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})):=\\{A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G):\sigma_{A}\equiv\widehat{A}(x,\xi)\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})\\},$

with

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

We will use the notation $\widehat{\mathcal{M}}$ for the matrix-valued symbol
of the operator $\mathcal{M}:=(1+\mathcal{L})^{\frac{1}{2}},$ and, for every
$[\xi]\in\widehat{G}$ and $s\in\mathbb{R},$ we define

$\widehat{\mathcal{M}}(\xi)^{s}:=\textnormal{diag}[(1+\nu_{ii}(\xi)^{2})^{\frac{s}{2}}]_{1\leqslant
i\leqslant d_{\xi}},$

where
$\widehat{\mathcal{L}}(\xi)=:\textnormal{diag}[\nu_{ii}(\xi)^{2}]_{1\leqslant
i\leqslant d_{\xi}}$ is the symbol of the sub-Laplacian $\mathcal{L}$ at
$[\xi].$

######  Definition 2.7 (Subelliptic amplitudes).

A function $a:G\times
G\times\widehat{G}\rightarrow\cup_{[\xi]\in\widehat{G}}\mathbb{C}^{d_{\xi}\times
d_{\xi}}$ is an amplitude symbol if for every $[\xi]\in\widehat{G},$
$a(\cdot,\cdot,[\xi])$ is smooth. In addition, $a$ belongs to the subelliptic
amplitude class of order $m$ and type $(\rho,\delta)$,
$\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times G\times\widehat{G})$ if

$\sup_{(x,y,[\xi])\in G\times
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta(|\beta|+|\gamma|)-m)}\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}a(x,y,\xi)\|_{\textnormal{op}}<\infty,$
(2.10)

and

$\sup_{(x,y,[\xi])\in G\times
G\times\widehat{G}}\|(\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}a(x,y,\xi))\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta(|\beta|+|\gamma|)-m)}\|_{\textnormal{op}}<\infty.$
(2.11)

The amplitude operator associated with an amplitude
$a\in\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times G\times\widehat{G})$
is defined via

$Af(x)\equiv\textnormal{AOp}(a)f(x):=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{{Tr}}\left(\xi(x)\smallint\limits_{G}a(x,y,\xi)\xi(y)^{*}f(y)dy\right),$

where $f\in C^{\infty}(G).$

The decay properties of subelliptic symbols are summarized in the following
lemma (see [3, Chapter 4]).

######  Lemma 2.8.

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ If
$a\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ then for every
$\alpha,\beta\in\mathbb{N}_{0}^{n},$ there exists $C_{\alpha,\beta}>0$
satisfying the estimates

$\|\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}a(x,\xi)\|_{\textnormal{op}}\leqslant
C_{\alpha,\beta}\sup_{1\leqslant i\leqslant
d_{\xi}}(1+\nu_{ii}(\xi)^{2})^{\frac{m-\rho|\alpha|+\delta|\beta|}{2}},$

uniformly in $(x,[\xi])\in G\times\widehat{G}.$

In the next theorem we describe some fundamental properties of the subelliptic
calculus [3].

######  Theorem 2.9.

Let $0\leqslant\delta<\rho\leqslant 1,$ and let
$\Psi^{m,\,\mathcal{L}}_{\rho,\delta}:=\textnormal{Op}({S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})),$
for every $m\in\mathbb{R}.$ Then,

  * -

The mapping $A\mapsto
A^{*}:\Psi^{m,\,\mathcal{L}}_{\rho,\delta}\rightarrow\Psi^{m,\,\mathcal{L}}_{\rho,\delta}$
is a continuous linear mapping between Fréchet spaces and the symbol of
$A^{*},$ $\sigma_{A^{*}}(x,\xi)$ satisfies the asymptotic expansion,

$\widehat{A^{*}}(x,\xi)\sim\sum_{|\alpha|=0}^{\infty}\Delta_{\xi}^{\alpha}\partial_{X}^{(\alpha)}(\widehat{A}(x,\xi)^{*}).$

This means that, for every $N\in\mathbb{N},$ and for all $\ell\in\mathbb{N},$

$\footnotesize{\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\widehat{A^{*}}(x,\xi)-\sum_{|\alpha|\leqslant
N}\Delta_{\xi}^{\alpha}\partial_{X}^{(\alpha)}(\widehat{A}(x,\xi)^{*})\right)\in{S}^{m-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})},$

where $|\alpha_{\ell}|=\ell.$

  * -

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1},\,\mathcal{L}}_{\rho,\delta}\times\Psi^{m_{2},\,\mathcal{L}}_{\rho,\delta}\rightarrow\Psi^{m_{3},\,\mathcal{L}}_{\rho,\delta}$
is a continuous bilinear mapping between Fréchet spaces, and the symbol of
$A=A_{1}\circ A_{2}$ satisfies is given by the asymptotic formula

$\sigma_{A}(x,\xi)\sim\sum_{|\alpha|=0}^{\infty}(\Delta_{\xi}^{\alpha}\widehat{A}_{1}(x,\xi))(\partial_{X}^{(\alpha)}\widehat{A}_{2}(x,\xi)),$

which, in particular, means that, for every $N\in\mathbb{N},$ and for all
$\ell\in\mathbb{N},$

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma_{A}(x,\xi)-\sum_{|\alpha|\leqslant
N}(\Delta_{\xi}^{\alpha}\widehat{A}_{1}(x,\xi))(\partial_{X}^{(\alpha)}\widehat{A}_{2}(x,\xi))\right)$
$\displaystyle\hskip
56.9055pt\in{S}^{m_{1}+m_{2}-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$

for all $\alpha_{\ell}\in\mathbb{N}_{0}^{n}$ with $|\alpha_{\ell}|=\ell.$

  * -

For $0\leqslant\delta<\rho\leqslant 1,$ (or for $0\leq\delta\leq\rho\leq 1,$
$\delta<1/\kappa$) let us consider a continuous linear operator
$A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ with symbol
$\sigma\in{S}^{0,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then $A$
extends to a bounded operator from $L^{2}(G)$ to $L^{2}(G).$

Here and in the rest of the paper we shall denote by $H^{s,\,\mathcal{L}}(G)$,
for $s\in\mathbb{R}$, the subelliptic Sobolev space of order $s$ associated
with a fixed positive sub-Laplacian $\mathcal{L}$, that is, the space defined
as the completion of $C^{\infty}(G)$ with respect to the norm

$\|u\|_{H^{s,\,\mathcal{L}}(G)}:=\|(1+\mathcal{L})^{\frac{s}{2}}u\|_{L^{2}(G)}.$

######  Remark 2.10.

The last assertion in Theorem 2.9 remains valid if we consider
$0\leq\delta\leq\rho\leq 1,$ $\delta<1/\kappa.$ This is the subelliptic
Calderón-Vaillancourt theorem proved in [3], which gives the boundedness of
pseudo-differential operators in the subelliptic calculus in subelliptic
Sobolev spaces.

######  Proposition 2.11.

Let $A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a continuous linear
operator with symbol $a\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ $0\leq\delta<\rho\leq
1.$ Then $A:H^{s,\,\mathcal{L}}(G)\rightarrow H^{s-m,\,\mathcal{L}}(G)$
extends to a bounded operator for all $s\in\mathbb{R}.$

Next, we state the Gårding inequality and the Sharp Gårding inequality proved
for the subelliptic Hörmander classes in [3, 5].

######  Theorem 2.12 (Subelliptic Gårding inequality).

For $0\leqslant\delta<\rho\leqslant 1,$ let
$a(x,D):C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be an operator with
symbol $a\in{S}^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$,
$m\in\mathbb{R}$. Let us assume that

$A(x,\xi)=\frac{1}{2}(a(x,\xi)+a(x,\xi)^{*})\geqslant
C\widehat{\mathcal{M}}(\xi)^{m},$ (2.12)

for all $(x,[\xi])\in G\times\widehat{G},$ for some $C>0.$ Then, there exist
$C_{1},C_{2}>0,$ such that the lower bound

$\displaystyle\mathsf{Re}(a(x,D)u,u)\geqslant
C_{1}\|u\|_{{L}^{2,\mathcal{L}}_{\frac{m}{2}}(G)}-C_{2}\|u\|_{L^{2}(G)}^{2},$
(2.13)

holds true for every $u\in C^{\infty}(G).$

Theorem 2.12 motivates to define the sub-family of $\mathcal{L}$-elliptic
operators, namely, the class of strongly $\mathcal{L}$-elliptic operators.

######  Definition 2.13 (Strongly $\mathcal{L}$-elliptic operators).

We will say that an operator
$A\in\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ with
$0\leq\delta<\rho\leq 1,$ is strongly $\mathcal{L}$-elliptic or order $m$ if
there exists $C>0,$ such that

$\mathsf{Re}(a(x,\xi))=\frac{1}{2}(a(x,\xi)+a(x,\xi)^{*})\geqslant
C\widehat{\mathcal{M}}(\xi)^{m},$ (2.14)

for all $(x,[\xi])\in G\times\widehat{G}.$

Now, we are going to present the corresponding lower bound for subelliptic
operators with non-negative symbols on any representation space, as in [5].

######  Theorem 2.14 (Subelliptic sharp Gårding inequality).

For $0<\rho\leqslant 1$ and $0\leq\delta<(2\kappa-1)^{-1}\rho$, and for
$m\in\mathbb{R}$, let

$A\equiv a(x,D):C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$

be a pseudo-differential operator with global symbol
$a\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then, if
$a(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times\widehat{G}$, there exists a
positive constant $C$ such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{m-\varkappa}{2},\,\mathcal{L}}(G)}^{2},\quad\varkappa={\rho}/{\kappa}-\left(2-{1}/{\kappa}\right)\delta,$
(2.15)

for all $u\in C^{\infty}(G).$

##  3\. Analysis of diffusion problems on compact Lie groups

Let $T>0.$ In this section we will study the regularity, existence and
uniqueness for the Cauchy problem

$(\textnormal{IVP}):\begin{cases}\frac{\partial v}{\partial t}=K(t)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G)\\\ v(0)=u_{0},&\text{ }\end{cases}$
(3.1)

associated to a (possibly) time-dependent continuous family of pseudo-
differential operators $K(t).$ According to the literature, we start by
considering the general case where
$K(t)\in\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ belongs to the
subelliptic Hörmander classes on $G,$ with the initial data $u_{0}\in
H^{s,\mathcal{L}}(G),$ and $f\in L^{2}([0,T],H^{s,\mathcal{L}}(G)),$ where
$m>0,$ and $s\in\mathbb{R}.$

The generality of this consideration is justified by the consideration of
(possibly) continuous time-dependent family of subellitic pseudo-differential
operators $K(t)$ on $G$ with a fixed positive order $m$. We split our analysis
in two cases.

  * •

Case 1. We consider that a family of continuous time-dependent subellitic
pseudo-differential operators $K(t)$ is strongly $\mathcal{L}$-elliptic, that
is, for any $0\leqslant t\leqslant T,$ the estimate

$\sigma_{\mathsf{Re}(-K(t))}(x,[\xi])\geq\widehat{\mathcal{M}}(\xi)^{m},\,\,(x,[\xi])\in
G\times\widehat{G},$ (3.2)

holds on any representation space. In this case, the Gårding inequality for
the subelliptic classes

$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$

allows us to consider a general non-negative order $m>0.$

  * •

Case 2. The time-dependent family of symbols associated to the family of
continuous time-dependent subellitic pseudo-differential operators $K(t)$ is
negative definite on every representation space and for any $0\leqslant
t\leqslant T,$ that is,

$\sigma_{\mathsf{Re}(-K(t))}(x,[\xi])\geq 0,\,\,(x,[\xi])\in
G\times\widehat{G}.$ (3.3)

In this case, the sharp-Gårding inequality for the subelliptic classes

$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$

allows us to analyse the strictly fractional case

$0\leq m\leq\varkappa={\rho}/{\kappa}-\left(2-{1}/{\kappa}\right)\delta.$
(3.4)

In the case where the family of operators $K(t)$ is contained in the elliptic
Hörmander classes $\Psi^{m}_{\rho,\delta}(G\times\widehat{G}),$ the order
restriction becomes to be

$0\leq m\leq\rho-\delta,$

for all $0\leq\delta<\rho\leq 1,$ which recaptures (3.4) for $\kappa=1.$

The next subsection will be dedicated to stating the technical results in the
analysis of the well-posedness of (3.1).

###  3.1. Preliminary results

One of the facts of central difference between the subelliptic pseudo-
differential calculus in [3], and then of the elliptic calculus in [33] with
the Hörmander calculus via local coordinate systems is that, in general, the
order of the commutator of two operators in both calculus does not decrease,
which is a natural consequence of the non-commutativity of the products of
arbitrary matrices.

This point makes the techniques under which the theory of initial value
problems of pseudo-differential operators in Taylor [38] based on commutator
techniques (and several of the argument there involving commutators of pseudo-
differential operators) inoperable in the setting of the subelliptic classes
$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}).$ The following lemma
will be useful in the proof of the well posedness of (3.1) without the use of
commutator techniques.

######  Lemma 3.1.

Let $A\in\Psi^{\nu,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ be a
$\mathcal{L}$-elliptic operator for some $\nu\in\mathbb{R}$. Then, for all
$s\in\mathbb{R}$ there exists a constant $C>0$ such that

$\|A\mathcal{M}_{s}v\|_{L^{2}(G)}\leq
C[\|Av\|_{H^{s,\mathcal{L}}(G)}+\|v\|_{H^{s,\mathcal{L}}(G)}],$

for all $v\in H^{s,\mathcal{L}}(G)$. Here
$\mathcal{M}_{s}:=(1+\mathcal{L})^{\frac{s}{2}}$ is the subelliptic Bessel
potential of order $s\in\mathbb{R}.$

###### Proof.

Since
$A\mathcal{M}_{s}\in\Psi_{\rho,\delta}^{\nu+s,\mathcal{L}}(G\times\widehat{G}),$
in view of its Sobolev boundedness, there exists a constant $C_{1}>0$ such
that

$\|A\mathcal{M}_{s}v\|_{L^{2}(G)}\leq C_{1}\|v\|_{H^{\nu+s,\mathcal{L}}(G)}$
(3.5)

for all $v\in H^{\nu+s,\mathcal{L}}(G)$.

By the $\mathcal{L}$-ellipticity of $\mathcal{M}_{s}A$, there exists a
subelliptic parametrix
$T\in\Psi_{\rho,\delta}^{-(\nu+s),\mathcal{L}}(G\times\widehat{G})$ of
$\mathcal{M}_{s}A$. Then, there exists a constant $C_{2}>0$ such that

$\|Tu\|_{H^{\nu+s,\mathcal{L}}(G)}\leq C_{2}\|u\|_{L^{2}(G)},\,\,\textnormal{
for all }u\in L^{2}(G).$ (3.6)

On the other hand, let
$R=R(x,D)\in\Psi_{\rho,\delta}^{-\infty,\mathcal{L}}(G\times\widehat{G})$ be
such that

$T\mathcal{M}_{s}A=I+R(x,D),$

and let $v\in H^{\nu+s,\mathcal{L}}(G)$ and $u=\mathcal{M}_{s}Av$. Then $u\in
L^{2}(G)$ and

$Tu=T\mathcal{M}_{s}Av=v+R(x,D)v.$

Hence and by (3.6) we have

$\|v\|_{H^{\nu+s,\mathcal{L}}(G)}-\|R(x,D)v\|_{H^{\nu+s,\mathcal{L}}(G)}\leq\|v+R(x,D)v\|_{H^{\nu+s,\mathcal{L}}(G)}=\|T\mathcal{M}_{s}Au\|_{H^{\nu+s,\mathcal{L}}(G)}$
$\leq C_{2}\|\mathcal{M}_{s}Av\|_{L^{2}(G)}.$

Thus

$\|v\|_{H^{\nu+s,\mathcal{L}}(G)}\leq
C_{2}\|\mathcal{M}_{s}Av\|_{L^{2}(G)}+\|R(x,D)v\|_{H^{\nu+s,\mathcal{L}}(G)}.$

Now, since $R(x,D)$ belongs in particular to
$\Psi_{\rho,\delta}^{-\nu,\mathcal{L}}(G\times\widehat{G})$, we have

$\|R(x,D)v\|_{H^{\nu+s,\mathcal{L}}(G)}\leq
C_{3}\|v\|_{H^{s,\mathcal{L}}(G)}.$

Then

$\|v\|_{H^{\nu+s,\mathcal{L}}(G)}\leq
C_{2}\|\mathcal{M}_{s}Av\|_{L^{2}(G)}+\|v\|_{H^{s,\mathcal{L}}(G)}.$ (3.7)

Therefore, by (3.5) and (3.7) we obtain

$\|A\mathcal{M}_{s}v\|_{L^{2}(G)}\leq
C(\|\mathcal{M}_{s}Av\|_{L^{2}(G)}+\|v\|_{H^{s}(G)}),$

which concludes the proof of the lemma. ∎

In the following lemma we prove an energy identity that allows us to apply the
lower bounds for the Gårding inequality and the sharp Gårding inequality in
the form of Theorems 2.12 and 2.14.

######  Lemma 3.2.

Let $G$ be a compact Lie group. Let us assume that $f\in
C([0,T],\mathscr{D}(G))$ and $v\in C^{1}([0,T],\mathscr{D}(G))$ are
distributions such that

$\frac{\partial v}{\partial t}=K(t)v+f$

where, for any $t\in[0,T],$ $K(t):C([0,T],\mathscr{D}(G))\rightarrow
C([0,T],\mathscr{D}(G))$ is a continuous linear operator with respect to the
standard Fréchet structure on $C([0,T],\mathscr{D}(G)).$ Then,

$\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}=2\mathsf{Re}(K(t)v(t),v(t))_{L^{2}(G)}+2\mathsf{Re}(f(t),v(t))_{L^{2}(G)},$
(3.8)

for any $t\in[0,T].$

###### Proof.

Indeed, observe that

$\displaystyle\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}$
$\displaystyle=\frac{d}{dt}\left(v(t),v(t)\right)_{L^{2}(G)}=\left(\frac{dv(t)}{dt},v(t)\right)_{L^{2}(G)}+\left(v(t),\frac{dv(t)}{dt}\right)_{L^{2}(G)}$
$\displaystyle=\left(K(t)v(t)+f(t),v(t)\right)_{L^{2}(G)}+\left(v(t),K(t)v(t)+f(t)\right)_{L^{2}(G)}$
$\displaystyle=\left((K(t)+K(t)^{*})v(t),v(t)\right)_{L^{2}(G)}+2\mathsf{Re}(f(t),v(t))_{L^{2}(G)}$
$\displaystyle=2\mathsf{Re}(K(t)v(t),v(t))_{L^{2}(G)}+2\mathsf{Re}(f(t),v(t))_{L^{2}(G)},$

for any $t\in[0,T].$ The proof of the identity (3.8) is complete. ∎

The next lemma proves that the energy estimates on subelliptic Sobolev spaces
imply the existence and uniqueness of the IVP problem (3.10) via the Hanh-
Banach theorem.

######  Lemma 3.3.

Let $s\in\mathbb{R}$ and let $\mathbb{X}=C^{1}([0,T],H^{-s,\mathcal{L}}(G)).$
Assume that for any $v\in\mathbb{X}$ the following energy estimate

$\|v(t)\|^{2}_{H^{-s,\mathcal{L}}(G)}\leqslant\left(C\|v(0)\|^{2}_{H^{-s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau))v(\tau)\|^{2}_{H^{-s,\mathcal{L}}(G)}d\tau\right),$
(3.9)

holds for every $0\leqslant t\leqslant T,$ where $K(t)\in
C([0,T],\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})).$ Then, for
any $f\in C([0,T],\mathscr{D}(G))$ and $u_{0}\in C^{1}([0,T],\mathscr{D}(G))$
there exists a unique $v\in L^{2}([0,T],H^{s,\mathcal{L}}(G))$ solving the
following initial value problem

$(\textnormal{IVP}):\begin{cases}\frac{\partial v}{\partial t}=K(t)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G)\\\ v(0)=u_{0},&\text{ }\end{cases}$
(3.10)

and satisfying the energy inequality

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|f(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}d\tau\right),$
(3.11)

for every $0\leqslant t\leqslant T,$ between the solutions and the initial
data.

###### Proof.

Now, we will use the Hanh-Banach theorem to deduce that the solution to (3.10)
indeed, belongs to $\mathbb{X}$. So, let us denote $Q=\partial_{t}-K(t),$ and
let us introduce the spaces

$\displaystyle E:=\\{\phi\in\mathbb{X}:\phi(T)=0\\},$

and $Q^{*}E:=\\{Q^{*}\phi:\phi\in E\\}.$ Let us define the linear form
$\beta\in(Q^{*}E)^{\prime}$ by

$\beta(Q^{*}\phi):=\smallint\limits_{0}^{T}(f(\tau),\phi(\tau))d\tau+\frac{1}{i}(u_{0},\phi(0)).$

From (3.9) we have that for every $\phi\in E,$

$\displaystyle\|\phi(t)\|^{2}_{H^{-s,\mathcal{L}}(G)}$
$\displaystyle\leqslant\left(C\|\phi(T)\|^{2}_{H^{-s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau)^{*})\phi(\tau)\|^{2}_{H^{-s,\mathcal{L}}(G)}d\tau\right)$
$\displaystyle=C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau))^{*}\phi(\tau)\|^{2}_{H^{-s,\mathcal{L}}(G)}d\tau$
$\displaystyle=C^{\prime}\|(\partial_{t}-K(\tau))^{*}\phi(\tau)\|^{2}_{L^{2}([0,T],H^{-s,\mathcal{L}}(G))}.$

The previous analysis for implies the inequality

$\|\phi(t)\|^{2}_{H^{-s,\mathcal{L}}}\lesssim\|(\partial_{t}-K(\tau))^{*}\phi(\tau)\|^{2}_{L^{2}([0,T],H^{-s,\mathcal{L}})}=\|Q^{*}\phi(\tau)\|^{2}_{L^{2}([0,T],H^{-s,\mathcal{L}})}.$
(3.12)

So, in view of (3.12) we have

$\displaystyle|\beta(Q^{*}\phi)|$
$\displaystyle\leqslant\smallint\limits_{0}^{T}\|f(\tau)\|_{H^{s,\mathcal{L}}(G)}\|\phi(\tau)\|_{H^{-s,\mathcal{L}}(G)}d\tau+\|u_{0}\|_{{H^{s,\mathcal{L}}(G)}}\|\phi(0)\|_{H^{-s,\mathcal{L}}(G)}$
$\displaystyle\leqslant\|f\|_{L^{2}([0,T],H^{s,\mathcal{L}}(G))}\|\phi\|_{L^{2}([0,T],H^{-s,\mathcal{L}}(G))}+\|u_{0}\|_{H^{s,\mathcal{L}}(G)}\|\phi\|_{H^{-s,\mathcal{L}}(G)}$
$\displaystyle\lesssim_{T}(\|f\|_{L^{2}([0,T],H^{s,\mathcal{L}}(G)}+\|u_{0}\|_{H^{s,\mathcal{L}}(G)})\|Q^{*}\phi(\tau)\|_{L^{2}([0,T],H^{-s,\mathcal{L}}(G))},$

which shows that $\beta$ is a bounded functional on

$\mathscr{T}:=Q^{*}E\cap L^{2}([0,T],H^{-s,\mathcal{L}}(G)),$

with $\mathscr{T}$ endowed with the topology induced by the norm of
$L^{2}([0,T],H^{-s,\mathcal{L}}(G)).$ By using the Hahn-Banach extension
theorem, we can extends $\beta$ to a bounded functional $\tilde{\beta}$ on
$L^{2}([0,T],H^{-s,\mathcal{L}}(G)),$ and by using the Riesz representation
theorem, there exists
$v\in(L^{2}([0,T],H^{-s,\mathcal{L}}(G))^{\prime}=L^{2}([0,T],H^{s,\mathcal{L}}(G)),$
such that

$\tilde{\beta}(\psi)=(v,\psi),\quad\psi\in
L^{2}([0,T],H^{-s,\mathcal{L}}(G)).$

In particular, for $\psi=Q^{*}\phi\in\mathscr{T},$ we have

$\tilde{\beta}(Q^{*}\phi)={\beta}(Q^{*}\phi)=(v,Q^{*}\phi),$

Because, we can identify $C^{\infty}_{0}((0,T),\mathscr{D}(G))$ as a subspace
of $E$

$C^{\infty}_{0}((0,T),C^{\infty}(G))\subset E=\\{\phi\in
C^{1}([0,T],H^{s,\mathcal{L}}(G)):\phi(T)=0\\},$

we have the identity

$\displaystyle(f,\phi)=\smallint\limits_{0}^{T}(f(\tau),\phi(\tau))d\tau=\smallint\limits_{0}^{T}(f(\tau),\phi(\tau))d\tau+\frac{1}{i}(u_{0},\phi(0))=(v,Q^{*}\phi),$

for every $\phi\in C^{\infty}_{0}((0,T),C^{\infty}(G)).$ So, this implies that
$v\in\textnormal{Dom}(Q^{**}).$ Because $Q^{**}=Q,$ we have that

$\displaystyle(v,Q^{*}\phi)=(Qv,\phi)=(f,\phi),\,\,\forall\phi\in
C^{\infty}_{0}((0,T),C^{\infty}(G)),$

which implies that $Qv=f.$ A routine argument of integration by parts shows
that $v(0)=u_{0}.$ Now, in order to show the uniqueness of $v,$ let us assume
that $u\in\mathbb{X}$ is a solution of the problem

$\begin{cases}\frac{\partial u}{\partial t}=K(t,x,D)u+f,&\text{
}u\in\mathscr{D}^{\prime}((0,T)\times G)\\\ u(0)=u_{0}.&\text{ }\end{cases}$

Then $\omega:=v-u\in\mathbb{X}$ solves the problem

$\begin{cases}\frac{\partial\omega}{\partial t}=K(t,x,D)\omega,&\text{
}\omega\in\mathscr{D}^{\prime}((0,T)\times G)\\\ \omega(0)=0,&\text{
}\end{cases}$

and from Proposition 3.7, $\|\omega(t)\|_{H^{s,\mathcal{L}}(G)}=0,$ for all
$0\leqslant t\leqslant T,$ and consequently, from the continuity in $t$ of the
functions we have that $v(x,t)=u(x,t)$ for all $t\in[0,T]$ and a.e. $x\in G.$
∎

We will make use of the following lemma (see Lemma 4.4 of [7, Chapter 6]).
Indeed, in view of Lemma 3.4 the terms
$y(\tau)=\|v(\tau)\|_{H^{s,\mathcal{L}}}$ inside of the integrals in our
energy estimates can be neglected (compare e.g. (3.23) with (3.24)).

######  Lemma 3.4.

Suppose that $y:[0,T]\rightarrow\mathbb{R}^{+}_{0}$ and
$\phi:[0,T]\rightarrow\mathbb{R}^{+}_{0}$ are continuous and positive
functions such that

$\forall t\in[0,T],\,y(t)\leq
C\left(\alpha+\smallint\limits_{0}^{T}[y(\tau)+\phi(\tau)]d\tau\right).$
(3.13)

Then,

$\forall t\in[0,T],\,y(t)\leq C_{T}\left(\alpha
e^{Ct}+\smallint\limits_{0}^{T}e^{C(t-\tau)}\phi(\tau)d\tau\right).$ (3.14)

###  3.2. Case I: Diffusion problems for strongly elliptic operators with
positive order

Now, we are ready for the proof of our energy estimate when $K(t)$ is strongly
elliptic for all $0\leq t\leq T.$

######  Proposition 3.5.

Let $0\leq\delta<\rho\leqslant 1,$ and let $K(t)\in
C([0,T],\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}))$ be a
continuous family of pseudo-differential operators of order $m>0.$ Assume that
for all $t\in[0,T],$ $K(t)$ is strongly $\mathcal{L}$-elliptic. If $v\in
C^{1}([0,T],H^{s}(G))\cap C([0,T],H^{s+\frac{m}{2},\mathcal{L}}(G))$ then
there exist $C,C^{\prime}>0,$ such that

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|v(0)\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau))v(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}d\tau\right),$
(3.15)

holds for every $0\leqslant t\leqslant T.$ Moreover, we also have the estimate

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|v(T)\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau)^{*})v(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}d\tau\right).$
(3.16)

###### Proof.

Let $v\in C^{1}([0,T],H^{s}(G))\cap
C([0,T],H^{s+\frac{m}{2},\mathcal{L}}(G)).$ This fact will be useful later
because we will use the Gårding inequality in Theorem 2.12 applied to the
operator $\mathsf{Re}(-K(t)).$ For instance we will use that for $0\leq t\leq
T,$ $u_{t}\in L^{2}(G).$ So, $v\in\textnormal{Dom}(\partial_{\tau}-K(\tau))$
for every $0\leqslant\tau\leqslant T.$

Let us define $f(\tau):=Q(\tau)v(\tau),$ with
$Q(\tau):=(\partial_{\tau}-K(\tau)),$ for every $0\leqslant\tau\leqslant T.$
Using Lemma 3.2 we deduce that

$\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}=2\mathsf{Re}(K(t)v(t),v(t))_{L^{2}(G)}+2\mathsf{Re}(f(t),v(t))_{L^{2}(G)},$
(3.17)

for any $t\in[0,T].$ Now, from the subelliptic Gårding inequality,

$\displaystyle\mathsf{Re}(-K(t)v(t),v(t))\geqslant
C_{1}\|v(t)\|_{H^{\frac{m}{2},\mathcal{L}}(G)}-C_{2}\|v(t)\|_{L^{2}(G)}^{2},$
(3.18)

and the parallelogram law, we have

$\displaystyle 2\mathsf{Re}(f(t),v(t))_{L^{2}(G)}$ $\displaystyle\leqslant
2\mathsf{Re}(f(t),v(t))_{L^{2}(G)}+\|f(t)\|_{L^{2}(G)}^{2}+\|v(t)\|_{L^{2}(G)}^{2}$
$\displaystyle=\|f(t)+v(t)\|^{2}\leqslant\|f(t)+v(t)\|^{2}+\|f(t)-v(t)\|^{2}$
$\displaystyle=2\|f(t)\|^{2}_{L^{2}(G)}+2\|v(t)\|^{2}_{L^{2}(G)},$

and consequently

$\displaystyle\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}$ $\displaystyle\leqslant
2\left(C_{2}\|v(t)\|_{L^{2}(G)}^{2}-C_{1}\|v(t)\|_{H^{\frac{m}{2},\mathcal{L}}(G)}\right)+2\|f(t)\|^{2}_{L^{2}(G)}+2\|v(t)\|^{2}_{L^{2}(G)}$
$\displaystyle\lesssim\|f(t)\|^{2}_{L^{2}(G)}+\|v(t)\|^{2}_{L^{2}(G)}.$

By using Gronwall's Lemma we obtain the energy estimate

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|v(0)\|_{L^{2}(G)}^{2}+C^{\prime}\smallint\limits_{0}^{T}\|f(\tau)\|_{L^{2}(G)}^{2}d\tau\right),$
(3.19)

for every $0\leqslant t\leqslant T,$ and $T>0.$ By replacing $v(t)$ by
$\mathcal{M}_{s}v(t),$ we have for every $0\leqslant\tau\leqslant T,$ that

$\|\mathcal{M}_{s}v\|^{2}_{L^{2}(G)}\leqslant\left(C\|\mathcal{M}_{s}v(0)\|_{L^{2}(G)}^{2}+C^{\prime}\smallint\limits_{0}^{T}\|Q(\tau)\mathcal{M}_{s}v(\tau)\|_{L^{2}(G)}^{2}d\tau\right).$
(3.20)

So, we have proved that

$\|v\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|v(0)\|_{H^{s,\mathcal{L}}(G)}^{2}+C^{\prime}\smallint\limits_{0}^{T}\|Q(\tau)\mathcal{M}_{s}v(\tau)\|_{L^{2}(G)}^{2}d\tau\right).$
(3.21)

Now, by using Lemma 3.1 with $A=Q(\tau),$ we have that

$\|Q(\tau)\mathcal{M}_{s}v\|_{L^{2}(G)}\leq
C[\|Q(\tau)v\|_{H^{s,\mathcal{L}}(G)}+\|v\|_{H^{s,\mathcal{L}}(G)}]=C[\|f(\tau)\|_{H^{s,\mathcal{L}}(G)}+\|v\|_{H^{s,\mathcal{L}}(G)}],$

that indeed implies the estimate

$\|Q(\tau)\mathcal{M}_{s}v\|^{2}_{L^{2}(G)}\lesssim[\|f(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}+\|v\|^{2}_{H^{s,\mathcal{L}}(G)}].$
(3.22)

In view of (3.21) we have for $0\leq t\leq T$ the inequality

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\lesssim\|v(0)\|_{H^{s,\mathcal{L}}(G)}^{2}+\smallint\limits_{0}^{T}[\|v(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}+\|f(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}]d\tau.$
(3.23)

Using the integral Lemma 3.4 to (3.23), we deduce that

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\lesssim_{T}\|v(0)\|_{H^{s,\mathcal{L}}(G)}^{2}+\smallint\limits_{0}^{T}\|f(\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}d\tau,$
(3.24)

for $0\leq t\leq T.$ To finish the proof, we can change the analysis above
with $v(T-\cdot)$ instead of $v(\cdot),$ $f(T-\cdot)$ instead of $f(\cdot)$
and $Q^{*}=-\partial_{t}-K(t)^{*},$ (or equivalently $Q=\partial_{t}-K(t)$ )
instead of $Q^{*}=-\partial_{t}+K(t)^{*}$ (or equivalently
$Q=\partial_{t}-K(t)$) using that
$\mathsf{Re}(K(T-t)^{*})=\mathsf{Re}(K(T-t))$ to deduce that

$\displaystyle\|v(T-t)\|^{2}_{H^{s,\mathcal{L}}(G)}$
$\displaystyle\leqslant\left(C\|v(T)\|^{2}_{L^{s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(-\partial_{t}+K(T-t)^{*})v(T-\tau)\|^{2}_{H^{s,\mathcal{L}}(G)}d\tau\right)$
$\displaystyle=\left(C\|v(T)\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(-\partial_{t}-K(t)^{*})v(s)\|^{2}_{H^{s,\mathcal{L}}(G)}ds\right).$

So, we conclude the proof. ∎

Now, we will prove the existence and uniqueness for (3.1) when the initial
data are taken in $L^{2}(G).$

######  Theorem 3.6.

Let $0\leq\delta<\rho\leq 1$, $T>0$ and
$K(t)\in\Psi_{\rho,\delta}^{m,\mathcal{L}}(G\times\widehat{G})$ be strongly
elliptic for some $m>0$. Let $s\in\mathbb{R}$, $u_{0}\in
H^{s,\mathcal{L}}(G)$, $f\in L^{2}([0,T],H^{s,\mathcal{L}}(G))$ . Then, there
exists a unique solution $v$ in the space

$C^{1}([0,T],H^{s,\mathcal{L}}(G))\bigcap
C([0,T],H^{s+\frac{m}{2},\mathcal{L}}(G))$

of the Cauchy problem:

$(\textnormal{IVP}):\begin{cases}\frac{\partial v}{\partial t}=K(t)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G),\\\ v(0)=u_{0}.&\text{ }\end{cases}$
(3.25)

Moreover, $v$ satisfies the energy estimate

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],H^{s,\mathcal{L}}(G))}\right),$
(3.26)

for every $0\leqslant t\leqslant T.$

###### Proof.

We have proved the a priori estimate in Proposition 3.5 for

$v\in\mathbb{X}:=C^{1}([0,T],H^{-s}(G))\bigcap
C([0,T],H^{-s+\frac{m}{2}}(G)).$

Essentially the condition $v\in C([0,T],H^{-s+m/2}(G))$ was used when applying
the Gårding inequality in (3.29). Then, the existence of $v\in
C^{1}([0,T],H^{s}(G))$ solving (3.35) now follows from Lemma 3.3. ∎

###  3.3. Case II: Diffusion problems for operators with positive symbols on
the phase space

We will start with the following energy estimate. Since we will apply the
Sharp Gårding inequality proved for the subelliptic Hörmander classes in [5],
we will require the condition $0\leq\delta<(2\kappa-1)^{-1}\rho,$ with
$0<\rho\leq 1.$

######  Proposition 3.7.

For $0<\rho\leqslant 1,$ let $0\leq\delta<(2\kappa-1)^{-1}\rho.$ Let
$K(t)=K(t,x,D)\in
C([0,T],\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}))$ be a
continuous family of pseudo-differential operators of order $m\in\mathbb{R},$
such that

$\forall\,(x,[\xi])\in G\times\widehat{G},\quad\sigma_{-\textnormal{Re
}K(t)}(x,[\xi])\geq 0,\textnormal{ and }\,\quad 0\leq
m\leq\varkappa={\rho}/{\kappa}-\left(2-{1}/{\kappa}\right)\delta.$

If $v\in C^{1}([0,T],L^{2}(G)),$ then there exist $C,C^{\prime}>0,$ such that

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|v(0)\|^{2}_{L^{2}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau))v(\tau)\|^{2}_{L^{2}(G)}d\tau\right),$
(3.27)

holds for every $0\leqslant t\leqslant T.$ Moreover, we also have the estimate

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|v(T)\|^{2}_{L^{2}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(\partial_{t}-K(\tau)^{*})v(\tau)\|^{2}_{L^{2}(G)}d\tau\right).$
(3.28)

###### Proof.

Let $v\in C^{1}([0,T],L^{2}(G))).$ This fact will be useful later because we
will use the subelliptic sharp Gårding inequality applied to the operator
$\mathsf{Re}(-K(t)).$ For instance we will use that for $0\leq t\leq T,$
$u_{t}\in L^{2}(G).$ So, $v\in\textnormal{Dom}(\partial_{\tau}-K(\tau))$ for
every $0\leqslant\tau\leqslant T.$

Let us define $f(\tau):=Q(\tau)v(\tau),$ with
$Q(\tau):=(\partial_{\tau}-K(\tau)),$ for every $0\leqslant\tau\leqslant T.$
Observe that in view of Lemma 3.2 we have that

$\displaystyle\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}=2\mathsf{Re}(K(t)v(t),v(t))_{L^{2}(G)}+2\mathsf{Re}(f(t),v(t))_{L^{2}(G)}.$

Now, from the sharp Gårding inequality (2.15) we deduce that

$\displaystyle\mathsf{Re}(-K(t)v(t),v(t))\leqslant
C_{1}\|v(t)\|_{H^{\frac{m-\varkappa}{2}}(G)},\quad\varkappa={\rho}/{\kappa}-\left(2-{1}/{\kappa}\right)\delta,$
(3.29)

and in view of the parallelogram law, we have

$\displaystyle 2\mathsf{Re}(f(t),v(t))_{L^{2}(G)}$ $\displaystyle\leqslant
2\mathsf{Re}(f(t),v(t))_{L^{2}(G)}+\|f(t)\|_{L^{2}(G)}^{2}+\|v(t)\|_{L^{2}(G)}^{2}$
$\displaystyle=\|f(t)+v(t)\|^{2}\leqslant\|f(t)+v(t)\|^{2}+\|f(t)-v(t)\|^{2}$
$\displaystyle=2\|f(t)\|^{2}_{L^{2}(G)}+2\|v(t)\|^{2}_{L^{2}(G)}.$

Putting all the estimates above together we have proved that

$\displaystyle\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}\leqslant
2\left(C_{1}\|v(t)\|_{H^{\frac{m-\varkappa}{2}}(G)}\right)+2\|f(t)\|^{2}_{L^{2}(G)}+2\|v(t)\|^{2}_{L^{2}(G)}.$

Using that $m-\varkappa\leq 0,$ we have the embedding $L^{2}(G)\hookrightarrow
H^{\frac{m-\varkappa}{2}}(G),$ and consequently the comparison of norms
$\|v(t)\|_{H^{\frac{m-\varkappa}{2}}}\lesssim\|v(t)\|_{L^{2}(G)}.$ So, we have
the inequality

$\displaystyle\frac{d}{dt}\|v(t)\|^{2}_{L^{2}(G)}\lesssim\|f(t)\|^{2}_{L^{2}(G)}+\|v(t)\|^{2}_{L^{2}(G)}.$

By using Gronwall's Lemma we obtain the energy estimate

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|v(0)\|_{L^{2}(G)}^{2}+C^{\prime}\smallint\limits_{0}^{T}\|f(\tau)\|_{L^{2}(G)}^{2}d\tau\right),$
(3.30)

for every $0\leqslant t\leqslant T,$ and $T>0.$ To finish the proof, we can
change the analysis above with $v(T-\cdot)$ instead of $v(\cdot),$
$f(T-\cdot)$ instead of $f(\cdot)$ and $Q^{*}=-\partial_{t}-K(t)^{*},$ (or
equivalently $Q=\partial_{t}-K(t)$ ) instead of $Q^{*}=-\partial_{t}+K(t)^{*}$
(or equivalently $Q=\partial_{t}-K(t)$) using that
$\mathsf{Re}(K(T-t)^{*})=\mathsf{Re}(K(T-t))$ to deduce that

$\displaystyle\|v(T-t)\|^{2}_{L^{2}(G)}$
$\displaystyle\leqslant\left(C\|v(T)\|^{2}_{L^{2}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(-\partial_{t}+K(T-t)^{*})v(T-\tau)\|^{2}_{L^{2}(G)}d\tau\right)$
$\displaystyle=\left(C\|v(T)\|^{2}_{L^{2}(G)}+C^{\prime}\smallint\limits_{0}^{T}\|(-\partial_{t}-K(t)^{*})v(s)\|^{2}_{L^{2}(G)}ds\right).$

So, we conclude the proof. ∎

Now, we will prove the existence and uniqueness for (3.1) when the initial
data are taken in $L^{2}(G).$

######  Theorem 3.8.

For $0<\rho\leqslant 1,$ let $0\leq\delta<(2\kappa-1)^{-1}\rho.$ Let
$K(t)=K(t,x,D)\in
C([0,T],\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}))$ be a
continuous family of pseudo-differential operators of order $m\in\mathbb{R},$
such that

$\forall\,(x,[\xi])\in G\times\widehat{G},\quad\sigma_{-\textnormal{Re
}K(t)}(x,[\xi])\geq 0,\textnormal{ and }\,\quad 0\leq
m\leq\varkappa={\rho}/{\kappa}-\left(2-{1}/{\kappa}\right)\delta.$

Let $u_{0}\in L^{2}(G)$, and let $f\in L^{2}([0,T],L^{2}(G)).$ Then there
exists a unique $v\in C^{1}([0,T],L^{2}(G))$ solving (3.1). Moreover, $v$
satisfies the energy estimate

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|u_{0}\|^{2}_{L^{2}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],L^{2}(G))}\right),$
(3.31)

for every $0\leqslant t\leqslant T.$

###### Proof.

We have proved the a priori estimate in Proposition 3.5 for

$v\in\mathbb{X}:=C^{1}([0,T],L^{2}(G)).$

Essentially the condition $v\in C([0,T],L^{2}(G))$ was used when applying the
sharp Gårding inequality in Theorem 2.14. Then, the existence of
$v\in\mathbb{X}$ solving (3.35) now follows from Lemma 3.3. ∎

Now, we extend Theorem 3.6 to the case where the initial data are considered
in general Sobolev spaces.

######  Theorem 3.9.

For $0<\rho\leqslant 1,$ let $0\leq\delta<(2\kappa-1)^{-1}\rho,$ and let
$s\in\mathbb{R}.$ Let $K(t)=K(t,x,D)\in
C([0,T],\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}))$ be a
continuous family of pseudo-differential operators of order $m\in\mathbb{R},$
such that

$\forall\,(x,[\xi])\in G\times\widehat{G},\quad\sigma_{-\textnormal{Re
}\tilde{K}(t)}(x,[\xi])\geq 0,$

where

$\tilde{K}(t)=(1+\mathcal{L})^{\frac{s}{2}}K(t)(1+\mathcal{L})^{-\frac{s}{2}}\textnormal{
and }\,0\leq
m\leq\varkappa={\rho}/{\kappa}-\left(2-{1}/{\kappa}\right)\delta.$

Let $u_{0}\in H^{s,\mathcal{L}}(G)$, and let $f\in
L^{2}([0,T],H^{s,\mathcal{L}}(G)).$ Then there exists a unique $v\in
C^{1}([0,T],H^{s,\mathcal{L}}(G)))$ solving (3.1). Moreover, $v$ satisfies the
energy estimate

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],H^{s,\mathcal{L}}(G))}\right),$
(3.32)

for every $0\leqslant t\leqslant T.$

###### Proof.

Observe that if $v$ is a solution for the Cauchy problem

$\begin{cases}\frac{\partial v}{\partial t}=K(t,x,D)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G)\\\ v(0)=u_{0},&\text{ }\end{cases}$

where the initial data $u_{0}\in H^{s,\mathcal{L}}(G),$
$K(t):=K(t,x,D)\in\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$
$f\in L^{2}([0,T],H^{s}(G)),$ then we have

$\displaystyle\frac{\partial}{\partial t}(1+\mathcal{L})^{\frac{s}{2}}v$
$\displaystyle=(1+\mathcal{L})^{\frac{s}{2}}\frac{\partial v}{\partial
t}=(1+\mathcal{L})^{\frac{s}{2}}K(t,x,D)v+(1+\mathcal{L})^{\frac{s}{2}}f$
$\displaystyle=\tilde{K}(t,x,D)(1+\mathcal{L})^{\frac{s}{2}}v+(1+\mathcal{L})^{\frac{s}{2}}f,$

showing that $v^{\prime}:=(1+\mathcal{L})^{\frac{s}{2}}v\in L^{2}(G)$ and
$f^{\prime}:=(1+\mathcal{L})^{\frac{s}{2}}f\in L^{2}([0,T],L^{2}(G))$ satisfy
the initial boundary value problem

$\begin{cases}\frac{\partial v^{\prime}}{\partial
t}=\tilde{K}(t,x,D)v^{\prime}+f^{\prime},&\text{
}v^{\prime}\in\mathscr{D}^{\prime}((0,T)\times G)\\\
v^{\prime}(0)=u^{\prime}_{0}:=(1+\mathcal{L})^{\frac{s}{2}}u_{0}\in
L^{2}(G).&\text{ }\end{cases}$

So, by applying Proposition 3.6 to the family of operators $\tilde{K}(t)$ we
get the estimate

$\|v^{\prime}(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|u^{\prime}_{0}\|^{2}_{L^{2}(G)}+C^{\prime}\|f^{\prime}\|^{2}_{L^{2}([0,T],L^{2}(G))}\right),$
(3.33)

for every $0\leqslant t\leqslant T.$ Finally, observe that (3.33) is
equivalent to the following energy inequality

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\|f\|^{2}_{H^{s,\mathcal{L}}([0,T],L^{2}(G))}\right),$
(3.34)

for every $0\leqslant t\leqslant T,$ proving Theorem 3.9. ∎

###  3.4. Consequences: Well posedness for diffusion equations in elliptic
Hörmander classes

This subsection will be dedicated to presenting the corresponding statements
of the well-posedness for the diffusion problem where $K(t)$ belongs to the
global Hörmander classes $\Psi^{m}_{\rho,\delta}(G\times\widehat{G}),$ for
$0\leq\delta<\rho\leq 1$ developed in [33]. So, in particular, the results of
this sub-section also cover the analysis when $K(t)$ belongs to the Hörmander
classes $\Psi^{m}_{\rho,\delta}(G,\textnormal{loc})$ defined by local
coordinate systems, see Hörmander [25].

####  3.4.1. The strongly elliptic case

We start with the well posedness of diffusion problems for strongly elliptic
(and then, for elliptic) pseudo-differential operators.

The prototype of the operators covered by Corollary 3.10 are the ones with
positive order $m>0$ and given by
$K(t)=a(x,t)(1+\mathcal{L}_{G})^{\frac{m}{2}},$ and
$K(t)=a(x,t)\mathcal{L}_{G}^{\frac{m}{2}},$ with $a(x,t)$ being smooth in
$(x,t)$ and positive.

######  Corollary 3.10.

Let $0\leq\delta<\rho\leq 1$, $T>0$ and
$K(t)\in\Psi_{\rho,\delta}^{m}(G\times\widehat{G})$ be strongly elliptic for
some $m>0$. Let $s\in\mathbb{R}$, $u_{0}\in H^{s}(G)$, $f\in
L^{2}([0,T],H^{s}(G))$ . Then, there exists a unique solution $v$ in the space
$C^{1}([0,T],H^{s}(G))\bigcap C([0,T],H^{s+\frac{m}{2}}(G))$ of the Cauchy
problem:

$(\textnormal{IVP}):\begin{cases}\frac{\partial v}{\partial t}=K(t)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G),\\\ v(0)=u_{0},&\text{ }\end{cases}$
(3.35)

Moreover, $v$ satisfies the energy estimate

$\|v(t)\|^{2}_{H^{s}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],H^{s}(G)))}\right),$
(3.36)

for every $0\leqslant t\leqslant T.$

We also analyse the previous statement when $K(t)$ in Corollary 3.10 is
perturbed by a drift

$D=\sum_{j=1}^{n}a_{j}(x,t)X_{j}\in
C^{\infty}([0,T],TG^{\mathbb{C}}),\,\,a_{i}\in
C^{\infty}(G\times[0,T],\mathbb{R}),\,X_{j}\in i\mathfrak{g}.$

For this, see Subsection 3.6. Now, we are going to analyse the case where
$-K(t)$ has a matrix-valued positive symbol and the order $m$ is inside of a
interval determined by the subelliptic sharp Gårding inequality [5].

####  3.4.2. The case of non-negative symbols

Observe that in the case where the family of operators
$K(t)\in\Psi^{m}_{\rho,\delta}(G\times\widehat{G}),$ that is, if we replace
the sub-Laplacian $\mathcal{L}$ by the Laplace-Beltrami operator
$\mathcal{L}_{G}$ in Theorem 3.8, the condition

$0\leq\delta<(2\kappa-1)^{-1}\rho$

is less restrictive since $\kappa=1$ and we gain the complete range
$0\leq\delta<\rho\leq 1.$ So, we have the following well-posedness result.

######  Corollary 3.11.

For $0\leq\delta<\rho\leq 1,$ let $K(t)=K(t,x,D)\in
C([0,T],\Psi^{m}_{\rho,\delta}(G\times\widehat{G}))$ be a continuous family of
pseudo-differential operators of order $m\in\mathbb{R},$ such that

$\forall\,(x,[\xi])\in G\times\widehat{G},\quad\sigma_{-\textnormal{Re
}K(t)}(x,[\xi])\geq 0,\textnormal{ and }\,\quad 0\leq m\leq\rho-\delta.$

Let $u_{0}\in L^{2}(G)$, and let $f\in L^{2}([0,T],L^{2}(G)).$ Then there
exists a unique $v\in C^{1}([0,T],L^{2}(G))$ solving (3.1). Moreover, $v$
satisfies the energy estimate

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|u_{0}\|^{2}_{L^{2}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],L^{2}(G))}\right),$
(3.37)

for every $0\leqslant t\leqslant T.$

Moreover, we have the following extension of Corollary 3.11 to Sobolev spaces
as a consequence of Theorem 3.8.

######  Corollary 3.12.

For $0\leq\delta<\rho\leq 1,$ let $K(t)=K(t,x,D)\in
C([0,T],\Psi^{m}_{\rho,\delta}(G\times\widehat{G}))$ be a continuous family of
pseudo-differential operators of order $m\in\mathbb{R},$ such that

$\forall\,(x,[\xi])\in G\times\widehat{G},\quad\sigma_{-\textnormal{Re
}\tilde{K}(t)}(x,[\xi])\geq 0,$

where

$\tilde{K}(t)=(1+\mathcal{L}_{G})^{\frac{s}{2}}K(t)(1+\mathcal{L}_{G})^{-\frac{s}{2}}\textnormal{
and }\,0\leq m\leq\varkappa={\rho}-\delta.$

Let $s\in\mathbb{R},$ $u_{0}\in H^{s}(G)$, and let $f\in
L^{2}([0,T],H^{s}(G)).$ Then there exists a unique $v\in
C^{1}([0,T],H^{s}(G)))$ solving (3.1). Moreover, $v$ satisfies the energy
estimate

$\|v(t)\|^{2}_{H^{s}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],H^{s}(G))}\right),$
(3.38)

for every $0\leqslant t\leqslant T.$

###  3.5. Analysis of diffusion equation for fractional sub-Laplacians

Now, let us use the analytic functional calculus in [3, Chapter 8] to obtain a
significant consequence of Corollary 3.10.

######  Corollary 3.13.

Assume that $a(x,t)\in C^{\infty}(G\times[0,T],\mathbb{R}^{+})$ does not have
zeros and let $m>0$. Then, for any $s\in\mathbb{R}$ the diffusion problem

$\begin{cases}\frac{\partial v}{\partial
t}=a(x,t)\mathcal{L}^{\frac{m}{2}}v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G),\\\ v(0)=u_{0},&\text{ }\end{cases}$
(3.39)

for initial data $u_{0}\in H^{s,\mathcal{L}}(G)$ and $f\in
L^{2}([0,T],H^{s,\mathcal{L}}(G))$ admits a unique solution $v$ in the space
$C^{1}([0,T],H^{s,\mathcal{L}}(G))\bigcap
C([0,T],H^{s+\frac{m}{2},\mathcal{L}}(G)).$ Moreover, $v$ satisfies the energy
estimate

$\|v(t)\|^{2}_{H^{s,\mathcal{L}}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s,\mathcal{L}}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],H^{s,\mathcal{L}}(G)))}\right),$
(3.40)

for every $0\leqslant t\leqslant T.$

###### Proof.

In view of the subelliptic functional calculus [3, Chapter 8] we have that
$\forall
z\in\mathbb{C},\mathcal{L}^{\frac{z}{2}}\in\Psi^{\mathsf{Re}(z),\mathcal{L}}_{1,0}(G\times\widehat{G}),$
and in consequence for $z=m+0i>0,$
$\mathcal{L}^{\frac{m}{2}}\in\Psi^{m,\mathcal{L}}_{1,0}(G\times\widehat{G}).$
Since the smooth function $a(x,t)$ does not have zeroes, the
$\mathcal{L}$-ellipticity of the operator $\mathcal{L}^{\frac{m}{2}}$ is
preserved when multiplying it on the left with the function $a(x,t).$ So, the
operator $K(t)=a(x,t)\mathcal{L}^{\frac{m}{2}}$ is $\mathcal{L}$-elliptic and
the statement in Corollary 3.13 now follows from Corollary 3.10. ∎

###  3.6. Analysis of diffusion equations for fractional Laplacians with drift

We now state the well-posedness for the fractional diffusion with drift on
compact Lie groups. To do so, let us consider the fractional diffusion problem

$\begin{cases}\frac{\partial v}{\partial
t}=(a(x,t)\mathcal{L}_{G}^{\frac{m}{2}}+D)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G),\\\ v(0)=u_{0},&\text{ }\end{cases}$
(3.41)

where $0\leq m\leq 1.$ If $D\equiv 0,$ we are in the case of a fractional
diffusion equation for the fractional Laplacian. However, if

$D=\sum_{j=1}^{n}a_{j}(x,t)X_{j}\in
C^{\infty}([0,T],TG^{\mathbb{C}}),\,\,a_{i}\in
C^{\infty}(G\times[0,T],\mathbb{R}),\,X_{j}\in i\mathfrak{g},$

is a differential operator of first order we are in the case of a diffusion
equation with drift $D$. We have denoted by $TG^{\mathbb{C}}$ to the
complexification of the tangent space

$TG^{\mathbb{C}}\cong TG\oplus iTG\cong(G\times G)\times(\mathfrak{g}\otimes
i\mathfrak{g}).$

For $m=1,$ the previous model is an analogue on compact Lie groups of the
quasi-geostrophic Euclidean case that have been considered e.g. by Caffarelli
and Vasseur in [6], where it was considered the case of a differential
operator

$D=v\cdot\nabla=\sum_{i=1}^{n}{v_{i}}\partial_{x_{i}},$

of first order with minimal smoothness condition on $v$.

######  Remark 3.14.

The matrix-valued symbol of $K(t)=a(x,t)\mathcal{L}_{G}^{\frac{m}{2}}+D$ is
given by

$\sigma_{K(t)}(x,[\xi])=a(x,t)\lambda_{[\xi]}^{\frac{m}{2}}I_{d_{\xi}\times
d_{\xi}}+\sum_{i=1}a_{i}(x)(X_{i}\xi)(e).$ (3.42)

The hypothesis $X_{j}\in i\mathfrak{g}$ implies that we can choose an element
$\xi\in[\xi]$ such that $\sigma_{X_{j}}(\xi)=X_{j}\xi(e)$ can be diagonalised
with diagonal entries determining the spectrum of the symbol
$\sigma_{X_{j}}(\xi)$ at any representation space. The smooth functions
$a_{i}$ have been considered real-valued, and if we want that the symbol

$\sigma_{-K(t)}(x,[\xi])=-a(x,t)\lambda_{[\xi]}^{\frac{m}{2}}I_{d_{\xi}\times
d_{\xi}}-\sum_{i=1}a_{i}(x,t)(X_{i}\xi)(e),$ (3.43)

becomes a positive matrix on any representation space is natural to assume
that $-a(x,t)\in C^{\infty}(G,\mathbb{R}^{+}_{0})$ is smooth and bounded from
below, that is $-a(x,t)\geq 0.$

In view of Corollaries 3.11 and 3.12 we have the following well posedness
result.

######  Corollary 3.15.

Let $0\leq m\leq 1.$ Let $a(x,t)\in C^{\infty}(G,\mathbb{R}^{+}_{0})$ be a
smooth function such that $a(x,t)\leq 0.$ Let $X_{j}\in i\mathfrak{g}$ be a
family of symmetric vector-fields and for for any $1\leq i\leq n,$ let
$a_{i}\in C^{\infty}(G\times[0,T],\mathbb{R})$ be a set of real-valued smooth
functions. If, for any $0\leq t\leq T,$ the (drift) first-order differential
operator

$D=\sum_{j=1}^{n}a_{j}(x,t)X_{j}\in TG^{\mathbb{C}}\cong(G\times
G)\times\mathfrak{g}\otimes i\mathfrak{g},$ (3.44)

is such that

$\sigma_{-K(t)}(x,[\xi])=-a(x,t)\lambda_{[\xi]}^{\frac{m}{2}}I_{d_{\xi}\times
d_{\xi}}-\sum_{i=1}a_{i}(x,t)(X_{i}\xi)(e)\geq 0,$ (3.45)

for any $(x,[\xi])\in G\times\widehat{G}.$ Then, for any $s\in\mathbb{R},$
$T>0,$ $u_{0}\in L^{2}(G)$, and $f\in L^{2}([0,T],L^{2}(G)),$ there exists a
unique $v\in C^{1}([0,T],L^{2}(G)))$ solving (3.41). Moreover, $v$ satisfies
the energy estimate

$\|v(t)\|^{2}_{L^{2}(G)}\leqslant\left(C\|u_{0}\|^{2}_{L^{2}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],L^{2}(G))}\right),$
(3.46)

for every $0\leqslant t\leqslant T.$ Moreover, the existence of the solution
$v$ in (3.41) with $u_{0}\in H^{s}(G)$, and $f\in L^{2}([0,T],H^{s}(G)),$ and
the energy estimate in (3.38) still are valid if $D$ is an element of the
complex universal enveloping algebra $\mathfrak{U}(\mathfrak{g})\oplus
i\mathfrak{U}(\mathfrak{g})$ and the positivity condition in (3.45) holds.

###### Proof.

Observe that if $0\leq m\leq 1,$ and $D$ is a left-invariant differential
operator of first order as in (3.44), under the positivity condition in (3.45)
the well posedness of (3.41) and the validity of the energy estimate in (3.46)
follow from Corollary 3.11.

In the case where $D$ is an element of the universal enveloping algebra
$\mathfrak{U}(\mathfrak{g})\oplus i\mathfrak{U}(\mathfrak{g}),$ that is, $D$
has constant coefficients, $K(t)$ commutes with the Laplacian
$\mathcal{L}_{G},$ indeed,

$\mathcal{L}_{G}\in\textnormal{center}[\mathfrak{U}(\mathfrak{g})\oplus
i\mathfrak{U}(\mathfrak{g})],$

and the well posedness of (3.41) with $u_{0}\in H^{s}(G)$, and $f\in
L^{2}([0,T],H^{s}(G)),$ and the validity of the energy estimate in (3.38)
remain valid in view of Corollary 3.12. ∎

The generalization for all $m>1,$ in the Corollary 3.16 removes the symmetry
condition $X_{j}\in i\mathfrak{g},$ and the positivity condition in (3.45),
allowing an arbitrary and smooth drift $D$ on $(0,T)$ and taking values on
$TG^{\mathbb{C}}.$ This application of Corollary 3.10 can be announced as
follows.

######  Corollary 3.16.

Let $D\equiv D(t)=\sum_{j=1}^{n}a_{j}(x,t)X_{j}:[0,T]\rightarrow
TG^{\mathbb{C}},$ be a smooth function and let us consider the (modified)
fractional Laplacian with drift $D$

$K(t)=a(x,t)\mathcal{L}_{G}^{\frac{m}{2}}+D,$ (3.47)

where $m>1.$ Assume that $a(x,t)\in C^{\infty}(G\times[0,T],\mathbb{R}^{+})$
does not have zeros. Then, for any $s\in\mathbb{R}$ the diffusion problem

$\begin{cases}\frac{\partial v}{\partial
t}=(a(x,t)\mathcal{L}_{G}^{\frac{m}{2}}+D)v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times G)\\\ v(0)=u_{0},&\text{ }\end{cases},$
(3.48)

for initial data $u_{0}\in H^{s}(G)$ and $f\in L^{2}([0,T],H^{s}(G))$ admits a
unique solution $v$ in the space $C^{1}([0,T],H^{s}(G))\bigcap
C([0,T],H^{s+\frac{m}{2}}(G)).$ Moreover, $v$ satisfies the energy estimate

$\|v(t)\|^{2}_{H^{s}(G)}\leqslant\left(C\|u_{0}\|^{2}_{H^{s}(G)}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],H^{s}(G)))}\right),$
(3.49)

for every $0\leqslant t\leqslant T.$

###  3.7. Examples: diffusion problems with drift for the Laplacian on
$\textnormal{SU}(2)$

Let us consider the compact Lie group of complex unitary $2\times 2$-matrices

$\textnormal{SU}(2)=\\{X=[X_{ij}]_{i,j=1}^{2}\in\mathbb{C}^{2\times
2}:X^{*}=X^{-1}\\},\,X^{*}:=\overline{X}^{t}=[\overline{X_{ji}}]_{i,j=1}^{2}.$

Let us consider the left-invariant first-order differential operators

$\partial_{+},\partial_{-},\partial_{0}:C^{\infty}(\textnormal{SU}(2))\rightarrow
C^{\infty}(\textnormal{SU}(2)),$

called creation, annihilation, and neutral operators respectively, (see
Definition 11.5.10 of [33]) and let us define

$X_{1}=-\frac{i}{2}(\partial_{-}+\partial_{+}),\,X_{2}=\frac{1}{2}(\partial_{-}-\partial_{+}),\,X_{3}=-i\partial_{0},$
(3.50)

where $X_{3}=[X_{1},X_{2}],$ based on the commutation relations
$[\partial_{0},\partial_{+}]=\partial_{+},$
$[\partial_{-},\partial_{0}]=\partial_{-},$ and
$[\partial_{+},\partial_{-}]=2\partial_{0},$ the system $X=\\{X_{1},X_{2}\\}$
satisfies the Hörmander condition at step $\kappa=2.$

We record that the unitary dual of $\textnormal{SU}(2)$ (see [33]) can be
identified as

$\widehat{\textnormal{SU}}(2)\equiv\\{[\xi^{l}]:2l\in\mathbb{N},d_{l}:=\dim\xi^{l}=(2l+1)\\}\sim({1}/{2})\mathbb{N}.$
(3.51)

There are explicit formulae for $\xi^{l}$ as functions of Euler angles in
terms of the so-called Legendre-Jacobi polynomials, see [33].

Consider the basis $\\{X_{1},X_{2},X_{3}=[X_{1},X_{2}]\\}$ of the Lie agebra
$\mathfrak{su}(2).$ The sum of squares

$\mathcal{L}_{\textnormal{SU}(2)}=-X_{1}^{2}-X_{2}^{2}-X_{3}^{2}=-\partial_{0}^{2}-\frac{1}{2}[\partial_{+}\partial_{-}+\partial_{-}\partial_{+}],$
(3.52)

is the positive Laplacian on $\textnormal{SU}(2).$ Again, by following e.g.
[33], the spectrum of the positive Laplacian
$\mathcal{L}_{\textnormal{SU}(2)}$ can be indexed by the sequence

$\lambda_{\ell}:=\ell(\ell+1),\quad\ell\in({1}/{2})\mathbb{N}.$

Next, we illustrate that the positivity condition in Corollary 3.44. Consider
the problem with drift

$\begin{cases}\frac{\partial v}{\partial
t}=(a(x,t)\mathcal{L}_{\textnormal{SU}(2)}^{\frac{m}{2}}+ia_{3}(x,t)X_{3})v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times\textnormal{SU}(2)),\\\
v(0)=u_{0}.&\text{ }\end{cases}$ (3.53)

For $m>1,$ if $a(x,t)\in
C^{\infty}(\textnormal{SU}(2)\times[0,T],\mathbb{R}^{+})$ does not have zeros,
Corollary 3.16 implies that (3.53) is well posed if
$u_{0}\in{H}^{s}(\textnormal{SU}(2))$ and $f\in
L^{2}([0,T],{H}^{s}(\textnormal{SU}(2))$ with arbitrary regularity order
$s\in\mathbb{R}.$

Now, if $0\leq m\leq 1,$ note that the symbol of

$-K(t)=a(x,t)\mathcal{L}_{\textnormal{SU}(2)}^{\frac{m}{2}}+ia_{3}(x,t)X_{3}$

can be computed in diagonal form into a representative
$\xi^{\ell}\in[\xi^{\ell}],$ as

$\displaystyle\sigma_{-K(t)}(x,\xi^{\ell})=\textnormal{diag}[-a(x,t)(\ell^{2}+\ell)^{\frac{m}{2}}-a_{3}(x,t)j_{1}]_{-\ell\leq
j_{1},j_{2}\leq\ell,\,j_{i}\in(1/2)\mathbb{N}}.$ (3.54)

Now, note that $\sigma_{-K(t)}(x,\xi^{\ell})\geq 0$ for all
$\ell\in(1/2)\mathbb{N},$ if $a_{3}\equiv 0$ and $0\leq m\leq 1,$ or for
$m=1,$ if

$a(x,t)\leq 0,\,|a_{3}(x,t)|+a(x,t)\leq 0.$ (3.55)

Indeed, the condition in (3.54) with $\ell=0=j_{1}$ implies that $-a(x,t)\geq
0$ for all $(x,t)\in G\times[0,T].$ If $0\leq m<1,$ note that

$-a(x,t)(\ell^{2}+\ell)^{\frac{m}{2}}-a_{3}(x,t)j_{1}\geq-a(x,t)(\ell^{2}+\ell)^{\frac{m}{2}}-|a_{3}(x,t)|\ell\geq
0$ (3.56)

for all $\ell\in(1/2)\mathbb{N}$ if and only if $a_{3}\equiv 0.$ On the other
hand, for $m=1,$

$\forall\ell,-a(x,t)(\ell^{2}+\ell)^{\frac{m}{2}}-|a_{3}(x,t)|\ell\geq 0$

if and only if

$\forall\ell,\,-a(x,t)(1+(1/\ell))^{\frac{1}{2}}\geq|a_{3}(x,t)|,$

from which is required that $|a_{3}(x,t)|+a(x,t)\leq 0.$ The analysis above
and Corollary 3.16 implies the following well posedness result for (3.59)
showing that the positivity criterion in (3.45) is in certain cases
verifiable.

######  Corollary 3.17.

Let $0\leq m\leq 1.$ Let $a(x,t)\in
C^{\infty}(\textnormal{SU}(2),\mathbb{R}^{+}_{0})$ be a smooth function such
that $a(x,t)\leq 0.$ Let $a_{3}\in
C^{\infty}(\textnormal{SU}(2)\times[0,T],\mathbb{R})$ be a real-valued smooth
function. If $m=1$, and for any $0\leq t\leq T,$ and all
$x\in\textnormal{SU}(2),$

$|a_{3}(x,t)|+a(x,t)\leq 0,$ (3.57)

then, for any $u_{0}\in L^{2}(\textnormal{SU}(2))$, and $f\in
L^{2}([0,T],L^{2}(\textnormal{SU}(2))),$ there exists a unique $v\in
C^{1}([0,T],L^{2}(\textnormal{SU}(2))))$ solving the problem

$\begin{cases}\frac{\partial v}{\partial
t}=(a(x,t)\mathcal{L}_{\textnormal{SU}(2)}^{\frac{m}{2}}+a_{3}(x,t)iX_{3})v+f,&\text{
}v\in\mathscr{D}^{\prime}((0,T)\times\textnormal{SU}(2)),\\\
v(0)=u_{0},&\text{ }\end{cases}$ (3.58)

where $X_{3}=-i\partial_{0}$ is as in (3.50). Moreover, $v$ satisfies the
energy estimate

$\|v(t)\|^{2}_{L^{2}(\textnormal{SU}(2))}\leqslant\left(C\|u_{0}\|^{2}_{L^{2}(\textnormal{SU}(2))}+C^{\prime}\|f\|^{2}_{L^{2}([0,T],L^{2}(\textnormal{SU}(2))}\right),$
(3.59)

for every $0\leqslant t\leqslant T.$ Moreover, the existence of the solution
$v$ in (3.53) with $u_{0}\in H^{s}(\textnormal{SU}(2))$, and $f\in
L^{2}([0,T],H^{s}(\textnormal{SU}(2))),$ and the energy estimate in (3.59)
still are valid if $a(x,t)\equiv c_{1}$ and $a_{3}(x,t)\equiv a_{3}$ are (non-
trivial) constant functions (at the same time) such that $|a_{3}|+c_{1}\leq 0$
with $c_{1}\leq 0.$ On the other hand, the conclusions above still remain
valid when $0\leq m<1$ if $a_{3}\equiv 0.$

## References

  * [1] S. Bhattacharyya, T. Ghosh, G. Uhlmann.  Inverse problems for the fractional-Laplacian with lower order non-local perturbations.  Trans. Amer. Math. Soc., 374(5), 3053–3075, 2021.
  * [2] J. M. Bismut. The hypoelliptic Laplacian on a compact Lie group, J. Funct. Anal. 255, 2190–2232, 2008.
  * [3] D. Cardona, M. Ruzhansky. Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups. arXiv:2008.09651.
  * [4] D. Cardona, J. Delgado, M. Ruzhansky.  Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations, to appear in J. Geom. Anal.
  * [5] D. Cardona, S. Federico, M. Ruzhansky. Subelliptic sharp Gårding inequality on compact Lie groups, submitted, arXiv:2110.00838
  * [6] L. A. Caffarelli, A. Vasseur.  Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation.  Ann. of Math., 171(3), 1903–1930, 2010.
  * [7] J. Chazarain, A. Piriou.  Introduction to the Theory of Linear Partial Differential Equations.  North-Holland, 1982.
  * [8] O. Ciaurri, T. Alastair, L. Roncal, J. Torrea, J. Varona.  Harmonic analysis associated with a discrete Laplacian.  J. Anal. Math., 132, 109–131, 2017.
  * [9] O. Ciaurri, L. Roncal, P. Stinga, J. Torrea, J. Varona.  Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications.  Adv. Math., 330:688–738, 2018.
  * [10] P. Constantin.  Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, 1.  43 Lecture Notes in Mathematics 1871, Springer. Springer, Berlin, 2006\.
  * [11] R. Cont, P. Tankov.  Financial Modelling with Jump Processes.  Financial Mathematics Series. Chapmann and Hall/CRC, Boca Raton, FL, 2004\.
  * [12] M. Dabkowski.  Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equations.  Geom. Funct. Anal. 21, 1–13, 2011.
  * [13] J. Delgado.  On a class of hyperbolic equations in Weyl-Hörmander calculus.  J. Math. Anal. Appl., 436(1), 339–354, 2016.
  * [14] J. Delgado.  On the well-posedness of the Cauchy problem for a class of degenerate parabolic equations.  Int. Equ. Oper. Theor. 90(1), 19pp, 2018.
  * [15] E. Di Nezza, G. Palatucci, E. Valdinoci.  Hitchhiker's guide to the fractional Sobolev spaces.  Bull. Sci. Math., 136(5), 521–573, 2012.
  * [16] G. Duvaut, J. L. Lions. Les inéquations en méecanique et en physique. Dunod, Paris, 1972. Travaux et Recherches Mathéematiques, No. 21.
  * [17] Ch. Epstein, C. Pop.  Regularity for the supercritical fractional Laplacian with drift.  J. Geom. Anal. 26(2), 1231–1268, 2016.
  * [18] C. Fefferman, D. H. Phong,. On Positivity of Pseudo-Differential Operators, Proceedings Nat. Acad. Sci. USA, 75, 4673–4674, 1978.
  * [19] V. Fischer. Intrinsic pseudo-differential calculi on any compact Lie group, J. Funct. Anal. 268, 3404–3477, 2015.
  * [20] R. Frank, P. Ivanisvili.  Hypercontractivity of the semigroup of the fractional Laplacian on the $n$-sphere.  J. Funct. Anal., 281(8), 109–145, 2021.
  * [21] C. Garetto, Ruzhansky, M. Wave equation for sum of squares on compact Lie groups, J. Differential Equations. 258, 4324–4347, 2015.
  * [22] L. Gårding. Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. (1), 55–72, 1953.
  * [23] G. Grubb.  Exact Green's formula for the fractional Laplacian and perturbations.  Math. Scand., 126(3), 568–592, 2020.
  * [24] L. Hörmander. Hypoelliptic second order differential equations, Acta Math. 119, 147–171, 1967.
  * [25] L. Hörmander. The Analysis of the linear partial differential operators. Vol. III-IV. Springer-Verlag, 1985.
  * [26] F. Nazarov, A. Kiselev, A. Volberg.  Global well-posedness for the critical 2D dissipative quasi-geostrophic equation.  Invent. Math. (167), 445–453, 2007.
  * [27] J. Klafter, I. M. Sokolov, A. Blumen.  Fractional kinetics.  Physics Today, November, 48–54, 2002.
  * [28] N. Laskin.  Principles of fractional quantum mechanics.  Fractional dynamics, World Sci. Publ. 393–427, 2012.
  * [29] A. J. Majda, A. L. Bertozzi.  Vorticity and Incompressible Flow.  Cambridge Texts in Applied Mathematics. Cambridge Univ. Press, Cambridge, 2002.
  * [30] A. Petrosyan, C.A. Pop.  Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift.  J. Funct. Anal. 268(2), 417–472, 2015.
  * [31] S. Raffaella, E. Valdinoci.  Fractional Laplacian equations with critical Sobolev exponent.  Rev. Mat. Complut., 28(3), 655–676, 2015.
  * [32] L. Roncal, P. Stinga.  Fractional Laplacian on the torus.  Commun. Contemp. Math. 18(3), 2016.
  * [33] M. Ruzhansky, V. Turunen. Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhäuser-Verlag, Basel, 2010.
  * [34] M. Ruzhansky, V. Turunen, J. Wirth. Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity, J. Fourier Anal. Appl. 20, 476–499, 2014.
  * [35] M. Ruzhansky, J. Wirth. Global functional calculus for on compact Lie groups, J. Funct. Anal. 267(1), 144–172, 2014.
  * [36] M. Ruzhansky, J. Wirth. $L^{p}$ Fourier multipliers on compact Lie groups, Math. Z. 280, 21–642, 2015.
  * [37] R. Servadei, E. Valdinoci.  The Brezis-Nirenberg result for the fractional Laplacian.  Trans. Amer. Math. Soc., 367(1), 67–102, 2015.
  * [38] M. Taylor. Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J. 1981.