# Spectral inequalities for elliptic pseudo-differential operators on closed
manifolds

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University. Krijgslaan 281, Building S8, 9000
Ghent, Belgium. E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove
the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-
differential operator $E(x,D)$ on $M,$ of order $\nu>0,$ in the Hörmander
class $\Psi^{\nu}_{\rho,\delta}(M).$ In control theory this has been an open
problem prior to this work. As an application of this fundamental result, we
establish the null-controllability of the (fractional) heat equation
associated with $E(x,D).$ The sensor $\omega\subset M$ in the observability
inequality is an open subset of $M.$ The obtained results (that are, the
corresponding spectral inequality for an elliptic operator and the null-
controllability for its diffusion model) extend in the setting of closed
manifolds, classical results of the control theory, as the spectral inequality
due to Lebeau and Robbiano and their result on the null-controllability of the
heat equation giving a complete picture of the subject in the setting of
closed manifolds. As a consequence of our analysis we prove the Donnelly-
Fefferman inequality for positive pseudo-differential operators for sums of
eigenfunctions. For the proof of the spectral inequality we introduce a
periodization approach in time inspired by the global pseudo-differential
calculus due to Ruzhansky and Turunen.

###### Key words and phrases:

Pseudo-differential operator, Null-controllability, Fractional diffusion
model, Microlocal Analysis, Spectral Inequality

###### 2020 Mathematics Subject Classification:

42B20, 42B37

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement NO: 694126-DyCon). This work also has been 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).

###### Contents

  1. 1 Introduction
    1. 1.1 Outline and main results
    2. 1.2 Structure of the work
  2. 2 Null-controllability of the fractional heat equation and observability inequalities
    1. 2.1 An interpolation inequality
    2. 2.2 Proof of the spectral inequality
      1. 2.2.1  An auxiliar interpolation inequality on $[0,T+\varepsilon)$.
      2. 2.2.2  Construction of a suitable function $\phi$:
      3. 2.2.3 Analysis of the term $\|\partial_{t}\phi(x,0)\|_{L^{2}(\omega)}$
      4. 2.2.4 Embedding of $M_{T}$ in a closed manifold $M\times\mathbb{T}_{T,\varepsilon},$ $\mathbb{T}_{T,\varepsilon}\cong\mathbb{S}^{1}$:
      5. 2.2.5 Proof of the interpolation inequality
      6. 2.2.6 Estimate of $Z_{1}$
      7. 2.2.7 Estimate for $I$
      8. 2.2.8 Estimate for $II$:
      9. 2.2.9 Estimate for $Z_{1}$
      10. 2.2.10 Final Analysis
    3. 2.3 Null-controllability for the diffusion model
    4. 2.4 The controllability cost for the diffusion model over short times
  3. 3 Conclusions and some open problems
  4. 4 Appendix I: Pseudo-differential operators and null-controllability on Hilbert spaces
    1. 4.1 Hörmander classes of pseudo-differential operators on compact manifolds
    2. 4.2 Complex powers of an elliptic pseudo-differential operator
    3. 4.3 Ruzhansky-Turunen classes of pseudo-differential operators on the torus
    4. 4.4 Null-controllability of diffusion problems on Hilbert spaces
  5. 5 Appendix II: Construction of the regularising function $\psi$
  6. 6 Appendix III: Computing the inverse of $E(x,t,D,\partial_{t})$
  7. 7 Appendix IV: $L^{2}$-theory of the operator $(-\partial_{t}^{2}-\Delta_{g})E(x,t,D,\partial_{t})^{-1}$ and the Calderón-Vaillancourt theorem
  8. 8 Proof of the Donnelly-Fefferman type inequality for pseudo-differential operators

##  1\. Introduction

In 1995 Labeau and Robbiano [15] proved the null-controllability of the heat
equation for the Laplacian on a compact manifold (with or without smooth
boundary). An instrumental tool in the proof of Labeu-Robbiano’s result is a
spectral inequality for the wave packets of eigenfunctions of the Laplacian
originated from [15, 13]. In the case of a single eigenfuction $\rho_{j},$ the
Lebeau-Robbiano spectral inequality implies the doubling property proved by
Donnelly and Fefferman [6]. The spectral inequality from [15, 13] has been
used in many areas of analysis, for instance, in the aforementioned result of
the control theory, or as in [6] in the geometric analysis of the Hausdorff
measure of nodal sets. In this work we extend such a spectral inequality to
the case of a positive elliptic pseudo-differential operator $E(x,D)$ of
arbitrary order $\nu>0$ and we apply this spectral inequality to the null-
controllability of the heat equation (and also of the fractional heat
equation) associated to $E(x,D).$

The analysis here includes the family of elliptic differential operators of
arbitrary order with smooth coefficients where the main obstruction in proving
the Lebeau-Robbiano spectral inequality is the lack of Carleman estimates.
Indeed, in the high-dimensional case, the so-called Hörmander subellipticity
condition does not hold, which is typical for product operators. Our approach
compensates it with the use of the calculus of (possibly non-local) pseudo-
differential operators on compact manifolds due to Hörmander [12]. Moreover,
we prove that any positive elliptic pseudo-differential operator of positive
order in the standard $(\rho,\delta)$-Hörmander class on a closed manifold $M$
satisfies the Lebeau-Robbiano spectral inequality.

In general pseudo-differential operators are non-local and the use of Carleman
estimates, which is the analytical tool by excellence in the proof of the
Lebeau-Robbiano spectral inequality and their subsequent generalisations, are
not valid. This work introduces a new approach in the proof of these spectral
inequalities for elliptic differential operators, embedding this problem in
the general setting of the pseudo-differential calculus due to Hörmander where
we exploit e.g. the Calderón-Vaillancourt theorem and other microlocal
properties of the pseudo-differential calculus.

The validity of the Lebeau-Robbiano spectral inequality for an elliptic
pseudo-differential operator implies the null-controllability for its
corresponding diffusion model. This is a very well known consequence of the
equivalence between the null-controllability of a linear model and its
corresponding observability inequality due to the Hilbert Uniqueness Method
developed by J. -L. Lions [18]. The null-controllability for the diffusion
model associated to the operator $E(x,D)^{\alpha}$ as well as its
corresponding spectral inequality are presented in Theorem 1.2 and in Theorem
1.4 below.

###  1.1. Outline and main results

To discuss the contributions of this work we precise some notations. Let
$(M,g)$ be a compact Riemannian manifold without boundary and of dimension
$n$. Let $\omega$ be a non-empty open subset of $M$ and let $1_{\omega}$ be
its characteristic function. For any $\nu>0,$ let $E(x,D)$ be a positive
elliptic pseudo-differential operator of order $\nu$ in the Hörmander class
$\Psi^{\nu}_{\rho,\delta}(M),$ see [12]. The conditions $\rho\geq 1-\delta$
and $0\leq\delta<\rho\leq 1,$ imply that the class
$\Psi^{\nu}_{\rho,\delta}(M)$ is invariant under changes of coordinates and in
particular, for $\nu\in 2\mathbb{N},$ the family of elliptic operators in the
class $\Psi^{\nu}_{1,0}(M)$ contains any positive elliptic differential
operator with smooth coefficients of order $\nu=2m.$ One of the goals of this
work is to establish the null controllability for the model of fractional
diffusion

$\begin{cases}u_{t}(x,t)+E(x,D)^{\alpha}u(x,t)=g(x,t)\cdot
1_{\omega}(x),&(x,t)\in M\times(0,T),\\\ u(0,x)=u_{0},\end{cases}$ (1.1)

at any time $T>0,$ that is, to investigate the conditions on the pair
$(\alpha,\nu),$ in order that for any initial condition $u_{0},$ there is an
input function $g\in L^{2}(M\times(0,T))$ such that the solution to (1.1)
vanishes in time $T,$ that is $u(x,T)=0,$ $x\in M.$

Going to the historical aspects of the control theory for the heat equation,
in the context of compact manifolds, it started with the aforementioned result
by Lebeau and Robbiano [15] where they proved the null-controllability of
(1.1) in the case of the Laplacian $E(x,D)=-\Delta_{g},$ and for $\alpha=1$
(even if the manifold has a smooth non-empty boundary). Then the general
result by Miller [21] proved in the setting of separable Hilbert spaces allows
to use the spectral inequality in [15] to get the same result for
$E(x,D)=(-\Delta_{g})^{\alpha}$ for all $\alpha>1/2.$ Moreover, the sharpness
of the inequality $\alpha>1/2$ was analysed in [19].

One of the main tools in proving the null-controllability result in the case
of the Laplacian is the following spectral inequality for wave packets of
eigenfunctions, see Jerison and Lebeau [13], and [15, 16].

######  Theorem 1.1.

Let $M$ be a compact Riemmanian manifold with (or without) smooth boundary
$\partial M.$ Let $(\rho_{j},\lambda_{j}^{2})$ be the corresponding spectral
data of the Laplacian $-\Delta_{g},$ determined by the eigenvalue problem
$-\Delta_{g}\rho_{j}=\lambda_{j}^{2}\rho_{j}.$

Then, for any non-empty open subset $\omega\subset M,$ we have the loss of
orthogonality estimate

$\|\varkappa\|_{L^{2}(M)}\leq
C_{1}e^{C_{2}{\lambda}}\|\varkappa\|_{L^{2}(\omega)},\,\,\,\varkappa\in\textnormal{span}\\{\rho_{j}:\lambda_{j}\leq\lambda\\}.$
(1.2)

Moreover, the growth constant $e^{C_{2}{\lambda}}$ is sharp.

Observation estimates of the type (1.2) have many applications in control
theory and in geometric analysis. For instance, in control theory only to
mention a few, it was used in the aforementioned result of null-
controllability of the heat equation [15], in the null-controllability of the
thermoelasticity system [16], the null-controllability of the thermoelastic
plate system [2, 22], the null-controllability for the elasticity model of the
bi-Laplacian $\Delta_{g}^{2}$ (endowed with “clampled” boundary conditions)
[24], and the null-controllability of some systems of parabolic PDE [14]. On
the other hand, in geometric analysis it can be used to estimate the Hausdorff
measure of the nodal set of finite sums of eigenfunctions of $-\Delta_{g},$
when the manifold is analytic, see [17, 13] generalizing a result by Donnelly
and Fefferman [6]. The Lebeau-Robbiano spectral inequality has shown to be
valid even when $\omega$ is a measurable set and satisfies some additional
geometric conditions, see e.g. [1] and references therein.

Let us consider a positive elliptic pseudo-differential operator
$A\in\Psi^{\nu}_{\rho,\delta}(M)$ with positive order $\nu>0$ on a compact
Riemannian manifold $(M,g).$ The sequence
$\\{(\mu_{j}=\lambda_{j}^{\nu},\rho_{j}):j\in\mathbb{N}\\}$ determines the
spectral data of the operator $E(x,D)$ that are determined by the eigenvalue
problem:

$E(x,D)\rho_{j}=\lambda_{j}^{\nu}\rho_{j},$
$0\leq\lambda_{j}\leq\lambda_{j+1}\rightarrow\infty.$

The contributions of this work start with the following null-controllability
statement of the fractional heat equation for elliptic operators on compact
manifolds. Here, $0\leq\delta<\rho\leq 1,$ and $\rho\geq 1-\delta,$ are the
required conditions in order that the classes $\Psi^{\nu}_{\rho,\delta}(M)$
became invariant under changes of local coordinate systems.

######  Theorem 1.2.

Let $E(x,D)\in\Psi^{\nu}_{\rho,\delta}(M)$ be a positive elliptic pseudo-
differential operator of order $\nu>0.$ Let $(x,\xi)\in T^{*}M,$ and assume
that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly positive. Then, for any
$\alpha>1/\nu,$ the fractional diffusion problem in (1.1) is null-controllable
at any time $T>0.$

As a consequence of Theorem 1.2 we obtain the null-controllability of the
fractional diffusion model (1.1) for a class of elliptic differential
operators of arbitrary (even) order and with smooth coefficients.

######  Corollary 1.3.

Let $\nu\in 2\mathbb{N},$ and let $E(x,D)$ be a positive elliptic differential
operator of order $\nu=2m$ on $M$ with smooth coefficients. Let $(x,\xi)\in
T^{*}M,$ and assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly
positive. Then, for any $\alpha>1/\nu,$ the fractional diffusion problem in
(1.1) is null-controllable at any time $T>0.$

By following the strategy of Lebeau and Robbiano [15], we we shall derive the
proof of the controllability Theorem 1.2, from the corresponding spectral
inequality for the elliptic pseudo-differential operator $E(x,D).$

######  Theorem 1.4.

Let $\nu>0,$ and let $0\leq\delta<\rho\leq 1$ be such that $\rho\geq
1-\delta.$ Let $E(x,D)\in\Psi^{\nu}_{\rho,\delta}(M)$ be an elliptic positive
pseudo-differential operator of order $\nu>0.$ Let $(x,\xi)\in T^{*}M,$ and
assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly positive. Then, for
any non-empty open subset $\omega\subset M,$ we have

$\|\varkappa\|_{L^{2}(M)}\leq
C_{1}e^{C_{2}{\lambda}}\|\varkappa\|_{L^{2}(\omega)},\,\,\,\varkappa\in\textnormal{span}\\{\rho_{j}:\lambda_{j}\leq\lambda\\}.$
(1.3)

As a consequence of Theorem 1.4 we obtain in the case of elliptic differential
operators the following spectral inequality.

######  Corollary 1.5.

Let $\nu\in 2\mathbb{N},$ and let $E(x,D)$ be a positive elliptic differential
operator of order $\nu$ and with smooth coefficients on $M.$ Let $(x,\xi)\in
T^{*}M,$ and assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly
positive. Then, for any non-empty open subset $\omega\subset M,$ we have

$\|\varkappa\|_{L^{2}(M)}\leq
C_{1}e^{C_{2}{\lambda}}\|\varkappa\|_{L^{2}(\omega)},\,\,\,\varkappa\in\textnormal{span}\\{\rho_{j}:\lambda_{j}\leq\lambda\\}.$
(1.4)

The point of view that we adopt, and that is one of the novelties of this
work, is the use of the pseudo-differential calculus on compact manifolds
without boundary (see Hörmander [12]) in the proof of the spectral inequality
(1.3). One of the reasons for introducing this technique comes from the lack
of Carleman estimates for pseudo-differential operators (which in general are
not local operators) (see e.g. the discusssion by Rousseau and Robbiano in
[24, Subsection 1.4]). Nevertheless, when using the theory of pseudo-
differential operators, other techniques of the modern microlocal analysis as
the Calderón-Vaillancourt theorem for closed manifolds, and the analytic
functional calculus of elliptic operators can be applied in order to prove an
interpolation inequality of the form
$\|F\|_{H^{1}(M\times(\alpha,T-\alpha))}\lesssim\|F\|_{H^{1}(M\times(0,T)}^{\kappa}\|F\|^{1-\kappa}_{L^{2}(\omega)}$
for a suitable wave function $F$ and that is a crucial tool in the proof of
(1.3). One of the difficulties in the proof of this inequality is that the
operators $E(x,D)$ could be in general of non-local type.

We also observe that the main reason in considering only in our analysis the
case of a compact manifold without boundary is the use of the pseudo-
differential calculus by itself. Indeed, when the manifold $M$ has a non-empty
smooth boundary $\partial M$, the interior $\textnormal{int}(M)$ of $M$ is an
open submanifold, and the pseudo-differential calculus for $M$ requires a more
delicate analysis. In this setting, pseudo-differential operators are matrices
acting from $C^{\infty}(\textnormal{int}(M))\oplus C^{\infty}(\partial M)$
into itself and the calculus uses the transmission property introduced by
Hörmander. We refer the reader to the seminal work of Boutet de Monvel in [4]
for details.

Now, we briefly discuss our main results in relation with some known results.

######  Remark 1.6.

In the case where $E(x,D)=-\Delta_{g}$ is the positive Laplacian, the symbol
$E(x,\xi)=\|\xi\|_{g}>0$ is strictly positive on $T^{*}M\setminus\\{0\\}.$ So,
our Theorem 1.4 recovers the Lebeau-Robianno inequality and then Theorem 1.2
recovers the null controllability of the heat equation for the Laplacian. We
observe that in the case of fractional powers $E(x,D)=(-\Delta_{g})^{\alpha},$
of order $\nu=2\alpha,$ $\alpha>0,$ the heat equation for the fractional
operator becomes to be null-controllable in any time $T>0,$ always that
$\alpha>1/2.$ This recover the fractional result due to Miller [21].

######  Remark 1.7.

If
$E(x,D)=-\sum_{i,j=1}^{n}a_{ij}(x)\partial^{2}_{x_{i}x_{j}}+\textnormal{lower
terms},$ is a positive elliptic second order differential operator with real
smooth coefficients $a_{ij}\in C^{\infty}(M),$ and $a_{ij}=a_{ji}$ for any
$(i,j),$ the ellipticity condition implies that the principal symbol
$E(x,\xi)=\sum_{i,j=1}^{n}a_{ij}(x)\xi_{i}\xi_{j}\geq C\|\xi\|_{g}^{2}$ is
strictly positive on $T^{*}M\setminus\\{0\\}.$ In this setting, the positivity
condition on $E(x,D)$ and on its symbol $E(x,\xi)$ implies the null-
controllability of (1.1) for all $\alpha>1/2.$ This fact follows from the
validity of the spectral inequality for $E(x,D),$ see e.g. [23] and Theorem 1
in Miller [21, Page 263] (see Theorem 4.17 below). This situation is also
covered by our main results.

######  Remark 1.8.

If $E(x,D)=\sum_{|\alpha|\leq\nu}a_{\alpha}(x)D_{x}^{\alpha}$ is a positive
elliptic differential operator with smooth coefficients, necessarily $\nu\in
2\mathbb{N},$ and Corollary 1.5 gives the Lebeau-Robbiano spectral inequality
(1.4). Different to the case $\nu=2,$ where the spectral inequality (1.4) can
be derived by the use of Carleman inequalities, the case of higher-order
differential operators (that is when $\nu\geq 4$) there is a lack of Carleman
estimates since the Hörmander subellipticity condition for the weight function
could be not valid. This is the case of product operators $E(x,D)=Q_{1}Q_{2},$
see Rousseau and Robbiano [24, Subsection 1.4]). Recently, for the case of the
bi-Laplacian $-\Delta_{g}^{2},$ the Lebeau-Robianno spectral inequality was
proved in [24] endowed of the so called clamped boundary conditions leaving
open the question for similar boundary conditions for higher-dimensional
powers $(-\Delta_{g})^{k},$ $k\geq 3,$ on a manifold with non-empty smooth
boundary and with the operator endowed with the natural boundary conditions,
e.g. $u|_{\partial M}=\cdots=\partial_{\nu}^{k-1}u|_{\partial M}=0,$ or more
general Lopatinskii-type conditions.

######  Remark 1.9.

The Lebeau-Robbiano spectral inequality remains valid even if the coefficients
of a second-order elliptic differential operators are not smooth. One allows
the case of Lipchitz coefficients, coefficients with jumps at an interface,
etc. We refer the reader to [10, Pages 55-56] and the references discussed
there for details.

######  Remark 1.10.

In the case of a single eigenfuction $\rho_{j},$ the estimate in (1.2) can be
derived from the doubling property proved by Donnelly and Fefferman [6], see
also [17, 13]. The sharpness of the estimate (1.2) was proved by Jerison and
Lebeau [13].

In the general case of a positive pseudo-differential operator we have the
following Donnelly-Fefferman type inequality (extending the previous result in
[6]). The proof of Theorem 1.11 will be given in Section 8.

######  Theorem 1.11.

Let $\nu>0,$ and let $0\leq\delta<\rho\leq 1$ be such that $\rho\geq
1-\delta.$ Let $E(x,D)\in\Psi^{\nu}_{\rho,\delta}(M)$ be an elliptic positive
pseudo-differential operator of order $\nu>0.$ Let $(x,\xi)\in T^{*}M,$ and
assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly positive. For any
$R>0$ let $B(x,R)$ be a ball defined by the geodesic distance, of radius $R>0$
and centred at $x.$ Then,

$\sup_{B(x,2R)}|\varkappa|\leq
e^{C_{1}^{\prime}{\lambda}+C_{2}^{\prime}}\sup_{B(x,R)}|\varkappa|,\,\,\,\varkappa\in\textnormal{span}\\{\rho_{j}:\lambda_{j}\leq\lambda\\},$
(1.5)

with $C_{1}^{\prime}=C_{1}^{\prime}(R)$ and $C_{2}^{\prime}=C_{2}^{\prime}(R)$
are dependent only on the radius $R>0$ but not on $\varkappa.$

######  Remark 1.12.

We observe that the case of the 1D spectral fractional heat equation was
analysed in [20]. Moreover, the analysis of null-controllability for the heat
equation for the 1D-integral fractional Laplacian has been considered in [3].

###  1.2. Structure of the work

This article is organised as follows. In Section 4 we present the basics on
pseudo-differential operators and the null-controllability results for
fractional differential problems on Hilbert spaces. In particular we present
the basics about the construction of complex powers of pseudo-differential
operators. Then, the spectral inequality in Theorem 1.4 and the null-
controllability result in Theorem 1.2 are proved in Section 2.

##  2\. Null-controllability of the fractional heat equation and observability
inequalities

Let us consider an orientable compact Riemannian manifold $(M,g)$ (without
boundary) of dimension $n.$ Let us consider:

  * -

The $L^{2}$-space $L^{2}(M)=L^{2}(M,d\textnormal{Vol}_{g})$ associated with
the volume form $d\textnormal{Vol}_{g},$ for the metric $g.$ For simplicity we
always write $dx:=d\textnormal{Vol}_{g}(x).$

  * -

The inner product $(f,g)=\smallint_{M}f(x)\overline{g(x)}dx$ on $L^{2}(M).$

  * -

For any pair $(\rho,\delta)\in[0,1]^{2},$ such that $0\leq\delta<\rho\leq 1$
and $\rho\geq 1-\delta,$ we consider the class $\Psi^{m}_{\rho,\delta}(M)$ of
pseudo-differential operators of order $m\in\mathbb{R}.$ We refer the reader
to Appendix I in Section 4 for the definition of these classes.

If $E(x,D)\in\Psi^{\nu}_{\rho,\delta}(M)$ of positive order $\nu>0,$ in this
section we establish the null controllability for the model of fractional
diffusion

$\begin{cases}u_{t}(x,t)+E(x,D)^{\alpha}u(x,t)=g(x,t)\cdot
1_{\omega}(x),&(x,t)\in M\times(0,T),\\\ u(0,x)=u_{0},\end{cases}$ (2.1)

at any time $T>0,$ that is, we will prove that the solution to (2.1) satisfies
$u(x,T)=0,$ if $\alpha>1/\nu.$ As described in the introduction, first, we
require validity of the spectral inequality (1.3). We employ the following
notation.

  * -

We denote by $\omega\neq\emptyset$ an open subset in $M,$ as well as a generic
compact subset in $M$ will be denoted by $K.$

  * -

For any $\lambda>0,$ we denote by $\textnormal{E}_{\lambda}$ the spectral
projection on $L^{2}(M)$ defined by

$\textnormal{E}_{\lambda}f=\sum_{\lambda_{j}\leq\lambda}(f,\rho_{j})\rho_{j},\,\,f\in
L^{2}(M),$ (2.2)

where $\\{\mu_{j}:=\lambda_{j}^{\nu},\rho_{j}\\}$ are the corresponding
spectral data

$E(x,D)\rho_{j}=\lambda_{j}^{\nu}\rho_{j},\,\,\lambda_{j}\geq 0,$

with the eigenfunctions $\rho_{j}$ being $L^{2}$-normalised.

  * -

For any $T>0,$ let us consider the space-time manifold

$M_{T}:=M\times(0,T),$

and the Sobolev space $H^{s}(M_{T})$ of order $s\in\mathbb{N},$ which consists
of all distribution $f$ on $M$ such that,

$\|f\|_{H^{s}(M_{T})}^{2}=\sum_{0\leq j\leq
s}\smallint\limits_{0}^{T}\smallint\limits_{M}\left[\left|\partial_{t}^{j}f(x,t)\right|^{2}+|(1-\Delta_{g})^{\frac{j}{2}}f(x,t)|^{2}\right]dxdt<\infty,$
(2.3)

where $-\Delta_{g}$ is the positive Laplacian associated to the metric $g.$

  * -

Observe that under the identification $-(T+\varepsilon)\sim T+\varepsilon,$
$\varepsilon>0,$ the manifold $[-(T+\varepsilon),T+\varepsilon)$ can be
identified with the dilated torus
$\mathbb{T}_{T,\varepsilon}:=\mathbb{R}/2(T+\varepsilon)\mathbb{Z}\cong[-(T+\varepsilon),T+\varepsilon)$
and the resulting manifold is diffeomorphic to the circle $\mathbb{S}^{1}$,
and then, it is compact and without boundary. Note also, that the operator
$-\partial_{t}^{2}$ became, up to a constant, the (positive) Laplace-Beltrami
operator on $\mathbb{T}_{T,\varepsilon}.$ Finally, note that
$-\partial_{t}^{2}\in\Psi^{2}_{1,0}(\mathbb{T}_{T,\varepsilon})$ is a positive
and elliptic differential operator of second order on
$\mathbb{T}_{T,\varepsilon}$ and then

$E(x,t,D,\partial_{t})=-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}}\in\Psi^{2}_{1,\delta}({M}\times\mathbb{T}_{T,\varepsilon})$

is also a positive and elliptic pseudo-differential operator on the product
manifold ${M}\times\mathbb{T}_{T,\varepsilon}.$ Note that $M_{T}$ can be
viewed as an open sub-manifold of ${M}\times\mathbb{T}_{T,\varepsilon}.$

Our analysis for the null-controllability of the heat equation starts with the
spectral inequality in Proposition 2.1 below that corresponds in Theorem 1.4
to the case where $E(x,D)$ satisfies the lower bound $E(x,D)\geq cI,$ for some
$c>0.$ Then, the proof of Theorem 1.4 will be deduced from this particular
situation.

######  Proposition 2.1.

Let $(M,g)$ be a compact Riemannian manifold without boundary. Let
$0\leq\delta<\rho\leq 1,$ be such that $1-\delta\geq\rho$. Let
$E(x,D)\in\Psi^{\nu}_{\rho,\delta}(M)$ be an elliptic positive pseudo-
differential operator of order $\nu>0,$ and assume that for some $c>0,$
$E(x,D)\geq cI$ in $L^{2}(M)$ and that for any $(x,\xi)\in T^{*}M,$
$E(x,\xi)\geq c$ is strictly positive.

Then, for any open subset $\omega\subset M,$ any $a_{j}\in\mathbb{R},$ and all
$\lambda>0,$ the following observability inequality holds

$\left(\sum_{\lambda_{j}\leq\lambda}a_{j}^{2}\right)^{\frac{1}{2}}\leq
C_{1}e^{C_{2}{\lambda}}\left\|\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x)\right\|_{L^{2}(\omega)}.$
(2.4)

We postpone the proof of Proposition 2.1 for a moment. Indeed, for the proof
of this statement we require a delicate microlocal analysis. First, we recall
an interpolation inequality for the Laplacian in the next subsection.

###  2.1. An interpolation inequality

For our further analysis we require the following interpolation inequality.

######  Lemma 2.2.

Let us consider the operator
$A(x,t,D,\partial_{t})=-\partial_{t}^{2}-\Delta_{g}\in\Psi^{2}_{1,0}(M_{T})$
with $M_{T}=M\times(0,T)$. Let $\omega$ be a non-empty open subset in $M.$

Then, for any $T>0$ and all $\alpha\in(0,T/2),$ there exists $\delta\in(0,1)$
such that

$\|\phi\|_{H^{1}(M\times(\alpha,T-\alpha))}\leq
C\|\phi\|_{H^{1}(M_{T})}^{\delta}\left(\|A(x,t,D,\partial_{t})\phi\|_{L^{2}(M\times(0,T))}+\|\partial_{t}\phi\|_{L^{2}(\omega)}\right)^{1-\delta},$
(2.5)

for all $\phi\in H^{2}(M\times(0,T))$ such that $\phi=0$ in $M\times\\{0\\}.$

The inequality (2.5) in Lemma 2.5 has been obtained in [16] in the case of a
compact manifold without boundary. We also refer the reader to Rousseau and
Lebeau [23] for alternative proofs of this result even, for second-order
elliptic operators with smooth coefficients.

###  2.2. Proof of the spectral inequality

For any $\lambda>0,$ let
$\varkappa\in\textnormal{Im}(\textnormal{E}_{\lambda}),$ that is $\varkappa$
can be written as

$\varkappa(x)=\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x).$ (2.6)

Observe that for the proof of the spectral inequality (2.4), is enough to
prove that the function

$F(x,t):=\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}a_{j}\rho_{j}(x),\,\,(x,t)\in
M_{T}:=M\times[0,T],$ (2.7)

satisfies the interpolation inequality

$\|F\|_{H^{1}(M\times(\alpha,T-\alpha))}\leq
C\|F\|_{H^{1}(M_{T})}^{\kappa}\|\varkappa\|_{L^{2}(\omega)}^{1-\kappa}.$ (2.8)

Indeed, one can follows e.g. the analysis in [16]. It starts using the
Parseval theorem in order to have

$\displaystyle\|F\|^{2}_{H^{1}(M\times(\alpha,T-\alpha))}$ $\displaystyle\geq
C\|F\|^{2}_{L^{2}(M\times(\alpha,T-\alpha))}$
$\displaystyle=\smallint_{\alpha}^{T-\alpha}\smallint\limits_{M}\left|\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}a_{j}\rho_{j}(x)\right|^{2}dx\,dt$
$\displaystyle=\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}\smallint_{\alpha}^{T-\alpha}\left|\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}\right|^{2}dt$
$\displaystyle\geq\sum_{\lambda_{j}\leq
C\lambda}|a_{j}|^{2}\smallint_{\alpha}^{T-\alpha}t^{2}dt$
$\displaystyle=C_{\alpha}\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}.$

Observing that

$\partial_{t}F(x,0)=\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x),\textnormal{
and that, }\|F\|_{H^{1}(M_{T})}^{2}\lesssim
e^{2T\lambda}\lambda^{2}\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2},$ (2.9)

we deduce that

$C_{\alpha}\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}\lesssim_{\alpha,T}\left(e^{2T\lambda}\lambda^{2}\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}\right)^{\kappa}\left\|\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x)\right\|_{L^{2}(\omega)}^{2(1-\kappa)},$
(2.10)

from which we deduce that

$\displaystyle\left(\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}\right)^{1-\kappa}\lesssim
e^{2T\lambda}\lambda^{2}\left\|\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x)\right\|_{L^{2}(\omega)}^{2(1-\kappa)},$

and in consequence

$\displaystyle\left(\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}\right)^{\frac{1}{2}}\lesssim
e^{{T\lambda/(1-\kappa)}}\lambda^{{1/(1-\kappa)}}\left\|\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x)\right\|_{L^{2}(\omega)},$

proving (2.4). Indeed, one can estimate
$e^{{T\lambda/(1-\kappa)}}\lambda^{{1/(1-\kappa)}}\lesssim
e^{{T\lambda/(1-\kappa)}}.$

###### Proof of Proposition 2.1.

We will procced with the proof of (2.8). Note that, by normalising $\varkappa$
on $L^{2}(M)$ we can assume without loss of generality that
$\|\varkappa\|_{L^{2}(M)}=1.$

####  2.2.1.  An auxiliar interpolation inequality on $[0,T+\varepsilon)$.

Let $\varepsilon\in(0,1)$ be a positive parameter whose conditions will be
imposed later. Firstly, by replacing in Lemma 2.2 the open interval
$I_{T}:=(0,T)$ by $I_{T+\varepsilon}:=(0,T+\varepsilon),$ and with
$M_{T+\varepsilon}:=M\times(0,T+\varepsilon),$ we shall make use of the
following interpolation inequality:

For any $T>0$ and all $\alpha\in(0,T/2),$ there exists $\kappa\in(0,1)$ such
that $\|\phi\|_{H^{1}(M\times(\alpha,T-\alpha))}\leq
C\|\phi\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}\left(\|(-\partial_{t}^{2}-\Delta_{g})\phi\|_{L^{2}(M_{T+\varepsilon})}+\|\partial_{t}\phi(x,0)\|_{L^{2}(\omega)}\right)^{1-\kappa}$
(2.11) for all $\phi\in H^{2}(M_{T+\varepsilon})$ such that $\phi=0$ in
$M\times\\{0\\}.$

####  2.2.2.  Construction of a suitable function $\phi$:

Let us apply (2.11) with $\phi$ defined as follows. Consider $\psi\in
C^{\infty}(M\times[0,T+\varepsilon])$ satisfying that

$\displaystyle\psi(t):=\begin{cases}C,&\text{ }t\in[0,T],\,x\in M,\\\
0,&\text{ }t\in[T+\frac{3\varepsilon}{4},T+\varepsilon],\,x\in M,\end{cases}$
(2.12)

where $0<C\leq\varepsilon.$ Assume that there exists $M_{0}>0,$ independent of
$\varepsilon$ such that

$\|\psi^{(i)}\|_{L^{\infty}}\leq M_{0},$ (2.13)

for all $i\in\\{1,2,3,4\\}.$ We construct a function with these properties in
Lemma 5.1 of Appendix II in Section 5. Then, by considering the function

$F(x,t):=\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}a_{j}\rho_{j}(x),\,\,(x,t)\in
M_{T}:=M\times[0,T],$ (2.14)

and its extension to the whole interval $I_{T+\varepsilon}=[0,T+\varepsilon]$
by the constant function equal to $F(x,T)$, that is

$\displaystyle
F(x,t)=\begin{cases}\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}a_{j}\rho_{j}(x),\,\,(x,t),&\text{
}t\in[0,T],\,x\in M,\\\ F(x,T),&\text{ }t\in[T,T+\varepsilon],\,x\in
M,\end{cases}$ (2.15)

we consider

$\phi(x,t):=F(x,t)\psi(t),\,\,(x,t)\in M_{T+\varepsilon}.$ (2.16)

Note that $\phi(x,0)=F(x,0)=0=\phi(x,T+\varepsilon)=0.$ Note also that $\phi$
is an extension of $F$ from $M_{T}$ to $M_{T+\varepsilon}.$

Now, let us consider the odd extension of $\phi$ to the whole interval
$[-(T+\varepsilon),T+\varepsilon],$ that is,

$\phi(x,t)=-\phi(x,-t)=-\psi(-t)F(x,-t),\,-(T+\varepsilon)\,\leq t\leq 0.$
(2.17)

####  2.2.3. Analysis of the term $\|\partial_{t}\phi(x,0)\|_{L^{2}(\omega)}$

Note that $\psi$ has been defined on $[0,T+\varepsilon]$ and it has been
extended to $[-(T+\varepsilon),0]$ using its odd extension, that is, the one
defined by $\psi(-t)=-\psi(t),$ for $t\in[0,T+\varepsilon].$

An elementary calculation gives for any $t$ in a neigboorhood of $t=0,$ that

$\partial_{t}\phi(x,t)=\psi^{\prime}(t)F(x,t)+\psi(t)\partial_{t}F(x,t).$

At $t=0$ we have

$\partial_{t}\phi(x,0)=\psi^{\prime}(0)F(x,0)+\psi(0)\partial_{t}F(x,0)=\psi(0)\partial_{t}F(x,0),$
(2.18)

and then

$\|\partial_{t}\phi(x,0)\|_{L^{2}(\omega)}=|\psi(0)|\times\|\partial_{t}F(x,0)\|_{L^{2}(\omega)}=|\psi(0)|\times\|\varkappa\|_{L^{2}(\omega)}.$
(2.19)

####  2.2.4. Embedding of $M_{T}$ in a closed manifold
$M\times\mathbb{T}_{T,\varepsilon},$
$\mathbb{T}_{T,\varepsilon}\cong\mathbb{S}^{1}$:

Now, we will proceed with a topological construction.

It is clear that in the variable $t\in[-(T+\varepsilon),T+\varepsilon]$ the
function $\phi$ can be extended periodically way to the whole line
$\mathbb{R},$ or in other words, we can identify $\phi$ with a distribution
(which we also denote by $\phi$) on the dilated torus

$\mathbb{T}_{T,\varepsilon}=\mathbb{R}/(2(T+\varepsilon)\mathbb{Z})=[-(T+\varepsilon),T+\varepsilon),$
(2.20)

where in the resulting manifold $[-(T+\varepsilon),T+\varepsilon)$ we identify
the end points $-(T+\varepsilon)\sim T+\varepsilon.$ This construction allows
the manifold $\mathbb{T}_{T,\varepsilon}$ to be diffeomorphic to the circle
$\mathbb{S}^{1},$ and in consequence the function
$\phi\in\mathscr{D}^{\prime}(M\times\mathbb{T}_{T,\varepsilon})$ is smooth on
the product space $M\times\mathbb{T}_{T,\varepsilon}$ which is a compact
manifold of $C^{\infty}$-class without boundary. In particular, we have the
inclusion: $\forall T,\varepsilon>0,\,M_{T}\subset
M\times\mathbb{T}_{T,\varepsilon}.$

####  2.2.5. Proof of the interpolation inequality

Now, we are ready for the proof of the interpolation inequality

$\|F\|_{H^{1}(M\times(\alpha,T-\alpha))}\leq
C_{s_{0},s_{00}}\|F\|_{H^{1}(M_{T})}^{\kappa}\|\varkappa\|_{L^{2}(\omega)}^{1-\kappa}.$

In the identity (2.24) below, we will prove that with $\phi$ defined in
(2.16), we have that

$\displaystyle\|(-\partial_{t}^{2}-\Delta_{g})\phi(x,t)\|_{L^{2}(M_{T+\varepsilon})}=1/\sqrt{2}\|(-\partial_{t}^{2}-\Delta_{g})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}.$

The positivity condition

$E(x,D)\geq cI,\,\,\,c>0,$

and the spectral mapping theorem gives the lower bound

$E(x,D)^{\frac{2}{\nu}}\geq c^{\frac{2}{\nu}}I,\,\,\,c>0,$

which gives the invertibility of the pseudo-differential operator

$E(t,x,D,\partial_{t})=-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}}:H^{2}(M\times\mathbb{T}_{T,\varepsilon})\rightarrow
L^{2}(M\times\mathbb{T}_{T,\varepsilon}),$

that is, the operator ,

$E(t,x,D,\partial_{t})^{-1}:L^{2}(M\times\mathbb{T}_{T,\varepsilon})\rightarrow
H^{2}(M\times\mathbb{T}_{T,\varepsilon})$

is bounded, see Lemma 6.1. Moreover,

$\displaystyle\|(-\partial_{t}^{2}-\Delta_{g})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}\leq
C\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})},$

where the constant $C$ is independent of $\varepsilon>0.$ Indeed,

$\|(-\partial_{t}^{2}-\Delta_{g})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}$
$=\|(-\partial_{t}^{2}-\Delta_{g})(-\partial_{t}^{2}+E(x,D)^{2/\nu})^{-1}(-\partial_{t}^{2}+E(x,D)^{2/\nu})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}.$

From Lemma 7.1, the operator
$(-\partial_{t}^{2}-\Delta_{g})(-\partial_{t}^{2}+E(x,D)^{2/\nu})^{-1}$
belongs to the class $\Psi^{0}_{1,\delta}(M)$ and the Calderón-Vaillancourt
theorem gives its boundedness on $L^{2},$ with its operator norm bounded by a
constant $C>0,$ independent of $\varepsilon>0.$ Consequently,

$\|(-\partial_{t}^{2}-\Delta_{g})(-\partial_{t}^{2}+E(x,D)^{2/\nu})^{-1}(-\partial_{t}^{2}+E(x,D)^{2/\nu})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}$
$\leq
C\|(-\partial_{t}^{2}+E(x,D)^{2/\nu})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}.$

In what follows we estimate this norm. Let us denote

$Z_{1}:=\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})},$

and let us keep in mind that the partial analysis above gives us the
inequality

$\|\phi\|_{H^{1}(M\times(\alpha,T-\alpha))}\lesssim\|\phi\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}(Z_{1}+|\psi(0)|\|\varkappa\|_{L^{2}(\omega)})^{1-\kappa},$
(2.21)

where we have used that

$\psi(0)\varkappa=\partial_{t}\phi(x,0)=\psi(0)\partial_{t}F(x,0)=\psi(0)\sum_{\lambda_{j}\leq\lambda}a_{j}\rho_{j}(x).$

Indeed, we record that for $t\in[0,T],$ we have the identities

$\partial_{t}F(x,t)=\sum_{\lambda_{j}\leq\lambda}{\sinh(\lambda_{j}t)}a_{j}\rho_{j}(x),$
$\partial_{t}^{2}F(x,t)=\sum_{\lambda_{j}\leq\lambda}{\sinh(\lambda_{j}t)}\lambda_{j}a_{j}\rho_{j}(x).$

####  2.2.6. Estimate of $Z_{1}$

Observe that

$E(x,D)^{\frac{2}{\nu}}F(x,t)=\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}a_{j}E(x,D)^{\frac{2}{\nu}}(\rho_{j})(x)=\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}a_{j}\lambda_{j}^{2}\rho_{j}(x)$
$=\partial_{t}^{2}F(x,t),\,\,t\in[0,T].$

Since $\phi(x,t)=\psi(t)F(x,t)$ on $[0,T],$ and $\psi$ is constant on $[0,T]$
we have that

$\forall x\in M,\,\forall
t\in(0,T),\,(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)=0.$ (2.22)

First, note that the following symmetry property is valid due to the identity
$\phi(x,t)=-\phi(x,-t),$

$\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}^{2}=2\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|^{2}_{L^{2}(M_{T+\varepsilon})}.$
(2.23)

Moreover, once proved (2.23), we have in the particular case (where
$E(x,D)=-\Delta_{g}$) of the Laplacian, the following inequality.

$\|(-\partial_{t}^{2}-\Delta_{g})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}^{2}=2\|(-\partial_{t}^{2}-\Delta_{g})\phi(x,t)\|^{2}_{L^{2}(M_{T+\varepsilon})}.$
(2.24)

Indeed, for the proof of (2.23) observe that

$\displaystyle\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}^{2}$
$\displaystyle=\smallint\limits_{M}\smallint\limits_{-T-\varepsilon}^{0}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)|^{2}dt\,dx+\smallint\limits_{M}\smallint\limits_{0}^{T+\varepsilon}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)|^{2}dt\,dx$
$\displaystyle=\smallint\limits_{M}\smallint\limits_{-T-\varepsilon}^{0}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})(-\phi(x,-t)|^{2}dt\,dx+\smallint\limits_{M}\smallint\limits_{0}^{T+\varepsilon}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)|^{2}dt\,dx$
$\displaystyle=\smallint\limits_{M}\smallint\limits_{-T-\varepsilon}^{0}|\phi_{tt}(x,-t)-E(x,D)^{\frac{2}{\nu}}\phi(x,-t)|^{2}dt\,dx+\smallint\limits_{M}\smallint\limits_{0}^{T+\varepsilon}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)|^{2}dt\,dx$
$\displaystyle=\smallint\limits_{M}\smallint\limits_{0}^{T+\varepsilon}|-\phi_{tt}(x,t)+E(x,D)^{\frac{2}{\nu}}\phi(x,t)|^{2}dt\,dx+\smallint\limits_{M}\smallint\limits_{0}^{T+\varepsilon}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)|^{2}dt\,dx$
$\displaystyle=2\smallint\limits_{M}\smallint\limits_{0}^{T+\varepsilon}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)|^{2}dt\,dx.$

The cancellation property in (2.22), the symmetry property
$\phi(x,t)=-\phi(x,-t),$ and the positivity of the operator
$(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})$ on
$L^{2}(M\times\mathbb{T}_{T,\varepsilon})$ (that is, making integration by
parts with respect to the operator
$(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})$) imply that

$\displaystyle\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})$
$\displaystyle\phi(x,t)\|_{L^{2}(M_{T+\varepsilon})}^{2}=\frac{1}{2}\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}^{2}$
$\displaystyle=\frac{1}{2}|((-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi),(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi)_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}|$
$\displaystyle=\frac{1}{2}|((-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi),\phi)_{L^{2}(M\times\mathbb{T}_{T,\varepsilon})}|$
$\displaystyle\leq\smallint\limits_{M}\smallint\limits_{[0,T+\varepsilon)}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi(x,t)||\overline{\phi(x,t)}|dxdt$
$\displaystyle=\smallint\limits_{M}\smallint\limits_{[T,T+\varepsilon)}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi(x,t)||{\phi(x,t)}|dxdt.$

Therefore, we have the estimate

$\displaystyle\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M_{T+\varepsilon})}^{2}$
$\displaystyle\leq\smallint\limits_{M}\smallint\limits_{[T,T+\varepsilon)}|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi(x,t)||\phi(x,t)|dxdt$
$\displaystyle\leq\smallint\limits_{M}\smallint\limits_{[T,T+\varepsilon)}|\phi(x,t)|dxdt\times\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi\|_{L^{\infty}}$
$\displaystyle=I\times II,$

where

$I=\smallint\limits_{M}\smallint\limits_{[T,T+\varepsilon)}|\phi(x,t)|dx\,dt,\,II=\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi\|_{L^{\infty}(M\times[T,T+\varepsilon))}.$

Now, we will estimate each one of these norms.

####  2.2.7. Estimate for $I$

Note that

$\displaystyle
I\leq\textnormal{Vol}(M)\times\varepsilon\times\|\phi\|_{L^{\infty}(M\times[T,T+\varepsilon])}=\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}\|F(x,t)\|_{L^{\infty}}.$

Now, for any $t$ fixed, and $s_{0}\in\mathbb{N}$ observe that

$|F(x,t)|\leq\sup_{0\leq s\leq
s_{0}}\|(1+E(x,D))^{\frac{s}{\nu}}F(\cdot,t)\|_{L^{\infty}(M)}.$ (2.25)

Using the Sobolev embedding theorem, we have that for any $s_{00}>n/2,$

$\displaystyle\sup_{0\leq s\leq
s_{0}}\|(1+E(x,D))^{\frac{s}{\nu}}F(\cdot,t)\|_{L^{\infty}(M)}$
$\displaystyle\leq\sup_{0\leq s\leq
s_{0}}\|(1+E(x,D))^{\frac{s+s_{00}}{\nu}}F(\cdot,t)\|_{L^{2}(M)}$
$\displaystyle\leq\sup_{0\leq s\leq
s_{0}+s_{00}}\|(1+E(x,D))^{\frac{s}{\nu}}F(\cdot,t)\|_{L^{2}(M)}.$

Now, let us use the spectral theory of the operator
$(1+E(x,D))^{\frac{s}{\nu}}$ on $L^{2}(M).$ Since

$\displaystyle\|(1+E(x,D))^{\frac{s}{\nu}}F(\cdot,t)\|_{L^{2}(M)}^{2}$
$\displaystyle=\left\|\sum_{\lambda_{j}\leq\lambda}\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}(1+\lambda_{j}^{\nu})^{\frac{s}{\nu}}a_{j}\rho_{j}(x)\right\|^{2}_{L^{2}(M)}$
$\displaystyle=\sum_{\lambda_{j}\leq\lambda}\left|\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}\right|^{2}(1+\lambda_{j}^{\nu})^{\frac{2s}{\nu}}|a_{j}|^{2}$
$\displaystyle\lesssim\sum_{\lambda_{j}\leq\lambda}\left|\frac{\sinh(\lambda_{j}t)}{\lambda_{j}}\right|^{2}\lambda_{j}^{2s}|a_{j}|^{2}\lesssim\sum_{\lambda_{j}\leq\lambda}e^{\lambda_{j}t}\lambda_{j}^{2s-1}|a_{j}|^{2}$
$\displaystyle\lesssim_{s_{0},s_{00}}e^{\lambda
T}\sum_{\lambda_{j}\leq\lambda}|a_{j}|^{2}$ $\displaystyle=e^{\lambda
T}\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)}^{2}.$

In consequence, we deduce the inequality

$\forall s\in[0,s_{0}],\,\forall
s_{00}>n/2,\,\,\|(1+E(x,D))^{\frac{s}{\nu}}F\|_{L^{\infty}}\lesssim_{s_{0},s_{00}}e^{T\lambda/2}\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)},$
(2.26)

as well as the Sobolev estimate

$\forall s_{00}>n/2,\,\forall
s\in[0,s_{0}+s_{00}],\,\,\,\|(1+E(x,D))^{\frac{s}{\nu}}F\|_{L^{2}}\lesssim_{s_{0},s_{00}}e^{T\lambda/2}\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)}.$
(2.27)

With $s_{0}=0,$ we have that
$\|F\|_{L^{\infty}}\lesssim_{s_{0},s_{00}}e^{T\lambda/2}\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)}.$
The analysis above gives us the inequality:

$I\lesssim\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda/2}\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)}=\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda/2},$
(2.28)

where we have used that
$\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)}=\|\varkappa\|_{L^{2}(M)}=1.$
Summarising

$I\lesssim\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda/2}.$

####  2.2.8. Estimate for $II$:

To estimate the second term, we start by observing the inequality

$II=\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}\phi\|_{L^{\infty}(M\times[T,T+\varepsilon))}=\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}[\psi(t)F(x,T)]\|_{L^{\infty}(M\times[T,T+\varepsilon))}.$

Since

$\displaystyle(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}$
$\displaystyle[\psi(t)F(x,T)]=(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})[\psi(t)F(x,T)]$
$\displaystyle=(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})[-\psi_{tt}(t)F(x,T)+\psi(t)E(x,D)^{\frac{2}{\nu}}F(x,T)]$
$\displaystyle=\psi^{(4)}(t)F(x,T)-2\psi_{tt}(t)E(x,D)^{\frac{2}{\nu}}F(x,T)$
$\displaystyle\,\,\,\,+\psi(t)E(x,D)^{\frac{4}{\nu}}(F(x,T)),$

for $s_{0}\geq 4,$ and with $s_{00}>n/2,$ the Sobolev inequality in (2.26)
implies that

$\displaystyle\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})^{2}[\psi(t)F(x,T)]\|_{L^{\infty}(M\times[T,T+\varepsilon))}$
$\displaystyle\leq\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}\|F(x,T)\|_{L^{\infty}}$
$\displaystyle+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}\|E(x,D)^{\frac{2}{\nu}}F(x,T)\|_{L^{\infty}}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}}\|E(x,D)^{\frac{4}{\nu}}F(x,T)\|_{L^{\infty}}$
$\displaystyle\leq\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}\|F(x,T)\|_{L^{\infty}}$
$\displaystyle+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}\|(1+E(x,D))^{\frac{s_{00}+2}{\nu}}F(x,T)\|_{L^{2}}$
$\displaystyle+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}}\|(1+E(x,D))^{\frac{s_{00}+4}{\nu}}F(x,T)\|_{L^{2}}$
$\displaystyle\lesssim_{s_{0},s_{00}}e^{{T}{\lambda}/2}\|\partial_{t}F(\cdot,0)\|_{L^{2}(M)}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}})$
$\displaystyle=e^{{T}{\lambda}/2}\|\varkappa\|_{L^{2}(M)}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}})$
$\displaystyle=e^{{T}{\lambda}/2}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}}).$

####  2.2.9. Estimate for $Z_{1}$

Having proved these estimates for $I$ and $II,$ we have that

$\displaystyle\|(-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}})\phi(x,t)\|_{L^{2}(M_{T+\varepsilon})}^{2}$
$\displaystyle\lesssim\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda/2}\times
e^{{T}{\lambda}/2}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}})$
$\displaystyle=\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}}).$

####  2.2.10. Final Analysis

The estimates above for $Z_{1}$ lead to the following inequality in view of
the interpolation inequality

$\displaystyle\|\phi\|_{H^{1}(M\times(\alpha,T-\alpha))}$
$\displaystyle\lesssim\|\phi\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}((\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}}))^{\frac{1}{2}}$
$\displaystyle+|\psi(0)|\|\varkappa\|_{L^{2}(\omega)})^{1-\kappa}.$

Now, dividing both sides of this inequality by $|\phi(0)|$ and using that
$\psi(0)=\psi(T)=\psi(t),$ $0\leq t\leq T,$ we get

$\displaystyle\|F(x,t)\|_{H^{1}(M\times(\alpha,T-\alpha))}=\left\|\frac{\psi(t)F(x,t)}{\psi(0)}\right\|_{H^{1}(M\times(\alpha,T-\alpha))}$
$\displaystyle\lesssim\left\|\frac{\psi(t)F(x,t)}{\psi(0)}\right\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}$
$\displaystyle\times\frac{1}{|\psi(0)|^{1-\kappa}}((\textnormal{Vol}(M)\times\varepsilon\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}e^{T\lambda}(\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}+2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}+\|\psi\|_{{L^{\infty}[T,T+\varepsilon]}}))^{\frac{1}{2}}$
$\displaystyle+|\psi(0)|\|\varkappa\|_{L^{2}(\omega)})^{1-\kappa}$
$\displaystyle=\left\|\frac{\psi(t)F(x,t)}{\psi(0)}\right\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}$
$\displaystyle\times(\left(\textnormal{Vol}(M)\times\varepsilon
e^{T\lambda}\frac{\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(0)|}\left(\frac{\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}}{\|\psi(0)|}+\frac{2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(0)|}+1\right)\right)^{\frac{1}{2}}$
$\displaystyle+\|\varkappa\|_{L^{2}(\omega)})^{1-\kappa}$
$\displaystyle=\left\|\frac{\psi(t)F(x,t)}{\psi(T)}\right\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}$
$\displaystyle\times(\left(\textnormal{Vol}(M)\times\varepsilon
e^{T\lambda}\frac{\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(T)|}\left(\frac{\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(T)|}+\frac{2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(T)|}+1\right)\right)^{\frac{1}{2}}$
$\displaystyle+\|\varkappa\|_{L^{2}(\omega)})^{1-\kappa}.$

Consequently, the previous analysis leads to the auxiliary inequality

$\displaystyle\|F(x,t)\|_{H^{1}(M\times(\alpha,T-\alpha))}\lesssim\left\|\frac{\psi(t)F(x,t)}{\psi(T)}\right\|_{H^{1}(M_{T+\varepsilon})}^{\kappa}$
$\displaystyle\times(\left(\textnormal{Vol}(M)\times\varepsilon
e^{T\lambda}\frac{\|\psi(t)\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(T)|}\left(\frac{\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(T)|}+\frac{2\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}}{|\psi(T)|}+1\right)\right)^{\frac{1}{2}}$
$\displaystyle+\|\varkappa\|_{L^{2}(\omega)})^{1-\kappa},$

and taking the limit when $\varepsilon\rightarrow 0^{+}$ in both sides of this
estimate we conclude the expected inequality,

$\|F\|_{H^{1}(M\times(\alpha,T-\alpha))}\leq
C_{s_{0},s_{00}}\|F\|_{H^{1}(M_{T})}^{\kappa}\|\varkappa\|_{L^{2}(\omega)}^{1-\kappa},$
(2.29)

where we have used that $\psi(t)/\psi(0)=\psi(t)/\psi(T)=1,$ $0\leq t\leq T,$
the smoothness of $\psi,$ the fact that (see Lemma 5.1)

$\lim_{\varepsilon\rightarrow
0^{+}}\|\psi^{(4)}\|_{L^{\infty}[T,T+\varepsilon]}=\psi_{tttt}(T)=\lim_{\varepsilon\rightarrow
0^{+}}\|\psi_{tt}\|_{L^{\infty}[T,T+\varepsilon]}=\psi_{tt}(T)=0,$

and the properties

  * •

$\lim_{\varepsilon\rightarrow
0^{+}}\left\|{\psi(t)}/{\psi(T)}\right\|_{L^{\infty}[T,T+\varepsilon]}=1,$
(2.30)

and

  * •

$\lim_{\varepsilon\rightarrow
0}\left\|{\psi(t)F(x,t)}/{\psi(0)}\right\|_{H^{1}(M_{T+\varepsilon})}=\|F(x,t)\|_{H^{1}(M_{T})}.$
(2.31)

For the proof of (2.31) note that for $\tilde{\psi}:=\psi(t)/\psi(0),$ and
using that $F(x,t)=F(x,T)$ if $0\leq t\leq T+\varepsilon,$ we have

$\lim_{\varepsilon\rightarrow
0}\left\|{\psi(t)F(x,t)}/{\psi(0)}\right\|^{2}_{H^{1}(M_{T+\varepsilon})}=\lim_{\varepsilon\rightarrow
0}\sum_{j=0,1}\smallint\limits_{0}^{T+\varepsilon}\|\partial_{t}^{(j)}(\tilde{\psi}(t)F(x,t))\|^{2}_{H^{1}(M)}dt$
$=\lim_{\varepsilon\rightarrow
0}\sum_{j=0,1}\smallint\limits_{0}^{T}\|\partial_{t}^{(j)}(\tilde{\psi}(t)F(x,t))\|^{2}_{H^{1}(M)}dt+\lim_{\varepsilon\rightarrow
0}\sum_{j=0,1}\smallint\limits_{T}^{T+\varepsilon}\|\partial_{t}^{(j)}(\tilde{\psi}(t)F(x,t))\|^{2}_{H^{1}(M)}dt$
$=\sum_{j=0,1}\smallint\limits_{0}^{T}\|\partial_{t}^{(j)}(F(x,t))\|^{2}_{H^{1}(M)}dt+\lim_{\varepsilon\rightarrow
0}\sum_{j=0,1}\smallint\limits_{T}^{T+\varepsilon}\|\tilde{\psi}^{(j)}(t)F(x,T)\|^{2}_{H^{1}(M)}dt$
$=\|F(x,t)\|_{H^{1}(M_{T})}^{2},$

where we have used that when $t\rightarrow T,$
$\tilde{\psi}^{(j)}(t)\rightarrow 0.$ Now, from (2.29) we can follow the
standard Lebeau-Robbiano argument that has been described at the beginning of
the section to conclude the proof of the spectral inequality. Having proved
(2.4), the proof of Proposition 2.1 is complete. ∎

###### Proof of Theorem 1.4.

Let $\mu,c>0$ be two positive parameters and assume that $0<c<\mu^{\nu}$.
Define

$\tilde{E}(x,D)=E(x,D)+c.$

Observe that the principal symbol $\tilde{E}(x,\xi)$ of the operator
$\tilde{E}(x,D)$ satisfies the lower bound

$\tilde{E}(x,\xi)\geq c,\,\,\xi\in T^{*}_{x}M,$

and also that $\tilde{E}(x,D)\geq cI.$ If
$\\{\mu_{j}:=\lambda_{j}^{\nu},\rho_{j}\\}$ are the corresponding spectral
data

$E(x,D)\rho_{j}=\lambda_{j}^{\nu}\rho_{j},\,\,\lambda_{j}\geq 0,$

of $E(x,D),$ then $\\{\mu_{j}+c:=\lambda_{j}^{\nu}+c,\rho_{j}\\}$ are the
corresponding spectral data

$\tilde{E}(x,D)\rho_{j}=(\lambda_{j}^{\nu}+c)\rho_{j},\,\,\lambda_{j}\geq 0$

of the operator $\tilde{E}(x,D).$ Let
$\lambda:=(\mu^{\nu}+c)^{\frac{1}{\nu}}.$ From Proposition 2.1 we deduce the
spectral inequality

$\left(\sum_{(\lambda_{j}^{\nu}+c)^{\frac{1}{\nu}}\leq\lambda}a_{j}^{2}\right)^{\frac{1}{2}}\leq
C_{1}e^{C_{2}{\lambda}}\left\|\sum_{{(\lambda_{j}^{\nu}+c)^{\frac{1}{\nu}}\leq\lambda}}a_{j}\rho_{j}(x)\right\|_{L^{2}(\omega)}.$
(2.32)

Note that $(\lambda_{j}^{\nu}+c)^{\frac{1}{\nu}}\leq\lambda$ becomes
equivalent to the inequality $\lambda_{j}\leq\mu$ and since $0<c<\mu^{\nu},$
then $\lambda<2\mu.$ Thus, we have proved the spectral inequality

$\left(\sum_{\lambda_{j}\leq\mu}a_{j}^{2}\right)^{\frac{1}{2}}\leq
C_{1}e^{2C_{2}{\mu}}\left\|\sum_{\lambda_{j}\leq\mu}a_{j}\rho_{j}(x)\right\|_{L^{2}(\omega)}.$
(2.33)

In consequence the proof of Theorem 1.4 is complete. ∎

###  2.3. Null-controllability for the diffusion model

Now, we present the proof of Theorem 1.2 which we present in the following
way.

######  Theorem 2.3.

Let $E(x,D)$ be a positive and elliptic pseudo-differential operator of order
$\nu>0$ in the Hörmander class $\Psi^{\nu}_{\rho,\delta}(M).$ Let $(x,\xi)\in
T^{*}M,$ and assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly
positive.

Then, for any $\alpha>1/\nu,$ the fractional diffusion model

$\begin{cases}u_{t}(x,t)+E(x,D)^{\alpha}u(x,t)=g(x,t)\cdot
1_{\omega}(x),&(x,t)\in M\times(0,T),\\\ u(0,x)=u_{0},\end{cases}$ (2.34)

is null-controllable at any time $T>0,$ that is, there exists an input
function $g\in L^{2}(M)$ such that for any $x\in M,$ $u(x,T)=0.$

###### Proof.

The spectral inequality in (2.4) allows us to make use of Theorem 4.17 with
$A=E(x,D)^{\alpha}$. Indeed, since
$A^{\gamma}=E(x,D)^{\alpha\gamma}=E(x,D)^{\frac{1}{\nu}},$ satisfies (2.4)
(that is, the inequality (4.27) holds) for $\alpha\gamma=1/\nu.$ Because
$\gamma\in(0,1)$ if an only if $\alpha>1/\nu,$ Theorem 4.17 guarantees that
this inequality on the fractional order $\alpha$ is a sufficient condition in
order that (2.34) will be null-controllable in time $T>0.$ The proof of
Theorem 2.3 is complete. ∎

###  2.4. The controllability cost for the diffusion model over short times

In the following result we analyse the controllability cost of the model
(2.34) when the time is small.

######  Corollary 2.4.

The controllability cost $C_{T}$ for the fractional heat equation (2.34) over
short times $T\in(0,1)$ satisfies

$C_{T}\leq C_{1}e^{C_{2}T^{-\beta}},$ (2.35)

where $\beta>1/(\alpha\nu-1).$

###### Proof.

For the proof, note that
$A^{\gamma}=E(x,D)^{\alpha\gamma}=E(x,D)^{\frac{1}{\nu}},$ satisfies (2.4) for
$\alpha\gamma=1/\nu.$ Then, from Theorem 4.17 we have the estimate $C_{T}\leq
C_{1}e^{C_{2}T^{-\beta}}$ for any $\beta>\gamma/(\gamma-1)=1/(\alpha\nu-1).$
The proof of Corollary 2.4 is complete. ∎

##  3\. Conclusions and some open problems

This work deals with the null-controllability of the fractional heat equation
associated to an invertible positive elliptic pseudo-differential operator on
a compact Riemannian manifold. Concrete examples are indeed invertible
positive elliptic partial differential operators of (necessarily) even order.

In view of the difficulties when working with Carleman estimates for high-
order differential operators or even for pseudo-differential operators which
in general are non-local operators, we have shown that the theory of pseudo-
differential operators can be effectively used. Moreover, this theory was
strongly used in deriving the interpolation inequality in (2.29), that is

$\|F\|_{H^{1}(M\times(\alpha,T-\alpha))}\leq
C_{s_{0},s_{00}}\|F\|_{H^{1}(M_{T})}^{\kappa}\|\varkappa\|_{L^{2}(\omega)}^{1-\kappa}.$
(3.1)

In conclusion with this work

  * •

we show that the theory of pseudo-differential operators is an effective
microlocal tool for the control theory of parabolic problems, even when we
have the lack of Carleman estimates.

The following are some open problems that arise from this investigation.

  * •

To extend the spectral inequality in (2.29) to the case of suitable boundary
value problem on compact manifolds with smooth boundary. Just to mention a
few, the case of the Laplacian $-\Delta_{g}$ with the corresponding Dirichlet
boundary conditions was derived in the classical Lebeau-Robianno result in
[15], and the case of the bi-Laplacian $(-\Delta_{g})^{2}$ endowed with the so
called clamped boundary conditions was analysed in the recent work of Rousseau
and Robbiano [24]. Even, in [24] the authors leave open the question of the
validity of the spectral inequality for the powers $(-\Delta_{g})^{k},$ $k\geq
3,$ when $\partial M\neq 0.$

  * •

The following problem could be of interest in global analysis: to analyse for
which kind of manifolds $(M,g)$ the condition $\rho\geq 1-\delta$ can be
removed in Theorem 1.4. Indeed, the inequality $\rho\geq 1-\delta$ is the
required condition in order to have the invariance of the pseudo-differential
calculus under changes of coordinates. We propose the following conjecture: in
the case where $M$ is a manifold with symmetries, and under a suitable
positivity condition on the symbol of the operator $E(x,D)$, for any
$0\leq\delta<\rho\leq 1,$ the spectral inequality in Theorem 1.4 is still
valid.

##  4\. Appendix I: Pseudo-differential operators and null-controllability on
Hilbert spaces

In this section we present the preliminaries about the theory of pseudo-
differential operators used in this work as well as some results on the null-
controllability for the Cauchy problem on Hilbert spaces. We use the following
notation.

  * •

For any pair $(X,Y)$ of Hilbert spaces, we denote by $\mathscr{B}(X,Y)$ the
family of bounded and linear operators $T:X\rightarrow Y.$

  * •

The spectrum of a densely defined linear operator
$A:\textnormal{Dom}(A)\subset{X}\rightarrow X$ will be denoted by $\sigma(A).$

###  4.1. Hörmander classes of pseudo-differential operators on compact
manifolds

In this subsection we present the construction of the classes of pseudo-
differential operators on compact manifolds by using local coordinate systems
(see Hörmander [12] and e.g. M. Taylor [27] for an introductory presentation).

Let us briefly introduce these classes starting with the definition in the
Euclidean setting.

######  Definition 4.1 (Symbol classes).

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

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leqslant
C_{\alpha,\beta,K}(1+|\xi|)^{m-\rho|\alpha|+\delta|\beta|},$ (4.1)

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 $\xi\in\mathbb{R}^{n}.$ In this
case we denote the class of pseudo-differential operators with symbols in the
family $S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n})$ as
$\Psi^{m}_{\rho,\delta}(U).$

Once the definition of Hörmander classes on open subsets of $\mathbb{R}^{n}$
is given, it can be extended to smooth manifolds as follows.

######  Remark 4.2 (Pseudo-differential operators on compact manifolds).

Given a compact manifold without boundary $M,$ 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 $a_{T}\in
S^{m}_{\rho,\delta}(U_{\omega}\times\mathbb{R}^{n}).$ As usually, $\omega^{*}$
and $(\omega^{-1})^{*}$ are the pullbacks, induced by the maps $\omega$ and
$\omega^{-1}$ respectively. In this case we write
$A\in\Psi^{m}_{\rho,\delta}(M).$

######  Remark 4.3 (The principal symbol).

The symbol of a pseudo-differential operator $A$ is unique as an element in
$\Psi^{m}_{\rho,\delta}(M)/\Psi^{m}_{\rho,\delta}(M).$ To this element we will
call the principal symbol of $A.$ We denote it by

$a_{m}(x,\xi),\,\,(x,\xi)\in T^{*}M.$ (4.2)

The main feature of the principal symbol of a pseudo-differential operator is
that it remains invariant under coordinates changes.

In the next theorem we describe some fundamental properties of the Hörmander
calculus [12].

######  Theorem 4.4.

Let $0\leqslant\delta<\rho\leqslant 1,$ be such that $\rho\geq 1-\delta.$ Then
$\Psi^{\infty}_{\rho,\delta}(M):=\cup_{m\in\mathbb{R}}\Psi^{m}_{\rho,\delta}(M)$
is an algebra of operators stable under compositions and adjoints, that is:

  * -

the mapping $A\mapsto
A^{*}:\Psi^{m}_{\rho,\delta}(M)\rightarrow\Psi^{m}_{\rho,\delta}(M)$ is a
continuous linear mapping between Fréchet spaces.

  * -

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1}}_{\rho,\delta}(M)\times\Psi^{m_{2}}_{\rho,\delta}(M)\rightarrow\Psi^{m_{1}+m_{2}}_{\rho,\delta}(M)$
is a continuous bilinear mapping between Fréchet spaces.

Moreover, any operator in the class $\Psi^{0}_{\rho,\delta}(M)$ admits a
bounded extension from $L^{2}(G)$ to $L^{2}(G).$

######  Remark 4.5.

The $L^{2}$-boundedness result in Theorem 4.4 is the microlocalised version of
the Calderón-Vaillancourt theorem, see [7]. Moreover, if
$A\in\Psi^{0}_{\rho,\delta}(\mathbb{R}^{n})$ is such that
$0\leq\delta\leq\rho\leq 1,$ $\rho\neq 1,$ then

$\|A\|_{\mathscr{B}(L^{2})}\lesssim\sup_{|\alpha|+|\beta|\leq[\frac{n}{2}]+1}C_{\alpha,\beta},$
(4.3)

where

$C_{\alpha,\beta}:=\sup_{(x,\xi)\in\mathbb{R}^{2n}}(1+|\xi|)^{\rho|\alpha|-\delta|\beta|}|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|.$

######  Remark 4.6.

If $U$ is an open subset of a close manifold $M$ of dimension $n,$ and $f\in
C^{\infty}_{0}(U),$ then, by microlocalising $A\in\Psi^{0}_{\rho,\delta}(M)$
when $0\leq\delta<\rho\leq 1,$ and $\rho\geq 1-\delta,$ one has

$\|Af\|_{L^{2}}\lesssim_{U}\sup_{|\alpha|+|\beta|\leq[\frac{n}{2}]+1}C_{\alpha,\beta,U}\|f\|_{L^{2}(M)},$
(4.4)

where

$C_{\alpha,\beta,U}:=\sup_{(x,\xi)\in
U\times\mathbb{R}^{n}}(1+|\xi|)^{\rho|\alpha|-\delta|\beta|}|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|,$

by making the identification of $U$ with an open subset of $\mathbb{R}^{n}.$

In this work we will use the functional calculus for elliptic pseudo-
differential operators on $M.$ We record that a pseudo-differential operator
$A\in\Psi^{m}_{\rho,\delta}(M)$ is elliptic if in any local coordinate system
$U$, there exists $R>0,$ such that its symbol satisfies uniformly on any
compact subset $K\subset{U}$ the growth estimate

$C_{1}(1+|\xi|)^{m}\leq|a(x,\xi)|\leq C_{2}(1+|\xi|)^{m},\,|\xi|\geq R,$ (4.5)

uniformly in $x\in K,$ and $\xi\in\mathbb{R}^{n}.$ The ellipticity condition
(4.5) is not enough for constructing the complex powers of an elliptic
operator. We present such a construction in the following sub-section.
However, one of the main aspects of the spectral theory of an elliptic
operator is that it spectrum is purely discrete [12].

###  4.2. Complex powers of an elliptic pseudo-differential operator

In this subsection to a complex sector

$\Lambda\subset\mathbb{C},$

we will associate a class class of elliptic operators
$\Psi^{m}_{\rho,\delta}(M;\Lambda)$. In the applications this will, as a rule,
be an angle with the vertex at point $z_{0}\in\mathbb{C}$. For this we will
follow Shubin [26]. For a closed manifold $M$ we require the conditions
$\rho\geq 1-\delta,$ and $0\leq\delta<\rho\leq 1.$ We define these classes as
follows. For an open subset $U\subset\mathbb{R}^{n},$ let
$a(x,\theta,\lambda)$ be a function on $U\times\mathbb{R}^{n}\times\Lambda.$
For $m\in\mathbb{R},$ and $d_{\Lambda}>,0$ we say that $a$ belongs to the
class

$S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n};\Lambda)=S^{m}_{\rho,\delta,d_{\Lambda}}(U\times\mathbb{R}^{n};\Lambda)$

if the following conditions are satisfied,

  * •

$\forall\lambda_{0}\in\Lambda,\,a(x,\theta,\lambda_{0})\in
C^{\infty}(U\times\mathbb{R}^{n}).$

  * •

For every compact subset $K\subset U$ and for all
$\alpha,\beta\in\mathbb{N}_{0}^{n}$, the symbol inequalities

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

hold true uniformly in $x\in K,$ for all $\theta\in\mathbb{R}^{n},$ and all
$\lambda\in\Lambda.$

By simplicity of the notation we have omitted the dependence of these classes
with respect to the parameter $d_{\Lambda}.$

Then, to any $a\in S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n};\Lambda)$ we can
associate a pseudo-differential operator $A_{\lambda},$ depending on the
parameter $\lambda\in\Lambda,$ such that

$A_{\lambda}f(x)=\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
ix\cdot\theta}a(x,\theta,\lambda)(\mathscr{F}_{\mathbb{R}^{n}}{f})(\theta)d\theta,\,f\in
C^{\infty}_{0}(\mathbb{R}^{n}).$

In this case we will write $A_{\lambda}\in\Psi^{m}_{\rho,\delta}(U;\Lambda).$
Now, to make the extension of this definition to the setting of compact
manifolds without boundary, we will say that a family $A_{\lambda},$
$\lambda\in\Lambda,$ belongs to the class $\Psi^{m}_{\rho,\delta}(M;\Lambda)$
if for any $\lambda\in\Lambda,$ the linear continuous operator
$A_{\lambda}:C^{\infty}_{0}(M)\rightarrow C^{\infty}(M)$ is a pseudo-
differential operator of order $m$ of $(\rho,\delta)$-type, with
$\rho\geqslant 1-\delta,$ and $0\leq\delta<\rho\leq 1,$ and 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

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

is a pseudo-differential operator with symbol $a\in
S^{m}_{\rho,\delta}(U_{\omega}\times\mathbb{R}^{n};\Lambda).$ Generalising the
notion of ellipticity in 4.5 we will consider the following notion of
parameterellipticity. We say that an operator
$A\in\Psi^{m}_{\rho,\delta}(M,\Lambda)$ is parameter-elliptic of order $m,$ if
in any local coordinate system $U,$ and for any compact subset $K\subset{U},$
its symbol satisfies the inequality

$C_{1}(1+|\theta|+|\lambda|^{\frac{1}{d_{\Lambda}}})^{m}\leq|a(x,\theta)|\leq
C_{2}(1+|\theta|+|\lambda|^{\frac{1}{d_{\Lambda}}})^{m},|\theta|+|\lambda|\geq
R,$ (4.6)

for some $R>0,$ uniformly in $x\in K,$ and $\theta\in\mathbb{R}^{n}.$ Now, we
present one of the main applications of the notion of parameter-ellipticity.
For $R>0,$ we denote

$\Lambda_{R}:=\\{z\in\mathbb{C}:|z|\geq R\\}\cap\Lambda.$ (4.7)

######  Theorem 4.7.

Let $A_{\lambda}\in\Psi^{m}_{\rho,\delta}(M,\Lambda)$ be parameter-elliptic of
order $m.$ Let $s\in\mathbb{R}.$ Then, there exists $R>0,$ such that for any
$|\lambda|\geq R,$ $A_{\lambda}:H^{s}(M)\rightarrow H^{s}(M)$ is an invertible
operator and $A_{\lambda}^{-1}\in\Psi^{-m}_{\rho,\delta}(M,\Lambda_{R}).$

The main example of elliptic operators depending on a parameter is the
resolvent of a elliptic pseudo-differential operator as we will discuss in the
following remark.

######  Remark 4.8 (The resolvent of an elliptic operator).

Let us consider a Sobolev space $H^{s}(M).$ Let
$A\in\Psi^{m}_{\rho,\delta}(M)$ be an elliptic pseudo-differential operator of
order $m.$ Define

$A_{\lambda}:=A-\lambda I,$ (4.8)

where $\lambda$ belongs to $\Lambda,$ that is a closed angle in $\mathbb{C}$
with vertex at $0.$

If the principal symbol $a_{m}(x,\xi)$ of $A$ does not take values in
$\Lambda$ for all $\xi\neq 0,$ then
$A_{\lambda}\in\Psi^{m}_{\rho,\delta}(M,\Lambda)$ is parameter-elliptic of
order $m.$ So, there is $R=R_{A}>0,$ such that for $|\lambda|\geq R,$

$A_{\lambda}:H^{s}(M)\rightarrow H^{s}(M)$

is an invertible operator and
$A_{\lambda}^{-1}\in\Psi^{-m}_{\rho,\delta}(M,\Lambda_{R}).$

Now, we present the construction of a holomorphic family of operators $A^{z},$
$z\in\mathbb{C},$ and to this family we will called the complex powers of the
elliptic pseudo-differential operator $A,$ see Shubin [26, Chapter II].

######  Theorem 4.9 (Complex powers of elliptic operators).

Let $\varepsilon_{0}\geq 0.$ Let $A\in\Psi^{m}_{\rho,\delta}(M)$ be an
elliptic pseudo-differential operator of order $m>0,$ and let $a_{m}(x,\xi)$
be its principal symbol. Assume that

$a_{m}(x,\xi)-\lambda\neq
0,\,\lambda\in(-\infty,-\varepsilon_{0}),\,\,\sigma(A)\cap(-\infty,-\varepsilon_{0}]=\emptyset,$
(4.9)

that is the ray $L_{\varepsilon_{0}}:=(-\infty,-\varepsilon_{0}]$ belongs to
the resolvent set of $A.$

Then, for some closed angle $\Lambda(\varepsilon_{0})$ with vertex at
$-\varepsilon_{0}$ and containing $L_{0}$ in its interior,

$a_{m}(x,\xi)-\lambda\neq
0,\,\lambda\in\Lambda(\varepsilon_{0}),\,\sigma(A)\cap\Lambda(\varepsilon_{0})=\emptyset.$
(4.10)

Moreover, for $\lambda\in\Lambda(\varepsilon_{0}),$ $A_{\lambda}:=A-\lambda
I,$ is invertible on $H^{s}(M),$ for any $s\in\mathbb{R},$ parameter-elliptic
of order $m,$
$A_{\lambda}\in\Psi^{m}_{\rho,\delta}(M,\Lambda(\varepsilon_{0})),$
$A_{\lambda}^{-1}\in\Psi^{-m}_{\rho,\delta}(M,\Lambda(\varepsilon_{0})),$ and
the mapping

$z\in\mathbb{C}\mapsto A^{z}:=-\frac{1}{2\pi
i}\smallint\limits_{\partial\Lambda(\varepsilon_{0})}\lambda^{z}(A-\lambda
I)^{-1}d\lambda\in\Psi^{\textnormal{Re}(z)m}_{\rho,\delta}(M),$ (4.11)

is a holomorphic function from $\mathbb{C}$ into the family of bounded
operators $\mathscr{B}(H^{s}(M))$ when $\textnormal{Re}(z)\leq 0,$ and for
$\textnormal{Re}(z)>0,$ it is a holomorphic function from $\mathbb{C}$ into
the family of bounded operators
$\mathscr{B}(H^{s}(M),H^{s-m\textnormal{Re}(z)}(M)).$

######  Corollary 4.10 (Inverse of positive pseudo-differential operators).

Let Let $A\in\Psi^{m}_{\rho,\delta}(M)$ be a positive elliptic pseudo-
differential operator of order $m>0,$ and let $a_{m}(x,\xi)$ be its principal
symbol. Assume that

$a_{m}(x,\xi)-\lambda\neq 0,\,\lambda\in(-\infty,-\varepsilon_{0}).$ (4.12)

Define $A^{z}$ via the contour integral (4.11). Let
$E_{0}:=\textnormal{Ker}(A)$ and let $E_{0}^{\prime}$ be its orthogonal
complement in $L^{2}(M).$ Then,

  * •

$A^{z}(E_{0})=\\{0\\},$ and $A^{z}(E_{0}^{\prime})\subset E_{0}^{\prime},$ for
any $z\in\mathbb{C},$ such that $\textnormal{Re}(z)<0.$

  * •

For any $z,w\in\mathbb{C}$ such that
$\textnormal{Re}(z),\textnormal{Re}(w)<0,$ $A^{z+w}=A^{z}A^{w}.$

  * •

If $P_{0}:L^{2}(M)\rightarrow E_{0}$ is the orthogonal projection on the
subspace $E_{0},$ then $A^{0}=I-P_{0}.$

  * •

For any $k\in\mathbb{N},$ $k\geq 1,$ the operator

$A^{-k}:=-\frac{1}{2\pi
i}\smallint\limits_{\partial\Lambda(\varepsilon_{0})}\lambda^{-k}(A-\lambda
I)^{-1}d\lambda,$

is the inverse of the operator $A^{k}$ on $E_{0}^{\prime}$ and equals to the
null operator on $E_{0}.$ Summarising, $A^{-k}A^{k}=I-P_{0}.$

###  4.3. Ruzhansky-Turunen classes of pseudo-differential operators on the
torus

Let us consider the torus $\mathbb{T}^{n}\cong\mathbb{R}^{n}/\mathbb{Z}^{n}.$
In order to construct our periodization approach for the proof of the spectral
inequalities, we will use the Ruzhanksy-Turunen global calculus on the torus.

We will use the standard notation for this family of periodic pseudo-
differential operators taken from [25]. The discrete Schwartz space
$\mathcal{S}(\mathbb{Z}^{n})$ denote the space of discrete functions
$\phi:\mathbb{Z}^{n}\rightarrow\mathbb{C}$ verifying the estimate

$\forall M\in\mathbb{R},\exists C_{M}>0,\,|\phi(\xi)|\leq
C_{M}\langle\xi\rangle^{M},$ (4.13)

where $\langle\xi\rangle=(1+|\xi|^{2})^{\frac{1}{2}},$ denotes the Japanese
bracket of $\xi.$ The periodic Fourier transform is defined for any $u\in
C^{\infty}(\mathbb{T}^{n})$ by

$\widehat{u}(\xi)=\smallint\limits_{\mathbb{T}^{n}}e^{-i2\pi\langle
x,\xi\rangle}u(x)dx,\,\,\xi\in\mathbb{Z}^{n}.$

Here, $dx$ stands for the normalised Haar measure on the torus. The Fourier
inversion formula is given by

$u(x)=\sum_{\xi\in\mathbb{Z}^{n}}e^{i2\pi\langle
x,\xi\rangle}\widehat{u}(\xi),\,\,x\in\mathbb{T}^{n}.$

The periodic Hörmander class
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n}),\,\,0\leq\rho,\delta\leq
1,$ consists of those functions $a(x,\xi)$ which are smooth in
$(x,\xi)\in\mathbb{T}^{n}\times\mathbb{R}^{n}$ and which satisfy toroidal
symbols inequalities

$|\partial^{\beta}_{x}\partial^{\alpha}_{\xi}a(x,\xi)|\leq
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}.$ (4.14)

Note that symbols in $S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n})$
are symbols in $S^{m}_{\rho,\delta}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ (see
[25]) of order $m$ which are 1-periodic in $x.$ Then, if $a(x,\xi)\in
S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n}),$ the corresponding
pseudo-differential operator is defined by the quantisation formula

$a(X,D_{x})f(x)=\smallint\limits_{\mathbb{T}^{n}}\smallint\limits_{\mathbb{R}^{n}}e^{i2\pi\langle
x-y,\xi\rangle}a(x,\xi)f(y)d\xi dy.$ (4.15)

The class
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),\,0\leq\rho,\delta\leq
1,$ consists of those functions $a(x,\xi)$ which are smooth in
$x\in\mathbb{T}^{n},$ for all $\xi\in\mathbb{Z}^{n}$ and which satisfy the
symbol inequalities

$\forall\alpha,\beta\in\mathbb{N}^{n},\exists
C_{\alpha,\beta}>0,\,\,|\Delta^{\alpha}_{\xi}\partial^{\beta}_{x}a(x,\xi)|\leq
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}.$ (4.16)

The operator $\Delta$ is the standard difference operator defined in
$\mathbb{Z}^{n},$ [25]. The toroidal operator with symbol $a(x,\xi)$ is
defined as

$a(x,D)u(x)=\sum_{\xi\in\mathbb{Z}^{n}}e^{i2\pi\langle
x,\xi\rangle}a(x,\xi)\widehat{u}(\xi),\,\,u\in C^{\infty}(\mathbb{T}^{n}).$
(4.17)

######  Remark 4.11.

The corresponding class of operators with symbols in
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ (resp.
$S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n})$) will be denoted by
$\Psi^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),$ (resp.
$\Psi^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n})$).

There exists a process to interpolate the second argument of symbols on
$\mathbb{T}^{n}\times\mathbb{Z}^{n}$ in a smooth way to get a symbol defined
on $\mathbb{T}^{n}\times\mathbb{R}^{n}.$ It leads to the following
equivalence-of-classes-theorem.

######  Theorem 4.12.

Let $(\rho,\delta)\in[0,1]^{2}$ be such that $0\leq\delta\leq 1,$ $0<\rho\leq
1.$ Then, the symbol $a\in
S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ if only if there
exists an Euclidean symbol $a^{\prime}\in
S^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n})$ such that
$a=a^{\prime}|_{\mathbb{T}^{n}\times\mathbb{Z}^{n}}.$ Moreover, we have

$\Psi^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})=\Psi^{m}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{R}^{n}).$
(4.18)

Moreover, any $A\in\Psi^{0}_{\rho,\delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$
is bounded on $L^{2}(\mathbb{T}^{n}),$ and

$\|A\|_{\mathscr{B}(L^{2})}\lesssim\sup_{|\alpha|+|\beta|\leq[n/2]+1}\sup_{(x,\xi)}\langle\xi\rangle^{\rho|\alpha|-\delta|\beta|}|\Delta^{\alpha}_{\xi}\partial^{\beta}_{x}a(x,\xi)|.$
(4.19)

###### Proof.

The proof of (4.18) can be found in [25]. The proof of the $L^{2}$-estimate in
(4.19) can be found in [5]. ∎

###  4.4. Null-controllability of diffusion problems on Hilbert spaces

In this section we present a functional analysis tool that provides the null
controllability of fractional problems in the setting of Hilbert spaces. We
use the following notation:

  * •

the norm of a Hilbert space $H$ will be denoted by $\|\cdot\|$ without using
subscript.

Let $H$ be a separable Hilbert space and let $\mathcal{A}$ be a positive self-
adjoint operator with dense domain $\textnormal{Dom}(\mathcal{A})\subset{H}.$
In what follows, any Hilbert space will be identified with its topological
dual in the canonical way. Consider $H_{1}$ the Hilbert space obtained by
choosing on the domain $\textnormal{Dom}(\mathcal{A})$ the graph norm. We
extend $\\{e^{-t\mathcal{A}}:t>0\\}$ to a semigroup on the dual space
$H_{1}^{*}.$ Let $S$ be an observation operator from $H$ to a Hilbert space of
inputs $U,$ and let us consider the control operator
$B\in\mathscr{B}(U,H_{1}^{*})$ be its adjoint. Assume the following properties
on $S$ and $B.$

######  Assumption 4.13.

We assume that $B\in\mathscr{B}(U,H_{1}^{*})$ and that $S$ are such that, for
some $T>0$ (and hence, for any $T>0$), the following estimates hold.

  * •

There exists $K_{T}>0,$ such that

$\forall
v_{0}\in\textnormal{Dom}(\mathcal{A}),\,\,\smallint\limits_{0}^{T}\|Se^{-t\mathcal{A}}v_{0}\|^{2}\leq
K_{T}\|v_{0}\|^{2}.$ (4.20)

  * •

$\forall u\in
L^{2}_{\textnormal{loc}}(\mathbb{R},U),\,\|\smallint\limits_{0}^{T}e^{-t\mathcal{A}}Bu(t)\|^{2}dt\leq
K_{T}\smallint\limits_{0}^{T}\|u(t)\|^{2}dt.$ (4.21)

Assumption 4.13 identifies the necessary hypotheses for the existence and
uniqueness of the solution of the model

$\phi_{t}+\mathcal{A}\phi=Bu,\phi(0)=\phi_{0}\in{H},\,u\in
L^{2}_{\textnormal{loc}}(\mathbb{R},U).$ (4.22)

We precise this in the following result.

######  Proposition 4.14.

Under the hypothesis (4.20) and (4.21), for any input $u\in
L^{2}_{\textnormal{loc}}(\mathbb{R},U),$ there exists a unique solution $u\in
C(\mathbb{R}^{+}_{0},U)$ to (4.22) such that

$\phi(t)=e^{-t\mathcal{A}}\phi_{0}+\smallint\limits_{0}^{T}e^{(s-t)}Bu(s)ds.$
(4.23)

We precise the notion of null-controllability in the following definition.

######  Definition 4.15.

The model (4.22) is null-controllable in time $T>0,$ if for any initial state
$\phi_{0}\in H,$ there exists an input function $u\in
L^{2}_{\textnormal{loc}}(\mathbb{R}^{+}_{0},H)$ such that its solution (4.23)
satisfies $\phi(T)=0.$

######  Remark 4.16.

Let us consider the adjoint model to (4.22) without the source term, that is

$v_{t}+\mathcal{A}v=0.$ (4.24)

By duality, the null-controllability of (4.22) is equivalent to the following
observability inequality: there exists $C_{T}>0,$ such that

$\forall v_{0}\in H,\,\|e^{-T\mathcal{A}}v_{0}\|\leq
C_{T}\|Se^{-t\mathcal{A}}v_{0}\|_{L^{2}((0,T),U)}.$ (4.25)

The smallest constant $C_{T}>0$ is called the cost of controllability in time
$T>0$. Note that by the duality argument, the cost of controllability in time
$T>0,$ is the smallest constant $C_{T}>0$ satisfying that

$\forall\phi_{0}\in H,\exists u\textnormal{ in Definition \eqref{defi} such
that }\|u\|_{L^{2}((0,T),U)}\leq C_{T}\|\phi_{0}\|.$ (4.26)

The following theorem implies that a spectral inequality for the power
$\mathcal{A}^{\gamma}$ defined by the functional calculus of the operator
$\mathcal{A},$ is a sufficient condition for the null-controllability of the
model (4.22). We give the precise statement in the following way. Here,
$E_{\lambda}^{\mathcal{A}}:=E(0,\lambda):H\rightarrow H$ denotes the spectral
measure of the positive operator $\mathcal{A}.$

######  Theorem 4.17 (Miller [21]).

Assume that for some $\gamma\in(0,1),$ the fractional operator
$\mathcal{A}^{\gamma}$ satisfies the spectral inequality

$\forall\lambda>0,\,\forall u\in E_{\lambda}^{\mathcal{A}^{\gamma}}(H),\exists
d_{1},d_{2}>0,\,\,\,\|v\|\leq d_{1}e^{d_{2}\lambda}\|Sv\|.$ (4.27)

Then, the problem (4.22) is null-controllable in time $T>0.$ Moreover, the
controllability cost $C_{T}$ over short times $T,$ satisfies the inequality

$\forall\beta>\frac{\gamma}{1-\gamma},\,\exists C_{1},C_{2},\,\forall
T\in(0,1),\,C_{T}\leq C_{1}e^{C_{2}T^{-\beta}}.$ (4.28)

##  5\. Appendix II: Construction of the regularising function $\psi$

This appendix is dedicated to the construction of the regularising function
$\psi$ used in the proof of Proposition 2.1. For any $\varepsilon\in(0,1),$
let $a:={3\varepsilon}/4.$ Define the function

$E(t)=\begin{cases}e^{-\frac{1}{a^{2}-t^{2}}}(a^{2}-t^{2})^{10},&\text{
}t\in[0,a],\\\ 0,&\text{ }t\in[a,\varepsilon].\end{cases}$ (5.1)

By straightforward computation one can show that for any $t\in[0,a],$

  * •

$E_{t}(t)=2t\exp(-1/(a^{2}-t^{2}))(a^{2}-t^{2})^{8}(-10a^{2}+10t^{2}-1).$

  * •

$E_{tt}(t)=-2\exp(-1/(a^{2}-t^{2}))(a^{2}-t^{2})^{6}(10a^{6}+a^{4}(1-210t^{2})+a^{2}(390t^{4}-38t^{2})-190t^{6}+37t^{4}-2t^{2}).$

  * •

$E^{(3)}(t)=4t\exp(-1/(a^{2}-t^{2}))(a^{2}-t^{2})^{4}(270a^{8}+a^{6}(54-2520t^{2})+a^{4}(5940t^{4}-594t^{2}+3)-54a^{2}(100t^{6}-19t^{4}+t^{2})+t^{2}(1710t^{6}-486t^{4}+51t^{2}-2)).$

  * •

$E^{(4)}(t)=4\exp(-1/(a^{2}-t^{2}))(a^{2}-t^{2})^{2}(270a^{12}-54a^{10}(190t^{2}-1)+3a^{8}(22470t^{4}-954t^{2}+1)-12a^{6}t^{2}(14370t^{4}-1581t^{2}+25)+6a^{4}t^{2}(35235t^{6}-6714t^{4}+371t^{2}-2)-2a^{2}t^{4}(62730t^{6}-17415t^{4}+1782t^{2}-68)+t^{4}(29070t^{8}-10710t^{6}+1635t^{4}-124t^{2}+4)).$

These explicit formulae, allow us to write the first fourth derivatives of
$\psi$ in the form

$E^{(i)}(t)=\exp(-1/(a^{2}-t^{2}))(a^{2}-t^{2})^{10-2i}P_{i}(t,a),\,i\in\\{1,2,3,4\\}.$
(5.2)

where the functions $P_{i}(t,a)\in\mathbb{C}[t,a]$ are polynomials in two
variables. By evaluating the functions $E^{(i)},$ $i\in\\{0,1,2,3,4\\},$ at
$t=0,$ we get

  * •

$E(0)=a^{20}e^{-1/a^{2}}.$

  * •

$E^{\prime}(0)=0.$

  * •

$E^{(3)}(0)=0.$

  * •

$E^{(4)}(0)=4a^{4}(270a^{12}+54a^{10}+3a^{8})e^{-1/a^{2}}.$

Now, consider the function

$\tilde{\eta}(t):=E(t)(1-bt^{2}+ct^{4}),\,\,0\leq t\leq a,$ (5.3)

where $b$ and $c$ are real parameters. Then, straightforward computation shows
that for

  * •

$b=(E^{(2)}(0)-E(0))/2.$

  * •

$c=(6(E^{(2)}(0)-E(0))E^{(2)}(0)-E^{(4)}(0))/12E(0),$

the function $\tilde{\eta}$ satisfies the following properties

$\tilde{\eta}(0)=E(0),\,\eta^{(i)}(0)=0,\,\,i=1,2,3,4.$

Let $T>0.$ The analysis above shows that the function

$\displaystyle\psi(t):=\begin{cases}E(0)\tilde{\eta}(t-T),&\text{
}t\in[T,T+a],\\\ E(0){\tilde{\eta}}(0),&\text{ }t\in[0,T],\\\ 0,&\text{
}t\in[T+a,T+\varepsilon],\end{cases}$ (5.4)

satisfies the required properties in the proof of Proposition 2.1. We
summarise the analysis above and some their straightforward consequences in
the following lemma.

######  Lemma 5.1.

The function $\psi$ defined in (5.4) satisfies the following properties.

  * •

$0<\psi(0)<\varepsilon.$

  * •

$\psi^{(i)}(T)=0,$ for $i\in{\\{1,2,3,4\\}}.$

  * •

For $i\in{1,2,3,4},$ $\psi^{(i)}\in C^{\infty}(0,T+\varepsilon),$

and there is a constant $M_{0}>0,$ independent of $\varepsilon\in(0,1),$ such
that

$\|\psi^{(i)}\|_{L^{\infty}}\leq M_{0},$ (5.5)

for all $i=1,2,3,4.$

##  6\. Appendix III: Computing the inverse of $E(x,t,D,\partial_{t})$

Let us consider the operator $E(x,D)\geq cI$ of Proposition 2.1. We will
analyse the mapping properties of its inverse, when $E(x,D)$ is considered
continuously acting as follows:

$E(x,t,D,\partial_{t})=-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}}:H^{2}(M\times\mathbb{T}_{T,\varepsilon})\rightarrow
L^{2}(M\times\mathbb{T}_{T,\varepsilon}).$

Note that we have embedded the manifold $M_{T}$ (with lateral boundary
$\partial M_{T}=(M\times\\{0\\})\cup(M\times\\{T\\})$) on the closed manifold
$M\times\mathbb{T}_{T,\varepsilon}.$

In view of our assumptions, there exists $\varepsilon_{0}>0,$ such that the
principal symbol $E(x,\xi)$ of $E(x,D)$ and its spectrum $\sigma(A)$ satisfy

$\textnormal{(H): }E(x,\xi)-\lambda\neq
0,\,\lambda\in(-\infty,-\varepsilon_{0}),\,\,\sigma(E(x,D))\cap(-\infty,c/2]=\emptyset,$
(6.1)

that is, the ray $L_{\varepsilon_{0}}:=(-\infty,-\varepsilon_{0}]$ is
contained in the resolvent set of $E(x,D).$ Note that the principal symbol of
$-\partial_{t}^{2}+E(x,D)^{2/\nu}$ is given by

$\tau^{2}+E(x,\xi)^{2/\nu},$

and satisfies that

$\textnormal{(H'): }\tau^{2}+E(x,\xi)^{2/\nu}-\lambda\neq
0,\,\lambda\in(-\infty,-\varepsilon_{0}^{2/\nu}].$ (6.2)

Moreover, the spectral mapping theorem implies that

$\sigma\left(-\partial_{t}^{2}+E(x,D)^{2/\nu}\right)\cap(-\infty,-\varepsilon_{0}^{2/\nu}]=\emptyset,$

that is the ray

$L_{\varepsilon_{0}^{2/\nu}}:=(-\infty,-\varepsilon_{0}^{2/\nu}]$

is contained in the resolvent set of the operator
$-\partial_{t}^{2}+E(x,D)^{2/\nu}\in\Psi^{2}_{1,\delta}(M).$

This analysis implies that
$E(x,t,D,\partial_{t}):=-\partial_{t}^{2}+E(x,D)^{2/\nu}$ is parameter
elliptic of order two with respect to a closed angle
$\Lambda(\varepsilon_{0}^{2/\nu})$ of the complex plane with vertex at
$z_{0}:=-\varepsilon_{0}^{2/\nu},$ and containing the ray
$(-\infty,-\varepsilon_{0}^{2/\nu}].$ With the property

$A_{\lambda}:=E(x,t,D,\partial_{t})-\lambda
I\in\Psi^{2}_{1,\delta}(M,\Lambda(\varepsilon_{0}^{2/\nu})),\,\,\,A_{\lambda}^{-1}\in\Psi^{-2}_{1,\delta}(M,\Lambda(\varepsilon_{0}^{2/\nu})),$

in view of the existence-of-complex-powers Theorem 4.9, we have that

$z\in\mathbb{C}\mapsto G^{z}:=E(x,t,D,\partial_{t})^{z},$ (6.3)

is a holomorphic family of pseudo-differential operators, that maps any
$z\in\mathbb{C}$ into the class
$\Psi^{2\textnormal{Re}(z)}_{\rho,\delta}(M\times\mathbb{T}_{T,\varepsilon}),$
where

$G^{z}f(x)=-\frac{1}{2\pi
i}\smallint\limits_{\partial\Lambda\left(\varepsilon_{0}^{2/\nu}\right)}{\lambda^{z}}(E(x,t,D,\partial_{t})-\lambda
I)^{-1}f(x)d\lambda,\,\,f\in C^{\infty}(M\times\mathbb{T}_{T,\varepsilon}).$
(6.4)

In particular, with $z=-1,$ we have the inverse $G^{-1},$

$G^{-1}f(x)=-\frac{1}{2\pi
i}\smallint\limits_{\partial\Lambda\left(\varepsilon_{0}^{2/\nu}\right)}{\lambda^{-1}}(E(x,t,D,\partial_{t})-\lambda
I)^{-1}f(x)d\lambda,\,\,f\in C^{\infty}(M\times\mathbb{T}_{T,\varepsilon}),$
(6.5)

of $E(x,t,D,\partial_{t})$ on the orthogonal complement of its kernel. This
is, if $P_{0}$ is the orthogonal projection on the subspace
$\textnormal{Ker}(E(x,t,D,\partial_{t})),$ $G^{-1}G=I-P_{0},$ see Corollary
4.10. In view of the lower bound $E(x,D)\geq cI,$ we deduce that $P_{0}$ is
the null operator. Moreover, we have the following property.

######  Proposition 6.1.

Let $0<\varepsilon<1,$ and let us consider the operator norm

$B_{\varepsilon}=\|E(t,x,D,\partial_{t})^{-1}\|_{\mathscr{B}(L^{2}(M\times\mathbb{T}_{T,\varepsilon}),H^{2}(M\times\mathbb{T}_{T,\varepsilon}))}.$

Then

$B:=\sup_{0<\varepsilon<1}B_{\varepsilon}\leq 1+1/c,$ (6.6)

where $c>0$ in the constant is the positivity condition $E(x,D)\geq cI$ of
Proposition 2.1.

###### Proof.

Let us consider the orthogonal basis of $L^{2}(\mathbb{T}_{T,\varepsilon})$
formed by the exponential functions

$t\in\mathbb{T}_{T,\varepsilon}\mapsto\tilde{e}^{\varepsilon}_{k}(t)=\exp\left(\frac{2\pi
itk}{2(T+\varepsilon)}\right),\,\,k\in\mathbb{Z},$ (6.7)

and let us consider the $L^{2}$-normalised system of
$2(T+\varepsilon)$-periodic eigenfunctions

${e}^{\varepsilon}_{k}:=\tilde{e}^{\varepsilon}_{k}/\sqrt{2(T+\varepsilon)},$

of the Laplacian $-\partial_{t}^{2}.$ The corresponding eigenvalues of
$-\partial_{t}^{2}$ are given by

$\mu_{k,\varepsilon}=\frac{4\pi^{2}k^{2}}{4(T+\varepsilon)^{2}}=\left(\frac{\pi
k}{T+\varepsilon}\right)^{2},\,k\in\mathbb{Z}.$

Since $\\{{e}^{\varepsilon}_{k}\otimes\rho_{j}\\}$ is a basis for
$L^{2}(M\times\mathbb{T}_{T,\varepsilon})$ the spectrum of the operator
$-\partial_{t}^{2}+E(x,D)^{\frac{2}{\nu}}$ is determined by the sequence

$\mu_{k,\varepsilon}+\lambda_{j}^{2}=\left(\frac{\pi
k}{T+\varepsilon}\right)^{2}+\lambda_{j}^{2},\,\,k\in\mathbb{Z},\,j\in\mathbb{N}_{0}.$

Since $E(x,D)\geq cI,$ we have the eigenvalue inequality $\lambda_{k}\geq c,$
and then for any $f\in L^{2}(M\times\mathbb{T}_{T,\varepsilon})$ we have that

$\displaystyle\|E(t,x,D,\partial_{t})^{-1}f\|_{H^{2}}^{2}$
$\displaystyle=\left\|\sum_{k,j}(\mu_{k,\varepsilon}+\lambda_{j}^{2})^{-1}(f,{e}^{\varepsilon}_{k}\otimes\rho_{j}){e}^{\varepsilon}_{k}\otimes\rho_{j}\right\|_{H^{2}}^{2}$
$\displaystyle=\sum_{k,j}(1+\mu_{k,\varepsilon}+\lambda_{j}^{2})^{2}(\mu_{k,\varepsilon}+\lambda_{j}^{2})^{-2}|(f,{e}^{\varepsilon}_{k}\otimes\rho_{j})|^{2}$
$\displaystyle\leq\left(\frac{1}{c}+1\right)^{2}\sum_{k,j}|(f,{e}^{\varepsilon}_{k}\otimes\rho_{j})|^{2}=\left(\frac{1}{c}+1\right)^{2}\|f\|_{L^{2}}^{2}.$

From the previous analysis we deduce that $B\leq 1+\frac{1}{c}$ as desired. ∎

##  7\. Appendix IV: $L^{2}$-theory of the operator
$(-\partial_{t}^{2}-\Delta_{g})E(x,t,D,\partial_{t})^{-1}$ and the Calderón-
Vaillancourt theorem

The Calderón-Vaillancourth theorem is a sharp $L^{2}$-estimate for pseudo-
differential operators. We have applied it in the proof of our spectral
inequality. Indeed, let us consider the operator $E(x,D)\geq cI$ of
Proposition 2.1. Observe that

$(-\partial_{t}^{2}-\Delta_{g})\in\Psi^{2}_{1,0}(M\times\mathbb{T}_{T,\varepsilon}),$

belongs to the Hörmander class of order 2 and the inverse

$E(x,t,D,\partial_{t})^{-1}\in\Psi^{-2}_{1,\delta}(M\times\mathbb{T}_{T,\varepsilon}).$

The pseudo-differential calculus implies that

$F(x,t,D,\partial_{t}):=(-\partial_{t}^{2}-\Delta_{g})E(x,t,D,\partial_{t})^{-1}\in\Psi^{0}_{1,\delta}(M\times\mathbb{T}_{T,\varepsilon}).$

The celebrated Calderón-Vaillancourt theorem implies that
$F(x,t,D,\partial_{t})$ is a bounded operator on
$L^{2}(M\times\mathbb{T}_{T,\varepsilon}).$ Note that in any local coordinate
system $U\subset M,$ where $U$ then, is diffeomorphic to an open subset
$\tilde{U}\subset\mathbb{R}^{n},$ the operator has the form

$F(x,t,D,\partial_{t})u(t,x)=\sum_{k\in\mathbb{Z}^{n}}\smallint\limits_{\mathbb{R}^{n}}e^{\frac{i2\pi(t,k)}{2(T+\varepsilon)}+2\pi(x,\xi)}\sigma(t,x,k,\xi)\widehat{u}(k,\xi)d\xi,$
(7.1)

where, for any $u\in
C^{\infty}_{0}(\mathbb{T}_{T,\varepsilon}\times\tilde{U}),$ the Fourier
transform of $u$ is defined by

$\widehat{u}(k,\xi)=\smallint\limits_{\mathbb{T}_{T,\varepsilon}}\smallint\limits_{\mathbb{R}^{n}}e^{-\frac{i2\pi(t,k)}{2(T+\varepsilon)}-2\pi(x,\xi)}{u}(t,x)dtdx,\,k\in\mathbb{Z},\,\xi\in\mathbb{R}^{n}.$
(7.2)

For the toroidal variable, we have used the toroidal Ruzhansky-Turunen
calculus, see Subsection 4.3 or [25] for details. From now, $\Delta_{k}$
denotes the difference operator on a lattice. Note that the principal symbol
of the operator $F(x,t,D,\partial_{t})$ viewed on $\tilde{U}$ is given by

$\sigma_{0}(t,x,k,\xi)=\frac{\left(\frac{\pi
k}{T+\varepsilon}\right)^{2}+4\pi^{2}|\xi|^{2}}{\left(\frac{\pi
k}{T+\varepsilon}\right)^{2}+E(x,\xi)^{\frac{2}{\nu}}},\,\,|\xi|\geq 1,$ (7.3)

where $E(x,\xi)^{\frac{2}{\nu}}$ denotes the symbol of
$E(x,D)^{\frac{2}{\nu}},$ when microlocalised to the open $\tilde{U}.$ Note
that the ellipticity of $E(x,D)^{\frac{2}{\nu}},$ that is

$C_{1}|\xi|^{2}\leq E(x,\xi)^{\frac{2}{\nu}}\leq C_{2}|\xi|^{2},\,|\xi|\geq
1,$

and the positivity condition $E(x,\xi)\geq c,$ implies that the symbol
$\sigma_{0}$ satisfies the estimates

$\tilde{C}_{1}\leq|\sigma_{0}(t,x,k,\xi)|\leq\tilde{C}_{2},$

with $C_{1}$ and $C_{2}$ independent of $\varepsilon\in(0,1).$ Moreover, a
detailed analysis of the derivatives of $\sigma_{0},$ shows that the symbol
inequalities

$|\partial_{x,t}^{\beta}\partial_{\xi}^{\alpha}\Delta_{k}^{\gamma}\sigma_{0}(t,x,k,\xi)|\leqslant
C_{\alpha,\beta,\gamma,K}(1+|k|+|\xi|)^{-|\alpha|-|\gamma|+\delta|\beta|},$

hold true uniformly in $\varepsilon\in(0,1),$ $x\in K$ for all
$\xi\in\mathbb{R}^{n},$ where $\Delta_{k}$ stands for the standard difference
operator on $\mathbb{Z}.$ Since the Calderón-Vaillancourt estimates the
$L^{2}$-boundendess of $F$ when it is microlocalized to $\tilde{U},$ in terms
of the constants $C_{\alpha,\beta,\gamma,K}$ and of $\tilde{C}_{2},$ (see
Remark 4.6) that is, for any $u\in
C^{\infty}_{0}(\mathbb{T}_{T,\varepsilon}\times\tilde{U}),$

$\|F(x,t,D,\partial_{t})u\|_{L^{2}}\leq\left(\sup_{|\alpha|+|\beta|+|\gamma|\leq[\frac{n+1}{2}]+1}{\\{C_{\alpha,\beta,\gamma,K},\tilde{C_{2}}\\}}\right)\|u\|_{L^{2}}.$
(7.4)

Since $M$ is a compact manifold, the estimate in (7.4) can be extended from
local coordinate systems of $M$ to the whole manifold. We have proved the
following lemma.

######  Lemma 7.1.

Let $0<\varepsilon<1,$ and let us consider the operator norm

$C_{\varepsilon}=\|(-\partial_{t}^{2}-\Delta_{g})E(t,x,D,\partial_{t})^{-1}\|_{\mathscr{B}(L^{2}(M\times\mathbb{T}_{T,\varepsilon}))}.$

Then

$C:=\sup_{0<\varepsilon<1}C_{\varepsilon}<\infty.$ (7.5)

Moreover, there is a finite family of compact subsets
$K_{1},K_{2},\cdots,K_{\ell}$ covering $M,$ and such that

$C\lesssim\left(\max_{1\leq
i\leq\ell}\sup_{|\alpha|+|\beta|+|\gamma|\leq[\frac{n+1}{2}]+1}{\\{C_{\alpha,\beta,\gamma,K_{i}},\tilde{C_{2}}\\}}\right).$
(7.6)

###### Proof.

Let $U_{i}$ be an atlas on $M$ in such a way that $M$ can be covered by a
finite family of compact subsets $K_{i}\subset U_{i}.$ Using a partition of
the unity $\phi_{i}$ subordinated to the $U_{i},$’s with
$\textnormal{supp}(\psi_{i})\subset K_{i},$ we can microlocalise the operator
$F(x,t,D,\partial_{t})$ on any $K_{i}.$ Using (7.4), we have that

$C_{\varepsilon}=\|(-\partial_{t}^{2}-\Delta_{g})E(t,x,D,\partial_{t})^{-1}\|_{\mathscr{B}(L^{2}(M\times\mathbb{T}_{T,\varepsilon}))}\lesssim\sum_{i=1}^{\ell}\left(\sup_{|\alpha|+|\beta|+|\gamma|\leq[\frac{n+1}{2}]+1}{\\{C_{\alpha,\beta,\gamma,K_{i}},\tilde{C_{2}}\\}}\right)$
$\lesssim\ell\left(\max_{1\leq
i\leq\ell}\sup_{|\alpha|+|\beta|+|\gamma|\leq[\frac{n+1}{2}]+1}{\\{C_{\alpha,\beta,\gamma,K_{i}},\tilde{C_{2}}\\}}\right).$

Thus, the proof of Lemma 7.1 is complete. ∎

##  8\. Proof of the Donnelly-Fefferman type inequality for pseudo-
differential operators

###### Proof of Theorem 1.5.

For the proof of (1.5) we can use (1.3) and the Sobolev embedding theorem.
Indeed, let $R>0$ and let us consider $s\in\mathbb{R}$ such that $s>n/2.$ With
$\omega=B(x,R)$ a ball of radius $R>0$ we have that

$\|\varkappa\|_{L^{\infty}(B(x,2R))}\leq\|\varkappa\|_{L^{\infty}(G)}.$ (8.1)

Now, the Sobolev embedding theorem and the inequality in (1.3) imply that

$\displaystyle\|\varkappa\|_{L^{\infty}(G)}\lesssim\|(1+E(x,D))^{\frac{s}{\nu}}\varkappa\|_{L^{2}(G)}\lesssim(1+\lambda)^{s}\|\varkappa\|_{L^{2}(G)}$
$\lesssim(1+\lambda)^{s}C_{1,R}e^{C_{2,R}\lambda}\|\varkappa\|_{L^{2}(B(x,R))}.$

By using (8.1) we conclude this analysis with the inequality

$\displaystyle\|\varkappa\|_{L^{\infty}(B(x,2R))}\lesssim\|\varkappa\|_{L^{\infty}(G)}\leq
e^{C_{2,R}^{\prime}+C_{1,R}^{\prime}\lambda}\|\varkappa\|_{L^{2}(B(x,2R))}\leq
e^{C_{2,R}^{\prime}+C_{1,R}^{\prime}\lambda}\|\varkappa\|_{L^{\infty}(B(x,2R))},$

for some $C_{1,R}^{\prime}>C_{1,R}$ and $C_{2,R}^{\prime}>C_{2,R}.$ The proof
of Theorem 1.2 is complete. ∎

Acknowledgments. I would like to thank Enrique Zuazua and Umberto Biccari for
discussions about an earlier version of this work.

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