# The $L^{2}$-unique continuation property on manifolds with bounded geometry
and the deformation operator

Nadine Große  Mathematisches Institut, Universität Freiburg, 79104 Freiburg,
Germany<EMAIL_ADDRESS>,  Mirela Kohr  Faculty of
Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu
Str., 400084 Cluj-Napoca, Romania<EMAIL_ADDRESS>and  Victor Nistor
Université de Lorraine, UFR MIM, Ile du Saulcy, CS 50128, 57045 METZ, France
and Inst. Math. Romanian Acad. PO BOX 1-764, 014700 Bucharest Romania
<EMAIL_ADDRESS>In memory of Professor Gabriela Kohr, with deep
respect

###### Abstract.

A differential operator $T$ satisfies the $L^{2}$-unique continuation property
if every $L^{2}$-solution of $T$ that vanishes on an open subset vanishes
identically. We study the $L^{2}$-unique continuation property of an operator
$T$ acting on a manifold with bounded geometry. In particular, we establish
some connections between this property and the regularity properties of $T$.
As an application, we prove that the deformation operator on a manifold with
bounded geometry satisfies regularity and $L^{2}$-unique continuation
properties. As another application, we prove that suitable elliptic operators
are invertible (Hadamard well-posedness). Our results apply to compact
manifolds, which have bounded geometry.

###### Key words and phrases:

Sobolev spaces, elliptic and uniformly strongly elliptic operators,
invertibility, Hadamard well-posedness, regularity estimates, unique
continuation, Fredholm operators.

###### 2000 Mathematics Subject Classification:

Primary 35R01; Secondary 35B65, 76M99.

M.K. has been partially supported by the Babeş-Bolyai University Grant
SEED/2022. N.G. has been partially supported by SPP2026 (DFG)

###### Contents

  1. 1 Introduction
  2. 2 Background material and notation
    1. 2.1 Vector bundles, connections, and tensor products
    2. 2.2 Covariant Sobolev spaces
      1. 2.2.1 Positive integer order covariant Sobolev spaces
      2. 2.2.2 Negative order covariant Sobolev spaces
    3. 2.3 Differential operators
  3. 3 Manifolds with bounded geometry
    1. 3.1 Definition of bounded geometry
    2. 3.2 Sobolev spaces via coverings and partitions of unity
  4. 4 Regularity estimates for second order strongly elliptic operators
    1. 4.1 Regularity conditions
    2. 4.2 Consequences of regularity
  5. 5 Application: $L^{2}$-unique continuation and the deformation operator
    1. 5.1 An invertibility result
    2. 5.2 Vanishing of Killing vector fields
    3. 5.3 Applications to the deformation operator

##  1\. Introduction

Let $M$ be a Riemannian manifold and $E,F\to M$ be Hermitian vector bundles
and $T$ be differential operator acting from sections of $E$ to sections of
$F$. We shall study the following _weak form_ of the unique continuation
property (see below for references to the usual unique continuation property):

######  Definition 1.1.

We shall say that $T$ has the  $L^{2}$-unique continuation property if, given
$u\in L^{2}(M;E)$ that vanishes in an open subset $U\subset M$ and satisfies
$Tu\,=\,0$, then $u\,=\,0$ _everywhere_ on $M$.

In our results we shall assume that the manifold $M$ has _bounded geometry_
(Definition 3.3). The main results of this paper relate are:

  * •

Results relating the $L^{2}$-unique continuation property of $T$ to the
regularity properties of the operator $T$.

  * •

A proof that the deformation operator $\operatorname{Def}\,$ and the
deformation Laplacian
$\mathbf{L}:=2\operatorname{Def}^{*}\operatorname{Def}\,$ have the
$L^{2}$-unique continuation property.

  * •

A proof that certain PDEs (including some of the form $\mathbf{L}+V$) are
well-posed in the sense of Hadamard.

Let us first discuss _the connections between the $L^{2}$-unique continuation
property and regularity._ The result that we need and prove is the following
extension of a regularity result from [18], which in turn extends the
classical regularity result on compact manifolds (our convention is that
$\mathbb{N}=\\{1,2,\ldots\\}$ and $\mathbb{Z}_{+}=\mathbb{N}\cup\\{0\\}$):

######  Theorem 1.2.

Let $s,\rho\in\mathbb{Z}_{+}$, $s\leq\rho$, and $t\in\mathbb{R}$. Let $M$ be a
manifold with bounded geometry and $E,F\to M$ be two Hermitian vector bundles
with totally bounded curvature (Definition 3.1). Let $P$ be a second order,
uniformly strongly elliptic differential operator with coefficients in
$W^{\rho+3,\infty}$ acting on sections of $E$ to sections of $F$. Then there
exists $c_{s,t}>0$ such that, if $u\in H^{t}(M;E)$ and $Pu\in H^{s}(M;F)$,
then $u\in H^{s+2}(M;E)$ and

$\|u\|_{H^{s+2}(M;E)}\leq
c_{s,t}\left(\|Pu\|_{H^{s}(M;F)}+\|u\|_{H^{t}(M;E)}\right).$

Related results for _scalar differential operators with smooth coefficients_
were obtained in [29, 30] using the calculus of uniform pseudodifferential
operators on manifolds with bounded geometry. (More regularity results will be
included in [16], so we are not striving in this paper for the greatest
generality.) The proof of Theorem 1.2 is based on the regularity result in
[18] and a generalization of the classical Rellich-Kondrachov Theorem, which,
in the case of manifolds with bounded geometry, is an embedding result, see
Proposition 3.9. Our proof uses a description of Sobolev spaces using
partitions of unity [19], which we also recall. The regularity result allows
us to prove that certain differential operators are closed or that they are
self-adjoint.

The main application in this paper is to use Theorem 1.2 to prove that the
_deformation operator_ on a manifold with bounded geometry satisfies
_regularity and $L^{2}$-unique continuation properties._ Another application
is to prove that certain elliptic operators are invertible (i.e. are well-
posed in the sense of Hadamard). In turn, these results on the deformation
operator are useful for the study of the Stokes system on manifolds with
bounded geometry [27].

As we have alluded to above, this paper uses a weaker version of the unique
continuation property than the usual one. Thus, in fact, this paper is more
about regularity, self-adjointness, and well-posedness of elliptic operators
than about their unique continuation property. The unique continuation
property (in its strong form) is a _very important topic_ in mathematics. Its
origins go back at least to classical results of Holmgren and Carleman at the
very beginning of the twentieth century. It is hard to give a complete list of
references. See however the following very incomplete lists [23, 32, 33, 42,
44, 45, 47] for some general references as well as the following research
papers [8, 11, 13, 14, 24, 25, 31, 34, 43] and the references therein for more
information.

The paper is organized as follows. In Section 2, we set up notation and we
recall the definition of the Sobolev spaces and of the differential operators
that we will use. That sections contains only standard material. In Section 3,
we recall the definition of manifolds with bounded geometry as well as the
equivalent definition of Sobolev spaces on such manifolds using suitable
partitions of unity. We also prove our Rellich-Kondrachov embedding result,
Proposition 3.9. Section 4 is devoted to Theorem 1.2 and some direct
applications. Section 5 contains our application, which is that the
deformation operator $\operatorname{Def}\,$ and the deformation Laplacian
$\mathbf{L}\mathrel{\ :=\ }2\operatorname{Def}^{*}\operatorname{Def}\,$ have
the $L^{2}$-unique continuation property. We also include there an
invertibility result for certain classes of operators that include certain
operators of the form $\mathbf{L}+V$.

##  2\. Background material and notation

We begin with some background material, for the benefit of the reader. More
precisely, we introduce here the needed function spaces and their associated
differential operators with the definitions used in this paper. Since we are
interested in systems, all our functions spaces and differential operators
will be defined using sections of vector bundles. We also use this opportunity
to fix the notation. This section contains no new results. See [18] and [26]
for more details.

_Throughout this paper, $M$ will be a (usually non-compact) connected smooth
Riemannian manifold without boundary with metric $g$._ We use the convention
that ${\mathbb{Z}}_{+}=\\{0,1,2,\ldots,\\}$ and
${\mathbb{N}}={\mathbb{Z}}_{+}\smallsetminus\\{0\\}$.

###  2.1. Vector bundles, connections, and tensor products

Let $E\to M$ denote a smooth (real or complex) vector bundle endowed with a
fiberwise metric $(\,,\,)_{E}$ over our Riemannian manifold $M$ and let
${\mathcal{C}}^{\infty}(M;E)$ be the set of _smooth_ sections of $E$. Unless
otherwise stated, we assume that $E$ is endowed with a metric preserving
connection

$\nabla^{E}\colon{\mathcal{C}}^{\infty}(M;E)\to{\mathcal{C}}^{\infty}(M;E\otimes
T^{*}M)\,.$

Let $F\to M$ be another vector bundle with connection $\nabla^{F}$. Then
$E\otimes F$ will be endowed with the product connection $\nabla^{E\otimes
F}(\xi\otimes\eta)\mathrel{\ :=\
}(\nabla^{E}\xi)\otimes\eta+\xi\otimes\nabla^{F}\eta$. We will use the
notation $V^{\otimes k}\mathrel{\ :=\ }V\otimes V\otimes\ldots\otimes V$ for
the tensor product of $V$ with itself of $k$-times. In particular, all the
tensor product bundles $E\otimes T^{*\otimes k}M\mathrel{\ :=\
}E\otimes(T^{*}M)^{\otimes k}$ will be endowed with the induced tensor product
connections from $E$ and from the Levi-Civita connection on $TM$. (Recall that
the Levi-Civita connection on $TM$ is the unique metric preserving torsion
free connection on $TM$ [37].)

When there is no risk of confusion, we shall drop the notation for the vector
bundle $E$ from the notation of its connection $\nabla^{E}$ and write simply
$\nabla$ instead.

###  2.2. Covariant Sobolev spaces

Let us recall now the definition of the covariant Sobolev spaces (or
$\nabla$-Sobolev spaces) on Riemannian manifolds. These spaces are defined
using connections, see, for instance [5, 21, 22, 26].

####  2.2.1. Positive integer order covariant Sobolev spaces

Let $p\in[1,\infty]$ and $E\to M$ be a given Hermitian vector bundle with
$\|\cdot\|_{E}$ denoting the fiberwise norm on $E$. Let
$\,d\operatorname{vol}_{g}$ denote the induced volume form on $M$ (defined
using the given metric $g$ on $M$). We then have the usual formula for the
$L^{p}$-norm:

$\|u\|_{L^{p}(M;E)}\mathrel{\ :=\ }\,\begin{cases}\
\left(\int_{M}\|u(x)\|_{E}^{p}\,d\operatorname{vol}_{g}(x)\right)^{1/p}\,,&\quad\mbox{if
}\ p<\infty\\\ \ \,\operatorname{ess-sup}_{x\in
M}\,\|u(x)\|_{E}\,,&\quad\mbox{if }\ p=\infty\,.\end{cases}$

Identifying two sections of $E$ that are equal on $M$ except on a zero measure
set, we then obtain the usual Lebesgue spaces

$L^{p}(M;E)\mathrel{\ :=\ }\\{u\colon M\to
E\mid\|u\|_{L^{p}(M;E)}<+\infty\\}/\ker(\|\cdot\|_{L^{p}(M;E)})\,.$

Let $(f,g)=(f,g)_{L^{2}(M;E)}$ denote the scalar (inner) product on
$L^{2}(M;E)$. It is linear in the first variable and conjugate linear in the
second one.

######  Definition 2.1.

Let $E\to M$ be a finite dimensional, Hermitian vector bundle with metric
preserving connection $\nabla=\nabla^{E}$. Let $k\in{\mathbb{Z}}_{+}$ and
$p\in[1,\infty]$. Then

$W^{k,p}(M;E)\mathrel{\ :=\ }\\{u\in L^{p}(M;E)\mid\nabla^{j}u\in
L^{p}(M;E\otimes T^{*\otimes j}M)\,,\,\mbox{ for }\,1\leq j\leq k\\}$

is the _order $k$, $L^{p}$-type covariant Sobolev space_ of sections of $E$
with the norm

$\|u\|_{W^{k,p}(M;E)}\mathrel{\ :=\ }\ell^{p}\mbox{--norm of
}\\{\|\nabla^{j}(u)\|_{L^{p}(M;E\otimes T^{*\otimes j}M)}\,,\ 0\leq j\leq
k\\}\,.$

We shall write also $H^{k}(M;E)\mathrel{\ :=\ }W^{k,2}(M;E)$.

The adjective “covariant” in “covariant Sobolev spaces” (a term coined in
[26]) serves to remind us that these spaces depend on the choice of the
covariant derivative $\nabla^{E}$ on the coefficient bundle $E$. In [26], the
covariant Sobolev spaces are also called $\nabla$-Sobolev spaces. Since we use
only canonical connections and work on manifolds with bounded geometry, these
spaces coincide with the usual Sobolev spaces, so we shall drop the connection
$\nabla$ or the adjective “covariant” when talking about these spaces.

####  2.2.2. Negative order covariant Sobolev spaces

We proceed as usual and we define the negative order (covariant) Sobolev
spaces for $p\in[1,\infty)$ by

$W^{-k,p^{\prime}}(M;E^{*})\mathrel{\ :=\ }W^{k,p}(M;E)^{*}\,,$

where $V^{*}$ is the complex conjugate dual of $V$, $k\in\mathbb{N}$, and
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$. For simplicity, in the following, we
shall identify $E^{*}$ with $E$ using the Hermitian metric on $E$, so
$W^{-k,p^{\prime}}(M;E^{*})=W^{-k,p^{\prime}}(M;E)$. We set
$W^{\infty,p}(M;E)\mathrel{\ :=\ }\cap_{k\in\mathbb{N}}W^{k,p}(M;E)$ (which we
shall use only for $p=\infty$ in this paper). The spaces $H^{s}(M;E)$,
$s\in\mathbb{R}$, are defined by complex interpolation.

###  2.3. Differential operators

Let $E,F\to M$ be vector bundles, with $E$ endowed with a connection. A
_differential operator acting on sections of $E$ with values sections of $F$_
is an expression of the form

(2.1) $Pu\mathrel{\ :=\ }\sum_{j=0}^{\mu}a_{j}\nabla^{j}u\,,$

with $a_{j}$ a section of $\operatorname{Hom}(E\otimes T^{*\otimes j}M;F)$.
Let us assume $E=F$ in what follows, for simplicity. Then
$\operatorname{Hom}(E\otimes T^{*\otimes
j}M;E)\simeq\operatorname{End}(E)\otimes TM^{\otimes j}$, which simplifies the
notation, so, in the following, we shall identify these spaces. Let
$\rho\in\mathbb{Z}_{+}\cup\\{\infty\\}$. We shall say that the operator $P$ of
Equation (2.1) _has coefficients in_ $W^{\rho,\infty}$ if $a_{j}\in
W^{\rho,\infty}(M;\operatorname{End}(E)\otimes TM^{\otimes j})$ for all $0\leq
j\leq\mu$. A differential operator $P=\sum_{j=0}^{\mu}a_{j}\nabla^{j}$ with
coefficients in $W^{\rho,\infty}$ defines a _continuous map_

$\displaystyle P\,=\,\sum_{j=0}^{\mu}a_{j}\nabla^{j}\colon
W^{s+\mu,p}(M;E)\,\to\,W^{s,p}(M;E),\quad 0\leq s\leq\rho\ \mbox{ and }\
p\in[1,\infty]\,.$

See [5, 18, 26] for details and further results. It will be convenient to use
the following notation for our sets of differential operators.

######  Definition 2.2.

Let $\mu\in\mathbb{Z}_{+}$, $\rho\in\mathbb{Z}_{+}\cup\\{\infty\\}$, and let
$E,F\to M$ be Hermitian vector bundles with metric preserving connections.
Then $\mathcal{D}^{\mu,\rho}(M;E,F)$ will denote the set of differential
operators of order $\leq\mu$ with coefficients in $W^{\rho,\infty}$ acting
from sections of $E$ to sections of $F$. We let
$\mathcal{D}^{\mu,\rho}(M;E)\mathrel{\ :=\ }\mathcal{D}^{\mu,\rho}(M;E,E)$.

In [26], the differential operators introduced in this section were also
called “$\nabla$-differential operators.” Locally, there is no difference
between the $\nabla$-differential operators and the “usual” differential
operators. In particular, the principal symbol is defined in the same way.

##  3\. Manifolds with bounded geometry

We now recall some basic material on manifolds with bounded geometry following
[5], to which we refer for more details and references. Once we will have
recalled the definition of manifolds with bounded geometry, we will start
assuming that our given Riemannian manifold $(M,g)$ is of bounded geometry.

###  3.1. Definition of bounded geometry

We now recall the definition of a “manifold with bounded geometry.” Let $M$ be
a Riemannian manifold with metric $g$, as usual. We let
$\operatorname{dist}_{g}(x,y)$ denote the distance between two points $x,y\in
M$.

######  Definition 3.1.

A Hermitian vector bundle $E\to M$ with metric preserving connection is said
to have _totally bounded curvature_ if its curvature $R^{E}$ and all covariant
derivatives $\nabla^{k}R^{E}$, $k\in\mathbb{N}$, are bounded on $M$. Also, we
shall say that $M$ has _totally bounded curvature_ if the tangent bundle
$TM\to M$ has totally bounded curvature.

We shall use the following standard notation.

######  Notation 3.2.

Let us assume $M$ to be complete in its metric $g$, for simplicity.

  1. (i)

$\exp_{x}^{M}\colon T_{x}M\to M$ will be the geodesic exponential map at $x$.

  2. (ii)

$B_{x}^{M}(r)\mathrel{\ :=\ }\\{y\in M\mid\operatorname{dist}_{g}(x,y)<r\\}$,
the ball of radius $r$ in $M$ centered at $x$.

  3. (iii)

$\mathop{r_{\mathrm{inj}}}(x)\\!\mathrel{\ :=\
}\\!\\!\sup\\{r\mid\exp_{x}^{M}\colon B_{r}^{T_{x}M}(0)\to M\text{ is a
diffeomorphism onto its image}\\}$.

  4. (iv)

$\mathop{r_{\mathrm{inj}}}(M)\mathrel{\ :=\ }\inf_{x\in
M}\mathop{r_{\mathrm{inj}}}(x)$, the _injectivity radius_ of $M$.

The concept of manifolds with bounded geometry (see, for instance, [7, 4, 10,
30, 38, 41, 39, 48]) is classical and fundamental for this paper.

######  Definition 3.3.

Let $(M,g)$ be a boundaryless Riemannian manifold. Then $M$ has bounded
geometry if it has totally bounded curvature and
$\mathop{r_{\mathrm{inj}}}(M)>0$ (see 3.2).

We shall need the following remark.

######  Remark 3.4.

Our definition of differential operators coincides with the one in [29, 30,
38] for manifolds with bounded geometry since in any normal coordinate chart
defined on a ball of radius $r<\mathop{r_{\mathrm{inj}}}(M)$, the coefficient
of our differential operator can be expressed in terms of the Christoffel
coefficients and of the bundle morphisms $a_{j}$ of Equation 2.1 and hence
they and their derivatives are uniformly bounded. This will be discussed in
greater detail in our forthcoming paper [17].

Of course, a smooth, compact manifold (without boundary) will have bounded
geometry. Another useful class of manifolds with bounded geometry is the class
of manifolds with cylindrical ends, see Figure 1 for an example of such a
manifold.

Figure 1. A manifold with cylindrical ends

Some of the results that hold for these manifolds (compact or with cylindrical
ends) hold also for manifolds with bounded geometry, most notably the ones
related regularity, which are exactly the results that we will need next.
Before discussing regularity, we need to introduce, however, an alternative
definition of Sobolev spaces on manifolds with bounded geometry.

###  3.2. Sobolev spaces via coverings and partitions of unity

_From now on, we will assume that $M$ is a manifold with bounded geometry and
that the vector bundles $E,F\to M$ have totally bounded curvature._ With these
assumptions, we now recall an alternative definition of the Sobolev spaces on
$M$ with coefficients in $E$, see [19, 48]. See also [6].

Let ${n}$ denote the dimension of $M$. We will write $B_{r}^{{n}}(x)$ instead
of $B_{r}^{\mathbb{R}^{{n}}}(x)$, for simplicity. Identifying the space
$T_{x}M$ with $\mathbb{R}^{{n}}$ via some choice of an orthonormal basis, we
obtain, for $r<\mathop{r_{\mathrm{inj}}}(M)$, the _geodesic normal
coordinates_ diffeomorphisms

(3.1) $\kappa_{x}\mathrel{\ :=\ }\exp^{M}_{x}\colon B^{{n}}_{r}(0)\to
B^{M}_{r}(x)\subset M\,.$

######  Definition 3.5.

Let $M$ be a manifold with bounded geometry and let
$0<r\leq\frac{1}{2}\mathop{r_{\mathrm{inj}}}(M)$. An _$r$ -covering subset of
$M$_ is a subset $\\{x_{\gamma}\\}_{\gamma\in\mathbb{N}}\subset M$ that
satisfies the following conditions:

  1. (i)

$M\subset\cup_{\gamma\in\mathbb{N}}B_{r}^{M}(x_{\gamma})$.

  2. (ii)

For any $R>0$, there exists $N_{R}\in\mathbb{N}$ with the property that, for
any $y\in M$, the set
$\\{\gamma\in\mathbb{N}\mid\operatorname{dist}_{g}(x_{\gamma},y)<R\\}$
contains at most $N_{R}$ elements.

Note that it is always possible to construct an $r$-covering set
$\\{x_{\gamma}\\}$ and the corresponding covering
$\\{B_{r}^{M}(x_{\gamma})\\}$ of $M$ is strongly locally finite (see, for
instance, [6, 19] and the references therein). For further purposes, we need
the following class of partitions of unity defined using $r$-covering sets.

######  Definition 3.6.

A partition of unity $\\{\phi_{\gamma}\\}_{\gamma\in\mathbb{N}}$ of $M$ is
called _an $r$-uniform partition of unity associated to the $r$-covering set
$\\{x_{\gamma}\\}\subset M$_ (Definition 3.5) if, for each
$\rho\in\mathbb{Z}_{+}$, we have
$\sup_{\gamma}\|\phi_{\gamma}\|_{W^{\rho,\infty}(M)}<\infty$ and, for each
$\gamma\in\mathbb{N}$, the support of $\phi_{\gamma}$ is contained in
$B_{r}^{M}(x_{\gamma})$.

In order to deal with the coefficient vector bundles $E,F\to M$, we will also
need to locally trivialize these bundles.

######  Definition 3.7.

Let $M$ be a Riemannian manifold with bounded geometry, $x\in M$,
$r<\mathop{r_{\mathrm{inj}}}(M)$, and $E\to M$ be a Hermitian vector bundle
with metric preserving connection. Then the _synchronous trivialization_ of
$E$ above $B_{x}^{M}(r)$ is the vector bundle isomorphism $\xi_{x}\colon
E|_{B_{r}^{M}(x)}\to B_{r}^{M}(x)\times E_{x}$ induced by parallel transport
along the geodesic rays emanating from $x$. By abuse of notation, we shall
also denote by

$\xi_{x}\colon E|_{B_{r}^{M}(x)}\to B_{r}^{{n}}(0)\times\mathbb{C}^{\tau}$

the resulting isomorphism obtained using also the geodesic coordinates
diffeomorphism $\kappa_{x}\colon B_{r}^{{n}}(0)\to B_{r}^{M}(x)$ (of Equation
(3.1)) and some choice of an isomorphism $E_{x}\simeq\mathbb{C}^{\tau}$.

We shall need a definition of Sobolev spaces using partitions of unity [19]
and a few standard results. The following proposition is a direct consequence
of Theorems 14 and 26 in [19].

######  Proposition 3.8.

Let $M$ be a Riemannian manifold with bounded geometry. Let $E\to M$ be a
vector bundle over $M$ with totally bounded curvature. Let
$\\{\phi_{\gamma}\\}$ be an $r$-uniform partition of unity associated to an
$r$-covering set $\\{x_{\gamma}\\}\subset M$ (Definition 3.6) and let
$\kappa_{\gamma}\mathrel{\ :=\ }\kappa_{x_{\gamma}}\colon B_{r}^{{n}}(0)\to
B_{r}^{M}(x_{\gamma})$ and $\xi_{\gamma}\mathrel{\ :=\ }\xi_{x_{\gamma}}\colon
E|_{B_{r}^{M}(x)}\to B_{r}^{{n}}(0)\times\mathbb{C}^{\tau}$ be as in
Definition 3.7. Let $1<p<\infty$ and $s\in\mathbb{R}$. Then

$|||u|||_{s,p}^{p}\mathrel{\ :=\
}\sum_{\gamma}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}(B_{r}^{{n}}(0);\mathbb{C}^{\tau})}^{p}$

defines a norm equivalent to the norm on $W^{s,p}(M;E)$ introduced in
Definition 2.1 using connections.

Let $\|\cdot\|_{\ell^{p}}$ be the norm on $\ell^{p}(\mathbb{N})$. That is, if
$a=(a_{n})_{n\in\mathbb{N}}$ is a sequence of complex numbers, then

$\|a\|_{\ell^{p}}\,=\,\|(a_{n})_{n\in\mathbb{N}}\|_{\ell^{p}}\mathrel{\ :=\
}\begin{cases}\ \big{(}\sum_{n\in\mathbb{N}}|a_{n}|^{p}\big{)}^{1/p}&\mbox{ if
}p<\infty\\\ \ \sup_{n\in\mathbb{N}}|a_{n}|&\mbox{ if }p=\infty\,.\end{cases}$

Then $\|a\|_{p}$ is _decreasing_ in $p\in[1,\infty]$. It will be convenient to
also write

(3.2)
$|||u|||_{s,p}\,=\,\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}\,.$

with
$\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}}=\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}(B_{r}^{{n}}(0);\mathbb{C}^{\tau})}$.

The Rellich-Kondrachov compact embedding theorem [1, Theorem 6.3] (see also
[2, 3, 28, 46]) gives then the following extension to manifolds with bounded
geometry of the results of Rellich-Kondrachov and Lions. Related results were
obtained in [41] (but using weighted spaces) and in [15, Appendix A] for
$p=2$.

######  Proposition 3.9.

Let $p,p^{\prime},q\in(1,\infty)$, $s,s^{\prime}\in\mathbb{Z}_{+}$,
$t\in\mathbb{Z}$, satisfy the Rellich-Kondrachov conditions:

$t<s<s^{\prime}\quad\mbox{ and }\quad{\max\\{t,0\\}}-n/q\leq s-n/p\leq
s^{\prime}-n/p^{\prime}\,.$

Let also $M$ be a manifold with bounded geometry and $E\to M$ be a hermitian
vector bundle with totally bounded curvature.

  1. (i)

If also $p^{\prime}\leq p$, then we have
$W^{s^{\prime},p^{\prime}}(M;E)\subset W^{s,p}(M;E)$ (continuous embedding).

  2. (ii)

Assume furthermore that $\max\\{p^{\prime},q\\}\leq p$ and
$s-n/p<s^{\prime}-n/p^{\prime}$. Then, for every $\epsilon>0$, there exists
$C_{\epsilon}>0$ such that, for all $u\in W^{s^{\prime},p^{\prime}}(M;E)$, we
have

$\|u\|_{W^{s,p}(M;E)}\leq\epsilon\|u\|_{W^{s^{\prime},p^{\prime}}(M;E)}+C_{\epsilon}\|u\|_{W^{t,q}(M;E)}\,.$

###### Proof.

Both statements follows from the analogous statements over an open ball
$M_{0}\subset\mathbb{R}^{n}$ together with Proposition 3.8.

Let us see how this is done. We first review the case when $M$ is replaced
with an open ball $M_{0}$ in a euclidean spaces. We may assume then that $E$
is replaced with a trivial bundle $E_{0}\to M_{0}$. The first statement for
$(M_{0},E_{0})$ is then nothing but the classical Sobolev embedding, [1,
Theorem 5.4].

Let us recall next a proof of the second statement for $(M_{0},E_{0})$ (recall
that $M_{0}$ is an open ball). Let us assume that the statement is not true.
Let us fix $\epsilon>0$. Then, there exists a sequence $u_{n}\in
W^{s^{\prime},p^{\prime}}(M_{0},E_{0})$ such that

$\|u_{n}\|_{W^{s,p}(M_{0},E_{0})}>\epsilon\|u_{n}\|_{W^{s^{\prime},p^{\prime}}(M_{0},E_{0})}+n\|u_{n}\|_{W^{t,q}(M_{0},E_{0})}\,.$

We can assume that $\|u_{n}\|_{W^{s^{\prime},p^{\prime}}(M_{0},E_{0})}=1$. In
view of the Rellich-Kondrachov compactness theorem, the natural inclusions
$W^{s^{\prime},p^{\prime}}(M_{0},E_{0})\to W^{s,p}(M_{0},E_{0})$ is compact,
and hence (eventually after replacing the sequence $(u_{n})_{n\in\mathbb{N}}$
with a subsequence), we may also assume that the sequence $u_{n}$ converges in
norm to $w\in W^{s,p}(M_{0},E_{0})$. Passing to the limit in the relation

$\|u_{n}\|_{W^{s,p}(M_{0},E_{0})}>\epsilon\|u_{n}\|_{W^{s^{\prime},p^{\prime}}(M_{0},E_{0})}=\epsilon,$

we obtain $\|w\|_{W^{s,p}(M_{0},E_{0})}\geq\epsilon>0$. Hence $w\neq 0$. Let
us notice that the assumptions also give that $W^{s,p}(M_{0},E_{0})\to
W^{t,q}(M_{0},E_{0})$ is continuous. Indeed, if $t\geq 0$, this follows
directly from the Rellich-Kondrachov embedding. If $t<0$, then it follows from
the continuity of $W^{s,p}(M_{0},E_{0})\to W^{0,q}(M_{0},E_{0})$ and of
$W^{0,q}(M_{0},E_{0})\to W^{t,q}(M_{0},E_{0})$. The continuity of
$W^{s,p}(M_{0},E_{0})\to W^{t,q}(M_{0},E_{0})$ gives that $u_{n}$ converges to
$w$ also in $W^{t,q}(M_{0},E_{0})$. Let $R\mathrel{\ :=\
}\sup_{n\in\mathbb{N}}\|u_{n}\|_{W^{s,p}(M_{0},E_{0})}$. We have $R<\infty$
since $u_{n}$ converges in $W^{s,p}(M_{0},E_{0})$. Then, passing to the limit
in the relation

$R/n\geq\|u_{n}\|_{W^{s,p}(M_{0},E_{0})}/n>\|u_{n}\|_{W^{t,q}(M_{0},E_{0})}\,,$

we obtain $\|w\|_{W^{t,q}(M_{0},E_{0})}=0$. This is in contradiction with the
previously proved relation $w\neq 0$. Thus the statement (ii) is true if
$M=M_{0}$, an open ball in $\mathbb{R}^{n}$.

Let us next prove (i) in general. Let $u\in W^{s^{\prime},p^{\prime}}(M;E)$.
We shall write $A\lesssim B$ if there is a bound $C\geq 0$ independent of the
functions involved and $\epsilon>0$ such that $A\leq CB$. We shall use below
also Equation (3.2). Then
$\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}}\lesssim\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s^{\prime},p^{\prime}}}$
by the case of a ball in $\mathbb{R}^{n}$ (the Rellich-Kondrachov inequality),
and hence

$\displaystyle\|u\|_{W^{s,p}(M;E)}\ $
$\displaystyle\lesssim|||u|||_{s,p}\mathrel{\ :=\
}\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}$
$\displaystyle\lesssim\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s^{\prime},p^{\prime}}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}$
$\displaystyle\leq\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s^{\prime},p^{\prime}}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p^{\prime}}}\
=:\
|||u|||_{s^{\prime},p^{\prime}}\,\lesssim\,\|u\|_{W^{s^{\prime},p^{\prime}}(M;E)}\,,$

where we have used also the fact that $\|\cdot\|_{\ell^{p}}$ is decreasing in
$p$. This proves (i).

Let us now prove (ii) in the general case. We have successively:

$\displaystyle\|u\|_{W^{s,p}(M;E)}$
$\displaystyle\lesssim\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s,p}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}$
$\displaystyle\leq\big{\|}\big{(}\epsilon\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s^{\prime},p^{\prime}}}+C_{\epsilon}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{t,q}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}$
$\displaystyle\leq\epsilon\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s^{\prime},p^{\prime}}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}+C_{\epsilon}\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{t,q}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p}}$
$\displaystyle\leq\epsilon\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{s^{\prime},p^{\prime}}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{p^{\prime}}}+C_{\epsilon}\big{\|}\big{(}\|\xi_{\gamma}(\phi_{\gamma}u)\|_{W^{t,q}}\big{)}_{\gamma\in\mathbb{N}}\big{\|}_{\ell^{q}}$
$\displaystyle\lesssim\epsilon\|u\|_{W^{s^{\prime},p^{\prime}}(M;E)}+C_{\epsilon}\|u\|_{W^{t,q}(M;E)}\,,$

where we have used again the decreasing monotony of the $\ell^{p}$-norm and
$p\geq p^{\prime}$ and $p\geq q$. This means $\|u\|_{W^{s,p}(M;E)}\leq
C\epsilon\|u\|_{W^{s^{\prime},p^{\prime}}(M;E)}+CC_{\epsilon}\|u\|_{W^{t,q}(M;E)}$.
To obtain the desired result, one needs only to rescale $\epsilon$ by
replacing it with $\epsilon/C$. ∎

Of course, if $p=p^{\prime}=q=2$, then one can prove the above result using
also interpolation inequalities. See also [4, 6, 7, 21, 22, 48] for related
results, in particular, for the use of the partitions of unity. See [9, 35,
45, 49, 20] for the general results on Sobolev spaces not proved above.

##  4\. Regularity estimates for second order strongly elliptic operators

We shall need the regularity result for uniformly strongly elliptic operators
on manifolds bounded geometry $M$ stated in the Introduction (and which is a
slight generalization of a result in [18]). Note, however, that in [18], the
authors have used operators in _divergence form_ (unlike in this paper), so
the regularity requirements on the coefficients in that and this paper are
(slightly) different.

###  4.1. Regularity conditions

Recall the notation $\mathcal{D}^{\mu,\rho}(M;E,F)$ for our order $\mu$
differential operators with coefficients in $W^{\rho,\infty}$ (Definition
2.2).

######  Definition 4.1.

Let $P\in\mathcal{D}^{2,0}(M;E)$. We shall say that $P$ is _uniformly strongly
elliptic_ if there exists $\gamma_{P}>0$ such that, for all $x\in M$ and all
$\eta\in T_{x}^{*}M$ and $\zeta\in E_{x}$, we have that its principal symbol
$\sigma_{2}(P)\colon T^{*}M\to\operatorname{End}(E)$ satisfies the inequality

(4.1)
$\operatorname{Re}(\sigma_{2}(P)(\eta)\zeta,\zeta)_{E_{x}}\geq\gamma_{P}\|\eta\|^{2}\|\zeta\|^{2}\,.$

Of course, the defining relation (4.1) is equivalent to

(4.2) $\sigma_{2}(P)(\eta)+\sigma_{2}(P)(\eta)^{*}\geq
2\gamma_{P}\|\eta\|^{2}$

(with the inequality in operator sense). Let us now prove our regularity
result stated in the Introduction.

###### Proof of Theorem 1.2.

A differential operator with coefficients in $W^{\rho+3,\infty}$ can be
written as a differential operator in divergence form with coefficients in
$W^{\rho+2,\infty}$. (Indeed, in local coordinates, an operator in divergence
form is a linear combination of operators of the form
$\partial_{i}a\partial_{j}$. The claim is obtained by writing
$\partial_{i}a\partial_{j}=a\partial_{i}\partial_{j}+\partial_{i}(a)\partial_{j}$,
with the right hand side consisting of operators written in the standard form.
The loss of one order of regularity is due to the appearance of
$\partial_{i}(a)$.) For $t=1$ and operators in divergence form with
coefficients in $W^{\rho+2,\infty}$, this result was proved in [18, Theorem
5.12]. This completes our proof if $t=1$. Proposition 3.9 for $\epsilon
c_{s,1}<1$, $p=p^{\prime}=q=2$, and $s^{\prime}=s+2$ then allows us to extend
to the case $t<1$. The result for $t>1$ follows from the result for $t=1$. The
proof is thus complete. ∎

A more general result than this theorem will be provided in [16]. We
immediately obtain the following result (which eliminates the restriction
$s\leq\rho$ of the above theorem).

######  Corollary 4.2.

We use the notation and assumptions of Theorem 1.2, except that we assume that
$P$ has coefficients in $W^{\infty,\infty}$, but allow $s\in\mathbb{N}$. Then
there exists $c_{s,t}>0$ such that, if $u\in H^{t}(M;E)$ and $Pu\in
H^{s}(M;F)$, it follows that $u\in H^{s+2}(M;E)$ and

$\|u\|_{H^{s+2}(M;E)}\leq
c_{s,t}\left(\|Pu\|_{H^{s}(M;F)}+\|u\|_{H^{t}(M;E)}\right).$

The above result will be used (mostly) for $s=t=0$, which explains why we
needed to include the case $t=0$ in our regularity result, Theorem 1.2. As
already alluded to above, these regularity results are closely related to the
results in [18], but also to the ones in [29, 30, 40].

######  Remark 4.3.

As noticed already in Remark 3.4, a differential operator in our sense (i.e. a
$\nabla$-differential operator) is also a uniform differential operator in the
sense of [29, 30]. Thus, if $P$ is _scalar_ , then Corollary 4.2 was stated
also in [29, 30], where it was also noticed that it can be proved as in the
classical case of compact manifolds. See also Proposition 4 in [41]. Clearly,
these classical proofs are also valid for uniformly strongly elliptic _systems
of operators._ On a different note, the restriction $t\geq 1$ in [18] was due
to the fact that, in that paper, the authors worked on manifolds with
boundary. In principle, the method of that paper could also give the result
for $t\in\mathbb{Z}$, when there is no boundary.

###  4.2. Consequences of regularity

We continue to assume that $M$ is a manifold with bounded geometry and that
$E,F\to M$ are two Hermitian vector bundles with totally bounded curvature. We
shall need the following standard consequences of our regularity results. For
simplicity, we assume that all our differential operators $P$ have smooth
coefficients. Let then

(4.3) $\mathcal{D}^{\mu,\infty}(M;E,F)\mathrel{\ :=\
}\bigcap_{\rho\in\mathbb{N}}\mathcal{D}^{\mu,\rho}(M;E,F).$

(Compare with Definition 2.2, where the spaces $\mathcal{D}^{\mu,\rho}(M;E,F)$
of differential operators were introduced.) Corollary 4.2 yields the following
results.

######  Corollary 4.4.

Let $P\in\mathcal{D}^{2,\infty}(M;E,F)$ be a uniformly strongly elliptic
differential operator (see Equation 4.3 for notation). We regard $P$ as an
_unbounded_ operator on $H^{s}(M;E)$ with domain $H^{s+2}(M;E)\subset
H^{s}(M;E)$ and with values in $H^{s}(M;F)$. Then $P$ is closed.

###### Proof.

Let $\eta_{n}\in H^{s+2}(M;E)$, $\eta\in H^{s}(M;E)$, and $\zeta\in
H^{s}(M;F)$ be such that $\eta_{n}\to\eta$ and $P\eta_{n}\to\zeta$, the first
convergence being in $H^{s}(M;E)$ and the second one being in $H^{s}(M;F)$.
Then, since $P$ is continuous in distribution sense, we have $P\eta_{n}\to
P\eta$ in distribution sense, and hence $P\eta=\zeta$ in distribution sense.
Theorem 1.2 for $u=\eta$ and $t=s$ gives that $\eta\in H^{s+2}(M;E)$. Hence
the graph of $P$ is closed. ∎

##  5\. Application: $L^{2}$-unique continuation and the deformation operator

We continue to assume that $M$ is a manifold with bounded geometry and that
$E\to M$ is a Hermitian vector bundle with totally bounded curvature.

###  5.1. An invertibility result

The next result states that if $T\in\mathcal{D}^{2,\infty}(M;E)$ is a
uniformly strongly elliptic symmetric operator on $L^{2}(M;E)$, on which we
consider the inner product $(\cdot,\cdot)_{L^{2}(M;E)}$, then $T$ is
essentially self-adjoint, that is, its closure is a self-adjoint operator on
$L^{2}(M;E)$. Its proof is standard.

######  Lemma 5.1.

Let $M$ be a manifold with bounded geometry and $E\to M$ be a Hermitian vector
bundle with totally bounded curvature. Let $T\in\mathcal{D}^{2,\infty}(M;E)$
be a uniformly strongly elliptic differential operator and consider the
setting of Corollary 4.4 for $s=0$. If $T\colon H^{2}(M;E)\to L^{2}(M;E)$ is
symmetric $($that is, $(Tf,g)=(f,Tg)$, for all $f,g\in H^{2}(M;E)$$)$, then
$T$ is self-adjoint.

###### Proof.

Let
$T^{\dagger}\colon{\mathcal{C}}^{\infty}_{\text{c}}(M;E)^{*}\to{\mathcal{C}}^{\infty}_{\text{c}}(M;E)^{*}$
be the adjoint of $T$ in distribution sense (with
$V^{*}:=\overline{V}^{\prime}$ the conjugate linear dual of $V$). Our
assumption is that $T^{\dagger}=T$ as _differential operators acting on
${\mathcal{C}}^{\infty}_{\text{c}}(M;E)$_, and hence also as _differential
operators acting on distributions_ , since $T$ and $T^{\dagger}$ are
continuous on distributions and ${\mathcal{C}}^{\infty}_{\text{c}}(M;E)$ is
dense in the space ${\mathcal{C}}^{\infty}_{\text{c}}(M;E)^{*}$ of
distributions. Let $T^{*}$ be the adjoint of $T$ in $L^{2}$-sense. (The
pairing between $V$ and $V^{*}$ is sesquilinear.) Since we know that $T$ is
closed, by Corollary 4.4, it is enough to show that $T^{*}\pm\imath$ are both
injective (where $\imath\in\mathbb{C}$ is such that $\imath^{2}=-1$). Let
$\phi$ in the domain of $T^{*}$ be such that $(T^{*}+\imath)\phi=0$. Since
$\phi\in L^{2}(M;E)$, it is a distribution, and $T^{\dagger}\phi$ is defined.
By taking inner products with test functions
$\psi\in{\mathcal{C}}^{\infty}_{\text{c}}(M;E)$ (which are in the domain of
$T$), we see that

$(T+\imath)\phi=(T^{\dagger}+\imath)\phi=(T^{*}+\imath)\phi=0\,,$

in distribution sense. Since $T$ is elliptic of order $2>0$, we obtain that
$T+\imath$ is also elliptic of order $2$. The regularity theorem, Theorem 1.2,
gives that $\phi\in H^{2}(M;E)$. Hence $\phi$ is in the domain of $T$. But
then the usual $L^{2}$-estimate

$0=\|(T+\imath)\phi\|_{L^{2}}^{2}\,=\,((T+\imath)\phi,(T+\imath)\phi)\,=\,\|T\phi\|_{L^{2}}^{2}+\|\phi\|_{L^{2}}^{2}$

gives that $\phi=0$. ∎

Recall the definition of the $L^{2}$-unique continuation property from
Definition 1.1. The most basic example is the following.

######  Example 5.2.

Let $M_{0}$ be a connected Riemannian manifold. The exterior derivative
$d\colon
L^{2}(M_{0})\to{\mathcal{C}}^{\infty}_{\text{c}}(M_{0},T^{*}M_{0})^{\prime}$
has the $L^{2}$-unique continuation property. Indeed, if $du=0$, then $u$ is
locally constant, and hence constant, since $M_{0}$ was assumed to be
connected. If, furthermore, $u$ vanishes on some non-empty open subset of
$M_{0}$, then this constant must be zero.

The following result is a key technical application of $L^{2}$-unique
continuation property that allows us to establish the invertibility of our
operators in applications to the Stokes system [27]. It motivates the
consideration of the $L^{2}$-unique continuation property. Let $X$ and $Y$ be
Banach spaces. Recall that a (possibly unbounded) closed operator
$T:\mathcal{D}(T)\subset X\to Y$ is _Fredholm_ if $T^{-1}(0)$ and $Y/TX$ are
finite dimensional. Then its _index_ is $\dim T^{-1}(0)-\dim(Y/TX)$.

######  Proposition 5.3.

Let $M$ be a manifold with bounded geometry and $E\to M$ be a Hermitian vector
bundle with totally bounded curvature. Let $L\in\mathcal{D}^{2,\infty}(M;E)$
be a differential operator (so of order $2$) and $V\in
W^{\infty,\infty}(M;\operatorname{End}(E))$ (so of order zero) with the
following properties:

  1. (i)

$V,L\geq 0$, more precisely, for all
$v\in{\mathcal{C}}^{\infty}_{\text{c}}(M;E)$,

$(Lv,v)\geq 0\ \mbox{ and }\ (Vv,v)\geq 0\,.$

  2. (ii)

$L$ is uniformly strongly elliptic and has the $L^{2}$-unique continuation
property (Definition 1.1).

  3. (iii)

There exists a non-empty open set ${\mathcal{O}}\subset M$ such that, if $v\in
L^{2}(M;E)$ and $(Vv,v)=0$, then $v|_{{\mathcal{O}}}\,=\,0.$

  4. (iv)

$L+V\colon H^{2}(M;E)\to L^{2}(M;E)$ is a Fredholm operator.

Then $L+V\colon H^{2}(M;E)\to L^{2}(M;E)$ is invertible.

###### Proof.

Since $V$ has order smaller than the order of $L$ and $L$ is uniformly
strongly elliptic, by assumption, it follows that $L+V$ is also uniformly
strongly elliptic. We then notice that, since $L$ and $L+V$ are both uniformly
strongly elliptic, both have domain $H^{2}(M;E)$ and are closed as unbounded
operators on $L^{2}(M;E)$ by Corollary 4.4. The positivity of both $L$ and $V$
implies also the positivity of $L+V$. (We obtain positivity on the whole
domain since ${\mathcal{C}}^{\infty}_{\text{c}}(M;E)$ is dense in $H^{2}(M;E)$
in the graph norm.) In turn, positivity implies that $L$ and $L+V$ are
symmetric (see, for instance, Proposition 2.12 in [12]). Lemma 5.1 then gives
that both $L$ and $L+V$ are self-adjoint. Since $L+V$ is Fredholm and self-
adjoint, its index will be 0.

Let us assume, by contradiction, that $L+V$ is _not_ invertible. Since it is
Fredholm of index zero, it must have a non-zero kernel. Thus, there exists
$u\in H^{2}(M;E)$, $u\neq 0$, such that $(L+V)u=0$. Then

$(Lu,u)+(Vu,u)=0\,.$

The relations $(Lu,u)\geq 0$, $(Vu,u)\geq 0$, and the self-adjointness of $L$,
as unbounded operator on $L^{2}(M;E)$, yield

$\|L^{\frac{1}{2}}u\|^{2}=(Lu,u)=0\ \mbox{ and }\ (Vu,u)=0\,.$

Hence, we obtain $Lu=0$ on $M$. Moreover, assumption (iii) and the relation
$(Vu,u)=0$ imply that $u=0$ on the open subset ${\mathcal{O}}$ of $M$. The
unique continuation property satisfied by the operator $L$ finally shows that
$u=0$ on $M$, which is a contradiction with the initial assumption that $u\neq
0$ is in the kernel of $L+V$. The proof is complete. ∎

See [36] for a motivation and references related to the above result. We shall
need the following simple lemma.

######  Lemma 5.4.

Let $M$ be a manifold with bounded geometry and $E,F\to M$ be Hermitian vector
bundles with totally bounded curvature. Let
$T\in\mathcal{D}^{1,\infty}(M;E,F)$ have the $L^{2}$-unique continuation
property. If $T^{*}T$ is uniformly strongly elliptic, then it also has the
$L^{2}$-unique continuation property.

###### Proof.

Let $u\in L^{2}(M;E)$ be such that $T^{*}Tu=0$ and $u=0$ on some non-empty
open set $\mathcal{O}\subset M$. Since $T^{*}T$ is uniformly strongly
elliptic, we obtain that $u\in H^{2}(M;E)$ by elliptic regularity (see Theorem
1.2 or Corollary 4.2). Therefore, we have

$\|Tu\|^{2}_{L^{2}}\,=\,(Tu,Tu)\,=\,(T^{*}Tu,u)\,=\,0\,.$

Hence $Tu=0$. The $L^{2}$-unique continuation property for $T$ then gives
$u=0$. ∎

###  5.2. Vanishing of Killing vector fields

We shall need the following lemma (a stronger result can be found in [37,
Prop. 8.1.4], we include, nevertheless a simple proof, for the benefit of the
reader):

######  Lemma 5.5.

Let $M_{0}$ be a connected Riemannian manifold with positive injectivity
radius (and hence complete). Let $\mathcal{O}\subset M_{0}$ be a closed subset
with non-empty interior and with the following property:

_“If $\gamma\colon\mathbb{R}\to M_{0}$ is a (complete) geodesic that is
minimizing between two points $a,b\in\mathcal{O}$, then
$\gamma\subset\mathcal{O}$.”_

Then $\mathcal{O}=M_{0}$.

###### Proof.

Let $\mathcal{O}_{0}\subset\mathcal{O}$ be the interior of $\mathcal{O}$,
which is known to be non-empty by hypothesis. Let us show that
$\mathcal{O}_{0}$ is also closed in $M_{0}$. For $x\in M_{0}$, we shall let
$B_{r}^{M_{0}}(x)\subset M_{0}$ denote the open ball with center $x$ and
radius $r$ (the set of points $y\in M_{0}$ at distance $<r$ to $x$). Let
$r_{0}>0$ be the injectivity radius of $M_{0}$, we claim that

(5.1) $x_{0}\in\mathcal{O}_{0}\ \Rightarrow\
B_{r_{0}}^{M_{0}}(x_{0})\subset\mathcal{O}_{0}\,.$

Indeed, to prove the claim (5.1), let $y\in B_{r_{0}}^{M_{0}}(x_{0},r_{0})$.
Then there is a minimizing geodesic $\gamma\colon\mathbb{R}\to M_{0}$ joining
$y$ to $x_{0}$. We can assume $\gamma(0)=x_{0}$. Then, for $t$ small enough,
$\gamma(t)\in\mathcal{O}_{0}$ and $\gamma$ is minimizing between
$x_{0}=\gamma(0)$ and $\gamma(t)$. It follows by assumption that
$\gamma\subset\mathcal{O}$ and hence, in particular, $y\in\mathcal{O}$. This
shows that, for every $x_{0}\in\mathcal{O}_{0}$, the ball
$B_{r_{0}}^{M_{0}}(x_{0})\subset\mathcal{O}$. Hence also
$B_{r_{0}}^{M_{0}}(x_{0})\subset\mathcal{O}_{0}$ since
$B_{r_{0}}^{M_{0}}(x_{0})$ is open. We have thus proved our claim, that is,
Equation (5.1).

Let us next show that the proven claim implies that $\mathcal{O}_{0}$ is
closed. Indeed, let $y\in\overline{\mathcal{O}}_{0}$. Then
$B^{M_{0}}_{r_{0}}(y)\cap\mathcal{O}_{0}$ is non-empty. Let $x_{0}\in
B_{r_{0}}^{M_{0}}(y)\cap\mathcal{O}_{0}$. Then $y\in
B_{r_{0}}^{M_{0}}(x_{0})\subset\mathcal{O}_{0}$. Thus $\mathcal{O}_{0}$ is
closed, as stated.

This completes the proof. Indeed, $M_{0}$ is connected and
$\mathcal{O}_{0}\subset M_{0}$ is open, closed, and non-empty, it follows that
$\mathcal{O}_{0}=M_{0}$. This proves that $\mathcal{O}=M_{0}$ as well, and
concludes the proof. ∎

Let $\mathfrak{L}_{X}$ denote the Lie derivative in the direction of a vector
field $X$. Then the deformation tensor $\operatorname{Def}\,X$ is defined as

(5.2) $\displaystyle\operatorname{Def}\,X=\frac{1}{2}\mathfrak{L}_{X}g$

(see [45, Chapter 2 §3]). Thus, $\operatorname{Def}\,X$ is a symmetric tensor
field of type $(0,2)$, and $\operatorname{Def}\,$ will be viewed as an
unbounded operator from $L^{2}(TM)$ to $L^{2}(T^{*}M\otimes T^{*}M)$. Let also
$\operatorname{Def}\,^{*}$ denote the adjoint of the deformation operator
$\operatorname{Def}\,$. If $\operatorname{Def}\,X=0$, then $X$ is called a
_Killing vector field._

The next corollary will provide us with the $L^{2}$-unique continuation
property for the deformation operator $\operatorname{Def}\,$.

######  Corollary 5.6.

Let $M_{0}$ be a connected Riemannian manifold with positive injectivity
radius. Let $X$ be a smooth _Killing_ vector field on $M_{0}$ that vanishes on
a non-empty open subset of $M_{0}$. Then $X=0$ $($i.e. it vanishes on
$M_{0}$$)$.

###### Proof.

Let $\mathcal{O}\subset M_{0}$ be the set of points where $X=0$, which
contains an open subset of $M_{0}$, by assumption. Since $X$ is smooth and
$M_{0}$ is complete, $X$ generates a _local_ flow $\phi_{t}\colon M_{0}\to
M_{0}$, with $\phi_{t}(x)$ defined for $t\in(-\epsilon_{x},\epsilon_{x})$, for
some $\epsilon_{x}>0$. Since $X$ is a Killing vector field, the local flow
$\phi_{t}$ consists of (partially defined) isometries. Moreover, $\phi_{t}=id$
on $\mathcal{O}$ (since $X=0$ there). We conclude that $\phi_{t}$ fixes any
minimizing geodesic between two points of $\mathcal{O}$. Hence any (complete)
geodesic minimizing the distance between two points of $\mathcal{O}$ is
contained in $\mathcal{O}$. Lemma 5.5 then gives that $\mathcal{O}=M_{0}$.
Hence $X=0$ everywhere on $M$. ∎

###  5.3. Applications to the deformation operator

We continue to assume that $M$ is a manifold with bounded geometry. Let $g$ be
a Riemannian metric tensor on $M$. We now prove our main application to the
deformation operator.

######  Theorem 5.7.

Let $M$ be a manifold with bounded geometry. Then the following statements
hold.

  1. (i)

We have $\operatorname{Def}\,\in\mathcal{D}^{1,\infty}(M;TM,T^{*}M\otimes
T^{*}M)$.

  2. (ii)

The deformation Laplacian $\mathbf{L}\mathrel{\ :=\
}2\operatorname{Def}^{*}\operatorname{Def}\,$ is uniformly strongly elliptic.

  3. (iii)

If $M$ is connected, then $\operatorname{Def}\,$ and $\mathbf{L}\mathrel{\ :=\
}2\operatorname{Def}^{*}\operatorname{Def}\,$ have the $L^{2}$-unique
continuation property on $M$.

###### Proof.

(i) follows from formula (5.2) and the results in [26] (see, for instance,
[26, Proposition 5.12], noting that manifolds with bounded geometry satisfy
the (FFC) condition by the results in [18], Lemma 3.1).

(ii) The operator $\mathbf{L}\mathrel{\ :=\
}2\operatorname{Def}^{*}\operatorname{Def}\,$ is in
$\mathcal{D}^{2,\infty}(M;\operatorname{End}(TM))$ by point (i) already proved
and [26, Proposition 4.5]. The principal symbol of $\mathbf{L}$ (a second
order differential operator) is given by [45, Chapter 5 §12, (12.20)]

$\sigma_{2}(\mathbf{L})(\eta)=|\eta|^{2}({1}+P_{\eta})\geq|\eta|^{2}\,,\
\forall\,\eta\in T_{x}^{*}M\setminus\\{0\\},$

where $P_{\eta}(u):=|\eta|^{-2}(\eta,u)\eta$ is the orthogonal projection of
$u$ along $\eta$ (regarded as a vector) and we use the notation $1$ for the
identity operator. Hence $\mathbf{L}$ is a uniformly strongly elliptic
operator, as asserted.

(iii) Let us assume that $M$ is a connected manifold and let $X\in
L^{2}(M,TM)$ be such that $\operatorname{Def}\,X=0$ on $M$. Then
$\operatorname{Def}^{*}\operatorname{Def}\,X=0$. Since
$\operatorname{Def}^{*}\operatorname{Def}\,$ is uniformly strongly elliptic by
point (ii) already proved, elliptic regularity (Corollary 4.2) gives that $X$
is smooth. Since $M$ has positive injectivity radius, Lemma 5.5 and, more
precisely, Corollary 5.6, give that $X=0$. Hence $\operatorname{Def}\,$ has
the $L^{2}$ unique continuation property. Lemma 5.4, in conjunction again with
point (ii) already proved, gives then that
$\operatorname{Def}^{*}\operatorname{Def}\,$ also has the $L^{2}$ unique
continuation property. ∎

Combining with Proposition 5.3, we obtain the following consequence.

######  Corollary 5.8.

Let $M$ be a _connected_ manifold with bounded geometry and
$\mathbf{L}:=2\operatorname{Def}^{*}\operatorname{Def}\,$, as before. Let
$V\in W^{\infty,\infty}(M;\operatorname{End}(E))$ with the following
properties:

  1. $(i)$

$V\geq 0$.

  2. $(ii)$

There exists a non-empty open subset ${\mathcal{O}}\subset M$ such that, if
$v\in L^{2}(M;E)$ and $(Vv,v)=0$, then $v|_{{\mathcal{O}}}\,=\,0.$

  3. $(iii)$

$\mathbf{L}+V\colon H^{2}(M;E)\to L^{2}(M;E)$ is a Fredholm operator.

Then $\mathbf{L}+V\colon H^{2}(M;E)\to L^{2}(M;E)$ is invertible.

There is an analogous definition of $W^{s,p}$-unique continuation property:

######  Definition 5.9.

Let $T$ be pseudodifferential operator acting from sections of $E$ to sections
of $F$. Let us assume that $T$ extends by continuity to a map
$T:W^{s,p}(M;E)\to\mathcal{D}(M;F)^{\prime}$. We shall say that $T$ has the
$W^{s,p}$-unique continuation property if, given $u\in W^{s,p}(M;E)$ that
vanishes in an open subset $U\subset M_{0}$ and satisfies $Tu\,=\,0$, then
$u\,=\,0$.

Then the same proof gives as the proof of the $L^{2}$-unique continuation
property for $\operatorname{Def}\,$ in Theorem 5.7(iii) gives the following.

######  Corollary 5.10.

Let $M_{0}$ be a connected manifold with bounded geometry and
$s\in\mathbb{R}_{+}$, then $\operatorname{Def}\,$ has the $H^{s}$-unique
continuation property.

Generalizations of this kind are possible for other statements, when suitable
reformulated. For instance, the problem in Lemma 5.4 is to specify which
adjoint one uses. One can still use the adjoint with respect to the
$L^{2}$-inner product (not with respect to $H^{s}$), then, as a consequence,
one would obtain also that
$\mathbf{L}:=\operatorname{Def}^{*}\operatorname{Def}\,$ also has the
$H^{s}$-unique continuation property.

## References

  * [1] Adams, R. Sobolev spaces.  Pure and Applied Mathematics, 65. A Series of Monographs and Textbooks. New York-San Francisco-London: Academic Press, Inc., a subsidiary of Harcourt Brace Jovanovich, Publishers. XVIII, 268 p. $ 24.50 (1975)., 1975.
  * [2] Adams, R., and Fournier, J. Sobolev spaces, 2nd ed. ed.  New York, NY: Academic Press, 2003.
  * [3] Amann, H. Compact embeddings of vector-valued Sobolev and Besov spaces.  Glasnik Matematicki 35 (2000), 161–177.
  * [4] Amann, H. Function spaces on singular manifolds.  Math. Nachr. 286, 5-6 (2013), 436–475.
  * [5] Ammann, B., Große, N., and Nistor, V. Well-posedness of the Laplacian on manifolds with boundary and bounded geometry.  Math. Nachr. 292, 6 (2019), 1213–1237.
  * [6] Ammann, B., Ionescu, A. D., and Nistor, V. Sobolev spaces on Lie manifolds and regularity for polyhedral domains.  Doc. Math. 11 (2006), 161–206 (electronic).
  * [7] Aubin, T. Espaces de Sobolev sur les variétés Riemanniennes.  Bull. Sc. math. 100 (1976), 149–173.
  * [8] Barcelo, B., Kenig, C. E., Ruiz, A., and Sogge, C. D. Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms.  Ill. J. Math. 32, 2 (1988), 230–245.
  * [9] Brezis, H. Functional analysis, Sobolev spaces and partial differential equations.  Universitext. Springer, New York, 2011.
  * [10] Browder, F. E. On the spectral theory of elliptic differential operators. I.  Math. Ann. 142 (1960/61), 22–130.
  * [11] Choulli, M. The unique continuation property for second order evolution PDEs.  SN Partial Differ. Equ. Appl. 2, 5 (2021), 46.  Id/No 67.
  * [12] Conway, J. A course in functional analysis., 2nd ed. ed., vol. 96 of Grad. Texts Math. New York etc.: Springer-Verlag, 1990.
  * [13] Dos Santos Ferreira, D., Kenig, C., Salo, M., and Uhlmann, G. Limiting Carleman weights and anisotropic inverse problems.  Invent. Math. 178, 1 (2009), 119–171.
  * [14] Garofalo, N., and Lin, F.-H. Unique continuation for elliptic operators: A geometric-variational approach.  Commun. Pure Appl. Math. 40, 3 (1987), 347–366.
  * [15] Große, N. Solutions of the equation of a spinorial Yamabe-type problem on manifolds of bounded geometry.  Commun. Partial Differ. Equations 37, 1-3 (2012), 58–76.
  * [16] Große, N., Kohr, M., and Nistor, V. General regularity on manifolds with bounded geometry and applications to the deformation operator.  work in progress.
  * [17] Große, N., Kohr, M., and Nistor, V. Sobolev spaces and $\nabla$-differential operators on manifolds II: Christoffel polynomials.  work in progress.
  * [18] Große, N., and Nistor, V. Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry.  Potential Anal. 53, 2 (2020), 407–447.
  * [19] Große, N., and Schneider, C. Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces.  Math. Nachr. 286, 16 (2013), 1586–1613.
  * [20] Haroske, D., and Triebel, H. Distributions, Sobolev spaces, elliptic equations.  EMS Textb. Math. Zürich: European Mathematical Society, 2008.
  * [21] Hebey, E. Sobolev spaces on Riemannian manifolds, vol. 1635 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1996.
  * [22] Hebey, E., and Robert, F. Sobolev spaces on manifolds.  In Handbook of global analysis. Elsevier Sci. B. V., Amsterdam, 2008, pp. 375–415, 1213.
  * [23] Hörmander, L. The analysis of linear partial differential operators. I.  Classics in Mathematics. Springer, Berlin.
  * [24] Kenig, C. E., and Sogge, C. D. A note on unique continuation for Schrödinger’s operator.  Proc. Am. Math. Soc. 103, 2 (1988), 543–546.
  * [25] Koch, H., and Tataru, D. Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients.  Commun. Partial Differ. Equations 34, 4 (2009), 305–366.
  * [26] Kohr, M., and Nistor, V. Sobolev spaces and $\nabla$-differential operators on manifolds I: basic properties and weighted spaces.  Annals of Global Analysis and Geometry 61 (2022), 721–758.
  * [27] Kohr, M., Nistor, V., and Wendland, W. Essentially invariant pseudodifferential operators on domains with cylindrical ends and the well-posedness of the Stokes system.  Preprint, 2023.
  * [28] Kondrachov, V. I. On certain properties of functions in the space ${L}_{p}$.  Dokl. Akad. Nauk SSSR 48 (1945), 563–566.
  * [29] Kordyukov, Y. A. $L^{p}$-theory of elliptic differential operators with bounded coefficients.  Vestnik Moskov. Univ. Ser. I Mat. Mekh., 4 (1988), 98–100.
  * [30] Kordyukov, Y. A. $L^{p}$-theory of elliptic differential operators on manifolds of bounded geometry.  Acta Appl. Math. 23, 3 (1991), 223–260.
  * [31] Le Rousseau, J., and Lebeau, G. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations.  ESAIM, Control Optim. Calc. Var. 18, 3 (2012), 712–747.
  * [32] Le Rousseau, J., Lebeau, G., and Robbiano, L. Elliptic Carleman estimates and applications to stabilization and controllability. Volume II. General boundary conditions on Riemannian manifolds, vol. 98 of Prog. Nonlinear Differ. Equ. Appl. Cham: Birkhäuser, 2022.
  * [33] Lerner, N. Carleman inequalities. An introduction and more, vol. 353 of Grundlehren Math. Wiss. Cham: Springer, 2019.
  * [34] Lerner, N. Unique continuation through transversal characteristic hypersurfaces.  J. Anal. Math. 138, 1 (2019), 135–156.
  * [35] Lions, J.-L., and Magenes, E. Non-homogeneous boundary value problems and applications. Vol. I.  Springer-Verlag, New York-Heidelberg, 1972.  Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.
  * [36] Mitrea, M., and Nistor, V. Boundary value problems and layer potentials on manifolds with cylindrical ends.  Czechoslovak Math. J. 57(132), 4 (2007), 1151–1197.
  * [37] Petersen, P. Riemannian geometry, 2nd ed. ed., vol. 171 of Grad. Texts Math. New York, NY: Springer, 2006.
  * [38] Shubin, M. A. Spectral theory of elliptic operators on noncompact manifolds.  Astérisque 207 (1992), 5, 35–108.  Méthodes semi-classiques, Vol. 1 (Nantes, 1991).
  * [39] Skrzypczak, L. Atomic decompositions on manifolds with bounded geometry.  Forum Math. 10, 1 (1998), 19–38.
  * [40] Skrzypczak, L. Mapping properties of pseudodifferential operators on manifolds with bounded geometry.  J. Lond. Math. Soc., II. Ser. 57, 3 (1998), 721–738.
  * [41] Skrzypczak, L. Wavelet frames, Sobolev embeddings and negative spectrum of Schrödinger operators on manifolds with bounded geometry.  J. Fourier Anal. Appl. 14, 3 (2008), 415–442.
  * [42] Sogge, C. Fourier integrals in classical analysis, vol. 105 of Camb. Tracts Math. Cambridge: Cambridge University Press, 1993.
  * [43] Tataru, D. Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem.  Commun. Partial Differ. Equations 20, 5-6 (1995), 855–884.
  * [44] Tataru, D. Unique continuation problems for partial differential equations.  In Geometric methods in inverse problems and PDE control. Selected articles presented at the 2001 IMA summer program, Minneapolis, MN, USA. New York, NY: Springer, 2003, pp. 239–255.
  * [45] Taylor, M. Partial differential equations I. Basic theory, second ed., vol. 115 of Applied Mathematical Sciences.  Springer, New York, 2011.
  * [46] Taylor, M. E. Partial differential equations. III: Nonlinear equations., 2nd ed. ed., vol. 117 of Appl. Math. Sci. New York, NY: Springer, 2011.
  * [47] Trèves, F. Basic linear partial differential equations, reprint of the 1975 original ed.  Mineola, NY: Dover Publications, 2006.
  * [48] Triebel, H. Characterizations of function spaces on a complete Riemannian manifold with bounded geometry.  Math. Nachr. 130 (1987), 321–346.
  * [49] Triebel, H. Theory of function spaces.  Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010\.  Reprint of 1983 edition [MR0730762], Also published in 1983 by Birkhäuser Verlag [MR0781540].