Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp.

DOI 10.15781/c93s-pk62

ISSN: 2769-8505

00footnotetext: © Herbert Koch, Angkana Rüland, and Mikko Salo00footnotetext:
Licensed under a Creative Commons Attribution License (CC-BY).

On instability mechanisms
for inverse problems

Herbert Koch

Universität Bonn

Angkana Rüland

Ruprecht-Karls-Universität Heidelberg

Mikko Salo

University of Jyvaskyla

Communicated by Gunther Uhlmann

Abstract. In this article we present three robust instability mechanisms for
linear and nonlinear inverse problems. All of these are based on strong
compression properties (in the sense of singular value or entropy number
bounds) which we deduce through either strong global smoothing, only weak
global smoothing or microlocal smoothing for the corresponding forward
operators, respectively. As applications we for instance present new
instability arguments for unique continuation, for the backward heat equation
and for linear and nonlinear Calderón type problems in general geometries,
possibly in the presence of rough coefficients. Our instability mechanisms
could also be of interest in the context of control theory, providing
estimates on the cost of (approximate) controllability in rather general
settings.

Keywords. Inverse problems, instability mechanisms, entropy numbers.

##  1\. Introduction

The purpose of this article is to revisit and to systematically extend
instability arguments for certain classes of inverse problems. In such
problems the forward operator does not have a continuous inverse, and these
problems are thus automatically unstable. In fact instability, or ill-
posedness, lies at the heart of inverse problems research and much of the
related literature addresses various aspects of it. We mention very briefly a
few selected aspects:

  * •

Inverting the two-dimensional Radon transform, or the geodesic X-ray transform
in geometries without conjugate points, is a mildly ill-posed inverse problem
[Na01, AS20]. Variants involving limited data or geometries with conjugate
points may be severely ill-posed [Na01, SU04, MSU15].

  * •

For the boundary rigidity problem a stability theory has been developed in
[SU04, SU08]. A main feature is that suitable stability for the linearized
problem implies local uniqueness and stability for the corresponding nonlinear
problem. Corresponding abstract results are given in [SU09].

  * •

In the Calderón problem one has logarithmic stability [Al88] and this is
optimal in general [Ma01]. Stability improves if the unknowns are restricted
to a finite dimensional space [AV05]. For certain partial data or anisotropic
variants only double logarithmic stability is known [CS14, CDR16].

  * •

For the wave equation, the problem of determining coefficients from the
hyperbolic Dirichlet-to-Neumann map is often quite stable in geometries
without conjugate points [SU05, Mo14]. In the presence of conjugate points
only weak stability results are known [AKK+04, BKL17].

  * •

Even in the presence of ill-posedness, regularization can be used to reduce
the effect of measurement errors and to obtain more stable approximate
reconstructions [EHN96].

Many of the above results focus on showing that the inverse problem in
question has at least a certain (weak) degree of stability. In this article we
consider corresponding _instability_ results, showing that a certain type of
instability is inherent to the inverse problem and cannot be improved. In the
literature several arguments deducing the degree of instability are known also
in cases where direct singular value computations are not immediately
feasible. Prominent examples for linear inverse problems are based on
(microlocal) regularization for the forward operator e.g. in the setting of
the geodesic X-ray transform and related problems, see [Qu93, Qu06, SU09,
SU12, SU13, MSU15, HU18]. For nonlinear Calderón type inverse problems there
is a method due to Mandache [Ma01] based on the construction of certain
exponentially decaying bases and entropy arguments, see also [DR03, Is11,
Is13a, Is13b, Al07, ZZ19]. However, the latter results rely on strong
assumptions on the operators (e.g. constant coefficients in the principal
part) or on the geometry in which the problem is phrased (e.g. in the form of
symmetry of the underlying domain such that separation of variables arguments
can be used). Even for linear inverse problems the available results showing
that logarithmic stability is optimal often rely on explicit computations for
special operators and geometries, see e.g. [AB18] or the classical works
[Ha23, Jo60] for instability of the unique continuation principle. We further
remark that, in general, there is no direct relation between the instability
of a nonlinear inverse problem and its linearization [EKN89, Appendix].

As a key novelty of this article, we present three robust instability
mechanisms which can be applied to many different inverse problems involving
general variable coefficient operators in general domains. A prototypical
setting to which this applies are Calderón type problems (both linearized and
nonlinear) in nonsmooth domains with coefficients having low, scaling critical
regularity. As other examples we consider instability in general settings for
the unique continuation principle for elliptic and hyperbolic equations,
instability for the backward heat equation, the cost of approximate
controllability in various problems from control theory (of elliptic,
parabolic and hyperbolic type), and examples involving microlocal smoothing
such as Radon transforms with limited data. An important aspect of these
instability mechanisms is that they allow us to read off the degree of
instability of an inverse problem from directly accessible properties of the
forward operator (which may now lack symmetry or high regularity). In spite of
their wide applicability, a central feature of our arguments which is shared
by all three mechanisms is that while a stability result for an inverse
problem is “hard” (one needs a quantitative version of a uniqueness result),
the corresponding instability result is often rather “soft”.

Before turning to the explicit mechanisms, let us formulate the abstract set-
up of the problem we are interested in. To this end, consider a (possibly
nonlinear) map

$\displaystyle F:X\rightarrow Y,$

where $X,Y$ are metric spaces (often Banach spaces) and $F$ is continuous
injective map modelling the forward operator of the inverse problem under
consideration. We then seek to investigate the stability properties of the
associated inverse problem: given $y\in F(X)$, find $x\in X$ so that

$F(x)=y.$

In typical inverse problems the map $F:X\to F(X)$ does not have a continuous
inverse. Yet, it is well-known that when restricted to a compact set $K\subset
X$ the operator $F|_{K}:K\mapsto F(K)$ is a homeomorphism whose inverse has
some modulus of continuity $\omega$ (see Lemma 2.1). It is the purpose of this
article to identify general properties of the forward operator $F$ yielding
information on the possible moduli $\omega$ in dependence of the choice of
$X,Y,K$.

In the sequel, we discuss the three instability mechanisms and address
prominent examples to which these apply. We however emphasise that the methods
which we introduce in this article are applicable to a much larger class of
problems and that we have only selected a number of expository examples to
illustrate these.

###  1.1. Instability by strong (analytic) smoothing properties of the forward
operator

In our first instability mechanism we are mainly motivated by inverse problems
for elliptic or parabolic equations. Here one of the most prominent examples
which had been studied by Mandache [Ma01] and Di Cristo-Rondi [DR03] is the
instability of the Calderón problem. In this problem which goes back to work
of Calderón [Ca80] one is interested in recovering a positive conductivity
function $\gamma$ in the equation

(1.1) $\displaystyle\begin{split}\nabla\cdot\gamma\nabla u&=0\mbox{ in
}\Omega\subset\varmathbb{R}^{n},\\\ u&=f\mbox{ on }\partial\Omega,\end{split}$

from the knowledge of the Dirichlet-to-Neumann map

$\displaystyle\Lambda_{\gamma}:f\mapsto\gamma\partial_{\nu}u|_{\partial\Omega},$

where $\partial_{\nu}u|_{\partial\Omega}$ denotes the normal derivative of $u$
on the boundary (viewed in an appropriate weak sense). This problem is further
closely related to the problem of recovering the potential $q$ in the
Schrödinger equation

(1.2) $\displaystyle\begin{split}-\Delta u+qu&=0\mbox{ in }\Omega,\\\
u&=f\mbox{ on }\partial\Omega,\end{split}$

from the knowledge of the Dirichlet-to-Neumann operator

$\displaystyle\Lambda_{q}:f\mapsto\partial_{\nu}u|_{\partial\Omega}.$

The Calderón problem with a sufficiently regular scalar conductivity can be
reduced to this problem by a Liouville transform. For simplicity, we assume
that $q$ is such that zero is not a Dirichlet eigenvalue of the equation
(1.2). Both problems are well-studied non-linear inverse problems with (for
$n\geqslant 3$) uniqueness proofs at different regularities [SU87, No88, Ch90,
LN91, Ha15, Ha17], stability estimates [Al88] and reconstruction algorithms
[Na88] available (we refer to [Uh14] for an overview of the known results and
further references). We are mainly interested in the stability properties of
these problems. By virtue of the work of Alessandrini [Al88] it is known that
both problems (in suitable regularity scales) enjoy logarithmic stability
estimates under mild a priori assumptions on the data, e.g. if $q_{1},q_{2}\in
L^{\infty}(\Omega)$ with $\|q_{j}\|_{L^{\infty}(\Omega)}\leqslant M$, there
exists a constant $C=C(\Omega,n,M)>0$ such that

$\displaystyle\|q_{1}-q_{2}\|_{H^{-1}(\Omega)}\leqslant\omega(\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{H^{\frac{1}{2}}(\partial\Omega)\rightarrow
H^{-\frac{1}{2}}(\partial\Omega)}),$

where $\omega(t)\leqslant C\lvert\log\,t\rvert^{-\frac{2}{n+2}}$ for $t$
small. A natural question then concerns the optimality of these estimates.
This was studied by Mandache [Ma01] who proved the following necessity of the
exponential instability:

######  Theorem 1.1 ([Ma01]).

Let $B_{1}(0)\subset\varmathbb{R}^{n}$ for $n\geqslant 2$ denote the unit
ball. Let $q_{0}\in L^{\infty}(B_{1})\cap C^{m}(\overline{B}_{1})$ with
$\mathrm{supp}(q_{0})\subset B_{1/2}(0)$. Then there is $\varepsilon_{0}>0$
such that for any $\varepsilon\in(0,\varepsilon_{0})$ and $m\geqslant 1$ there
exist potentials $q_{1},q_{2}\in C^{m}(\overline{B}_{1})\cap
L^{\infty}(B_{1})$ with $\mathrm{supp}(q_{j})\subset B_{1/2}(0)$ and constants
$C>0$, $\gamma>0$ such that we have

$\displaystyle\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{H^{\frac{1}{2}}(\partial
B_{1})\rightarrow H^{-\frac{1}{2}}(\partial B_{1})}$
$\displaystyle\leqslant\exp(-C\varepsilon^{-\gamma}),$
$\displaystyle\|q_{1}-q_{2}\|_{L^{\infty}(B_{1})}$
$\displaystyle=\varepsilon,$
$\displaystyle\|q_{j}-q_{0}\|_{L^{\infty}(B_{1})}$
$\displaystyle\leqslant\varepsilon,\ j\in\\{1,2\\},$
$\displaystyle\|q_{j}-q_{0}\|_{C^{m}(\overline{B}_{1})}$
$\displaystyle\leqslant 1.$

We emphasize that bounds as in Theorem 1.1 directly imply conditions on
possible moduli of continuity for the Calderón problem showing that for the
function spaces from the theorem the modulus of continuity _cannot_ be better
than of logarithmic type. More precisely, if for all potentials $q$ with
$\|q\|_{C^{m}(\overline{B}_{1})}\leqslant 1$ an estimate of the form

$\displaystyle\|q_{1}-q_{2}\|_{L^{\infty}(B_{1})}\leqslant\omega(\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{H^{\frac{1}{2}}(\partial\Omega)\rightarrow
H^{-\frac{1}{2}}(\partial\Omega)})$

holds, then necessarily $\omega(t)\geqslant C|\log(t)|^{-\frac{1}{\gamma}}$.

Mandache’s argument relied on two key observations which were later
systematically investigated and extended by Di Cristo-Rondi [DR03]:

  * (i)

For the setting of the unit ball $B_{1}(0)$ it is possible to explicitly
construct a basis $(f_{j})_{j=1}^{\infty}\subset H^{\frac{1}{2}}(\partial
B_{1}(0))$ such that

$\displaystyle|((\Lambda_{q}-\Lambda_{0})f_{j_{1}},f_{j_{2}})_{L^{2}(\partial
B_{1})}|\leqslant Ce^{-c\max\\{j_{1},j_{2}\\}},$

for some constants $c,C>0$.

  * (ii)

Using (i), on the one hand, Mandache proved an entropy estimate showing that
the map $q\mapsto\Gamma(q):=\Lambda_{q}-\Lambda_{0}$ is highly compressing
(with respect to suitable Hilbert-Schmidt type norms). This means that the
image of a ball in $L^{\infty}(B_{1})$ under the operator $\Gamma$ can be
covered by a relatively small $\delta$-net. On the other hand, the metric
space of admissible potentials has a large $\varepsilon$-discrete set, which
corresponds to a capacity estimate (see Section 3 for the corresponding
definitions). Relying on a pigeonhole argument, the comparison of the entropy
vs the capacity estimate thus leads to the result of Theorem 1.1.

While the argument in (ii) is rather general (and mainly based on entropy and
capacity estimates), (i) is a rather problem specific ingredient in Mandache’s
original work. It highly uses the interplay of the geometry of $B_{1}(0)$ and
the constant coefficient Laplacian in separating variables and working with
spherical harmonics.

Thus, natural questions which arise are:

  * (Q1)

Does the exponential instability remain true in more general geometries which
enjoy less symmetry than the unit ball?

  * (Q2)

Do the instability arguments persist for more general elliptic operators with
variable coefficients (i.e., with
$L_{0}=\partial_{i}a^{ij}\partial_{j}+\partial_{j}b_{j}^{1}+b_{j}^{2}\partial_{j}+c$)?

Our first instability mechanism gives a positive answer to these questions in
the presence of _strong (analytic) smoothing properties_ : We prove that if
the forward operator is analytically smoothing, then exponential instability
in the associated inverse problem is unavoidable. This in particular allows
us, for instance, to reprove Mandache type instability results for Calderón
type inverse problems in rather general (analytic) geometries with analytic
background metrics. One of the novelties here is that we obtain instability
results for general geometries and coefficients under real-analyticity
assumptions, whereas most earlier results have been restricted to special
geometries that allow explicit computations based on separation of variables.
It is possible to establish a rather general, abstract framework for this (see
Section 3).

As an example of this abstract scheme, we present a new proof of the
exponential instability of the Calderón problem relying on abstract smoothing
arguments (see Section 4 for the details and proofs).

######  Theorem 1.2.

Let $(M,g)$ be a compact Riemannian $n$-manifold with smooth boundary
$\partial M$. Let $M^{\prime}\Subset M^{\mathrm{int}}$. Fix $s>\frac{n}{2}$
and $\delta>0$, and suppose that

$\lVert
q_{1}-q_{2}\rVert_{H^{s}(M)}\leqslant\omega(\lVert\Lambda_{q_{1}}-\Lambda_{q_{2}}\rVert_{H^{1/2}(\partial
M)\to H^{-1/2}(\partial M)}),$

when $\lVert q_{j}\rVert_{H^{s+\delta}(M)}\leqslant 1$,
$\mathrm{supp}(q_{j})\subset M^{\prime}$, and $0$ is not a Dirichlet
eigenvalue of $-\Delta_{g}+q_{j}$. Then $\omega(t)$ cannot be a Hölder modulus
of continuity. Moreover, if $M$, $g$ and $\partial M$ are real-analytic, then
$\omega(t)\gtrsim\lvert\log t\rvert^{-\frac{\delta(2n-1)}{n}}$ for $t$ small.

######  Remark 1.3.

We highlight that while the formulation of the instability statement in
Theorem 1.1 may appear to be more quantitative at first sight than the
formulation of our Theorem 1.2, under very mild conditions both types
instability formulations are indeed equivalent. We refer to Remark 3.4 for a
detailed discussion of this.

In contrast to the argument of Mandache, a key feature of this first
instability mechanism is the robustness of our arguments with respect to
(analytic) domain or coefficient perturbations. Instead of constructing
explicit singular value bases by hand as in the first step of Mandache’s
argument, we use certain analytic smoothing properties and abstract
entropy/capacity number arguments. These allow us to conclude the existence of
corresponding singular value bases for which the operator has exponential
decay. We remark that this is _not_ restricted to linear problems and that
similar arguments apply to any inverse problem where the forward operator has
suitable analytic smoothing properties.

###  1.2. Instability through an iterated small regularity gain

While our first instability mechanism from Section 1.1 allows us to generalize
the Mandache type exponential instability results to more general geometries,
it still relies on very strong (real-analytic) smoothing properties of the
forward operators and can only be used if the underlying structures
(coefficients and boundaries) are real-analytic. A further natural question
thus addresses the necessary regularity of the forward operator.

  * (Q3)

Are the analytic smoothing properties of the forward operator necessary for
exponential instability of the inverse problem? Can one find operators with
possibly irregular coefficients so that there are stronger stability results?

In the case of the Calderón problem there are particularly three points which
have an influence on the smoothing properties of the forward operator: These
are the regularity of the underlying domain, the regularity of the
coefficients of the operator $L_{0}$ and the fact that the potential $q_{0}$
is still chosen to vanish near $\partial\Omega$. It is thus natural to wonder
whether it is possible to weaken these assumptions and whether this has an
effect on the instability properties of the inverse problem.

As our second mechanism we show that (in mainly elliptic and parabolic inverse
problems) it is _not_ necessary to have a high degree of smoothing for
instability. In these cases it is not possible to construct operators at lower
regularity for which better stability estimates hold. On the contrary, we show
that by a suitable iteration already a very small amount of regularity in the
equation suffices to deduce (exponential) instability in the inverse problem.
As an example of this type of instability mechanism, we present instability
results for the Calderón problem with scaling critical low regularity metrics
and potentials which are non-smooth up to the boundary and prove that even in
this framework (in which e.g. the unique continuation principle in general
fails) one can prove logarithmic instability.

######  Theorem 1.4.

Let $\Omega\subset\varmathbb{R}^{n}$ with $n\geqslant 2$ be a bounded
Lipschitz domain and let

$L=-\partial_{j}a^{jk}\partial_{k}+b_{j}\partial_{j}+\partial_{j}c_{j}+q_{0}$

with $(a^{jk})\in L^{\infty}(\Omega,\varmathbb{R}^{n\times n})$ symmetric and
uniformly elliptic, $b_{j},c_{j}\in L^{n}(\Omega)$ and $q_{0}\in
L^{\frac{n}{2}}(\Omega)$. Assume that zero is not a Dirichlet eigenvalue of
$L$. Denote the associated Dirichlet-to-Neumann map by $\Lambda_{L}$. Then,
for any $\delta>0$ there are $\varepsilon_{0}>0$ and $C>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$, there exist potentials $q_{1},q_{2}\in
W^{\delta,\frac{n}{2}}(\Omega)$ with

$\displaystyle\|\Lambda_{L+q_{1}}-\Lambda_{L+q_{2}}\|_{H^{\frac{1}{2}}(\partial\Omega)\rightarrow
H^{-\frac{1}{2}}(\partial\Omega)}$
$\displaystyle\leqslant\exp(-C\varepsilon^{-\frac{n}{\delta(2n+1)}}),$
$\displaystyle\|q_{1}-q_{2}\|_{L^{\frac{n}{2}}(\Omega)}$
$\displaystyle\geqslant\varepsilon,$
$\displaystyle\|q_{j}\|_{W^{\delta,\frac{n}{2}}(\Omega)}$
$\displaystyle\leqslant 1,\ j\in\\{1,2\\}.$

In the setting of the Calderón problem this gives a complete answer to the
questions (Q1)-(Q3) showing that even in settings in which the unique
continuation property fails, one can at best hope for logarithmic stability in
the inverse problem.

In contrast to the arguments in the previous section, where we deduced the
decay of the singular values of the relevant maps through a high degree of
regularity, we here deduce the decay of the singular values through an
iteration of only a very small amount of regularity. This improvement allows
us to treat equations and systems with rough coefficients in (relatively)
rough domains. Related observations which however build on the separability of
Green’s functions have been used in the numerical analysis literature to prove
(quantitative) approximation properties by means of hierarchical matrices
[BH03, EZ18]. Also, in these works only minimal regularity – encoded in the
validity of Caccioppoli inequalities – is required.

While the Calderón problem is a prototypical example of a nonlinear inverse
problem to which these ideas apply, we emphasize that they are not restricted
to this specific problem but could also be used to explain the instability of
related inverse problems such as for instance the fractional Calderón problem
[RS18], inverse inclusion or scattering problems as treated in [DR03] and the
deterioration of Lipschitz stability estimates for finite dimensional problems
depending on the available degrees of freedom [Ro06]. As further applications,
we employ these ideas to prove that the unique continuation property for
elliptic equations up to the boundary can at best have a logarithmic modulus
of continuity (see Theorem 5.7). This shows that the classical instability
result of Hadamard [Ha23] (see also [ARRV09]), involving the constant
coefficient Laplacian in a flat domain, remains valid for quite general
coefficients and geometries. The instability of unique continuation is also
closely related to estimates on the _cost of approximation_ in control
theoretic problems. For instance, in Section 5.4 we present a sample result of
this for lower bounds on the cost of controllability for the heat equation
under low regularity assumptions, see Theorem 5.11.

###  1.3. Instability through microlocal smoothing properties

As a last instability mechanism, we discuss microlocal smoothing properties.

In contrast to the previous two mechanisms, this type of instability mechanism
does not require the forward operator to be _globally_ smoothing.
Instabilities of different type can already be deduced from microlocal
smoothing properties. This includes for instance instability properties of the
limited data Radon transform [Qu93, Qu06] as well as the super-polynomial
instability of the geodesic X-ray transform [SU12, MSU15, HU18]. We will
discuss these examples and give an abstract framework for this in Section 6.

A further example to which this applies, and which might be of interest also
to the control theory community, is the logarithmic instability of the unique
continuation property for the wave equation in the absence of the geometric
control condition of Bardos-Lebeau-Rauch [BLR92]:

######  Theorem 1.5.

Let $(M,g)$ be an analytic, closed Riemannian manifold. Let $\Omega\subset M$
be an open set. Assume that $T>2\max\\{\operatorname{dist}(x,\Omega),\ x\in
M\\}$ and assume that there exists a ray of geometric optics which does not
intersect $\Omega\times[0,T]$. Consider the solution to

$\displaystyle\partial_{t}^{2}u-\Delta_{g}u$ $\displaystyle=0\mbox{ in
}M\times(0,T),$ $\displaystyle(u,\partial_{t}u)$
$\displaystyle=(u_{0},u_{1})\mbox{ on }M\times\\{0\\}.$

Suppose that there is an inequality of the form

$\displaystyle\|(u_{0},u_{1})\|_{L^{2}(M)\times
H^{-1}(M)}\leqslant\omega\left(\frac{\|u\|_{L^{2}((0,T),H^{1}(\Omega))}}{\|(u_{0},u_{1})\|_{H^{1}(M)\times
L^{2}(M)}}\right)\|(u_{0},u_{1})\|_{H^{1}(M)\times L^{2}(M)}$

for some modulus of continuity $\omega$. Then for any $\varepsilon>0$ there
exists $C_{\varepsilon}>0$ such that we have $\omega(t)\geqslant
C_{\varepsilon}|\log(t)|^{-\frac{n+2}{n}-\varepsilon}$ for $t$ small.

Although this is a known result in the control theory community (see for
instance [Le92] or the recent article [LL18] providing corresponding stability
estimates), our argument presents a simple and conceptually very clear way of
obtaining this instability property. It shows that in particular our third
mechanism applies in non-elliptic and non-parabolic contexts.

All in all, the three outlined mechanisms provide methods of deducing
instability in inverse and control theory problems in a “soft” but robust way.
In particular our second instability mechanism shows that high order
regularity is not the only mechanism leading to instability. Instability
should rather be viewed as relying on a _strong compression mechanism_.

###  1.4. Outline of the article

The remainder of the article is organised as follows. Section 2 includes some
useful facts related to abstract inverse problems, singular values, Weyl
asymptotics, and Sobolev and Gevrey/real-analytic spaces. In Section 3 we
formulate the abstract instability framework based on entropy and capacity
estimates. Entropy numbers for various embeddings between spaces of functions
or operators are also discussed. Next, in Section 4, we present our first
robust instability mechanism, the “analytic smoothing implies exponential
instability” argument outlined in Section 1.1. As examples, we discuss the
instability of unique continuation and the linearized and nonlinear Calderón
problems for $C^{\infty}$ or analytic coefficients. In Section 5 we then
address the instability arguments in the presence of low regularity as
outlined in Section 1.2. In addition to the low regularity Calderón problem we
also investigate here linear problems such as the backward heat equation or
unique continuation at low regularity. In Section 6 we discuss our last
instability mechanism based on microlocal smoothing properties, and apply this
to the limited data Radon transform, geodesic X-ray transform and an inverse
problem for the transport equation.

The article includes four appendices. Appendix A presents a framework for
instability properties of linear inverse problems based on singular value
arguments. In the linear case this gives an alternative, more direct approach
than the entropy/capacity approach in Section 3. Appendix B includes some
(technical) proofs from Section 2. Appendix C discusses optimal stability for
interior unique continuation, and shows how Carleman estimates can lead to
complementary lower bounds on the singular values of the relevant operators.
Finally, Appendix D includes some facts related to Gevrey/analytic
pseudodifferential operators.

### Notation

We will write $A\lesssim B$ (resp. $A\gtrsim B$) if $A\leqslant CB$ (resp.
$A\geqslant cB$) for some constants $C,c>0$ which do not depend on asymptotic
parameters. We also write $A\sim B$ if $cA\leqslant B\leqslant CA$ for such
constants $C,c>0$. For example, the singular value estimate $\sigma_{k}\sim
k^{-s}$ means that $ck^{-s}\leqslant\sigma_{k}\leqslant Ck^{-s}$ uniformly
over $k\geqslant 1$, and the modulus of continuity estimate $\omega(t)\gtrsim
t^{\alpha}$ for small $t$ means that $\omega(t)\geqslant ct^{\alpha}$ when
$0\leqslant t\leqslant t_{0}$ for some small $t_{0}>0$. We will also use the
Einstein summation convention where a repeated index in upper and lower
position is summed, e.g. $\partial_{j}(a^{jk}\partial_{k}u)$ denotes
$\sum_{j,k=1}^{n}\partial_{j}(a^{jk}\partial_{k}u)$ if $u$ is a function in a
domain in $\varmathbb{R}^{n}$.

### Acknowledgements

The authors are deeply grateful to Luca Rondi, who has made important
contributions to this article. Conversations between the last named author and
Rondi initiated this research, and the presentation has improved considerably
thanks to several helpful comments from Rondi. The authors are also grateful
to Thorsten Hohage, Barbara Kaltenbacher, Richard Nickl and Hongkai Zhao for
providing relevant references, and to the anonymous referees for helpful
suggestions. H.K. was partially supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the Hausdorff
Center for Mathematics under Germany’s Excellence Strategy - EXC-2047/1 -
390685813 and through CRC 1060 - project number 211504053. A.R. was supported
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg
STRUCTURES Cluster of Excellence). She would also like to thank the MPI MIS
where part of this work was carried out. M.S. was supported by the Academy of
Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, grant
numbers 312121 and 309963) and by the European Research Council under Horizon
2020 (ERC CoG 770924).

##  2\. Preliminaries

###  2.1. Abstract setup for stability

Let $F:X\to Y$ be a continuous map (linear or nonlinear) between two metric
spaces. We also call $F$ the _forward operator_. We consider the basic inverse
problem for $F$: given $y\in Y$, find $x\in X$ with

$F(x)=y.$

In this section, we will make the following simplifying assumptions (which
will be true in most of our examples):

  * •

$y\in F(X)$, so that there exists at least one $x\in X$ with $F(x)=y$ (more
generally, one could consider approximate solutions such as regularized or
least squares solutions);

  * •

$F$ is injective, so that there exists at most one $x\in X$ with $F(x)=y$
(more generally, one could consider equivalence classes of solutions such as
those related by some gauge invariance, or minimal norm solutions).

We are interested in cases where the inverse problem for $F$ is _ill-posed_ ,
meaning that the continuous bijective map $F:X\to F(X)$ does not have a
continuous inverse.

It is well known that one can restore some stability in ill-posed problems
when the unknowns are restricted to a compact subset of $X$ (see for instance
[Ti43, Jo60]). This corresponds to imposing _a priori bounds_ on the unknowns,
and leads to _conditional stability results_ which are often formulated in
terms of a _modulus of continuity_ , i.e. an increasing function
$\omega:[0,\infty]\to[0,\infty]$ which is continuous at $0$ with
$\omega(0)=0$. The basis for this is the following simple topological result.

######  Lemma 2.1.

Let $F:X\to Y$ be a continuous injective map between two metric spaces. If $K$
is any compact subset of $X$, the map $F|_{K}:K\to F(K)$ is a homeomorphism
and there is a modulus of continuity $\omega$ such that

$d_{X}(x_{1},x_{2})\leqslant\omega(d_{Y}(F(x_{1}),F(x_{2}))),\qquad
x_{1},x_{2}\in K.$

###### Proof.

The map $F|_{K}:K\to F(K)$ is a continuous bijective map from a compact space
to a Hausdorff space, hence it is a homeomorphism. Its inverse $G:F(K)\to K$
is uniformly continuous since $F(K)$ is compact, and thus

$d_{X}(G(y_{1}),G(y_{2}))\leqslant\omega(d_{Y}(y_{1},y_{2})),\qquad y_{j}\in
F(K),$

for some modulus of continuity $\omega$. The result follows since
$y_{j}=F(x_{j})$ for a unique $x_{j}\in K$. ∎

With the previous observation in hand, we thus define the modulus of stability
for an inverse problem given the spaces $X,Y$ and the compact set $K$:

######  Definition 2.2.

Let $F:X\to Y$ be a continuous map between metric spaces, let $K$ be a subset
of $X$ (usually compact), and let $\omega$ be a modulus of continuity. We say
that the _inverse problem for $F$ is $\omega$-stable in $K$_ if

$d_{X}(x_{1},x_{2})\leqslant\omega(d_{Y}(F(x_{1}),F(x_{2}))),\qquad
x_{1},x_{2}\in K.$

We note that the modulus of continuity $\omega$ in the above definition
depends on

  * •

the forward operator $F$;

  * •

the spaces $X$ and $Y$ and their topologies (there could be several reasonable
choices); and

  * •

the set $K$ (again there could be many possible choices).

In the following sections, given $F$, $X$, $Y$, $K$, we seek to find necessary
conditions for $\omega$ (thus providing bounds on the instability of an
inverse problem).

In many places of this article, we will consider a less general set-up than
the one outlined in Definition 2.2: Often the relevant operator
$A:X\rightarrow Y$ will be linear with $X$, $Y$ normed spaces (or even
separable Hilbert spaces) and $K:=\\{x\in X:\ \|x\|_{X_{1}}\leqslant 1\\}$
where $X_{1}$ is a normed space (or even separable Hilbert space) which embeds
into $X$ compactly. In this setting we will consider two closely related
notions of $\omega$-stability:

  * (a)

$\|x\|_{X}\leqslant\omega(\|Ax\|_{Y})$ for all $x\in X$ with
$\|x\|_{X_{1}}\leqslant 1$,

  * (b)

$\|x\|_{X}\leqslant\omega\left(\frac{\|Ax\|_{Y}}{\|x\|_{X_{1}}}\right)\|x\|_{X_{1}}$
for all $x\in X_{1}\setminus\\{0\\}$.

We note that, on the one hand, by linearity of $A$ and scaling, the stability
condition (a) always implies the condition (b). If, on the other hand, (b) is
satisfied and if $\omega(t)$ is such that “a concavity condition at zero”
holds, i.e.

(2.1) $\displaystyle r\omega\left(\frac{t}{r}\right)\leqslant\omega(t)\mbox{
whenever }0<r\leqslant 1,$

then also the condition (a) holds.

We remark that the condition (2.1) does not pose any essential restrictions,
since any continuous map on a compact metric space admits a concave modulus of
continuity satisfying (2.1). Under these conditions one thus has the
equivalence of (a) and (b).

###  2.2. Singular value decomposition

If the forward map $F$ is a linear, compact operator between separable Hilbert
spaces (in which case we often write $F=A$), then it has a singular value
decomposition and the decay of singular values plays a crucial role in the
instability properties. We recall the properties of singular values that will
be useful for our purposes.

######  Proposition 2.3.

Let $A:X\to Y$ be a compact linear operator between separable Hilbert spaces
with $X$ infinite dimensional. Let
$\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant 0$ be the eigenvalues
of $A^{*}A$, and let $(\varphi_{j})_{j=1}^{\infty}$ be a corresponding
orthonormal basis of $X$ consisting of eigenfunctions. Let
$\sigma_{j}:=\sqrt{\lambda_{j}}$ for $j\geqslant 1$, and let
$\psi_{j}:=\frac{1}{\sigma_{j}}A\varphi_{j}$ when $\sigma_{j}\neq 0$. Then
$\psi_{j}$ are orthonormal in $Y$, and $A$ has the singular value
decomposition

$Au=\sum_{\sigma_{j}\neq 0}\sigma_{j}(u,\varphi_{j})_{X}\psi_{j},\qquad u\in
X.$

Here $(\varphi_{j})_{j=1}^{\infty}$ is called a singular value basis for $A$.
In particular one has

$A\varphi_{j}=\sigma_{j}\psi_{j},\ \ A^{*}\psi_{j}=\sigma_{j}\varphi_{j}\ \
\text{ when $\sigma_{j}\neq 0$}.$

The singular values $\sigma_{j}=\sigma_{j}(A)=\sigma_{j}(A^{*})$ have the
following properties:

  * (a)

$\lVert A\rVert=\sigma_{1}(A)\geqslant\sigma_{2}(A)\geqslant\ldots\geqslant
0$.

  * (b)

If $A$ is injective, then $\sigma_{j}(A)>0$ for all $j\geqslant 1$.

  * (c)

If $B:X\to Y$ is compact, then for all $j,k\geqslant 1$ one has

$\displaystyle\sigma_{j+k-1}(A+B)\leqslant\sigma_{j}(A)+\sigma_{k}(B).$

  * (d)

If $C:X_{1}\to X$ is compact, then for all $j,k\geqslant 1$ one has

$\displaystyle\sigma_{j+k-1}(A\circ C)\leqslant\sigma_{j}(A)\sigma_{k}(C).$

  * (e)

If $\Phi:X_{1}\to X$ and $\Psi:Y\to Y_{1}$ are bounded linear operators, one
has

$\sigma_{j}(A\circ\Phi)\leqslant\sigma_{j}(A)\lVert\Phi\rVert,\qquad\sigma_{j}(\Psi\circ
A)\leqslant\lVert\Psi\rVert\sigma_{j}(A).$

Moreover, if $\Phi$ and $\Psi$ are isometries, then

$\sigma_{j}(A\circ\Phi)=\sigma_{j}(\Psi\circ A)=\sigma_{j}(A).$

###### Proof.

Since $A^{*}A:X\to X$ is compact and self-adjoint, by the spectral theorem
there is an orthonormal basis $(\varphi_{j})_{j=1}^{\infty}$ of $X$ and
eigenvalues $\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant 0$ with
$\lambda_{j}\to 0$ so that $A^{*}A\varphi_{j}=\lambda_{j}\varphi_{j}$. The
expression for $Au$ and the equations $A\varphi_{j}=\sigma_{j}\psi_{j}$ and
$A^{*}\psi_{j}=\sigma_{j}\varphi_{j}$ follow immediately, and one also has
$\sigma_{j}(A)=\sigma_{j}(A^{*})$.

For (a) it is enough to note that
$\sigma_{1}(A)^{2}=\lambda_{1}(A^{*}A)=\lVert A^{*}A\rVert=\lVert
A\rVert^{2}$, and (b) is clear. The standard Weyl inequality in (c) follows
from the Courant minimax principle

$\sigma_{j+k-1}(A+B)=\lambda_{j+k-1}((A+B)^{*}(A+B))^{1/2}=\min_{S}\max_{u\perp
S,\lVert u\rVert=1}\lVert Au+Bu\rVert$

where the minimum is over all subspaces $S\subset X$ with $\dim(S)\leqslant
j+k-2$. Applying the minimax principle to $A^{*}A$ and $B^{*}B$, there are
subspaces $U$ and $V$ of $X$ of dimension $j-1$ and $k-1$, respectively, so
that

$\lVert Au\rVert\leqslant\sigma_{j}(A)\lVert u\rVert,\qquad\lVert
Bv\rVert\leqslant\sigma_{k}(B)\lVert v\rVert$

when $u\perp U$ and $v\perp V$. Let $S$ be a subspace of $X$ containing $U$
and $V$ with $\dim(S)=j+k-2$. Then

$\sigma_{j+k-1}(A+B)\leqslant\max_{u\perp S,\lVert u\rVert=1}\lVert
Au+Bu\rVert\leqslant\sigma_{j}(A)+\sigma_{k}(B).$

For (d) we argue as in (c) and find subspaces $U$ and $V$ of $X$ having
dimension $j-1$ and $k-1$, respectively, so that when $u\perp U$ and $v\perp
V$ one has

$\lVert Au\rVert\leqslant\sigma_{j}(A)\lVert u\rVert,\qquad\lVert
Cv\rVert\leqslant\sigma_{k}(C)\lVert v\rVert.$

Let $S$ be a subspace of $X$ having dimension $j+k-2$ and containing both
$C^{*}U$ and $V$. If $w\perp S$, one has $Cw\perp U$ and

$\lVert ACw\rVert\leqslant\sigma_{j}(A)\lVert
Cw\rVert\leqslant\sigma_{j}(A)\sigma_{k}(C)\lVert w\rVert.$

The minimax principle proves (d). Finally, for (e) note that

$\displaystyle\sigma_{j}(\Psi\circ A)=\min_{S}\max_{u\perp S,\lVert
u\rVert=1}\lVert\Psi(Au)\rVert\leqslant\lVert\Psi\rVert\sigma_{j}(A)$

where the minimum is over subspaces $S$ with dimension $j-1$. It also follows
that $\sigma_{j}(A\circ\Phi)=\sigma_{j}(\Phi^{*}\circ
A^{*})\leqslant\lVert\Phi^{*}\rVert\sigma_{j}(A^{*})=\lVert\Phi\rVert\sigma_{j}(A)$.
The last part of (e) follows by looking at $A^{*}A$ and using that eigenvalues
are preserved under isometries. ∎

###  2.3. Weyl asymptotics

Most of the instability results proved in this article involve some form of
Weyl asymptotics for the eigenvalues or singular values of certain operators.
We will state three such results. The first one is just the usual Weyl law for
the Laplace-Beltrami operator on a compact manifold without boundary.

Let $(M,g)$ be a closed (i.e. compact with no boundary) oriented smooth
Riemannian manifold with $\dim(M)=n$. Then there is a natural $L^{2}$ space
$L^{2}(M)$ equipped with the measure induced by the volume form on $(M,g)$.
The Laplace-Beltrami operator $\Delta_{g}$ of $(M,g)$ is given in local
coordinates by

$\Delta_{g}=\lvert g\rvert^{-1/2}\partial_{j}(\lvert
g\rvert^{1/2}g^{jk}\partial_{k}u)$

where $(g_{jk})$ is the metric $g$ in local coordinates, $(g^{jk})$ is the
inverse matrix of $(g_{jk})$, and $\lvert g\rvert=\det(g_{jk})$. The operator
$-\Delta_{g}$ (with domain $C^{\infty}(M)$) is an unbounded self-adjoint
operator on $L^{2}(M)$ with compact resolvent and hence has a discrete
spectrum. The eigenvalues have the following Weyl asymptotics (see e.g. [Ta96,
Section 8.3 in vol. II]).

######  Theorem 2.4.

Let $(M,g)$ be a closed smooth $n$-manifold, and let
$0=\lambda_{1}\leqslant\lambda_{2}\leqslant\ldots$ be the eigenvalues of
$-\Delta_{g}$ on $L^{2}(M)$. The eigenvalues of $-\Delta_{g}$ satisfy

$\lim_{\lambda\to\infty}\frac{\\#\\{j\,;\,\lambda_{j}\leqslant\lambda\\}}{\lambda^{n/2}}=\frac{\mathrm{Vol}(M)}{\Gamma(\frac{n}{2}+1)(4\pi)^{n/2}}.$

In particular, the nonzero eigenvalues satisfy

$\lambda_{j}\sim j^{\frac{2}{n}}.$

The next result is a Weyl law for the Dirichlet Laplacian with bounded
measurable coefficients [BS72]. It is sufficient for us to state this result
in a bounded Euclidean domain with smooth boundary, but various
generalizations are available (see e.g. [BNR18]).

######  Theorem 2.5.

Let $\Omega\subset\varmathbb{R}^{n}$ be a bounded domain with smooth boundary,
and let $g$ be a symmetric matrix function with $g\in
L^{\infty}(\Omega,\varmathbb{R}^{n\times n})$ and
$g^{jk}(x)\xi_{j}\xi_{k}\geqslant c\lvert\xi\rvert^{2}$ for a.e. $x\in\Omega$
where $c>0$. If $0<\lambda_{1}\leqslant\lambda_{2}\leqslant\ldots$ are the
eigenvalues of the Dirichlet Laplacian $-\Delta_{g}$ on $\Omega$, then

$\lambda_{j}\sim j^{2/n}.$

Finally we state a Weyl law for classical pseudodifferential operators of
negative order on a closed manifold $(M,g)$ that have nonvanishing principal
symbol (i.e. are _elliptic_ or _noncharacteristic_) at some point of $T^{*}M$.
These operators are compact on $L^{2}(M)$, and the behaviour of their singular
values will be useful when studying instability in the presence of microlocal
smoothing effects. The result essentially follows from [BS77b, BS79, Ka15] but
for completeness we give a proof in Appendix B. We refer to [Hö85, Chapter 18]
for the notation and basic facts related to pseudodifferential operators
($\Psi$DOs).

######  Theorem 2.6.

Let $(M,g)$ be a closed smooth $n$-manifold and let
$A\in\Psi^{-m}_{\mathrm{cl}}(M)$ have nonvanishing principal symbol at some
$(x_{0},\xi_{0})\in T^{*}M\setminus\\{0\\}$, where $m>0$. The compact operator
$A:L^{2}(M)\to L^{2}(M)$ satisfies

$\sigma_{j}(A)\sim j^{-m/n}.$

###  2.4. Sobolev spaces

In many inverse problems the forward operator acts between Sobolev type
function spaces. Here we will set up certain spaces that will be relevant for
this article, mostly in the $L^{2}$ setting on $C^{\infty}$ manifolds (more
general $L^{p}$ based spaces will be considered in the low regularity setting
in Section 5).

Let first $(M,g)$ be a closed smooth $n$-manifold. For any $s\in\varmathbb{R}$
one can define the $L^{2}$ based Sobolev space $H^{s}(M)$ using local
coordinates and the space $H^{s}(\varmathbb{R}^{n})$, see [Ta96, Section 4.3
in vol. I]. This is a Hilbert space and there are many equivalent definitions.
For example, if $k\geqslant 0$ is an integer one has the equivalent norm (see
[Be87, Appendix B])

(2.2) $\lVert
u\rVert_{H^{k}(M)}^{2}=\sum_{j=0}^{k}\lVert\nabla^{j}u\rVert_{L^{2}(M)}^{2}$

where $\nabla$ is the total covariant derivative induced by the Levi-Civita
connection on $(M,g)$, and the $L^{2}$ norms on the right are norms on tensor
fields induced by the metric $g$. Moreover, for any $s\in\varmathbb{R}$ we may
consider the Bessel potential

$J^{s}u=(1-\Delta_{g})^{s/2}u.$

Here $J^{s}$ is an elliptic pseudodifferential operator of order $s$ on $M$,
and we may use the equivalent norm $\lVert u\rVert_{H^{s}(M)}=\lVert
J^{s}u\rVert_{L^{2}(M)}$ on $H^{s}(M)$ (see e.g. [Sh87]).

It will be particularly convenient for us that $H^{s}(M)$ can be identified
with a sequence space with polynomial weights. To see this, let
$(\varphi_{j})_{j=1}^{\infty}$ be an orthonormal basis of $L^{2}(M)$
consisting of eigenfunctions of the Laplace-Beltrami operator $-\Delta_{g}$,
corresponding to eigenvalues
$0=\lambda_{1}\leqslant\lambda_{2}\leqslant\ldots\to\infty$. Then

$J^{s}u=\sum_{j=1}^{\infty}(1+\lambda_{j})^{s/2}(u,\varphi_{j})_{L^{2}(M)}\varphi_{j}.$

By Theorem 2.4 one has the Weyl law $1+\lambda_{j}\sim j^{2/n}$ for
$j\geqslant 1$. This gives the following equivalent norm on $H^{s}(M)$:

######  Proposition 2.7.

Let $(M,g)$ be a closed smooth $n$-manifold, and let
$(\varphi_{j})_{j=1}^{\infty}$ be an orthonormal basis of $L^{2}(M)$
consisting of eigenfunctions of $-\Delta_{g}$ as above. Then

$\lVert
u\rVert_{H^{s}(M)}^{2}\sim\sum_{j=1}^{\infty}j^{\frac{2s}{n}}\lvert(u,\varphi_{j})\rvert^{2}.$

The previous result shows that indeed $H^{s}(M)$ is isomorphic to a sequence
space. We will also need such a space in an abstract setting.

######  Definition 2.8.

Let $X$ be a separable Hilbert space and let
$\varphi=(\varphi_{j})_{j=1}^{\infty}$ be an orthonormal basis of $X$. Let
also $n\geqslant 1$. For any $s\in\varmathbb{R}$ define

$J^{s}u=\sum_{j=1}^{\infty}j^{\frac{s}{n}}(u,\varphi_{j})\varphi_{j}$

and the norm

$\lVert u\rVert_{h^{s}}=\lVert
J^{s}u\rVert_{\ell^{2}}=\left(\sum_{j=1}^{\infty}j^{\frac{2s}{n}}\lvert(u,\varphi_{j})\rvert^{2}\right)^{1/2}.$

For $s\geqslant 0$ let $h^{s}=h^{s}_{n,X,\varphi}$ be the subspace of $X$
consisting of elements with finite $h^{s}$ norm, and for $s<0$ let $h^{s}$ be
the completion of $X$ under the $h^{s}$ norm.

Thus Proposition 2.7 states that $H^{s}(M)=h^{s}$ with equivalent norms when
$n=\dim(M)$, $X=L^{2}(M)$, and $\varphi=(\varphi_{j})_{j=1}^{\infty}$ is an
orthonormal basis of $L^{2}(M)$ consisting of eigenfunctions of the Laplace-
Beltrami operator $-\Delta_{g}$. Note that we slightly abuse notation and
write $J^{s}$ both for the Bessel potential $(1-\Delta_{g})^{s/2}$, which is
convenient when we want to use $\Psi$DO properties, and for the sequence space
operator in Definition 2.8.

We will also need Sobolev spaces $H^{s}(M)$ when $(M,g)$ is a compact manifold
with smooth boundary. In this case the $H^{k}(M)$ spaces can be defined using
the norm (2.2), and more generally $H^{s}(M)$ is defined as in [Ta96, Section
4.4 in vol. I]. In the low regularity setting we will use the spaces
$H^{s}(\Omega)$ when $\Omega\subset\varmathbb{R}^{n}$ is a bounded open set
with Lipschitz boundary and also the spaces $H^{t}(\partial\Omega)$ for
$-1\leqslant t\leqslant 1$, see e.g. [Mc00].

For $M$ a smooth, not necessarily compact manifold, we define “localized”
Sobolev spaces:

######  Definition 2.9.

For $M$ a smooth manifold, the space $H^{s}_{L}(M)$ with $L\subset
M^{\text{int}}$ compact will be identified with the space
$H^{s}_{L}(M_{1}):=\\{u\in H^{s}(M_{1})\,;\,\mathrm{supp}(u)\subset L\\}$,
where $M_{1}$ is a (fixed) closed manifold containing an open neighbourhood of
$L$ in $M$.

###  2.5. Normal derivatives of solutions

We record here a standard fact regarding weak normal derivatives of solutions
of elliptic equations, which will be used several times later. Let
$\Omega\subset\varmathbb{R}^{n}$ be a bounded open set with Lipschitz
boundary, and consider the operator

$L=-\partial_{j}a^{jk}\partial_{k}+b_{j}\partial_{j}+\partial_{j}c_{j}+q_{0}$

with $(a^{jk})\in L^{\infty}(\Omega,\varmathbb{R}^{n\times n})$ symmetric and
uniformly elliptic, $b_{j},c_{j}\in L^{n}(\Omega)$ and $q_{0}\in
L^{\frac{n}{2}}(\Omega)$. In terms of scaling this is the roughest possible
framework in which the Dirichlet problem for $L$ is well-posed outside its
spectrum and satisfies the Fredholm alternative. This also includes the case
where $q_{0}\in W^{-1,n}(\Omega)$ (see Remark 5.15).

If $u\in H^{1}(\Omega)$ is a weak solution of $Lu=0$ in $\Omega$, the normal
derivative $\partial_{\nu}^{L}u|_{\partial\Omega}$ is defined weakly as an
element of $H^{-1/2}(\partial\Omega)$ by

(2.3)
$\displaystyle\langle\partial_{\nu}^{L}u|_{\partial\Omega},h\rangle_{\partial\Omega}:=\int\limits_{\Omega}(a^{jk}\partial_{j}u\partial_{k}v+vb_{j}\partial_{j}u-uc_{j}\partial_{j}v+q_{0}uv)\,dx,\
h\in H^{\frac{1}{2}}(\partial\Omega),$

where $v\in H^{1}(\Omega)$ is a function such that $v|_{\partial\Omega}=h$. We
remark that the right hand side of (2.3) is well-defined by the Hölder
inequality and Sobolev embedding and that the weak definition of the normal
derivative is independent of the choice of the extension $v\in H^{1}(\Omega)$
of $h\in H^{\frac{1}{2}}(\partial\Omega)$. Indeed, the latter follows from the
fact that $u$ is a weak solution of the equation $Lu=0$, which means that the
right hand side of (2.3) vanishes if $v\in H^{1}_{0}(\Omega)$.

The definition (2.3) is justified by the fact that when $u$ and the
coefficients are sufficiently regular, an integration by parts shows that
$\partial_{\nu}^{L}u=(a^{jk}\partial_{j}u-c_{k}u)\nu_{k}$. Similar
considerations are valid for second order elliptic operators on a compact
Riemannian manifold $(M,g)$ with smooth boundary.

###  2.6. Gevrey spaces

In cases where the forward operator is analytic smoothing, one needs to
consider spaces of real analytic functions. It is convenient to work more
generally with Gevrey functions. We recall the necessary definitions following
[Gr87, Ro93, HR01, Tr80] and [Hö85, Chapter 8.4] (and use the convention that
$0^{0}=1$).

######  Definition 2.10.

Let $U\subset\varmathbb{R}^{n}$ be an open set, $\sigma\geqslant 1$ and
$f:U\rightarrow\varmathbb{C}$. We say that $f$ is _Gevrey- $\sigma$ regular_,
or $f\in G^{\sigma}(U)$, iff $f\in C^{\infty}(U)$ and for each compact set
$K\subset U$ there exists a constant $C_{K}>1$ such that

$\displaystyle\sup\limits_{x\in K}|\partial^{\alpha}f(x)|\leqslant
C_{K}^{|\alpha|+1}|\alpha|^{\sigma|\alpha|}\mbox{ with
}\alpha\in(\varmathbb{N}\cup\\{0\\})^{n}.$

Further, $G^{\sigma}_{c}(U):=G^{\sigma}(U)\cap C^{\infty}_{c}(U)$.

We note that the space $G^{1}(U)$ corresponds to the real analytic functions
and thus $G_{c}^{1}(U)=\emptyset$. This leads to a number of technical
difficulties, since partitions of unity are not available and one needs to
work with sequences of cutoff functions that are “analytic up to a finite
order” (see Lemma D.4). However, for $\sigma>1$, we have that
$G_{c}^{\sigma}(U)\neq\emptyset$ (even with $G_{c}^{\sigma}(U)$ dense in
function spaces like $C_{c}^{\infty}(U)$ or in $L^{1}_{\mathrm{loc}}(U)$, see
[Ro93, Section 1.4]). In the sequel, for simplicity, we will in certain
applications restrict our attention to the case $\sigma>1$. We remark that
Gevrey spaces for $\sigma>1$ allow for the construction of partitions of
unity, implying that all local definitions can also be transferred to
manifolds with $G^{\sigma}$ atlases.

Next we consider certain Hilbert spaces of Gevrey functions on closed
manifolds, defined in terms of Fourier coefficients. We first define an
abstract sequence space. Here it is natural to consider sequences over
$j\geqslant 0$ instead of $j\geqslant 1$.

######  Definition 2.11.

Let $X$ be a separable Hilbert space and let
$\varphi=(\varphi_{j})_{j=0}^{\infty}$ be an orthonormal basis of $X$. Let
also $n\geqslant 1$. For $1\leqslant\sigma<\infty$ and $\rho>0$, define

$\lVert u\rVert_{a^{\sigma,\rho}}=\left(\sum_{j=0}^{\infty}e^{2\rho
j^{\frac{1}{n\sigma}}}\lvert(u,\varphi_{j})\rvert^{2}\right)^{1/2}.$

Let $a^{\sigma,\rho}=a^{\sigma,\rho}_{n,X,\varphi}$ be the subspace of $X$
consisting of elements with finite $a^{\sigma,\rho}$ norm.

Now consider a closed smooth $n$-manifold $(M,g)$. Let
$\varphi=(\varphi_{j})_{j=0}^{\infty}$ be an orthonormal basis of
eigenfunctions for $-\Delta_{g}$ with eigenvalues
$0=\lambda_{0}\leqslant\lambda_{1}\leqslant\lambda_{2}\leqslant\ldots$. By
Weyl asymptotics, $\lambda_{j}\sim j^{2/n}$ for $j\geqslant 1$. Given
$1\leqslant\sigma<\infty$ and $\rho>0$, consider the Hilbert space

(2.4) $\displaystyle A^{\sigma,\rho}(M)=a^{\sigma,\rho}_{n,L^{2}(M),\varphi}$

with the norm

$\lVert u\rVert_{A^{\sigma,\rho}}:=(\sum_{j=0}^{\infty}e^{2\rho
j^{\frac{1}{n\sigma}}}\lvert(u,\varphi_{j})\rvert^{2})^{1/2}.$

Clearly $A^{\sigma,\rho}(M)\subset C^{\infty}(M)$. For $(M,g)$ analytic, we
also define the Gevrey space (with the convention $0^{0}=1$)

$G^{\sigma}(M)=\\{u\in C^{\infty}(M)\,;\,\text{there is $C>0$ with
$\lVert\nabla^{k}u\rVert_{L^{\infty}(M)}\leqslant C^{k+1}k^{\sigma k}$ for
$k\geqslant 0$}\\}.$

Then $G^{1}(M)$ is the space of real-analytic functions.

The connection of $G^{\sigma}(M)$ with the Hilbert spaces $A^{\sigma,\rho}(M)$
is given in Lemma B.1, which states that

$G^{\sigma}(M)=\cup_{\rho>0}A^{\sigma,\rho}(M).$

The point is roughly that functions satisfying
$\lvert\nabla^{k}u\rvert\leqslant CR^{k}k^{\sigma k}$ for _fixed_ $R>0$ form a
Banach space (where $C>0$ corresponds to the norm), and the Fourier
coefficients of these functions satisfy $\lvert(u,\varphi_{j})\rvert\leqslant
C^{\prime}e^{-cj^{\frac{1}{n\sigma}}}$ for some _fixed_ $c>0$ leading to the
spaces $A^{\sigma,\rho}(M)$ above. We refer to Appendix B for more details.

Similarly as in Definition 2.9 we also use the following localized
$A^{\sigma,\rho}$ spaces:

######  Definition 2.12.

Let $(M,g)$ be a smooth manifold and $L\subset M^{\text{int}}$ compact. We
define

$A^{\sigma,\rho}_{L}(M):=\\{u\in
A^{\sigma,\rho}(M_{1})\,;\,\mathrm{supp}(u)\subset L\\}\,.$

Here $M_{1}$ is a (fixed) closed manifold containing a neighbourhood of $L$.

We remark that by virtue of Lemma B.1 and the connection to the Gevrey spaces,
the choice of $M_{1}$ has an influence on the value of $\rho$ but not the one
of $\sigma$.

##  3\. Abstract framework for instability

In this section, following ideas introduced by Mandache [Ma01] in the context
of the Calderón problem (see also [SV93] for an even earlier application of
these ideas in the specific context of analytic continuation), we exploit the
observation that the instability of a map can be encoded in entropy and
capacity estimates in suitable function spaces. These quantities measure
complexity in an information-theoretic sense (see the Appendix II in [KT59]).
These and related concepts have also been used in the context of statistical
minimax theory (see [GN21, Chapter 6.3 and Theorem 6.3.2] and the references
therein) and, more recently, in machine learning contexts [BGKP19]. While
using the same notions and objects as in these works, in our context and
applications, we have to make problem-dependent choices and develop the theory
for these (e.g. including general complexity estimates on certain operator
spaces, see Section 3.5). We stress that this approach works equally well for
linear and nonlinear inverse problems.

We recall that the argument of Mandache relied on two main steps:

  * (i)

the construction of an explicit orthonormal basis with (exponential) decay,

  * (ii)

general entropy and capacity estimates.

While (ii) consisted of a very general argument, Mandache strongly exploited
symmetries of the domain and operator to infer (i).

In the sequel, we argue that step (i) is in fact not required: instability can
be deduced from pure singular value, entropy and capacity considerations. In
particular, this allows us to discard the strong symmetry assumptions that
have been used in Mandache’s argument and all adaptations of it (see for
instance [DR03, Is11, Is13a, Is13b]). Examples for applications of this in
inverse problems will be discussed in Section 4.

In Section 5 we extend this idea even further and prove that it is also
applicable if only a minimal amount of regularization is available.

###  3.1. General principle

First we address step (ii) in the argument of Mandache and formulate this as a
general principle. To this end, we begin by recalling the notions of
$\varepsilon$-discrete sets and $\delta$-nets.

######  Definition 3.1.

Let $(X,d_{X})$ denote a metric space and let $\delta>0$. A set $X_{1}\subset
X$ is a $\delta$-net for $X$, if for every $x\in X$ there exists $\tilde{x}\in
X_{1}$ such that $d_{X}(x,\tilde{x})\leqslant\delta$.

Let $\varepsilon>0$. A subset $X_{2}\subset X$ is an $\varepsilon$-discrete
subset of $X$, if for each pair $x,\tilde{x}\in X_{2}$ it holds
$d_{X}(x,\tilde{x})\geqslant\varepsilon$.

Informally speaking, on the one hand, $\varepsilon$-discrete sets measure how
large a function space is, by providing lower bounds on its “extendedness”. On
the other hand, $\delta$-nets measure how compact a space is by yielding upper
bounds on its size. In order to deduce instability results for a forward map
$F:X\to Y$ restricted to a compact set $K\subset X$, it is enough to show that
the image $F(K)$ in $Y$ is “compressed” (meaning that it can be covered by
relatively few $\delta$-balls), while $K$ is sufficiently “extended” (meaning
that it has many points at least $\varepsilon$ apart).

The two notions are closely related:

######  Lemma 3.2.

Let $X$ be a metric space and let $\delta>0$.

  * (a)

Let $A$ be a $\delta$-net and $B$ be $\varepsilon$-discrete where
$\varepsilon>2\delta$. Then $\\#B\leqslant\\#A$.

  * (b)

A maximal $\delta$-discrete set is a $\delta$-net.

  * (c)

If $f:X\to Y$ is continuous with modulus of continuity $\mu(r)$, then a
$\delta$-net of $X$ is mapped to a $\mu(\delta)$-net of $f(X)$. If $\rho$ is a
monotonically increasing function and

$d(x,y)\leqslant\rho(d(f(x),f(y))),$

the image of a $\delta$-discrete set in $X$ is $\rho^{-1}(\delta)$-discrete in
$f(X)$.

###### Proof.

For (a), it suffices to observe that for any point $x\in A$ there is at most
one point $y\in B$ with $d(x,y)\leqslant\delta$. For (b) we note that if $S$
is a maximal $\delta$-discrete set, then $S\cup\\{x\\}$ cannot be
$\delta$-discrete for any $x\notin S$, which means that for any $x\notin S$
there is $y\in S$ with $d(x,y)<\delta$. Thus $S$ is a $\delta$-net. The claims
in (c) also follow directly from the definition. The claim on the
$\delta$-nets follows by the definition of continuity. The claim on the
$\delta$-discrete sets follow from the fact that if $x,y$ are elements in a
$\delta$-discrete set, then

$\displaystyle\delta\leqslant d(x,y)\leqslant\rho(d(f(x),f(y))).$

Inverting $\rho$ yields the claim. ∎

The closely related concepts of _entropy_ and _capacity_ of a metric space are
discussed in [KT59, ET08].

With the notions of $\delta$-nets and $\varepsilon$-discrete sets in hand, we
formulate the second step (ii) of the instability argument of Mandache [Ma01]
in a general framework. This framework has the advantage that one can consider
mappings between general metric spaces (it is not necessary to work with
Sobolev type spaces).

######  Theorem 3.3 ([Ma01]).

Let $X$ and $Y$ be metric spaces, let $K\subset X$ be compact, and let
$F:K\rightarrow Y$ be an injective map. Let $f,g$ be strictly decreasing
continuous functions on $\varmathbb{R}_{+}$, with $f(0+)=g(0+)=+\infty$, so
that for any sufficiently small $\delta,\varepsilon>0$,

  * (i)

there is a $\delta$-net $Y_{\delta}\subset Y$ for $F(K)$ with $\leqslant
f(\delta)$ elements,

  * (ii)

there is a $\varepsilon$-discrete set $X_{\varepsilon}\subset K$ with
$>g(\varepsilon)$ elements.

Then the following statements hold:

  1. (a)

For any small $\varepsilon>0$ there are $x_{1},x_{2}$ such that

(3.1) $\displaystyle x_{1},x_{2}\in K,\qquad
d_{X}(x_{1},x_{2})\geqslant\varepsilon,\qquad
d_{Y}(F(x_{1}),F(x_{2}))\leqslant 2f^{-1}(g(\varepsilon)).$

  2. (b)

If $\omega$ is a modulus of continuity such that

(3.2) $d_{X}(x_{1},x_{2})\leqslant\omega(d_{Y}(F(x_{1}),F(x_{2}))),\qquad
x_{1},x_{2}\in K,$

then $\omega(t)\geqslant g^{-1}(f(t/2))$ for $t$ small.

We deduce the result as a consequence of the pigeonhole principle.

###### Proof.

Let $\delta>0$ be small, and let $Y_{\delta}\subset Y$ be a $\delta$-net for
$F(K)$ with $\lvert Y_{\delta}\rvert\leqslant f(\delta)$. Let $\varepsilon>0$
be such that $f(\delta)=g(\varepsilon)$, i.e. $\varepsilon=g^{-1}(f(\delta))$.
Then there is a $\varepsilon$-discrete set $X_{\varepsilon}\subset K$ with
$\lvert X_{\varepsilon}\rvert>g(\varepsilon)$. It follows that

(3.3) $\displaystyle\lvert Y_{\delta}\rvert\leqslant
f(\delta)=g(\varepsilon)<\lvert X_{\varepsilon}\rvert.$

By the pigeonhole principle, there exist $x_{1},x_{2}\in X_{\varepsilon}$ such
that

$\displaystyle d_{Y}(F(x_{1}),F(x_{2}))\leqslant 2\delta,\mbox{ but
}d_{X}(x_{1},x_{2})\geqslant\varepsilon.$

This proves (a) since $\delta=f^{-1}(g(\varepsilon))$. If (3.2) holds, by the
monotonicity of $\omega$, one has

$\varepsilon\leqslant\omega(2\delta).$

Thus $\omega(t)\geqslant g^{-1}(f(t/2))$ for $t>0$ small, proving (b). ∎

######  Remark 3.4.

It is easy to see that (a) and (b) above are equivalent, if $F$ is required to
be continuous. Indeed, assume that (a) is valid. If (3.2) holds, then choosing
$\varepsilon>0$ small and $x_{1},x_{2}\in K$ as in (a) gives that
$\varepsilon\leqslant\omega(2f^{-1}(g(\varepsilon))$, which implies (b).
Conversely, assume that $F$ is continuous and (b) is valid. By Lemma 2.1 there
is a modulus of continuity $\omega$ such that

$d_{X}(x_{1},x_{2})\leqslant\omega(d_{Y}(F(x_{1}),F(x_{2}))),\qquad
x_{1},x_{2}\in K.$

Here we may replace $\omega$ by the corresponding minimal modulus of
continuity. Then (b) gives that $\omega(t)\geqslant g^{-1}(f(t/2))$. On the
other hand, the minimal modulus satisfies

$\omega(t)=\sup\\{d_{X}(x_{1},x_{2})\,;\,d_{Y}(F(x_{1}),F(x_{2}))\leqslant t,\
x_{j}\in K\\}.$

By compactness the supremum is in fact a maximum, and hence given $t>0$ there
are $x_{1},x_{2}\in K$ with $d_{Y}(F(x_{1}),F(x_{2}))\leqslant t$ and
$d_{X}(x_{1},x_{2})\geqslant\omega(t)$. Since $\omega(t)\geqslant
g^{-1}(f(t/2))$, the claim in (a) follows.

We emphasise that the assumption on the continuity of $F$ is natural and is
always true in our applications (see for instance Section 2 where our general
setting is described). Thus it makes no difference if the instability results
are formulated in the form of (a) or in the form of (b) above.

###  3.2. Entropy numbers

Our first instability mechanism is based on global smoothing properties of the
forward map $F$. Here we assume that $F$ maps into a Banach space $Y$. If
$F(K)$ is contained in a “smooth” or “compressed” subspace $Y_{1}$ of $Y$, we
may write

$F|_{K}=i\circ\tilde{F}$

where $\tilde{F}:K\to Y_{1}$ satisfies $\tilde{F}(x)=F(x)$, and $i$ is the
inclusion $Y_{1}\to Y$. If $\tilde{F}$ is continuous, then $F(K)\subset i(B)$
where $B$ is a ball in $Y_{1}$. To show that $F(K)$ is compressed, it is
enough to show that $i(B)$ can be covered by relatively few $\delta$-balls in
$Y$.

The notion of _entropy numbers_ (see [CS90, ET08]), which measure the
compactness of a linear operator, is ideally suited to the setup described
above. We also introduce the related _capacity numbers_ [CS90].

######  Definition 3.5.

Let $X,Y$ be Banach spaces and let $A:X\to Y$ be a bounded linear operator.
Put $U_{X}:=\\{x\in X:\|x\|_{X}\leqslant 1\\}$. Then for any $k\geqslant 1$,
the $k$th entropy number $e_{k}(A)$ of $A$ is defined by

$\displaystyle e_{k}(A)$
$\displaystyle:=\inf\left\\{\varepsilon>0:A(U_{X})\subset\bigcup\limits_{j=1}^{2^{k-1}}(y_{j}+\varepsilon
U_{Y})\mbox{ for some }y_{1},\dots,y_{2^{k-1}}\in Y\right\\}$
$\displaystyle=\inf\\{\varepsilon>0:\text{ there exists an $\varepsilon$-net
for $A(U_{X})$ of cardinality $2^{k-1}$}\\}.$

The $k$th capacity number $c_{k}(A)$ of $A$ is

$\displaystyle c_{k}(A)$ $\displaystyle:=\sup\\{\varepsilon>0:\ \text{there
are $x_{1},\ldots,x_{N}\in U_{X}$ with $N>2^{k-1}$ and}$
$\displaystyle\qquad\qquad\qquad\qquad\text{$d_{Y}(Ax_{j},Ax_{k})\geqslant
2\varepsilon$ for $j\neq k$}\\}$ $\displaystyle=\sup\\{\varepsilon>0:\text{
there is a $2\varepsilon$-discrete set for $A(U_{X})$ of cardinality
$2^{k-1}+1$}\\}.$

The entropy and capacity numbers are very similar (see also [CS90, Section
1.1]):

######  Lemma 3.6.

One has

(3.4) $\frac{1}{2}e_{k}(A)\leqslant c_{k}(A)\leqslant e_{k}(A).$

###### Proof.

Let $\varepsilon>e_{k}(A)$, so that
$A(U_{X})\subset\cup_{j=1}^{2^{k-1}}\overline{B(y_{j},\varepsilon)}$. Let also
$\rho<c_{k}(A)$, so that there is a $2\rho$-discrete set $S$ in $A(U_{X})$
with cardinality $2^{k-1}+1$. Then some two points $z,w\in S$ must lie in some
ball $\overline{B}(y_{j},\varepsilon)$. One has $2\rho\leqslant
d(z,w)\leqslant 2\varepsilon$, which proves that $c_{k}(A)\leqslant e_{k}(A)$.

On the other hand, if $c_{k}(A)<\rho$, then there is a maximal
$2\rho$-discrete set $S$ in $A(U_{X})$ with cardinality $\leqslant 2^{k-1}$.
By Lemma 3.2 $S$ is also a $2\rho$-net, showing that $e_{k}(A)\leqslant
2\rho$. Thus $e_{k}(A)\leqslant 2c_{k}(A)$. ∎

By (3.4) it will be sufficient to focus on entropy number estimates. The
following simple result shows how typical entropy number bounds could be used
for showing instability based on Theorem 3.3.

######  Lemma 3.7.

Let $X,Y$ be Banach spaces, and let $X_{1}\subset X$ and $Y_{1}\subset Y$ be
closed subspaces so that the inclusions $i:X_{1}\to X$ and $j:Y_{1}\to Y$ are
compact. Let $K=\\{x\in X\,;\,\lVert x\rVert_{X_{1}}\leqslant r\\}$ for some
$r>0$, and assume that $F:K\to Y$ is a map such that $F(K)\subset\\{y\in
Y_{1}\,;\,\lVert y\rVert_{Y_{1}}\leqslant R\\}$ for some $R>0$. Suppose that,
for some function $\omega:\varmathbb{R}_{+}\to\varmathbb{R}_{+}$,

$\lVert x_{1}-x_{2}\rVert_{X}\leqslant\omega(\lVert
F(x_{1})-F(x_{2})\rVert_{Y}),\qquad x_{1},x_{2}\in K.$

  1. (a)

If $e_{k}(i)\gtrsim k^{-m}$ and $e_{k}(j)\lesssim k^{-s}$ for some $m,s>0$,
then for $t$ small

$\omega(t)\gtrsim t^{m/s}.$

  2. (b)

If $e_{k}(i)\gtrsim k^{-m}$ and $e_{k}(j)\lesssim e^{-ck^{\alpha}}$ for some
$m,c,\alpha>0$, then for $t$ small

$\omega(t)\gtrsim\lvert\log t\rvert^{-m/\alpha}.$

  3. (c)

If $e_{k}(i)\gtrsim e^{-dk^{\beta}}$ and $e_{k}(j)\lesssim e^{-ck^{\alpha}}$
for some $c,d,\alpha,\beta>0$, then for $t$ small

$\omega(t)\gtrsim e^{-c_{1}\lvert\log t\rvert^{\beta/\alpha}}.$

###### Proof.

The main point is to use the entropy bounds to estimate the functions
$f(\delta)$ and $g(\varepsilon)$ in Theorem 3.3.

(a) If $e_{k}(i)\geqslant c_{0}k^{-m}$, by (3.4) one has
$c_{k}(i)\geqslant\frac{c_{0}}{2}k^{-m}$, and thus for any
$\varepsilon<\frac{c_{0}}{2}rk^{-m}$ there is a set $X_{\varepsilon}\subset K$
so that $\lvert X_{\varepsilon}\rvert>2^{k-1}$ and $X_{\varepsilon}$ is
$\varepsilon$-discrete in $X$. Now

$\varepsilon=\frac{c_{0}}{2}rk^{-m}\quad\Longleftrightarrow\quad
2^{k-1}=\frac{1}{2}e^{((\frac{c_{0}r}{2})^{1/m}\log 2)\varepsilon^{-1/m}}.$

Thus one may choose

$g(\varepsilon)=e^{c^{1/m}\varepsilon^{-1/m}},\qquad
c=\frac{c_{0}r}{2^{m+1}}.$

The inverse function is $g^{-1}(\eta)=c(\log\eta)^{-m}$.

If $e_{k}(j)<C_{0}k^{-s}$, by the bound on $F(K)$, the set $F(K)$ can be
covered with $2^{k-1}$ balls of radius $RC_{0}k^{-s}$. Now

$\delta=RC_{0}k^{-s}\quad\Longleftrightarrow\quad
2^{k-1}=\frac{1}{2}e^{((C_{0}R)^{1/s}\log 2)\delta^{-1/s}}.$

Hence, if $\delta\geqslant RC_{0}k^{-s}$, there is a $\delta$-net $Y_{\delta}$
for $F(K)$ with $\lvert Y_{\delta}\rvert\leqslant
e^{(C_{0}R)^{1/s}\delta^{-1/s}}$. This proves that one may choose

$f(\delta)=e^{C\delta^{-1/s}},\qquad C=(C_{0}R)^{1/s}.$

By Theorem 3.3, for $t$ small one has

$\displaystyle\omega(t)$ $\displaystyle\geqslant
g^{-1}(f(t/2))=c(C(t/2)^{-1/s})^{-m}\gtrsim t^{m/s}.$

(b) We can use the same $g(\varepsilon)$ as in part (b). If $e_{k}(j)\leqslant
C_{0}e^{-ck^{\alpha}}$ for some $C_{0},c,\alpha>0$, then a computation shows
that one may choose for $\delta$ small

$f(\delta)=e^{C\lvert\log\delta\rvert^{1/\alpha}}.$

It follows that $\omega(t)\geqslant g^{-1}(f(t/2))\gtrsim\lvert\log
t\rvert^{-m/\alpha}$.

(c) We can take $f(\delta)$ as in part (b), and a computation shows that one
may take

$g(\varepsilon)=e^{C_{1}\lvert\log\varepsilon\rvert^{1/\beta}}.$

The inverse function is $g^{-1}(\eta)=e^{-c_{0}(\log\eta)^{\beta}}$, and
$\omega(t)\geqslant g^{-1}(f(t/2))\gtrsim e^{-c_{1}\lvert\log
t\rvert^{\beta/\alpha}}$. ∎

###  3.3. Properties of entropy numbers

In the remainder of this section we will study estimates for entropy numbers.
We note that entropy numbers have the following properties (cf. Chapter 1.3 in
[ET08]).

######  Lemma 3.8.

Let $X,Y,Z$ be Banach. For bounded linear operators $A,B:X\to Y$ and $C:Y\to
Z$ with $A\not\equiv 0$, we have the following properties:

  * (i)

$\|A\|=e_{1}(A)\geqslant e_{2}(A)\geqslant\ldots>0$.

  * (ii)

For all $j,k\geqslant 1$ one has

$\displaystyle e_{j+k-1}(C\circ A)\leqslant e_{j}(C)e_{k}(A).$

  * (iii)

For all $j,k\geqslant 1$ one has

$\displaystyle e_{j+k-1}(A+B)\leqslant e_{j}(A)+e_{k}(B).$

  * (iv)

$A$ is compact if and only if $e_{k}(A)\to 0$ as $k\to\infty$.

  * (v)

$A$ has finite rank if and only if $e_{k}(A)\lesssim e^{-ck}$ for some $c>0$.

  * (vi)

If $X$ and $Y$ are Hilbert spaces, then
$e_{k}(A)=e_{k}(A^{*})=e_{k}(\sqrt{A^{*}A})$.

The above result shows that entropy numbers are indeed related to compactness.
Singular values also encode compactness properties of linear operators. There
are various relations between these two notions (see [CS90, ET08]). For us,
the following result stating the equivalence of typical decay rates will be
sufficient.

######  Lemma 3.9.

Let $A:X\to Y$ be a compact operator between separable Hilbert spaces. For any
$s>0$, one has

$\displaystyle\sigma_{k}(A)\lesssim k^{-s}\quad$
$\displaystyle\Longleftrightarrow\quad e_{k}(A)\lesssim k^{-s},$
$\displaystyle\sigma_{k}(A)\gtrsim k^{-s}\quad$ $\displaystyle\implies\quad
e_{k}(A)\gtrsim k^{-s}.$ Moreover, if $\mu>0$, then
$\displaystyle\sigma_{k}(A)\lesssim e^{-ck^{\mu}}\text{ for some $c>0$}\quad$
$\displaystyle\Longleftrightarrow\quad e_{k}(A)\lesssim
e^{-\tilde{c}k^{\frac{\mu}{1+\mu}}}\text{ for some $\tilde{c}>0$},$
$\displaystyle\sigma_{k}(A)\gtrsim e^{-ck^{\mu}}\text{ for some $c>0$}\quad$
$\displaystyle\implies\quad e_{k}(A)\gtrsim
e^{-\tilde{c}k^{\frac{\mu}{1+\mu}}}\text{ for some $\tilde{c}>0$}.$

###### Proof.

The argument is based on [CS90, Proposition 1.3.2], which states that for any
$N\geqslant 1$,

(3.5) $\sup_{k\geqslant
1}2^{-\frac{N-1}{2k}}(\sigma_{1}(A)\cdots\sigma_{k}(A))^{1/k}\leqslant
e_{N}(A)\leqslant 6\,\sup_{k\geqslant
1}2^{-\frac{N-1}{2k}}(\sigma_{1}(A)\cdots\sigma_{k}(A))^{1/k}.$

In fact this is stated in [CS90] for diagonal operators $\ell^{2}\to\ell^{2}$,
but $\sqrt{A^{*}A}$ is unitarily equivalent to such an operator $D$ and
unitary equivalence does not affect singular values or entropy numbers. Thus
the result holds first for $D$, and then also for $A$ since
$\sigma_{j}(\sqrt{A^{*}A})=\sigma_{j}(A)$ and $e_{j}(\sqrt{A^{*}A})=e_{j}(A)$.
We rewrite (3.5) in the more convenient form, with $c_{1}>0$ an absolute
constant,

(3.6) $\sup_{k\geqslant
1}e^{-\frac{N}{2k}}(\sigma_{1}\cdots\sigma_{k})^{1/k}\leqslant
e_{N}\lesssim\,\sup_{k\geqslant
1}e^{-c_{1}N/k}(\sigma_{1}\cdots\sigma_{k})^{1/k}.$

If $\sigma_{j}\lesssim j^{-s}$, then
$(\sigma_{1}\cdots\sigma_{k})^{1/k}\lesssim(k!)^{-s/k}\sim k^{-s}$ by
Stirling’s formula. Now optimizing the function $f(k):=ck^{-s}e^{-c_{1}N/k}$
in $k$ yields that $k\sim c_{s}N$ and thus (3.6) gives $e_{N}\lesssim N^{-s}$.
Conversely, if $e_{N}\lesssim N^{-s}$, then choosing $k=N$ on the left of
(3.6) and using $\sigma_{1}\geqslant\sigma_{2}\geqslant\ldots$ gives
$\sigma_{N}\lesssim N^{-s}$. The statement that $\sigma_{j}\gtrsim j^{-s}$
implies $e_{N}\gtrsim N^{-s}$ follows from the Stirling formula as above.

Now if $\sigma_{k}\lesssim e^{-ck^{\mu}}$, then evaluating a Riemann sum gives

$(\sigma_{1}\cdots\sigma_{k})^{1/k}\lesssim e^{-\frac{c}{\mu+1}k^{\mu}}.$

Given $N$, the expression $e^{-c_{1}N/k}e^{-\frac{c}{\mu+1}k^{\mu}}$ is
maximal when $k\sim N^{\frac{1}{1+\mu}}$. Choosing this value of $k$ on the
right of (3.6) gives $e_{N}\lesssim e^{-\tilde{c}N^{\frac{\mu}{1+\mu}}}$.
Conversely, assume $e_{N}\lesssim e^{-\tilde{c}N^{\frac{\mu}{1+\mu}}}$. Fixing
$j\geqslant 1$, choosing $k=j$ on the left of (3.6) and using that
$\sigma_{1}\geqslant\sigma_{2}\geqslant\ldots$ gives

$\sigma_{j}\lesssim e^{c_{0}N/j-\tilde{c}N^{\frac{\mu}{1+\mu}}},\qquad
N\geqslant 1.$

The right hand side is minimal when $N\sim j^{1+\mu}$, and this choice gives
$\sigma_{j}\lesssim e^{-cj^{\mu}}$ for some $c>0$ as required. Finally, if
$\sigma_{j}\gtrsim e^{-cj^{\mu}}$, then a Riemann sum argument gives
$(\sigma_{1}\cdots\sigma_{k})^{1/k}\gtrsim
e^{-\frac{c}{\mu+1}\frac{(k+1)^{\mu+1}}{k}}\gtrsim e^{-\tilde{c}k^{\mu}}$. By
(3.6) one has $e_{N}\gtrsim\sup_{k\geqslant
1}e^{-c_{0}N/k}e^{-\tilde{c}k^{\mu}}$, and choosing $k\sim
N^{\frac{1}{1+\mu}}$ gives $e_{N}\gtrsim e^{-\tilde{c}N^{\frac{\mu}{1+\mu}}}$.
∎

###  3.4. Embeddings between Sobolev or Gevrey spaces

Recall from the beginning of this subsection that we are interested in entropy
numbers of inclusions $i:Y_{1}\to Y$, where $Y_{1}$ is a subspace of a Banach
space $Y$. The case of Sobolev type spaces, at least on domains in
$\varmathbb{R}^{n}$, is well understood [ET08]. The estimates on manifolds are
similar, and we give a proof for completeness.

######  Theorem 3.10.

Let $M$ be a compact smooth $n$-manifold with or without smooth boundary, and
let $s_{1},s_{2}\in\varmathbb{R}$ with $s_{1}>s_{2}$. The embedding
$i:H^{s_{1}}(M)\to H^{s_{2}}(M)$ satisfies

$\displaystyle\sigma_{k}(i)\sim k^{-(s_{1}-s_{2})/n},$ $\displaystyle
e_{k}(i)\sim k^{-(s_{1}-s_{2})/n}.$

The same bounds hold for the embedding $i:h^{s_{1}}_{n,X,\varphi}\to
h^{s_{2}}_{n,X,\varphi}$ (see Definition 2.8).

###### Proof.

By Lemma 3.9, it is enough to prove the statement about singular values.
Assume first that $M$ has no boundary, and consider the norm $\lVert
f\rVert_{H^{s}}=\lVert J^{s}f\rVert_{L^{2}}$ where

$J^{s}f=\sum_{j=1}^{\infty}j^{\frac{s}{n}}(f,\varphi_{j})_{L^{2}}\varphi_{j}$

as in Definition 2.8. Note that

$(i^{*}i(f),h)_{H^{s_{1}}}=(f,h)_{H^{s_{2}}}=(J^{-2(s_{1}-s_{2})}f,h)_{H^{s_{1}}}.$

Thus $i^{*}i=J^{-2(s_{1}-s_{2})}$ on $H^{s_{1}}(M)$, and using the isometries
$J^{\pm s_{1}}$ we have

$\displaystyle\sigma_{k}(i)^{2}$
$\displaystyle=\sigma_{k}(i^{*}i)=\sigma_{k}(J^{s_{1}}\circ i^{*}i\circ
J^{-s_{1}})=\sigma_{k}(J^{-2(s_{1}-s_{2})}:L^{2}(M)\to L^{2}(M))$
$\displaystyle=k^{-\frac{2(s_{1}-s_{2})}{n}}.$

In the last step we used that $J^{-2(s_{1}-s_{2})}:L^{2}(M)\to L^{2}(M)$ is a
diagonal operator.

Next assume that $M$ is compact with smooth boundary. Let $N$ be a closed
manifold containing $M$, let $E:H^{s_{1}}(M)\to H^{s_{1}}(N)$ be a bounded
extension operator, and let $R:H^{s_{2}}(N)\to H^{s_{2}}(M),Ru=u|_{M}$ be the
restriction operator. Then $i=R\circ\iota\circ E$, where $\iota$ is the
inclusion $H^{s_{1}}(N)\to H^{s_{2}}(N)$. Consequently we have the upper bound

$\sigma_{k}(i)\leqslant\lVert R\rVert\sigma_{k}(\iota)\lVert E\rVert\lesssim
k^{-(s_{1}-s_{2})/n}.$

For the lower bound on $\sigma_{k}(i)$, we choose a compact set $K\subset
M^{\mathrm{int}}$ and a cutoff function $\chi\in C^{\infty}(N)$ with
$\mathrm{supp}(\chi)\subset K$ and $\chi\neq 0$ somewhere. Consider the
operator

$A:H^{s_{1}}(N)\to H^{s_{2}}(N),\ \ A=E_{0}\circ i\circ R\circ m_{\chi}$

where $E_{0}$ denotes extension by zero, and $m_{\chi}$ denotes multiplication
by $\chi$. Then

$\sigma_{k}(A)\leqslant\lVert E_{0}\rVert\sigma_{k}(i)\lVert R\circ
m_{\chi}\rVert\lesssim\sigma_{k}(i).$

It is enough to prove a lower bound for $\sigma_{k}(A)$. We compute $A^{*}A$
as

$(A^{*}Af,h)_{H^{s_{1}}}=(\chi f,\chi h)_{H^{s_{2}}}=(\chi J^{2s_{2}}\chi
f,h)_{L^{2}}=(J^{-2s_{1}}\chi J^{2s_{2}}\chi f,h)_{H^{s_{1}}}$

where we now choose $J^{s}=(1-\Delta_{g})^{s/2}$. Thus
$A^{*}Af=J^{-2s_{1}}\chi J^{2s_{2}}\chi f$ on $H^{s_{1}}(N)$. Using the
isometries $J^{\pm s_{1}}$ yields

$\sigma_{k}(A)^{2}=\sigma_{k}(A^{*}A:H^{s_{1}}\to
H^{s_{1}})=\sigma_{k}(J^{-s_{1}}\chi J^{2s_{2}}\chi J^{-s_{1}}:L^{2}\to
L^{2}).$

Now $J^{-s_{1}}\chi J^{2s_{2}}\chi J^{-s_{1}}$ is a classical
pseudodifferential operator of order $-2(s_{1}-s_{2})$ on $N$ and it has
nonvanishing principal symbol at points where $\chi\neq 0$. Thus by Theorem
2.6 its singular values on $L^{2}(N)$ satisfy $\sigma_{k}\gtrsim
k^{-\frac{2(s_{1}-s_{2})}{n}}$. This concludes the proof. ∎

By using similar localization arguments as in Theorem 3.10, one obtains the
following localized bounds which will be used in Section 6.1.

######  Proposition 3.11.

Let $N$ be a smooth $n$-dimensional manifold. Let $Q\subset N$ be compact and
let $t\in\varmathbb{R}$, $m>0$. Then for $H^{t+m}_{Q}(N):=\\{f\in H^{t+m}(N):\
\mathrm{supp}(f)\subset Q\\}$ and for the embedding $i:H^{t+m}_{Q}(N)\to
H^{t}_{Q}(N)$ we have

$\displaystyle e_{k}(i)\sim k^{-m/n}.$

For later purposes we note that similar estimates also hold under weaker
assumptions on the domain (no high smoothness necessary), see [Tr97, Theorem
23.2].

######  Proposition 3.12.

Let $\Omega$ be a bounded Lipschitz domain in $\varmathbb{R}^{n}$. If
$s_{1}>s_{2}$, then the embedding $i:H^{s_{1}}(\Omega)\to H^{s_{2}}(\Omega)$
has entropy numbers satisfying

$e_{k}(i)\sim k^{-\frac{s_{1}-s_{2}}{n}},\qquad k\geqslant 1.$

Next we consider spaces of real-analytic or more generally Gevrey functions.
Let $(M,g)$ be a compact connected smooth $n$-manifold without boundary, and
recall the spaces $A^{\sigma,\rho}(M)$ of Gevrey functions introduced in (2.4)
and below in Section 2.6. These Hilbert spaces have the following entropy
properties.

######  Theorem 3.13.

Let $(M,g)$ be a closed smooth $n$-manifold, let $1\leqslant\sigma<\infty$,
$\rho>0$ and $s\in\varmathbb{R}$. The embedding $i:A^{\sigma,\rho}(M)\to
H^{s}(M)$ satisfies for some $\tilde{\rho}>0$

$\displaystyle\sigma_{k}(i)$ $\displaystyle\sim
k^{s/n}\exp(-\rho(k-1)^{\frac{1}{n\sigma}}),$ $\displaystyle e_{k}(i)$
$\displaystyle\lesssim\exp(-\tilde{\rho}k^{\frac{1}{n\sigma+1}}).$

The same bounds hold for the embedding $i:a^{\sigma,\rho}_{n,X,\varphi}\to
h^{s}_{n,X,\varphi}$ (see Definitions 2.8 and 2.11).

###### Proof.

By Lemma 3.9, it is enough to prove the statement about singular values. Since
$M$ is closed, $H^{s}(M)$ is isomorphic to the space $h^{s}$ of sequences
$x=(x_{0},x_{1},\ldots)$ with norm $\lVert
x\rVert_{h^{s}}=(\sum_{j=0}^{\infty}(1+j)^{\frac{2s}{n}}\lvert
x_{j}\rvert^{2})^{1/2}$ (see Proposition 2.7), and $A^{\sigma,\rho}(M)$ is
isomorphic to the space

$a^{\sigma,\rho}=\\{x=(x_{0},x_{1},\ldots)\in\ell^{2}\,;\,\lVert
x\rVert_{a^{\sigma,\rho}}:=(\sum_{j=0}^{\infty}e^{2\rho
j^{\frac{1}{n\sigma}}}\lvert x_{j}\rvert^{2})^{1/2}<\infty\\}.$

Then $\sigma_{k}(i)\sim\sigma_{k}(\iota)$ where $\iota$ is the inclusion
$a^{\sigma,\rho}\to h^{s}$. We compute

$\sum_{j=0}^{\infty}e^{2\rho
j^{\frac{1}{n\sigma}}}(\iota^{*}\iota(x))_{j}\overline{y}_{j}=(\iota^{*}\iota(x),y)_{a^{\sigma,\rho}}=(\iota(x),\iota(y))_{h^{s}}=\sum_{j=0}^{\infty}(1+j)^{2s/n}x_{j}\overline{y}_{j}.$

Thus $\iota^{*}\iota$ is the diagonal operator with
$(\iota^{*}\iota(x))_{j}=(1+j)^{2s/n}e^{-2\rho j^{\frac{1}{n\sigma}}}x_{j}$
for $j\geqslant 0$, which shows that
$\sigma_{k}(\iota)=k^{s/n}e^{-\rho(k-1)^{\frac{1}{n\sigma}}}$ for $k$ large. ∎

The above results immediately yield decay rates for singular values of
smoothing operators. Further results of this type may be found in [BS77a].

######  Theorem 3.14.

Let $M$ and $N$ be compact $C^{\infty}$ manifolds, having dimensions $n_{M}$
and $n_{N}$. Let $A:H^{s_{2}}(N)\to H^{s_{1}}(M)$ be a compact linear operator
with singular values $\sigma_{j}(A)$.

  1. (i)

If $A$ extends as a bounded operator $A_{1}:H^{s_{2}-m_{2}}(N)\to
H^{s_{1}}(M)$ such that

$A_{1}(H^{s_{2}-m_{2}}(N))\subset H^{s_{1}+m_{1}}(M)$

where $m_{1},m_{2}\geqslant 0$ with $m_{1}+m_{2}>0$, then

$\sigma_{j}(A)\lesssim j^{-m_{1}/n_{M}-m_{2}/n_{N}}.$

In particular, if $M$ is a closed $n$-manifold and $B\in\Psi^{-s}(M)$, where
$B$ is considered as a compact operator $H^{t}(M)\to H^{t}(M)$ for some
$t\in\varmathbb{R}$, then

$\sigma_{j}(B)\lesssim j^{-s/n}.$

  2. (ii)

If $(M,g)$ is a closed analytic $n$-manifold and if $A(H^{s_{2}}(N))\subset
A^{\sigma,\rho}(M)$, then for some $c>0$

$\sigma_{j}(A)\lesssim\exp(-cj^{\frac{1}{n\sigma}}).$

In particular if $B\in\Psi^{-\infty}_{\sigma}(M)$, i.e. $B$ is Gevrey
$G^{\sigma}$-smoothing on $M$, and $B$ is considered as an operator
$L^{2}(M)\to L^{2}(M)$, then

$\sigma_{j}(B)\lesssim\exp(-cj^{\frac{1}{n\sigma}}).$

We refer to Appendix D.1 for definitions and results on Gevrey smoothing
operators.

###### Proof of Theorem 3.14.

(i) We write $A=i_{1}\circ\tilde{A}\circ i_{2}$, where
$\tilde{A}:H^{s_{2}-m_{2}}(N)\to H^{s_{1}+m_{1}}(M)$ is bounded (this holds by
the closed graph theorem since $A_{1}(H^{s_{2}-m_{2}}(N))\subset
H^{s_{1}+m_{1}}(M)$) and $i_{1}:H^{s_{1}+m_{1}}(M)\to H^{s_{1}}(M)$,
$i_{2}:H^{s_{2}}(N)\to H^{s_{2}-m_{2}}(N)$ are the natural inclusions. Write
$j+1=2\ell+r$ where $\ell\geqslant 1$ and $r\in\\{0,1\\}$. Then

$\displaystyle\sigma_{j}(A)=\sigma_{2\ell+r-1}(i_{1}\circ\tilde{A}\circ
i_{2})\leqslant\sigma_{\ell}(i_{1})\sigma_{\ell+r}(\tilde{A}\circ
i_{2})\leqslant\sigma_{\ell}(i_{1})\lVert\tilde{A}\rVert\sigma_{\ell}(i_{2}).$

Next we use that

$\displaystyle\sigma_{\ell}(i_{1})\lesssim\ell^{-m_{1}/n_{M}},$
$\displaystyle\sigma_{\ell}(i_{2})\lesssim\ell^{-m_{2}/n_{N}}.$

Thus, since $\ell\geqslant j/2$, it follows that

$\sigma_{j}(A)\lesssim\ell^{-m_{1}/n_{M}-m_{2}/n_{N}}\lesssim
j^{-m_{1}/n_{M}-m_{2}/n_{N}}.$

The result for $B$ follows immediately.

(ii) Write $A=i\circ\tilde{A}$ where $\tilde{A}:H^{s_{2}}(N)\to
A^{\sigma,\rho}(M)$ is bounded (by the closed graph theorem since
$A(H^{s_{2}}(N))\subset A^{\sigma,\rho}(M)$) and $i:A^{\sigma,\rho}(M)\to
H^{s_{1}}(M)$ is the natural inclusion. Then

$\sigma_{j}(A)\leqslant\sigma_{j}(i)\lVert\tilde{A}\rVert\lesssim\lVert\tilde{A}\rVert
e^{-cj^{\frac{1}{n\sigma}}}.$

Now if $B$ is $G^{\sigma}$ smoothing on $M$, then $B$ has a $G^{\sigma}$
integral kernel $K_{B}(x,y)$ so that

$Bu(x)=\int_{M}K_{B}(x,y)u(y)\,dV_{g}(y).$

The Hilbert-Schmidt bound gives

$\lVert(-\Delta_{g})^{k}Bu\rVert_{L^{2}(M)}\leqslant\lVert(-\Delta_{g,x})^{k}K_{B}\rVert_{L^{2}(M\times
M)}\lVert u\rVert_{L^{2}(M)}.$

Since $K_{B}$ is a $G^{\sigma}$ function, there are $C,R>0$ so that

$\lVert(-\Delta_{g})^{k}Bu\rVert_{L^{2}(M)}\leqslant
CR^{2k}(2k)^{2k\sigma}\lVert u\rVert_{L^{2}(M)}.$

Lemma B.1(c) implies that for $\rho=c_{0}R^{-1/\sigma}$, one has

$\lVert Bu\rVert_{A^{\sigma,\rho}(M)}\leqslant c_{1}C\lVert
u\rVert_{L^{2}(M)}.$

Thus $B$ induces a bounded map $L^{2}(M)\to A^{\sigma,\rho}(M)$, and the
result follows from the first part of (ii). ∎

###  3.5. Embeddings between spaces of operators

Finally, in order to deal with inverse problems where the measurement is a
Dirichlet-to-Neumann type operator, we discuss entropy bounds between spaces
of operators. In this case the range of the forward map lies in the Banach
space $B(H^{s}(M),H^{-s}(M))$ of bounded linear operators from a Sobolev space
$H^{s}(M)$ to its dual $H^{-s}(M)$, where $M$ is a closed $n$-manifold.

We first consider the case $s=0$ and prove the following entropy bound when
embedding smoothing operators of order $m$ into the space of bounded operators
on $L^{2}$.

######  Theorem 3.15.

Let $(M,g)$ be a closed smooth $n$-manifold. If $m>0$, then the embedding
$i:B(H^{-m},L^{2})\cap B(L^{2},H^{m})\to B(L^{2},L^{2})$ satisfies

(3.7) $e_{k}(i)\lesssim k^{-\frac{m}{2n}+\delta}$

for any $\delta>0$.

In the proof we will work with Hilbert spaces instead of the Banach space
$B(X,Y)$. This amounts to replacing $B(X,Y)$ by the space $HS(X,Y)$ of
Hilbert-Schmidt operators (i.e. replacing the Schatten class $S_{\infty}$ by
$S_{2}$). We first recall some definitions. Let $X$ and $Y$ be separable
Hilbert spaces. The space $HS(X,Y)$ consists of those compact linear operators
$T:X\to Y$ for which the quantity

(3.8) $\lVert
T\rVert_{HS(X,Y)}^{2}=\sum_{j=1}^{\infty}\sigma_{j}(T)^{2}=\mathrm{tr}(T^{*}T)=\sum_{j=1}^{\infty}\lVert
T\varphi_{j}\rVert_{Y}^{2}=\sum_{j,k=1}^{\infty}\lvert(T\varphi_{j},\psi_{k})_{Y}\rvert^{2}$

is finite and independent of the choice of orthonormal bases $(\varphi_{j})$
and $(\psi_{k})$ of $X$ and $Y$, respectively. This is a Hilbert space with
inner product

$(S,T)_{HS(X,Y)}=\operatorname{tr}(T^{*}S)=\sum_{j=1}^{\infty}(S\varphi_{j},T\varphi_{j})_{Y}.$

Recall also that when $X=Y=L^{2}(M)$, any $T\in HS(L^{2},L^{2})$ has a
Schwartz kernel $K_{T}\in L^{2}(M\times M)$ and

(3.9) $\lVert T\rVert_{HS(L^{2},L^{2})}=\lVert K_{T}\rVert_{L^{2}(M\times
M)}.$

###### Proof of Theorem 3.15.

We argue in two steps, first proving an initial (suboptimal) entropy bound
when $m>n/2$, and then improving on this in the second step by using a
decomposition of operators into smooth and small parts. In the proof we fix an
orthonormal basis $(\varphi_{j})_{j=1}^{\infty}$ of $L^{2}(M)$ consisting of
eigenfunctions of $-\Delta_{g}$, and use the Sobolev norm $\lVert
f\rVert_{H^{s}(M)}=\lVert J^{s}f\rVert_{L^{2}(M)}$ where
$J^{s}f=\sum_{j=1}^{\infty}j^{\frac{s}{n}}(f,\varphi_{j})\varphi_{j}$. We will
also write $\lVert T\rVert_{*}=\lVert T\rVert_{B(H^{-m},L^{2})}+\lVert
T\rVert_{B(L^{2},H^{m})}$.

_Step 1: Initial entropy bound._ Assuming $m>n/2$, we will prove that whenever
$0\leqslant r<m-n/2$ one has

(3.10) $e_{k}(i:B(H^{-m},L^{2})\cap B(L^{2},H^{m})\to B(L^{2},L^{2}))\lesssim
k^{-\frac{r}{2n}}.$

We first claim that there is a continuous embedding

(3.11) $B(H^{-m},L^{2})\cap B(L^{2},H^{m})\subset HS(H^{-r},L^{2})\cap
HS(L^{2},H^{r}).$

In fact, by (3.8) and interpolation we have

$\displaystyle\lVert T\rVert_{HS(H^{-r},L^{2})}^{2}$ $\displaystyle=\sum\lVert
TJ^{r}\varphi_{j}\rVert_{L^{2}}^{2}\leqslant\sum\lVert T\rVert_{*}^{2}\lVert
J^{r}\varphi_{j}\rVert_{H^{-m}}^{2}\leqslant\lVert T\rVert_{*}^{2}\sum
j^{-\frac{2(m-r)}{n}},$ $\displaystyle\lVert T\rVert_{HS(L^{2},H^{r})}^{2}$
$\displaystyle=\sum\lVert T\varphi_{j}\rVert_{H^{r}}^{2}\leqslant\sum\lVert
T\varphi_{j}\rVert_{L^{2}}^{\frac{2(m-r)}{m}}\lVert
T\varphi_{j}\rVert_{H^{m}}^{\frac{2r}{m}}\leqslant\lVert T\rVert_{*}^{2}\sum
j^{-\frac{2(m-r)}{n}}.$

Since $m-r>n/2$, this proves (3.11).

The next step is to show that the embedding $\iota:HS(H^{-r},L^{2})\cap
HS(L^{2},H^{r})\rightarrow HS(L^{2},L^{2})$ satisfies

$e_{k}(\iota)\lesssim k^{-\frac{r}{2n}}.$

Using (3.8) and (3.9), it is easy to check that

$\displaystyle\lVert T\rVert_{HS(H^{-r},L^{2})\cap HS(L^{2},H^{r}))}$
$\displaystyle=\lVert TJ^{r}\rVert_{HS(L^{2},L^{2})}+\lVert
J^{r}T\rVert_{HS(L^{2},L^{2})}$ $\displaystyle=\lVert
J_{y}^{r}K_{T}\rVert_{L^{2}(M\times M)}+\lVert
J_{x}^{r}K_{T}\rVert_{L^{2}(M\times M)}.$

Since $\lVert K_{T}\rVert_{H^{r}}\leqslant\lVert
J_{y}^{r}K_{T}\rVert_{L^{2}}+\lVert J_{x}^{r}K_{T}\rVert_{L^{2}}$, by Theorem
3.10 one has

$e_{k}(\iota)\leqslant e_{k}(i:H^{r}(M\times M)\to L^{2}(M\times M))\lesssim
k^{-\frac{r}{2n}}.$

We have proved that one has continuous embeddings

(3.12) $\displaystyle B(H^{-m},L^{2})\cap B(L^{2},H^{m})\subset
HS(H^{-r},L^{2})\cap HS(L^{2},H^{r})\subset HS(L^{2},L^{2})\subset
B(L^{2},L^{2})$

where the middle embedding has entropy numbers $e_{k}\lesssim
k^{-\frac{r}{2n}}$. This implies (3.10).

_Step 2: Improvement and derivation of the claimed entropy bound._ Now we
assume $m>0$ and improve the bound (3.10) to the claimed bound
$e_{k}(i)\lesssim k^{-\frac{m}{2n}+\delta}$.

Our strategy is to fix $N\geqslant 1$ and split the map $i$ as

$i(T)=\alpha_{N}(T)+\beta_{N}(T)$

where, using the projection
$P_{N}f=\sum_{j=1}^{N}(f,\varphi_{j})_{L^{2}}\varphi_{j}$, one has

$\alpha_{N}(T)=P_{N}TP_{N},\qquad\beta_{N}(T)=T-P_{N}TP_{N}.$

Note that for any $s\geqslant t$

$\lVert P_{N}f\rVert_{H^{s}}\leqslant N^{\frac{s-t}{n}}\lVert
f\rVert_{H^{t}},\qquad\lVert(I-P_{N})f\rVert_{H^{t}}\leqslant
N^{-\frac{s-t}{n}}\lVert f\rVert_{H^{s}}.$

These estimates imply that

$\lVert\beta_{N}(T)f\rVert_{L^{2}}\leqslant\lVert(I-P_{N})Tf\rVert_{L^{2}}+\lVert
P_{N}T(I-P_{N})f\rVert_{L^{2}}\leqslant N^{-\frac{m}{n}}\lVert
T\rVert_{*}\lVert f\rVert_{L^{2}}.$

On the other hand, for any $s>\max(m,n/2+1)$ one has

$\displaystyle\lVert\alpha_{N}(T)f\rVert_{H^{s}}$ $\displaystyle\leqslant
N^{\frac{s-m}{n}}\lVert TP_{N}f\rVert_{H^{m}}\leqslant N^{\frac{s-m}{n}}\lVert
T\rVert_{*}\lVert f\rVert_{L^{2}},$
$\displaystyle\lVert\alpha_{N}(T)f\rVert_{L^{2}}$
$\displaystyle\leqslant\lVert T\rVert_{*}\lVert P_{N}f\rVert_{H^{-m}}\leqslant
N^{\frac{s-m}{n}}\lVert T\rVert_{*}\lVert f\rVert_{H^{-s}}.$

Thus $\alpha_{N}$ maps $B(H^{-m},L^{2})\cap B(L^{2},H^{m})$ to
$B(H^{-s},L^{2})\cap B(L^{2},H^{s})$ with norm $\leqslant N^{\frac{s-m}{n}}$.
Combining this with (3.10) using the choice $r=s-n/2-1/2$, we have

$e_{k}(\alpha_{N}:B(H^{-m},L^{2})\cap B(L^{2},H^{m})\to
B(L^{2},L^{2}))\lesssim N^{\frac{s-m}{n}}k^{-\frac{r}{2n}}.$

On the other hand, since $e_{1}(\beta_{N})\leqslant\lVert\beta_{N}\rVert$ one
has

$\displaystyle e_{1}(\beta_{N}:B(H^{-m},L^{2})\cap B(L^{2},H^{m})\to
B(L^{2},L^{2}))\leqslant N^{-\frac{m}{n}}.$

It follows from the above estimates and Lemma 3.8(iii) that

$e_{k}(i)\leqslant e_{k}(\alpha_{N})+e_{1}(\beta_{N})\lesssim
N^{\frac{s-m}{n}}k^{-\frac{r}{2n}}+N^{-\frac{m}{n}}.$

Choosing $N\sim k^{\frac{r}{2s}}$ yields that $e_{k}(i)\lesssim
k^{-\frac{m}{2n}\frac{r}{s}}=k^{-\frac{m}{2n}(1-\frac{n+1}{2s})}$. This proves
the bound $e_{k}(i)\lesssim k^{-\frac{m}{2n}+\delta}$ for any $\delta>0$ by
taking $s$ large. ∎

We next give a more general version of Theorem 3.15. It is related to
embedding the space of operators which are smoothing of order $m$ (resp.
Gevrey smoothing), and whose adjoints are also smoothing, into
$B(H^{s},H^{-s})$. Recall that any $T\in B(H^{s},H^{-s})$ has a formal adjoint
$T^{\prime}\in B(H^{s},H^{-s})$ defined by

$(T^{\prime}u,v)=(u,Tv),\qquad u,v\in H^{s},$

using the standard distributional (or $L^{2}$) pairing.

######  Theorem 3.16.

Let $M$ be a closed smooth $n$-manifold, let $s\in\varmathbb{R}$, and let
$H^{t}=H^{t}(M)$.

  1. (a)

Let $m>0$, and define

$Z^{m}(H^{s},H^{-s}):=\\{T\in B(H^{s},H^{-s})\,;\,T(H^{s})\subset
H^{-s+m}\text{ and }T^{\prime}(H^{s})\subset H^{-s+m}\\}.$

Then $Z^{m}(H^{s},H^{-s})$ is a Banach space with norm

$T\mapsto\max(\sup_{\lVert u\rVert_{H^{s}}=1}\lVert
Tu\rVert_{H^{-s+m}},\sup_{\lVert u\rVert_{H^{s}}=1}\lVert
T^{\prime}u\rVert_{H^{-s+m}}).$

If $m>0$, the embedding $i:Z^{m}(H^{s},H^{-s})\to B(H^{s},H^{-s})$ satisfies
for any $\delta>0$

$e_{k}(i)\lesssim k^{-\frac{m}{2n}+\delta}.$

  2. (b)

Let $\rho>0$ and $1\leqslant\sigma<\infty$, and define

$W^{\sigma,\rho}(H^{s},H^{-s}):=\\{T\in B(H^{s},H^{-s})\,;\,T(H^{s})\subset
A^{\sigma,\rho}\text{ and }T^{\prime}(H^{s})\subset A^{\sigma,\rho}\\}.$

Then $W^{\sigma,\rho}(H^{s},H^{-s})$ is a Banach space with norm

$T\mapsto\max(\sup_{\lVert u\rVert_{H^{s}}=1}\lVert
Tu\rVert_{A^{\sigma,\rho}},\sup_{\lVert u\rVert_{H^{s}}=1}\lVert
T^{\prime}u\rVert_{A^{\sigma,\rho}}).$

For some $c>0$, the embedding $i:W^{\sigma,\rho}(H^{s},H^{-s})\to
B(H^{s},H^{-s})$ satisfies

$e_{k}(i)\lesssim e^{-ck^{\frac{1}{2n\sigma+1}}}.$

######  Remark 3.17.

In Theorem 3.16 one needs to consider operators that are smoothing and also
their adjoints are smoothing. In fact, if one does not consider the adjoints,
the result fails since the embedding

$B(H^{s},H^{-s+m})\to B(H^{s},H^{-s})$

is not even compact. Then by Lemma 3.8 its entropy numbers cannot go to zero.
To see this, let $(\varphi_{j})$ and $(\psi_{k})$ be orthonormal bases of
$H^{s}$ and $H^{-s}$, respectively, so that $H^{-s+m}$ has equivalent norm

$\lVert
u\rVert_{H^{-s+m}}=(\sum_{k=1}^{\infty}k^{2m}\lvert(u,\psi_{k})_{H^{-s}}\rvert^{2})^{1/2}.$

Let $T_{l}\in B(H^{s},H^{-s+m})$ with
$T_{l}(u)=(u,\varphi_{l})_{H^{s}}\psi_{1}$. Then $\lVert T_{l}\rVert_{H^{s}\to
H^{-s+m}}=1$, so $(T_{l})$ is a bounded sequence in $B(H^{s},H^{-s+m})$. But
if some subsequence $(T_{l_{j}})$ converges in $B(H^{s},H^{-s})$, the limit
must be $0$ since one has $T_{l_{j}}(u)\to 0$ as $j\to\infty$ for any $u\in
H^{s}$. This contradicts the fact that $\lVert T_{l}\rVert_{H^{s}\to
H^{-s}}=1$.

We begin by proving the first part of Theorem 3.16(a) which we reduce to the
estimates from Theorem 3.15.

###### Proof of Theorem 3.16(a).

It follows from the closed graph theorem that

$Z^{m}(H^{s},H^{-s})\subset B(H^{s},H^{-s+m}).$

For any $T\in Z^{m}(H^{s},H^{-s})$ the closed graph theorem also gives that
$T^{\prime}$ is bounded as a map from $H^{s}\to H^{-s+m}$, hence by duality
$T$ is bounded as a map from $H^{s-m}\to H^{-s}$. Thus we also have

$Z^{m}(H^{s},H^{-s})\subset B(H^{s-m},H^{-s}).$

It follows that $Z^{m}(H^{s},H^{-s})$ is a Banach space with the given norm
and

$Z^{m}(H^{s},H^{-s})\subset B(H^{-s},H^{-s+m})\cap B(H^{s-m},H^{-s}).$

The proof of the estimate in Theorem 3.16(a) is now a direct consequence of
the estimate from Theorem 3.15. Indeed, we simply factor the mapping as

$\displaystyle Z^{m}(H^{s},H^{-s})\rightarrow B(L^{2},H^{m})\cap
B(H^{-m},L^{2})\rightarrow B(L^{2},L^{2})\rightarrow B(H^{s},H^{-s}),$
$\displaystyle T\mapsto J^{-s}TJ^{-s}\mapsto J^{-s}TJ^{-s}\mapsto T,$

where as above $J^{\alpha}$ is an isometry $H^{s}\to H^{s-\alpha}$. By virtue
of the result of Theorem 3.15 we have that the middle mapping has entropy
numbers satisfying $e_{k}\lesssim k^{-\frac{m}{2n}+\delta}$ for any
$\delta>0$. Together with the properties of entropy numbers of compositions
(Lemma 3.8(ii)) this concludes the proof for (a). ∎

In order to deduce the remaining part (b) of Theorem 3.16, we first provide
the following abstract lemma which gives bounds for the singular values (hence
also entropy numbers) of embeddings between spaces of Hilbert-Schmidt
operators. It will also be used later in our study of instability of the
Calderón problem in low regularity.

######  Lemma 3.18.

Let $X$ and $Z$ be separable Hilbert spaces with orthonormal bases
$(\varphi_{j})$ and $(\psi_{k})$, respectively. Define subspaces $X_{1}\subset
X$ and $Y\subset Z$ by the norms

$\lVert
x\rVert_{X_{1}}=(\sum_{j=1}^{\infty}\alpha_{j}^{2}\lvert(x,\varphi_{j})_{X}\rvert^{2})^{1/2},\qquad\lVert
y\rVert_{Y}=(\sum_{k=1}^{\infty}\beta_{k}^{2}\lvert(y,\psi_{k})_{Z}\rvert^{2})^{1/2}$

where $0<\alpha_{1}\leqslant\alpha_{2}\leqslant\ldots$ and
$0<\beta_{1}\leqslant\beta_{2}\leqslant\ldots$. The embedding

$\iota:HS(X,Y)\to HS(X_{1},Z)$

satisfies for any $M,N\geqslant 1$

$\sigma_{MN+1}(\iota)\leqslant\max(\frac{1}{\alpha_{1}\beta_{N+1}},\frac{1}{\alpha_{M+1}\beta_{1}}).$

###### Proof.

By the minimax principle

$\sigma_{MN+1}^{2}(\iota)=\lambda_{MN+1}(\iota^{*}\iota)=\min_{E}\max_{T\perp
E,\lVert T\rVert_{HS(X,Y)}=1}\lVert\iota(T)\rVert_{HS(X_{1},Z)}^{2},$

where the minimum is over subspaces $E\subset HS(X,Y)$ with $\dim(E)=MN$. We
choose

$E=\\{T\in HS(X,Y)\,;\,(T\varphi_{j},\psi_{k})_{Y}=0\text{ for }j\geqslant
M+1\text{ or }k\geqslant N+1\\}.$

Note that $(\alpha_{j}^{-1}\varphi_{j})$ and $(\beta_{k}^{-1}\psi_{k})$ are
orthonormal bases of $X_{1}$ and $Y$, respectively. Thus for any $T\perp E$
one has

$\lVert\iota(T)\rVert_{HS(X_{1},Z)}^{2}=\sum_{j\geqslant M+1\text{ or
}k\geqslant N+1}\alpha_{j}^{-2}\lvert(T\varphi_{j},\psi_{k})_{Z}\rvert^{2}.$

Now $(y,\psi_{k})_{Y}=\beta_{k}^{2}(y,\psi_{k})_{Z}$ for any $y\in Y$, so for
$T\in E$

$\lVert\iota(T)\rVert_{HS(X_{1},Z)}^{2}=\sum_{j\geqslant M+1\text{ or
}k\geqslant
N+1}\alpha_{j}^{-2}\beta_{k}^{-2}\lvert(T\varphi_{j},\beta_{k}^{-1}\psi_{k})_{Y}\rvert^{2}.$

The last quantity is
$\leqslant\max((\alpha_{1}\beta_{N+1})^{-2},(\alpha_{M+1}\beta_{1})^{-2})$ if
$\lVert T\rVert_{HS(X,Y)}=1$. ∎

###### Proof of Theorem 3.16(b).

First note that for $r>0$, the space $A^{\sigma,-r}$ defined as the completion
of $L^{2}(M)$ with respect to the $A^{\sigma,-r}$ norm is a Hilbert space that
contains all $H^{t}$ spaces (it can be considered as a space of
ultradistributions on $M$). If $T\in W^{\sigma,\rho}(H^{s},H^{-s})$ and $u\in
H^{s}$, we have for any $\varepsilon>0$

$\lVert Tu\rVert_{A^{\sigma,\rho}}\lesssim\lVert u\rVert_{H^{s}}\lesssim\lVert
u\rVert_{A^{\sigma,\varepsilon}}.$

Since $T^{\prime}$ is bounded as a map from $H^{s}\to A^{\sigma,\rho}$ we have
by duality

$\lVert Tu\rVert_{A^{\sigma,-\varepsilon}}\lesssim\lVert
Tu\rVert_{H^{-s}}\lesssim\lVert u\rVert_{A^{\sigma,-\rho}}.$

Choosing $\varepsilon=\rho/3$ and using the Stein-Weiss interpolation theorem,
we have

$\lVert Tu\rVert_{A^{\sigma,\rho/3}}\lesssim\lVert
u\rVert_{A^{\sigma,-\rho/3}}.$

This shows that $W^{\sigma,\rho}(H^{s},H^{-s})\subset
B(A^{\sigma,-\rho/3},A^{\sigma,\rho/3})$. As in the proof of Theorem 3.13, it
is easy to check that the embedding $A^{\sigma,\rho}\subset
A^{\sigma,\rho-\varepsilon}$ has singular values $\sigma_{k}\lesssim
e^{-\varepsilon j^{\frac{1}{n\sigma}}}$. Thus as in (3.12), we have a sequence
of continuous embeddings

(3.13) $W^{\sigma,\rho}(H^{s},H^{-s})\subset
HS(A^{\sigma,-\rho/4},A^{\sigma,\rho/4})\subset
HS(A^{\sigma,-\rho/8},A^{\sigma,\rho/8})\subset B(H^{s},H^{-s}).$

Lemma 3.18 with $\alpha_{j}=\beta_{j}=e^{\frac{\rho}{8}j^{\frac{1}{n\sigma}}}$
shows that the middle embedding has singular values $\sigma_{N^{2}+1}\lesssim
e^{-\frac{\rho}{8}(N+1)^{\frac{1}{n\sigma}}}$. Thus by Lemma 3.9 the middle
embedding has entropy numbers $e_{k}\lesssim e^{-ck^{\frac{1}{2n\sigma+1}}}$.
This proves the bound for $e_{k}(i)$. ∎

######  Remark 3.19.

In inverse problems where the measurement is a Dirichlet-to-Neumann type
operator, the forward operator actually maps to a subspace of
$B(H^{s},H^{-s})$ consisting of pseudodifferential operators. In this remark
we show that in a related case one can improve the exponents in Theorems 3.15
and 3.16 and give a sharp decay rate.

Let $(M,g)$ be a closed smooth $n$-manifold and consider the space
$\Psi^{-m}_{F}$ of Fourier multipliers of order $-m$, consisting of operators
$T=T_{a}$ where $(a_{j})_{j=1}^{\infty}$ is a sequence satisfying $\lvert
a_{j}\rvert\lesssim j^{-m}$, and $T_{a}$ is defined by

$T_{a}f=\sum_{j=1}^{\infty}a_{j}(f,\varphi_{j})\varphi_{j}.$

Here $\lambda_{1}\leqslant\lambda_{2}\leqslant\ldots$ are the eigenvalues of
$-\Delta_{g}$, and $(\varphi_{j})$ is an orthonormal basis of $L^{2}(M)$
consisting of eigenfunctions. We think of $\Psi^{-m}_{F}$ as a subspace of
$B(H^{-m/2},H^{m/2})$. Using the Sobolev norm $\lVert
f\rVert_{H^{s}}^{2}=\sum_{j=1}^{\infty}j^{2s/n}\lvert(f,\varphi_{j})\rvert^{2}$,
it is easy to see that

$\displaystyle\lVert T_{a}\rVert_{L^{2}\to L^{2}}$
$\displaystyle=\sup_{j\geqslant 1}\,\lvert a_{j}\rvert,$ $\displaystyle\lVert
T_{a}\rVert_{H^{-m/2}\to H^{m/2}}$ $\displaystyle=\sup_{j\geqslant
1}\,j^{m/n}\lvert a_{j}\rvert.$

It follows that the embedding $i:\Psi^{-m}_{F}\to B(L^{2},L^{2})$ may be
identified with the embedding $w^{m,\infty}\to\ell^{\infty}$, where
$\lVert(a_{j})\rVert_{w^{m,\infty}}=\sup_{j\geqslant 1}\,j^{m/n}\lvert
a_{j}\rvert$. Writing $i=A\circ J^{m}$ where
$J^{m}:(a_{j})\mapsto(j^{m/n}a_{j})$ is an isometry between $w^{m,\infty}$ and
$\ell^{\infty}$, the entropy numbers of $i$ are the same as the entropy
numbers of

$A:\ell^{\infty}\to\ell^{\infty},\ A((a_{j}))=(j^{-m/n}a_{j}).$

This is a diagonal operator whose entropy numbers are estimated in [CS90,
Proposition 1.3.2]. As in the proof of Lemma 3.9 it follows that $e_{k}(A)\sim
k^{-m/n}$, and thus also $e_{k}(i:\Psi^{-m}_{F}\to B(L^{2},L^{2}))\sim
k^{-m/n}$.

##  4\. Instability in the presence of global smoothing

In this section we give some examples of instability in inverse problems based
on global smoothing properties when the coefficients are smooth or real-
analytic. We will be quite brief, since in Section 5 stronger results will be
given for low regularity coefficients.

We first combine Lemma 3.7 with Theorems 3.10 and 3.13 and state a result for
the case where the forward map acts between Sobolev spaces. Part (a) shows
that if the forward map is $C^{\infty}$ smoothing then the inverse problem
cannot be Hölder stable. Part (b) shows that a Gevrey/analytic smoothing
forward map leads to at best logarithmic stability.

######  Theorem 4.1.

Let $M$ and $N$ be compact smooth manifolds with or without smooth boundary,
let $s,t\in\varmathbb{R}$, and let $\delta>0$. Let $K$ be a closed and bounded
set in $H^{s+\delta}(M)$, and let $F$ be a map $K\to H^{t}(N)$. Suppose that
$\omega$ is a modulus of continuity such that

$\lVert f_{1}-f_{2}\rVert_{H^{s}(M)}\leqslant\omega(\lVert
F(f_{1})-F(f_{2})\rVert_{H^{t}(N)}),\qquad f_{1},f_{2}\in K.$

  1. (a)

If $F$ maps $K$ into a bounded set of $H^{t+m}(N)$ where $m>0$, then
$\omega(t)\gtrsim t^{\frac{\delta\dim(N)}{m\dim(M)}}$. In particular, if $F$
maps $K$ into a bounded set of $H^{t+m}(N)$ for any $m>0$, then for any
$\alpha\in(0,1)$ one has $\omega(t)\gtrsim t^{\alpha}$ for $t$ small.

  2. (b)

If $N$ is closed and $F$ maps $K$ into a bounded set of $A^{\sigma,\rho}(N)$
for some $1\leqslant\sigma<\infty$ and $\rho>0$, then
$\omega(t)\gtrsim\lvert\log
t\rvert^{-\frac{\delta(\sigma\dim(N)+1)}{\dim(M)}}$ for $t$ small.

Next we state an analogous result where the range of the forward map is in the
space $B(H^{s},H^{-s})$ as in the case of Dirichlet-to-Neumann type operators.
This follows immediately from Lemma 3.7 and Theorem 3.16 (we also use the
notation $Z^{m}(H^{s},H^{-s})$ and $W^{\sigma,\rho}(H^{s},H^{-s})$ from that
theorem).

######  Theorem 4.2.

Let $M$ and $N$ be compact smooth manifolds so that $N$ has no boundary, let
$r,s\in\varmathbb{R}$, and let $\delta>0$. Let $K$ be a closed and bounded set
in $H^{r+\delta}(M)$, and let $F$ be a map $K\to B(H^{s}(N),H^{-s}(N))$.
Suppose that $\omega$ is a modulus of continuity such that

$\lVert f_{1}-f_{2}\rVert_{H^{r}}\leqslant\omega(\lVert
F(f_{1})-F(f_{2})\rVert_{H^{s}\to H^{-s}}),\qquad f_{1},f_{2}\in K.$

  1. (a)

If $F$ maps $K$ into a bounded set of $Z^{m}(H^{s},H^{-s})$ for any $m>0$,
then for any $\alpha\in(0,1)$ one has $\omega(t)\gtrsim t^{\alpha}$ for $t$
small.

  2. (b)

If $F$ maps $K$ into a bounded set of $W^{\sigma,\rho}(H^{s},H^{-s})$ for some
$1\leqslant\sigma<\infty$ and $\rho>0$, then $\omega(t)\gtrsim\lvert\log
t\rvert^{-\frac{\delta(2\sigma\dim(N)+1)}{\dim(M)}}$ for $t$ small.

###  4.1. Unique continuation

Let $M$ be a compact $n$-manifold with smooth boundary, and let $P$ be an
elliptic second order operator on $M$ having the form

$P=\Delta_{g}u+Bu+cu$

where $g$ is a smooth Riemannian metric on $M$, $B$ is a smooth vector field
on $M$ and $c\in C^{\infty}(M)$. As in Section 2.5, for any $u\in H^{1}(M)$
solving $Pu=0$ in $M$ there is a normal derivative $\partial_{\nu}u|_{\partial
M}$ defined weakly as an element of $H^{-1/2}(\partial M)$. If $u$ is smooth,
one has in local coordinates
$\partial_{\nu}u=g^{jk}\partial_{j}u\nu_{k}|_{\partial M}$ where $\nu$ is the
unit outer conormal to $\partial M$.

The unique continuation principle states that if $\Gamma$ is a nonempty open
subset of $\partial M$, then any $u\in H^{1}(M)$ solving $Pu=0$ in $M$ and
satisfying $u|_{\Gamma}=\partial_{\nu}u|_{\Gamma}=0$ must be identically zero.
This can be made quantitative, and one has (conditional) logarithmic stability
[ARRV09]. The following result shows that logarithmic stability is optimal, at
least when the underlying structures are real-analytic.

######  Theorem 4.3.

Let $M$ and $P$ be as above, and let $\Gamma$ be a nonempty open subset of
$\partial M$ so that $\partial M\setminus\overline{\Gamma}\neq\emptyset$. Let
$\delta>0$, and suppose that $\omega$ is a modulus of continuity so that

(4.1) $\lVert u\rVert_{H^{1}(M)}\leqslant\omega(\lVert
u\rVert_{H^{1/2}(\Gamma)}+\lVert\partial_{\nu}u\rVert_{H^{-1/2}(\Gamma)})$

whenever $Pu=0$ and $\lVert u\rVert_{H^{1+\delta}(M)}\leqslant 1$.Then
$\omega(t)\gtrsim t^{\alpha}$ for any $\alpha\in(0,1)$ when $t$ is small.
Moreover, if $M$, the coefficients of $P$ and $\partial M$ are real-analytic,
then $\omega(t)\gtrsim\lvert\log t\rvert^{-\mu}$ for $t$ small whenever
$\mu>\frac{\delta n}{n-1}$.

###### Proof.

We rewrite (4.1) in a form where a smoothing operator appears. We assume that
$0$ is not a Dirichlet eigenvalue for $P$ in $M$, i.e. for any $f\in
H^{1/2}(\partial M)$ there is a unique solution $u=Sf\in H^{1}(M)$ of $Pu=0$
in $M$ with $u|_{\partial M}=f$. (Otherwise one can argue with $f$ replaced by
$f-Qf$, where $Q$ is a projection to a finite dimensional space.) Elliptic
regularity gives that $\lVert u\rVert_{H^{1+t}(M)}\sim\lVert
f\rVert_{H^{1/2+t}(\partial M)}$ for any $t\geqslant 0$. Thus (4.1) implies
that for some $r_{0}>0$ one has

(4.2) $\lVert f\rVert_{H^{1/2}(\partial M)}\lesssim\omega(\lVert
f\rVert_{H^{1/2}(\Gamma)}+\lVert\partial_{\nu}Sf\rVert_{H^{-1/2}(\Gamma)}),\qquad\lVert
f\rVert_{H^{1/2+\delta}(\partial M)}\leqslant r_{0}.$

We wish to get rid of the $\lVert f\rVert_{H^{1/2}(\Gamma)}$ term on the
right. This can be done by restricting to functions $f$ that vanish near
$\Gamma$. Using the condition $\partial
M\setminus\overline{\Gamma}\neq\emptyset$, there is a neighborhood
$\Gamma_{1}$ of $\overline{\Gamma}$ in $\partial M$ and a compact domain
$\Sigma\subset\partial M$ with smooth boundary so that
$\Sigma\cap\overline{\Gamma}_{1}=\emptyset$. Let $E$ be a bounded Sobolev
extension operator from $H^{t}(\Sigma)$ to $H^{t}(\partial M)$, chosen so that
$Eh|_{\Gamma_{1}}=0$. Applying (4.2) to $f=Eh$, it follows that for some $r>0$

$\lVert h\rVert_{H^{1/2}(\Sigma)}\lesssim\omega(\lVert
Ah\rVert_{H^{-1/2}(\partial M)}),\qquad\lVert
h\rVert_{H^{1/2+\delta}(\Sigma)}\leqslant r/2,$

where $A$ is the linear operator

$A:H^{1/2}(\Sigma)\to H^{-1/2}(\partial M),\ Ah=\chi\partial_{\nu}SEh$

and $\chi\in C^{\infty}(\partial M)$ satisfies $\chi=1$ near
$\overline{\Gamma}$ and $\mathrm{supp}(\chi)\subset\Gamma_{1}$.

Now $u=SEh$ satisfies $Pu=0$ in $M$ and $u|_{\Gamma_{1}}=0$. By elliptic
regularity it follows that $u$ is smooth near $\Gamma_{1}$, showing that $A$
maps into $H^{m}(\partial M)$ for any $m$ (continuously, by the closed graph
theorem). Theorem 4.1(a) shows that $\omega(t)$ cannot be a Hölder modulus of
continuity.

Suppose now that all the structures are real-analytic. Since $u=SEh$ satisfies
$Pu=0$ in $M$ and $u|_{\Gamma_{1}}=0$, elliptic regularity gives that $u$ must
be real-analytic near $\Gamma_{1}$. Moreover, since $\lVert
u\rVert_{L^{2}(M)}$ is uniformly bounded, we have uniform bounds in the Cauchy
estimates for $u$ by [LM12a, Theorem 1.3 in Chapter 8 of vol. III]. It follows
that there are uniform bounds in the Cauchy estimates for
$\partial_{\nu}u|_{\partial M}$ in any compact subset of $\Gamma_{1}$. Now fix
$\sigma>1$ and choose $\chi\in C^{\infty}_{c}(\Gamma_{1})\cap
G^{\sigma}(\partial M)$ so that $\chi=1$ near $\overline{\Gamma}$. Then $A$
maps $H^{1/2}(\Sigma)$ to $A^{\sigma,\rho}(\partial M)$ for some fixed
$\rho>0$ (continuously, by the closed graph theorem). It follows from Theorem
4.1(b) that $\omega(t)\gtrsim\lvert\log
t\rvert^{-\frac{\delta(\sigma(n-1)+1}{n-1}}$. Since this is true for any
$\sigma>1$, the result follows. ∎

###  4.2. Linearized Calderón problem

Let $(M,g)$ be a compact $n$-manifold with smooth boundary. We consider the
Dirichlet problem

$(-\Delta_{g}+q)u=0\text{ in $M$},\qquad u|_{\partial M}=f,$

where $q\in L^{\infty}(M)$ (lower regularity coefficients will be considered
in Section 5). Assuming that $0$ is not a Dirichlet eigenvalue, for any $f\in
H^{1/2}(\partial M)$ there is a unique weak solution $u\in H^{1}(M)$. Consider
the DN map

$\Lambda_{q}:H^{1/2}(\partial M)\to H^{-1/2}(\partial M),\
f\mapsto\Lambda_{q}f:=\partial_{\nu}u|_{\partial M},$

where the normal derivative $\partial_{\nu}u|_{\partial M}$ is defined in the
weak sense as in Section 2.5.

The following standard result computes the Fréchet derivative of the map

$\Lambda:L^{\infty}(M)\to B(H^{1/2}(\partial M),H^{-1/2}(\partial M)).$

We give the proof for completeness.

######  Lemma 4.4.

Let $q\in L^{\infty}(M)$ be such that $0$ is not a Dirichlet eigenvalue of
$-\Delta_{g}+q$ in $M$, let $P_{q}:H^{1/2}(\partial M)\to H^{1}(M)$ be the
solution operator for the Dirichlet problem

$(-\Delta_{g}+q)P_{q}f=0\text{ in $M$},\qquad P_{q}f|_{\partial M}=f,$

and let $G_{q}:H^{-1}(M)\to H^{1}_{0}(M)$ be the Green operator with vanishing
Dirichlet boundary values,

$(-\Delta_{g}+q)G_{q}F=F\text{ in $M$},\qquad G_{q}F|_{\partial M}=0.$

Then the linearized DN map $A_{q}=(D\Lambda)_{q}$ is the operator

$\displaystyle A_{q}:L^{\infty}(M)\rightarrow B(H^{1/2}(\partial
M),H^{-1/2}(\partial M)),$ $\displaystyle
A_{q}(h)f=\partial_{\nu}G_{q}(-hP_{q}f)|_{\partial M}.$

###### Proof.

If $\lVert h\rVert_{L^{\infty}}$ is small then $\Lambda_{q+h}$ is well
defined. Given $f\in H^{1/2}(\partial M)$ one has

$\Lambda_{q+h}f-\Lambda_{q}f=\partial_{\nu}(P_{q+h}f-P_{q}f)|_{\partial M}.$

The function $w:=P_{q+h}f-P_{q}f\in H^{1}_{0}(M)$ solves

$(-\Delta_{g}+q)w=-hw-hP_{q}f\mbox{ in }M,\ w=0\mbox{ on }\partial M,$

which implies that

$w=G_{q}(-hP_{q}f)+G_{q}(-hw).$

If $\lVert h\rVert_{L^{\infty}}$ is chosen small enough, one has $\lVert
G_{q}(-hw)\rVert_{H^{1}}\leqslant\frac{1}{2}\lVert w\rVert_{H^{1}}$ and hence

$\lVert w\rVert_{H^{1}}\lesssim\lVert h\rVert_{L^{\infty}}\lVert
f\rVert_{H^{1/2}}.$

It follows that

$(\Lambda_{q+h}-\Lambda_{q}-A_{q}(h))f=\partial_{\nu}(w-G_{q}(-hP_{q}f))|_{\partial
M}=\partial_{\nu}(G_{q}(-hw))|_{\partial M}.$

The $H^{-1/2}(\partial M)$ norm of the last quantity is $\lesssim\lVert
h\rVert_{L^{\infty}}\lVert w\rVert_{H^{1}}\lesssim\lVert
h\rVert_{L^{\infty}}^{2}\lVert f\rVert_{H^{1/2}}$. This proves that the
Fréchet derivative of $q\mapsto\Lambda_{q}$ at $q$ is $A_{q}$. ∎

We note that if $q$ is smooth and if $h$ vanishes near $\partial M$, then
$A_{q}(h)$ is a smoothing operator. This implies strong instability properties
for the linearized Calderón problem where one would like to determine $h$ from
the knowledge of $A_{q}(h)$.

######  Theorem 4.5.

Let $M^{\prime}\Subset M^{\mathrm{int}}$, let $s>n/2$, and let $\delta>0$.
Consider $A_{q}$ as an operator $H^{s}_{0}(M^{\prime})\to B(H^{1/2}(\partial
M),H^{-1/2}(\partial M))$ and suppose that the linearized Calderón problem has
the stability estimate

$\lVert h\rVert_{H^{s}}\leqslant\omega(\lVert
A_{q}(h)\rVert_{B(H^{1/2},H^{-1/2})}),\qquad\lVert
h\rVert_{H^{s+\delta}}\leqslant 1.$

If $q\in C^{\infty}(M)$, then $\omega(t)$ cannot be a Hölder modulus of
continuity. Moreover, if $M$, $g$, $\partial M$ and $q$ are real-analytic,
then $\omega(t)\gtrsim\lvert\log t\rvert^{-\frac{\delta(2n-1)}{n}}$ for $t$
small.

###### Proof.

Since $h$ vanishes in $M\setminus M^{\prime}$, the function
$u=G_{q}(-hP_{q}f)$ solves $(-\Delta_{g}+q)u=0$ in $M\setminus M^{\prime}$
with $u|_{\partial M}=0$. If $q\in C^{\infty}(M)$ it follows that $A_{q}(h)$
maps $H^{1/2}(\partial M)$ boundedly to $H^{m}(\partial M)$ for any
$m\geqslant 0$. By Theorem 4.2(a) $\omega(t)$ cannot be a Hölder modulus of
continuity. Similarly, if all the structures are real-analytic then $u$ is
real-analytic near $\partial M$ with $\lVert u\rVert_{H^{1}}\lesssim\lVert
h\rVert_{L^{\infty}}\lVert f\rVert_{H^{1/2}}$, so $u$ satisfies uniform Cauchy
estimates by [LM12a, Theorem 1.3 in Chapter 8 of vol. III]. Consequently
$A_{q}(h)$ maps $H^{s}_{0}(M^{\prime})$ boundedly into
$W^{1,\rho}(H^{\frac{1}{2}},H^{-\frac{1}{2}})$ for some $\rho>0$ (this uses
that $A_{q}(h)$ is formally self-adjoint). By Theorem 4.2(b) $\omega(t)$ is at
best logarithmic with the given exponent. ∎

###  4.3. Calderón problem

We next turn to the instability of the classical Calderón problem in smooth
and analytic settings (again, results in a low regularity framework will be
presented in the next section) and thus prove Theorem 1.2.

###### Proof of Theorem 1.2.

We seek to apply Theorem 4.2 to the map

$\displaystyle F:K\rightarrow B(H^{\frac{1}{2}}(\partial
M),H^{-\frac{1}{2}}(\partial M)),\
q\mapsto\Gamma(q):=\Lambda_{q}-\Lambda_{0},$

where $K=\\{q\in H^{s}(M):\ \mathrm{supp}(q)\subset M^{\prime},\
\|q\|_{H^{s+\delta}(M)}\leqslant 1\\}$. Now by elliptic regularity, for any
$q\in K$ the map $F$ indeed maps into $B(H^{\frac{1}{2}}(\partial
M),H^{-\frac{1}{2}}(\partial M))$ as for $f\in
H^{\frac{1}{2}}(\partial\Omega)$ we have that
$\Gamma(q)f=\partial_{\nu}u_{f}-\partial_{\nu}u_{f}^{0}\in
H^{-\frac{1}{2}}(\partial M)$, where $u_{f},u_{f}^{0}$ are solutions to

$\displaystyle-\Delta_{g}u+Vu$ $\displaystyle=0\mbox{ in }M,$ $\displaystyle
u$ $\displaystyle=f\mbox{ on }\partial M,$

with $V=q$ for $u_{f}$ and $V=0$ for $u_{f}^{0}$. Further, we claim that $F\in
Z^{m}(H^{\frac{1}{2}},H^{-\frac{1}{2}})$ for any $m>0$. Indeed, this follows
from the fact that $v:=u_{f}-u_{0}$ satisfies the equation

$\displaystyle-\Delta_{g}v+qv$ $\displaystyle=-qu_{0}\mbox{ in }M,$
$\displaystyle v$ $\displaystyle=0\mbox{ on }\partial M.$

The fact that $\mathrm{supp}(q)\subset M^{\prime}$ and elliptic regularity
give that $\partial_{\nu}v\in C^{\infty}(\partial M)$. In particular,
$\Gamma(q)(H^{\frac{1}{2}}(\partial M))\subset H^{-\frac{1}{2}+m}(\partial M)$
for any $m>0$. Since $\Gamma(q)$ is a self-adjoint operator, this implies that
$\Gamma(q)\in Z^{m}(H^{\frac{1}{2}},H^{-\frac{1}{2}})$ for each $m>0$.
Invoking Theorem 4.2(a) then implies the impossibility of Hölder estimates.

If moreover, $M,g$ and $\partial M$ are real analytic, then by [LM12a, Theorem
1.2 in Chapter 8 of vol. III] $\partial_{\nu}v$ is analytic and hence
$\Gamma(q)\subset W^{1,\rho}(H^{\frac{1}{2}},H^{-\frac{1}{2}})$ for some
$\rho>0$ for which we again use the self-adjointness of $\Gamma(q)$. The
logarithmic lower bound then follows from Theorem 4.2(b). ∎

##  5\. Instability at low regularity

In the sequel we seek to provide a further instability mechanism, related to
direct singular value or entropy bounds, showing that a strong regularity
improvement (as in our analyticity arguments from Section 3) is not the only
mechanism leading to exponential instability. The main, common mechanism of
all our results should rather be regarded as a “compressing mechanism”. In
order to prove this, we estimate the associated singular values and exploit a
balance between gaining some decay from regularity and loosing some control
through growing constants. We illustrate this argument by applying it to a
number of model problems including the backward heat equation with low
regularity space-time dependent coefficients (Section 5.2), the unique
continuation property (Section 5.3) and the Calderón problem (thus proving
Theorem 1.4 in Section 5.5). In all these settings our arguments imply that in
spite of the low regularity of the coefficients of the problem, one can prove
the same instability results for the associated inverse problems. This gives a
complete answer to the question (Q3) from the introduction for the discussed
model problems. Most of these instability results are (possibly up to the
precise exponents) sharp.

###  5.1. Abstract setup for linear inverse problems

We begin with instability results for linear inverse problems. The following
variant of Lemma 3.7 shows that it is sufficient to find some way of proving
decay for the singular values of the forward operator. We only state a version
related to exponential instability, which will be sufficient below.

######  Theorem 5.1.

Let $A:X\to Y$ be a compact injective linear operator between separable
Hilbert spaces with $X$ infinite dimensional. Let $X_{1}\subset X$ be a closed
subspace so that $i:X_{1}\to X$ is compact with $e_{k}(i)\gtrsim k^{-m}$ for
some $m>0$. Let $K=\\{u\in X\,;\,\lVert u\rVert_{X_{1}}\leqslant r\\}$ for
some $r>0$. Assume that the singular values of $A:X\to Y$ satisfy for some
$\rho,\mu>0$

$\sigma_{k}(A)\lesssim e^{-\rho k^{\mu}},\qquad k\geqslant 1.$

Then there is $c>0$ with the following property: for any $\varepsilon>0$ small
enough there is $u=u_{\varepsilon}$ such that

(5.1) $\|u\|_{X}\geqslant\varepsilon,\quad\|u\|_{X_{1}}\leqslant
r,\quad\|Au\|_{Y}\leqslant\exp(-c\varepsilon^{-\frac{\mu}{m(\mu+1)}}).$

In particular, if one has the stability property

$\lVert u\rVert_{X}\leqslant\omega(\lVert Au\rVert_{Y}),\qquad u\in K,$

then necessarily $\omega(t)\gtrsim\lvert\log\,t\rvert^{-\frac{m(\mu+1)}{\mu}}$
for $t$ small.

###### Proof.

Let $(\varphi_{j})_{j=1}^{\infty}$ be a singular value orthonormal basis of
$X$, so that $A\varphi_{j}=\sigma_{j}\psi_{j}$ where
$\psi=(\psi_{j})_{j=1}^{\infty}$ is orthonormal in $Y$. Let $Y^{\prime}$ be
the span of $\psi$ in $Y$. We define a subspace
$a^{\sigma,\rho}=a^{\sigma,\rho}_{1,Y^{\prime},\psi}$ of $Y^{\prime}$ using
the norm

$\lVert v\rVert_{a^{\sigma,\rho}}=\left(\sum_{j=1}^{\infty}e^{2\rho
j^{\frac{1}{\sigma}}}\lvert(v,\psi_{j})\rvert^{2}\right)^{1/2}.$

Since $Au=\sum_{j=1}^{\infty}(u,\varphi_{j})\sigma_{j}\psi_{j}$, we have
$(Au,\psi_{j})=\sigma_{j}(u,\varphi_{j})$ and

$\lVert Au\rVert_{a^{1/\mu,\rho}}^{2}=\sum_{j=1}^{\infty}e^{2\rho
j^{\mu}}\sigma_{j}^{2}\lvert(u,\varphi_{j})\rvert^{2}\leqslant C^{2}\lVert
u\rVert_{X}^{2}.$

Thus $A(K)$ is contained in a bounded subset of $Y_{1}:=a^{1/\mu,\rho}$. The
embedding $j_{1}:Y_{1}\to Y^{\prime}=h^{0}_{1,Y^{\prime},\psi}$ satisfies
$e_{k}(j_{1})\lesssim e^{-ck^{\frac{\mu}{\mu+1}}}$ by Theorem 3.13, and thus
also the embedding $j:Y_{1}\to Y$ has these entropy bounds. It follows from
Lemma 3.7 that $\omega(t)\gtrsim\lvert\log\,t\rvert^{-\frac{m(\mu+1)}{\mu}}$
for $t$ small. By Remark 3.4 this conclusion can be rewritten as (5.1). ∎

###  5.2. The backward heat equation

As a first model case for the type of arguments that we have in mind, we
consider parabolic systems of the form

(5.2) $\displaystyle\begin{split}(\partial_{t}-\nabla\cdot a\nabla)u&=0\mbox{
in }\Omega\times[0,1],\\\ u&=0\mbox{ on }\partial\Omega\times[0,1],\\\
u&=u_{0}\mbox{ on }\Omega\times\\{0\\},\end{split}$

where $\Omega\subset\varmathbb{R}^{n}$ is an open, bounded $C^{1}$ domain,
$u:\Omega\rightarrow\varmathbb{C}^{m}$ and
$a=(a_{ij}^{\alpha\beta}(x,t))^{\alpha,\beta\in\\{1,\dots,m\\}}_{i,j\in\\{1,\dots,n\\}}$
are bounded functions which satisfy a coercivity condition, i.e. for which
there exist constants $\lambda>0,\kappa>0$ such that

(5.3) $\displaystyle\operatorname{Re}\int_{\Omega}a\nabla
v\cdot\overline{\nabla
v}\,dx\geqslant\lambda\|v\|_{H^{1}(\Omega)}^{2}-\kappa\|v\|_{L^{2}(\Omega)}^{2}\mbox{
for all }v\in H^{1}(\Omega)\mbox{ and a.e.\ }t\in[0,1].$

Here and in the following discussion $\nabla$ always denotes the spatial
gradient, $\overline{f}$ denotes the complex conjugate of $f$ and

(5.4) $\displaystyle a\nabla u\cdot\overline{\nabla
u}:=\sum\limits_{i,j\in\\{1,\dots,n\\},\
\alpha,\beta\in\\{1,\dots,m\\}}a^{\alpha\beta}_{ij}\partial_{j}u^{\beta}\overline{\partial_{i}u^{\alpha}}.$

Here and in the sequel, with slight abuse of notation, we write
$\|u\|_{X(\Omega)}$ instead of $\|u\|_{X(\Omega,\varmathbb{C}^{m})}$. We
remark that all the following results are in particular valid for scalar
parabolic equations (which just corresponds to the case $m=1$). As there is no
difference in the argument with respect to the systems case, we have opted to
formulate the results in the systems case directly. We refer to [Mc00] for
some background on energy estimates for elliptic systems.

In this set-up we are interested in the inverse problem associated with the
forward map

(5.5) $\displaystyle i_{1,0}:L^{2}(\Omega)\ni u_{0}\mapsto u(1)\in
L^{2}(\Omega).$

It is well-known that the backward heat equation is highly ill-posed. In spite
of this for $C^{1}$ regular metrics $a$ the map $i_{1,0}$ is injective. We
refer to [Is90, Chapter 3.1] and [Ya09, Section 9.1] for a discussion of the
scalar backward heat equation. The backward uniqueness property of the heat
equation can be quantified to yield quantitative backward uniqueness results
in compact sets. Under suitable regularity assumption on the coefficients in
the equation, for the recovery of $u(\cdot,t_{0})$ with $t_{0}\in(0,T)$ this
turns into a _Hölder_ stability estimate:

######  Proposition 5.2 ([Ve09], estimate (3.65)).

Let $u$ be a solution to (5.2) with $m=1$ and $a\in
C^{1}(\overline{\Omega}\times[0,T],\varmathbb{R}^{n\times n})$ uniformly
elliptic. Let $t_{0}\in(0,T)$. Then there exists $\theta=\theta(t)\in(0,1)$
and $C>0$ depending on $\|a\|_{C^{1}(\overline{\Omega}\times[0,T])}$ such that

(5.6) $\displaystyle\|u(t_{0})\|_{L^{2}(\Omega)}\leqslant
C\|u_{0}\|_{L^{2}(\Omega)}^{1-\theta}\|u(T)\|_{L^{2}(\Omega)}^{\theta}.$

For isotropic and time-independent metrics $a$, the function $\theta(t)$ can
for instance be chosen to be $\theta(t)=\frac{t}{T}$ (see [Is90, (3.1.9)]).
For $t_{0}\rightarrow 0$, this Hölder estimate degenerates, resulting in an
only _logarithmic_ bound (see [Ya09, equation (9.2)], [Is90, Chapter 3] or
[Kl06]) of the following type: If $\|u(0)\|_{H^{2}(\Omega)}\leqslant M$, there
exist $\nu>0$, $C_{M}\geqslant 1$ such that

(5.7) $\displaystyle\|u(0)\|_{L^{2}(\Omega)}\leqslant
C_{M}|\log(\|u(T)\|_{L^{2}(\Omega)})|^{-\nu}.$

These quantitative backward uniqueness estimates can be obtained through
various methods with Carleman estimates possibly providing the most robust
arguments (e.g. allowing for low regularity coefficients). We refer to Section
9.1 of the survey article [Ya09], to [Kl06, Theorems 3, 4] and the references
therein for more background on the positive results in this direction.

It is the _optimality_ of the logarithmic stability estimate of the type (5.7)
that we are investigating in this section (we refer to Section C.1 for a
discussion of an elliptic analogue of the optimality of the Hölder estimates
in (5.6)). If $a(x,t)$ is independent of $t$ this follows easily from
eigenvalue estimates (see Lemma 5.6), but in the time-dependent case a
different argument is needed. As our main result, we show that the logarithmic
moduli of continuity for the inversion of the map $i_{1,0}$ are optimal.
Further, we prove that this behaviour persists in the low regularity setting
for parabolic systems which was described above, see (5.2), e.g. in settings
in which the metric $a$ is only bounded and no $C^{\infty}$ or analytic
smoothing properties can be used.

######  Theorem 5.3.

Let $\Omega$ and $a$ be as above, and let $\ell>0$. If for some modulus of
continuity $\omega$ one has

(5.8)
$\displaystyle\|u_{0}\|_{L^{2}(\Omega)}\leqslant\omega(\|u(1)\|_{L^{2}(\Omega)}),\qquad\|u_{0}\|_{H^{\ell}(\Omega)}\leqslant
1,$

where $u$ is the solution to (5.2), then
$\omega(t)\gtrsim\lvert\log\,t\rvert^{-\frac{\ell(n+4)}{2n}}$ for $t$ small.

For higher regularity metrics $a$ a similar result can be obtained. We remark
that without any major modification it is possible to replace the
$L^{2}(\Omega)$ norm $\|u(1)\|_{L^{2}(\Omega)}$ in the estimate (5.8) by an
$H^{\delta}(\Omega)$ norm for $\delta\in(0,\delta_{0})$, where $\delta_{0}>0$
denotes the regularity exponent from the Sneiberg-type Lemma 5.5 below.
Indeed, in order to obtain this, we just apply the result from (5.8) with
$t=1$ replaced by $t=\frac{1}{2}$ and then use the smoothing property from
Lemma 5.5 to infer that $\|u(1)\|_{H^{\delta}(\Omega)}\leqslant
C\|u(\frac{1}{2})\|_{L^{2}(\Omega)}$.

Theorem 5.3 will follow from Theorem 5.1 together with the fact that the
singular values associated with the mapping (5.5) decay exponentially. In
spite of the low regularity set-up for the coefficients, we prove that the
remaining little bit of regularisation leads to exponentially decaying
singular value estimates.

######  Theorem 5.4.

Let $\Omega$, $a$ be as above. There is a constant $c>0$ so that the singular
values of the map $i_{1,0}$ in (5.5) satisfy

$\sigma_{k}(i_{1,0})\leqslant e^{-ck^{\frac{2}{n+2}}}.$

We further recall the validity of entropy estimates in irregular domains,
which had been stated in Proposition 3.12 and which will be used in the proof
of Theorem 5.3.

###### Proof of Theorem 5.3 using Theorem 5.4.

We use Theorem 5.1 with $X=L^{2}(\Omega)$, $X_{1}=H^{\ell}(\Omega)$, and
$A=i_{1,0}$. One has $e_{k}(i)\sim k^{-\frac{\ell}{n}}$ by Proposition 3.12
and $\sigma_{k}(A)\lesssim e^{-ck^{\mu}}$ with $\mu=\frac{2}{n+2}$ by Theorem
5.4. Then Theorem 5.1 yields
$\omega(t)\gtrsim\lvert\log\,t\rvert^{-\frac{\ell(n+4)}{2n}}$. ∎

Thus it remains to prove Theorem 5.4. The first step is a regularity result.

######  Lemma 5.5.

Let $a$, $\Omega$ be as above and $s\in[0,1)$, $t>0$, and $s+2t\leqslant 1$.
Then, there exist $\delta>0$ and $C>0$ depending only on $\lambda$, $n$ and
$\Omega$ so that

$\|u(s+t)\|_{H^{\delta}(\Omega)}\leqslant
Ct^{-\frac{\delta}{2}}\|u(s)\|_{L^{2}(\Omega)}.$

###### Proof.

We argue in three steps: First we extend the problem to a heat equation with
controlled right hand side on $\varmathbb{R}^{n}\times(0,4)$ and $L^{2}$
bounded initial data. Then, we prove an interpolation result. Finally, we
return to the desired estimate.

_Step 1: Extension to the whole space and energy estimates._ We recall that in
[ABES19, Theorem 8.1] it is proved that for solutions to

(5.9) $\displaystyle\partial_{t}w-\nabla\cdot a\nabla w$
$\displaystyle=f+\partial_{i}F^{i}\mbox{ in }\varmathbb{R}^{n}\times(0,4)$

the following a priori estimate holds for some $\alpha\in(0,1)$ and $p>2$:

(5.10) $\displaystyle\begin{split}&\|\nabla
w\|_{L^{p}(\Omega_{1})}+\sup\limits_{t\in
I_{1}}\|w\|_{L^{p}(B_{1})}+\sup\limits_{s,t\in
I_{1}}\frac{\|u(t,\cdot)-u(s,\cdot)\|_{L^{2}(B_{1})}}{|t-s|^{\alpha}}\\\
&\leqslant
C\left(\|w\|_{L^{2}(2\Omega_{1})}+\|f\|_{L^{p}(2\Omega_{1})}+\|F\|_{L^{p*}(2\Omega_{1})}\right),\end{split}$

where $\Omega_{r}=B_{r}\times I_{r}$, $I_{r}=[r/2,r]$ and
$2\Omega_{r}=B_{2r}\times[r/4,2r]$.

Now in our setting we consider solutions $u$ to (5.2) in the domain $\Omega$.
Thus, by localizing by means of a partition of unity the equation is locally
of the form (5.10) with $f$ and $F$ depending on $u$, its first derivatives
and the cut-off functions. Moreover, in the interior of $\Omega$ we may assume
that $f=0=F$ and thus obtain estimates in (5.10) which only involve quantities
controlled by the usual energy estimates on the right hand side. We seek to
obtain similar bounds in the neighbourhood of $\partial\Omega$.

To this end, it suffices to consider solutions $\bar{u}$ which are supported
in a patch in a neighbourhood $U$ of $\partial\Omega$. After a change of
coordinates and by choosing the support of $u$ possibly even smaller, it is
possible to assume that $U=B_{2}(0)$ and that
$U\cap\partial\Omega\subset\varmathbb{R}^{n-1}=B_{2}(0)\cap\varmathbb{R}^{n-1}$.
Now using the homogeneous Dirichlet conditions and reflecting $u$ oddly across
the boundary $\varmathbb{R}^{n-1}$ again gives a solution $v$ to the heat
equation with a divergence form right hand side which is now defined in the
whole space. The resulting inhomogeneities $f,F$ depend on $u$ and $\nabla u$,
the coefficient matrix $a$, the cut-off functions and the change of
coordinates. In $B_{1/2}(0)$ we may, however, assume that $f=0$ and $F=0$ as
the cut-offs are not active there. Hence, a (rescaled) version of the energy
estimate (5.10) holds there with a right hand side which only depends on $v$
(with $f=0,F=0$). In $B_{2}(0)\setminus B_{1/2}$ we may estimate $f,F$ in
terms of $u$ and $\nabla u$ in the interior of some other boundary patches
(thus corresponding to the $B_{1/2}$ setting for this other patch in which we
have $f=0=F$) or in some interior patches (where also $f=0=F$). Thus, in the
original coordinates, only space-time $L^{2}$ norms of $u$ in some of the
other patches appear as right hand sides for these estimates as well.

Thus, patching together all the local estimates for the functions $\bar{u}$ on
the different supports of the partition of unity (using that only finitely
many of them overlap), we obtain a new function $\tilde{u}$ which satisfies

(5.11)
$\displaystyle\begin{split}&\|\nabla\tilde{u}\|_{L^{p}(\varmathbb{R}^{n}\times[1/2,1])}+\sup\limits_{t\in[1/2,1]}\|\tilde{u}\|_{L^{p}(\varmathbb{R}^{n})}+\sup\limits_{s,t\in
I_{1}}\frac{\|u(t,\cdot)-u(s,\cdot)\|_{L^{2}(\varmathbb{R}^{n})}}{|t-s|^{\alpha}}\\\
&\leqslant C\|\tilde{u}\|_{L^{2}(\varmathbb{R}^{n+1})}.\end{split}$

Extending $u_{0}$ by zero outside of $\Omega$, energy estimates further yield
that

$\displaystyle\|\tilde{u}\|_{L^{2}(\varmathbb{R}^{n+1})}\leqslant
C\|u_{0}\|_{L^{2}(\varmathbb{R}^{n})}.$

_Step 2: Interpolation on $\varmathbb{R}^{n}\times[1/2,1]$._

We next prove the central interpolation result, starting from the observation
(see (5.11)) that

$\displaystyle\tilde{u}\in L^{2}(I,H^{1}(\varmathbb{R}^{n}))\mbox{ and
}\sup\limits_{s,t\in
I}\frac{\|u(t,\cdot)-u(s,\cdot)\|_{L^{2}(\varmathbb{R}^{n})}}{|t-s|^{\alpha}}\leqslant
C<\infty,$

where $I=[\frac{1}{2},1]$. Moreover, using the arguments from Step 1, without
loss of generality, we may assume that $\tilde{u}$ and $\tilde{u}_{0}$ are
compactly supported in space and time. We seek to prove that under these
conditions $\tilde{u}\in L^{\infty}(I,H^{\delta}(\varmathbb{R}^{n}))$ for some
small $\delta>0$ (depending on $\alpha\in(0,1)$).

In order to prove the claim, we consider the temporal convolution of
$\tilde{u}$ with a one-dimensional standard mollifier
$\eta:\varmathbb{R}\rightarrow\varmathbb{R}$. For $j\in\varmathbb{N}$, we set
$\eta_{j}(t):=2^{j}\eta(2^{j}t)$ and note that it suffices to prove that

(5.12)
$\displaystyle\|\tilde{u}\ast\eta_{j}-\tilde{u}\ast\eta_{j+1}\|_{L^{\infty}(I,H^{\delta}(\varmathbb{R}^{n}))}\leqslant
C2^{-\gamma j},\ j\in\varmathbb{N},$

where $\gamma>0$ depends on $\delta$ and where $\tilde{u}\ast\eta_{j}$ denotes
mollification in $t$. Indeed, by a telescope sum, from (5.12) we obtain that
$(\tilde{u}_{j}(t))_{j\in\varmathbb{N}}:=(\tilde{u}\ast\eta_{j}(t))_{j\in\varmathbb{N}}$
is a Cauchy sequence in $L^{\infty}(I,H^{\delta}(\varmathbb{R}^{n}))$, hence a
limit exists as $j\rightarrow\infty$. Since it holds that also pointwise
almost everywhere $\tilde{u}_{j}(t)\rightarrow\tilde{u}(t)$, we have that
$\tilde{u}_{j}(t)\rightarrow\tilde{u}(t)$ in