$L^{\infty}(I,H^{\delta}(\varmathbb{R}^{n}))$.

Defining $\varphi(t)=\eta(t)-2\eta(2t)$, our claim (5.12) reduces to proving
that

(5.13) $\displaystyle\sup\limits_{t\in
I}\|(\tilde{u}\ast\varphi_{j})(t)\|_{H^{\delta}(\varmathbb{R}^{n})}\leqslant
C2^{-\gamma j}.$

We deduce this by interpolating between an $L^{2}(\varmathbb{R}^{n})$ bound
and a $H^{1}(\varmathbb{R}^{n})$ estimate. To this end, on the one hand, we
note that by virtue of the fact that $\sup\limits_{s,t\in
I_{1}}\frac{\|u(t,\cdot)-u(s,\cdot)\|_{L^{2}(\varmathbb{R}^{n})}}{|t-s|^{\alpha}}\leqslant
C<\infty$ we have

$\displaystyle\|(\tilde{u}\ast\varphi_{j})(t)\|_{L^{2}(\varmathbb{R}^{n})}$
$\displaystyle\leqslant 2^{-j\alpha}C.$

On the other hand, by Cauchy-Schwarz we obtain that

$\displaystyle\|(\tilde{u}\ast\varphi_{j})(t)\|_{H^{1}(\varmathbb{R}^{n})}$
$\displaystyle\leqslant\|\|\tilde{u}\|_{L^{2}(I)}(\cdot)+\|\nabla\tilde{u}\|_{L^{2}(I)}(\cdot)\|_{L^{2}(\varmathbb{R}^{n})}\|\eta_{j}\|_{L^{2}(\varmathbb{R})}$
$\displaystyle\leqslant
2^{j/2}\|\tilde{u}\|_{L^{2}(I,H^{1}(\varmathbb{R}^{n}))}.$

Using that

$\displaystyle\|(\tilde{u}\ast\varphi_{j})(t)\|_{H^{\delta}(\varmathbb{R}^{n})}\leqslant\|(\tilde{u}\ast\varphi_{j})(t)\|_{H^{1}(\varmathbb{R}^{n})}^{\delta}\|(\tilde{u}\ast\varphi_{j})(t)\|_{L^{2}(\varmathbb{R}^{n})}^{1-\delta},$

choosing $\delta\in(0,1)$ (depending on $\alpha$) sufficiently small and
inserting the $L^{\infty}L^{2}$ and $L^{\infty}H^{1}$ bounds from above as
well as the bounds from (5.11), then implies (5.13).

_Step 3: The higher regularity estimate._ Last but not least, we return to the
estimate of the lemma. By the previous two steps, we infer that for some
constant $C>0$ and for any solution $\tilde{u}$ obtained from a solution $u$
to (5.2) as described in Step 1 it holds

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

Rescaling parabolically and using that $t\in(0,1)$, we thus obtain that

$\displaystyle\|\tilde{u}(t)\|_{H^{\delta}(\varmathbb{R}^{n})}\leqslant
Ct^{-\frac{\delta}{2}}\|u_{0}\|_{L^{2}(\varmathbb{R}^{n})}.$

Noting that $\|u(t)\|_{H^{\delta}(\Omega)}\leqslant
C\|\tilde{u}(t)\|_{H^{\delta}(\varmathbb{R}^{n})}$, then concludes the
argument. ∎

With the regularity from Lemma 5.5 at our disposal, we address the proof of
Theorem 5.4.

###### Proof of Theorem 5.4.

We first use Proposition 3.12 and note that the compact embedding
$\iota:H^{\delta}(\Omega)\to L^{2}(\Omega)$ has bounds for its singular values
of the form

(5.14) $\displaystyle\sigma_{k}(\iota)\leqslant Ck^{-\frac{\delta}{n}}.$

Let $i_{s+t,s}:L^{2}(\Omega)\to H^{\delta}(\Omega)\to L^{2}(\Omega)$,
$s\in[0,1),t,s+t\in(0,1]$, be the map which takes $u(s)$ to $u(s+t)$ where $u$
solves (5.2) in $(s,s+t)\times\Omega$. Then, by the bound from Lemma 5.5 and
the estimate (5.14) we have

$\sigma_{k}(i_{s+t,s})\leqslant Ck^{-\frac{\delta}{n}}t^{-\frac{\delta}{2}}.$

Using the behaviour of singular values under composition of maps, we iterate
this $N$ times (with $N\leqslant k$ to be determined below) to infer

$\sigma_{k}(i_{1,0})\leqslant\prod_{j=1}^{N}\sigma_{k/N}(i_{j/N,(j-1)/N})\leqslant(C^{\frac{1}{\delta}}(k/N)^{-\frac{1}{n}}N^{\frac{1}{2}})^{\delta
N}.$

Roughly minimizing the right hand side by choosing $N$ as a function of $k$,
we obtain

$N(k)=\rho k^{\frac{2}{n+2}}$

with $\rho$ a small positive constant to be determined below. Indeed, this
follows from computing

$0=\frac{d}{dN}\exp(N(\ln C+(\frac{1}{2}+\frac{1}{n})\delta\ln
N-\frac{1}{n}\delta\ln k))$

which is equivalent to

$\ln C+\frac{n+2}{2n}\delta\ln N-\frac{1}{n}\delta\ln
k+\delta(\frac{n+2}{2n})=0$

or respectively,

$\ln N=\frac{2}{n+2}\ln k-1-\frac{2n}{n+2}\frac{1}{\delta}\ln C.$

Then,

$\sigma_{k}(i_{1,0})\leqslant\left(C\rho^{\delta\frac{n+2}{2n}}\right)^{\rho
k^{\frac{2}{n+2}}}.$

Choosing $\rho>0$ such that $\rho^{\frac{n+2}{2n}\delta}<C^{-1}$, we obtain
that with a constant $c>0$

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

This concludes the argument. ∎

As an instructive comparison, we note that in the time-independent case for
$m=1$, a different exponent appears in Theorem 5.3:

######  Lemma 5.6.

Let $\Omega$ and $a$ be as in (5.2), (5.3) and (5.4) with $m=1$ and assume
that $a$ does not depend on $t$. If for some modulus of continuity one has

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

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

###### Proof.

The argument follows in exactly the same way as the proof of Theorem 5.4,
however noting that in the time-independent case
$\sigma_{k}(i_{1,0})=e^{-\lambda_{k}}\leqslant c^{k^{\frac{2}{n}}}$ for some
$c\in(0,1)$, where $\lambda_{k}$ denote the eigenvalues of the operator
$-\nabla\cdot a\nabla$. ∎

###  5.3. Instability of unique continuation up to the boundary

We study the instability of unique continuation up to the boundary. It is well
known (see for instance [ARRV09, Theorem 1.9]) that for coefficients and
domains having certain minimal regularity one has logarithmic stability
estimates for this problem, i.e. given a relatively open subset
$\Gamma\subset\partial\Omega$ there exists $\mu>0$ such that for solutions $u$
of an elliptic equation satisfying $\lVert u\rVert_{H^{1}(\Omega)}\leqslant 1$
one has

$\displaystyle\|u\|_{L^{2}(\Omega)}\leqslant\omega(\|u\|_{H^{\frac{1}{2}}(\Gamma)}+\|\partial_{\nu}u\|_{H^{-\frac{1}{2}}(\Gamma)}),$

where $\omega$ is a modulus of continuity satisfying
$\omega(t)\lesssim\lvert\log\,t\rvert^{-\mu}$ for some $\mu>0$.

We show that this type of stability is optimal and that the optimality, in the
sense that no modulus of continuity can be bettern than logarithmic, can be
proved also in the low regularity setting (where the UCP in general fails when
$n\geqslant 3$). More precisely, we consider the following set-up: Let
$\Omega\subset\varmathbb{R}^{n}$ with $n\geqslant 2$ be a bounded open set
with Lipschitz boundary. Let $\Gamma\subset\partial\Omega$ be a (relatively)
open set with $\partial\Omega\setminus\overline{\Gamma}\neq\emptyset$. We are
interested in the continuity properties of the inverse of the mapping

$\displaystyle L^{2}(\Omega)\ni
u\mapsto(u|_{\Gamma},\partial_{\nu}^{L}u|_{\Gamma})\in
H^{\frac{1}{2}}(\Gamma)\times H^{-\frac{1}{2}}(\Gamma),$

where $u$ is a solution to the equation

(5.16) $\displaystyle
Lu:=-\partial_{i}a^{ij}\partial_{j}u+b_{j}\partial_{j}u+cu$
$\displaystyle=0\text{ in $\Omega$},$

with $a^{ij}\in L^{\infty}(\Omega,\varmathbb{R}^{n\times n})$ bounded,
symmetric and uniformly elliptic, i.e.
$0<\lambda|\xi|^{2}\leqslant\xi_{i}a^{ij}\xi_{j}\leqslant\lambda^{-1}|\xi|^{2}<\infty$
for some $\lambda\in(0,1]$, $b_{j}\in L^{n}(\Omega,\varmathbb{C})$, $c\in
L^{n/2}(\Omega,\varmathbb{C})$. The normal derivative
$\partial_{\nu}^{L}u|_{\partial\Omega}$ is defined as an element of
$H^{-1/2}(\partial\Omega)$ as in Section 2.5.

We show that even in the presence of low regularity coefficients, so that no
$C^{\infty}$ or analytic smoothing is available, one can prove that no
stability estimate can be better than logarithmic:

######  Theorem 5.7.

Let $\Omega$, $\Gamma\subset\partial\Omega$,L be as above. There is $c>0$ so
that for $\varepsilon>0$ small there exists a solution $u$ to (5.16) such that

$\displaystyle\|u\|_{H^{1}(\Omega)}\geqslant\varepsilon,\qquad\|u|_{\partial\Omega}\|_{H^{1}(\partial\Omega)}\leqslant
1,\qquad\|u\|_{H^{\frac{1}{2}}(\Gamma)}+\|\partial_{\nu}^{L}u\|_{H^{-\frac{1}{2}}(\Gamma)}\leqslant
e^{-c\varepsilon^{-\frac{2(n-1)}{n+1}}}.$

Moreover, if $\partial\Omega$ is $C^{1,1}$, $a^{ij}\in W^{1,\infty}$,
$b^{j}\in L^{\infty}$, $c\in L^{n}$, then there is $c>0$ so that for
$\varepsilon>0$ small there exists a solution $u$ to (5.16) such that

$\displaystyle\|u\|_{L^{2}(\Omega)}\geqslant\varepsilon,\qquad\|u\|_{H^{1}(\Omega)}\leqslant
1,\qquad\|u\|_{H^{\frac{1}{2}}(\Gamma)}+\|\partial_{\nu}^{L}u\|_{H^{-\frac{1}{2}}(\Gamma)}\leqslant
e^{-c\varepsilon^{-\frac{n-1}{n}}}.$

Thus logarithmic stability is optimal for these unique continuation problems
in the sense that no modulus of continuity can be better than logarithmic.

######  Remark 5.8.

Note that for $n\geqslant 3$ the case of low regularity metrics ($a^{ij}\in
C^{0,\alpha}$ with $\alpha\in(0,1)$) is of particular interest, as the unique
continuation property fails in this low regularity regime in general.

######  Remark 5.9.

We point out that in the first instability estimate in the low regularity
setting in Theorem 5.7 the condition that
$\|u|_{\partial\Omega}\|_{H^{1}(\partial\Omega)}\leqslant 1$ may seem rather
restrictive given the low regularity assumptions. In this low regularity
context, a condition of the form
$\|u|_{\partial\Omega}\|_{H^{\frac{1}{2}}(\partial\Omega)}\leqslant 1$ or –
since we deal with compactness conditions – an assumption of the form
$\|u|_{\partial\Omega}\|_{H^{\frac{1}{2}+\delta}(\partial\Omega)}\leqslant 1$,
for $\delta>0$, may appear most natural. We stress that since Theorem 5.7 is
an _existence result_ and since the inclusions $H^{1}(\partial\Omega)\subset
H^{\frac{1}{2}+\delta}(\partial\Omega)\subset H^{\frac{1}{2}}(\partial\Omega)$
for $\delta\in(0,1)$ hold, for the constructed functions $u$ these possibly
“more natural” bounds are indeed satisfied in our context.

In order to obtain the desired result, we argue similarly as in the previous
section exploiting higher integrability results. We only give the proof when
$n\geqslant 3$ using the estimate of Meyers [Me63] for the necessary small
regularity gain. For $n=2$ the proof is the same, except that one uses the
Gehring lemma instead.

######  Lemma 5.10.

Let $\Omega,L$ be as above. There are $C>0$ and $p>2$ so that whenever $u\in
H^{1}(\Omega)$ solves $Lu=0$ in $\Omega$ and $B(x_{0},3r)\subset\Omega$, then

$\lVert\nabla u\rVert_{L^{p}(B(x_{0},r))}\leqslant
Cr^{-(\frac{n}{2}-\frac{n}{p})}(\lVert\nabla
u\rVert_{L^{2}(B(x_{0},2r))}+\lVert u\rVert_{L^{2n/(n-2)}(B(x_{0},2r))}).$

Moreover, if $x_{0}\in\partial\Omega$ and $u=0$ on $\partial\Omega\cap
B(x_{0},3r)$, then

$\lVert\nabla u\rVert_{L^{p}(B(x_{0},r)\cap\Omega)}\leqslant
Cr^{-(\frac{n}{2}-\frac{n}{p})}(\lVert\nabla
u\rVert_{L^{2}(B(x_{0},2r)\cap\Omega)}+\lVert
u\rVert_{L^{2n/(n-2)}(B(x_{0},2r)\cap\Omega)}).$

###### Proof.

The first estimate follows from [Me63, Theorem 2]. The second estimate reduces
to the first one after flattening the boundary near $x_{0}$ by a bi-Lipschitz
map and reflecting the solution and coefficients to the other side in a
suitable way. ∎

Figure 1. A schematic illustration of the “foliation” of $\Gamma$ by the
domains $\Omega_{r}$.

###### Proof of Theorem 5.7.

Let $\Gamma$ be as above, and choose an open set
$\Gamma_{1}\subset\partial\Omega$ such that
$\overline{\Gamma}\cap\overline{\Gamma}_{1}=\emptyset$. Further, by possibly
making $\Gamma_{1}$ smaller, without loss of generality, we may assume that by
a bi-Lipschitz change of variables the boundary piece $\Gamma_{1}$ can be
flattened to become a ball in $\varmathbb{R}^{n-1}$. We assume the solvability
of the Dirichlet problem. If this is not the case, we argue by working in the
respective quotient spaces. We consider solutions $u$ of

(5.17) $\displaystyle\begin{split}Lu&=0\mbox{ in }\Omega,\\\ u&=g\mbox{ on
}\partial\Omega,\end{split}$

where $g\in H^{1/2}_{\overline{\Gamma}_{1}}(\partial\Omega)=\\{f\in
H^{1/2}(\partial\Omega)\,;\,\mathrm{supp}(f)\subset\overline{\Gamma}_{1}\\}$.
Note that for such boundary values $g$ one has $u|_{\Gamma}=0$, and the
instability of unique continuation can be studied by analyzing the operator

$T:H^{1/2}_{\overline{\Gamma}_{1}}(\partial\Omega)\to H^{-1/2}(\Gamma),\ \
g\mapsto\partial_{\nu}^{L}u|_{\Gamma}.$

We consider a “foliation” (see Figure 1) of $\Gamma$ by Lipschitz domains
$\Omega_{r}\subset\Omega$ with $r\in[0,1]$ such that

  * •

$\Omega_{1}=\Omega$,

  * •

$\Omega_{0}=\tilde{\Omega}$, where $\tilde{\Omega}\subset\Omega$ is a bounded,
very thin Lipschitz domain of thickness $\delta_{0}>0$ (depending on
$\|b_{j}\|_{L^{n}(\Omega)},\|c\|_{L^{\frac{n}{2}}(\Omega)}$ and $\Omega$) such
that $\Gamma\subset\partial\tilde{\Omega}$,

  * •

$\Omega_{s}\subset\Omega_{r}$ if $r>s$,

  * •

$\operatorname{dist}(\Omega_{s},\Omega_{0})\leqslant\delta_{0}$ for
$s\in(0,1)$ and for some small, fixed constant $\delta_{0}>0$ depending on
$\|b_{j}\|_{L^{n}(\Omega)},\|c\|_{L^{\frac{n}{2}}(\Omega)}$ and $\Omega$,

  * •

$\Gamma\subset\partial\Omega_{r}$ for all $r\in[0,1]$,

  * •

for all $r\in(0,1)$ we have $\Gamma_{1}\cap\partial\Omega_{r}=\emptyset$ and
$\partial\Omega_{r}\cap\partial\Omega_{s}\subset\partial\Omega$ (and thus
$u=0$ on $\partial\Omega_{r}\cap\partial\Omega_{s}$),

  * •

for $s\neq r$ and $s,r\in(0,1)$ we assume that
$\operatorname{dist}(\partial^{0}\Omega_{s},\partial^{0}\Omega_{r})=\frac{\delta_{0}|r-s|}{10}$
where $\partial^{0}\Omega_{t}$ for $t\in\\{s,r\\}$ denotes the part of
$\partial\Omega_{t}$ which is not contained in $\partial\Omega$.

The domains $\Omega_{r}$, $r\in(0,1)$, are chosen to be uniform in the sense
that for $r\in(0,1)$ it holds that the sets $\partial\Omega_{r}\cap\Omega$ are
uniform bi-Lipschitz deformations of a suitable neighbourhood of $\Gamma$.
Here the parameter $\delta_{0}>0$ is chosen such that the Dirichlet problem is
always solvable in the domains $\Omega_{r}$ with $r\in(0,1)$ (for $s=1$ this
solvability is either assumed or one works with associated quotient spaces).
If the domains are sufficiently thin, solvability of the Dirichlet problem is
guaranteed since the constants in the Poincaré and Sobolev inequalities become
very small and allow one to absorb all lower order terms in an existence proof
(through Lax-Milgram, for instance). Since the constant $\delta_{0}>0$ is
fixed throughout the argument, we do not track its dependence in the sequel.

For $s>r$ we then consider the maps

$\displaystyle i_{s,r}:H^{\frac{1}{2}}(\partial\Omega_{s})\rightarrow
H^{\frac{1}{2}}(\partial\Omega_{r}),\ u|_{\partial\Omega_{s}}\mapsto
u|_{\partial\Omega_{r}},$

where $u$ solves $Lu=0$ in $\Omega_{s}$.

We wish to estimate the singular values of $i_{s,r}$. Using Lemma 5.10
together with a suitable cover of $\overline{\Omega}_{r}$ by small balls,
Sobolev embedding, and the trace theorem, there are $C>0$ and $p>2$ so that

$\displaystyle\|u\|_{W^{1,p}(\Omega_{r})}\leqslant\frac{C}{|r-s|^{\delta}}\|u\|_{H^{1/2}(\partial\Omega_{s})},$

with $\delta=\frac{n}{2}-\frac{n}{p}$. We combine this with a trace estimate
and the Sobolev embedding

$W^{1,p}(\Omega_{r})\to
B^{1-\frac{1}{p}}_{pp}(\partial\Omega_{r})\hookrightarrow
H^{\frac{1}{2}+\delta}(\partial\Omega_{r}).$

The trace estimate is given e.g. in [JK95, Theorem 3.1] or [St70, Chapter VI],
and the Sobolev embedding follows from the corresponding embedding in
$\varmathbb{R}^{n-1}$ after flattening the boundary locally by bi-Lipschitz
maps. Thus, as by [Tr97, Theorem 20.6] the compact embedding
$H^{\frac{1}{2}+\delta}(\partial\Omega_{r})\hookrightarrow
H^{\frac{1}{2}}(\partial\Omega_{r})$ has singular values estimated by

$\displaystyle\sigma_{k}(id_{H^{\frac{1}{2}+\delta}(\partial\Omega_{r})\rightarrow
H^{\frac{1}{2}}(\partial\Omega_{r})})\leqslant Ck^{-\frac{\delta}{n-1}},$

we obtain for the singular values of the mapping $i_{s,r}$ the estimate

$\displaystyle\sigma_{k}(i_{s,r})\leqslant\frac{C}{|s-r|^{\delta}}k^{-\frac{\delta}{n-1}}.$

The constant $C>0$ can be chosen to be uniform over $r,s\in[0,1]$ with $r<s$
by the uniform choice of the domain geometry.

We now concatenate the mappings $i_{s,r}$ and consider the map

$\displaystyle T:=T_{0}\circ i_{\frac{1}{N},0}\circ
i_{\frac{2}{N},\frac{1}{N}}\circ\cdots\circ
i_{1-\frac{1}{N},1-\frac{2}{N}}\circ i_{1,1-\frac{1}{N}},$

where $T_{0}$ is the bounded map

$T_{0}:H^{1/2}(\partial\Omega_{0})\to H^{-1/2}(\Gamma),\ \
u|_{\partial\Omega_{0}}\mapsto\partial_{\nu}^{L}u|_{\Gamma}.$

Arguing as in the proof of Theorem 5.4, using the properties of the singular
values under concatenation of mappings, we obtain that

(5.18) $\displaystyle\sigma_{k}(T)\leqslant
C^{N}\left(N^{\frac{n}{n-1}}k^{-\frac{1}{n-1}}\right)^{\delta N},$

which is roughly optimized at $N(k)=\rho k^{\frac{1}{n}}$. With $\rho>0$
chosen appropriately depending on $C,\delta,n$, this yields

$\displaystyle\sigma_{k}(T)\leqslant e^{-ck^{\frac{1}{n}}},$

for some constant $c>0$ depending only on $C,\delta,n$.

Now applying Theorem 5.1 to $T$ with
$X=H^{1/2}_{\overline{\Gamma}_{1}}(\partial\Omega)$ and
$X_{1}=H^{1}_{\overline{\Gamma}_{1}}(\partial\Omega)$, so that
$e_{k}(i:X_{1}\to X)\sim k^{-\frac{1}{2(n-1)}}$ by [Tr97, Theorem 20.6],
Theorem 5.1 implies that for $\varepsilon>0$ small there is a solution $u\in
H^{1}(\Omega)$ with

$\lVert
u|_{\partial\Omega}\rVert_{H^{1/2}(\partial\Omega)}\geqslant\varepsilon,\quad\lVert
u|_{\partial\Omega}\rVert_{H^{1}(\partial\Omega)}\leqslant 1,\quad\lVert
u\rVert_{H^{1/2}(\Gamma)}+\lVert\partial_{\nu}^{L}u\rVert_{H^{-1/2}(\Gamma)}\leqslant
e^{-c\varepsilon^{-\frac{2(n-1)}{n+1}}}.$

Using the trace estimate $\lVert
u|_{\partial\Omega}\rVert_{H^{1/2}(\partial\Omega)}\leqslant C\lVert
u\rVert_{H^{1}(\Omega)}$ yields the first estimate in the theorem.

We now show that assuming a bit more regularity, one can change the spaces for
$u$ and prove the second estimate in the theorem. We will use the method in
Proposition A.6. Let $-\Delta_{\Gamma_{1}}$ be the Laplacian on
$\Gamma_{1}\subset\partial\Omega$, and let
$(\varphi_{j})_{j=1}^{\infty}\subset H^{1}_{0}(\Gamma_{1})$ be an orthonormal
basis of $L^{2}(\Gamma_{1})$ consisting of Dirichlet eigenfunctions of
$-\Delta_{\Gamma_{1}}$. We recall that $\Gamma_{1}$ was chosen such that there
is a bi-Lipschitz change of coordinates flattening it so that $\Gamma_{1}$
becomes a ball in $\varmathbb{R}^{n-1}$. Under this change of coordinates
$-\Delta_{\Gamma_{1}}$ becomes an elliptic operator with $L^{\infty}$
coefficients. Hence, by an application of Theorem 2.5, the corresponding
eigenvalues satisfy the Weyl asymptotics $\lambda_{j}\sim j^{\frac{2}{n-1}}$.
For $\lvert s\rvert\leqslant 1$, we will use the equivalent Sobolev norms

$\lVert\sum
a_{j}\varphi_{j}\rVert_{H^{s}(\partial\Omega)}=\left(\sum_{j=1}^{\infty}j^{\frac{2s}{n-1}}\lvert
a_{j}\rvert^{2}\right)^{1/2}.$

Consider the finite dimensional space (with $N$ to be determined later)

$W=\left\\{\sum_{j=1}^{N}a_{j}\varphi_{j}\,;\,a_{j}\in\varmathbb{C}\right\\}.$

For any $f=\sum_{j=1}^{N}a_{j}\varphi_{j}\in W$, we have

(5.19) $\lVert
f\rVert_{H^{1/2}(\partial\Omega)}^{2}=\sum_{j=1}^{N}j^{\frac{1}{n-1}}\lvert
a_{j}\rvert^{2}\leqslant N^{\frac{2}{n-1}}\lVert
f\rVert_{H^{-1/2}(\partial\Omega)}^{2}.$

By (A.5) and (5.18), there exists $g\in W\setminus\\{0\\}$ so that

$\lVert Tg\rVert_{H^{-1/2}(\Gamma)}\leqslant\sigma_{N}(T)\lVert
g\rVert_{H^{1/2}(\partial\Omega)}\leqslant e^{-cN^{\frac{1}{n}}}\lVert
g\rVert_{H^{1/2}(\partial\Omega)}.$

Let $u\in H^{1}(\Omega)$ solve $Lu=0$ in $\Omega$ with
$u|_{\partial\Omega}=g$. We multiply $u$ by a constant so that $\lVert
u\rVert_{L^{2}(\Omega)}=\varepsilon$. Now $\lVert
u\rVert_{H^{1}(\Omega)}\leqslant C\lVert g\rVert_{H^{1/2}(\partial\Omega)}$,
and by a duality argument (using the additional regularity of $\partial\Omega$
and the coefficients) one also has $\lVert
g\rVert_{H^{-1/2}(\partial\Omega)}\leqslant C\lVert u\rVert_{L^{2}(\Omega)}$.
Combining these facts with (5.19), we obtain that

$\lVert u\rVert_{H^{1}(\Omega)}\leqslant CN^{\frac{1}{n-1}}\lVert
g\rVert_{H^{-1/2}(\partial\Omega)}\leqslant
C^{\prime}N^{\frac{1}{n-1}}\varepsilon.$

Now choose $N=N(\varepsilon)=\rho\varepsilon^{-(n-1)}$ for suitable small
$\rho>0$, so that $C^{\prime}N^{\frac{1}{n-1}}\varepsilon=1$. Then

$\lVert u\rVert_{L^{2}(\Omega)}=\varepsilon,\quad\lVert
u\rVert_{H^{1}(\Omega)}\leqslant 1,\quad\lVert
Tg\rVert_{H^{-1/2}(\Gamma)}\leqslant e^{-cN^{\frac{1}{n}}}\lVert
g\rVert_{H^{1/2}(\partial\Omega)}\leqslant
Ce^{-c^{\prime}\varepsilon^{-\frac{n-1}{n}}}$

by the trace estimate $\lVert g\rVert_{H^{1/2}(\partial\Omega)}\leqslant
C\lVert u\rVert_{H^{1}(\Omega)}\leqslant C$. This proves the second estimate
in the theorem. ∎

###  5.4. Exponential cost in approximate controllability for the heat
equation

As an example of how our results can also be applied in the context of
problems from control theory, we rederive lower bounds on the cost of
approximate controllability for the variable coefficient heat equation at low
regularity. For constant coefficients in the principal symbol and $L^{\infty}$
potentials this was first treated by [FCZ00] and later revisited in [Ph04]
where the authors proved _upper_ bounds on the cost of controllability. While
proving upper bounds requires “hard” arguments such as Carleman estimates, we
use our strategy from the previous sections to obtain “soft” arguments for the
_lower_ bounds on the cost of controllability – even at rather low
regularities for the coefficients and with essentially optimal dependences
(modulo the exponents). To this end, for $\omega\Subset\Omega$ and
$\Omega\subset\varmathbb{R}^{n}$ open, bounded $C^{1,\alpha}$,
$\alpha\in(0,1)$, domains, consider the equation

(5.20)
$\displaystyle\begin{split}\partial_{t}u-\partial_{i}a^{ij}\partial_{j}u+b^{1}_{j}\partial_{j}u+\partial_{j}(b^{2}_{j}u)+cu&=f\chi_{\omega}\mbox{
in }\Omega\times(0,T),\\\ u&=0\mbox{ on }\partial\Omega\times(0,T),\\\
u(\cdot,0)&=u_{0}\mbox{ in }\Omega,\end{split}$

where

  * (i)

$\chi_{\omega}$ denotes the characteristic function of $\omega$,

  * (ii)

$f\in L^{2}(\omega\times(0,T))$, $u_{0}\in H^{1}_{0}(\Omega)$,

  * (iii)

$a^{ij}\in(C^{0,1}_{x}\cap C^{0,\frac{1}{2}}_{t})(\Omega\times(0,T))$ is a
uniformly elliptic matrix with $\lambda|\xi|^{2}\leqslant
a^{ij}\xi_{i}\xi_{j}\leqslant\lambda^{-1}|\xi|^{2}$ for some
$\lambda\in(0,1)$,

  * (iv)

$b_{j}^{1},b_{j}^{2}\in C^{0}((0,T),L^{n+2}(\Omega))$, $c\in
C^{0}((0,T),L^{\frac{n}{2}}(\Omega))$ with

$\|b_{j}^{(\ell)}\|_{C^{0}((0,T),L^{n+2}(\Omega))}+\|c\|_{C^{0}((0,T),L^{\frac{n}{2}}(\Omega))}\leqslant\mu\ll
1$

for $\ell\in\\{1,2\\}$. If $n=2$ we further assume that $c\in
C^{0}((0,T),L^{p}(\Omega))$ for some $p>1$.

Under these conditions the Cauchy problem (5.20) is well-posed. Here and in
the sequel it is always interpreted in its weak form, i.e. $u$ is a solution
to (5.20) if the following identity holds for all $v\in
C^{1}((0,1),H^{1}(\Omega))$

$\displaystyle-(u,\partial_{t}v)_{L^{2}(\Omega\times(0,T))}+(a^{ij}\partial_{i}u,\partial_{j}v)_{L^{2}(\Omega\times(0,T))}+(b_{j}^{1}\partial_{j}u,v)_{L^{2}(\Omega\times(0,T))}$
$\displaystyle\qquad-(b_{j}^{2}u,\partial_{j}v)_{L^{2}(\Omega\times(0,T))}+(cu,v)_{L^{2}(\Omega\times(0,T))}$
$\displaystyle=(f\chi_{\omega},v)_{L^{2}(\Omega\times(0,T))}-(u(T),v(T))_{L^{2}(\Omega)}+(u(0),v(0))_{L^{2}(\Omega)}.$

The coefficients are chosen sufficiently regular so that the adjoint equation
satisfies the unique continuation principle (see [KT09] for weaker conditions
guaranteeing this). By standard (duality) arguments from control theory one
thus obtains that the equation (5.20) is approximately controllable, i.e. for
every function $u_{d}\in H^{1}_{0}(\Omega)$ and each $\varepsilon>0$ there
exists a control $f\in L^{2}(\Omega\times(0,T))$ such that the associated
solution $u$ of (5.20) satisfies

$\displaystyle\|u(\cdot,T)-u_{d}\|_{L^{2}(\Omega)}\leqslant\varepsilon.$

A more precise question then deals with the _cost of control_ which estimates
the size of the norm of $f$: As the image at time $T>0$ of $H^{1}_{0}(\Omega)$
under the evolution of the equation (5.20) is in general _not_ the whole space
$H^{1}_{0}(\Omega)$, this cost has to diverge as $\varepsilon\rightarrow 0$ in
general. The _cost of control_ provides bounds on this in terms of the size of
$\varepsilon$. For the case that $a^{ij}=\delta^{ij}$ and
$b_{j}^{1}=b_{j}^{2}=0$ it was proved in [FCZ00, Ph04] that there exists $f\in
L^{2}(\Omega\times(0,T))$ such that

(5.21) $\displaystyle\|f\|_{L^{2}(\Omega\times(0,T))}\leqslant
c\exp\left(C\frac{\|u_{d}\|_{H^{1}_{0}(\Omega)}}{\varepsilon}\right)\|u_{d}\|_{L^{2}(\Omega)}.$

Using the methods introduced in the previous sections, we complement (5.21)
with lower bounds (in different function spaces). More precisely, we show that
also in the case of low regularity coefficients as in (5.20) the
$\varepsilon$-dependence in the cost of controllability must be exponential:

######  Theorem 5.11.

Let $\omega,\Omega$ and $L$ be as above and let
$\delta\in(0,\min\\{\frac{1}{2},\delta_{0}\\})$, where $\delta_{0}>0$ denotes
the exponent from Lemma 5.5. Assume that for any $\varepsilon>0$ and any
$u_{d}\in H^{\delta}(\Omega)$ with $\|u_{d}\|_{H^{\delta}(\Omega)}=1$ there
exists a solution $u$ to (5.20) such that

$\displaystyle\|u(\cdot,T)-u_{d}\|_{L^{2}(\Omega)}\leqslant\varepsilon,$

and

$\displaystyle\|f\|_{L^{2}(\Omega\times(0,T))}\leqslant
M(\varepsilon)\|u_{d}\|_{L^{2}(\Omega)}.$

Then there is a constant $C>0$ such that
$M(\varepsilon)\geqslant\exp(C\varepsilon^{-\frac{1}{\delta}\frac{n}{n+2}})$.

The instability of approximate controllability will be deduced as a
consequence of the instability of an associated unique continuation problem.
We thus next formulate this result:

######  Proposition 5.12.

Let $\Omega,\omega$ be as above and let
$L^{\prime}:=-\partial_{t}-\partial_{i}a^{ij}\partial_{j}-\partial_{j}b_{j}^{1}-b_{j}^{2}\partial_{j}+c$
be the (formal) adjoint operator associated with the operator in (5.20). Let
further $\delta\in(0,\min\\{\frac{1}{2},\delta_{0}\\})$, where $\delta_{0}>0$
denotes the exponent from Lemma 5.5. For $\varphi_{T}\in L^{2}(\Omega)$
consider the equation

(5.22) $\displaystyle\begin{split}L^{\prime}\varphi&=0\mbox{ in
}\Omega\times(0,T),\\\ \varphi&=0\mbox{ on }\partial\Omega\times(0,T),\\\
\varphi(\cdot,T)&=\varphi_{T}\mbox{ in }\Omega.\end{split}$

Let $\delta_{0}>0$ be the constant from Lemma 5.5. Then, for each
$\varepsilon>0$ there exists a function $\varphi_{T}^{\varepsilon}\in
L^{2}(\Omega)$ and a solution $\varphi^{\varepsilon}$ to (5.22) with data
$\varphi_{T}^{\varepsilon}$ such that for any $\delta\in(0,\delta_{0})$

(5.23) $\displaystyle\|\varphi_{T}^{\varepsilon}\|_{L^{2}(\Omega)}=1,\
\|\varphi_{T}^{\varepsilon}\|_{H^{-\delta}(\Omega)}=\varepsilon,\
\|\varphi^{\varepsilon}\|_{L^{2}(\omega\times(0,T))}\leqslant
C\exp(-C\varepsilon^{-\frac{1}{\delta}\frac{n}{n+2}}).$

Before turning to the proof of Proposition 5.12, as an auxiliary tool, we
provide Davies-Gaffney type Gaussian estimates for solutions to the heat
equation:

######  Lemma 5.13.

Let $\Omega,\omega$ be as above and let $u$ be a solution to (5.20) with
initial data $u_{0}$, where $L$ satisfies the conditions from above. Assume
that $\mathrm{supp}(u_{0})\cap\overline{\omega}=\emptyset$. Let
$d:=\operatorname{dist}(\mathrm{supp}(u_{0}),\omega)>0$. Then, there exist
constants $C,c>0$ such that

$\displaystyle\|u(t)\|_{L^{2}(\omega)}\leqslant
Ce^{-c\frac{d^{2}}{t}}\|u_{0}\|_{L^{2}(\Omega)}.$

In the proof of Proposition 5.12, we will reverse time and will apply this to
solutions of (5.22).

###### Proof of Lemma 5.13.

The proof follows from exponentially weighted estimates. Let
$\psi:\Omega\rightarrow\varmathbb{R}$ be a Lipschitz function which is still
to be determined below. Then,

$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\int\limits_{\Omega}e^{2\psi}u^{2}dx\right)=\int\limits_{\Omega}e^{2\psi}u\partial_{t}udx$
$\displaystyle=-\int\limits_{\Omega}\partial_{i}(e^{2\psi}u)a^{ij}\partial_{j}udx-\int\limits_{\Omega}b_{j}^{1}\partial_{j}ue^{2\psi}udx+\int\limits_{\Omega}b_{j}^{2}u\partial_{j}(e^{2\psi}u)dx+\int\limits_{\Omega}cu^{2}e^{2\psi}dx$
$\displaystyle=-\int\limits_{\Omega}\partial_{i}(e^{\psi}u)a^{ij}\partial_{j}(e^{\psi}u)dx+\int\limits_{\Omega}\partial_{i}\psi
a^{ij}\partial_{j}\psi(e^{\psi}u)^{2}dx$
$\displaystyle\quad-\int\limits_{\Omega}b_{j}^{1}\partial_{j}(ue^{\psi})(e^{\psi}u)dx+\int\limits_{\Omega}b_{j}^{1}(ue^{\psi})(\partial_{j}\psi)(e^{\psi}u)dx$
$\displaystyle\quad+\int\limits_{\Omega}b_{j}^{2}(ue^{\psi})\partial_{j}(e^{\psi}u)dx+\int\limits_{\Omega}b_{j}^{2}(e^{\psi}u)\partial_{j}\psi\partial_{j}(e^{\psi}u)dx+\int\limits_{\Omega}c(e^{\psi}u)^{2}dx.$

Using the ellipticity of $a^{ij}$, Poincaré’s inequality and Sobolev
embedding, this leads to the differential inequality

$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\|e^{\psi}u\|_{L^{2}(\Omega)}^{2}\right)$
$\displaystyle\leqslant-\lambda\|\nabla(e^{\psi}u)\|_{L^{2}(\Omega)}^{2}+\lambda^{-1}\|e^{\psi}u\nabla\psi\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\quad+(\|b_{j}^{1}\|_{L^{n+2}(\Omega)}+\|b_{j}^{2}\|_{L^{n+2}(\Omega)})(\epsilon\|\nabla(e^{\psi}u)\|_{L^{2}(\Omega)}^{2}+C\epsilon^{-1}\|ue^{\psi}\|_{L^{2}(\Omega)}^{2})$
$\displaystyle\quad+\|c\|_{L^{\frac{n}{2}}(\Omega)}\|\nabla(e^{\psi}u)\|_{L^{2}(\Omega)}.$

Choosing $0<\epsilon=\frac{\lambda}{10}$ and using the smallness conditions on
the norms of the potentials imposed in condition (iv) above, we hence arrive
at

$\displaystyle\frac{d}{dt}\frac{1}{2}\|e^{\psi}u\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leqslant-\frac{\lambda}{10}\|\nabla(e^{\psi}u)\|_{L^{2}(\Omega)}^{2}+C\lambda^{-1}(1+\|\nabla\psi\|_{L^{\infty}(\Omega)}^{2})\|e^{\psi}u\|_{L^{2}(\Omega)}^{2},$

where $C>0$ depends on the norms of the coefficients. Invoking Gronwall’s
lemma, we then obtain that for some constant $C>0$ (depending on $\lambda$ and
the coefficient bounds)

(5.24) $\displaystyle\|e^{\psi}u\|_{L^{2}(\Omega)}\leqslant
e^{C(\|\nabla\psi\|_{L^{\infty}(\Omega)}^{2}+1)t}\|e^{\psi}u_{0}\|_{L^{2}(\Omega)}.$

Next, we fix $\psi$ to be Lipschitz continuous in such a way that $\psi=L$ in
$\omega$, $\psi=0$ on $\mathrm{supp}(u_{0})=0$ and such that
$\|\nabla\psi\|_{L^{\infty}(\Omega)}\leqslant\frac{L}{d}$. As a consequence,
from (5.24) and the bounds on $\psi$ we obtain

(5.25) $\displaystyle\|u\|_{L^{2}(\omega)}\leqslant
e^{C\frac{L^{2}}{d^{2}}t-L}\|u_{0}\|_{L^{2}(\Omega)}.$

Optimizing this in the choice of $L>0$ yields $L=\tilde{C}\frac{d^{2}}{t}$
which concludes the argument. ∎

With Lemma 5.13 in hand and recalling the regularity estimate (5.5), we
proceed to the proof of Proposition 5.12:

###### Proof of Proposition 5.12.

_Step 1: A regularity estimate._ We first note that by duality the estimate in
(5.5) yields that for any $t\in(0,T)$ and any $\delta\in(0,\delta_{0})$ (where
$\delta_{0}>0$ denotes the constant from Lemma 5.5)

$\displaystyle\|\varphi(T-t)\|_{L^{2}(\Omega)}\leqslant
Ct^{-\delta/2}\|\varphi_{T}\|_{H^{-\delta}(\Omega)}.$

Indeed, given $s\in(0,T)$, we consider a solution $\varphi$ of (5.22) and a
solution $u$ to (5.20), the latter with initial data at time $T-s$. Then, by
definition of solutions, we obtain that

$\displaystyle(u_{T-s},\varphi(T-s))_{L^{2}(\Omega)}=(u(T),\varphi_{T})_{L^{2}(\Omega)}.$

As a consequence,

$\displaystyle\|\varphi(T-s)\|_{L^{2}(\Omega)}$
$\displaystyle\leqslant\sup\limits_{\|u_{T-s}\|_{L^{2}(\Omega)}=1}(u_{T-s},\varphi(T-s))_{L^{2}(\Omega)}=\sup\limits_{\|u_{T-s}\|_{L^{2}(\Omega)}=1}(u(T),\varphi_{T})_{L^{2}(\Omega)}$
$\displaystyle\leqslant\sup\limits_{\|u_{T-s}\|_{L^{2}(\Omega)}=1}\|u(T)\|_{H^{\delta}(\Omega)}\|\varphi_{T}\|_{H^{-\delta}(\Omega)}$
$\displaystyle\leqslant\sup\limits_{\|u_{T-s}\|_{L^{2}(\Omega)}=1}Cs^{-\delta/2}\|u_{T-s}\|_{L^{2}(\Omega)}\|\varphi_{T}\|_{H^{-\delta}(\Omega)}\leqslant
Cs^{-\delta/2}\|\varphi_{T}\|_{H^{-\delta}(\Omega)}.$

Here we used that $\delta\in(0,\frac{1}{2})$ and the Sneiberg type result of
Lemma 5.5.

_Step 2: Gaussian bounds._ Next, we claim that for any $t\in(0,T)$ there
exists a constant $\tilde{C}>0$ such that

(5.26)
$\displaystyle\|\varphi\|_{L^{2}(\omega\times(T-t,T))}\leqslant\tilde{C}e^{-\frac{\tilde{C}}{t}}\|\varphi_{T}\|_{H^{-\delta}(\Omega)}.$

Indeed, by the Gaussian bounds from Lemma 5.13 (together with a reversal of
time) and by Step 1, for any $s\in(0,T)$ we have that

$\displaystyle\|\varphi(T-s)\|_{L^{2}(\omega)}\leqslant
Ce^{-\frac{C}{s}}\|\varphi(T-s/2)\|_{L^{2}(\Omega)}\leqslant
Ce^{-\frac{C}{s}}s^{-\delta/2}\|\varphi_{T}\|_{H^{-\delta}(\Omega)}.$

Squaring and integrating this from $s\in(0,t)$, we obtain the desired bound
(5.26).

_Step 3: Conclusion._ We next consider the two maps

$\displaystyle
i_{1}(t):H^{-\delta}(\Omega)\ni\varphi_{T}\mapsto\varphi|_{\omega\times(T-t,T)}\in
L^{2}(\omega\times(T-t,T)),$ $\displaystyle
i_{2}(t):H^{-\delta}(\Omega)\ni\varphi_{T}\mapsto\varphi|_{\omega\times(0,T-t)}\in
L^{2}(\omega\times(0,T-t)).$

Now, on the one hand, by Lemma 5.13 and by Theorem 5.4 (with a time-step
larger or equal to $t$ and not a time step of the order one) and an
integration argument as in Step 2, we obtain

(5.27) $\displaystyle\sigma_{k}(i_{2}(t))\leqslant
e^{-ck^{\frac{2}{n+2}}t^{\frac{n}{n+2}}}.$

On the other hand, Step 2 implies that

(5.28) $\displaystyle\sigma_{1}(i_{1}(t))\leqslant e^{-\frac{C}{t}}.$

Since the the map
$i:L^{2}(\Omega)\ni\varphi_{T}\mapsto\varphi|_{\omega\times(0,T)}\in
L^{2}(\omega\times(0,T))$ can we written as $i=i_{2}(t)+i_{1}(t)$, for the
singular values of $i$ we infer that

$\displaystyle\sigma_{k}(i)\leqslant\sigma_{1}(i_{1}(t))+\sigma_{k}(i_{2}(t))\leqslant
Ce^{-\frac{C}{t}}+e^{-ck^{\frac{2}{n+2}}t^{\frac{n}{n+2}}}.$

Optimizing this in $t$, we consider $t=\tilde{c}k^{-\frac{1}{n+1}}$ for some
$\tilde{c}>0$ and hence arrive at

$\displaystyle\sigma_{k}(i)\leqslant e^{-ck^{\frac{1}{n+1}}}.$

Invoking Theorem 5.1 with $X=L^{2}(\Omega)$, $X_{1}=H^{-\delta}(\Omega)$ we
thus deduce that

$\displaystyle\omega(t)\geqslant C|\log(t)|^{-\delta\frac{n+2}{n}}.$

This concludes the proof. ∎

With Proposition 5.12 in hand, the proof of Theorem 5.11 reduces to a duality
argument.

###### Proof of Theorem 5.11.

We prove that if there was a cost of control that was better than exponential,
then also the associated unique continuation estimate (5.23) would have to be
better than logarithmic (which by (5.23) cannot be the case). The following
duality argument is analogous to the strategy outlined in [Ph04]: Considering
a solution $u$ to (5.20) and a solution $\varphi$ to (5.22), integrating by
parts and using the Dirichlet boundary conditions, we obtain (5.22)

$\displaystyle\int\limits_{0}^{T}\int\limits_{\omega}f\varphi
dxdt=\int\limits_{\Omega}u(x,T)\varphi_{T}dx=\int\limits_{\Omega}(u(x,T)-u_{d}(x))\varphi_{T}(x)dx+\int\limits_{\Omega}u_{d}(x)\varphi_{T}(x)dx.$

Rearranging and assuming the approximation bounds stated in Theorem 5.11 then
yields

$\displaystyle\left|\int\limits_{\Omega}u_{d}\varphi_{T}dx\right|$
$\displaystyle\leqslant\|u(\cdot,T)-u_{d}\|_{L^{2}(\Omega)}\|\varphi_{T}\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\omega\times(0,T))}\|\varphi\|_{L^{2}(\omega\times(0,T))}$
$\displaystyle\leqslant\varepsilon\|u_{d}\|_{H^{\delta}(\Omega)}\|\varphi_{T}\|_{L^{2}(\Omega)}+M(\varepsilon)\|u_{d}\|_{L^{2}(\Omega)}\|\varphi\|_{L^{2}(\omega\times(0,T))}.$

By the definition of the $H^{-\delta}(\Omega)$ norm through duality (where we
use that $\delta\in(0,1/2)$), we hence infer that

(5.29)
$\displaystyle\|\varphi_{T}\|_{H^{-\delta}(\Omega)}\leqslant\varepsilon\|\varphi_{T}\|_{L^{2}(\Omega)}+M(\varepsilon)\|\varphi\|_{L^{2}(\omega\times(0,T))}.$

Inserting the functions $\varphi^{\tilde{\varepsilon}}$ with
$\tilde{\varepsilon}=2\varepsilon$ from Proposition 5.12 into (5.29) implies

(5.30) $\displaystyle
2\varepsilon\leqslant\varepsilon+M(2\varepsilon)C\exp(-C(2\varepsilon)^{-\frac{1}{\delta}\frac{n}{n+2}}).$

As a consequence, $M(2\varepsilon)$ has to be of the order of
$\exp(C(2\varepsilon)^{-\frac{1}{\delta}\frac{n}{n+2}})$ for some $C>0$. ∎

###  5.5. Instability of the Calderón problem at low regularity

In this section, we show that similarly as for the stability estimates in
unique continuation, also for the stability properties of the Calderón
problem, one cannot hope for a gain due to the presence of low regularity
metrics and potentials (thus proving Theorem 1.4 and answering question (Q3)
from the introduction for the example of the Calderón problem).

Let $\Omega\subset\varmathbb{R}^{n}$ be a bounded open set with Lipschitz
boundary. As in Section 2.5 we consider the operator
$L=-\partial_{j}a^{jk}\partial_{k}+b_{j}\partial_{j}+\partial_{j}c_{j}+q_{0}$
with $(a^{jk})\in L^{\infty}(\Omega,\varmathbb{R}^{n\times n})$ uniformly
elliptic, $b_{j},c_{j}\in L^{n}(\Omega)$ and $q_{0}\in
L^{\frac{n}{2}}(\Omega)$. If $u\in H^{1}(\Omega)$ is a weak solution of
$Lu=0$, we recall that the normal derivative
$\partial_{\nu}^{L}u|_{\partial\Omega}$ is defined weakly as an element of
$H^{-1/2}(\partial\Omega)$ by (2.3). If $L$ is such that zero is not a
Dirichlet eigenvalue, we may thus define the Dirichlet-to-Neumann operator

(5.31) $\displaystyle\Lambda_{L}:H^{\frac{1}{2}}(\partial\Omega)\rightarrow
H^{-\frac{1}{2}}(\partial\Omega),\
f\mapsto\partial_{\nu}^{L}u|_{\partial\Omega},$

where $u$ is a solution to $Lu=0$ in $\Omega$ with $u|_{\partial\Omega}=f$.

In the following, we seek to prove that for the Dirichlet-to-Neumann operator
the instability result from Theorem 1.4 holds. In this context, the main
novelty of the associated estimates with respect to the instability results
from Mandache [Ma01] and Di Cristo-Rondi [DR03] is that our mechanism is very
robust in that we can treat

  * •

potentials $q_{0}+q_{1},q_{0}+q_{2}$ which are _not necessarily compactly
supported_ ,

  * •

coefficients of _scaling critical low regularity_ ,

  * •

and consider the Calderón problem in _relatively rough and general domains_.

The proof of Theorem 1.4 will be based on the following abstract result.
Recall that $Y^{\prime}$ is the set of bounded linear functionals on $Y$, and
that for $Y$ Hilbert any $A\in B(Y,Y^{\prime})$ has a formal adjoint
$A^{\prime}\in B(Y,Y^{\prime})$.

######  Theorem 5.14.

Let $F:X\to B(Y,Y^{\prime})$ be a continuous map, where $X$ is a Banach space
and $Y$ is a separable Hilbert space. 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 there is an orthonormal basis
$(\varphi_{j})_{j=1}^{\infty}$ of $Y$ and constants $C,\rho,\mu>0$ so that
$F(u)$ and $F(u)^{\prime}$ satisfy

(5.32) $\lVert F(u)\varphi_{k}\rVert_{Y^{\prime}},\ \lVert
F(u)^{\prime}\varphi_{k}\rVert_{Y^{\prime}}\leqslant Ce^{-\rho k^{\mu}},\qquad
k\geqslant 1,$

uniformly over $u\in K$. Then there is $c>0$ with the following property: for
any $\varepsilon>0$ small enough there are $u_{1},u_{2}$ such that

(5.33)
$\|u_{1}-u_{2}\|_{X}\geqslant\varepsilon,\quad\|u_{j}\|_{X_{1}}\leqslant
r,\quad\|F(u_{1})-F(u_{2})\|_{B(Y,Y^{\prime})}\leqslant\exp(-c\varepsilon^{-\frac{\mu}{m(\mu+2)}}).$

In particular, if one has the stability property

$\lVert u_{1}-u_{2}\rVert_{X}\leqslant\omega(\lVert
F(u_{1})-F(u_{2})\rVert_{B(Y,Y^{\prime})}),\qquad u_{1},u_{2}\in K,$

then necessarily $\omega(t)\gtrsim\lvert\log\,t\rvert^{-\frac{m(\mu+2)}{\mu}}$
for $t$ small.

Let us comment on this abstract result which will be the main ingredient in
our discussion of the instability properties of the Calderón problem at low
regularity. The main point is that there is a fixed orthonormal basis
$(\varphi_{j})$ that gives the decay (5.32) uniformly over $u\in K$ (cf. also
[DR03]). As a sufficient condition for (5.32), it is enough to find a linear
operator $A:X\to Z$ which dominates $F(u)$ and $F(u)^{\prime}$ in the sense
that

$\lVert F(u)f\rVert_{Y^{\prime}},\ \lVert
F(u)^{\prime}f\rVert_{Y^{\prime}}\leqslant C\lVert Af\rVert_{Z}$

uniformly over $u\in K$, and that the singular values of $A$ satisfy
$\sigma_{k}(A)\leqslant Ce^{-\rho k^{\mu}}$ (one can then take $(\varphi_{j})$
as a singular value basis of $A$). In our discussion of the Calderón problem
below, we will follow this strategy by considering
$F(q)=\Lambda_{L+q}-\Lambda_{L}$ and by taking $A$ to be the operator which
maps $u|_{\partial\Omega}$ to $u|_{\partial\Omega^{\prime}}$ where $u$ solves
$Lu=0$ in $\Omega$ and $\Omega^{\prime}\Subset\Omega$. We already saw in the
proof of Theorem 5.7 that the singular values of $A$ decay exponentially.

###### Proof of Theorem 5.14.

Define the norm

$\lVert
f\rVert_{\tilde{Y}}=\left(\sum_{j=1}^{\infty}e^{-(\rho/2)j^{\mu}}\lvert(f,\varphi_{j})_{Y}\rvert^{2}\right)^{1/2}$

and let $\tilde{Y}$ be the completion of $Y$ with respect to this norm. We
claim that for any $u\in K$, the map $F(u)$ extends as a bounded operator
$\tilde{Y}\to\tilde{Y}^{\prime}$. In fact, write
$a_{jk}=a_{jk}(u)=(F(u)\varphi_{k})(\varphi_{j})$, so that

(5.34) $\lvert a_{jk}\rvert\leqslant Ce^{-\rho(\max(j,k))^{\mu}}$

by (5.32). For any $f,g\in Y$ we have $F(u)f\in Y^{\prime}$ and

$\displaystyle\lvert(F(u)f)(g)\rvert$
$\displaystyle\leqslant\lvert\sum_{j,k}a_{jk}(f,\varphi_{k})(g,\varphi_{j})\rvert\leqslant\left(\sum_{j,k}\lvert
a_{jk}\rvert^{2}e^{(\rho/2)(j^{\mu}+k^{\mu})}\right)^{1/2}\lVert
f\rVert_{\tilde{Y}}\lVert g\rVert_{\tilde{Y}}$ (5.35) $\displaystyle\leqslant
C\lVert f\rVert_{\tilde{Y}}\lVert g\rVert_{\tilde{Y}}.$

Let $\tilde{\varphi}_{j}=e^{(\rho/4)j^{\mu}}\varphi_{j}$, so that
$(\tilde{\varphi}_{j})$ is an orthonormal basis of $\tilde{Y}$. Then as in
(5.35)

$\lVert
F(u)\tilde{\varphi}_{l}\rVert_{\tilde{Y}^{\prime}}\leqslant\sup_{\lVert
g\rVert_{\tilde{Y}}=1}\lvert\sum_{j}a_{jl}e^{(\rho/4)l^{\mu}}(g,\varphi_{j})\rvert\leqslant\left(\sum_{j}\lvert
a_{jl}\rvert^{2}e^{(\rho/2)(j^{\mu}+l^{\mu})}\right)^{1/2}\leqslant
Ce^{-(\rho/4)l^{\mu}}$

where we used (5.34). It follows that

$\lVert
F(u)\rVert_{HS(\tilde{Y},\tilde{Y}^{\prime})}^{2}=\sum_{l=1}^{\infty}\lVert
F(u)\tilde{\varphi}_{l}\rVert_{\tilde{Y}^{\prime}}^{2}\leqslant C<\infty.$

The constants are uniform over $u\in K$.

Now $F(K)$ is contained in a bounded subset of
$HS(\tilde{Y},\tilde{Y}^{\prime})$, and the embedding

$\iota:HS(\tilde{Y},\tilde{Y}^{\prime})\to HS(Y,Y^{\prime})$

has singular values satisfying $\sigma_{k}(\iota)\lesssim Ce^{-ck^{\mu/2}}$ by
Lemma 3.18 (we can take $\alpha_{j}=\beta_{j}=e^{(\rho/4)j^{\mu}}$). Lemma 3.9
gives that $e_{k}(\iota)\lesssim e^{-ck^{\frac{\mu}{2+\mu}}}$. Now the theorem
follows from Lemma 3.7. ∎

###### Proof of Theorem 1.4.

Since $L$ has discrete spectrum and $0$ is not a Dirichlet eigenvalue of $L$
in $\Omega$, there is $\varepsilon_{0}>0$ so that $0$ is not a Dirichlet
eigenvalue of $L+q$ in $\Omega$ whenever $\lVert
q\rVert_{L^{n/2}(\Omega)}\leqslant\varepsilon_{0}$. Now choose a bounded
smooth domain $\Omega^{\prime}\subset\subset\Omega$, and let

$K:=\\{q\in W^{\delta,n/2}_{0}(\Omega^{\prime})\,;\,\lVert
q\rVert_{W^{\delta,n/2}(\Omega^{\prime})}\leqslant\varepsilon_{0}\\}.$

Identifying $q$ with its zero extension to $\Omega$, we have that $K$ is a
compact subset of $X:=L^{n/2}(\Omega)$.

Write $Y:=H^{1/2}(\partial\Omega)$, and define the map

$F:X\to B(Y,Y^{\prime}),\ \ F(q)=\Lambda_{L+q}-\Lambda_{L}.$

In fact $F$ is well defined near the set where $\lVert
q\rVert_{L^{n/2}(\Omega)}\leqslant\varepsilon_{0}$ which contains $K$, and
this is enough for our purposes. The embedding
$i:W^{\delta,n/2}_{0}(\Omega^{\prime})\to L^{n/2}(\Omega)$ has entropy numbers
$e_{k}(i)\gtrsim k^{-\delta/n}$ by [ET08, Theorem 1 in Section 3.3.3] (note
that the functions $f$ in that theorem are compactly supported). Given $q\in
K$ and $f\in Y$, let $u,u_{0}\in H^{1}(\Omega)$ solve $(L+q)u=Lu_{0}=0$ in
$\Omega$ with $u|_{\partial\Omega}=u_{0}|_{\partial\Omega}=f$. Writing
$w:=u-u_{0}$, we have

$F(q)f=\partial_{\nu}^{L}w|_{\partial\Omega}$

where $w\in H^{1}(\Omega)$ solves

$(L+q)w=-qu_{0}\text{ in $\Omega$},\qquad w|_{\partial\Omega}=0.$

Elliptic regularity and the fact that $q$ is supported in
$\overline{\Omega}^{\prime}$ give that

$\displaystyle\lVert\partial_{\nu}^{L}w\rVert_{H^{-1/2}(\partial\Omega)}$
$\displaystyle\lesssim\lVert w\rVert_{H^{1}(\Omega)}\lesssim\lVert
qu_{0}\rVert_{H^{-1}(\Omega)}\lesssim\lVert
q\rVert_{L^{n/2}(\Omega^{\prime})}\lVert u_{0}\rVert_{H^{1}(\Omega^{\prime})}$
$\displaystyle\lesssim\lVert
u_{0}|_{\partial\Omega^{\prime}}\rVert_{H^{1/2}(\partial\Omega^{\prime})}.$

Writing $Z:=H^{1/2}(\partial\Omega^{\prime})$, we thus have

$\lVert F(q)f\rVert_{Y^{\prime}}\leqslant C\lVert Af\rVert_{Z}$

uniformly over $q\in K$, where $A:Y\to Z$ maps $u_{0}|_{\partial\Omega}$ to
$u_{0}|_{\partial\Omega^{\prime}}$. The constant is uniform over $q\in K$. By
the argument in Theorem 5.7 there is $c>0$ so that

$\sigma_{k}(A)\leqslant e^{-ck^{1/n}},\qquad k\geqslant 1.$

Choosing $(\varphi_{j})$ to be a singular value basis for $A$ and using that
$F(q)$ is self-adjoint, we see that (5.32) holds with $\mu=1/n$. The result
now follows from Theorem 5.14. ∎

######  Remark 5.15 (Possible extensions).

Finally, we comment on possible extensions of this result:

  * •

_Cauchy data._ Instead of working with the Dirichlet-to-Neumann map a perhaps
more natural setting is that of working with Cauchy data

$\displaystyle\mathcal{C}_{q}=\left\\{\left(u|_{\partial\Omega},\partial_{\nu}^{L}u|_{\partial\Omega}\right)\in
H^{\frac{1}{2}}(\partial\Omega)\times H^{-\frac{1}{2}}(\partial\Omega):\
u\mbox{ is a solution to }Lu=0\mbox{ in }\Omega\right\\},$

endowed with the distance

$\displaystyle\operatorname{dist}(\mathcal{C}_{q_{1}},\mathcal{C}_{q_{2}})$
$\displaystyle=\max\left\\{\max\limits_{(f,g)\in\mathcal{C}_{q_{1}}}\min\limits_{(\tilde{f},\tilde{g})\in\mathcal{C}_{q_{2}}}\frac{\|(f,g)-(\tilde{f},\tilde{g})\|_{H^{\frac{1}{2}}(\partial\Omega)\times
H^{-\frac{1}{2}}(\partial\Omega)}}{\|(f,g)\|_{H^{\frac{1}{2}}(\partial\Omega)\times
H^{-\frac{1}{2}}(\partial\Omega)}},\right.$
$\displaystyle\quad\left.\max\limits_{(f,g)\in\mathcal{C}_{q_{2}}}\min\limits_{(\tilde{f},\tilde{g})\in\mathcal{C}_{q_{1}}}\frac{\|(f,g)-(\tilde{f},\tilde{g})\|_{H^{\frac{1}{2}}(\partial\Omega)\times
H^{-\frac{1}{2}}(\partial\Omega)}}{\|(f,g)\|_{H^{\frac{1}{2}}(\partial\Omega)\times
H^{-\frac{1}{2}}(\partial\Omega)}}\right\\},$

where $\|(f,g)\|_{H^{\frac{1}{2}}(\partial\Omega)\times
H^{-\frac{1}{2}}(\partial\Omega)}=\|f\|_{H^{\frac{1}{2}}(\partial\Omega)}+\|g\|_{H^{-\frac{1}{2}}(\partial\Omega)}$.
In order to avoid technicalities we did not formulate the Calderón problem in
this setting above. Similar arguments as above could however be used also in
the framework involving Cauchy data (dealing additionally with finite
dimensional subspaces).

  * •

_Potentials in lower regularity classes._ We emphasize that our Schrödinger
operators also include the setting of potentials $q_{0}\in W^{-1,n}(\Omega)$.
Indeed, any such potential can be represented in the form
$q_{0}=F_{0}+\partial_{i}F_{i}$ with $F_{0}\in L^{\frac{n}{2}}(\Omega)$ and
$F_{i}\in L^{n}(\Omega)$ for $i\in\\{1,\dots,n\\}$. In order to prove an
instability result for the Calderón problem in these spaces, one would have to
vary not only the potentials $q_{0}$ but also the drift terms $b_{j},c_{j}$.
Such an instability result would follow using our strategy in a similar way as
the result from Theorem 1.4.

##  6\. Instability in the presence of microlocal smoothing

In the previous sections we considered forward operators that have _global_
smoothing properties, in the sense that they either erase all singularities of
any function that they are applied to, or at least provide a certain (possibly
small) amount of global regularization. In this section, we will now consider
forward operators that are only _microlocally_ smoothing at certain points. If
$A$ is linear and microlocally smoothing at $(x_{0},\xi_{0})$, then no
singularity of $u$ at $(x_{0},\xi_{0})$ can be recovered from the knowledge of
$Au$. It is known (see e.g. [MSU15, Theorem 4.4]) that in such cases a
stability estimate of the type

(6.1) $\lVert u\rVert_{H^{s_{1}}}\leqslant C(\lVert
Au\rVert_{H^{s_{2}}}+\lVert u\rVert_{H^{s_{3}}})$

cannot hold for any choices of $s_{1},s_{2},s_{3}\in\varmathbb{R}$ for which
$s_{1}>s_{3}$. This situation is relevant for many inverse problems, including
limited data X-ray tomography [Qu93, Qu06] or the study of geodesic X-ray
transforms [MSU15].

Let $A$ be linear and microlocally smoothing at $(x_{0},\xi_{0})$. Then the
fact that an estimate like (6.1) cannot be valid can be proved by testing
(6.1) with wave packets, or “coherent states", $u$ that concentrate in a conic
neighborhood of $(x_{0},\xi_{0})$ in phase space (for such functions $Au$ is
very small because of microlocal smoothing). See e.g. [SU09, Proposition 5]
for an argument of this type. From the point of view of the entropy/capacity
approach in Section 3 this means that not all of $A(K)$ is contained in a
compressed space, but there is a large enough set $K_{1}\subset K$ so that
$A(K_{1})$ lies in a compressed space. The set $K_{1}$ contains functions that
concentrate in a conic neighborhood of $(x_{0},\xi_{0})$, and there are
relatively large $\varepsilon$-discrete sets contained in $K_{1}$ (in fact
such sets could be constructed directly from wave packets using almost
orthogonality). Replacing $K$ by $K_{1}$ in the analysis in Section 3 then
leads to instability.

We can use the remarks in the preceding paragraph to give a more general
instability argument that also applies to nonlinear inverse problems. The idea
is that finding $\varepsilon$-discrete sets in $K_{1}$ can be reduced to
obtaining lower bounds for entropy/capacity numbers of microlocal cutoffs to a
conic neighborhood of $(x_{0},\xi_{0})$. This in turn reduces to a standard
Weyl law for pseudodifferential operators (Theorem 2.6).

######  Theorem 6.1.

Let $(M,g)$ be a closed smooth $n$-manifold and let
$P\in\Psi^{0}_{\mathrm{cl}}(M)$ be noncharacteristic (i.e. have nonvanishing
principal symbol) at some $(x_{0},\xi_{0})\in T^{*}M\setminus\\{0\\}$. For any
$s_{1}>s_{2}$, the compact operator $P:H^{s_{1}}(M)\to H^{s_{2}}(M)$ satisfies

$\sigma_{k}(P),\ e_{k}(P)\sim k^{-\frac{s_{1}-s_{2}}{n}}.$

###### Proof.

By Lemma 3.9 it is enough to prove the claim for $\sigma_{k}(P)$. We seek to
reduce the claim to Theorem 2.6. To this end, let
$J^{\alpha}=(1-\Delta_{g})^{\alpha/2}$ be the Bessel potential which induces
an isomorphism $H^{s}(M)\to H^{s-\alpha}(M)$ for any $s$, write
$m=s_{1}-s_{2}>0$, and let
$A=J^{s_{2}}PJ^{-s_{1}}\in\Psi^{-m}_{\mathrm{cl}}(M)$. Then the theorem
follows if we can show that any $A\in\Psi^{-m}_{\mathrm{cl}}(M)$ which is
noncharacteristic at some $(x_{0},\xi_{0})$, considered as an operator
$A:L^{2}(M)\to L^{2}(M)$, has singular values satisfying

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

This, however, follows from Theorem 2.6. ∎

Combining the Weyl law of Theorem 6.1 with Theorem 3.3 and Lemma 3.7 we obtain
the following abstract instability result. We will later apply it to the Radon
transform in $\varmathbb{R}^{2}$, and hence we will allow noncompact
manifolds. For this we recall the localized Sobolev spaces $H^{s}_{L}(M)$ from
Definition 2.9.

######  Theorem 6.2.

Let $M,N$ be smooth manifolds, let $L\subset M$ and $Q\subset N$ be compact,
let $s,t\in\varmathbb{R}$, and let $\delta>0$. Let $K_{r}=\\{f\in
H^{s+\delta}_{L}(M)\,;\,\lVert f\rVert_{H^{s+\delta}}\leqslant r\\}$, and let
$F$ be a map $K_{r}\to H^{t}_{Q}(N)$. Suppose that $\omega$ is a modulus of
continuity such that

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

  1. (a)

If there is $P\in\Psi^{0}_{\mathrm{cl}}(M)$, noncharacteristic at some point
of $T^{*}L^{\mathrm{int}}\setminus\\{0\\}$, and $r_{0}>0$ so that
$P(K_{r_{0}})\subset K_{r}$ and $F\circ P$ maps $K_{r_{0}}$ into a bounded set
of $H^{t+m}_{Q}(N)$ for any $m>0$, then for any $\alpha\in(0,1)$ one has
$\omega(t)\gtrsim t^{\alpha}$ for $t$ small.

  2. (b)

If there is $P\in\Psi^{0}_{\mathrm{cl}}(M)$, noncharacteristic at some point
of $T^{*}L^{\mathrm{int}}\setminus\\{0\\}$, and $r_{0}>0$ so that
$P(K_{r_{0}})\subset K_{r}$ and $F\circ P$ maps $K_{r_{0}}$ into a bounded set
of $A^{\sigma,\rho}_{Q}(N)$ 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.

###### Proof.

We first give the proof under the assumption that $L=M$, so that $M$ is
compact. By Theorem 6.1 and (3.4), the capacity numbers of
$P:H^{s+\delta}(M)\to H^{s}(M)$ satisfy $c_{k}(P)\gtrsim k^{-\delta/n_{M}}$.
Arguing as in the proof of Lemma 3.7, we see that for $\varepsilon>0$ small
there is an $\varepsilon$-discrete set in
$(P(K_{r_{0}}),\lVert\,\cdot\,\rVert_{H^{s}})$ having $>g(\varepsilon)$
elements, where

$g(\varepsilon)=e^{c\varepsilon^{-n_{M}/\delta}}.$

On the other hand, since $F\circ P$ maps $K_{r_{0}}$ into a bounded set of
$H^{t+m}_{Q}(N)$ and since $j:H^{t+m}_{Q}(N)\to H^{t}_{Q}(N)$ has entropy
numbers satisfying $e_{k}(j)\lesssim j^{-m/n_{N}}$, we see that for $\delta>0$
small there is a $\delta$-net covering $F(P(K_{r_{0}}))$ with $\leqslant
f(\delta):=e^{C\delta^{-n_{N}/m}}$ elements. Part (a) now follows from Theorem
3.3 where $K$ is replaced by $P(K_{r_{0}})$. Part (b) follows similarly.

We now consider the case where $L\neq M$. Then there is a smooth compact
subdomain $D$ of $M$ so that $L\subset D^{\mathrm{int}}$. Let $M_{1}$ be a
closed manifold containing $D$, so that we may identify functions in
$H^{s}_{L}(M)$ with functions in $H^{s}_{L}(M_{1})$. By assumption there is
$(x,\xi)\in T^{*}L^{\mathrm{int}}\setminus\\{0\\}$ so that $P$ is
noncharacteristic at $(x,\xi)$. Let $\tilde{P}=P\chi$ where $\chi\in
C^{\infty}_{c}(L^{\mathrm{int}})$ satisfies $\chi=1$ near $x$. Also choose
$r_{1}$ so that $\lVert\chi u\rVert_{H^{s+\delta}(M_{1})}\leqslant r_{0}$ when
$\lVert u\rVert_{H^{s+\delta}(M_{1})}\leqslant r_{1}$.

By the assumption $P(K_{r_{0}})\subset K_{r}$ and linearity of $P$, we see
that $\tilde{P}$ maps functions in $H^{s+\delta}_{L}(M)$ to functions
supported in $L$. Hence we may consider $\tilde{P}$ as a $\Psi$DO on $M_{1}$.
By Theorem 6.1 and (3.4), the capacity numbers of
$\tilde{P}:H^{s+\delta}(M_{1})\to H^{s}(M_{1})$ satisfy
$c_{k}(\tilde{P})\gtrsim k^{-\delta/n_{M}}$. Then there is an
$\varepsilon$-discrete set in $(\tilde{P}(\\{u\in
H^{s+\delta}(M_{1})\,;\,\lVert u\rVert_{H^{s+\delta}}\leqslant
r_{1}\\}),\lVert\,\cdot\,\rVert_{H^{s}})$ having
$>g(\varepsilon)=e^{c\varepsilon^{-n_{M}/\delta}}$ elements. Since
$\tilde{P}u=P(\chi u)$ and $\chi$ is supported in $L$, this yields an
$\varepsilon$-discrete set in $(P(K_{r_{0}}),\lVert\,\cdot\,\rVert_{H^{s}})$
having $>g(\varepsilon)$ elements. The rest of the proof proceeds as before. ∎

Our discussion of microlocal smoothing effects continues as follows. In
Section 6.1 we begin by deducing instability results for general linear
operators under the assumption of microlocal $C^{\infty}$ or Gevrey smoothing.
In the next Section 6.2 we then discuss explicit inverse problems with
$C^{\infty}$ and real-analytic (or Gevrey) microlocal smoothing. Here we show
that using our abstract results from the first part of this section one can
actually prove that there is super-polynomial or exponential instability in
these applications.

###  6.1. Instability for microlocally smoothing linear operators

We now discuss the microlocal smoothing properties of general linear operators
based on wave front sets of their Schwartz kernels. Let $X$ and $Y$ be open
sets in Euclidean spaces. Recall from [Hö85, Section 8.2] that for a linear
operator $\mathcal{K}$ from $C_{c}^{\infty}(Y)$ to $\mathcal{D}^{\prime}(X)$
having Schwartz kernel $K\in\mathcal{D}^{\prime}(X\times Y)$, we have a unique
way of defining $\mathcal{K}u$ for any $u\in\mathcal{E}^{\prime}(Y)$ with
$WF(u)\cap WF^{\prime}(K)_{Y}=\emptyset$ (see [Hö85, Theorem 8.2.13]). In this
case

$\displaystyle WF(\mathcal{K}u)\subset WF(K)_{X}\cup WF^{\prime}(K)\circ
WF(u).$

Here we have used the notations

$\displaystyle WF^{\prime}(K)$ $\displaystyle=\\{(x,y,\xi,\eta):\
(x,y,\xi,-\eta)\in WF(K)\\},$ $\displaystyle WF(K)_{X}$
$\displaystyle=\\{(x,\xi):\ (x,y,\xi,0)\in WF(K)\mbox{ for some }y\in Y\\},$
$\displaystyle WF^{\prime}(K)_{Y}$ $\displaystyle=\\{(y,\eta):\
(x,y,0,-\eta)\in WF(K)\mbox{ for some }x\in X\\}.$

Moreover, let $\mathcal{K}_{2}$ be a linear operator with kernel
$K_{2}\in\mathcal{D}^{\prime}(Y\times Z)$ so that the projection
$\mathrm{supp}(K_{2})\ni(y,z)\mapsto z$ is proper (then we say that
$\mathcal{K}_{2}$ _preserves compact supports_), and let $\mathcal{K}_{1}$ be
an operator with kernel $K_{1}\in\mathcal{D}^{\prime}(X\times Y)$ such that
$WF^{\prime}(K_{1})_{Y}\cap WF(K_{2})_{Y}=\emptyset$. Then the composition
$\mathcal{K}_{1}\circ\mathcal{K}_{2}$ is defined as a map from
$C_{c}^{\infty}(Z)$ to $\mathcal{D}^{\prime}(X)$ (see [Hö85, Theorem 8.2.14]).
Its wave front set satisfies

$\displaystyle WF^{\prime}(K)$ $\displaystyle\subset WF^{\prime}(K_{1})\circ
WF^{\prime}(K_{2})\cup(WF(K_{1})_{X}\times Z\times\\{0\\})$ (6.2)
$\displaystyle\quad\cup(X\times\\{0\\}\times WF^{\prime}(K_{2})_{Z}).$

The same statements are valid on smooth manifolds if one works with half
densities (see [Hö71, Section 2.5]), or on smooth Riemannian manifolds if we
use the volume form to define Schwartz kernels and spaces of distributions.

We first discuss the absence of Hölder stability bounds in the presence of
$C^{\infty}$ microlocal smoothing.

######  Theorem 6.3.

Let $M_{1},M_{2}$ be smooth manifolds, and let
$A:C^{\infty}_{c}(M_{1})\to\mathcal{D}^{\prime}(M_{2})$ be a continuous linear
operator which preserves compact supports. Suppose that $A$ is microlocally
smoothing at $(y_{0},\eta_{0})\in T^{*}M_{1}\setminus\\{0\\}$, in the sense
that

$WF(K_{A})_{M_{2}}=\emptyset$

and $(x,y,\xi,\eta)\notin WF^{\prime}(K_{A})$ for $(x,\xi)\in T^{*}M_{2}$ and
for $(y,\eta)\in V$, where $V\subset T^{\ast}M_{1}$ is some conic
neighbourhood of $(y_{0},\eta_{0})$. Let also $L_{1}\subset M_{1}$ be compact
with $y_{0}\in L_{1}^{\mathrm{int}}$. If for some
$s_{1},s_{2}\in\varmathbb{R}$ and $\delta,r>0$ one has the stability estimate

(6.3) $\lVert u\rVert_{H^{s_{1}}(M_{1})}\leqslant\omega(\lVert
Au\rVert_{H^{s_{2}}(M_{2})}),\qquad\lVert
u\rVert_{H^{s_{1}+\delta}_{L_{1}}(M_{1})}\leqslant r,$

then for any $\alpha\in(0,1)$ one has $\omega(t)\gtrsim t^{\alpha}$ for $t$
small.

###### Proof of Theorem 6.3.

Since $A$ preserves compact supports, there is a compact $L_{2}\subset M_{2}$
so that $A$ maps functions supported in $L_{1}$ to distributions supported in
$L_{2}$. We seek to apply Theorem 6.2 with

$\displaystyle K_{r}=\\{u\in H^{s_{1}+\delta}_{L_{1}}(M_{1});\
\|u\|_{H^{s_{1}+\delta}(M_{1})}\leqslant r\\}$

and a pseudodifferential operator $P\in\Psi^{0}_{\mathrm{cl}}(M_{1})$ with the
property that $P$ has nonvanishing principal symbol at $(y_{0},\eta_{0})$, $P$
is microlocally smoothing outside of $V$, and $P(K_{r_{0}})\subset K_{r}$ for
suitable $r_{0}>0$. In fact, $P$ can be constructed by left quantizing the
local coordinate symbol $\chi(y)\psi(\eta)$ where $\chi$ is a cutoff supported
near $y_{0}$ with $\chi(y_{0})\neq 0$, and $\psi$ is a Fourier cutoff
supported in $V$ with $\psi(\eta_{0})=1$ and $\psi$ is homogeneous of degree
$0$ for $\lvert\eta\rvert\geqslant 1$.

It is enough to show that the operator
$AP:C^{\infty}_{c}(M_{1})\to\mathcal{D}^{\prime}(M_{2})$ has a $C^{\infty}$
integral kernel. For if this is true, then for each $N\geqslant 0$ there is
$R=R_{N}>0$ such that

(6.4) $\displaystyle AP(K_{r_{0}})\subset\\{v\in H^{s_{2}+N}_{L_{2}}(M_{2}):\
\|v\|_{H^{s_{2}+N}(M_{2})}\leqslant R\\},$

using that $\|u\|_{H^{s_{1}+\delta}_{L_{1}}(M_{1})}\leqslant r_{0}$ for $u\in
K_{r_{0}}$. Then Theorem 6.2(a) implies that $\omega(t)\gtrsim t^{\alpha}$ for
all $\alpha\in(0,1)$. This proves the claimed result.

To show that $AP$ has smooth integral kernel, we note that
$WF(K_{A})_{M_{2}}=\emptyset$ by assumption and
$WF^{\prime}(K_{P})_{M_{1}}=\emptyset$, since $P$ is a $\Psi$DO and hence
$WF^{\prime}(K_{P})$ is contained in the diagonal
$\\{(y,y,\eta,\eta)\,;\,\eta\neq 0\\}$. Then (6.2) gives

(6.5) $\displaystyle WF^{\prime}(K_{AP})\subset WF^{\prime}(K_{A})\circ
WF^{\prime}(K_{P}).$

But $WF^{\prime}(K_{P})\subset\\{(y,y,\eta,\eta)\,;\,(y,\eta)\in V\\}$ since
$P$ is microlocally smoothing away from $V$, and by assumption
$WF^{\prime}(K_{A})$ does not have elements of the form $(x,y,\xi,\eta)$ with
$(y,\eta)\in V$. Thus $WF^{\prime}(K_{AP})=\emptyset$ and $AP$ is indeed
smoothing. ∎

######  Example 6.4.

As a special case of Theorem 6.3, let $A\in\Psi^{m}(M)$ be a
pseudodifferential operator on a closed manifold $M$. One says that $A$ is
microlocally smoothing at $(y_{0},\eta_{0})\in T^{*}M\setminus\\{0\\}$ if
$(y_{0},y_{0},\eta_{0},\eta_{0})\notin WF^{\prime}(K_{A})$, or equivalently if
the full symbol of $A$ in some local coordinates is of order $-\infty$ in some
conic neighborhood of $(y_{0},\eta_{0})$ (see [Hö85, Proposition 18.1.26]). In
such a case $A$ satisfies the conditions of the previous theorem.

We next seek to extend Theorem 6.3 to the setting of Gevrey regularizing
operators. To this end, we refer to Section D.1 for the calculus of Gevrey
pseudodifferential operators and the definition of the Gevrey wave front set
$WF_{G,\sigma}(u)$.

In the setting of Gevrey (and analytically) regularizing operators, we will
deduce that an inverse problem can at best have logarithmic stability.

######  Theorem 6.5.

Let $M_{1},M_{2}$ be $G^{\sigma}$ manifolds with Gevrey regularity index
$\sigma>1$, and let $A:C^{\infty}_{c}(M_{1})\to\mathcal{D}^{\prime}(M_{2})$ be
a continuous linear operator which preserves compact supports. Assume that $A$
is microlocally $G^{\sigma}$ smoothing at $(y_{0},\eta_{0})\in
T^{*}M_{1}\setminus\\{0\\}$, in the sense that

$WF_{G,\sigma}(K_{A})_{M_{2}}=\emptyset$

and $(x,y,\xi,\eta)\notin WF^{\prime}_{G,\sigma}(K_{A})$ for $(x,\xi)\in
T^{*}M_{2}$ and for $(y,\eta)\in V\subset T^{\ast}M_{1}$ where $V$ is a conic
neighbourhood of $(y_{0},\eta_{0})$. Let also $L_{1}\subset M_{1}$ be compact
with $y_{0}\in L_{1}^{\mathrm{int}}$. If for some
$s_{1},s_{2}\in\varmathbb{R}$ and $\delta,r>0$ one has the stability estimate

(6.6) $\lVert u\rVert_{H^{s_{1}}(M_{1})}\leqslant\omega(\lVert
Au\rVert_{H^{s_{2}}(M_{2})}),\qquad\lVert
u\rVert_{H^{s_{1}+\delta}_{L_{1}}(M_{1})}\leqslant r,$

then necessarily $\omega(t)\gtrsim\lvert\log
t\rvert^{-\frac{\delta(\sigma\dim(M_{2})+1)}{\dim(M_{1})}}$ for $t$ small.

###### Proof of Theorem 6.5.

We argue similarly as in the proof of Theorem 6.3 again relying on the
arguments from Theorem 6.2. As above, we set

$\displaystyle K_{r}=\\{f\in H^{s_{1}+\delta}_{L_{1}}(M_{1});\
\|f\|_{H^{s_{1}+\delta}(M_{1})}\leqslant r\\}$

and fix a Gevrey pseudodifferential operator
$P\in\Psi^{0}_{\mathrm{cl}}(M_{1})$ with the property that $P$ has
nonvanishing principal symbol at $(y_{0},\eta_{0})$ and is $G^{\sigma}$
microlocally smoothing outside of $V$, and $P(K_{r_{0}})\subset K_{r}$ for
suitable $r_{0}>0$. More precisely, $P$ can be constructed by quantizing the
symbol $\chi(y)\psi(\eta)$ where $\chi$ is a $G^{\sigma}$ cutoff function
supported near $y_{0}$, and $\psi(\eta)$ is a $G^{\sigma}$ Fourier cutoff near
$\eta_{0}$ as in [Ro93, Proposition 3.4.4].

We seek to prove that $AP$ is $G^{\sigma}$ regularizing which implies the
condition from Theorem 6.2(b). In order to infer this, we note that (6.2)
remains true in the $G^{\sigma}$ setting (see [Hö85, Section 8.5]). This
implies that (6.5) determines the Gevrey wave front set of $K_{AP}$, i.e. that

$\displaystyle WF_{G,\sigma}^{\prime}(K_{AP})\subset
WF_{G,\sigma}^{\prime}(K_{A})\circ WF_{G,\sigma}^{\prime}(K_{P}).$

The assumed regularity of $K_{A}$ and the construction of $P$ entail that as
in the proof of Theorem 6.3 we also infer that
$WF^{\prime}_{G,s}(K_{AP})=\emptyset$. Finally, we observe that this yields
the desired smoothing result

(6.7) $\displaystyle AP(K_{r_{0}})\subset\\{y\in A^{\sigma,\rho}_{L_{2}}:\
\|y\|_{A^{\sigma,\rho}}\leqslant R\\}\mbox{ for some }\rho,R>0,$

since elements in the set $K_{r_{0}}$ are uniformly bounded and the kernel
$K_{AP}$ satisfies uniform $G^{\sigma}$ bounds which combined yield (6.7) as
in the proof of Theorem 3.14. Hence, Theorem 6.2(b) can be invoked and it
implies the claimed estimate. ∎

###  6.2. Applications

In this section we discuss a number microlocal instability results. These
include variants of the Radon and geodesic X-ray transforms and applications
of these, for instance, in the inverse transport problem.

####  6.2.1. The Radon transform with limited data

As a first example of the microlocal regularizing argument from the previous
section, we consider the two-dimensional Radon transform. To this end, we
introduce the following coordinates: Let
$\theta=\theta(\varphi)=(\cos(\varphi),\sin(\varphi))$,
$\theta^{\perp}=\theta^{\perp}(\varphi)=(-\sin(\varphi),\cos(\varphi))$ with
$\varphi\in\varmathbb{T}$, where $\varmathbb{T}=[0,2\pi]$ with $0,2\pi$ being
identified. We consider a line perpendicular to $\theta=\theta(\varphi)$ and
with distance $s\in\varmathbb{R}$ to the origin:

$\displaystyle
L(\varphi,s):=\\{x\in\varmathbb{R}^{2}:x\cdot\theta(\varphi)=s\\}.$

For $f\in C^{\infty}_{c}(\varmathbb{R}^{2})$ we then define the Radon
transform as

(6.8) $\displaystyle R(f)(\varphi,s):=\int\limits_{x\in
L(\varphi,s)}f(x)\,dx=\int\limits_{-\infty}^{\infty}f(s\theta+t\theta^{\perp})\,dt.$

The operator $R$ extends by duality to
$\mathcal{E}^{\prime}(\varmathbb{R}^{2})$. By the definition of $R$ we obtain
that $R(f)(\varphi+\pi,-s)=R(f)(\varphi,s)$ since the symmetry
$S:(\varphi,s)\mapsto(\varphi+\pi,-s)$ is also present in the definition of
$L(\varphi,s)$.

It is well known that the _full data_ Radon transform is invertible in
suitable function spaces. For example, if $L\subset\varmathbb{R}^{2}$ is a
compact set, $s\in\varmathbb{R}$, and $H^{s}_{L}(\varmathbb{R}^{2})$ is the
space of those $f\in H^{s}(\varmathbb{R}^{2})$ with $\mathrm{supp}(f)\subset
L$, then one has [Na01, Theorem II.5.1]

$\lVert f\rVert_{H^{s}(\varmathbb{R}^{2})}\lesssim\lVert
Rf\rVert_{H^{s+1/2}(\varmathbb{T}\times\varmathbb{R})}\lesssim\lVert
f\rVert_{H^{s}(\varmathbb{R}^{2})},\qquad f\in H^{s}_{L}(\varmathbb{R}^{2}).$

However, it is also known that the _limited data_ Radon transform may have
severe instabilities. Here we are interested in deducing (exponential)
instability results for the limited data Radon transform from our microlocal
regularity arguments.

We consider the Radon transform acting on $f\in H^{s}_{L}(\varmathbb{R}^{2})$,
where $L$ is the closed unit ball in $\varmathbb{R}^{2}$. Then clearly
$Rf(\varphi,s)=0$ for $\lvert s\rvert>1$. We consider the situation in which
we only measure the Radon transform on the lines $L(\varphi,s)\in\Lambda$,
where $\Lambda\subset\varmathbb{T}\times(-1,1)$ is an open set with
$S(\Lambda)=\Lambda$ and such that
$(\varmathbb{T}\times(-1,1))\setminus\overline{\Lambda}$ is open and non-
empty. This means that we do not have measurements over some fixed open set of
lines.

In order to understand the microlocal regularizing properties of the Radon
transform, we recall the following result describing the singularities of the
Schwartz kernel of the Radon transform (see [Qu06], Theorem 2.8 and [Qu93],
Theorem 3.1):

######  Lemma 6.6.

Let $R$ be the Radon transform defined in (6.8) above. Then, $R$ has the
integral kernel

$\displaystyle
K_{R}(\varphi,s,x):=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}e^{i(s-\theta\cdot
x)\sigma}\,d\sigma,$

where the integral is understood as an oscillatory integral. As a consequence,
for any $\mu>1$ we have that

(6.9) $\displaystyle
WF(K_{R}),\,WF_{G,\mu}(K_{R})\subset\\{(\varphi,s,x,-(\theta^{\perp}\cdot
x)\sigma,\sigma,-\theta\sigma)\,;\,\ \sigma\neq 0,\ s=\theta\cdot x\\}.$

###### Proof.

The claim follows from the fact that by the Fourier slice theorem for
$f\in\mathcal{S}(\varmathbb{R}^{2})$ it holds

$\displaystyle
R(f)(\varphi,s)=\operatorname{\mathcal{F}}_{s}^{-1}\operatorname{\mathcal{F}}_{s}(f)(\varphi,s)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{\varmathbb{R}^{2}}e^{i(s-\theta\cdot
x)\sigma}f(x)\,dx\,d\sigma,$

where $\operatorname{\mathcal{F}}_{s}$ denotes the Fourier transform in the
$s$-variable, see [Qu06, equation (A.7)]. Interpreted as an oscillatory
integral, this proves the identity for the kernel. The expression for the wave
front set now follows from the usual decay estimates for oscillatory integrals
in the case of the $C^{\infty}$ wave front set and from similar considerations
for the case of the Gevrey wave front set, see for instance Lemma D.3 in the
appendix. ∎

With these properties in hand, we can formulate our main result on the
optimality of logarithmic stability estimates for the limited data Radon
transform.

######  Proposition 6.7.

Let $\Lambda\subset\varmathbb{T}\times(-1,1)$ be an open set with
$S(\Lambda)=\Lambda$ and
$(\varmathbb{T}\times(-1,1))\setminus\overline{\Lambda}$ nonempty. If there
exists a modulus of continuity $\omega$ such that for some
$s_{1},s_{2}\in\varmathbb{R}$, $\delta>0$ one has the stability estimate

$\displaystyle\|f\|_{H^{s_{1}}(\varmathbb{R}^{2})}\leqslant\omega(\|Rf\|_{H^{s_{2}}(\Lambda)}),\qquad\lVert
f\rVert_{H^{s_{1}+\delta}_{L}(\varmathbb{R}^{2})}\leqslant 1,$

then for any $\mu>1$ there exists some constant $C_{\mu}>0$ such that

$\displaystyle\omega(t)\geqslant
C_{\mu}|\log(t)|^{-\frac{\delta(2\mu+1)}{2}},\qquad\text{$t$ small}.$

######  Remark 6.8.

We remark that the result of Proposition 6.7 includes the limited angle
problem or the exterior problem as special cases (see for instance Chapter VI
in [Na01]).

###### Proof.

By the assumption on $\Lambda$, there exists an open set $U$ in
$\varmathbb{T}\times(-1,1)$ with
$\overline{U}\cap\overline{\Lambda}=\emptyset$ and $S(U)=U$. Fix $\mu>1$ and
let $\chi_{\Lambda}$ be a $G^{\mu}$ cutoff function in
$\varmathbb{T}\times\varmathbb{R}$ so that $\chi_{\Lambda}=1$ near
$\overline{\Lambda}$ and $\chi_{\Lambda}=0$ near $\overline{U}$. We wish to
use Theorem 6.5. Define

$A=\chi_{\Lambda}R:C^{\infty}_{c}(\varmathbb{R}^{2})\to
C^{\infty}_{c}(\varmathbb{T}\times\varmathbb{R}).$

Note that $A$ preserves compact supports. The stability condition in the
proposition implies that

$\lVert f\rVert_{H^{s_{1}}(\varmathbb{R}^{2})}\leqslant\omega(\lVert
Af\rVert_{H^{s_{2}}(\varmathbb{T}\times\varmathbb{R})}),\qquad\lVert
f\rVert_{H^{s_{1}+\delta}_{L}}\leqslant 1.$

We now analyze the wave front set of $K_{A}$. Since $\chi_{\Lambda}\in
G^{\mu}$, (6.9) gives that

$\displaystyle WF_{G,\mu}^{\prime}(K_{A})$
$\displaystyle\subset\\{(\varphi,s,x,\hat{\varphi},\hat{s},\xi)\in
WF_{G,\mu}^{\prime}(K_{R})\,;\,(\varphi,s)\in\mathrm{supp}(\chi_{\Lambda})\\}$
(6.10) $\displaystyle\subset\\{(\varphi,s,x,-(\theta^{\perp}\cdot
x)\sigma,\sigma,\theta\sigma)\,;\,\ \sigma\neq 0,\ s=\theta\cdot x,\
(\varphi,s)\in\mathrm{supp}(\chi_{\Lambda})\\}.$

Since $\theta\sigma\neq 0$ in the last expression, we always have
$WF_{G,\mu}(K_{A})_{\varmathbb{T}\times\varmathbb{R}}=\emptyset$.

Now fix some $(\varphi_{0},s_{0})\in U$, so that lines near
$L(\varphi_{0},s_{0})$ are not in the data. Let $x_{0}\in
L(\varphi_{0},s_{0})$ and $\xi_{0}=\theta(\varphi_{0})$. We claim that $A$ is
microlocally $G^{\mu}$ smoothing near $(x_{0},\xi_{0})$. For if

$(\varphi,s,x_{0},\hat{\varphi},\hat{s},\xi_{0})\in
WF_{G,\mu}^{\prime}(K_{A}),$

then (6.10) implies that $\theta(\varphi)\sigma=\xi_{0}$ and
$s=\theta(\varphi)\cdot x_{0}$, which yields $(\varphi,s)=(\varphi_{0},s_{0})$
or $(\varphi,s)=S(\varphi_{0},s_{0})$. However, we must also have
$(\varphi,s)\in\mathrm{supp}(\chi_{\Lambda})$, which is impossible since
$\chi_{\Lambda}=0$ near $\overline{U}=\overline{S(U)}$. Since $U$ is open, we
may vary $(x_{0},\xi_{0})$ slightly in the previous argument. This proves that
$A$ is microlocally $G^{\mu}$ smoothing near $(x_{0},\xi_{0})$. The result now
follows from Theorem 6.5. ∎

####  6.2.2. Superpolynomial instability of the geodesic X-ray transform in
the presence of conjugate points

As another example of an instability arising from microlocal smoothing, we
discuss the instability properties of the geodesic X-ray transform on two-
dimensional Riemannian surfaces $(M,g)$. Throughout this section, we assume
that $(M,g)$ is a compact Riemannian manifold with smooth boundary which is
_non-trapping_ , i.e. all geodesics starting within $M$ meet $\partial M$ in
finite time in both directions, and that its boundary $\partial M$ is
_strictly convex_. While it is known that for _simple_ manifolds, i.e. for
manifolds which are non-trapping, have strictly convex boundary and no
conjugate points, the X-ray transform is injective and Lipschitz stable in
suitable Sobolev norms [SU04, AS20], in our discussion we allow for the
presence of conjugate points in $(M,g)$.

Following the article [MSU15], the set-up here is the following: $M$ is two-
dimensional, $\gamma_{0}$ is a fixed directed maximal geodesic in $M$, and $f$
is a function supported away from the endpoints of $\gamma_{0}$. In this
setting, the article [MSU15] investigates which parts of the wave front set
$WF(f)$ of $f$ can be recovered from knowing the geodesic X-ray transform

$\displaystyle If(x,v):=\int_{0}^{\tau(x,v)}f(\pi_{1}\varphi_{s}(x,v))ds.$

Here, for $\nu_{+}(x)$ being the outward normal vector at $x\in\partial M$, we
consider

$\displaystyle(x,v)\in\partial_{+}SM:=\\{(x,v)\in SM:\ x\in\partial M,\
\langle v,\nu_{+}(x)\rangle<0\\},$

where $SM=\\{(x,v)\in TM\,;\,\lvert v\rvert_{g}=1\\}$ is the unit sphere
bundle, $\tau\in C^{\infty}(\partial_{+}SM)$ denotes the boundary hitting
time, and $\pi_{1}\varphi_{s}(x,v)$ denotes the projection onto the first
component of the geodesic flow $\varphi_{s}$ onto $(M,g)$. In order to focus
on a neighbourhood of $\gamma_{0}$ only, we assume that $(x,v)$ is such that
the associated geodesics are close enough to $\gamma_{0}$, and we identify
$\gamma_{0}$ with $(x_{0},v_{0})\in\partial_{+}SM$.

Below, we will use several basic properties of Fourier integral operators
(FIOs) which may be found in [Hö85, Chapter 25]. A first key observation (see
[MSU15, Proposition 3.1 and Theorem 3.2]) on the two-dimensional geodesic
X-ray transform is the following result:

  * (i)

$I$, restricted to functions supported near $\gamma_{0}$ and evaluated for
geodesics near $\gamma_{0}$, is an FIO of order $-\frac{1}{2}$ whose canonical
relation $C$ can be explicitly computed and which is _locally_ the graph of a
canonical map $\mathcal{C}$ which is locally a diffeomorphism (see [MSU15,
Theorem 3.2]). The map $\mathcal{C}$ becomes a _global_ diffeomorphism if no
conjugate points are present. The absence of conjugate points is known as the
Bolker condition.

The microlocal smoothing arguments from [MSU15] exploit the violation of the
Bolker condition, i.e. hold if conjugate points are present. To summarize the
results from [MSU15] let us assume that $x_{1},x_{2}\in M^{\mathrm{int}}$ are
conjugate points on the geodesic $\gamma_{0}$ from above with no further
conjugate points in between them. Let $v_{1},v_{2}$ be the tangent vectors to
$\gamma_{0}$ at the points $x_{1},x_{2}\in M$ and
$\xi_{j}=(R_{-\pi/2}v_{j})^{\flat}$ correspond to their rotations by
$-90^{\circ}$, respectively. Let $V_{1}$ and $V_{2}$ be small neighbourhoods
in $T^{*}M$ of $(x_{1},\xi_{1})$ and $(x_{2},-\xi_{2})$. Then, the following
assertion may be found in [MSU15, Theorem 4.2] (note that our $V_{1}$ and
$V_{2}$ are $V^{1}_{+}$ and $V^{2}_{-}$ in [MSU15]):

  * (ii)

If $V_{1}$, $V_{2}$ are sufficiently small and
$\mathcal{C}_{j}=\mathcal{C}|_{V_{j}}$, the restrictions of the canonical
relation $C$ of $I$ to $V_{1},V_{2}$ are canonical graphs
$C|_{V_{1}}=\\{(\mathcal{C}_{1}(x,\xi),x,\xi)\,;\,(x,\xi)\in V_{1}\\}$ and
$C|_{V_{2}}=\\{(\mathcal{C}_{2}(x,\xi),x,\xi)\,;\,(x,\xi)\in V_{2}\\}$ with
$\mathcal{C}_{1}(V_{1})=\mathcal{C}_{2}(V_{2})$. The map
$\mathcal{C}_{2}^{-1}\circ\mathcal{C}_{1}$ is a diffeomorphism $V_{1}\to
V_{2}$.

Further, in view of an application of Theorem 6.3 we let $L\subset
M^{\mathrm{int}}$ be a compact set containing the $x$-projection of $V_{1}\cup
V_{2}$.

We can now formally describe the microlocal smoothing argument of [MSU15]
based on cancellation of singularities. Suppose that $f_{1}\in C^{\infty}(M)$
is supported near $x_{1}$ and has wave front set in $V_{1}$. We wish to find a
function $f_{2}\in C^{\infty}(M)$, supported near $x_{2}$ with wave front set
in $V_{2}$, so that

$I(f_{1}+f_{2})\in C^{\infty}.$

Let $I_{j}$ denote $I$ acting on functions whose wave front set is contained
in $V_{j}$. Since the canonical relation of $I_{j}$ is the graph of
$\mathcal{C}_{j}$ and since $I_{j}$ is an elliptic FIO of order $-1/2$, see
[MSU15, proof of Theorem 4.3], $I_{j}$ has a parametrix $I_{j}^{-1}$ (an FIO
of order $1/2$ whose canonical relation is the graph of
$\mathcal{C}_{j}^{-1}$) so that $I_{j}\circ I_{j}^{-1}=\mathrm{Id}+R_{j}$
where $R_{j}$ is microlocally smoothing near $\mathcal{C}_{j}(x_{j},\xi_{j})$.
Since $I(f_{1}+f_{2})=I_{1}f_{1}+I_{2}f_{2}$, we may just define

$f_{2}:=-I_{2}^{-1}I_{1}f_{1}$

and then $I(f_{1}+f_{2})\in C^{\infty}$. In the last step we used that
$\mathcal{C}_{1}(V_{1})=\mathcal{C}_{2}(V_{2})$, which makes it possible to
apply $I_{2}^{-1}$ to $I_{1}f_{1}$.

We will next combine the idea above with our microlocal smoothing arguments to
show that in the presence of conjugate points, the two-dimensional geodesic
X-ray transform cannot be Hölder stable with respect to any Sobolev norms
(even if it were injective). The fact that Lipschitz stability is not possible
was already stated in [MSU15]. It is likely that if $(M,g)$ is
analytic/Gevrey, then the stability could not be better than logarithmic in
this setting. However, this would require some facts about analytic/Gevrey
FIOs which may not be easily available in the literature, and we will not
consider this further.

######  Theorem 6.9.

Let $(M,g)$ be a compact two-dimensional non-trapping Riemannian manifold with
interior conjugate points and strictly convex boundary $\partial M$. Let $I$
be as above. Assume that for some $s_{1},s_{2}\in\varmathbb{R}$ and $\delta>0$
and for some modulus of continuity $\omega$ it holds

$\displaystyle\|f\|_{H^{s_{1}}(M)}\leqslant\omega(\|If\|_{H^{s_{2}}(\partial_{+}SM)})$

for all $f$ with $\|f\|_{H^{s_{1}+\delta}(M)}\leqslant 1$. Then, for any
$\alpha\in(0,1]$ there exists a constant $C=C_{\alpha}>0$ such that
$\omega(t)\geqslant Ct^{\alpha}$ for $t$ small.

###### Proof.

We will follow the scheme from [MSU15] described above. In deducing the
instability property, we do not directly invoke Theorem 6.3, but instead
follow its main ideas in a slightly modified functional setting. In the
framework laid out in the proof of Theorem 6.2 and Lemma 3.7 we consider

$\displaystyle I:P(K)\rightarrow Y,$

with

(6.11) $\displaystyle\begin{split}&X=H^{s_{1}}_{L}(M),\
X_{1}:=H^{s_{1}+\delta}_{L}(M),\ K=\\{f\in H^{s_{1}+\delta}_{L}(M)\,;\,\lVert
f\rVert_{H^{s_{1}+\delta}}\leqslant 1\\},\\\ &Y=H^{s_{2}}(\partial_{+}SM),\
Y_{1}:=H^{s_{2}+N}(\partial_{+}SM)\mbox{ with }N\in\varmathbb{N}\mbox{
arbitrary but fixed},\\\ &P=P_{1}-P_{2}I_{2}^{-1}I_{1}P_{1}.\end{split}$

Here
$P_{j}=\chi_{j}(x)Op(\chi_{j}(x)\psi_{j}(\xi))\chi_{j}(x)\in\Psi^{0}_{\mathrm{cl}}(M^{\mathrm{int}})$
is a microlocal cutoff to $V_{j}$ obtained by quantizing the symbol
$\chi_{j}(x)\psi_{j}(\xi)$ and multiplying with $\chi_{j}(x)$ from the right
and left. Here $\chi_{j}$ is a smooth function supported near $x_{j}$ with
$\chi_{j}=1$ near $x_{j}$, and $\psi_{j}$ is homogeneous of degree $0$ for
$\lvert\xi\rvert\geqslant 1$ with $\psi_{j}(\xi_{j})=1$. Moreover, $I_{j}$ is
the microlocalized version $I$ as described after (ii) above. We also note
that functions in $P(K)$ are supported in $L$, and hence functions in
$I(P(K))$ are compactly supported in $(\partial_{+}SM)^{\mathrm{int}}$, see
e.g. [PSU20+] for this simple fact which uses strict convexity of $\partial
M$. We will prove below that $I$ indeed maps $P(K)$ to
$C^{\infty}_{c}((\partial_{+}SM)^{\mathrm{int}})$ and hence to $Y$.

The main difference in the functional setting with respect to the proof of
Theorem 6.2 is that $P$ is not a $\Psi$DO, but rather an FIO of order zero.
However, modulo smoothing operators, one has

$P^{*}P=P_{1}^{*}P_{1}+P_{1}^{*}(I_{2}^{-1}I_{1})^{*}P_{2}^{*}P_{2}I_{2}^{-1}I_{1}P_{1}.$

Since $I_{2}^{-1}I_{1}$ is an FIO associated with the canonical graph of
$\mathcal{C}_{2}^{-1}\circ\mathcal{C}_{1}$, its adjoint is associated with the
graph of $\mathcal{C}_{1}^{-1}\circ\mathcal{C}_{2}$. Here we use that
$\mathcal{C}_{2}^{-1}\circ\mathcal{C}_{1}$ is a diffeomorphism $V_{1}\to
V_{2}$. Thus by the composition calculus of FIOs, $P^{*}P$ is $\Psi$DO of
order zero, and one can check that its principal symbol is nonvanishing at
$(x_{1},\xi_{1})$. Now if $M$ were a closed manifold, we could use Theorem 6.1
to show that $P$ as an operator $H^{s_{1}+\delta}(M)\to H^{s_{1}}(M)$ would
have singular value bounds $\sigma_{k}(P)\sim k^{-\frac{\delta}{2}}$. Since
$M$ has a boundary, we consider $P$ as an operator $H^{s_{1}+\delta}_{L}(M)\to
H^{s_{1}}_{L}(M)$ instead, and argue as in the end of proof of Theorem 6.2 to
obtain the analogous capacity estimates.

In order to conclude the desired instability result, it remains to check the
conditions stated in Theorem 6.2 in our functional setting from (6.11). The
result then follows by invoking Lemma 3.7 and Theorem 3.3. However, by
construction of $P$ and the characterization of the wave front sets in [MSU15]
(see our discussion after (ii) above, where we may choose $f_{1}=P_{1}f$ for
$f\in K$) we infer that $I(P(K))\subset C^{\infty}(\partial_{+}SM)$ and hence
$I(P(K))\subset\\{y\in Y_{1}:\|y\|_{Y_{1}}\leqslant R\\}$ for some $R=R(N)>0$.
This concludes the proof. ∎

We remark that the conditions on the manifold are not void. For both analytic
and numerical examples of such manifolds, we refer to [MSU15, Section 6].
Partial results in higher dimensional settings are proved in [HU18].

####  6.2.3. Exponential instability in quantitative unique continuation for
the wave equation

As a further application of the microlocal analytic (or Gevrey) smoothing
mechanism to instability, we discuss the exponential instability of the unique
continuation property for the wave equation. To this end, consider $(M,g)$ an
analytic, closed $n$-dimensional manifold. Let $\Omega\subset M$ with
$\Omega\neq\emptyset$ be open and such that
$M\setminus\overline{\Omega}\neq\emptyset$ is also open. We are interested in
the quantitative unique continuation properties for the wave equation

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

on $(M,g)$ with observation data in $\Omega$. Due to the finite speed of
propagation of the wave equation, we assume that the observation time $T$
always satisfies

$\displaystyle T>2\max\\{\operatorname{dist}(x,\Omega),\ x\in M\\}.$

By duality to unique continuation, this guarantees that approximate
observability holds [Ta99, LL18]. In order to show the strength of our Gevrey
microlocal instability argument, we assume that there exists a unit speed
geodesic $\gamma(t)$ in $(M,g)$ which never meets $\overline{\Omega}$ for
$0\leqslant t\leqslant T$. This is a situation in which the geometric control
condition of [BLR92] is not satisfied.

Under this condition, we reprove the fact that the stability estimates which
were deduced in [LL18] and which are logarithmic are indeed optimal (similar
observations were already made in earlier works starting with Lebeau [Le92];
by discussing this example, it is not our main purpose to present a new
result, but to show the versatility of the instability ideas from the previous
sections):

######  Theorem 6.10.

Let $(M,g)$ and $\Omega$ be as above. Assume that there exists some modulus of
continuity $\omega$ satisfying (2.1) such that for all $(u_{0},u_{1})\in
H^{1}(M)\times L^{2}(M)$ and $u\in L^{2}((0,T),H^{1}(M))$ solving (6.12) it
holds

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

Then for any $\sigma>1$ there is $C_{\sigma}>0$ such that $\omega(t)\geqslant
C_{\sigma}|\log(t)|^{-\frac{\sigma(n+1)+1}{n}}$ for $t$ small.

###### Proof.

For any $(y,\eta)\in T^{*}M\setminus\\{0\\}$, denote by $\gamma_{y,\eta}$ the
geodesic in $(M,g)$ with $\gamma_{y,\eta}(0)=y$ and
$\dot{\gamma}(y,\eta)=\eta^{\sharp}$. By assumption, there exists
$(y_{0},\eta_{0})\in T^{*}M$ with $\lvert\eta_{0}\rvert_{g}=1$ so that the
geodesic $\gamma_{y_{0},\eta_{0}}(t)$ never meets $\overline{\Omega}$ for
$0\leqslant t\leqslant T$. By compactness, there is a positive distance
between the sets $\gamma_{y_{0},\eta_{0}}([-\varepsilon,T+\varepsilon])$ and
$\overline{\Omega}$ for some $\varepsilon>0$. Thus there is a small conic
neighborhood $C$ of $(y_{0},\eta_{0})$ so that no geodesic $\gamma_{y,\eta}$
meets $\overline{\Omega}$ for $-\varepsilon\leqslant t\leqslant T+\varepsilon$
when $(y,\eta)\in C$ and $\lvert\eta\rvert_{g}=1$.

In general, a singularity at $(y,\eta)$ at time $t=0$ for a solution of the
wave equation will propagate in two opposite directions. In order to prove our
statement we need to focus on forward propagating singularities. To this end,
we factorize the wave operator as

$\partial_{t}^{2}-\Delta_{g}=(\partial_{t}-iB)(\partial_{t}+iB)$

where $B=(-\Delta_{g})^{1/2}$ is an elliptic $\Psi$DO of order one. If
$(\partial_{t}+iB)w=0$ with $w(0)=u_{0}$, then $w$ also solves the wave
equation with $(w(0),\partial_{t}w(0))=(u_{0},-iBu_{0})$. For such solutions
one has (by the discussion around (2.1))

$\lVert
u_{0}\rVert_{L^{2}(M)}\leqslant\omega(\|w\|_{L^{2}((0,T),H^{1}(\Omega))}),\qquad\lVert
u_{0}\rVert_{H^{1}(M)}\leqslant 1.$

Moreover, the $G^{\sigma}$ singularities of solutions of
$(\partial_{t}+iB)w=0$ propagate along forward bicharacteristics in the
following sense (see e.g. [Mi14, Theorem 1 in Lecture VI]): one has
$(y,\eta)\notin WF_{G,\sigma}(u_{0})$ if and only if $\gamma_{y,\eta}(t)\notin
WF_{G,\sigma}(w(\,\cdot\,,t))$.

We wish to use Theorem 6.2 in this setting. Fix a Gevrey pseudodifferential
operator $P\in\Psi^{0}_{\mathrm{cl}}(M)$ obtained by quantizing the symbol
$\chi(y)\psi(\eta)$ where $\chi$ is a $G^{\sigma}$ cutoff function supported
near $y_{0}$, and $\psi(\eta)$ is a $G^{\sigma}$ Fourier cutoff near
$\eta_{0}$ as in [Ro93, Proposition 3.4.4]. We may assume that $P$ is
$G^{\sigma}$ smoothing away from $C$.

We next define an operator

$A:L^{2}(M)\to L^{2}(\varmathbb{R},H^{1}(M)),\ \
Au_{0}(x,t)=\psi_{1}(t)\psi_{2}(x)w(x,t)$

where $\psi_{1}$ is a $G^{\sigma}$ cutoff function supported in
$(-\varepsilon,T+\varepsilon)$ with $\psi_{1}(t)=1$ near $[0,T]$, and
$\psi_{2}$ is a $G^{\sigma}$ cutoff function in $M$ with $\psi_{2}=1$ near
$\overline{\Omega}$ so that the geodesics $\gamma_{y,\eta}$ for $(y,\eta)\in
C$ and $\lvert\eta\rvert_{g}=1$ never meet $\mathrm{supp}(\psi_{2})$ for
$-\varepsilon\leqslant t\leqslant T+\varepsilon$. It follows that we have

$\lVert
u_{0}\rVert_{L^{2}(M)}\leqslant\omega(\|Au_{0}\|_{L^{2}(\varmathbb{R},H^{1}(M))}),\qquad\lVert
u_{0}\rVert_{H^{1}(M)}\leqslant 1.$

Now let $u_{0}=Pv$ with $v\in L^{2}(M)$, and let $w$ solve
$(\partial_{t}+iB)w=0$ with $w(0)=u_{0}=Pv$ as above. By $G^{\sigma}$
propagation of singularities and the fact that $Pv$ is $G^{\sigma}$ away from
$C$ when $v\in L^{2}(M)$, it follows that $w(\,\cdot\,,t)$ is a $G^{\sigma}$
function near $\mathrm{supp}(\psi_{2})$ for any
$t\in[-\varepsilon,T+\varepsilon]$. Since $\partial_{t}w=-iBw$ is also
$G^{\sigma}$ near $\mathrm{supp}(\psi_{2})$ for any fixed $t$, it follows from
[Mi14, end of Lecture IV] and finite speed of propagation that the solution
$w(x,t)$ of the wave equation is $G^{\sigma}$ for $(x,t)$ near
$\mathrm{supp}(\psi_{2})\times\mathrm{supp}(\psi_{1})$. This proves that
$A(Pv)$ is $G^{\sigma}$ in $M\times\varmathbb{R}$ whenever $v\in L^{2}(M)$.
This remains true for any $v$ in the dual of $G^{\sigma}(M)$ (Gevrey
ultradistributions), since $P$ is a Gevrey $\Psi$DO and since propagation of
singularities is valid in this setting [CZ90]. This implies as in the proof of
Theorem 3.14(ii) that the map $v\mapsto A(Pv)$ has $G^{\sigma}$ integral
kernel and hence is bounded $H^{1}(M)\to
A^{\sigma,\rho}_{Q}(M\times\varmathbb{R})$ for some $\rho>0$ and for
$Q=M\times[-\varepsilon,T+\varepsilon]$. The result now follows from Theorem
6.2(b). ∎

######  Remark 6.11.

By a duality argument as outlined in Section 5.4, we also obtain that
exponential cost of control is optimal in the associated steering problem for
the wave equation.

As a corollary of Theorem 6.10 we obtain the logarithmic instability of unique
continuation in space-like directions for the wave equation. This had already
been noted earlier in [Jo60] in special geometries by a separations of
variables argument and an analysis of the asymptotic behaviour of Bessel
functions. More precisely, the result in [Jo60] states that if $0<\rho<1$ and
$k\geqslant 1$, and if for some modulus of continuity $\omega$ one has the
stability estimate

$\lVert u\rVert_{L^{\infty}(B_{1}\times\varmathbb{R})}\leqslant\omega(\lVert
u\rVert_{L^{\infty}(B_{\rho}\times\varmathbb{R})})$

for any solution of $(\partial_{t}^{2}-\Delta)u=0$ in
$\varmathbb{R}^{2}\times\varmathbb{R}$ with $\lVert
u\rVert_{W^{k,\infty}(\varmathbb{R}^{2}\times\varmathbb{R})}\leqslant 1$, then
$\omega(t)\gtrsim\lvert\log t\rvert^{-k}$ for $t$ small. Since rays of
geometric optics that are tangent to $\partial B_{1}\times\varmathbb{R}$ never
enter $\overline{B}_{\rho}\times\varmathbb{R}$, instability results of this
kind with $L^{2}$ based norms can be obtained from Theorem 6.10 together with
energy conservation and finite propagation speed.

######  Remark 6.12.

We remark that alternatively, at lower coefficient regularity, an analogue of
[Jo60] could also have been obtaind by a separation of variables argument in
combination with a “blow-up” argument as in Appendix C. This would then lead
to the construction of solutions to Helmholtz-like equations which are
perturbations of Bessel functions. Their asymptotic behaviour was analysed in
[Jo60] to infer the instability in the constant coefficient case.

####  6.2.4. An inverse problem for the transport equation

Up to now we discussed microlocal instability for linear inverse problems. As
a direct example in which the instability of the geodesic X-ray transform
translates into instability of a nonlinear inverse problem, we discuss an
inverse problem for the transport equation.

Let $(M,g)$ be a compact, non-trapping Riemannian manifold with smooth
strictly convex boundary, and consider a potential $q\in C^{\infty}(M)$. Let
$SM=\\{(x,v)\in TM\,;\,|v|=1\\}$ be the unit sphere bundle, let $X$ be the
geodesic vector field on $SM$, and consider the boundary value problem

$Xu+qu=0\text{ in $SM$},\qquad u|_{\partial_{+}(SM)}=h$

where $\partial_{\pm}(SM)=\\{(x,v)\in\partial(SM)\,;\,\mp\langle
v,\nu\rangle\geqslant 0\\}$ denote the incoming and outgoing boundary of $SM$
($\nu$ is the unit outer normal to $\partial M$). On a fixed geodesic the
equation $Xu+qu=0$ is an ODE
$\frac{d}{dt}(u(\varphi_{t}))+q(\varphi_{t})u(\varphi_{t})=0$ where
$\varphi_{t}$ denotes the geodesic flow on $SM$. The solution is

$u(\varphi_{t}(x,v))=\exp\left[-\int_{0}^{t}q(\varphi_{s}(x,v))\,ds\right]h(x,v),\qquad(x,v)\in\partial_{+}(SM).$

Consider the boundary map

$\Lambda_{q}:C^{\infty}_{c}(\partial_{+}(SM))\to
C^{\infty}(\partial_{-}(SM)),\ \ \Lambda_{q}h=u|_{\partial_{-}(SM)}.$

This map takes the value of $u$ on the incoming boundary to the value of $u$
in the outgoing boundary, and is an analogue of the Dirichlet-to-Neumann map
in the setting of transport equations. The map $\Lambda_{q}$ is equivalent
information with the scattering data for the potential $q$ (see e.g. [PSU12]
where $q$ corresponds to the Higgs field $\Phi$).

We have the explicit expression

(6.13)
$\displaystyle\Lambda_{q}h(\alpha(x,v))=\exp(-Iq(x,v))h(x,v),\qquad(x,v)\in\partial_{+}(SM),$

where $Iq$ is the geodesic X-ray transform of $q$, and
$\alpha:\partial(SM)\to\partial(SM)$ is the scattering relation defined so
that $\alpha^{2}=\mathrm{Id}$ and $\alpha$ is a $C^{\infty}$ diffeomorphism
which is an isometric map with respect to natural weighted $L^{2}$ spaces on
$\partial_{\pm}(SM)$, see [PU05]. In particular, the explicit expression shows
that $\Lambda_{q}$ is a FIO of order $0$ with canonical relation given by the
scattering relation $\alpha$. It also follows that

$\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{L^{2}\to
L^{2}}\sim\|\exp(-Iq_{1})-\exp(-Iq_{2})\|_{L^{\infty}(\partial_{+}(SM))}.$

This shows that, under a priori bounds on $q_{j}$, one has

(6.14) $\displaystyle\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{L^{2}\to
L^{2}}\sim\|I(q_{1}-q_{2})\|_{L^{\infty}(\partial_{+}(SM))}.$

Thus the (in)stability properties of the inverse problem of recovering $q$
from $\Lambda_{q}$ are the same as those for the geodesic X-ray transform.

######  Theorem 6.13.

Let $(M,g)$ be compact and non-trapping with strictly convex boundary and let
$q$ be as above. Then the following instability results hold:

  * •

If $(M,g)$ is not simple, i.e. if conjugate points are present, but has
injective X-ray transform, there is no Hölder stability in Sobolev norms, i.e.
for no choice of $s_{1},s_{2},s_{3}$ with $s_{3}>s_{2}$ and $\alpha\in(0,1]$
do we have

$\displaystyle\|q_{1}-q_{2}\|_{H^{s_{1}}(M)}\leqslant
C\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{H^{s_{2}}(\partial_{+}(SM))\rightarrow
H^{s_{2}}(\partial_{+}(SM))}^{\alpha}$

with $q_{1},q_{2}\in K:=\\{q\in L^{2}(M):\ \|q\|_{H^{s_{3}}(M)}\leqslant
1\\}$.

  * •

If $(M,g)$ does not have injective X-ray transform, then both uniqueness and
Hölder stability fail.

###### Proof.

We only prove the first claim, as by virtue of the representation (6.13) the
lack of injectivity of the X-ray transform also implies the lack of
injectivity (and hence stability) for the DtN map.

Assuming that we had Hölder stability in spite of working on a the non-simple
manifold would amount to the existence of $s_{1},s_{2},s_{3}\in\varmathbb{R}$
with $s_{3}>s_{2}$ and $\alpha\in(0,1]$ such that

(6.15) $\displaystyle\|q_{1}-q_{2}\|_{H^{s_{1}}(M)}\leqslant
C\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{H^{s_{2}}(\partial_{+}(SM))\rightarrow
H^{s_{2}}(\partial_{+}(SM))}^{\alpha}$

for $q_{1},q_{2}\in K:=\\{q\in L^{2}(M):\ \|q\|_{H^{s_{3}}(M)}\leqslant 1\\}$.
Without loss of generality, we may further assume that $s_{3}\gg 1$ is as
large as necessary (since, if a stability estimate were to hold for a small
$s_{3}$, it would also hold for a large value of $s_{3}$).

Now, using the representation formula (6.13), we show that a stability
estimate of the form (6.15) would imply a Hölder stability estimate for the
geodesic X-ray transform in the presence of conjugate points. This, however
would contradict Theorem 6.9.

For $s_{3}>0$ sufficiently large, by an expansion of the exponential function,
we obtain

$\displaystyle\|[\exp(-Iq_{1})-\exp(-Iq_{2})]h\|_{H^{s_{2}}(\partial_{+}SM)}$
$\displaystyle\leqslant\sum\limits_{k=1}^{\infty}\frac{C(\|q_{1}\|_{H^{s_{3}}(M)},\|q_{2}\|_{H^{s_{3}}(M)})}{(k-1)!}\|(Iq_{1}-Iq_{2})h\|_{H^{s_{2}}(\partial_{+}SM)}$
$\displaystyle\leqslant
C(\|q_{1}\|_{H^{s_{3}}(M)},\|q_{2}\|_{H^{s_{3}}(M)})\|(Iq_{1}-Iq_{2})h\|_{H^{s_{2}}(\partial_{+}SM)}.$

Since

$\displaystyle\|\Lambda_{q_{1}}-\Lambda_{q_{2}}\|_{H^{s_{2}}(\partial_{+}(SM))\rightarrow
H^{s_{2}}(\partial_{+}(SM))}$
$\displaystyle=C_{\alpha}\sup\limits_{\|h\|_{H^{s_{2}}(\partial_{+}SM)}=1}\|[\exp(-Iq_{1})-\exp(-Iq_{2})]h\|_{H^{s_{2}}(\partial_{+}SM)},$

(6.15) would thus imply the Hölder stability of the geodesic X ray transform
in the presence of conjugate points, a contradiction. ∎

######  Remark 6.14.

We note that if $(M,g)$ is simple, then the geodesic X-ray transform is
Lipschitz stable in suitable Sobolev spaces. Using (6.14) together with
Sobolev embedding, then also implies Lipschitz stability of the inverse
transport problem.

##  Appendix A Instability for linear inverse problems

In this appendix we complement the results from the main text by giving
several instability results for _linear_ inverse problems based on direct
arguments using singular value decay or abstract smoothing properties.
Compared to our strategy in the main part of the text, the final results will
be similar, though in some cases slightly sharper, to those obtained by using
the entropy/capacity estimates in Section 3 (which also apply to nonlinear
inverse problems).

Let $A:X\to Y$ be a bounded linear operator between two separable Hilbert
spaces. We consider the basic inverse problem for $A$: given
$g\in\mathrm{Ran}(A)$, find $f\in X$ with

$Af=g.$

In the linear inverse problems that we are interested in, the following
conditions will hold:

  * •

$A$ is injective. This means that the inverse problem for $A$ has a unique
solution $f\in X$ for any $g\in\mathrm{Ran}(A)$.

  * •

$X$ is infinite dimensional. This rules out trivial cases, and since $A$ is
injective it implies that $\mathrm{Ran}(A)$ (and hence also $Y$) is infinite
dimensional.

  * •

$A$ is compact. Then $A$ has a singular value decomposition (see Proposition
2.3) with singular values $(\sigma_{j})_{j=1}^{\infty}$ converging to zero as
$j\to\infty$, a singular value orthonormal basis $(\varphi_{j})$ of $X$, and
an orthonormal set $(\psi_{j})$ in $Y$ so that
$A\varphi_{j}=\sigma_{j}\psi_{j}$ and $A^{*}\psi_{j}=\sigma_{j}\varphi_{j}$.

The fact that the singular values converge to zero implies that the inverse
problem for $A$ is ill-posed (not Lipschitz stable) in $X$, i.e. there is no
constant $C>0$ such that

$\lVert f\rVert_{X}\leqslant C\lVert Af\rVert_{Y},\qquad f\in X.$

This follows just by taking $f=\varphi_{j}$ and letting $j\to\infty$.

In many cases, one expects that rapid decay of singular values of $A:X\to Y$
implies strong ill-posedness for the corresponding inverse problem. However,
the choice of the compact set $K$ also plays an important role. We remark that
for _instability results_ , one often does not need to know the decay of the
full sequence of singular values, but only the decay on a suitable subspace.
This will be exploited in the instability mechanism based on microlocal
smoothing, see Section 6. In contrast, for stability mechanisms lower bounds
are needed for all singular values.

We will next describe three approaches to instability for linear inverse
problems, related to different choices of the compact set $K$ and to smoothing
properties/singular value decay of $A$. The first two approaches consider
compact sets $K$ given in terms of some orthonormal basis
$\varphi=(\varphi_{j})$ of $X$ and an increasing sequence
$\kappa=(\kappa_{j})$ with
$0<\kappa_{1}\leqslant\kappa_{2}\leqslant\ldots\to\infty$. We define the set

(A.1) $\displaystyle K=K_{\kappa,\varphi}=\\{f\in X\ ;\ \lVert
f\rVert_{\kappa}:=\left(\sum_{j=1}^{\infty}\kappa_{j}^{2}|(f,\varphi_{j})|^{2}\right)^{1/2}\leqslant
1\\}.$

This set is compact in $X$, since the inclusion map
$j:(X,\lVert\,\cdot\,\rVert_{\kappa})\to X$ is easily shown to be approximated
by finite rank operators. Moreover, this setup is general in the sense that
any compact subset of $X$ is contained in some $K_{\kappa,\varphi}$, as proved
in the next lemma:

######  Lemma A.1.

Let $H$ be a separable Hilbert space, let $(e_{j})_{j\in\varmathbb{N}}$ be a
fixed (but else arbitrary) orthonormal basis of $H$, and let $K\subset H$.
Then, $K$ is compact if and only if K is closed and there exists an
increasing, positive sequence $(\kappa_{j})_{j\in\varmathbb{N}}$ with
$\kappa_{j}\rightarrow\infty$ such that for any $u\in K$

(A.2)
$\displaystyle\sum\limits_{j=1}^{\infty}|(u,e_{j})|^{2}\kappa_{j}^{2}\leqslant
1.$

######  Remark A.2.

Although the characterization of compact sets obtained in Lemma A.1 is very
useful if the singular value bases $\\{\varphi_{j}\\}$, $\\{\psi_{j}\\}$
associated with an operator $A$ are known, in general, there is the obvious
caveat, that these bases are not explicitly known (and a computation may not
be feasible). Hence, the set $K_{\kappa,\varphi}$ may not have an easy
characterization in terms of standard function spaces in general.

######  Remark A.3.

The condition (A.1) can also be viewed as a source type condition. Indeed,
this corresponds to a source condition $f=\varphi(A^{*}A)v$ with
$\|v\|_{X}\leqslant 1$, with the choice
$\varphi(\sigma_{j}^{2})=\kappa_{j}^{-1}$, where the numbers $\sigma_{j}$
correspond to the singular values of the operator $A$. We, for instance, refer
to [EHN96] for HÃ¶lder source type conditions, to [Ho00] for logarithmic ones
and to [MP03] for general ones. In many places, in this subsection, it would
be possible to formulate the results in terms of such source conditions.

###### Proof of Lemma A.1.

We first assume that $K$ is compact. Standard functional analysis arguments
then in particular entail that $K$ is closed and bounded. It thus suffices to
prove the existence of an increasing sequence
$\\{\kappa_{j}\\}_{j\in\varmathbb{N}}$ such that all elements $u\in K$ obey
the summability condition (A.2). However, compact sets in separable Hilbert
spaces can be characterized by having equi-small tails (see [Me, Proposition
25, Section 12, Chapter 3]), i.e. for each $\varepsilon>0$ there exists
$N\in\varmathbb{N}$ such that for any $u\in K$

$\displaystyle\sum\limits_{j\geqslant
N}|(u,e_{j})|^{2}\leqslant\varepsilon^{2}.$

From this we can however construct the desired sequence
$\\{\kappa_{j}\\}_{j\in\varmathbb{N}}$ (or rather one possible choice of this
sequence). Indeed, considering a sequence $\varepsilon_{k}:=\frac{1}{k^{2}}$,
there exists a corresponding sequence $N_{k}\in\varmathbb{N}$,
$N_{k+1}>N_{k}$, such that for all $u\in K$

$\displaystyle\sum\limits_{j=N_{k}}^{\infty}|(u,e_{j})|^{2}<\varepsilon_{k}^{2}=\frac{1}{k^{4}}.$

In particular, we have that

$\displaystyle\sum\limits_{k=1}^{\infty}\sum\limits_{j=N_{k}}^{N_{k+1}}k^{2}|(u,e_{j})|^{2}\leqslant\sum\limits_{k=1}^{\infty}k^{-2}\leqslant
C_{1}.$

Hence defining

$\displaystyle\kappa_{j}^{2}:=\left\\{\begin{array}[]{ll}&1/C_{1}\mbox{ if
}j\in\\{1,\dots,N_{2}\\},\\\ &l^{2}/C_{1}\mbox{ if
}j\in\\{N_{l},\dots,N_{l+1}\\}\mbox{ for }l\geqslant 2,\end{array}\right.$

then yields a possible choice for the increasing sequence $\kappa_{j}$. We
remark that the sequence $\kappa_{j}$ can also be chosen to be strictly
increasing by gradually growing from $l$ to $l+1$ in the interval
$j\in\\{N_{l},\dots,N_{l+1}\\}$.

Next we argue that closedness and the growth condition (A.2) characterize
compactness. To this end, we argue similarly as in the proof of [Me,
Proposition 25]: By boundedness (which follows from (A.2)) and closedness of
$K$, it suffices to show that any sequence
$\\{u_{m}\\}_{m\in\varmathbb{N}}\subset K$ has a convergent subsequence.
However, by boundedness of the sequence $\\{u_{m}\\}_{m\in\varmathbb{N}}$ we
have that for any $k\in\varmathbb{N}$ the sequence $(u_{m},e_{k})$ is bounded.
Hence, for each $k$ there exists a subsequence
$\\{u_{m_{k}}\\}_{m_{k}\in\varmathbb{N}}$ such that $(u_{m_{k}},e_{k})$ is
convergent. Moreover, we may assume that for each $k\in\varmathbb{N}$ it holds
$\\{u_{m_{k}}\\}_{m_{k}\in\varmathbb{N}}\subset\\{u_{m_{k-1}}\\}_{m_{k-1}\in\varmathbb{N}}$.
Using a Cantor diagonal argument and setting $v_{m}:=u_{m_{m}}$ then yields a
sequence such that $(v_{m},e_{k})$ converges for each $k\in\varmathbb{N}$. We
claim that $\\{v_{m}\\}_{m\in\varmathbb{N}}$ is the desired convergent
subsequence of $\\{u_{m}\\}_{m\in\varmathbb{N}}$. Indeed, let
$n,l\in\varmathbb{N}$ to be fixed later. Then,

$\displaystyle\|v_{n}-v_{n+l}\|^{2}\leqslant\sum\limits_{k\leqslant
N}|(v_{n}-v_{n+l},e_{k})|^{2}+2\sum\limits_{k>N}|(v_{n},e_{k})|^{2}+2\sum\limits_{k>N}|(v_{n+l},e_{k})|^{2}.$

By assumption, the condition (A.2) holds. In particular, we have that

$\sum\limits_{k>N}|(v_{n},e_{k})|^{2}+\sum\limits_{k>N}|(v_{n+l},e_{k})|^{2}\leqslant
2\kappa_{N}^{-2}\,.$

Now, let $\varepsilon>0$ be arbitary. Choosing $N\in\varmathbb{N}$ so large
that $\kappa_{N}^{-2}\leqslant\varepsilon^{2}/2$ and choosing
$n\in\varmathbb{N}$ so large that for $k\in\\{1,\dots,N\\}$ it holds that
$|(v_{n}-v_{n+l},e_{k})|\leqslant\frac{\varepsilon^{2}}{2N}$ (which is
possible by the convergence of $(v_{n},e_{k})$ for all $k\in\varmathbb{N}$),
we infer that

$\displaystyle\|v_{n}-v_{n+l}\|^{2}\leqslant\sum\limits_{k\leqslant
N}|(v_{n}-v_{n+l},e_{k})|^{2}+4\kappa_{N}^{-2}\leqslant\varepsilon^{2}.$

This proves the desired convergence of the subsequence under consideration. ∎

###  A.1. Compact sets described by a singular value basis

In cases where $K=K_{\kappa,\varphi}$ and where $\varphi=(\varphi_{j})$
happens to be a singular value basis for the operator $A$, the next result
completely characterizes the stability properties of the inverse problem (for
typical moduli of continuity) in terms of the sequences $(\sigma_{j})$ and
$(\kappa_{j})$.

######  Proposition A.4.

Let $A$, $\sigma_{j}$, $\varphi_{j}$ be as in Proposition 2.3, and let
$K=K_{\kappa,\varphi}$ for some sequence $(\kappa_{j})$ with
$0<\kappa_{1}\leqslant\kappa_{2}\leqslant\ldots\to\infty$ as in (A.1). Let
also $\eta(t)$ be a strictly increasing concave function so that $\eta(t)/t$
is nonincreasing.

Then the inverse problem for $A$ in $K$ is $\eta$-stable in the sense that

(A.3) $\lVert f\rVert_{X}^{2}\leqslant\eta(\lVert Af\rVert_{Y}^{2}),\qquad
f\in K,$

if and only if

(A.4) $\kappa_{j}^{-2}\leqslant\eta((\sigma_{j}/\kappa_{j})^{2}),\qquad
j\geqslant 1.$

###### Proof.

If (A.3) holds, then choosing $f=\kappa_{j}^{-1}\varphi_{j}\in K$ yields
(A.4). Conversely, assume that (A.4) holds for some strictly increasing
concave function $\eta$. Then $\eta^{-1}$ is convex. Let $f=\sum
a_{j}\varphi_{j}\in K$ where $\sum\kappa_{j}^{2}a_{j}^{2}=1$, and define
$c_{j}=\kappa_{j}^{2}a_{j}^{2}$. Since $\sum c_{j}=1$, Jensen’s inequality and
(A.4) yield

$\eta^{-1}(\lVert f\rVert_{X}^{2})=\eta^{-1}(\sum a_{j}^{2})=\eta^{-1}(\sum
c_{j}\kappa_{j}^{-2})\leqslant\sum
c_{j}\eta^{-1}(\kappa_{j}^{-2})\leqslant\sum\sigma_{j}^{2}a_{j}^{2}=\lVert
Af\rVert_{Y}^{2}.$

Thus (A.3) holds for any $f\in K$ with $\sum\kappa_{j}^{2}a_{j}^{2}=1$. If
instead $\sum\kappa_{j}^{2}a_{j}^{2}=\lambda^{2}\leqslant 1$, then (A.3) holds
for $f/\lambda$, so

$\lVert f\rVert_{X}^{2}\leqslant\frac{\eta(\lVert
Af\rVert_{Y}^{2}/\lambda^{2})}{1/\lambda^{2}}\leqslant\eta(\lVert
Af\rVert_{Y}^{2})$

using that $\eta(t)/t$ is nonincreasing. ∎

######  Example A.5.

We can draw some immediate conclusions:

  * •

The inverse problem for $A$ is $\alpha$-Hölder stable in $K$ (i.e.
$\eta(t)=Ct^{\alpha}$) if and only if
$\kappa_{j}^{1-\alpha}\sigma_{j}^{\alpha}\geqslant c>0$ for $j\geqslant 1$. In
particular, no matter what the singular values $\sigma_{j}$ of $A$ are, if one
chooses $\kappa_{j}=\sigma_{j}^{-s}$ for some $s>0$ then the inverse problem
for $A$ in $K$ will be Hölder stable. This is the case even if the singular
values of $A$ are exponentially decaying, however then the set $K$ may be
restrictively small (similar to a space of real-analytic functions).

  * •

As a second possible scenario, we consider the case in which the singular
values of $A$ satisfy $\sigma_{j}\leqslant e^{-cj^{\mu}}$, i.e. they are
exponentially decaying. Further we choose _$\kappa_{j}=j^{s}$_ for some $s>0$
(if $(\varphi_{j})$ by chance also corresponds to an eigenbasis of some
elliptic operator, as in Proposition 2.7, then this implies that $K$ is a
bounded set in a Sobolev type space). In this case, Proposition A.4 implies
that the inverse problem for $A$ in $K$ has at best a logarithmic modulus of
continuity. Moreover, if one also has lower bounds $\sigma_{j}\geqslant
e^{-c_{1}j^{\mu}}$ for the singular values, the proposition implies that the
inverse problem is indeed logarithmically stable.

The advantage of using a singular value basis to describe $K$ is the fact that
the action of $A$ on an element $f=\sum a_{j}\varphi_{j}$ can be easily
computed. However, the drawback of this approach is that the relation of the
norm used in the definition of $K$ and “standard norms" such as Sobolev norms
is not obvious in general. For example, if $X=L^{2}(M)$ for a compact
$n$-manifold $M$, then a more natural choice would be to consider $K=\\{f\in
H^{s}(M)\,;\,\lVert f\rVert_{H^{s}(M)}\leqslant 1\\}$ for some $s>0$, which
would correspond to (A.1) with $\kappa_{j}\sim j^{s/n}$ but where the basis
$(\varphi_{j})$ would consist of the eigenfunctions of the Laplacian on $M$.
However, the action of $A$ is much harder to compute on a general basis
$(\varphi_{j})$ than for a singular value basis.

Analogously to Proposition 2.7, a special case where the singular value basis
can be used to describe standard Sobolev spaces, so that Proposition A.4
applies, is the case where $A$ is an elliptic operator (see [Ta96, Section
7.10 in vol. II]).

It may also be possible to apply Proposition A.4 in cases where there is very
high symmetry. One such situation, known as “Slepian’s miracle", was
discovered by Landau, Pollak, Slepian, see [SP61] for more information and
references on this. However, we will not discuss these points further, since
the other two approaches described below will give more general instability
results.

###  A.2. Compact sets described by a reference basis

If one is only interested in an instability result, i.e. in showing that the
inequality

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

can only hold for certain moduli of continuity $\omega$, it is enough to
exhibit elements $f\in K$ so that $\lVert Af\rVert_{Y}$ is relatively small
but $\lVert f\rVert_{X}$ is not too small. The following result shows that
this is possible whenever $K$ is of the form (A.1) where $(\varphi_{j})$ can
be any fixed orthonormal basis (for instance a basis used for defining Sobolev
type spaces).

######  Proposition A.6.

Let $\varphi=(\varphi_{j})_{j=1}^{\infty}$ be a fixed orthonormal basis of
$X$, and let $K=K_{\kappa,\varphi}$ as in (A.1) for some sequence $\kappa$
with $0<\kappa_{1}\leqslant\kappa_{2}\leqslant\ldots$ and
$\kappa_{j}\to\infty$. Define

$N(\varepsilon)=\sup\\{j\geqslant 1\,;\,\kappa_{j}\leqslant 1/\varepsilon\\}.$

Let $A:X\to Y$ be a compact injective operator with singular values
$\sigma_{1}\geqslant\sigma_{2}\geqslant\ldots$. If

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

then

$\omega(t)\geqslant g^{-1}(t),\qquad\text{$t$ small},$

where $g(\varepsilon):=\varepsilon\sigma_{N(\varepsilon)}$ is a strictly
decreasing function.

###### Proof.

We first observe that if $W$ is any finite dimensional subspace of $X$, with
dimension $\dim(W)=N$, then

(A.5) $\text{there is $f\in W\setminus\\{0\\}$ with $\lVert
Af\rVert_{Y}\leqslant\sigma_{N}\lVert f\rVert_{X}$}.$

To prove this, let $P_{W}$ be the orthogonal projection to $W$ and note that
$Af=AP_{W}f$ for any $f\in W$. The operator $AP_{W}$ has at most $N$ nonzero
singular values $\mu_{1}\geqslant\ldots\geqslant\mu_{N}$, and they satisfy

$\mu_{j}=\sigma_{j}(AP_{W})\leqslant\sigma_{j}(A)\sigma_{1}(P_{W})\leqslant\sigma_{j}(A).$

Thus, choosing $f$ to be a singular vector of $AP_{W}$ corresponding to
$\mu_{N}$ yields (A.5).

Next we want to find a finite dimensional subspace $W$ of $X$, with as large
dimension as possible, so that

(A.6) $\lVert f\rVert_{\kappa}\leqslant\frac{1}{\varepsilon}\lVert
f\rVert_{X},\qquad f\in W.$

This is easily arranged by choosing $W=\\{\sum_{j=1}^{N}a_{j}\varphi_{j}\\}$
where $N=N(\varepsilon)$, since then

$\lVert
f\rVert_{\kappa}^{2}=\sum_{j=1}^{N}\kappa_{j}^{2}\lvert(f,\varphi_{j})\rvert^{2}\leqslant\frac{1}{\varepsilon^{2}}\sum_{j=1}^{N}\lvert(f,\varphi_{j})\rvert^{2}=\frac{1}{\varepsilon^{2}}\lVert
f\rVert_{X}^{2},\qquad f\in W.$

Now let $f$ be the vector in (A.5), scaled so that it satisfies $\lVert
f\rVert_{X}=\varepsilon$. Then (A.6) gives that $\lVert
f\rVert_{\kappa}\leqslant 1$, i.e. $f\in K$, and (A.5) gives that $\lVert
Af\rVert_{Y}\leqslant\sigma_{N(\varepsilon)}\lVert
f\rVert_{X}=\sigma_{N(\varepsilon)}\varepsilon$. The $\omega$-stability of the
inverse problem gives that

$\varepsilon\leqslant\omega(\varepsilon\sigma_{N(\varepsilon)}),$

which proves the result since
$g(\varepsilon)=\varepsilon\sigma_{N(\varepsilon)}$ is strictly decreasing and
hence has an inverse function. ∎

######  Example A.7.

In order to use Lemma A.6, one needs to know both upper bounds for
$\kappa_{j}$ and a decay rate for the singular values $\sigma_{j}$. Here are a
few examples:

  * •

If $\kappa_{j}\sim j^{s}$ for some $s>0$ (Sobolev type spaces) and
$\sigma_{j}\lesssim j^{-m}$, then $N(\varepsilon)\sim\varepsilon^{-1/s}$ and
$g(\varepsilon)\lesssim\varepsilon^{\frac{s+m}{s}}$. If the inverse problem
for $A$ in $K$ is $\omega$-stable, Lemma A.6 yields that $\omega(t)\geqslant
ct^{\frac{s}{s+m}}$ and thus $\alpha$-Hölder stability is only possible for
$\alpha\leqslant\frac{s}{s+m}$.

  * •

If $\kappa_{j}\sim j^{s}$ for some $s>0$ but the singular values of $A$ are
exponentially decaying, i.e. $\sigma_{j}\lesssim e^{-cj^{\mu}}$ for some
$c,\mu>0$, then $N(\varepsilon)\sim\varepsilon^{-1/s}$ and

$g(\varepsilon)=\varepsilon\sigma_{N(\varepsilon)}\lesssim\varepsilon
e^{-c\varepsilon^{-\mu/s}}\lesssim e^{-c_{1}\varepsilon^{-\mu/s}}$

for any $c_{1}<c$. It follows that $\omega(t)\geqslant
c\lvert\log\,t\rvert^{-s/\mu}$, i.e. logaritmic stability is best possible in
this setting.

The above examples confirm the expectation that rapidly decaying singular
values imply strong illposedness, at least when the compact set $K$ is related
to a standard Sobolev space.

###  A.3. Compact sets described by an interpolation property

Finally, we show instability results in cases where $A$ satisfies an abstract
smoothing property, and $K$ is a bounded set in a space satisfying certain
abstract interpolation estimates. The point is that in this setup, no
information about the decay rate of the singular values of $A$ is needed.

######  Proposition A.8.

Let $V\subset E\subset H$ be infinite dimensional separable Hilbert spaces
such that the inclusion $i:V\to H$ is compact, and one has the abstract
interpolation inequality

$\lVert u\rVert_{E}\leqslant\lVert u\rVert_{H}\,g(\lVert u\rVert_{V}/\lVert
u\rVert_{H}),\qquad u\in V\setminus\\{0\\},$

for some $C^{1}$ function $g:[1,\infty)\to[0,\infty)$ with $g^{\prime}>0$. Let
$A:H\to Y$ be bounded where $Y$ is a Banach space, and assume that $A$ is
smoothing in the sense that there is a bounded operator $\bar{A}:V^{\prime}\to
Y$ extending $A$ so that

$\bar{A}(\iota(u))=A(u),\qquad u\in H,$

where $\iota:H\to V^{\prime}$ is the natural inclusion with
$\iota(u)(v)=(u,i(v))_{H}$ for $u\in H$ and $v\in V$.

Under these assumptions, if $A$ satisfies the stability inequality

$\lVert u\rVert_{H}\leqslant\omega(\lVert Au\rVert_{Y}),\qquad\lVert
u\rVert_{E}\leqslant 1,$

for some modulus of continuity $\omega$, then there is a sequence $t_{j}\to 0$
with

$\omega(t_{j})\geqslant\frac{1}{g(1/f^{-1}(t_{j}))}$

where $f(\sigma):=\lVert\bar{A}\rVert\sigma/g(1/\sigma)$.

###### Proof.

Since $i:V\to H$ is compact and injective, there are orthonormal sets
$(\varphi_{j})\subset V$ and $(\psi_{j})\subset H$ such that

$i(\varphi_{j})=\sigma_{j}\psi_{j},\qquad
i^{*}(\psi_{j})=\sigma_{j}\varphi_{j},$

where the singular values $(\sigma_{j})$ of $i$ satisfy
$\sigma_{1}\geqslant\sigma_{2}\geqslant\ldots>0$ and $\sigma_{j}\to 0$. The
interpolation inequality implies

$\lVert\psi_{j}\rVert_{E}\leqslant
g(\lVert\psi_{j}\rVert_{V})=g(\frac{1}{\sigma_{j}}\lVert\varphi_{j}\rVert_{V})=g(\frac{1}{\sigma_{j}}).$

We define

$u_{j}:=\frac{\psi_{j}}{g(1/\sigma_{j})}.$

Then $\lVert u_{j}\rVert_{E}\leqslant 1$.

Applying the stability inequality to $u_{j}$ and writing
$C=\lVert\bar{A}\rVert$, we get

$\lVert
u_{j}\rVert_{H}\leqslant\omega(\lVert\bar{A}(\iota(u_{j}))\rVert_{Y})\leqslant\omega(C\lVert\iota(u_{j})\rVert_{V^{\prime}}).$

Now $\lVert u_{j}\rVert_{H}=\frac{1}{g(1/\sigma_{j})}$ and

$\lVert\iota(u_{j})\rVert_{V^{\prime}}=\sup_{\lVert
v\rVert_{V}=1}(u_{j},i(v))_{H}=\sup_{\lVert
v\rVert_{V}=1}\frac{1}{g(1/\sigma_{j})}(i^{*}(\psi_{j}),v)_{V}\leqslant\frac{\sigma_{j}}{g(1/\sigma_{j})}.$

Thus we have

$\frac{1}{g(1/\sigma_{j})}\leqslant\omega(C\sigma_{j}/g(1/\sigma_{j})).$

Writing $t_{j}:=f(\sigma_{j})$, it follows that

$\omega(t_{j})\geqslant\frac{1}{g(1/f^{-1}(t_{j}))}$

provided that $f^{-1}$ is well defined. But the derivative of $f$ is

$f^{\prime}(\sigma)=C\left(\frac{1}{g(1/\sigma)}+\frac{g^{\prime}(1/\sigma)}{\sigma
g(1/\sigma)^{2}}\right)>0,$

so $f$ is strictly increasing and $f^{-1}$ is well defined. ∎

######  Example A.9.

To illustrate the proposition, we have the following results:

  * •

Let $V=X^{s+m},E=X^{s+\delta},H=X^{s}$ where $X^{t}$ are Sobolev type spaces
satisfying the interpolation inequality

$\lVert u\rVert_{X^{s+\delta}}\leqslant\lVert
u\rVert_{X^{s}}^{1-\delta/m}\lVert u\rVert_{X^{s+m}}^{\delta/m}$

such that $X^{s+m}\subset X^{s}$ is compact. Then one can take
$g(t)=t^{\delta/m}$ and $f(\sigma)=\sigma^{1+\delta/m}$. The conclusion of
Proposition A.8 is that

$\omega(t_{j})\geqslant t_{j}^{\frac{\delta}{m+\delta}},\qquad t_{j}\to 0.$

The fact that $A$ extends as a map $V^{\prime}\to Y$, where the dual of
$V=X^{s+m}$ is taken with respect to $H=X^{s}$, can be interpreted so that $A$
is smoothing of order $m$. In particular, if $A$ is smoothing of infinite
order, no Hölder stability bound is possible.

  * •

Let $H=\ell^{2}=\\{x=(x_{1},x_{2},\ldots)\\}$, let $E$ be the Sobolev space
$\lVert x\rVert_{E}=(\sum j^{2s}x_{j}^{2})^{1/2}$, and let $V$ be the real
analytic type space

$\lVert x\rVert_{V}=(\sum e^{2\beta j}x_{j}^{2})^{1/2}$

where $s,\beta>0$. The inclusion $V\subset H$ is compact. If $\sum
x_{j}^{2}=1$, applying Jensen’s inequality with convex function
$h(t)=\exp(2\beta t^{1/(2s)})$, which has inverse
$h^{-1}(y)=(\frac{1}{2\beta}\log(y))^{2s}$, implies that

$h(\sum j^{2s}x_{j}^{2})\leqslant\sum x_{j}^{2}h(j^{2s})$

and therefore

$\lVert x\rVert_{E}^{2}\leqslant h^{-1}(\sum x_{j}^{2}h(j^{2s}))=h^{-1}(\sum
x_{j}^{2}e^{2\beta j}),\qquad\sum x_{j}^{2}=1.$

By homogeneity, this shows that

$\lVert x\rVert_{E}\leqslant\lVert x\rVert_{H}(\frac{1}{\beta}\log(\lVert
x\rVert_{V}/\lVert x\rVert_{H}))^{s}.$

Thus one can choose $g(t)=(\frac{1}{\beta}\log(t))^{s}$ and
$f(\sigma)=\sigma(\frac{1}{\beta}\log(1/\sigma))^{-s}$. Since $f(\sigma)$
behaves almost like $\sigma$, the conclusion of Proposition A.8 implies that
$\omega(t)$ cannot be much better than $\lvert\log t\rvert^{-s}$. We remark
that similar Jensen type inequalities had already been used in the 70s and
80s. As an non-exhaustive set of references we point to [TV82, LV85, LV85,
MP85] and the literature mentioned therein.

###  A.4. Comparison between different instability results

In the next example we compare the entropy based instability results from
Section 3 to the alternative linear instability results in Appendix A. We
consider an abstract inverse problem where the forward map
$A=(-\Delta_{g}+1)^{-m/2}$ is a linear elliptic pseudodifferential operator on
a closed manifold. It turns out that Proposition A.4 gives a sharp
characterization for $\alpha$-Hölder stability and instability, whereas
Propositions A.6 and A.8 and Lemma 3.7 give the same almost sharp instability
result. Thus at least in this example the entropy based methods of Section 3
give the optimal instability result except for missing the endpoint exponent.

Let $(M,g)$ be a closed connected smooth $n$-manifold, and let
$(\varphi_{j})_{j=0}^{\infty}$ be an orthonormal basis of $L^{2}(M)$
consisting of eigenfunctions of the Laplacian such that
$-\Delta_{g}\varphi_{j}=\lambda_{j}\varphi_{j}$ with $\lambda_{0}=0$ and
$\lambda_{j}\sim j^{2/n}$ for $j\geqslant 1$. Let $m>0$ and define

$A:L^{2}(M)\to L^{2}(M),\ \
Au=(-\Delta_{g}+1)^{-m/2}u=\sum_{j=0}^{\infty}(1+\lambda_{j})^{-m/2}(u,\varphi_{j})\varphi_{j}.$

Then $A$ has singular values $\sigma_{j}=(1+\lambda_{j-1})^{-m/2}\sim\langle
j\rangle^{-m/n}$ by Weyl asymptotics (Theorem 2.6), and the entropy numbers
satisfy $e_{j}\sim\langle j\rangle^{-m/n}$ using Lemma 3.9. Consider a compact
set

$K=\\{u\in H^{l}(M)\,;\,\lVert u\rVert_{H^{l}(M)}\leqslant 1\\}.$

Then $K=K_{\varphi,\kappa}$ with $\kappa_{j}=(1+\lambda_{j})^{l/2}\sim\langle
j\rangle^{l/n}$.

We can say the following about the $\alpha$-Hölder stability for the inverse
problem for $A$ in $K$:

  * •

By Proposition A.4 (see Example A.5) the problem is $\alpha$-Hölder stable if
and only if $l(1-\alpha)-m\alpha>0$, i.e. $\alpha<\frac{l}{l+m}$.

  * •

By Proposition A.6 (see Example A.7) the problem can only be $\alpha$-Hölder
stable if $\alpha\leqslant\frac{l}{l+m}$.

  * •

By Proposition A.8 (see Example A.9) the problem can only be $\alpha$-Hölder
stable if $\alpha\leqslant\frac{l}{l+m}$.

  * •

By Lemma 3.7, since $A(K)$ lies in a bounded subset of $H^{l+m}$, the problem
can only be $\alpha$-Hölder stable if $\alpha\leqslant\frac{l}{l+m}$.

##  Appendix B Proofs of some results in Section 2

In this section we present the proofs of some statements from Section 2. We
begin with a lemma related to Gevrey spaces.

######  Lemma B.1.

Let $(M,g)$ be a closed smooth $n$-manifold, let $1\leqslant\sigma<\infty$ and
$\rho>0$.

  1. (a)

$u\in G^{\sigma}(M)$ if and only if there are $C,R>0$ such that

(B.1) $\lVert(-\Delta_{g})^{t}u\rVert_{L^{2}}\leqslant
CR^{2t}(2t)^{2t\sigma},\qquad t\geqslant 0.$

  2. (b)

If $u\in A^{\sigma,\rho}(M)$, then $u\in G^{\sigma}(M)$ and

$\lVert(-\Delta_{g})^{t}u\rVert_{L^{2}}\leqslant\lVert
u\rVert_{A^{\sigma,\rho}(M)}R^{2t}(2t)^{2t\sigma},\qquad t\geqslant 0,$

for any $R\geqslant C_{0}(\sigma/\rho)^{\sigma}$ where $C_{0}>0$ only depends
on $(M,g)$.

  3. (c)

If $u\in G^{\sigma}(M)$ satisfies (B.1), then $u\in A^{\sigma,\rho}(M)$ for
any $\rho\leqslant c_{0}R^{-1/\sigma}$ and

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

where $c_{0},c_{1}>0$ depend on $(M,g)$, $n$ and $\sigma$, and $c_{1}$ also
depends on $R$.

###### Proof.

(a) Let first $k\geqslant 0$ and $0\leqslant l\leqslant k-1$. Recall that
$\nabla$ denotes the covariant derivative induced by the Levi-Civita
connection $(M,g)$ and that $\lvert\nabla^{l}u\rvert$ denotes the $g$-norm of
the $l$-tensor field $\nabla^{l}u$, see e.g. [Be87]. Using the facts that
$\Delta v=\operatorname{tr}_{g}\nabla^{2}v$ and that $\operatorname{tr}_{g}$
(the $g$-trace with respect to given two indices) commutes with the covariant
derivative $\nabla$, we have

$\lvert\nabla^{l}\Delta^{k-l}u\rvert=\lvert\nabla^{l}\operatorname{tr}_{g}\nabla^{2}\Delta^{k-l-1}u\rvert=\lvert\operatorname{tr}_{g}\nabla^{l+2}\Delta^{k-l-1}u\rvert\leqslant
n^{1/2}\lvert\nabla^{l+2}\Delta^{k-l-1}u\rvert.$

In the last step we used Cauchy-Schwarz. Using this repeatedly gives

$\lvert\Delta^{k}u\rvert\leqslant
n^{1/2}\lvert\nabla^{2}\Delta^{k-1}u\rvert\leqslant\ldots\leqslant
n^{k/2}\lvert\nabla^{2k}u\rvert.$

Now if $u\in G^{\sigma}(M)$ with
$\lVert\nabla^{k}u\rVert_{L^{\infty}(M)}\leqslant C_{0}R_{0}^{k}k^{\sigma k}$
for $k\geqslant 0$, the previous inequality gives

$\lVert\Delta^{k}u\rVert_{L^{2}}\leqslant
C_{0}\,\mathrm{Vol}_{g}(M)^{1/2}(n^{1/4}R_{0})^{2k}(2k)^{2\sigma k},\qquad
k\geqslant 0.$

This proves (B.1) when $t$ is an integer. If $0\leqslant t\leqslant 1$ then
(B.1) is trivial, and if $t\geqslant 1$ then $t=k\theta$ for some
$k\in\varmathbb{Z}_{+}$ and $\theta\geqslant 1/2$, so that

$\lVert\Delta^{t}u\rVert_{L^{2}}\leqslant\lVert
u\rVert_{L^{2}}^{1-\theta}\lVert\Delta^{k}u\rVert_{L^{2}}^{\theta}\leqslant
CR^{t}(2k)^{2\sigma t}\leqslant C(R\theta^{-2\sigma})^{t}(2t)^{2\sigma t}.$

We have proved (B.1) for all $t\geqslant 0$. Conversely, if (B.1) holds then
by Sobolev embedding and elliptic regularity

$\lVert\nabla^{2k}u\rVert_{L^{\infty}}\leqslant C_{0}\lVert
u\rVert_{H^{2k+2n}}\leqslant C_{0}C_{1}^{k}(\lVert
u\rVert_{L^{2}}+\lVert(-\Delta_{g})^{k+n}u\rVert_{L^{2}})$

where $C_{0},C_{1}>0$ only depend on $(M,g)$ and $n$. Thus one gets
$\lVert\nabla^{2k}u\rVert_{L^{\infty}}\leqslant\tilde{C}\tilde{R}^{2k}(2k)^{2\sigma
k}$ for some $\tilde{C},\tilde{R}>0$, showing that $u\in G^{\sigma}(M)$.

(b) Let $u\in A^{\sigma,\rho}(M)$. Then $u\in C^{\infty}(M)$, and for
$t\geqslant 0$

$(-\Delta_{g})^{t}u=\sum_{j=0}^{\infty}\lambda_{j}^{t}(u,\varphi_{j})\varphi_{j}.$

Consequently, using the Weyl asymptotics $\lambda_{j}\leqslant C_{0}j^{2/n}$
(Theorem 2.4),

$\lVert(-\Delta_{g})^{t}u\rVert_{L^{2}}^{2}=\sum_{j=0}^{\infty}\lambda_{j}^{2t}\lvert(u,\varphi_{j})\rvert^{2}\leqslant
C_{0}^{2t}(\sup_{j\geqslant 0}j^{2t/n}e^{-\rho j^{1/(n\sigma)}})^{2}\lVert
u\rVert_{A^{\sigma,\rho}}^{2}.$

The function $f(s)=s^{2t/n}e^{-\rho s^{1/(n\sigma)}}$ obtains its maximum over
$s\geqslant 0$ when $s=(2t\sigma/\rho)^{n\sigma}$. Thus we obtain

$\lVert(-\Delta_{g})^{t}u\rVert_{L^{2}}\leqslant
C_{0}^{t}(2t\sigma/\rho)^{2t\sigma}e^{-2t\sigma}\lVert
u\rVert_{A^{\sigma,\rho}}.$

(c) Let $u\in G^{\sigma}(M)$ satisfy (B.1). We observe that for any
$t\geqslant 0$

$(u,\varphi_{j})_{L^{2}}=\lambda_{j}^{-t}(u,(-\Delta_{g})^{t}\varphi_{j})_{L^{2}}=\lambda_{j}^{-t}((-\Delta_{g})^{t}u,\varphi_{j})_{L^{2}}.$

Using (B.1), the Weyl asymptotics $\lambda_{j}\geqslant c_{0}j^{2/n}$ and the
normalization $\lVert\varphi_{j}\rVert_{L^{2}}=1$, we get

$\lvert(u,\varphi_{j})\rvert_{L^{2}}\leqslant(c_{0}j^{2/n})^{-t}CR^{2t}(2t)^{2t\sigma}.$

The function $f(s)=(\frac{R}{c_{0}^{1/2}j^{1/n}})^{s}s^{s\sigma}$ is minimal
over $s\geqslant 0$ when
$s=\frac{1}{e}(\frac{c_{0}^{1/2}j^{1/n}}{R})^{1/\sigma}$. Thus choosing
$t=t_{0}:=\frac{1}{2e}(\frac{c_{0}^{1/2}j^{1/n}}{R})^{1/\sigma}$ gives

$\lvert(u,\varphi_{j})\rvert_{L^{2}}\leqslant Ce^{-2\sigma t_{0}}\leqslant
Ce^{-cj^{1/(n\sigma)}}$

with $c=\frac{\sigma}{e}(c_{0})^{\frac{1}{2\sigma}}R^{-1/\sigma}$. Thus if
$\rho\leqslant c/2$, one has

$\lVert u\rVert_{A^{\sigma,\rho}(M)}^{2}\leqslant
C^{2}\sum_{j=0}^{\infty}e^{2\rho
j^{1/(n\sigma)}}e^{-2cj^{1/(n\sigma)}}\leqslant
C^{2}\sum_{j=0}^{\infty}e^{-cj^{1/(n\sigma)}}=C^{2}c_{1}^{2}$

where $c_{1}$ only depends on $(M,g)$, $n$, $\sigma$ and $R$. ∎

As the final part of this section, we present a proof of the Weyl law in
Theorem 2.6 using semiclassical calculus following [Zw12, Section 14.3].

###### Proof of Theorem 2.6.

By passing to $A^{*}A$ we may assume that $A$ is self-adjoint and nonnegative.
Let $\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant 0$ be the
eigenvalues of $A$ on $L^{2}(M)$. We wish to count the number of eigenvalues
that are $\geqslant\varepsilon=h^{m}$, where $h>0$ is a semiclassical
parameter. In order to do this we will relate $A$ to the semiclassical
$\Psi$DO $h^{m}\mathrm{Op}_{h}(a)$ where $\mathrm{Op}_{h}$ denotes Weyl
quantization, and estimate the eigenvalues of the latter by using
semiclassical analysis.

First note that the operator $A$ is a classical self-adjoint $\Psi$DO of order
$-m$. Thus there is $\tilde{a}\in C^{\infty}(T^{*}M\setminus\\{0\\})$, real
and homogeneous of degree $-m$ in $\xi$, so that

$a(x,\xi):=(1-\psi(x,\xi))\tilde{a}(x,\xi)$

is a principal symbol for $A$. Here
$\psi(x,\xi)=\tilde{\psi}(\lvert\xi\rvert_{g(x)})$ where $\tilde{\psi}\in
C^{\infty}_{c}(\varmathbb{R})$ is a function with
$0\leqslant\tilde{\psi}\leqslant 1$, $\tilde{\psi}=1$ near $0$, and
$\tilde{\psi}(t)=0$ for $\lvert t\rvert\geqslant 1$. We now use the fact that
$\mathrm{Op}_{h}(b(x,\xi))=\mathrm{Op}(b(x,h\xi))$ together with the
homogeneity of $\tilde{a}$ and support properties of $\psi$, which yields that
for $h>0$ small

$h^{m}\mathrm{Op}_{h}(a)=\mathrm{Op}((1-\psi(x,h\xi))\tilde{a}(x,\xi))=\mathrm{Op}((1-\psi(x,h\xi))a(x,\xi)).$

This implies that

(B.2) $h^{m}\mathrm{Op}_{h}(a)=A-A\mathrm{Op}_{h}(\psi)+R$