Carleman estimates and anisotropic inverse problems · IntroductionCarleman weightsCarleman estimatesComplex geometrical optics Carleman estimates and anisotropic inverse problems

Introduction Carleman weights Carleman estimates Complex geometrical optics

Carleman estimates and anisotropic inverseproblems

David Dos Santos Ferreira

LAGA – Universite Paris 13

Heraklion — 05/31/2012


Collaborators

• Carlos Kenig (University of Chicago)

• Mikko Salo (University of Jyvaskyla)


Outline

Introduction

Carleman weights

Carleman estimates

Complex geometrical optics


An inverse problem on the Schrodinger equation

Let (M, g) be a compact Riemannian manifold with boundary ∂Mof dimension n ≥ 3 and q an unbounded potential in L

n2 (M).

Consider the Dirichlet problem(∆g + q)u = 0

u|∂M = f ∈ H12 (∂M)

and define the associated Dirichlet-to-Neumann map (under anatural spectral assumption)

Λg,qu = ∂νu|∂M

where ν is a unit normal to the boundary.If q = 0, we use Λg = Λg,0 as a short notation.


Riemannian rigidity

There is a gauge invariance, that is by isometries which leave theboundary points unchanged:

Λϕ∗g = Λg, ϕ|∂M = Id∂M

Inverse problem: Does the Dirichlet-to-Neumann map Λg,qdetermine the potential q and the metric g modulo such isometries?

If n ≥ 3 and q = 0 this is a generalization of the anisotropicconductivity problem and one passes from one to the other by

γjk =√

det g gjk, gjk = (det γ)−2

n−2 gjk.


Conformal metrics

There is a conformal gauge transformation

∆cgu = c−1(∆g + qc)(cn−24 u), qc = c

n+24 ∆g(c

n−24 )

which translates at the boundary into

Λcg,qf = c−n+24 Λg,q+qc(c

n−24 u) +

n− 2

4c−

12∂νcf.

So if one knows c at the boundary (boundary determination) thenone can deduce one DN map from the other.

A more reasonable inverse problem: Λcg = Λg ⇒ c=1.Note that there is no isometry gauge invariance in this case.


Some references (Isotropic Calderon problem)

1980 Calderon: linearized problem, introduction of harmonicexponentials

1984 Kohn-Vogelius: boundary determination, (piecewise) analyticcase

1987 Sylvester-Uhlmann: Case n ≥ 3, C2 conductivities, use ofcomplex geometrical optics with linear weights

1996 Nachman: Case n = 2, W 2,p conductivities, ∂-method

2003 Paivarinta-Panchenko-Uhlmann: Case n ≥ 3, W32,∞

conductivities, Brown-Torres, W32,p, p > 2n

2006 Astala-Paivarinta: Case n = 2, L∞ conductivities, use ofquasiconformal geometry

2007 Kenig-Sjostrand-Uhlmann: small subsets of the boundary,n ≥ 3, global Carleman estimates with logarithmic weights,introduction of limiting Carleman weights,

2011 Haberman-Tataru, Case n ≥ 3, C1 conductivities andLipschitz with a smallness condition.


Some references (Anisotropic Calderon problem n ≥ 3)

1989 Lee-Uhlmann: boundary determination, analytic metrics, nopotential, determination of the metric

2001 Lassas-Uhlmann: improvement on topological assumptions

2007 Kenig-Sjostrand-Uhlmann: small subsets of the boundary,n ≥ 3, global Carleman estimates with logarithmic weights,introduction of limiting Carleman weights.

2009 Guillarmou-Sa Baretto: Einstein manifolds, no potential,determination of the metric, unique continuation argument

2009 DSF-Kenig-Salo-Uhlmann: fixed admissible geometries,determination of a smooth potential, CGOs

2011 DSF-Kenig-Salo: fixed admissible geometries, determinationof an unbounded potential, CGOs


Remarks

1. Analytic metrics case fairly well understood. The smooth caseremains a challenging problem.

2. There are limitations in the method using CGO constructionas the following pages will show.

3. We will concentrate on the case of identifiability of the metricwithin a conformal class

Λcg = Λg ⇒ c = 1.

4. With boundary determination

Λcg = Λg ⇒ c|∂M = 1.

it is enough to solve the inverse problem on the Schrodingerequation with a fixed metric

Λg,q1 = Λg,q2 ⇒ q1 = q2.


An integration by parts

We will now focus on the Euclidean case, and present Sylvesterand Uhlmann’s method. Let u1, u2 be solutions to the Schrodingerequations

∆u1 + q1u1 = 0, ∆u2 + q2u2 = 0

then ∫Ω

(q1 − q2)u1u2 dx =

∫∂Ω

(Λq1 − Λq2)u1u2 dσ.

Hence the inverse problem is implied by the following densityproperty:

the set of products u1u2 of solutions to the Schrodinger equationswith respective potentials q1, q2 is dense in L1.


Complex geometrical optics with linear weights

Goal: Construct solutions to the Schrodinger equation

We start with harmonic exponentials (used by Calderon to dealwith the linearized problem)

e−ix·ζ , ζ2 = ζ21 + · · ·+ ζ2

n = 0.

Sylvester and Uhlmann constructed complex geometrical opticssolutions by perturbation of the form

e−ihx·ζ(1 +O(h)

).

where h is a small parameter. The correction term O(h) isconstructed using solvability properties of ∆ + q in L2 weightedspaces (with exponential weight e−

1h

Im ζ·x) provided | Im ζ| ≥ 1.


An important remark on the construction

In fact, there is some freedom in the CGO construction. Indeed

eihx·ζh2∆e−

ihx·ζ = −(hD + iζ)2 = −h2D2 − 2iζ · hD

hence if a satisfies the Cauchy-Riemann equation

ζ ·Da = 0

then eihx·ζa is still an approximate solution and the CGOs

construction work as before.For instance, we have solutions of the form

e−ihx·ζ(e−ix·ξ +O(h)

)with ξ ⊥ ζ and | Im ζ| ≥ 1.


Identifiability of the potential

Plugging our CGOs solutions in the integration by parts formula,we get ∫

Ωe−

ih

(ζ1+ζ2)e−ix·ξ(q1 − q2) dx = O(h)

provided Im(ζ1 + ζ2) = 0.If we can choose ζ1 = ζ, ζ2 = −ζ with

ζ2 = 0, ζ ⊥ ξ | Im ζ| ≥ 1,

then 1Ω(q1 − q2) = 0 leading to q1 = q2.

This is only possible in dimension n ≥ 3 !!

Note that there is some flexibility since by analiticity of the Fouriertransform we don’t need all frequencies ξ ∈ Rn.


Concluding remarks

The method to obtain the identifiability of the potential followingSylvester and Uhlmann is the following

1. Use integration by parts to relate the information on theboundary to the inside.

2. Construct an approximate solution to the Schrodingerequation using complex geometrical optics.

3. Construct a correction term (with corresponding estimates)using Carleman weights.

4. Use the injectivity of a certain functional transform (Fourier orRadon transforms in the Euclidean case).

This is our roadmap.


Outline

Introduction

Carleman weights

Carleman estimates



Construction of complex geometrical optics

Recall that the proof of the identifiability of the potential from theDN map was based on the construction of solutions to theSchrodinger equation by complex geometrical optics

e1h

(ϕ+iψ)(a(x) +O(h)

)Euclidean case:

1. Sylvester-Uhlmann: linear phase: ϕ+ iψ = −ix · ζ2. Kenig-Sjostrand-Uhlmann: nonlinear phases:

ϕ+ iψ = log |x|+ idSn−1

(x

|x|, N

)The question we will address is: when is this construction possible?


Formal WKB expansion

Conjugated operator : Pϕ = eϕ/hh2∆ge−ϕ/h P ∗ϕ = P−ϕ

Principal symbol : pϕ = |ξ|2 − |dϕ|2 + 2i〈ξ, dϕ〉

We have

∆g(e1h

(ϕ+iψ)a) = e1hϕPϕ(e

ihψa) = e

1h

(ϕ+iψ)

(h0pϕ(x,dψ)

+ 2h

[(gradgϕ+ igradgψ)a+

1

2∆g(ϕ+ iψ)a

]+ h2∆ga

).

Eikonal equation: pϕ(x,dψ) = 0

Transport equation: (gradgϕ+ igradgψ)a+1

2∆g(ϕ+ iψ)a = 0


A few important remarks

1. Remember that in the Sylvester-Uhlmann approach, we wereusing CGOs u1, u2 with antagonizing exponential behaviour inorder to have a product u1u2 without exponential growth.This has to do with the fact that the important object is theproduct of CGOs.This means here that we want to be able to construct CGOsof the form e

1h±(ϕ+iψ)

(a(x) +O(h)

)!

2. The construction of the construction terms necessitates thederivation of some weighted a priori estimates, namelyCarleman estimates. So we need to know when these are true.

3. Already the solvability of the eikonal equation impliesRe pϕ, Im pϕ

(x, dψ) = 0.


Hormander’s criterium of solvability

Our purpose is to determine for which type of weights ϕ, theCarleman estimate

‖u‖ ≤ Chs‖Pϕu‖, s ∈ R

might be true ?

Theorem

If the previous Carleman estimate is true then

Re pϕ, Im pϕ≥ 0

on p−1ϕ (0).

This is a variant or byproduct of Hormander’s criterium ofsolvability for non-selfadjoint operators.


Limiting Carleman weights

We want to be able to have Carleman estimates for P±ϕ byHormander’s criterium this means

1

i

p±ϕ, p±ϕ

= ±1

i

pϕ, pϕ

≥ 0

on p−1ϕ (0). This leads to

Definition

A limiting Carleman weight on an open Riemannian manifold issmooth real-valued function without critical points such that

1

i

pϕ, pϕ

= 0 on p−1

ϕ (0).

This notion was introduced by Kenig-Sjostrand-Uhlmann.


Geometrical characterization

Question: Do all manifolds admitt LCW? Are there many LCW?

Lemma

Being a LCW is a conformally invariant property.

Lemma

1

2i

pϕ, pϕ

= ∇2ϕ(ξ], ξ]) +∇2ϕ(gradϕ, gradϕ)

In dimension n = 2, LCW are harmonic functions without criticalpoints.


Geometrical characterization

Theorem

If (M, g) is an open manifold having a limiting Carleman weight,then some conformal multiple of the metric g admits a parallel unitvector field. For simply connected manifolds, the converse is alsotrue.

By a result of Salo and Liimatainen, this case is quite restrictive.

Level sets of LCW are totally umbilical hypersurfaces.


Outline

Introduction

Carleman weights

Carleman estimates



Spectral cluster estimates of SoggeWe define the spectral clusters as

χk =∑

k≤√λj<k+1

πj , k ∈ N.

Note that these are projection operators, χ2k = χk, and they

constitute a decomposition of the identity

Id =

∞∑k=0

χk.

The spectral cluster estimates of Sogge are

‖χku‖L

2nn−2 (M0)

≤ C(1 + k)12− 1n ‖u‖L2(M0),

‖χku‖L2(M0) ≤ C(1 + k)12− 1n ‖u‖

L2nn+2 (M0)

.


Carleman estimates

Theorem

Let (M0, g0) be an (n− 1)-dimensional compact manifold withoutboundary, and equip R×M0 with the metric g = e⊕ g0 where eis the Euclidean metric. The Euclidean coordinate is denoted byx1. For any compact interval I ⊂ R there exists a constantCI > 0 such that if h ≥ 1/4 then we have

‖e±1hx1u‖

L2nn−2 (R×M0)

≤ CI‖e±1hx1∆gu‖

L2nn+2 (R×M0)

when u ∈ C∞0 (I ×M0).


Short bibliography on Lp Carleman estimates on ellipticoperators

These works are in relation with unique continuation of solutionsto Schrodinger equation with unbounded potentials.

1985 Jerison-Kenig: first Lp Carleman estimates, logarithmicweights

1986 Jerison: simplification of the proof using spectral clusterestimates (see also Sogge’s book)

1987 Kenig-Ruiz-Sogge: Elliptic operators with constantcoefficients, linear weights

1989 Sogge: Elliptic operators with variable coefficients, non LCW

2001 Shen: Laplace operator on the torus

2005 Koch-Tataru: construction of parametrices in general context


Remarks

1. In unique continuation problems, one traditionally usesweights for which one has

Re pϕ, Im pϕ > 0 on p−1

ϕ (0), i.e.of the form x1 + x2

1/2.

2. These estimates can be seen as the anisotropic analogue ofthe estimates of Jerison and Kenig (with x1 = s = log r).

3. These estimates can also be seen as the anisotropic analogueof the estimates of Kenig, Ruiz and Sogge (who proved Lp

Carleman estimates for linear weights).

4. There are two proofs of those estimates: the first follows ideasof Jerison (see also Sogge, Shen) in relation with spectralcluster estimates, the second ideas of Kenig, Ruiz and Soggein relation with resolvent estimates

Proof of Carleman estimates Shortcut


Proof of Carleman estimatesMain goal:

‖u‖L

2nn−2 (R×M0)

≤ CI‖f‖L

2nn+2 (R×M0)

when u ∈ C∞0 (I ×M0) and

D2x1u+ 2iτDx1u−∆g0u− τ2u = f

with Dx1 = −i∂x1 .

(D2x1 + 2iτDx1 − τ2 + λj)πju = πjf

Symbol of the operator: ξ21 + 2iτξ1 − τ2 + λj 6= 0 if τ2 6= λj

Inverse operator:

Gτf(x1, x′) =

∞∑j=0

∫ ∞−∞

mτ

(x1 − y1,

√λj)πjf(y1, x

′) dy1

mτ (t, µ) =1

2π

∫ ∞−∞

eitη

η2 + 2iτη − τ2 + µ2dη, µ > 0.


Proof of Carleman estimates

Using the spectral cluster estimates we get

‖u‖L

2nn−2 (M0)

=

∥∥∥∥ ∞∑k=0

χ2ku

∥∥∥∥L

2nn−2 (M0)

.∞∑k=0

(1 + k)12− 1n ‖χku‖L2(M0).

Apply the estimate to u = Gτf(x1, ·),

‖Gτf(x1, ·)‖L

2nn−2 (M0)

.∞∑k=0

(1 + k)12− 1n

×( ∑k≤√λj<k+1

∣∣∣∣ ∫ ∞−∞

mτ

(x1 − y1,

√λj)f(y1, j) dy1

∣∣∣∣2) 12

.


Proof of Carleman estimate

By Minkowski’s inequality, we have

‖Gτf(x1, ·)‖L

2nn−2 (M0)

.∞∑k=0

(1 + k)12− 1n

×∫ ∞−∞

( ∑k≤√λj<k+1

∣∣∣mτ

(x1 − y1,

√λj)f(y1, j)

∣∣∣2) 12

dy1

and since∑k≤√λj<k+1

∣∣∣mτ

(x1 − y1,

√λj)f(y1, j)

∣∣∣2≤ sup

k≤√λj<k+1

∣∣mτ (x1 − y1,√λj)∣∣2 × ∥∥χkf(y1, ·)‖2L2(M0)


Proof of Carleman estimate

using once again the spectral cluster estimate we finally get

‖Gτf(x1, ·)‖L

2nn−2 (M0)

.∞∑k=0

(1 + k)1− 2n

×∫ ∞−∞

supk≤√λj<k+1

∣∣mτ (x1−y1,√λj)∣∣×∥∥f(y1, ·)‖

L2nn+2 (M0)

dy1.

We estimate

supk≤√λj<k+1

∣∣mτ (t,√λj)∣∣ ≤ 1

k

e−(k−τ)|t| when τ < k

1 when k ≤ τ < k + 1

e−(τ−k−1)|t| when τ ≥ k + 1

with k > 0.


Proof of Carleman estimateThis allows us to estimate the series

∞∑k=0

(1 + k)1− 2n supk≤√λj<k+1

∣∣mτ (t,√λj)∣∣ . 1 + |t|−1+ 2

n .

and we obtain

‖Gτf(x1, ·)‖L

2nn−2 (M0)

.∫ ∞−∞

(1 + |x1 − y1|−1+ 2

n

)∥∥f(y1, ·)‖L

2nn+2 (M0)

dy1

.∫ ∞−∞|x1 − y1|−1+ 2

n

∥∥f(y1, ·)‖L

2nn+2 (M0)

dy1

and we conclude using the Hardy-Littlewood-Sobolev inequality

‖Gτf‖L

2nn−2 (I×M0)

.∥∥f‖

L2nn+2 (I×M0)

.


Outline

Introduction

Carleman weights

Carleman estimates



Introduction

It is time now to state some precise uniqueness results. So far wehave seen that:

1. In order to have Carleman estimates with opposite weights,one has to work with limiting Carleman weights. This is inorder to comply with Hormander’s (necessary) criterium ofsolvability for non-selfadjoint operators.

2. On Riemannian manifolds, the existence of LCW is a limitingcondition. It implies that manifolds have to be locallyconformal to a product.

3. For reasons related to the global solvability of the transportequation, we will ask that the manifolds under scope beglobally conformal to a product.

4. In fact, we need more conditions: for reasons related to theglobal solvability of the eikonal eqution, we ask the cutlocusof the manifold to be empty.


Admissible geometries

The former remarks justify the introduction of:

Definition

A compact Riemannian manifold (M, g), with dimension n ≥ 3and with boundary ∂M , is called admissible if M b R×M0 forsome (n− 1)-dimensional simple manifold (M0, g0), and ifg = c(e⊕ g0) where e is the Euclidean metric on R and c is asmooth positive function on M .

Definition

Here, a compact manifold (M0, g0) with boundary is simple if forany p ∈M0 the exponential map expp with its maximal domain ofdefinition is a diffeomorphism onto M0, and if ∂M0 is strictlyconvex (that is, the second fundamental form of ∂M0 →M0 ispositive definite).


Main result (unbounded potentials)

Theorem

Let (M, g) be admissible and let q1, q2 be complex functions inLn/2(M). If Λg,q1 = Λg,q2 , then q1 = q2.

The above theorem was proved in a joint paper with Carlos Kenigand Mikko Salo.

From the point of view of unique continuation, the regularity Ln/2

is optimal. Shortcut


Reminder of the formal WKB expansion

Conjugated operator : Pϕ = eϕ/hh2∆ge−ϕ/h P ∗ϕ = P−ϕ

Principal symbol : pϕ = |ξ|2 − |dϕ|2 + 2i〈ξ, dϕ〉

We have

∆g(e1h

(ϕ+iψ)a) = e1hϕPϕ(e

ihψa) = e

1h

(ϕ+iψ)

(h0pϕ(x,dψ)

+ 2h

[(gradgϕ+ igradgψ)a+

1

2∆g(ϕ+ iψ)a

]+ h2∆ga

).

Eikonal equation: pϕ(x,dψ) = 0

Transport equation: (gradgϕ+ igradgψ)a+1

2∆g(ϕ+ iψ)a = 0


Complex geometrical optics (eikonal equation)

We suppose that the metric has the form

g(x) =

(1 00 g0(x′)

)and we choose ϕ(x) = x1. We have dϕ = dx1 and gradgϕ = ∂x1 .

Eikonal equation:

|dψ|2g = |dϕ|2g = 1

〈dϕ,dψ〉g = ∂x1ψ

We choose ψ to be independent of x1, and solution to

|dψ|g0 = 1.


Complex geometrical optics (eikonal equation)

Eikonal equation: |dψ|2g0 = 1

In simple manifolds, it is easy to give explicit solutions of thiseikonal equation

ψ(x) = dg0(x′, ω0), ω0 ∈ M0 \M0

where dg0 is the geodesical distance (M0 is a simple extension ofM0). In fact, one can use geodesical polar coordinates

x′ = expω0(rθ), r = dg0(x′, ω′0) > 0, θ ∈ Sω′0M0.

In those coordinates, the metric reads

g0 = dr2 + h0(r)


Complex geometrical optics (transport equation)

Transport equation: ∂x1a+ igradgψa+ 12∆g(ϕ+ iψ)a = 0

In polar coordinates

gradg0ψ = ∂r, Lgradg0ψ= ∂r,

∆g0ψ =1

|g0|1/2∂

∂r

(|g0|1/2

∂ψ

∂r

)= 1

2∂r log |g0|

hence the transport equation reads

∂x1a+ i∂ra+ 14∂r log |g0| a = 0

and can easily be solved

a = f(x1 + ir)|g0|−1/4b(θ)

where h is a holomorphic function.


Complex geometrical optics (Lp case)

Proposition

Assume that q ∈ Ln/2(M). Let ω′ ∈ M0 \M0 be a fixed point, letλ ∈ R be fixed, and let b ∈ C∞(Sn−2) be a function. Writex = (x1, r, θ) where (r, θ) are polar normal coordinates with centerω′ in (M0, g0). For h sufficiently small, there exists u0 ∈ H1(M)satisfying

(−∆g + q)u = 0 in M,

u = e±1hx1(e−

ihr|g|−1/4eiλ(x1+ir)b(θ) + r)

where r0 satisfies

h−1‖r‖L2(M) + ‖r‖H1(M) + ‖r‖L

2nn−2 (M)

. 1.

Shortcut


Attenuated X-ray transformThe unit sphere bundle :

SM0 =⋃x∈M0

Sx, Sx =

(x, ξ) ∈ TxM0 ; |ξ|g = 1.

Boundary: ∂(SM0) = (x, ξ) ∈ SM0 ; x ∈ ∂M0 union of inwardand outward pointing vectors:

∂±(SM0) =

(x, ξ) ∈ SM0 ; ±〈ξ, ν〉 ≤ 0.

Denote by t 7→ γ(t, x, ξ) the unit speed geodesic starting at x indirection ξ, and let τ(x, ξ) be the time when this geodesic exitsM0.Godesic ray transform with constant attenuation −λ:

Tλf(x, ξ) =

∫ τ(x,ξ)

0f(γ(t, x, ξ))e−λt dt, (x, ξ) ∈ ∂+(SM0).


Injectivity of the attenuated X-ray transform

Proposition

Let (M0, g0) be a simple manifold. There exists ε > 0 such that ifλ is a real number with |λ| < ε and if f ∈ C∞(M), then thecondition Tλf(x, ξ) = 0 for all (x, ξ) ∈ ∂+(SM0) implies thatf = 0.

This was known when λ = 0. The proof in the attenuated caseuses perturbation arguments.

What about nonsmooth functions ?


Normal operator

Notations: µ(x, ξ) = −〈ξ, ν(x)〉 and dN is the volume form on N .Scalar product:

(h, h)L2µ(∂+(SM0)) =

∫∂+(SM0)

hhµ d(∂(SM0))

Adjoint of the ray transform:

T ∗λh(x) =

∫Sx

e−λτ(x,−ξ)h(ϕ−τ(x,−ξ)(x, ξ)), dSx(ξ), x ∈M0.

where ϕt(x, ξ) = (γ(t, x, ξ), γ(t, x, ξ)) is the geodesic flow.

Lemma

T ∗λTλ is a self-adjoint elliptic pseudodifferential operator of order−1 in M int

0 .


Injectivity of the attenuated X-ray transform (non-smoothcase)

Lemma

Let (M0, g0) be an (n− 1)-dimensional simple manifold, and letf ∈ L1(M0). Consider the integrals∫

Sn−2

∫ τ(ω,θ)

0f(r, θ)e−λrb(θ) dr dθ

where (r, θ) are polar normal coordinates in (M0, g0) centered atsome ω ∈ ∂M0, and τ(ω, θ) is the time when the geodesicr 7→ (r, θ) exits M0. If |λ| is sufficiently small, and if these integralsvanish for all ω ∈ ∂M0 and all b ∈ C∞(Sn−2), then f = 0.

(End)


Using CGOs

Starting point: If q = q1 − q2∫Mqu1u2 dVg = 0

where

u1 = e−1h

(x1+ir)(|g|−1/4eiλ(x1+ir)b(θ) + r1),

u2 = e1h

(x1+ir)(|g|−1/4 + r2).

with b ∈ C∞(Sn−2) and

‖rj‖L

2nn−2 (M)

= O(1), ‖rj‖L2(M) = o(1)

as h→ 0.


Using CGOsNoting that dVg = |g|1/2dx1 dr dθ, we obtain that∫

Mqeiλ(x1+ir)b(θ) dx1 dr dθ =

∫Mq(a1r2 + a2r1 + r1r2) dV

The RHS converges to 0 as h→ 0.Taking the limit, we obtain that∫ ∞

−∞

∫ ∞0

∫Sn−2

q(x1, r, θ)eiλ(x1+ir)b(θ) dx1 dr dθ = 0.

This statement is true for all choices of ω ∈ M0 \M0, for all realnumbers λ, and for all functions b ∈ C∞(Sn−2). Hence∫

Sn−2

∫ ∞0

fλ(r, θ)e−λrb(θ) dr dθ = 0

where fλ ∈ L1(M0) is the function given by

fλ(r, θ) =

∫ ∞−∞

eiλx1q(x1, r, θ) dx1.


End of the proof

If |λ| is sufficiently small, it follows that fλ = 0.Attenuated geodesic transform

Since q( · , r, θ) is a compactly supported function in L1(R) fora.e. (r, θ), the Paley-Wiener theorem shows that q( · , r, θ) = 0 forsuch (r, θ).

Consequently q1 = q2.

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