A local estimate from Radon transform and stability of Inverse EIT with partial data

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1 A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid ge

2 Joint work with P. Caro (U. Helsinki) and D. Dos Santos Ferreira (Paris 13)

3 The purpose of this talk is to address the following points To prove a local control of the L norm of a function by its local Radon transform. Quantitative version of Helgason-Holmgren Thm. (Inspired by: DSF-K-Sj-U, Boman, Hörmander). To use the above to prove stability for Calderón Inverse problem in the context of Bukhgeim and Uhlmann (Recovering a potential from partial Cauchy data 2002)(semiglobal) Kenig, Sjoestrand and Uhlmann (The Calderon Problem for partial data 2007)

4 Previous results. Stability Full data: nd (n > 2) Alessandrini (1988) 2d (low a priori regularity) Liu (1997) B. Barcelo, J.A. Barcelo, R. (2001) B.Barcelo, D. Faraco and R.(2007) A. Clop, D. Faraco and R (discontinuous coefficients in Sobolev spaces)(2009) Semiglobal data: H. Heck and J.N. Wang (2006) L. Tzou (2008)

5 Theorem1 Theorem Consider q A λ for 0 < λ 1 and let y 0 R n and ω 0 S n 1 be such that y 0 supp q and supp q {x R n : (x y 0 ) ω 0 0, x y 0 R q }. Given α > 0 and β (0, 1] consider the sets I = {s R : s < α}, Γ = {ω S n 1 : ω ω 0 2 > 1 β 2 }. If R y0 q L 1 (I Γ) c 1(R q ), there exists a positive constant C such that [ ( ) ] 1 q L (G) C M q β n + Rn q α n + α + L q α λ 2 where G is a neighborhood of y 0. 1 log R y0 q L 1 (I Γ) λ 4

6 Constants and a-priori assumptions A λ : restrictions of λ-hölder functions to compacts R y0 q(s, ω) is the y 0 -centered Radon transform, integral in H = {x R n : ω (x y 0 ) = s} G = {x R n : x y 0 < α 8 cosh (8π/β) } M q := max(1, q L (R n ) ) and L q = inf{c > 0 : q(x) q(y) C x y λ x, y supp q}.

7 Sketch of proof 1 Qualitative case. Assume R y0 q L 1 (I Γ) = 0. Step 1 Microlocal Helgason s theorem: (y 0, ω 0 ) / WF a (q) Step 2 Microlocal Holmgren s theorem: From the conditions y 0 supp q and supp q {x R n : (x y 0 ) ω 0 0} one has (y 0, ω 0 ) WF a (q) The end Hence q = 0.

8 Sketch of proof: Step 1 Qualitative Microlocal Helgason s theorem: The WF a (u) is characterized be the exponential decay of the wave packet transform: For any u S (R n ) define the Wave packet (Segal-Bargmann) transform of u for ζ C n as T u(ζ) = u, e 1 2h (ζ )2. Where, stands for the duality between S (R n ) and S(R n ) the class of smooth rapidly decreasing functions.

9 Step 1:From Radon to wave packet Proposition Consider q A λ (R n ), y 0 R n and ω 0 S n 1. Let R q be a positive constant such that supp q {x R n : x y 0 R q }. Given α > 0 and β (0, 1] consider the sets I = {s R : s < α}, Γ = {ω S n 1 : ω ω 0 2 > 1 β 2 }. Then, there exists a positive constant C such that T q(ζ) C ( Im ζ n + α n + R n q) e 1 Im h n ζ 2 2h 2 [ ] R y0 q L 1 (I Γ) + q L (R n ) (e 1 α 2 2h 4 + e 1 γ 2 β 2 2h 16 ), for all h (0, 1], ζ C n such that Re ζ y 0 < α/2, Im ζ γ > 0 and ω 0 θ 2 > 1 β 2 /4 with θ = Im ζ 1 Im ζ.

10 Step 2 Step 2a (Key step) We tray to remove the condition Im ζ γ > 0. Try to go from the wave packets transform of q to the heat evolution of q (t = h) by allowing Im ζ = 0 Step 2b. Estimate the backward initial value problem for the heat equation under a priori regularity on q.

11 The key Proposition Proposition Under the same notation and assumptions of the Theorem. There exists a positive constant C such that e 1 2h Im ζ 2 T q(ζ) < CM q (( 2α β )n + α n + R n q) R y0 q κ L 1 (I Γ), (0.1) with κ := 1 ( ( )) 2, h := 4 cosh 8π β α 2 8 log R y0 q L 1 (I Γ), for all ζ C n such that Re ζ y 0 < α 8 cosh ( 8π β ), Im ζ < 2α (4 β 2 ) 1/2.

12 Proof of the Key proposition Estimates of the wave packet transform: (K1) (K2) T q(ζ) (2πh) n 2 q L (R n ) e 1 2h Im ζ 2, T q(ζ) (2πh) n 2 q L (R n ) e 1 2h Im ζ 2 e 1 2h ω 0 (Re ζ y 0 ) 2, for all ζ C n such that ω 0 (Re ζ y 0 ) 0 (follows from the supporting plane) (K3) T q(ζ) C h n 2 ( Im ζ n + α n + R n q) e 1 Im ζ 2 2h [ R y0 q L 1 (I Γ) + q L (R n ) e 1 2h ] α 2 4, for all h (0, 1], ζ C n such that Re ζ y 0 < α/2, Im ζ γ > 0 and ω 0 θ 2 > 1 β 2 /4 with θ = Im ζ 1 Im ζ. (Follows from quantitative Helgason-Holmgren with γ = 2α/β)

13 Barrier on a rectangle Lemma Let a, b be positive constants. Consider R = {z C : Re z < a, Im z < b + ε}, for some ε > 0. Let F be a sub-harmonic function in R such that F(z) < (min{0, Re z}) 2, for all z C. Assume that F(z) < 2a 2 for z R such that Im z b. Then ( ) F(z) < 1 2a 2 2 cosh ( 1 π b ) min a cosh ( π b a), 1, 3 for ( ) Im z < b, Re z < a 2 min 1 cosh ( π a), 1 b. 3

14 Proof of lemma Take the subharmonic function for G(x + iy) = 2a 2 cosh ( π a y) ( π ) cosh ( π sin a b) a (x + δ) + F(x + iy) δ 2, ( ) a δ = min cosh ( π b a), a 3 Check that G < 0 on the boundary of [ δ, a δ] [ b, b]. The estimate follows from the maximun principle when we restrict to the values x < δ/2.

15 End of proof of Key step α Choose h := 2, to obtain an appropriate negative 8 log R y0 q L 1 (I Γ) exponential in (K3). Let us denote z = ω 0 (ζ y 0 ) C and write ζ = (z + ω 0 y 0 )ω 0 + w with w C n such that Re w ω 0 = Im w ω 0 = 0. Use the Lemma for the function Φ(z) = Re z 2 Im z 2 + 2h log T q((z + ω 0 y 0 )ω 0 + w) ( ) e 1 Im w 2 2h +2h log CM q (ρ n + α n + R n, q) in the rectangle a = α/4, b = 2α/β

16 Step 2b.The backward estimate Lemma Consider q A λ and G an open set in R n such that supp q G. Then q L (G) C ( h n 2 T q R n L (G) + L qh λ 4 + Mq e 1 4h 1/2 ), where L q = inf{c > 0 : q(x) q(y) C x y λ x, y supp q}.

17 Stability of Calderón inverse problem with partial data The uniqueness Results. Locally vanishing Radon Transform. Stability: Hunk and Wang: Log-log type stability (log stable recovery of the Fourier tranform +log stable continuation of analytic functions) The Carleman estimates with signed partition of the boundary allow to partially recover the 2-plane Radon transform in some sets I Γ which are neighborhood of (y 0, ω 0 ) from the partial Dirichlet to Neuman map in a logarithmic stable way (following the approach of DSF, Kenig, Sjoestrand and Uhlmann). This allows to control the norm of q 1 q 2 in a neighborhood of y 0. In the case of BU ( semiglobal ) one can estimate q 1 q 2 L, since we have the control of R y0 (q 1 q 2 ) L 1 ((0, ) Γ). It remains open for us how to obtain, if it is possible, estimates for the whole q 1 q 2 in the case of KSU.

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