A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid ge
Joint work with P. Caro (U. Helsinki) and D. Dos Santos Ferreira (Paris 13)
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)
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)
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
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}.
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.
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.
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.
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.
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.
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)
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
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.
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
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}.
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.