Microlocal analysis and inverse problems Lecture 3 : Carleman estimates


 Julie Norman
 3 months ago
 Views:
Transcription
1 Microlocal analysis and inverse problems ecture 3 : Carleman estimates David Dos Santos Ferreira AGA Université de Paris 13 Monday May 16 Instituto de Ciencias Matemáticas, Madrid David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 1 / 38
2 Outline Introduction 1 Introduction 2 2 Carleman estimates 3 p Carleman estimates 4 Resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 2 / 38
3 Introduction Construction by complex geometrical optics In the former lecture, we saw that for our purpose, that is construction of solutions to the Schrödinger equation by means of complex geometrical optics with opposite exponential behaviours as in Sylvester and Uhlmann, one needs to use limiting Carleman weights. The purpose of this lecture is to indeed prove the corresponding Carleman estimates, both in the 2 setting (which corresponds to bounded potentials) and in the p setting (which corresponds to unbounded potentials). We will use alternatively h or τ = h 1 to denote the semiclassical parameter. Sorry for the change of notations! Recall that denotes the conjugated operator. P ϕ = e τϕ τ 2 2 ge τϕ David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 3 / 38
4 Introduction Construction by complex geometrical optics In the former lecture, we saw that for our purpose, that is construction of solutions to the Schrödinger equation by means of complex geometrical optics with opposite exponential behaviours as in Sylvester and Uhlmann, one needs to use limiting Carleman weights. The purpose of this lecture is to indeed prove the corresponding Carleman estimates, both in the 2 setting (which corresponds to bounded potentials) and in the p setting (which corresponds to unbounded potentials). We will use alternatively h or τ = h 1 to denote the semiclassical parameter. Sorry for the change of notations! Recall that denotes the conjugated operator. P ϕ = e τϕ τ 2 2 ge τϕ David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 3 / 38
5 Introduction Construction by complex geometrical optics In the former lecture, we saw that for our purpose, that is construction of solutions to the Schrödinger equation by means of complex geometrical optics with opposite exponential behaviours as in Sylvester and Uhlmann, one needs to use limiting Carleman weights. The purpose of this lecture is to indeed prove the corresponding Carleman estimates, both in the 2 setting (which corresponds to bounded potentials) and in the p setting (which corresponds to unbounded potentials). We will use alternatively h or τ = h 1 to denote the semiclassical parameter. Sorry for the change of notations! Recall that denotes the conjugated operator. P ϕ = e τϕ τ 2 2 ge τϕ David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 3 / 38
6 2 Carleman estimates Outline 1 Introduction 2 2 Carleman estimates 3 p Carleman estimates 4 Resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 4 / 38
7 2 Carleman estimates Carleman estimates Theorem et (U, g) be an open Riemannian manifold and (M, g) a compact Riemannian submanifold with boundary such that M U. Suppose that ϕ is a limiting Carleman weight on (U, g). et q be a smooth function on M. There exist two constants C > 0 and 0 < h 0 1 such that for all functions u C 0 (M ) and all 0 < h h 0, one has the inequality e ϕ h u H 1 scl (M) Ch e ϕ h ( + q)u 2 (M). We decompose P ϕ into its selfadjoint and skewadjoint parts P ϕ = A+iB, A = h 2 g grad g ϕ 2, B = 2i grad g ϕ, hgrad g ih g ϕ and we have by integration by parts P ϕ v 2 = Av 2 + Bv 2 + i([a, B]v v). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 5 / 38
8 2 Carleman estimates Carleman estimates Theorem et (U, g) be an open Riemannian manifold and (M, g) a compact Riemannian submanifold with boundary such that M U. Suppose that ϕ is a limiting Carleman weight on (U, g). et q be a smooth function on M. There exist two constants C > 0 and 0 < h 0 1 such that for all functions u C 0 (M ) and all 0 < h h 0, one has the inequality e ϕ h u H 1 scl (M) Ch e ϕ h ( + q)u 2 (M). We decompose P ϕ into its selfadjoint and skewadjoint parts P ϕ = A+iB, A = h 2 g grad g ϕ 2, B = 2i grad g ϕ, hgrad g ih g ϕ and we have by integration by parts P ϕ v 2 = Av 2 + Bv 2 + i([a, B]v v). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 5 / 38
9 2 Carleman estimates Convexification Convexification consists in replacing ϕ by ϕ = f ϕ. Add to denote all the corresponding symbols. Note that grad g (f ϕ) = (f ϕ) grad g ϕ 2 (f ϕ) = (f ϕ) dϕ dϕ + (f ϕ) 2 ϕ }{{} =0 therefore {ã, b}(x, ξ) = 4(f ϕ) (f ϕ) 2 grad g ϕ 4 + 4(f ϕ) grad g ϕ, ξ 2 = 4(f ϕ) (f ϕ) 2 + (f ϕ)(f ϕ) 2 }{{} b2. =β David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 6 / 38
10 2 Carleman estimates Convexification Convexification consists in replacing ϕ by ϕ = f ϕ. Add to denote all the corresponding symbols. Note that grad g (f ϕ) = (f ϕ) grad g ϕ 2 (f ϕ) = (f ϕ) dϕ dϕ + (f ϕ) 2 ϕ }{{} =0 therefore {ã, b}(x, ξ) = 4(f ϕ) (f ϕ) 2 grad g ϕ 4 + 4(f ϕ) grad g ϕ, ξ 2 = 4(f ϕ) (f ϕ) 2 + (f ϕ)(f ϕ) 2 }{{} b2. =β David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 6 / 38
11 2 Carleman estimates Convexification Convexification consists in replacing ϕ by ϕ = f ϕ. Add to denote all the corresponding symbols. Note that grad g (f ϕ) = (f ϕ) grad g ϕ 2 (f ϕ) = (f ϕ) dϕ dϕ + (f ϕ) 2 ϕ }{{} =0 therefore {ã, b}(x, ξ) = 4(f ϕ) (f ϕ) 2 grad g ϕ 4 + 4(f ϕ) grad g ϕ, ξ 2 = 4(f ϕ) (f ϕ) 2 + (f ϕ)(f ϕ) 2 }{{} b2. =β David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 6 / 38
12 2 Carleman estimates Convexification At the operator level, this gives i[ã, B] = 4h(f ϕ) (f ϕ) 2 + h Bβ B + h 2 R where R is a first order semiclassical differential operator. For the function f, we choose the following convex polynomial f(s) = s + h 2ε s2, f (s) = 1 + h ε s, f (s) = h ε. We choose h/ε small enough so that f > 1 2 Note that the coefficients of R, as well as β, are uniformly bounded with respect to h and ε. The estimate comes from the fact that the commutator is positive (by taking ε small enough to absorb error terms) and that e 1 h ϕ e 1 h ϕ. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 7 / 38
13 2 Carleman estimates Convexification At the operator level, this gives i[ã, B] = 4h(f ϕ) (f ϕ) 2 + h Bβ B + h 2 R where R is a first order semiclassical differential operator. For the function f, we choose the following convex polynomial f(s) = s + h 2ε s2, f (s) = 1 + h ε s, f (s) = h ε. We choose h/ε small enough so that f > 1 2 Note that the coefficients of R, as well as β, are uniformly bounded with respect to h and ε. The estimate comes from the fact that the commutator is positive (by taking ε small enough to absorb error terms) and that e 1 h ϕ e 1 h ϕ. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 7 / 38
14 p Carleman estimates Outline 1 Introduction 2 2 Carleman estimates 3 p Carleman estimates 4 Resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 8 / 38
15 p Carleman estimates Spectral cluster estimates of Sogge We denote by λ 0 = 0 < λ 1 λ 2... the sequence of eigenvalues of g0 on M 0 and (ψ j ) j 0 the corresponding sequence of eigenfunctions We denote by g0 ψ j = λ j ψ j. π j : 2 (M 0 ) 2 (M 0 ), u (u, ψ j )ψ j the projection on the linear space spanned by the eigenfunction ψ j so that π j = Id, j=0 λ j π j = g0 j=0 and by û(j) = u ψ j dv g0 M 0 the corresponding Fourier coefficients. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 9 / 38
16 p Carleman estimates Spectral cluster estimates of Sogge We denote by λ 0 = 0 < λ 1 λ 2... the sequence of eigenvalues of g0 on M 0 and (ψ j ) j 0 the corresponding sequence of eigenfunctions We denote by g0 ψ j = λ j ψ j. π j : 2 (M 0 ) 2 (M 0 ), u (u, ψ j )ψ j the projection on the linear space spanned by the eigenfunction ψ j so that π j = Id, j=0 λ j π j = g0 j=0 and by û(j) = u ψ j dv g0 M 0 the corresponding Fourier coefficients. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 9 / 38
17 p Carleman estimates Spectral cluster estimates of Sogge We denote by λ 0 = 0 < λ 1 λ 2... the sequence of eigenvalues of g0 on M 0 and (ψ j ) j 0 the corresponding sequence of eigenfunctions We denote by g0 ψ j = λ j ψ j. π j : 2 (M 0 ) 2 (M 0 ), u (u, ψ j )ψ j the projection on the linear space spanned by the eigenfunction ψ j so that π j = Id, j=0 λ j π j = g0 j=0 and by û(j) = u ψ j dv g0 M 0 the corresponding Fourier coefficients. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 9 / 38
18 p Carleman estimates Spectral cluster estimates of Sogge We define the spectral clusters as χ k = k λ j <k+1 π j, k N. Note that these are projection operators, χ 2 k = χ k, and they constitute a decomposition of the identity Id = χ k. k=0 The spectral cluster estimates of Sogge are χ k u 2n n 2 (M 0 ) C(1 + k) n u 2 (M 0 ), χ k u 2 (M 0 ) C(1 + k) n u 2n n+2 (M 0 ). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 10 / 38
19 p Carleman estimates Spectral cluster estimates of Sogge We define the spectral clusters as χ k = k λ j <k+1 π j, k N. Note that these are projection operators, χ 2 k = χ k, and they constitute a decomposition of the identity Id = χ k. k=0 The spectral cluster estimates of Sogge are χ k u 2n n 2 (M 0 ) C(1 + k) n u 2 (M 0 ), χ k u 2 (M 0 ) C(1 + k) n u 2n n+2 (M 0 ). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 10 / 38
20 p Carleman estimates Spectral cluster estimates of Sogge We define the spectral clusters as χ k = k λ j <k+1 π j, k N. Note that these are projection operators, χ 2 k = χ k, and they constitute a decomposition of the identity Id = χ k. k=0 The spectral cluster estimates of Sogge are χ k u 2n n 2 (M 0 ) C(1 + k) n u 2 (M 0 ), χ k u 2 (M 0 ) C(1 + k) n u 2n n+2 (M 0 ). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 10 / 38
21 p Carleman estimates Carleman estimates Theorem et (M 0, g 0 ) be an (n 1)dimensional compact manifold without boundary, and equip R M 0 with the metric g = e g 0 where e is the Euclidean metric. The Euclidean coordinate is denoted by x 1. For any compact interval I R there exists a constant C I > 0 such that if τ 4 and τ 2 / Spec( g0 ) then we have when u C 0 (I M 0). e τx 1 u 2n n 2 (R M 0 ) C I e τx 1 g u 2n n+2 (R M 0 ) David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 11 / 38
22 p Carleman estimates Short bibliography on p Carleman estimates on elliptic operators These works are in relation with unique continuation of solutions to Schrödinger equation with unbounded potentials JerisonKenig: first p Carleman estimates, logarithmic weights 1986 Jerison: simplification of the proof using spectral cluster estimates (see also Sogge s book) 1987 KenigRuizSogge: Elliptic operators with constant coefficients, linear weights 1989 Sogge: Elliptic operators with variable coefficients, non CW 2001 Shen: aplace operator on the torus 2005 KochTataru: construction of parametrices in general context David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 12 / 38
23 p Carleman estimates Remarks 1 In unique continuation problems, one traditionally uses weights for which one has { Re p ϕ, Im p ϕ } > 0 on p 1 ϕ (0), i.e. of the form x 1 + x 2 1 /2. 2 These estimates can be seen as the anisotropic analogue of the estimates of Jerison and Kenig (with x 1 = s = log r see ecture 1). 3 These estimates can also be seen as the anisotropic analogue of the estimates of Kenig, Ruiz and Sogge (who proved p Carleman estimates for linear weights). 4 There are two proofs of those estimates: the first follows ideas of Jerison (see also Sogge, Shen) in relation with spectral cluster estimates, the second ideas of Kenig, Ruiz and Sogge in relation with resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 13 / 38
24 p Carleman estimates Remarks 1 In unique continuation problems, one traditionally uses weights for which one has { Re p ϕ, Im p ϕ } > 0 on p 1 ϕ (0), i.e. of the form x 1 + x 2 1 /2. 2 These estimates can be seen as the anisotropic analogue of the estimates of Jerison and Kenig (with x 1 = s = log r see ecture 1). 3 These estimates can also be seen as the anisotropic analogue of the estimates of Kenig, Ruiz and Sogge (who proved p Carleman estimates for linear weights). 4 There are two proofs of those estimates: the first follows ideas of Jerison (see also Sogge, Shen) in relation with spectral cluster estimates, the second ideas of Kenig, Ruiz and Sogge in relation with resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 13 / 38
25 p Carleman estimates Remarks 1 In unique continuation problems, one traditionally uses weights for which one has { Re p ϕ, Im p ϕ } > 0 on p 1 ϕ (0), i.e. of the form x 1 + x 2 1 /2. 2 These estimates can be seen as the anisotropic analogue of the estimates of Jerison and Kenig (with x 1 = s = log r see ecture 1). 3 These estimates can also be seen as the anisotropic analogue of the estimates of Kenig, Ruiz and Sogge (who proved p Carleman estimates for linear weights). 4 There are two proofs of those estimates: the first follows ideas of Jerison (see also Sogge, Shen) in relation with spectral cluster estimates, the second ideas of Kenig, Ruiz and Sogge in relation with resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 13 / 38
26 p Carleman estimates Remarks 1 In unique continuation problems, one traditionally uses weights for which one has { Re p ϕ, Im p ϕ } > 0 on p 1 ϕ (0), i.e. of the form x 1 + x 2 1 /2. 2 These estimates can be seen as the anisotropic analogue of the estimates of Jerison and Kenig (with x 1 = s = log r see ecture 1). 3 These estimates can also be seen as the anisotropic analogue of the estimates of Kenig, Ruiz and Sogge (who proved p Carleman estimates for linear weights). 4 There are two proofs of those estimates: the first follows ideas of Jerison (see also Sogge, Shen) in relation with spectral cluster estimates, the second ideas of Kenig, Ruiz and Sogge in relation with resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 13 / 38
27 p Carleman estimates Remarks 1 In unique continuation problems, one traditionally uses weights for which one has { Re p ϕ, Im p ϕ } > 0 on p 1 ϕ (0), i.e. of the form x 1 + x 2 1 /2. 2 These estimates can be seen as the anisotropic analogue of the estimates of Jerison and Kenig (with x 1 = s = log r see ecture 1). 3 These estimates can also be seen as the anisotropic analogue of the estimates of Kenig, Ruiz and Sogge (who proved p Carleman estimates for linear weights). 4 There are two proofs of those estimates: the first follows ideas of Jerison (see also Sogge, Shen) in relation with spectral cluster estimates, the second ideas of Kenig, Ruiz and Sogge in relation with resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 13 / 38
28 p Carleman estimates Proof of Carleman estimates Main goal: when u C 0 (I M 0) and with D x1 = i x1. u 2n n 2 (R M 0 ) C I f 2n n+2 (R M 0 ) D 2 x 1 u + 2iτD x1 u g0 u τ 2 u = f (D 2 x 1 + 2iτD x1 τ 2 + λ j )π j u = π j f Symbol of the operator: ξ iτξ 1 τ 2 + λ j 0 if τ 2 λ j Inverse operator: G τ f(x 1, x ( ) = m τ x1 y 1, ) λ j πj f(y 1, x ) dy 1 m τ (t, µ) = 1 2π j=0 e itη η 2 + 2iτη τ 2 dη, µ > 0. + µ 2 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 14 / 38
29 p Carleman estimates Proof of Carleman estimates Main goal: when u C 0 (I M 0) and with D x1 = i x1. u 2n n 2 (R M 0 ) C I f 2n n+2 (R M 0 ) D 2 x 1 u + 2iτD x1 u g0 u τ 2 u = f (D 2 x 1 + 2iτD x1 τ 2 + λ j )π j u = π j f Symbol of the operator: ξ iτξ 1 τ 2 + λ j 0 if τ 2 λ j Inverse operator: G τ f(x 1, x ( ) = m τ x1 y 1, ) λ j πj f(y 1, x ) dy 1 m τ (t, µ) = 1 2π j=0 e itη η 2 + 2iτη τ 2 dη, µ > 0. + µ 2 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 14 / 38
30 p Carleman estimates Proof of Carleman estimates Main goal: when u C 0 (I M 0) and with D x1 = i x1. u 2n n 2 (R M 0 ) C I f 2n n+2 (R M 0 ) D 2 x 1 u + 2iτD x1 u g0 u τ 2 u = f (D 2 x 1 + 2iτD x1 τ 2 + λ j )π j u = π j f Symbol of the operator: ξ iτξ 1 τ 2 + λ j 0 if τ 2 λ j Inverse operator: G τ f(x 1, x ( ) = m τ x1 y 1, ) λ j πj f(y 1, x ) dy 1 m τ (t, µ) = 1 2π j=0 e itη η 2 + 2iτη τ 2 dη, µ > 0. + µ 2 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 14 / 38
31 p Carleman estimates Proof of Carleman estimates Main goal: when u C 0 (I M 0) and with D x1 = i x1. u 2n n 2 (R M 0 ) C I f 2n n+2 (R M 0 ) D 2 x 1 u + 2iτD x1 u g0 u τ 2 u = f (D 2 x 1 + 2iτD x1 τ 2 + λ j )π j u = π j f Symbol of the operator: ξ iτξ 1 τ 2 + λ j 0 if τ 2 λ j Inverse operator: G τ f(x 1, x ( ) = m τ x1 y 1, ) λ j πj f(y 1, x ) dy 1 m τ (t, µ) = 1 2π j=0 e itη η 2 + 2iτη τ 2 dη, µ > 0. + µ 2 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 14 / 38
32 p Carleman estimates Proof of Carleman estimates emma If τ > 0, µ > 0, τ µ and t R then m τ (t, µ) 1 µ e τ µ t, m τ (t, 0) t e τ t. Proof. This follows by writing 1 (iη (τ + µ))(iη (τ µ)) = 1 2µ [ ] 1 iη (τ + µ) 1 iη (τ µ) and by noting that for α > 0 F 1 η { 1 iη + α } { 0, t < 0 (t) = e αt, t > 0,. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 15 / 38
33 p Carleman estimates Proof of Carleman estimates emma If τ > 0, µ > 0, τ µ and t R then m τ (t, µ) 1 µ e τ µ t, m τ (t, 0) t e τ t. Proof. This follows by writing 1 (iη (τ + µ))(iη (τ µ)) = 1 2µ [ ] 1 iη (τ + µ) 1 iη (τ µ) and by noting that for α > 0 F 1 η { 1 iη + α } { 0, t < 0 (t) = e αt, t > 0,. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 15 / 38
34 p Carleman estimates Proof of Carleman estimates Using the spectral cluster estimates we get u 2n = χ 2 n 2 k u (M 0 ) 2n n 2 k=0 (M 0 ) (1 + k) n χk u 2 (M 0 ). k=0 Apply the estimate to u = G τ f(x 1, ), G τ f(x 1, ) 2n (1 + k) n n 2 (M 0 ) ( k λ j <k+1 k=0 m τ ( x1 y 1, λ j ) f(y1, j) dy 1 2) 1 2. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 16 / 38
35 p Carleman estimates Proof of Carleman estimates Using the spectral cluster estimates we get u 2n = χ 2 n 2 k u (M 0 ) 2n n 2 k=0 (M 0 ) (1 + k) n χk u 2 (M 0 ). k=0 Apply the estimate to u = G τ f(x 1, ), G τ f(x 1, ) 2n (1 + k) n n 2 (M 0 ) ( k λ j <k+1 k=0 m τ ( x1 y 1, λ j ) f(y1, j) dy 1 2) 1 2. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 16 / 38
36 p Carleman estimates Proof of Carleman estimate By Minkowski s inequality, we have G τ f(x 1, ) 2n (1 + k) n n 2 (M 0 ) and since k λ j <k+1 ( k=0 k λ j <k+1 ( m τ x1 y 1, ) λ j f(y1, j) ( m τ x1 y 1, ) λ j f(y1, j) 2) 1 2 dy1 sup mτ (x 1 y 1, ) λ j 2 χk f(y 1, ) 2 2 (M 0 ) k λ j <k+1 2 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 17 / 38
37 p Carleman estimates Proof of Carleman estimate using once again the spectral cluster estimate we finally get G τ f(x 1, ) 2n (1 + k) 1 2 n n 2 (M 0 ) k=0 sup m τ (x 1 y 1, ) λ j f(y 1, ) 2n dy 1. n+2 (M 0 ) k λ j <k+1 Using the emma, we estimate sup m τ (t, ) λ j 1 k k λ j <k+1 with k > 0. e (k τ) t when τ < k 1 when k τ < k + 1 e (τ k 1) t when τ k + 1 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 18 / 38
38 p Carleman estimates Proof of Carleman estimate using once again the spectral cluster estimate we finally get G τ f(x 1, ) 2n (1 + k) 1 2 n n 2 (M 0 ) k=0 sup m τ (x 1 y 1, ) λ j f(y 1, ) 2n dy 1. n+2 (M 0 ) k λ j <k+1 Using the emma, we estimate sup m τ (t, ) λ j 1 k k λ j <k+1 with k > 0. e (k τ) t when τ < k 1 when k τ < k + 1 e (τ k 1) t when τ k + 1 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 18 / 38
39 p Carleman estimates Proof of Carleman estimate This allows us to estimate the series (1 + k) 1 2 n sup k=0 Whence 1 k τ 2 τ 2 0 k λ j <k+1 m τ (t, λ j ) k 2 n e (τ k 1) t + τ 2 n + r 2 n e (τ r 2) t dr (1 + k) 1 2 n sup k=0 k λ j <k+1 k>τ+1 τ k 2 n e (k τ) t + e (τ/2) t r 2 n e (r τ) t dr. m τ (t, λ j ) 1 + t 1+ 2 n. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 19 / 38
40 p Carleman estimates Proof of Carleman estimate This allows us to estimate the series (1 + k) 1 2 n sup k=0 Whence 1 k τ 2 τ 2 0 k λ j <k+1 m τ (t, λ j ) k 2 n e (τ k 1) t + τ 2 n + r 2 n e (τ r 2) t dr (1 + k) 1 2 n sup k=0 k λ j <k+1 k>τ+1 τ k 2 n e (k τ) t + e (τ/2) t r 2 n e (r τ) t dr. m τ (t, λ j ) 1 + t 1+ 2 n. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 19 / 38
41 p Carleman estimates Proof of Carleman estimate We obtain G τ f(x 1, ) 2n n 2 (M 0 ) ( 1 + x 1 y n ) f(y1, ) 2n n+2 (M 0 ) dy 1 x 1 y n f(y1, ) 2n dy 1 + I n f 2n n+2 (M 0 ) n+2 (I M 0 ) and we conclude using the HardyittlewoodSobolev inequality G τ f 2n n 2 (I M 0 ) f 2n n+2 (I M 0 ). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 20 / 38
42 p Carleman estimates Proof of Carleman estimate We obtain G τ f(x 1, ) 2n n 2 (M 0 ) ( 1 + x 1 y n ) f(y1, ) 2n n+2 (M 0 ) dy 1 x 1 y n f(y1, ) 2n dy 1 + I n f 2n n+2 (M 0 ) n+2 (I M 0 ) and we conclude using the HardyittlewoodSobolev inequality G τ f 2n n 2 (I M 0 ) f 2n n+2 (I M 0 ). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 20 / 38
43 Outline Resolvent estimates 1 Introduction 2 2 Carleman estimates 3 p Carleman estimates 4 Resolvent estimates David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 21 / 38
44 Resolvent estimates Relation Carleman estimates / Resolvent estimates Here we follow Kenig, Ruiz and Sogge and relate our Carleman estimates in the product context to resolvent estimates by freezing derivatives. Inspired by Hähner s proof, we further conjugate the operator by an harmless oscillating factor e τx 1 i 2 x 1 P e τx 1+ i 2 x 1 = ( D x ( + 2iτ D x1 + 2) 1 ) τ 2 g0. 2 After translation and scaling I = [0, 2π] and use Fourier series. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 22 / 38
45 Resolvent estimates Relation Carleman estimates / Resolvent estimates Here we follow Kenig, Ruiz and Sogge and relate our Carleman estimates in the product context to resolvent estimates by freezing derivatives. Inspired by Hähner s proof, we further conjugate the operator by an harmless oscillating factor e τx 1 i 2 x 1 P e τx 1+ i 2 x 1 = ( D x ( + 2iτ D x1 + 2) 1 ) τ 2 g0. 2 After translation and scaling I = [0, 2π] and use Fourier series. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 22 / 38
46 Resolvent estimates KenigRuizSogge approach We denote by λ 0 = 0 < λ 1 λ 2... the sequence of eigenvalues of g0 on M 0 and (ψ k ) k 0 the corresponding sequence of eigenfunctions g0 ψ k = λ k ψ k. We denote by π k : 2 (M 0 ) 2 (M 0 ) the projection on the linear space spanned by the eigenfunction ψ k so that π k = Id, k=0 λ k π k = g0. k=0 Eigenvalues of the aplacian g : (j 2 + λ k ) Eigenfunctions: e ijx 1 ψ k. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 23 / 38
47 Resolvent estimates KenigRuizSogge approach We denote by λ 0 = 0 < λ 1 λ 2... the sequence of eigenvalues of g0 on M 0 and (ψ k ) k 0 the corresponding sequence of eigenfunctions g0 ψ k = λ k ψ k. We denote by π k : 2 (M 0 ) 2 (M 0 ) the projection on the linear space spanned by the eigenfunction ψ k so that π k = Id, k=0 λ k π k = g0. k=0 Eigenvalues of the aplacian g : (j 2 + λ k ) Eigenfunctions: e ijx 1 ψ k. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 23 / 38
48 Resolvent estimates KenigRuizSogge approach Denote by π j,k : 2 (M) 2 (M) the projection on the linear space spanned by e ijx 1 ψ k : ( 2π ) π j,k f(x) = e ijy 1 π k f(y 1, x ) dy 1 e ijx 1, and define the spectral clusters as χ m = 0 m j 2 +λ k <m+1 Note that these are projectors χ 2 m = χ m. Spectral cluster estimates of Sogge: π j,k, m N. χ m u 2n n 2 (M) C(1 + m) 1 2 u 2 (M) χ m u 2 (M) C(1 + m) 1 2 u 2n n+2 (M). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 24 / 38
49 Resolvent estimates KenigRuizSogge approach Denote by π j,k : 2 (M) 2 (M) the projection on the linear space spanned by e ijx 1 ψ k : ( 2π ) π j,k f(x) = e ijy 1 π k f(y 1, x ) dy 1 e ijx 1, and define the spectral clusters as χ m = 0 m j 2 +λ k <m+1 Note that these are projectors χ 2 m = χ m. Spectral cluster estimates of Sogge: π j,k, m N. χ m u 2n n 2 (M) C(1 + m) 1 2 u 2 (M) χ m u 2 (M) C(1 + m) 1 2 u 2n n+2 (M). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 24 / 38
50 Resolvent estimates KenigRuizSogge approach We are now ready to reduce the proof of Carleman estimates to resolvent estimates. u 2n n 2 (M) C f 2n n+2 (M) when ( D x1 + 2) 1 2 ( u + 2iτ D x1 + 1 ) u g0 u τ 2 u = f. 2 Inverse operator: G τ f = j= k=0 ( j π j,k f ) 2 ( + 2i j ) τ + λk τ 2. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 25 / 38
51 Resolvent estimates KenigRuizSogge approach We are now ready to reduce the proof of Carleman estimates to resolvent estimates. u 2n n 2 (M) C f 2n n+2 (M) when ( D x1 + 2) 1 2 ( u + 2iτ D x1 + 1 ) u g0 u τ 2 u = f. 2 Inverse operator: G τ f = j= k=0 ( j π j,k f ) 2 ( + 2i j ) τ + λk τ 2. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 25 / 38
52 Resolvent estimates KenigRuizSogge approach Use ittlewoodpaley theory to localize in frequency with respect to the Euclidean variable x 1 ; u = u ν, f = with ν=0 ( 2π u 0 = u(y 1, x ) dy 1 ), u ν = 0 2 ν 1 j <2 ν and similarly for f. It suffices to prove ν=0 ( 2π ) e ijy 1 u(y 1, x ) dy 1 e ijx 1, ν > 0 0 u ν 2n n 2 f ν C f ν 2n n+2. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 26 / 38
53 Resolvent estimates KenigRuizSogge approach Use ittlewoodpaley theory to localize in frequency with respect to the Euclidean variable x 1 ; u = u ν, f = with ν=0 ( 2π u 0 = u(y 1, x ) dy 1 ), u ν = 0 2 ν 1 j <2 ν and similarly for f. It suffices to prove ν=0 ( 2π ) e ijy 1 u(y 1, x ) dy 1 e ijx 1, ν > 0 0 u ν 2n n 2 f ν C f ν 2n n+2. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 26 / 38
54 Resolvent estimates KenigRuizSogge approach The conjugated operator and the localization in frequency commute u ν = G τ f ν. We denote R(z) = (( D x1 + 1 ) 2 ) 1 g0 2 + z the resolvent. The error made by replacing G τ with the resolvent is ( R( τ 2 + i(2 ν + 1)τ) G τ ) fν = a ν jk (τ) = j= k=1 a ν jk (τ) π j,kf ν iτ(2 ν 2j)1 [2 ν 1,2 ν )(j) ( j 2 + 2iτ j τ 2 + λ k )( j 2 + i(2 ν + 1)τ τ 2 + λ k ). with j = j David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 27 / 38
55 Resolvent estimates KenigRuizSogge approach The conjugated operator and the localization in frequency commute u ν = G τ f ν. We denote R(z) = (( D x1 + 1 ) 2 ) 1 g0 2 + z the resolvent. The error made by replacing G τ with the resolvent is ( R( τ 2 + i(2 ν + 1)τ) G τ ) fν = a ν jk (τ) = j= k=1 a ν jk (τ) π j,kf ν iτ(2 ν 2j)1 [2 ν 1,2 ν )(j) ( j 2 + 2iτ j τ 2 + λ k )( j 2 + i(2 ν + 1)τ τ 2 + λ k ). with j = j David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 27 / 38
56 Resolvent estimates KenigRuizSogge approach Using the spectral cluster estimates ( R( τ 2 + i2 ν τ) G ) τ fν n 2 2n (M) ( (1 + m) 1 2 χ m R( τ 2 + i(2 ν + 1)τ) G ) τ fν 2 (M) m=0 (1 + m) 1 2 sup m=0 and furthermore m=0 m j 2 +λ k <m+1 m j 2 +λ k <m+1 a ν jk (τ) χ m f ν 2 2 (M) ( R( τ 2 + i(2 ν + 1)τ) G ) τ fν n 2 2n (M) ( (1 + m) sup a ν jk (τ) ) f ν 2n n+2 (M). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 28 / 38
57 Resolvent estimates KenigRuizSogge approach Using the spectral cluster estimates ( R( τ 2 + i2 ν τ) G ) τ fν n 2 2n (M) ( (1 + m) 1 2 χ m R( τ 2 + i(2 ν + 1)τ) G ) τ fν 2 (M) m=0 (1 + m) 1 2 sup m=0 and furthermore m=0 m j 2 +λ k <m+1 m j 2 +λ k <m+1 a ν jk (τ) χ m f ν 2 2 (M) ( R( τ 2 + i(2 ν + 1)τ) G ) τ fν n 2 2n (M) ( (1 + m) sup a ν jk (τ) ) f ν 2n n+2 (M). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 28 / 38
58 Resolvent estimates KenigRuizSogge approach The above series converge and is uniformly bounded with respect to τ and ν sup a ν jk (τ) 2 ν τ (m 2 τ 2 ) ν+1 τ 2 as well as m=0 m j 2 +λ k <m+1 2 ν τ (1 + m) (m 2 τ 2 ) ν+1 τ 2 2 ν τ t 0 (t 2 τ 2 ) ν+1 τ 2 dt and if we perform the change of variables s = 4 ν 1 τ 2 (t 2 τ 2 ) in the righthand side integral, we obtain the bound m=0 2 ν τ (1 + m) (m 2 τ 2 ) ν+1 τ 2 ds s David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 29 / 38
59 Resolvent estimates KenigRuizSogge approach Summing up our computations, we have the error estimate ( R( τ 2 + i(2 ν + 1)τ) G τ ) fν 2n n 2 (M) f ν 2n n+2 (M), this means that it is enough to prove the resolvent estimate R( τ 2 + i(2 ν + 1)τ)f ν n 2 2n f ν 2n. (M) n+2 (M) Carleman estimates reduce to resolvent estimates of the form u 2n ((D n ) 2 ) g0 (M) 2 + z u 2n with z = τ 2 + iϱτ (ϱ 1). n+2 (M) David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 30 / 38
60 Resolvent estimates KenigRuizSogge approach Summing up our computations, we have the error estimate ( R( τ 2 + i(2 ν + 1)τ) G τ ) fν 2n n 2 (M) f ν 2n n+2 (M), this means that it is enough to prove the resolvent estimate R( τ 2 + i(2 ν + 1)τ)f ν n 2 2n f ν 2n. (M) n+2 (M) Carleman estimates reduce to resolvent estimates of the form u 2n ((D n ) 2 ) g0 (M) 2 + z u 2n with z = τ 2 + iϱτ (ϱ 1). n+2 (M) David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 30 / 38
61 Resolvent estimates Resolvent estimates Theorem et (M, g) be a compact Riemannian manifold (without boundary) of dimension n 3, and let δ be a positive number. There exists a constant C > 0 such that for all u C (M) and all z C : Re z + z δ the following resolvent estimate holds u 2n n 2 (M) C ( g z)u 2n n+2 (M). David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 31 / 38
62 Hadamard s parametrix Resolvent estimates Fundamental solution F 0 of the flat aplacian + z on R n : For a radial function F 0 ( x, z) = (2π) n R n g f(r) = f (r) dr 2 g + f (r) g r with dv g = r n 1 J(r, θ) dr dθ. Hence e ix ξ ξ 2 + z dξ, = f (r) + n 1 f (r) + rj r J f (r). ( g + z)f 0 = δ 0 2 rj 2J rf 0 }{{} error term David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 32 / 38
63 Hadamard s parametrix Resolvent estimates Fundamental solution F 0 of the flat aplacian + z on R n : For a radial function F 0 ( x, z) = (2π) n R n g f(r) = f (r) dr 2 g + f (r) g r with dv g = r n 1 J(r, θ) dr dθ. Hence e ix ξ ξ 2 + z dξ, = f (r) + n 1 f (r) + rj r J f (r). ( g + z)f 0 = δ 0 2 rj 2J rf 0 }{{} error term David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 32 / 38
64 Hadamard s parametrix Resolvent estimates Fundamental solution F 0 of the flat aplacian + z on R n : For a radial function F 0 ( x, z) = (2π) n R n g f(r) = f (r) dr 2 g + f (r) g r with dv g = r n 1 J(r, θ) dr dθ. Hence e ix ξ ξ 2 + z dξ, = f (r) + n 1 f (r) + rj r J f (r). ( g + z)f 0 = δ 0 2 rj 2J rf 0 }{{} error term David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 32 / 38
65 Hadamard s parametrix Resolvent estimates We now take r = d g (x, y). The Hadamard parametrix looks like ( T Had (z)u = χ(x, y)f 0 dg (x, y), z ) u(y) dv g (y) with χ a localizing function and one has M ( g + z)t Had (z)u = χ(x, x)u + S(z)u where S(z) is an error term. In fact, the construction has to be slightly refined. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 33 / 38
66 Hadamard s parametrix Resolvent estimates We now take r = d g (x, y). The Hadamard parametrix looks like ( T Had (z)u = χ(x, y)f 0 dg (x, y), z ) u(y) dv g (y) with χ a localizing function and one has M ( g + z)t Had (z)u = χ(x, x)u + S(z)u where S(z) is an error term. In fact, the construction has to be slightly refined. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 33 / 38
67 Hadamard s parametrix Resolvent estimates We now take r = d g (x, y). The Hadamard parametrix looks like ( T Had (z)u = χ(x, y)f 0 dg (x, y), z ) u(y) dv g (y) with χ a localizing function and one has M ( g + z)t Had (z)u = χ(x, x)u + S(z)u where S(z) is an error term. In fact, the construction has to be slightly refined. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 33 / 38
68 Hadamard s parametrix Resolvent estimates Theorem The Hadamard parametrix is a bounded operator z s 2 THad (z) : p (M) q (M) with a norm uniform with respect to the spectral parameter z C, z 1 when s 2, q p 2 1 p 1 q + s n = 2 n, and ( 1 min p 1 2, 1 q 1 ) 2 > 1 2n, 1 p 1 q < 1 n 1. David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 34 / 38
69 Hadamard s parametrix Resolvent estimates Theorem The Hadamard parametrix is a bounded operator when 1 p 2 q and with a norm bounded by T Had (z) : p (M) q (M) 1 p 1 q < 1 n 1, n 1 n + 1 T Had (z) ( p, q ) C z n p 1 q n + 1 n 1 ( ) 1 p 1 1 q 2. 1 p, David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 35 / 38
70 Resolvent estimates Hadamard s parametrix: admissible exponents 1 q = 1 1 = 1 q p q p F E E D C C 1 q = 1 p 2 n B n B A n 1 p David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 36 / 38
71 Resolvent estimates Decay of Hadamard s parametrix The combination of the two Theorems gives the following bound on the parametrix T Had (z)u ( p, q ) C z σ where the order σ is a piecewise linear function of δ = 1/p 1/q n 1 δ + 1 when δ 2 σ = 4 2 n + 1 n 2 δ + 1 when 2. n + 1 < 2 1 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 37 / 38
72 Resolvent estimates Decay of the parametrix: T Had (z)u (p, q ) = O( z σ ) σ 1 1 n 1 2 n n+1 1 p 1 q 1 n 2 1 n+1 2 n+1 David Dos Santos Ferreira (AGA) Inverse Problems 3 ICMAT 38 / 38