On L p resolvent and Carleman estimates on compacts manifolds

Transcription

1 On L p resolvent and Carleman estimates on compacts manifolds David Dos Santos Ferreira Institut Élie Cartan Université de Lorraine Three days on Analysis and PDEs ICMAT, Tuesday June 3rd

2 Collaborators Carlos Kenig (University of Chicago) Mikko Salo (University of Jyväskylä) (for the elliptic part) and Camille Laurent (Universtié Pierre et Marie Curie) (for the evolution part).

3 Outline Introduction Resolvent estimates for Laplace-Beltrami operators Relation with Carleman estimates Resolvent estimates for the Schrödinger equation

4 Outline Introduction Resolvent estimates for Laplace-Beltrami operators Relation with Carleman estimates Resolvent estimates for the Schrödinger equation

5 The Kenig-Ruiz-Sogge estimates In an article of 987, Kenig, Ruiz and Sogge proved resolvent estimates of the form ( z) u L 2n n 2 (R n ) C u L 2n n+2 (R n ) in dimension n 3, uniform in the spectral parameter z, for all values of z C \ R. As a consequence, they derived Carleman estimates with linear weights e τ x,ω u L 2n n 2 (R n ) C eτ x,ω u L 2n n+2 (R n ) and proved unique continuation across hyperplanes.

6 Resolvent estimates by Kenig-Sogge In an note of 988, Kenig and Sogge used similar ideas to prove Carleman estimates with linear weights for the (dynamical) Schrödinger equation e τ (t,x),ω u L 2(n+2) n (R n+ ) C eτ (t,x),ω (i t + )u L 2(n+2) n+4 (R n+ ) for n by reducing to an uniform resolvent estimate of a similar form u L 2(n+2) n (R n+ ) C (i t + + z)u L 2(n+2) n+4 (R n+ ) for all values of the spectral parameter z C.

7 Remarks on the proofs The reduction from Carleman estimates to resolvent estimates in both cases is done by freezing the derivative in the direction ω given by the weight after frequency localization in dyadic annuli thanks to a Littlewood-Paley decomposition. The proof of resolvent estimates is based on a parametrix construction involving Bessel functions and the study of its boundedness properties on Lebesgue spaces by the theory of oscillatory integral operators.

8 Our setting We want to investigate similar resolvent estimates which are uniform in the spectral parameter z C u L p (M) C ( g + z)u L p (M) u L p ([0,T ] M) C (i t + g + z)u L p ([0,T ] M) in a compact Riemannian manifold (M, g) without boundary.

9 Our setting Our interest for such estimates lies in the derivation of Carleman estimates with limiting weights of the form e τx u L p (M) C e τx g u L p (M) e τx u L p ([0,T ] M) C e τx (i t + g )u L p ([0,T ] M) in the case where the metric g presents a (warped) product structure g = dx 2 + g 0. The freezing procedure of Kenig, Ruiz and Sogge works similarly in that context.

10 Main motivations In the context of elliptic equations, our motivation is the construction of solutions the (static) Schrödinger equation with an unbounded potential through complex geometrical optics and the resolution of elliptic inverse problems with irregular potentials in certain admissible geometries. In the context of evolution equations, we are more motivated by unique continuation results in product manifolds and the application to control theory in the context where the geometrical control condition does not hold. There is also some interest in the study of product type manifolds with Euclidean factors (cf. work of Visciglia and Tzvetkov).

11 Outline Introduction Resolvent estimates for Laplace-Beltrami operators Relation with Carleman estimates Resolvent estimates for the Schrödinger equation

12 Setting Let (M, g) be a compact Riemannian manifold (without boundary) of dimension n 3. We consider the Laplace-Beltrami operator on (M, g) given in local coordinates by g = det g xj ( det g g jk xk ). This is a negative operator with discrete spectrum σ( g ) = { τ k, k N }, and we are interested in the boundedness properties of the resolvent on Lebesgue spaces. R(z) = ( g z), z C \ σ( g )

13 Preliminary remarks We are interested in proving estimates of the form u L 2n n 2 (M) C ( g + z)u L 2n n+2 (M) with a constant C uniform in the spectral parameter z C.. Taking u to be an eignefunction, it is clear that the spectral parameter z has to be away from the spectrum 2. In fact, we obtained those estimates, when z is outside of a parabola along the negative axis Parabola 3. These estimates don t seem to be a direct consequence of the spectral cluster estimates of Sogge 4. We got interested in those estimates because of the relation with Carleman estimates with limiting weights. Theorem

14 Admissible values of the spectral parameter Figure: Allowed values of the spectral parameter z Ξ δ = { z C \ R : Re z δ } = { z C \ R : (Im z) 2 4δ 2 (δ 2 Re z) }. Back to remarks

15 Resolvent estimates Theorem Let (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 Ξ δ = { z C \ R : Re z δ } the following resolvent estimate holds u L 2n n 2 (M) C ( g z)u L 2n n+2 (M). Joint work with Mikko Salo and Carlos Kenig

16 Short bibliography on L p spectral estimates on elliptic operators 97 Carleson-Sjölin: L p theory of oscillatory integral operators (2D), generalized by Stein (multi D), 987 Kenig-Ruiz-Sogge: Elliptic (in fact quadratic) operators with constant coefficients, 988 Sogge: Spectral cluster estimates for elliptic operators with variable coefficients, 200 Shen: Laplace operator on the torus 20 DSF, Kenig, Salo: compact manifolds 202 Bourgain, Shao, Sogge, Yao : sharpness of the estimates and improvement in certain cases 202 Guillarmou, Hassell: noncompact non-trapping manifolds 203 Krupchyk, Uhlmann: higher order elliptic operators

17 Hadamard s parametrix Fundamental solution F 0 of the flat Laplacian + 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 e ix ξ ξ 2 + z dξ, = f (r) + n f (r) + rj r J f (r). with dv g = r n J(r, θ) dr dθ. Hence ( g + z)f 0 = δ 0 2 rj 2J rf 0 }{{} error term

18 Hadamard s parametrix 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.

19 Bessel functions F 0 can be explicitly computed in terms of Bessel functions F 0 (r, z) = cr n 2 + z n 4 2 Kn/2 ( zr ) where K n/2 is a Bessel potential K n/2 (w) = 0 (( ) ) n e w cosh t cosh 2 t dt, Re w > 0. The qualitative properties of K n/2 are as follows K n/2 (w) C w n/2+ when w and Re w > 0, K n/2 (w) = a(w)w 2 e w where the functions a has a symbolic behaviour. when w and Re w > 0,

20 Theorem Hadamard s parametrix The Hadamard parametrix is a bounded operator z s T Had (z) : L p (M) L q (M) with a norm uniform with respect to the spectral parameter z C, z when 0 s, p 2 q p q + 2s n = 2 n, and ( min p 2, 2 ) > q 2n, 2 n + < p q 2 n.

21 Hadamard s parametrix: admissible exponents q 2 q = p F D C 0 2 B D C 2 + n q = p 2 n B 2 n A p

22 Hadamard s parametrix Theorem The Hadamard parametrix is a bounded operator when p 2 q and T Had (z) : L p (M) L q (M) p q < 2 n +, n n + with a norm bounded by T Had (z) L(L p,l q ) C z n 4 p q n + n ( ) p q 2. p,

23 Hadamard s parametrix: admissible exponents q 2 = = q p q p G E E F D C 0 2 B D C 2 + n q = p 2 n B 2 n A p

24 Decay of Hadamard s parametrix The combination of the two Theorems gives the following bound on the parametrix T Had (z)u L(L p,l q ) C z σ where the order σ is a piecewise linear function of d = /p /q n d + when 0 d 2 σ = 4 2 n + n 2 d + when 2. n + d

25 Decay of the parametrix: T Had (z)u L(Lp,L q ) = O( z σ ) σ n 2 n n+ p q n 2 n+ 2 n+

26 The Hadamard parametrix as an oscillatory integral With the asymptotic of the Bessel functions, the Hadamard parametrix takes the form T Had (z)u(x) = z n 4 2 e zd g(x,y) a(x, y, z) χ(x, y)u(y) dv M d g (x, y) n g (y). 2 This is an oscillatory integral operator for values of the spectral parmater z for which there is an oscillatory behaviour and no exponential decay (arg z / [ π/2, π/2]). This can be studied thanks to the Carleson-Sjölin theory of oscillatory integral operators. Final remarks Carleman estimates

27 Carleson-Sjölin theory We consider operators of the form T τ u(x) = e iτϕ(x,y) a(x, y)u(y) dy where a belongs to C 0 (U V ) with U, V open sets in Rn and ϕ is a smooth function on U V. Assumptions. corank 2 ϕ x y = on U V, 2. The hypersurfaces Σ U x 0 = { x ϕ(x 0, y) : y V } T x 0 U Σ V y 0 = { y ϕ(x, y 0 ) : x U } T y 0 V have everywhere non-vanishing Gaussian curvature for all x 0 U and y 0 V.

28 Carleson-Sjölin theory Theorem Under the previous non-degeneracy and curvature assumptions, for any compact set K V there exists a constant C > 0 such that for all τ and all u C 0 (K) with q = n+ n p and p 2. T τ u L q (R n ) Cτ n/q u L p (R n )

29 Carleson-Sjölin theory q q = p 2 2 n+ 2 n n p

30 Final remarks. The Carleson-Sjölin theory doesn t apply as such on the Hadamard parametrix because its kernel is singular on the diagonal x = y. This is taken care of by performing a dyadic decomposition for which d g (x, y) 2 ν z /2. Then one has to sum the corresponding series and this explains the different cases of boundedness of the parametrix. 2. The other issue is: how do we deal with the remainders? Unfortunately, the bounds on the parametrix are not strong enough to take care of the remainders (which behave like z /4 T Had (z)).the solution is to go through the L 2 space and use there improved bounds (i.e. gaining derivatives) which are valid because z lies outside of a parabola. This is the only place where this assumption is actually used.

31 Outline Introduction Resolvent estimates for Laplace-Beltrami operators Relation with Carleman estimates Resolvent estimates for the Schrödinger equation

32 Short bibliography on L p Carleman estimates on elliptic operators These works are in relation with unique continuation of solutions to Schrödinger equation with unbounded potentials. 985 Jerison-Kenig: first L p Carleman estimates, logarithmic weights 986 Jerison: simplification of the proof using spectral cluster estimates (see also Sogge s book) 987 Kenig-Ruiz-Sogge: Elliptic operators with constant coefficients, linear weights 989 Sogge: Elliptic operators with variable coefficients, non LCW 200 Shen: Laplace operator on the torus 2005 Koch-Tataru: construction of parametrices in general context

33 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 i 2 x P e τx + i 2 x = ( D x + 2 ( + 2iτ D x + 2) ) τ 2 g0. 2 After translation and scaling I = [0, 2π] and use Fourier series.

34 Kenig-Ruiz-Sogge approach We denote by τ 0 = 0 < τ τ 2... the sequence of eigenvalues of g0 on M 0 and (ψ k ) k 0 the corresponding sequence of eigenfunctions g0 ψ k = τ k ψ k. We denote by π k : L 2 (M 0 ) L 2 (M 0 ) the projection on the linear space spanned by the eigenfunction ψ k so that π k = Id, k=0 τ k π k = g0. k=0 Eigenvalues of the Laplacian g : (j 2 + τ k ) Eigenfunctions: e ijx ψ k.

35 Kenig-Ruiz-Sogge approach Denote by π j,k : L 2 (M) L 2 (M) the projection on the linear space spanned by e ijx ψ k : ( 2π ) π j,k f(x) = e ijy π k f(y, x ) dy e ijx, and define the spectral clusters as χ m = 0 m j 2 +τ k <m+ Note that these are projectors χ 2 m = χ m. Spectral cluster estimates of Sogge: π j,k, m N. χ m u L 2n n 2 (M) C( + m) 2 u L 2 (M) χ m u L 2 (M) C( + m) 2 u L 2n n+2 (M).

36 Kenig-Ruiz-Sogge approach We are now ready to reduce the proof of Carleman estimates to resolvent estimates. u L 2n n 2 (M) C f L 2n n+2 (M) when ( D x + 2) 2 ( u + 2iτ D x + ) u g0 u τ 2 u = f. 2 Inverse operator: G τ f = j= k=0 ( j + 2 π j,k f ) τ + τk τ. 2 ) 2 + 2i ( j + 2

37 Kenig-Ruiz-Sogge approach Use Littlewood-Paley theory to localize in frequency with respect to the Euclidean variable x ; u = u ν, f = ν=0 with ( 2π u 0 = u(y, x ) dy ), u ν = 0 2 ν j <2 ν and similarly for f. It suffices to prove ν=0 ( 2π ) e ijy u(y, x ) dy e ijx, ν > 0 0 u ν L 2n n 2 f ν C f ν L 2n n+2.

38 Kenig-Ruiz-Sogge approach The conjugated operator and the localization in frequency commute u ν = G τ f ν. We denote R(z) = (( D x + ) 2 ) g0 2 + z the resolvent. The error made by replacing G τ with the resolvent is ( R( τ 2 + i(2 ν + )τ) G τ ) fν = a ν jk (τ) = j= k= a ν jk (τ) π j,kf ν iτ(2 ν 2j) [2 ν,2 ν )(j) ( j 2 + 2iτ j τ 2 + τ k )( j 2 + i(2 ν + )τ τ 2 + τ k ). with j = j + 2. Error estimates

39 Kenig-Ruiz-Sogge approach Using the spectral cluster estimates ( R( τ 2 + i2 ν τ) G ) τ fν L n 2 2n (M) ( ( + m) 2 χ m R( τ 2 + i(2 ν + )τ) G ) τ fν L 2 (M) m=0 ( + m) 2 sup m=0 and furthermore m j 2 +τ k <m+ ( R( τ 2 + i(2 ν + )τ) G ) τ fν L n 2 2n (M) ( ( + m) sup a ν jk (τ) m=0 m j 2 +τ k <m+ a ν jk (τ) χ m f ν 2 L 2 (M) ) f ν L 2n n+2 (M).

40 Kenig-Ruiz-Sogge approach The above series converge and is uniformly bounded with respect to τ and ν as well as m=0 sup m j 2 +τ k <m+ a ν jk (τ) 2 ν τ ( + m) (m 2 τ 2 ) ν+ τ 2 2 ν τ (m 2 τ 2 ) ν+ τ ν τ t (t 2 τ 2 ) ν+ τ 2 dt and if we perform the change of variables s = 4 ν τ 2 (t 2 τ 2 ) in the right-hand side integral, we obtain the bound m=0 2 ν τ ( + m) (m 2 τ 2 ) ν+ τ 2 ds s 2 +.

41 Kenig-Ruiz-Sogge approach Summing up our computations, we have the error estimate ( R( τ 2 + i(2 ν + )τ) G ) τ fν L n 2 2n f ν 2n, (M) L n+2 (M) this means that it is enough to prove the resolvent estimate R( τ 2 + i(2 ν + )τ)f ν L n 2 2n f ν 2n. (M) L n+2 (M) Carleman estimates reduce to resolvent estimates of the form u 2n ((D L n 2 + ) 2 ) g0 (M) 2 + z u 2n L with z = τ 2 + iϱτ (ϱ ). n+2 (M)

42 Outline Introduction Resolvent estimates for Laplace-Beltrami operators Relation with Carleman estimates Resolvent estimates for the Schrödinger equation

43 A modified Christ-Kiselev lemma Lemma E and F two Banach spaces, I a compact interval. Let T be an operator of the form T u(t) = T (t, s)u(s) ds and suppose that T u L q (F ) M u L p (E) for some p < q. ϕ : I R is a non-decreasing function and T ϕ u(t) = H(t s)e ϕ(t)+ϕ(s) T (t, s)u(s) ds. I Then we have T ϕ u L q (F ) C p,q M u L p (E). I

44 The Christ-Kiselev dyadic decomposition s 0 t J ν,k = [(2k)2 ν, (2k + )2 ν ), L ν,k = ((2k + )2 ν, (2k + 2)2 ν ] { (s, t) [0, ] 2 : s < t } 2 ν = J ν,k L ν,k. Resolvent estimates ν= k=0

45 Normalization u L p (E) = Sketch of proof [0,] [0,] (s, t)h(t s) = Change of variables We get U(t) = t I I (s, t)h(t s) = with Q ν,k = U (J ν,k ) U (L ν,k ). 2 ν ν= k=0 t 0 u(s) p E ds. 2 ν ν= k=0 Jν,k L ν,k (s, t) Qν,k (s, t),

46 T ϕ u L q (F ) Sketch of proof 2 ν= 2 ν k=0 T ϕ ( Qν,k u) L q (F ). We set s ν,k = max U (J ν,k ), t ν,k = min U (L ν,k ) and observe that t ν,k s ν,k since U is increasing and We now write 2 ν k=0 max J ν,k min L ν,k. T ϕ ( Qν,k u) = e ϕ(t ν,k)+ϕ(s ν,k ) }{{} T ϕ ( Qν,k u) = L q (F ) bounded ( T (s, t) I ( 2 ν k=0 U (L ν,k )(t) e ϕ(t)+ϕ(t ν,k) }{{} bounded U (J ν,k )(s) e ϕ(s ν,k)+ϕ(s) }{{} T ϕ ( Qν,k u) q L q (F ) bounded ) q. ) u(s) d

47 We get ( T ϕ ( Qν,k u) L q (F ) M Sketch of proof 3 ( M Summation provides a bound T ϕ u L q (F ) M ν= e p(ϕ(s ν,k) ϕ(s)) }{{} U (J ν,k ) bounded ) U (s) ds U (J ν,k ) ( 2 ν k=0 ) 2 νq q p u(s) p E ds ) p p = M Jν,k p. ( ) = M 2 ν p q. ν= } {{ } C p,q

48 Resolvent estimates Theorem Let (M, g) be a Riemannian manifold and I be a compact interval. Let E, F be two Sobolev spaces on M, and p < q be two numbers for which the following inhomogeneous Strichartz estimate holds u L q (F ) C (i t + g )u L p (E), Then for all complex numbers z C, we have u L q (F ) C p,q C (i t + g z)u L p (E). Joint work with Camille Laurent

49 Parametrix construction Let z = τ + iµ, by conjugation we have i t + g τ iµ = e iτt (i t + g iµ)e iτt and we can disregard the oscillating factors e ±iτt. We further conjugate i t + g iµ = ie it g ( t µ)e it g and remind that 2π e itτ τ + iµ dτ = i { H(t)e µt when µ < 0 H( t)e µt when µ > 0. Thus if t u + g u iµu = f then u = i H(t s)e µ (t s) e} i(t s) g {{} f(s) ds =T (t,s) when µ < 0. We can apply the modified Christ-Kiselev lemma with ϕ(t) = µ t.

50 Carleman estimates Setting: product manifold M = R M endowed with g = dx 2 + g 0 Main assumption: Strichartz estimate holds where u L q (F ) C (i t + g )u L p (F ), F = W σ,r (M), r 2 or L r (R x, H s (M ) Then Carleman estimates of the form hold. e τx u L q (F ) C e τx (i t + g )u L p (F ), Comments on Strichartz estimates End

51 Parametrix Start with the conjugated operator e τx (i t + g )e τx = i t + 2 x 2τ x + g + τ 2 Oscillating factors are harmless. Goal: Write and parametrix = e iτ 2t (i t + 2 x 2τ x + g )e iτ 2t. u L q (F ) C (i t + g 2τ x )u L p (F ), i t 2τ x + g = e it g (i t 2τ x )e it g G τ f(t, x) = e i(t s)τ+i(x y )ξ 4π 2 e i(t s) g f(s, y, x ) dτ dξ ds dy. τ + 2iτξ

52 Freezing procedure Use a Littlewood-Paley dyadic decomposition in the Euclidean variable x : the argument works because the microlocalisation χ(2 ν D x ) commutes with the operator i t 2τ x + 2 x + g! Here we are using in an essential manner the product structure of the Riemannian manifold M. Once performed the microlocalisation, one freezes the derivative D x and replace it by its spectral equivalent 2 ν.hence replace the inverse G τ by the resolvent i t + g + z ν with and estimate the error. z ν = i2 ν+ τ

53 Strichartz estimates The question is now to see which Strichartz estimates hold on compact manifolds. Thamks to the previous procedure, they imediately imply corresponding Carleman estimates. So far, the generic Strichartz estimates are those of Burq, Gérard and Tzvetkov e it u L p (I,L q (M)) C u H p (M) where p, q are Strichartz admissible exponents, with a loss of derivatives of p, and the corresponding inhomogeneous estimates obtained by the T T argument. There are improved estimates on particular manifolds (such as the sphere, the torus).

54 Outline Introduction Laplace-Beltrami operators Relation with Carleman estimates Schro dinger equation iecl.univ-lorraine.fr/ David.Dos-Santos-Ferreira/ip205.html