Endpoint resolvent estimates for compact Riemannian manifolds

Transcription

1 Endpoint resolvent estimates for compact Riemannian manifolds joint work with R. L. Frank to appear in J. Funct. Anal. (arxiv: ) Lukas Schimmer California Institute of Technology 3 February 207 Schimmer (Caltech) Endpoint resolvent estimates 02/3/2067 / 5

2 Resolvent estimates on R n Investigate L p (R n ) L q (R n ) mapping properties of the resolvent ( z) on R n. Theorem (Kenig, Ruiz and Sogge, 987) For n 3 and z C \ (0, ) ( z) L Cp,n z n/2+n/p, p L p n + 3 2(n + ) p n + 2 2n. /q Im z 2 Re z /2 /p Figure : Admissible values of p Figure 2: Admissible values of z Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

3 Applications These inequalities and their extensions have found many applications in analysis and PDE, including: Unique continuation problems and absence of positive eigenvalues (Koch and Tataru 2005; 2006) Limiting absorption principles (Goldberg and Schlag 2004; Ionescu and Schlag 2006) Absolute continuity of the spectrum of periodic Schrödinger operators (Shen 200) Eigenvalue bounds for Schrödinger operators with complex potentials (Frank 20) Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

4 Resolvent estimates on compact Riemannian manifolds M Let M be a compact Riemannian manifold without boundary of dimension n 3. Investigate L p (M) L q (M) mapping properties of the resolvent ( z) on M. Theorem (Dos Santos Ferreira, Kenig and Salo, 204) For Im z δ with some arbitrary, but fixed δ > 0 ( z) L C p,n,δ z n/2+n/p, p L p n + 3 2(n + ) < p n + 2 2n. /q Im z 2 Re z /2 /p Figure 3: Admissible values of p Figure 4: Admissible values of z Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

5 Optimality of the parabolic area of exclusion Theorem (Bourgain, Shao, Sogge and Yao 205) Let M be a compact Riemannian manifold without boundary of dimension n 3 which is Zoll. Then there is a constant C > 0 such that for any function δ : R (0, ) with lim κ δ(κ) = 0 and lim inf κ κ δ(κ) C, lim sup κ (κ + iδ(κ)) 2 n/2 n/p+ ( (κ + iδ(κ)) 2) L =. p L p If M = S n (with the standard metric), this holds also with C = 0. Im z Note that Im z = δ can be written as (Im z) 2 = 4δ 2 (Re z + δ 2 ) and that z = (κ + iδ(κ)) 2 is on the curve Re z (Im z) 2 = 4δ(κ) 2 (Re z + δ(κ) 2 ) Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

6 Resolvent estimates on compact Riemannian manifolds M Let M be a compact Riemannian manifold without boundary of dimension n 2. Investigate L p (M) L q (M) mapping properties of the resolvent ( z) on M. Theorem (Frank and S., 206; Burq, Dos Santos Ferreira, Krupchyk 206) For Im z δ with some arbitrary, but fixed δ > 0 ( z) L C p,n,δ z n/2+n/p = C p,n,δ z n + 3 n+, p L p 2(n + ) = p. /q Im z 2 Re z /2 /p Figure 6: Admissible values of p Figure 7: Admissible values of z Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

7 Necessary parametrix and remainder bounds The proof relies on the construction of a parmetrix T (z) we obtain ( z)t (z) = I + S(z). ( z) = T (z) T (z) S(z) + S(z) ( z) S(z). and thus ( z) L T (z) p L p L p L + T p (z) L 2 L S(z) p L p L 2 + S(z) ( L 2 L z) p S(z) L 2 L 2 L p L 2 We need the following mapping properties: T (z) L p L p z /(n+) (Frank and S. 206) T (z) L p L 2 z (n+3)/(4(n+)) (n 3: Dos Santos Ferreira, Kenig and Salo 204) S(z) L p L 2 z (n )/(4(n+)) (n 3: Dos Santos Ferreira, Kenig and Salo 204) ( z) L 2 L 2 z 2 (Im z) z /2 δ Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

8 Necessary parametrix and remainder bounds The proof relies on the construction of a parmetrix T (z) ( z)t (z) = I + S(z). we obtain ( z) = T (z) T (z) S(z) + S(z) ( z) S(z). and thus ( z) L T (z) p L p L p L + T (z) p L p L S(z) 2 L p L 2 ( + S(z) L p L z) 2 S(z) L 2 L 2 L p L 2 We need the following mapping properties: T (z) L p L p z /(n+) (Frank and S. 206) T (z) L p L 2 z (n+3)/(4(n+)) (n 3: Dos Santos Ferreira, Kenig and Salo 204) S(z) L p L 2 z (n )/(4(n+)) (n 3: Dos Santos Ferreira, Kenig and Salo 204) ( z) L 2 L 2 z 2 (Im z) z /2 δ Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

9 The parametrix construction We will use the Hadamard parametrix T (z). The construction is local (T (z)u)(x) = χ(x)f (x, y, z)χ(y)u(y) dµ g (y) with dµ g denoting the volume form on M and, M F (x, y, z) = N α j (x, y)f j (d g (x, y), z) j=0 with smooth coefficients α j and the Bessel potentials F j (r, z) = j! Rn e ixξ dξ. (2π) n ( ξ 2 z) +j Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

10 Parametrix bounds Lemma (Frank and S., 206) Let δ > 0 and either (p, q) = ( 2n(n+), 2n 2n ) or (p, q) = (, 2n(n+) ). Then, if z δ, n 2 +4n n n+ n 2 2n+ T (z)u q, z n+ u p,. /q It is sufficient to consider characteristic functions u = I E. The statement is then equivalent to 2 /2 /p sup λµ g (A) q C z n+ µg (E) p λ>0 with A = { x M : ( ) } T (z)i E (x) > λ Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

11 Proof idea The kernel T (z) is decomposed dyadically z /2 d g (x, y) [ 2 ν, 2 ν ] [2 ν, 2 ν ] By the Carleson Sjölin theorem: T (z) = ν 0 T ν(z). Lemma (n 3: Dos Santos Ferreira, Kenig, Salo 204; n = 2: Frank and S. 206) T ν(z)u q C z n 2p n 2q 2 ν( n q n+ 2 ) u p if 2 q, p = n + n q, ν. 2 /q Decompose T (z) T 0(z) = T ν(z) =T () (z)+t (2) (z) ν ρt ν(z)+ ν>ρ and use Hölder s inequality to bound λµ g (A) T () (z)i E q µ g (A) q + T (2) (z)i E q2 µ g (A) q 2 /2 /p Applying the above bounds and optimising over ρ yields the desired bound. Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

12 Parametrix bounds Lemma (Frank and S., 206) Let δ > 0 and either (p, q) = ( 2n(n+), 2n 2n ) or (p, q) = (, 2n(n+) ). Then, if z δ, n 2 +4n n n+ n 2 2n+ T (z)u q, z n+ u p,. Corollary (Frank and S., 206) Let δ > 0 and let p 2 q with Then, if z δ, p q = 2 n +, n 2 2n + < 2n(n + ) q < n 2n. T (z)u q z n+ u p. In fact, the interpolation yields the inequality T (z)u q,s z n+ u p,s for s, which for p < s < q is stronger than the one stated above. Schimmer (Caltech) Endpoint resolvent estimates 02/3/2067 / 5

13 Optimality of the parabolic area of exclusion Theorem (Frank and S., 206) Let M be a compact Riemannian manifold without boundary of dimension n 2 which is Zoll. Then there is a constant C > 0 such that for any function δ : R (0, ) with lim κ δ(κ) = 0 and lim inf κ κ δ(κ) C, lim sup (κ + iδ(κ)) 2 n+ ( (κ + iδ(κ)) 2) L =. κ 2(n+)/(n+3) L 2(n+)/(n ) If M = S n (with the standard metric), this holds also with C = 0. Im z Note that Im z = δ can be written as (Im z) 2 = 4δ 2 (Re z + δ 2 ) and that z = (κ + iδ(κ)) 2 is on the curve Re z (Im z) 2 = 4δ(κ) 2 (Re z + δ(κ) 2 ) Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

14 Spectral clusters Resolvent norm bounds imply spectral cluster norm bounds. Lemma Let A be a self-adjoint real operator in L 2 and p. Let κ, δ > 0 and set P κ,δ = I [(κ δ) 2,(κ+δ) 2 ](A). Then P κ,δ 2 L p L 4δκ ( + δκ + (/2)δ 2 κ 2) ( A (κ + iδ) 2) L. 2 p L p Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

15 Spectral clusters The resolvent estimate implies mapping properties of P κ,δ = I [(κ δ) 2,(κ+δ) 2 ]( ) Pκ,δ(κ) L p L 2 2δ(κ) /2 κ /2 ( + ɛ (κ)) 2 with ɛ(κ) = δ(κ)/κ and ɛ (κ) = ɛ(κ) + ɛ(κ) 2 /2. ( (κ + iδ(κ)) 2) /2 L p L p Theorem (Sogge 988 & 989) For unit size spectral clusters L n P κ, Cκ 2(n+) 2 p L2 n + 3 2(n + ) = p The bounds are optimal. Since M is Zoll, all the eigenvalues of cluster around the values (k + α) 2 for k N, i.e. there is a constant C such that for κ = k + α and δ C/k, we have P κ,δ = P κ, 2. It follows that lim sup κ (κ + iδ(κ)) 2 n+ ( (κ + iδ(κ)) 2) L = +. p L p Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5

16 Thank you for your attention! Schimmer (Caltech) Endpoint resolvent estimates 02/3/ / 5