Endpoint resolvent estimates for compact Riemannian manifolds joint - - PowerPoint PPT Presentation

Endpoint resolvent estimates for compact Riemannian manifolds

joint work with R. L. Frank to appear in J. Funct. Anal. (arXiv:1611.00462)

Lukas Schimmer

California Institute of Technology

13 February 2017

Schimmer (Caltech) Endpoint resolvent estimates 02/13/20167 1 / 15

Resolvent estimates on Rn

Investigate Lp(Rn) → Lq(Rn) mapping properties of the resolvent (−∆ − z)−1 on Rn.

Theorem (Kenig, Ruiz and Sogge, 1987)

For n ≥ 3 and z ∈ C \ (0, ∞)

• (−∆ − z)−1
• Lp→Lp′ ≤ Cp,n|z|−n/2+n/p−1 ,

n + 3 2(n + 1) ≤ 1 p ≤ n + 2 2n . 1/p 1/q

1 2

1/2 1 1

Figure 1: Admissible values of p

Re z Im z

Figure 2: Admissible values of z

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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 Schr¨

• dinger operators (Shen 2001)

Eigenvalue bounds for Schr¨

• dinger operators with complex potentials (Frank 2011)

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Resolvent estimates on compact Riemannian manifolds M

Let M be a compact Riemannian manifold without boundary of dimension n ≥ 3. Investigate Lp(M) → Lq(M) mapping properties of the resolvent (−∆ − z)−1 on M.

Theorem (Dos Santos Ferreira, Kenig and Salo, 2014)

For Im √z ≥ δ with some arbitrary, but fixed δ > 0

• (−∆ − z)−1
• Lp→Lp′ ≤ Cp,n,δ|z|−n/2+n/p−1 ,

n + 3 2(n + 1) < 1 p ≤ n + 2 2n . 1/p 1/q

1 2

1/2 1 1

Figure 3: Admissible values of p

Re z Im z

Figure 4: Admissible values of z

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Optimality of the parabolic area of exclusion

Theorem (Bourgain, Shao, Sogge and Yao 2015)

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+1
• −∆ − (κ + iδ(κ))2−1
• Lp→Lp′ = ∞ .

If M = Sn (with the standard metric), this holds also with C = 0. Note that Im √z = δ can be written as (Im z)2 = 4δ2(Re z + δ2) and that z = (κ + iδ(κ))2 is on the curve (Im z)2 = 4δ(κ)2(Re z + δ(κ)2) Re z Im z

Schimmer (Caltech) Endpoint resolvent estimates 02/13/20167 5 / 15

Resolvent estimates on compact Riemannian manifolds M

Let M be a compact Riemannian manifold without boundary of dimension n ≥ 2. Investigate Lp(M) → Lq(M) mapping properties of the resolvent (−∆ − z)−1 on M.

Theorem (Frank and S., 2016; Burq, Dos Santos Ferreira, Krupchyk 2016)

For Im √z ≥ δ with some arbitrary, but fixed δ > 0

• (−∆ − z)−1
• Lp→Lp′ ≤ Cp,n,δ|z|−n/2+n/p−1 = Cp,n,δ|z|−

1 n+1 ,

n + 3 2(n + 1) = 1 p . 1/p 1/q

1 2

1/2 1 1

Figure 6: Admissible values of p

Re z Im z

Figure 7: Admissible values of z

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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)−1 = T (z) − T (z)∗S(z) + S(z)∗(−∆ − z)−1S(z) . and thus

• (−∆ − z)−1
• Lp→Lp′ ≤ T (z)Lp→Lp′ + T (z)∗L2→Lp′ S(z)Lp→L2

+ S(z)∗L2→Lp′

• (−∆ − z)−1
• L2→L2 S(z)Lp→L2

We need the following mapping properties: T (z)Lp→Lp′ |z|−1/(n+1) (Frank and S. 2016) T (z)Lp→L2 |z|−(n+3)/(4(n+1)) (n ≥ 3: Dos Santos Ferreira, Kenig and Salo 2014) S(z)Lp→L2 |z|(n−1)/(4(n+1)) (n ≥ 3: Dos Santos Ferreira, Kenig and Salo 2014)

• (−∆ − z)−1
• L2→L2 ≤ |z|− 1

2 (Im √z)−1 ≤ |z|−1/2δ−1 Schimmer (Caltech) Endpoint resolvent estimates 02/13/20167 7 / 15

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)−1 = T (z) − T (z)∗S(z) + S(z)∗(−∆ − z)−1S(z) . and thus

• (−∆ − z)−1
• Lp→Lp′ ≤ T (z)Lp→Lp′ + T (z)Lp→L2 S(z)Lp→L2

+ S(z)Lp→L2

• (−∆ − z)−1
• L2→L2 S(z)Lp→L2

We need the following mapping properties: T (z)Lp→Lp′ |z|−1/(n+1) (Frank and S. 2016) T (z)Lp→L2 |z|−(n+3)/(4(n+1)) (n ≥ 3: Dos Santos Ferreira, Kenig and Salo 2014) S(z)Lp→L2 |z|(n−1)/(4(n+1)) (n ≥ 3: Dos Santos Ferreira, Kenig and Salo 2014)

• (−∆ − z)−1
• L2→L2 ≤ |z|− 1

2 (Im √z)−1 ≤ |z|−1/2δ−1 Schimmer (Caltech) Endpoint resolvent estimates 02/13/20167 7 / 15

The parametrix construction

We will use the Hadamard parametrix T(z). The construction is local (T(z)u)(x) =

• M

˜ χ(x)F(x, y, z)χ(y)u(y) dµg(y) with dµg denoting the volume form on M and, F(x, y, z) =

N

• j=0

αj(x, y)Fj(dg(x, y), z) with smooth coefficients αj and the Bessel potentials Fj(r, z) = j! (2π)n

• Rn

eixξ (|ξ|2 − z)1+j dξ .

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Parametrix bounds

Lemma (Frank and S., 2016)

Let δ > 0 and either (p, q) = ( 2n(n+1)

n2+4n−1, 2n n−1) or (p, q) = ( 2n n+1, 2n(n+1) n2−2n+1). Then, if |z| ≥ δ,

T(z)uq,∞ |z|−

1 n+1 up,1 .

1/p 1/q

1 2

1/2 1 1 It is sufficient to consider characteristic functions u = IE. The statement is then equivalent to sup

λ>0

λµg(A)

1 q ≤ C|z|− 1 n+1 µg(E) 1 p

with A =

• x ∈ M :
• T(z)IE
• (x)
• > λ
• Schimmer (Caltech)

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Proof idea

The kernel T(z) is decomposed dyadically |z|1/2dg(x, y) ∈ [−2ν, −2ν−1] ∪ [2ν−1, 2ν] T(z) =

• ν≥0

Tν(z) . By the Carleson–Sj¨

• lin theorem:

Lemma (n ≥ 3: Dos Santos Ferreira, Kenig, Salo 2014; n = 2: Frank and S. 2016)

Tν(z)uq ≤ C|z|

n 2p − n 2q −12

−ν( n

q′ − n+1 2 ) up

if 2 ≤ q ≤ ∞, p′ = n + 1 n − 1q, ν ≥ 1 . 1/p 1/q

1 2

1/2 1 1 Decompose T(z)−T0(z) =

• 1≤ν≤ρ

Tν(z)+

• ν>ρ

Tν(z) =T (1)(z)+T (2)(z) and use H¨

• lder’s inequality to bound

λµg(A) ≤ T (1)(z)IEq1µg(A)

1 q′ 1 + T (2)(z)IEq2µg(A) 1 q′ 2

Applying the above bounds and optimising over ρ yields the desired bound.

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Parametrix bounds

Lemma (Frank and S., 2016)

Let δ > 0 and either (p, q) = ( 2n(n+1)

n2+4n−1, 2n n−1) or (p, q) = ( 2n n+1, 2n(n+1) n2−2n+1). Then, if |z| ≥ δ,

T(z)uq,∞ |z|−

1 n+1 up,1 .

Corollary (Frank and S., 2016)

Let δ > 0 and let 1 ≤ p ≤ 2 ≤ q with 1 p − 1 q = 2 n + 1 , n2 − 2n + 1 2n(n + 1) < 1 q < n − 1 2n . Then, if |z| ≥ δ, T(z)uq |z|−

1 n+1 up .

In fact, the interpolation yields the inequality T(z)uq,s |z|−

1 n+1 up,s

for 1 ≤ s ≤ ∞, which for p < s < q is stronger than the one stated above.

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Optimality of the parabolic area of exclusion

Theorem (Frank and S., 2016)

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|→∞

• (κ + iδ(κ))2
• 1

n+1

• −∆ − (κ + iδ(κ))2−1
• L2(n+1)/(n+3)→L2(n+1)/(n−1) = ∞ .

If M = Sn (with the standard metric), this holds also with C = 0. Note that Im √z = δ can be written as (Im z)2 = 4δ2(Re z + δ2) and that z = (κ + iδ(κ))2 is on the curve (Im z)2 = 4δ(κ)2(Re z + δ(κ)2) Re z Im z

Schimmer (Caltech) Endpoint resolvent estimates 02/13/20167 12 / 15

Spectral clusters

Resolvent norm bounds imply spectral cluster norm bounds.

Lemma

Let A be a self-adjoint real operator in L2 and 1 ≤ p ≤ ∞. Let κ, δ > 0 and set Pκ,δ = I[(κ−δ)2,(κ+δ)2](A). Then Pκ,δ2

Lp→L2 ≤ 4δκ

• 1 + δκ−1 + (1/2)δ2κ−2
• A − (κ + iδ)2−1
• Lp→Lp′ .

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Spectral clusters

The resolvent estimate implies mapping properties of Pκ,δ = I[(κ−δ)2,(κ+δ)2](−∆)

• Pκ,δ(κ)
• Lp→L2 ≤ 2δ(κ)1/2κ1/2 (1 + ǫ1(κ))

1 2

• −∆ − (κ + iδ(κ))2−1
• 1/2

Lp→Lp′

with ǫ(κ) = δ(κ)/κ and ǫ1(κ) = ǫ(κ) + ǫ(κ)2/2.

Theorem (Sogge 1988 & 1989)

For unit size spectral clusters

• Pκ, 1

2

• Lp→L2 ≤ Cκ

n−1 2(n+1)

n + 3 2(n + 1) = 1 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 Pκ,δ = Pκ, 1

2 . It follows that

lim sup

|κ|→∞

• (κ + iδ(κ))2
• 1

n+1

• −∆ − (κ + iδ(κ))2−1
• Lp→Lp′ = +∞ .

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Thank you for your attention!

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