On p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds Chris Sogge (Johns Hopkins University) Joint work with: Jean Bourgain (IAS) Peng Shao (JHU) Xiaohua Yao (JHU, Huazhong Normal U) July 16, 2012
Resolvent estimates: Earlier work Kenig-Ruiz-CS 1987: For the Euclidean case (R n, standard aplacian R n) you have uniform resolvent estimates: u n 2 (R n ) C ( R n + ζ)u n+2 (R n ), ζ C. lim Im ( ξ 2 + 1 iε) 1 = π ds (surface measure on S n 1 ), ε 0 + So letting ζ = 1 iε see that above results imply a special case of Stein-Tomas restriction theorem e ix ξˆf (ξ) ds C f S n 1 n 2 (R n ) n+2. (R n ) By a TT argument, above is equivalent to ( ) 1 ˆf 2 2 ds C f. S n 1 n+2 (R n )
Extensions to compact manifolds (discrete spectrum) Given a compact Riemannian manifold (M, g) of dimension n 3, interested in regions R(g) for which one can have uniform resolvent estimates: u n 2 (M) C ( g + ζ)u n+2 (M), ζ R(g), u C. Z. Shen (2001): For the torus T n = R n /Z n can take region to be {ζ C : Re ζ (Im ζ) 2, ζ 1} Dos Santos Ferreira, Kenig and Salo [DKS] (2011): Same results for any compact Riemannian manifold
Problem raised by DKS for ( g + ζ) 1 : n+2 n 2 Does the DKS-S theorem hold for a larger region, specifically the region outside the curve γ opt, which is Im ζ = 1 (unit distance from spectrum of g )? This would be natural Riemannian version of KRS results for R n. γ DKSS γ opt
Answer linked to Weyl aw (100th anniversary!!) In 1910 Sommerfeld, followed 3 months later by orentz, gave famous lectures inspiring Weyl s work. Sommerfeld interested in forced vibration problem in dimensions n = 1, 2, 3: ( + ζ)u(x) = f (x), x Ω R n, u Ω = 0 Asked how properties of solution operator ( + ζ) 1 related to solutions of the free vibration problem ( + λ 2 j )e j (x) = 0, e j Ω = 0, e j 2 dx = 1 Ω e j(x)e j(y) ζ λ 2 j Kernel of solution operator: S(x, y) = j Sommerfeld reasoned that ( + ζ)s(x, y) = e j (x)e j (y) is spike function Also conjectured that properties of S(x, y) should be related to distribution of eigenvalues {λ j }, and cancellation from numerator (oscillation of e.f. s)
Weyl aw orentz s subsequent 1910 lecture spelled out the eigenvalue problem more precisely and asked whether for the eigenvalues for the Dirichlet aplacian in smooth domains Ω R n one has for N(λ) = number λ j λ N(λ) = (2π) n (Vol B)(Vol Ω)λ n + o(λ n ) Hilbert: No way in my lifetime Weyl: Yes! (several proofs in 1911-12 heat kernel, Tauberian arguments...) Improvement by Carleman (1934): If e j = λ 2 j e j, λ 1 λ 2... e j (x) 2 = (2π) n (Vol B)λ n + o(λ n ) λ j λ
Manifold case Avakumovič (1956) (following earlier partial work of evitan (1952, 1955)): Compact boundaryless manifolds have Sharp Weyl formula: N(λ) = (2π) n (Vol B)(Vol g M)λ n + R(λ), R(λ) = O(λ n 1 ) Also Avakumovič: Sharp due to clustering of e.v. s on S n Hörmander (1968): Many important extensions/innovations Duistermaat and Guillemin (1975): Zero measure of periodic orbits = R(λ) = o(λ n 1 ) Extension to manifolds with boundary by Ivrii (1980) and verified conjecture of Weyl (d term here) (also Melrose) Bérard (1977) For boundaryless manifolds with negative curvature: R(λ) = O(λ n 1 / log λ) Hlawka (1950) For torus T n n 1 n 1 : R(λ) = O(λ n+1 ).
Return to DKS problem Which regions can you have u n 2 (M) C ( g + ζ)u n+2 (M), ζ R(g), u C? γ DKSS γ opt Figure: Earlier results and the problem
Sommerfeld s reasoning was correct Behavior of solution to forced membrane equation ( g + ζ)u = f is related to distribution of the spectrum of g : Theorem Suppose that 0 < ε(λ) 1 as λ and suppose further that ε(λ) 0 and ( lim sup ε(λ) λ n 1 ) 1 [ ] N(λ + ε(λ)) N(λ) =. λ + Then lim sup ( g + λ 2 + iλε(λ) ) 1 = +. n+2 λ + (M) n 2 (M) Earlier known resolvent bounds correspond to ζ = λ 2 ± iλ. Theorem says that to improve this to region bounded by curve ζ = λ 2 ± iε(λ)λ, need to have that the number of eigenvalues in [λ, λ + ε(λ)] band is proportional to volume of Euclidean annulus of width ε(λ) about sphere of radius λ.
Relevance to D-K-S problem and earlier results The curve γ opt in D-K-S problem corresponds to ε(λ) = 1/λ: γ DKSS γopt Seems impossibly difficult to hold for any (M, g). This would correspond to (mean level spacing) R(λ) = N(λ) (2π) n Vol g (B M)λ n = O(λ n 2 ) We earlier noted that R(λ) = O(λ n 1 ) is sharp for sphere. Correspondingly, above theorem implies that for S n, the best you can do is the earlier results of DKS and Shen, which is the region bounded by red curve. Same for Zoll manifolds due to eigenvalue clustering (Weinstein 1977).
Stein-Tomas versus Riemannian case redux Stein-Tomas implies there is a uniform constant C so that e ix ξ ˆf (ξ) dξ Cελ f n 2 (R n ) ξ [λ,λ+ε] n+2 (R n ). Can think of this as bounds for the projection of f onto the part of the spectrum of R n between λ and λ + ε. In 1988 CS proved a variant of this for the Riemannian case corresponding to ε = 1, unit spectral projection estimates : et E j : 2 (M) 2 (M) denote the projection onto jth eigenspace, i.e. E j f (x) = f, e j g e j (x). Then λ j [λ,λ+1] E j f Cλ f n 2 (M) n+2 (M).
Problem: To what extent can discrete spectrum (i.e., g ) case approach continuous spectrum case (i.e., R n)? Problem: Given (M, g), can you find 0 < ε(λ) < 1 with ε(λ) 0 and E j f Cε(λ) λ f n 2 (M) n+2? (M) λ j [λ,λ+ε(λ)] Facts and progress on this: NO! for S n with round metric or Zoll manifolds Yes for analogous 1 versus (easier) estimates for generic metrics on any M (CS-Zelditch 2002) Recent work by Bourgain related to above Problem for toral eigenfunctions (see below)
Conclusion: Extent you can approach the Euclidean resolvent estimates of KRS for a given (M, g) is exactly the same as the extent you can approach the Euclidean Fourier restriction estimates. Main Theorem (our joint work) Theorem et (M, g) be a compact manifold of dimension n 3. Suppose that 0 < ε(λ) 1 decreases to 0 as λ +. Then one has the uniform spectral projection estimates λ λ j ε(λ) E j f Cε(λ)λ f n 2 (M) n+2, λ 1, (1) (M) if and only if one has the uniform resolvent estimates u n 2 (M) C ( g + (λ + iµ) 2 )u n+2 (M), λ, µ R, λ 1, µ ε(λ), u C (M). (2)
Corollaries We obtain e(λ) improvements (1) of the ε = 1 of CS 87, and hence we obtain the improved bounds, (ζ = (λ + iε(λ)) 2 ) u n 2 (M) C ( g + (λ + iε(λ)) 2) u n+2 (M) in certain cases (each having spread out spectrum): ε(λ) = 1/ log λ works if nonpositive sectional curvatures Power improvements over Shen s work on T n, ε(λ) = λ εn, if ε n < 85 252 2(n 1) n(n+1), 0.337, n = 3, Re ζ Im ζ 6.146 2(n 1) n 2 + 2, n 5 odd n 4 even First result related to work of Bérard, latter to work of Bourgain-Guth (which also has differing even/odd numerology)
Our new positive results γ S n λ γ neg γ T n γ S n γ opt γ neg γ T n γ opt = {λ = 1/µ} µ Figure: Various regions R(M, g) for ( g + ζ) 1 Figure: Various regions for ( g + (λ + iµ) 2) 1, µ ε(λ)
Earlier resolvent estimate proofs of DKS-S Want: ( g + ζ) 1 : n+2 (M) n 2 (M) Earlier work of Shen as well as Dos Santos Ferreira, Kenig and Salo used Hadamard parametrix for forced membrane problem: ( g + ζ) N α ν (x, y)e ν (ζ, d g (x, y)) = g 1 2 δx (y) + R N (x, y). ν=0 E 0 (ζ, x ) = (2π) n R n e ix ξ ζ ξ 2 dξ. Gives rise to oscillatory integral operators for which you can use Stein s oscillatory integral theorem ( Beijing lectures in harmonic analysis 1986) (Similar arguments were used earlier by CS to obtain aforementioned ε = 1 spectral projection bounds.) Above formula only valid near the diagonal {x = y}. Difficulty: dealing with this local vs global issue ( global harmonic analysis )
Our approach: Use different Hadamard parametrix (the one for cos(t g ))
Formula for (even) functions of g Note that f ( g ) = 1 ˆf (t) cos(t g ) dt 2π u(t, x) = ( (cos t g )φ ) (x) is the solution of the Cauchy problem for (M, g) ( t 2 g )u(t, x) = 0 on R M u(0, ) = φ, t u(0, ) = 0. This wave equation approach (vs. Hadamard resolvent parametrix) makes it easier to do the global harmonic analysis required to exploit the various geometric assumptions, etc.
Derive formula we need: Complexify Poisson integral Want to apply above with f ( g ) = ( g + ζ) 1 If we write ζ = z 2, z = λ + iε, we need to compute ˆf if f (τ) = ( τ 2 + (λ + iε) 2) 1 = 1 τ 2 + ε 2 λ 2 2iλε If λ = 0 and ε 0 this is π/ ε times the Poisson kernel and so e iτt π τ 2 dτ = + ε2 ε e εt By analytic continuation, if ε 0, λ R ˆf (t) = e iτt (τ 2 +ε 2 λ 2 2iλε) 1 dτ = ( g + ζ) 1 = 1 2π π sgn ε i(λ + iε) ei(sgn ε)λ t e εt, ˆf (t) cos t g dt. (local) Headache due to lack of smoothness of the t s at t = 0.
Use of formula: Shrinking spectral proj = p -Resolvent Break up ( g + (λ + iε)) 1, ε = ε(λ), into three pieces using β C0 (R) with β(t) = 1, t 1, β(t) = 0, t 2 ( g + (λ + iε)) 1 = π sgn ε i(λ+iε) β(t)e i(sgn ε)λ t e εt cos t g dt + π sgn ε i(λ+iε) (1 β(εt) ) e i(sgn ε)λ t e εt cos t g dt + π sgn ε i(λ+iε) (β(εt) β(t) ) e i(sgn ε)λ t e εt cos t g dt = I + II + III. This splitting into small time, large time and medium time pieces is akin to what happens in Ivrii s proof of the Duistermaat-Guillemin theorem (although we don t use propagation of singularities but rather p -estimates to control large-time pieces).
First piece I = π sgn ε i(λ+iε) β(t)e i(sgn ε)λ t e εt cos t g dt This is a local term. Get uniform bounds λ, ε by using stationary phase and interpolation arguments. This piece agnostic to global dynamics of geodesic flow.
II = π sgn ε i(λ+iε) (1 β(εt) ) e i(sgn ε)λ t e εt cos t g dt = ρ(ε; g λ). Key remainder term sensitive to assumptions (and geometry). Easy to see that for τ > 0 ρ(ε; τ λ) λ 1 ε 1( 1 + ε 1 τ λ ) N, N = 1, 2, 3,.... Thus, since we re assuming ε-band estimates, λ j λ ε E jf ελ f n 2 n+2, get ρ(ε; g λ)f n 2 + k=1 [ (λ 1 ε 1 λε + λ j λ (kε,(k+1)ε] k=1 λ j λ ε ρ(ε; λ j λ)e j f n 2 ρ(ε; λ j λ)e j f n+2 [(λ 1 ε 1 λεk N ] ] f n+2 f n+2
Final piece III = π sgn ε i(λ+iε) (β(εt) β(t) ) e i(sgn ε)λ t e εt cos t g dt Use, if τ > 0 and N = 1, 2, 3,..., = ρ(ε; g λ) ρ(ε; τ λ) ε 1 λ 1 (1 + ε 1 τ λ ) N + λ 1 (1 + τ λ ) N, our argument for II along with the (old) unit-band estimates E j f λ f n 2 n+2, λ j λ 1 and our assumption about ε-band improvement E j f ελ f n 2 n+2 λ j λ ε
Power improvements of Shen s result for torus By above, proving resolvent estimates for Im ζ Re ζ 1 2 εn is equivalent to proving (with norms over T n [ 1 2, 1 2 )n )) ˆf (k)e 2πix k f. (3) n+2 Cλ1 εn n 2 {k Z n : k [λ,λ+λ εn ]} Hlawka s 1950 proof about lattice points yields ˆf (k)e 2πix k n 1 n 1 Cλ n+1 f 1. (4) k [λ,λ+λ n 1 n+1 ] Known also (Stein-Tomas/CS) (no ε-improvement) k [λ,λ+ε] ˆf (k)e 2πix k 2(n+1) n 1 Cλ n 1 n+1 f 2(n+1) n+3 Using (5) and the proof of (4) you can set up an interpolation argument to get (3) with ε n = 1 n + 1, (ε 3 = 1 4, i.e., Re ζ (Im ζ)4, n = 3) (5)
Further improvements using recent Harmonic Analysis of Bennett-Carbery-Tao, Bourgain-Guth, Bourgain Based on recent breakthroughs in harmonic analysis, using interpolation with certain n 1 (T n ) estimates can improve (5) to get bounds like ˆf (k)e 2πix k 2(n+1) Cλ n 1 n+1 δn(ε) f n 1 2(n+1), n+3 k [λ,λ+λ ε ] for certain functions δ n (ε) and ranges of ε (depending on n). In 3d, leads to ε 3 = 1/3, i.e., Re ζ (Im ζ) 6 (instead of earlier ε 3 = 1/4) Can also exploit oscillation in exponential sums used to prove the Hlawka-type bound (4) to get further improvement ε 3 = 85 = 0.337..., i.e., Re ζ Im ζ 6.146 252 For n > 3 the improvements are ε n 2/(n + 1) (instead of earlier 1/(n + 1))
Bourgain s recent related work Eigenfunctions on T n with eigenvalue λ are functions of the form e λ (x) = a k e 2πix k. {k Z n : k =λ} In 1974 Zygmund proved that for the 2-torus one has uniformly bounded 4 -norms of 2 -normalized eigenfunctions: e λ 4 (T 2 ) C e λ 2 (T 2 ). In 2011 (to appear in Israel Math J) Bourgain proved a deep and beautiful extension to higher dimensions, n 3: e λ n 1 (T n ) C ελ ε e λ 2 (T n ) Bourgain s bound is based on ideas from his earlier work with Guth and relies in a key way on a multilinear Fourier extension estimate of Bennett-Carbery-Tao (Acta 2006). The n 1 estimates we use to get εn 2/(n + 1) are proved in a way that s similar to the above inequality. Frustrating that, unlike Hlawka s theorem, the results get worse as n.
The case of negative curvature Just as Hlawka s 1950 theorem provides a template for toral arguments, there is one of Bérard 1977 providing one for manifolds with negative curvature Implicit in Bérard s proof of improved Weyl law here, N(λ) = (2π) n (Vol g B M) λ n + O(λ n 1 / log λ) is the following estimate λ j [λ,λ+1/ log λ] E j f (M) C λn 1 log λ f 1 (M). Use it and follow your nose through interpolations to obtain (as observed earlier by Hassell-Tacy) λ j [λ,λ+1/ log λ] E j f C λ n 2 (M) log λ f. n+2 (M) N.B. Unlike in the toral case, get exactly the same gain for n+2 n 2 as you do for 1. Due, in part, to the modest log-improvements, as opposed to the power-improvements for T n.