Global Harmonic Analysis and the Concentration of Eigenfunctions, Part III:

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1 Global Harmonic Analysis and the Concentration of Eigenfunctions, Part III: Improved eigenfunction estimates for the critical L p -space on manifolds of nonpositive curvature Christopher D. Sogge (Johns Hopkins University)

2 Setting and general problem Compact boundaryless manifold (M, g) of dimension n 2. Eigenfunctions: e j (x) = λ 2 j e j (x), e j 2 dv = 1 M Question: How can you detect and measure various types of concentration of eigenfunctions (or, more generally, quasi-modes)? Expect that extreme concentration might occur at certain points and near certain periodic geodesics. Can measure this in various ways...

3 Extreme behavior on round spheres, S n Consider the standard sphere S n = {x R n+1 : x x x 2 n+1 = 1} Eigenvalues of S n are λ = λ k = k(k + n 1) k, repeating with highest possible multiplicity (very non-generic). d k k n 1 Eigenfunctions are spherical harmonics, restrictions of homogeneous harmonic polynomials in R n+1 to S n.

4 Extreme concentration at points L 2 -normalized zonal functions Z k (x), by classical Darboux-Szegö formula: Z k (x) cos ( (k + n 1 2 )d(x, ±1) + σ ) ( ) n 1 n / (d(x, ±1) 2, if d(x, ±1) k 1 and Z k (x) = O(k n 1 2 ) if d(x, ±1) k 1, where 1 = (1, 0,..., 0) denotes north pole and d(x, y) distance on S n and σ n = (n 1)π/4 (Maslov factor). High concentration at poles ±1. Easy calculation using above: Z k L p (S n ) k n( p ) 1 2, p 2(n+1) n 1. (1)

5 Extreme concentration along periodic geodesics Highest weight spherical harmonics, Q k (x) k n 1 4 (x1 + ix 2 ) k have extreme concentration near equator (periodic geodesic) γ = {x S n : 0 = x = (x 3,..., x n+1 )}. Simplest example of Gaussian beams, Q k (x) = k n 1 k 4 e 2 ln(1 x 2) k n 1 4 e k 2 d(x,γ)2 k n T (γ), k 1 2 where T k 1 2 (γ) denotes a k 1 2 tubular neighborhood about γ. Since equator has codimension n 1 conclude Q k L p (S n ) k n 1 4 {x S n : d(x, γ) k } p k n 1 2 ( p ), p 2 (2)

6 Saturation of L p -norms of eigenfunctions In 1988 showed that one has the universal bounds e λ L p (M) λ σ(p) e λ L 2 (M) (3) σ(p) = { n( p ) 1 2, p 2(n+1) n 1 n 1 2 ( p ), 2 < p 2(n+1) n 1 (4) Bounds for large p sensitive to high concentration at points and small p to high concentration along periodic geodesics. The zonal functions Z λ saturate (3) when p p c = 2(n+1) n 1 The Q λ saturate the bounds for 2 < p p c They both saturate the critical norm L pc, in which case σ(p c ) = 1 p c. So this space, L pc, sensitive to both types of extreme concentration: at points and along periodic geodesics.

7 Exponents and Kink point

8 Improvements for large exponents (p > p c ) Implicit in Bérard 1978: If (M, g) has nonpositive sectional curvatures e λ L (M) = O(λ n 1 2 / log λ) (i.e., log-improvement). Hassell-Tacy 2011: Also get the same log-improvements over (5) for all exponents p > p c under this curvature assumption. Given x M and an initial unit direction ξ S x M over x, say that ξ L x if geodesic with initial direction ξ loops back through x in some positive time t. (CS, S. Zelditch 2002) If L x = 0 for all x M (a generic condition), then e λ L (M) = o(λ n 1 2 ), and hence e λ L p (M) = o(λ σ(p) ), p > p c = 2(n+1) n 1 (5) Much stronger results if one considers real analytic manifolds.

9 Other ways of measuring concentration along geodesics We ve argued that the L p -estimates (3) for 2 < p < p c are sensitive to concentration along geodesics as exhibited by the higher weight spherical harmonics, Q k. What about other estimates? Lower bounds for L 1 -norms (due to CS-Zelditch 2011 and relevant for nodal problems): λ n 1 4 e λ L 1 (M). (6) Extreme L 2 -concentration about λ 1 2 tubes about unit length geodesics γ Π: e λ L 2 (T λ 2 1 (γ)) 1 (7) Both are saturated by the Q k, λ = k(k + n 1) since Q k behaves like λ n 1 4 times indicator function of λ 1 2 neighborhood of equator.

10 L 1 lower bounds and nodal sets You can prove (6) using Hölder s inequality 1 = e λ 2 e λ θ 1 e λ 1 θ p c e λ θ 1 λ 1 θ pc By works with Zelditch and Hezari , if N λ = {x : e λ (x) = 0} is the nodal set of a real-valued eigenfunction and if N λ denotes its (n 1)-dimensional Hausdorff measure, have λ ( e λ dv ) 2 Nλ. M Can recover best known universal lower bounds of Colding-Minicozzi 2011 n 1 1 λ 2 N λ. By above, improvement of e λ L p (M), 2 < p p c, gives improved lower bounds. Recent work with Blair: When n = 3, (log λ) µ N λ for any µ < 16 if (M, g) has nonpositive sectional curvatures.

11 L 2 -mass on shrinking geodesic tubes Consider the Kakeya-Nikodym norms: e λ KN = sup e λ L 2 (T (γ)). γ Π λ 1 2 Since our e.f. s are L 2 -normalized, one has the trivial bounds e λ KN 1. Through the work of Bourgain, myself and joint work with Blair, now know that improving the L p -bounds in (3) for 2 < p < p c is equivalent to showing that e λ KN = o(1). Very recently, Blair and I showed that on manifolds of nonpositive curvature one has e λ KN = O((log λ) δn ) We also showed that this yields e λ L p = O(λ σ(p) (log λ) δp,n ). Get logarithmic improvements of Colding-Minicozzi lower bounds for size of nodal sets under this curvature assumption. Also, for manifolds of negative curvature, Hezari and Rivière showed that there is a density one subsequence of e.f. s with log-improvement of L pc -norms.

12 L p, 2 < p < p c bounds and KN estimates Recently, Blair and CS showed that if 2 < p < p c there is a θ p,n > 0 so that e λ L p (M) λ n 1 2 ( p ) e λ 1 θp,n L 2 (M) e λ θp,n KN, and so e λ KN = O((log λ) δn ) leads to above log-improvements of L p -norms, 2 < p < p c. e λ 2 KN = sup e λ (x) 2 dv. γ Π T λ 1 (γ) 2

13 Improving L p c -bounds Very recently showed that for manifolds of nonpositive curvature and all e.f. s one has e λ L pc = O(λ 1 pc (log log λ) δn ). (8) Note, that, by Chebyshev, this implies the weak-type bounds: {x M : e λ (x) > α} λ(log log λ) pcδn α pc. (9) For if Ω α denotes this set then Ω α α pc M e λ pc dv. Fortunately, by Lorentz-space improvements of the old 1988 spectral projection estimates, which are due to Bak and Seeger (2011), the weak-type bounds (9) actually imply (8) (at the cost of a smaller δ in (8)).

14 Respecting the weak-type bounds: Two dangerous heights Since e λ pc = O(λ 1 pc ), by Chebyshev, of course always have {x M : e λ (x) > α} λ α pc. (10) The highest weight spherical harmonics Q λ behave like λ n 1 4 times indicator function of λ 1 2 neighborhood of equator and therefore {x S n : Q λ (x) > α} λ n 1 2 λα pc, if α λ n 1 4. Zonal fncts, Z λ, of size λ n 1 2 in λ 1 ball about poles on S n, and, hence, {x S n : Z λ (x) > α} λ n λα pc, if α λ n 1 2. The two dangerous heights are in blue. Expect that in proving the log log-improvements in (9) have to take them into account.

15 Critical weak-type saturation on S n

16 Avoiding dangerous heights on nonpos curved manifolds We wish to get favorable estimates for Ω α where Ω α = {x M : e λ (x) > α}. Fortunately, by the work of Bérard, under our curvature assumption, e λ (x) λ n 1 2 (log λ) 1 2. Thus Ω α = for α λ n 1 2 (log λ) 1 2. (So get trivial bounds for Ω α if λ n 1 2 (log λ) 1 2 α) Fortunately, by recent joint work with Blair, e λ L p = O(λ σ(p) (log λ) δ ), 2 < p < p c and Chebyshev, get improvements logarithmically past other dangerous value, α λ n 1 4 : Ω α λ(log λ) δ α pc, if α λ n 1 4 (log λ) δ. Conclusion: Free of charge get log-improvements (better than log log λ) for all α / [λ n 1 4 (log λ) δ, λ n 1 2 (log λ) 1 2 ]

17 Handling Ω α, α [λ n 1 4 (log λ) δ, λ n 1 2 (log λ) 1 2] Using argument of Bourgain 1991, for above α can get if one knows and if ρ S(R) has ˆρ C 0, Ω α λ(log log λ) δ α pc e λ L 2 (B r ) = O(r 1 2 ), r 1, (11) ρ(t (λ g ))(x, y) = T 1 O ( (λ/d g (x, y)) n 1 ) 2 + λ n 1 2 O ( exp(ct ) ). (12) (11) is easy; was proved in (12) follows from arguments of Bérard. The 2nd term in the right of (12) accounts for why we only get log log-improvements. (Can t be O(λ n 1 2 ) by Jakobson-Polterovich 2007.)

18 Picture of the proof for manifolds of nonpositive curvature

19 You just witnessed Global harmonic analysis We succeeded in obtaining our improved critical L pc by combining local harmonic analysis techniques (always true), with global ones that used features of the dynamics of geodesic flow (based on curvature assumption). The local results were earlier works of Bourgain and Bak and Seeger, as well as the L p vs KN bounds of Blair and CS. The global estimates which made use of curvature assumptions were that we had improved Kakeya-Nikodym bounds and hence improved L p -bounds, 2 < p < p c (due to joint work w/ Zelditch and Blair), and the kernel estimate (12) for the smoothed out spectral projection operators: ρ(t (λ g ))(x, y) = O ( (log log λ) 1 (λ/d g (x, y)) n 1 ) 2, T log log λ, d g (x, y) (log λ) δ. Would likely get improvements of L pc -bounds if knew that e λ L 2 (B r ) = O(r 1 2 (log λ) ε ) for r < (log λ) δ.

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