Lp Bounds for Spectral Clusters. Compact Manifolds with Boundary
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1 on Compact Manifolds with Boundary Department of Mathematics University of Washington, Seattle Hangzhou Conference on Harmonic Analysis and PDE s
2 (M, g) = compact 2-d Riemannian manifold g = Laplacian ( Dirichlet or Neumann if M ) Eigenbasis: g φ j = λ 2 j φ j ( λ j = frequency ) Spectral Cluster, frequency λ : f = c j φ j λ j [λ,λ+1] Goal: find sharp powers δ(p) such that f L p (M) f L 2 (M) λ δ(p) ( p 2 )
3 (M, g) = compact 2-d Riemannian manifold g = Laplacian ( Dirichlet or Neumann if M ) Eigenbasis: g φ j = λ 2 j φ j ( λ j = frequency ) Spectral Cluster, frequency λ : f = c j φ j λ j [λ,λ+1] Goal: find sharp powers δ(p) such that f L p (M) f L 2 (M) λ δ(p) ( p 2 )
4 (M, g) = compact 2-d Riemannian manifold g = Laplacian ( Dirichlet or Neumann if M ) Eigenbasis: g φ j = λ 2 j φ j ( λ j = frequency ) Spectral Cluster, frequency λ : f = c j φ j λ j [λ,λ+1] Goal: find sharp powers δ(p) such that f L p (M) f L 2 (M) λ δ(p) ( p 2 )
5 Saturating examples on (M, g) = 2-sphere S 2 Example 1: f = highest weight spherical harmonic. f ( 1 + λ 1 2 sin(φ) ) N f L p f L 2 λ 1 2 ( p ) Lower bound (critical region) δ(p) 1 2 ( p )
6 Saturating examples on (M, g) = 2-sphere S 2 Example 1: f = highest weight spherical harmonic. f ( 1 + λ 1 2 sin(φ) ) N f L p f L 2 λ 1 2 ( p ) Lower bound (critical region) δ(p) 1 2 ( p )
7 Saturating examples on (M, g) = 2-sphere S 2 Example 2: f = zonal spherical harmonic, rotation invariant. f ( 1 + λ cos(φ) ) 1/2 f L p f L 2 λ p = λ 2( p ) 1 2 Lower bound (sub-critical region) δ(p) 2( p ) 1 2
8 Saturating examples on (M, g) = 2-sphere S 2 Example 2: f = zonal spherical harmonic, rotation invariant. f ( 1 + λ cos(φ) ) 1/2 f L p f L 2 λ p = λ 2( p ) 1 2 Lower bound (sub-critical region) δ(p) 2( p ) 1 2
9 Theorem: Sogge [1988] For compact n-dimensional manifold without boundary { n 1 2 δ(p) = ( p ), 2 p 2(n+1) n 1 n( p ) 1 2, 2(n+1) n 1 p 1/2 subcritical n = 2 1/6 critical 0 0 1/6 1/2
10 Grieser [1992] Sogge s spectral cluster estimates fail on D = { x 1 } R 2 Example: f (x) = e inθ J n (c o r), J n (c o ) = 0. f (x) concentrated in dist(x, D) n 2 3 } Vol( support (f )) 1 n 2 3 f L p f L 2 n 2 3 ( p ) Lower bound on manifolds with boundary δ(p) 2 3 ( p )
11 Grieser [1992] Sogge s spectral cluster estimates fail on D = { x 1 } R 2 Example: f (x) = e inθ J n (c o r), J n (c o ) = 0. f (x) concentrated in dist(x, D) n 2 3 } Vol( support (f )) 1 n 2 3 f L p f L 2 n 2 3 ( p ) Lower bound on manifolds with boundary δ(p) 2 3 ( p )
12 Multiple reflections / Gliding rays! Nondispersive region: angular spread λ 1 3 physical spread λ 2 3 Smith-Sogge [1995] M strictly concave spectral cluster estimates hold
13 Multiple reflections / Gliding rays! Nondispersive region: angular spread λ 1 3 physical spread λ 2 3 Smith-Sogge [1995] M strictly concave spectral cluster estimates hold
14 Smith-Sogge [2007]: M=2d manifold with boundary Spectral cluster estimates hold with { 2 3 δ(p) = ( p ), 6 p 8 2( p ) 1 2, 8 p 1/2 1/ /8 1/2
15 Key step in proof: Eliminate the boundary Geodesic normal coordinates along M : M = {x 2 0} g = d 2 x 2 + a 11 (x 1, x 2 ) d 2 x 1, smooth on x 2 0 Extend g across M to be even in x 2 g = d 2 x 2 + a 11 (x 1, x 2 ) d 2 x 1 Odd extension of Dirichlet eigenfunction: is eigenfunction for g x 2 x 2 φ j(x 1, x 2 ) Even extension of Neumann eigenfunction: φ j (x 1, x 2 ) is eigenfunction for g
16 Key step in proof: Eliminate the boundary Geodesic normal coordinates along M : M = {x 2 0} g = d 2 x 2 + a 11 (x 1, x 2 ) d 2 x 1, smooth on x 2 0 Extend g across M to be even in x 2 g = d 2 x 2 + a 11 (x 1, x 2 ) d 2 x 1 Odd extension of Dirichlet eigenfunction: is eigenfunction for g x 2 x 2 φ j(x 1, x 2 ) Even extension of Neumann eigenfunction: φ j (x 1, x 2 ) is eigenfunction for g
17 No more boundary / reflected geodesics: Disc: r 1 g = d 2 r + 1 r 2 d 2 θ Normal coordinates: x 2 = 1 r g = d 2 x (1 x 2 ) 2 d 2 x 1 But metric is Lipschitz.
18 No more boundary / reflected geodesics: Disc: r 1 g = d 2 r + 1 r 2 d 2 θ Normal coordinates: x 2 = 1 r g = d 2 x (1 x 2 ) 2 d 2 x 1 But metric is Lipschitz.
19 Metric g is of special Lipschitz type: d 2 x g δ(x 2) is integrable along non-tangential geodesics. "! dx 2 dt θ on γ dx 2 g(γ(t)) dt θ 1
20 Dispersive type estimates hold for g if d 2 g L 1 t L x Consider time dependent metric g(t, x) on M 2 t u(t, x) gu(t, x) = 0 u(0, x) = f (x), t u(0, x) = g(x) Tataru [2002] : Strichartz estimates If 2 t,x g L 1 t L x 1, then u L p t Lq x ([ 1,1] M) f H s + g H s 1 for same p, q, s as smooth manifolds, Euclidean space.
21 In our case d 2 g θ 1 Rescaled metric g(θt, θx) L 1 t L x norm 1: Tataru [2002] : Strichartz estimates If 2 t,x g L 1 t L x θ 1, then u L p t Lq x ([ θ,θ] M) f H s + g H s 1 for same p, q, s as smooth manifolds, Euclidean space.
22 Angular localization = subcritical gain Higer dimensions cos( t g )f (x) L p x L 2 t (M [ 1,1]) D δ(p) f L 2 (M), p 6 spectral cluster bounds: For spectral cluster f : cos( t g )f (x) cos(tλ)f (x) f L p (M) cos( t g )f (x) L p x L 2 t (M [ 1,1]) λδ(p) f L 2 (M) [Mockenhaupt-Seeger-Sogge (1993)] [S., d 2 g L 1 t L x (2006)]
23 Angular localization = subcritical gain Higer dimensions cos( t g )f (x) L p x L 2 t (M [ 1,1]) D δ(p) f L 2 (M), p 6 spectral cluster bounds: For spectral cluster f : cos( t g )f (x) cos(tλ)f (x) f L p (M) cos( t g )f (x) L p x L 2 t (M [ 1,1]) λδ(p) f L 2 (M) [Mockenhaupt-Seeger-Sogge (1993)] [S., d 2 g L 1 t L x (2006)]
24 Angular localization = subcritical gain Higer dimensions cos( t g )f (x) L p x L 2 t (M [ 1,1]) D δ(p) f L 2 (M), p 6 spectral cluster bounds: For spectral cluster f : cos( t g )f (x) cos(tλ)f (x) f L p (M) cos( t g )f (x) L p x L 2 t (M [ 1,1]) λδ(p) f L 2 (M) [Mockenhaupt-Seeger-Sogge (1993)] [S., d 2 g L 1 t L x (2006)]
25 Angular localization = subcritical gain Higer dimensions Phase-space localized spectral clusters: If ˆf (ξ 1, ξ 2 ) is localized to ξ 2 /ξ 1 [θ, 2θ], then we can prove good bounds on f L p over slabs S of size θ in x 1 direction. "! #! Problem: add up over θ 1 slabs lose θ 1/p for f L p (M).
26 Angular localization = subcritical gain Higer dimensions Phase-space localized spectral clusters: If ˆf (ξ 1, ξ 2 ) is localized to ξ 2 /ξ 1 [θ, 2θ], then we can prove good bounds on f L p over slabs S of size θ in x 1 direction. "! #! Problem: add up over θ 1 slabs lose θ 1/p for f L p (M).
27 Angular localization = subcritical gain Higer dimensions For subcritical p > 6, gain from small angle localization If ˆf θ is localized to a cone of angle θ, then f θ L p (S) θ p λ δ(p) f θ L 2 (M) Combined gain loss for f θ f θ L p (M) θ p λ δ(p) f θ L 2 (M) Sum over dyadic decomp in θ 1 yields f L p (M) λ δ(p) f L 2 (M), p 8
28 Angular localization = subcritical gain Higer dimensions For subcritical p > 6, gain from small angle localization If ˆf θ is localized to a cone of angle θ, then f θ L p (S) θ p λ δ(p) f θ L 2 (M) Combined gain loss for f θ f θ L p (M) θ p λ δ(p) f θ L 2 (M) Sum over dyadic decomp in θ 1 yields f L p (M) λ δ(p) f L 2 (M), p 8
29 Angular localization = subcritical gain Higer dimensions For subcritical p > 6, gain from small angle localization If ˆf θ is localized to a cone of angle θ, then f θ L p (S) θ p λ δ(p) f θ L 2 (M) Combined gain loss for f θ f θ L p (M) θ p λ δ(p) f θ L 2 (M) Sum over dyadic decomp in θ 1 yields f L p (M) λ δ(p) f L 2 (M), p 8
30 Gliding modes: θ = λ 1/3 Angular localization = subcritical gain Higer dimensions On slab S size λ 1/3 in x 1 : Sum over slabs: f L 6 (S) λ1/6 f L 2 (M) f L 6 (M) λ1/6+1/18 f L 2 (M) #!!1/3 "!!2/3
31 Gliding modes: θ = λ 1/3 Angular localization = subcritical gain Higer dimensions On slab S size λ 1/3 in x 1 : Sum over slabs: f L 8 (S) λ1/4 λ 1/24 f L 2 (M) f L 8 (M) λ1/4 f L 2 (M) #!!1/3 "!!2/3
32 Higher dimensional results: n 3 Angular localization = subcritical gain Higer dimensions Smith-Sogge [2007]: M = n dimensional manifold with boundary No-loss square function / spectral cluster estimates hold with { δ(p) = n( p ) 1 5 p, n = p, n 4 Result non-optimal: ignores dispersion tangent to M.
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