Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

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Heuristics We will study the remainder in the pointwise Weyl law rather than eigenfunctions per se. Since eigenfunctions are more intuitive, here is a heuristic guide to what we prove: ◮ |ϕ j (x)| 2 can only achieve the maximum possible size at or very near a self-focal point. ◮ If the first return map has no invariant L 1 measures at the self-focal point, then |ϕ j (x)| 2 cannot achieve the maximum possible size right at the self-focal point. ◮ Using the smoothess of the remainder in Weyl’s law, we also prove that if it does not have the maximal order of magnitude at the self-focal point, then it cannot be maximally large very near it either.
Local Weyl law and L ∞ estimates of eigenfunctions E λ (x, x) = ∑ λ ν≤λ |ϕ ν(x)| 2 with uniform remainder bounds = (2π) −n ∫ p(x,ξ)≤λ dξ + R(λ, x) |R(λ, x)| ≤ Cλ n−1 , x ∈ M. Since the integral in the local Weyl law is a continuous function of λ and since the spectrum of the Laplacian is discrete, this immediately gives ∑ |ϕ ν (x)| 2 ≤ 2Cλ n−1 which in turn yields λ ν=λ on any compact Riemannian manifold. L ∞ (λ, g) = 0(λ n−1 2 ) (9)
Page 1 and 2: Park City<
Page 3 and 4: L p norms We recall that ∆ϕ λ =
Page 5 and 6: Sogge upper bounds on L p norms of
Page 7 and 8: Extremals The upper bounds are shar
Page 9 and 10: Spherical harmonics of degree k Let
Page 11 and 12: Oscillation of a spherical harmonic
Page 13 and 14: Coherent state Φ x k at x For each
Page 15 and 16: Normalized zonal harmonics On S n ,
Page 17 and 18: Gaussian Beam: maximizes L p norms
Page 19 and 20: Intuitive principle The fact that t
Page 21 and 22: Lagrangian distributions The Lagran
Page 23 and 24: What is the Lagrangian submanifold?
Page 25 and 26: Intensity plot of an ergodic eigenf
Page 27 and 28: First condition for maximal eigenfu
Page 29 and 30: Special points and submanifolds ◮
Page 31 and 32: Additional necessary conditions For
Page 33 and 34: Pole of a surface of revolution and
Page 35: Sketch of proof ◮ Suffices to sho
Page 39 and 40: Notation for smoothed remainder Let
Page 41 and 42: Many charts, many oscillatory integ
Page 43 and 44: Pointwise Weyl asymptotics: Safarov
Page 45 and 46: Decomposition of R 1 into loops and
Page 47 and 48: Ergodic theory at the self-focal po
Page 49 and 50: Uniform o(λ n−1 ) outside of sma
Page 51: Summing up ◮ No eigenfunction ϕ

Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

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