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

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Spectral projections kernel = zonal harmonic A key object in the theory of spherical harmonics is the orthogonal projector Π k : L 2 (S 2 ) → H k . It has a kernel Π k (x, y) defined by ∫ Π k f (x) = Π k (x, y)f (y)dS(y). S n If {Y k m} is an orthonormal basis of H k then d k ∑ Π k (x, y) = Ym(x)Y k m(y). k j=1
Coherent state Φ x k at x For each y, Π k (x, y) ∈ H k . We can L 2 normalize this function by dividing by the square root of ∫ ||Π k (·, y)|| 2 L = Π 2 k (x, y)Π k (y, x)dS(x) = Π k (x, x). S n We get Φ x k (y) = Π k(x, y) √ Πk (x, x) .
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: Oscillation of a spherical harmonic
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 and 36: Sketch of proof ◮ Suffices to sho
Page 37 and 38: Local Weyl law and L ∞ estimates
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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