L p eigenfunction estimates and directional oscillation Melissa Tacy Department of Mathematics Northwestern University mtacy@math.northwestern.edu 20 June 2012
Eigenfunction Concentration Would like to understand behaviour of eigenfunctions of Laplace-Beltrami and similar operators. Let M be a compact Riemannian manifold without boundary − ∆ u j = λ 2 j u j How large can u j be ? Where can u j be large? What do concentrations look like?
L p Eigenfunction Estimates Seek estimates of the form | | u j | | L p � f ( λ j , p ) | | u j | | L 2 and sharp examples Expect properties of classical flow to be evident in estimates. Not easy to study eigenfunctions directly. Therefore we will study sums (clusters) of eigenfunctions.
Spectral Windows We study norms of spectral clusters on windows of width w � E λ = E j λ j ∈ [ λ − w ,λ + w ] E j projection onto λ j eigenspace. Obviously include eigenfunctions but also can include sums of eigenfunctions if w is large enough. Shrinking the window avoids pollution of estimates by eigenfunctions of similar eigenvalue.
Smoothed Spectral Clusters Pick χ smooth such that χ (0) = 1 and ˆ χ is supported in [ ǫ, 2 ǫ ]. We will study √ χ λ, A = χ ( A ( − ∆ − λ )) Write � 2 ǫ √ − ∆ e − itA λ ˆ e itA χ λ, A = χ ( t ) dt ǫ √ − ∆ as an integral operator with kernel If we can write e itA e ( x , y , t , A ) we can write � 2 ǫ � e ( x , y , t , A ) e − itA λ ˆ χ λ, A u = χ ( t ) u ( y ) dtdy ǫ M
Spectral Window Width One √ − ∆ is the fundamental solution to The operator e it � √ ( i ∂ t + − ∆) U ( t ) = 0 U (0) = δ y We can build a parametrix for this propagator and write its kernel as � ∞ e i θ ( d ( x , y ) − t ) a ( x , y , t , θ ) d θ e ( x , y , t ) = 0 where a ( x , y , t , θ ) has principal symbol n − 1 2 a 0 ( x , y , t ) θ
Expression for χ λ, 1 Substituting into the expression for χ λ � 2 ǫ � � ∞ n − 1 e i θ ( d ( x , y ) − t ) e − it λ θ 2 ˜ χ λ, 1 u = a ( x , y , t , θ ) u ( y ) d θ dydt 0 ǫ M Change of variables θ → λθ � 2 ǫ � � ∞ n +1 n − 1 e i λθ ( d ( x , y ) − t ) e − it λ θ 2 ˜ χ λ, 1 u = λ a ( x , y , t , θ ) u ( y ) d θ dydt 2 ǫ M 0 Now use stationary phase in ( t , θ ). Nondegenerate critical points when d ( x , y ) = t θ = 1 � n − 1 e i λ d ( x , y ) a ( x , y ) u ( y ) dy χ λ, 1 = λ 2 M where a ( x , y ) is supported away from the diagonal.
Estimates for χ λ, 1 In 1988 Sogge obtained a full set of L 2 → L p estimates for χ λ, 1 . Technique depends on TT ⋆ method, need to estimate � λ n − 1 e i λ ( d ( x , z ) − d ( z , y )) a ( x , z )¯ a ( z , y ) dz M Bound depends on | x − y | Interpolate with L 2 estimates and apply Hardy-Littlewood-Sobolev
Decaying Spectral Window Width Assume the window width w = 1 / A → 0 as λ → ∞ . We need to evaluate � √ − ∆ e it λ dt e it t < A Cannot achieve this on any manifold but for manifolds without conjugate point we can use the universal cover. If M has no conjugate points its universal cover � M is a manifold with infinite injectivity radius. Therefore we can find a solution for � √ ( i ∂ t + − ∆ � M ) U ( t ) = 0 U (0) = δ y for all time on � M
Expression for Propagator Kernel √ − ∆ has kernel e it � e ( x , y , t ) = ˜ e ( x , γ y , t ) γ ∈ Γ where Γ is the group of automorphisms of the covering π : � M → M and the fundamental solution of � √ ( i ∂ t + − ∆ � M ) U ( t ) = 0 U (0) = δ y is given by � U ( t ) u = ˜ e ( x , y , t ) u ( y ) dy � M
Technical Difficulties No longer have strong relationships between distance and time. Cannot use HLS as before.
Directionally Localised Examples Quasimode localised in direction ξ Well defined direction of oscillation for short time. Can obtain “good” L p bounds. Consider general quasimode as a sum of directionally localised ones.
Directional Localisation Return to window width one for simplicity Let { ξ j } be a set of λ − 1 2 separated directions in S n − 1 . ζ ( η ) a smooth, cut off function of scale supported when | η | ≤ 2 λ − 1 2 . x i a set of λ − 1 separated points in M . β ( x ) a cut off function supported when | x | ≤ 2 λ − 1 . � x − y � � n − 1 e i λ d ( x , y ) ζ 2 β ( x − x i ) χ λ, 1 ( ξ j , x i ) = λ | x − y | − ξ j a ( x , y ) u ( y ) dy M
Interaction of Directional Oscillation Let K i , j , l , m ( x , z ) be the kernel of χ λ, 1 ( ξ j , x i )( χ λ, 1 ( ξ m , x l )) ⋆ . Then for x i far enough from x l , | K i , j , l , m ( x , z ) | decays if x i − x l | x i − x l | � = ξ j ξ j � = ξ m Otherwise non-stationary phase d ( x , z ) − d ( z , y ) Take geometric averages localised at different points and many directions.
Image due to Alex Barnett
Image due to Alex Barnett
Application to Clusters with Decaying Window Width Can still define directionally localised projectors and these give good L 2 → L p estimates. Can get cancellation as long as frequency localisation is not too large Means we can run up to Ehrenfest time
Recommend
More recommend
