A note on constructing families of sharp examples for 𝐿^{𝑝} growth of eigenfunctions and quasimodes

Abstract

In this note we analyse L p estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition we produce a family flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check L p growth estimates against the simple flat model examples. Finally we present a heuristic that for any subset determines which quasimode in the family is expected to produce sharp examples.

Eigenfunction concentration via geodesic beams

“…Note that in this paper we study averages of relatively weak quasimodes for the Laplacian with no additional assumptions on the functions. This is in contrast with results which impose additional conditions on the functions such as: that they be Laplace eigenfunctions that simultaneously satisfy additional equations [IS95, GT18b,Tac18]; that they be eigenfunctions in the very rigid case of the flat torus [Bou93,Gro85]; or that they form a density one subsequence of Laplace eigenfunctions [JZ16].…”

Logarithmic improvements in $L^{p}$ bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature

“…However, by carefully splitting into oscillatory regions where stationary phase can be applied and θ-dependent non-oscillatory regions similar to (1.12), it can be seen that at p = p c , f λ,θ L pc λ 1/pc for any θ ∈ [λ −1/2 , 1], hence its designation as the "critical" exponent. These computations were carried out rigorously in [Tac18].…”