A note on constructing sharp examples for $L^{p}$ norms of eigenfunctions and quasimodes near submanifolds

TL;DR

This paper analyzes $L^{p}$ estimates for Laplacian eigenfunctions and quasimodes, introduces new restriction estimates near submanifolds, and constructs flat model examples to determine the sharpness of these estimates.

Contribution

It provides new restriction estimates for eigenfunctions near submanifolds and introduces flat model quasimodes to assess the sharpness of $L^{p}$ bounds.

Findings

New restriction estimates for eigenfunctions on thickened neighborhoods of submanifolds.

Construction of flat model quasimodes matching growth properties of spherical harmonics.

Heuristic for identifying sharp examples among quasimodes for arbitrary subsets.

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.