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

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.