A note on $L^p$-norms of quasi-modes

Abstract

In this note we show how improved $L^p$-estimates for certain types of quasi-modes are naturally equaivalent to improved operator norms of spectral projection operators associated to shrinking spectral intervals of the appropriate scale. Using this, one can see that recent estimates that were stated for eigenfunctions also hold for the appropriate types of quasi-modes.

11 pages

References in corpus (3)

• Concerning the $L^4$ norms of typical eigenfunctions on compact surfaces
• Focal points and sup-norms of eigenfunctions on analytic Riemannian manifolds II: the two-dimensional case
• Logarithmic-scale Quasimodes that do not Equidistribute

Cited by in corpus (2)

• Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions
• Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams