Defect measures of eigenfunctions with maximal $L^{\infty }$ growth

Abstract

We study the relationship between $L^{\infty }$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we characterize the defect measures of any sequence of eigenfunctions with maximal $L^{\infty }$ growth, showing that they must be neither more concentrated nor more diffuse than the zonal harmonics. As a consequence, we obtain new proofs of results on the geometry manifolds with maximal eigenfunction growth obtained by Sogge--Zelditch, and Sogge--Toth--Zelditch.

29 pages

References in corpus (2)

• Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions
• Eigenfunction scarring and improvements in $L^{\infty }$ bounds