PDE Seminar Abstracts

Spherical harmonics with maximal norm growth

Xiaolong Han
Australian National University, Australia
Tuesday 21 April 2015, 12-1pm, Carslaw Room 829 (AGR)


Sogges Lp-estimates give upper bounds to the Lp-norms of L2-normalized eigenfunctions on smooth and compact manifolds. They are also sharp on the sphere, where Gaussian beams saturate the maximal Lp-norm growth for small p and zonal harmonics for large p. But these maximisers are very sparse in the orthonormal basis of eigenfunctions (ONBE); their densities are both zero. Sogge and Zelditch proposed the question of whether there exists a manifold supporting an ONBE that contains a positive density subsequence of maximisers. In this talk, I will give the positive answer to this question and construct such example for all small p.