Lp bounds for eigenfunctions at the critial exponent

Abstract: We consider upper bounds on the growth of $L^pa$ norms of eigenfunctions of the Laplacian on a compact Riemannian manifold in the high frequency limit. In particular, we seek to identify geometric or dynamical conditions on the manifold which yield improvements on the universal $L^p$ bounds of C. Sogge. The emphasis will be on bounds at the "critical exponent", where a spectrum of scenarios for phase space concentration must be considered. We then discuss a recent work with C. Sogge which shows that when the sectional curvatures are nonpositive, there is a logarithmic type gain in the known $L^p$ bounds at the critical exponent.

analysis of PDEsnumber theoryrepresentation theory

Audience: researchers in the topic

Harmonic Analysis and Symmetric Spaces 2021