Localized $L^{p}$-estimates for eigenfunctions: II (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

If (M; g) is a compact Riemannian manifold of dimension n ≥ 2 we give necessary and sufficient conditions for improved Lp(M)-norms of eigenfunctions for all 2 < p ≠ pc = 2(n+1)/n-1, the critical exponent. Since improved Lpc (M) bounds imply improvement all other exponents, these conditions are necessary for improved bounds for the critical space. We also show that improved Lpc (M) bounds are valid if these conditions are met and if the half-wave operators, U(t), have no caustics when t ≠ 0. The problem of finding a necessary and sufficient condition for Lpc (M) improvement remains an interesting open problem.