Localized $L^p$-estimates for eigenfunctions: II

Abstract

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

12 pages

References in corpus (1)

• Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions

Cited by in corpus (2)

• Logarithmic improvements in $L^p$ bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature
• Endpoint resolvent estimates for compact Riemannian manifolds