Localized $L^p$-estimates for eigenfunctions: II
  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 $2<p\ne p_c=\tfrac{2(n+1)}{n-1}$, 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. 
Christopher D. Sogge
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