Improvement of eigenfunction estimates on manifolds of nonpositive curvature

Let $(M,g)$ be a compact, boundaryless manifold of dimension $n$ with the property that either (i) $n=2$ and $(M,g)$ has no conjugate points, or (ii) the sectional curvatures of $(M,g)$ are nonpositive. Let $\mathrm{\Delta }$ be the positive Laplacian on $M$ determined by $g$. We study the $L^2\to L^p$ mapping properties of a spectral cluster of $\sqrt{\mathrm{\Delta }}$ of width $1\mathrm{/}\log \lambda$. Under the geometric assumptions above, \cite{berard77} B'{e}rard obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a $(\log \lambda {)}^{1\mathrm{/}2}$ improvement for H"ormander’s estimate on the $L^{\mathrm{\infty }}$ norms of eigenfunctions. In this paper we extend this improvement to the $L^p$ estimates for all $p>\frac{2(n+1)}{n-1}$.