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 $Δ$ be the positive Laplacian on $M$ determined by $g$. We study the $L^{2}\to{}L^{p}$ mapping properties of a spectral cluster of $\sqrtΔ$ of width $1/\logλ$. Under the geometric assumptions above, \cite{berard77} Bérard obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a $(\logλ)^{1/2}$ improvement for Hörmander's estimate on the $L^{\infty}$ norms of eigenfunctions. In this paper we extend this improvement to the $L^p$ estimates for all $p>\frac{2(n+1)}{n-1}$.

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