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 Delta be the positive Laplacian on M determined by g. We study the L-2 -> L-p mapping properties of a spectral cluster of root Delta of width 1/log lambda. Under the geometric assumptions above, Berard [Math. Z. 155 (1977), 249-276] obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a (log lambda)(1/2) improvement for Hormander's estimate on the L-infinity norms of eigenfunctions. In this paper we extend this improvement to the L-p estimates for all p > 2(n+1)/n-1.