## Abstract

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} → *L ^{p}* mapping properties of a spectral cluster of (Δ)

^{1/2}of width 1/log λ. Under the geometric assumptions above, Bérard [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 λ)

^{1/2}improvement for Hörmander's estimate on the

*L*

^{∞}norms of eigenfunctions. In this paper we extend this improvement to the

*L*estimates for all

^{p}*p*> 2(

*n*+1)/(

*n*-1).

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