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 p estimates for all p > 2( n +1)/( n -1).
Forum Mathematicum – de Gruyter
Published: May 1, 2015
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