Improvement of eigenfunction estimates on manifolds of nonpositive curvature

  • 1 Mathematical Sciences Institute, Australian National University, Canberra 0200 ACT, Australia
  • 2 Department of Mathematics, Northwestern University, Evanston 60208 IL, USA


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 L2Lp 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 Lp estimates for all p > 2(n+1)/(n-1).

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