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

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 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).

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


Journal + Issues

Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

Search