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Forum Mathematicum

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Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 27, Issue 3


Improvement of eigenfunction estimates on manifolds of nonpositive curvature

Andrew Hassell / Melissa Tacy
Published Online: 2013-04-03 | DOI: https://doi.org/10.1515/forum-2012-0176


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).

Keywords: Eigenfunction estimates; nonpositive curvature; manifolds without conjugate points; logarithmic improvement; finite propagation speed

MSC: 35P99

About the article

Received: 2012-12-11

Revised: 2013-01-30

Published Online: 2013-04-03

Published in Print: 2015-05-01

Funding Source: Australian Research Council

Award identifier / Grant number: Future Fellowship FT0990895

Funding Source: Discovery

Award identifier / Grant number: DP1095448

Funding Source: Discovery

Award identifier / Grant number: DP120102019

Citation Information: Forum Mathematicum, Volume 27, Issue 3, Pages 1435–1451, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2012-0176.

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