Improved critical eigenfunction estimates on manifolds of nonpositive curvature

We prove new improved endpoint, $L^{p_c} , p_c = \dfrac{2(n+1)}{n-1}$, estimates (the “kink point”) for eigenfunctions on manifolds of nonpositive curvature. We do this by using energy and dispersive estimates for the wave equation as well as new improved $L^p, 2 \lt p \lt p_c$, bounds of Blair and the author [4], [6] and the classical improved sup-norm estimates of Bérard [3]. Our proof uses Bourgain’s [7] proof of weak-type estimates for the Stein–Tomas Fourier restriction theorem [42]–[43] as a template to be able to obtain improved weaktype $L^{p_c}$ estimates under this geometric assumption. We can then use these estimates and the (local) improved Lorentz space estimates of Bak and Seeger [2] (valid for all manifolds) to obtain our improved estimates for the critical space under the assumption of nonpositive sectional curvatures.

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The author was supported in part by the NSF grant DMS-1361476.

Received 14 March 2016

Accepted 30 June 2016

Published 24 July 2017