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# Mathematical Research Letters

## Volume 24 (2017)

### Number 2

### Improved critical eigenfunction estimates on manifolds of nonpositive curvature

Pages: 549 – 570

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n2.a14

#### Author

#### Abstract

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.

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

Received 14 March 2016

Published 24 July 2017