Restriction and spectral multiplier theorems on asymptotically conic manifolds

Guillarmou, Colin; Hassell, Andrew; Sikora, Adam

doi:10.2140/apde.2013.6.893

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Restriction and spectral multiplier theorems on asymptotically conic manifolds

Colin Guillarmou, Andrew Hassell and Adam Sikora

Vol. 6 (2013), No. 4, 893–950

DOI: 10.2140/apde.2013.6.893

Abstract

The classical Stein–Tomas restriction theorem is equivalent to the fact that the spectral measure $d “ E (λ)$ of the square root of the Laplacian on $ℝ^{n}$ is bounded from $L^{p} (ℝ^{n})$ to $L^{p^{'}} (ℝ^{n})$ for $1 \leq p \leq 2 (n + 1) ∕ (n + 3)$ , where $p^{'}$ is the conjugate exponent to $p$ , with operator norm scaling as $λ^{n (1 ∕ p - 1 ∕ p^{'}) - 1}$ . We prove a geometric, or variable coefficient, generalization in which the Laplacian on $ℝ^{n}$ is replaced by the Laplacian, plus a suitable potential, on a nontrapping asymptotically conic manifold. It is closely related to Sogge’s discrete $L^{2}$ restriction theorem, which is an $O (λ^{n (1 ∕ p - 1 ∕ p^{'}) - 1})$ estimate on the $L^{p} \to L^{p^{'}}$ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner–Riesz summability results, which are sharp for $p$ in the range above.

The paper divides naturally into two parts. In the first part, we show at an abstract level that restriction estimates imply spectral multiplier estimates, and are implied by certain pointwise bounds on the Schwartz kernel of $λ$ -derivatives of the spectral measure. In the second part, we prove such pointwise estimates for the spectral measure of the square root of Laplace-type operators on asymptotically conic manifolds. These are valid for all $λ > 0$ if the asymptotically conic manifold is nontrapping, and for small $λ$ in general. We also observe that Sogge’s estimate on spectral projections is valid for any complete manifold with $C^{\infty}$ bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on $d “ E (λ)$ may blow up exponentially as $λ \to \infty$ when trapping is present.

Keywords

restriction estimates, spectral multipliers, Bochner–Riesz summability, asymptotically conic manifolds

Mathematical Subject Classification 2010

Primary: 35P25, 42BXX

Secondary: 58J50, 47A40

Milestones

Received: 1 December 2011

Revised: 23 August 2012

Accepted: 23 September 2012

Published: 21 August 2013

Authors

Colin Guillarmou
DMA, U.M.R. 8553 CNRS École Normale Supérieure 45 rue d’Ulm 75005 Paris CEDEX 05 France

Andrew Hassell
Department of Mathematics Australian National University Canberra ACT 0200 Australia

Adam Sikora
Department of Mathematics Macquarie University Sydney NSW 2109 Australia

Vol. 6, No. 4, 2013