Vol. 6, No. 4, 2013

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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

The classical Stein–Tomas restriction theorem is equivalent to the fact that the spectral measure dE(λ) of the square root of the Laplacian on n is bounded from Lp(n) to Lp (n) for 1 p 2(n + 1)(n + 3), where p is the conjugate exponent to p, with operator norm scaling as λn(1p1p)1 . We prove a geometric, or variable coefficient, generalization in which the Laplacian on n is replaced by the Laplacian, plus a suitable potential, on a nontrapping asymptotically conic manifold. It is closely related to Sogge’s discrete L2 restriction theorem, which is an O(λn(1p1p)1 ) estimate on the Lp Lp 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 bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on dE(λ) may blow up exponentially as λ when trapping is present.

restriction estimates, spectral multipliers, Bochner–Riesz summability, asymptotically conic manifolds
Mathematical Subject Classification 2010
Primary: 35P25, 42BXX
Secondary: 58J50, 47A40
Received: 1 December 2011
Revised: 23 August 2012
Accepted: 23 September 2012
Published: 21 August 2013
Colin Guillarmou
DMA, U.M.R. 8553 CNRS
École Normale Supérieure
45 rue d’Ulm
75005 Paris CEDEX 05
Andrew Hassell
Department of Mathematics
Australian National University
Canberra ACT 0200
Adam Sikora
Department of Mathematics
Macquarie University
Sydney NSW 2109