Analysis & PDE
- Anal. PDE
- Volume 6, Number 4 (2013), 893-950.
Restriction and spectral multiplier theorems on asymptotically conic manifolds
The classical Stein–Tomas restriction theorem is equivalent to the fact that the spectral measure of the square root of the Laplacian on is bounded from to for , where is the conjugate exponent to , with operator norm scaling as . We prove a geometric, or variable coefficient, generalization in which the Laplacian on is replaced by the Laplacian, plus a suitable potential, on a nontrapping asymptotically conic manifold. It is closely related to Sogge’s discrete restriction theorem, which is an estimate on the 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 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 -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 if the asymptotically conic manifold is nontrapping, and for small in general. We also observe that Sogge’s estimate on spectral projections is valid for any complete manifold with bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on may blow up exponentially as when trapping is present.
Anal. PDE, Volume 6, Number 4 (2013), 893-950.
Received: 1 December 2011
Revised: 23 August 2012
Accepted: 23 September 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35P25: Scattering theory [See also 47A40] 42BXX
Secondary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
Guillarmou, Colin; Hassell, Andrew; Sikora, Adam. Restriction and spectral multiplier theorems on asymptotically conic manifolds. Anal. PDE 6 (2013), no. 4, 893--950. doi:10.2140/apde.2013.6.893. https://projecteuclid.org/euclid.apde/1513731384