Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 4 (2013), 893-950.

Restriction and spectral multiplier theorems on asymptotically conic manifolds

Colin Guillarmou, Andrew Hassell, and Adam Sikora

Full-text: Open access

Abstract

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 1p2(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 LpLp 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.

Article information

Source
Anal. PDE, Volume 6, Number 4 (2013), 893-950.

Dates
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
https://projecteuclid.org/euclid.apde/1513731384

Digital Object Identifier
doi:10.2140/apde.2013.6.893

Mathematical Reviews number (MathSciNet)
MR3092733

Zentralblatt MATH identifier
1293.35187

Subjects
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]

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

Citation

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


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