TY - JOUR

T1 - Restriction and spectral multiplier theorems on asymptotically conic manifolds

AU - Guillarmou, Colin

AU - Hassell, Andrew

AU - Sikora, Adam

PY - 2013

Y1 - 2013

N2 - 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 ℝn is bounded from Lp(ℝn) to Lp′(ℝn) for 1 ≤ p ≤ 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 ℝn 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(1/p-1/p′)-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.

AB - 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 ℝn is bounded from Lp(ℝn) to Lp′(ℝn) for 1 ≤ p ≤ 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 ℝn 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(1/p-1/p′)-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.

KW - Asymptotically conic manifolds

KW - Bochner-Riesz summability

KW - Restriction estimates

KW - Spectral multipliers

UR - http://www.scopus.com/inward/record.url?scp=84886948277&partnerID=8YFLogxK

U2 - 10.2140/apde.2013.6.893

DO - 10.2140/apde.2013.6.893

M3 - Article

SN - 2157-5045

VL - 6

SP - 893

EP - 950

JO - Analysis and PDE

JF - Analysis and PDE

IS - 4

ER -