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 -