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


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

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]

restriction estimates spectral multipliers Bochner–Riesz summability asymptotically conic manifolds


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.

Export citation


  • J.-P. Anker, “${\bf L}\sb p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type”, Ann. of Math. $(2)$ 132:3 (1990), 597–628.
  • V. M. Babich and V. F. Lazutkin, “Eigenfunctions concentrated near a closed geodesic”, pp. 9–18 in Topics in Mathematical Physics, vol. 2, edited by M. S. Birman, Consultant's Bureau, New York, 1968.
  • N. Burq, C. Guillarmou, and A. Hassell, “Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics”, Geom. Funct. Anal. 20:3 (2010), 627–656.
  • F. Cardoso and G. Vodev, “Uniform estimates of the resolvent of the Laplace–Beltrami operator on infinite volume Riemannian manifolds. II”, Ann. Henri Poincaré 3:4 (2002), 673–691.
  • J. Cheeger, M. Gromov, and M. Taylor, “Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds”, J. Differential Geom. 17:1 (1982), 15–53.
  • S. Y. Cheng, P. Li, and S. T. Yau, “On the upper estimate of the heat kernel of a complete Riemannian manifold”, Amer. J. Math. 103:5 (1981), 1021–1063.
  • F. M. Christ and C. D. Sogge, “The weak type $L\sp 1$ convergence of eigenfunction expansions for pseudodifferential operators”, Invent. Math. 94:2 (1988), 421–453.
  • J. L. Clerc and E. M. Stein, “$L\sp{p}$-multipliers for noncompact symmetric spaces”, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911–3912.
  • M. Cowling and A. Sikora, “A spectral multiplier theorem for a sublaplacian on $\rm SU(2)$”, Math. Z. 238:1 (2001), 1–36.
  • X. T. Duong, E. M. Ouhabaz, and A. Sikora, “Plancherel-type estimates and sharp spectral multipliers”, J. Funct. Anal. 196:2 (2002), 443–485.
  • C. Fefferman, “Inequalities for strongly singular convolution operators”, Acta Math. 124 (1970), 9–36.
  • C. Fefferman, “A note on spherical summation multipliers”, Israel J. Math. 15 (1973), 44–52.
  • C. Guillarmou and A. Hassell, “Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds, I”, Math. Ann. 341:4 (2008), 859–896.
  • C. Guillarmou and A. Hassell, “Uniform Sobolev estimates for non-trapping metrics”, preprint, 2012.
  • C. Guillarmou, A. Hassell, and A. Sikora, “Resolvent at low energy, III: The spectral measure”, preprint, 2012. To appear in Trans. Amer. Math. Soc.
  • A. Hassell and A. Vasy, “The spectral projections and the resolvent for scattering metrics”, J. Anal. Math. 79 (1999), 241–298.
  • A. Hassell and A. Vasy, “The resolvent for Laplace-type operators on asymptotically conic spaces”, Ann. Inst. Fourier $($Grenoble$)$ 51:5 (2001), 1299–1346.
  • A. Hassell and J. Wunsch, “The semiclassical resolvent and the propagator for non-trapping scattering metrics”, Adv. Math. 217:2 (2008), 586–682.
  • L. H örmander, “Estimates for translation invariant operators in $L\sp{p}$ spaces”, Acta Math. 104 (1960), 93–140.
  • L. H örmander, The analysis of linear partial differential operators, I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften 256, Springer, Berlin, 1983.
  • L. H örmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften 274, Springer, Berlin, 1985.
  • G. E. Karadzhov, “Riesz summability of multiple Hermite series in $L\sp p$ spaces”, C. R. Acad. Bulgare Sci. 47:2 (1994), 5–8.
  • C. E. Kenig, A. Ruiz, and C. D. Sogge, “Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators”, Duke Math. J. 55:2 (1987), 329–347.
  • H. Koch and F. Ricci, “Spectral projections for the twisted Laplacian”, Studia Math. 180:2 (2007), 103–110.
  • H. Koch and D. Tataru, “$L\sp p$ eigenfunction bounds for the Hermite operator”, Duke Math. J. 128:2 (2005), 369–392.
  • S. Lee, “Improved bounds for Bochner–Riesz and maximal Bochner-Riesz operators”, Duke Math. J. 122:1 (2004), 205–232.
  • G. Mauceri and S. Meda, “Vector-valued multipliers on stratified groups”, Rev. Mat. Iberoamericana 6:3-4 (1990), 141–154.
  • R. B. Melrose, “Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces”, pp. 85–130 in Spectral and scattering theory (Sanda, Japan, 1992), edited by M. Ikawa, Lecture Notes in Pure and Appl. Math. 161, Dekker, New York, 1994.
  • S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford, 1965.
  • D. Müller and E. M. Stein, “On spectral multipliers for Heisenberg and related groups”, J. Math. Pures Appl. $(9)$ 73:4 (1994), 413–440.
  • S. Nonnenmacher and M. Zworski, “Quantum decay rates in chaotic scattering”, Acta Math. 203:2 (2009), 149–233.
  • J. V. Ralston, “Approximate eigenfunctions of the Laplacian”, J. Differential Geometry 12:1 (1977), 87–100.
  • A. Seeger and C. D. Sogge, “On the boundedness of functions of (pseudo-) differential operators on compact manifolds”, Duke Math. J. 59:3 (1989), 709–736.
  • A. Sikora, “Riesz transform, Gaussian bounds and the method of wave equation”, Math. Z. 247:3 (2004), 643–662.
  • C. D. Sogge, “On the convergence of Riesz means on compact manifolds”, Ann. of Math. $(2)$ 126:2 (1987), 439–447.
  • C. D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics 105, Cambridge University Press, 1993.
  • E. M. Stein, “Interpolation of linear operators”, Trans. Amer. Math. Soc. 83 (1956), 482–492.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.
  • K. Stempak and J. Zienkiewicz, “Twisted convolution and Riesz means”, J. Anal. Math. 76 (1998), 93–107.
  • T. Tao, “Recent progress on the restriction conjecture”, preprint, 2003.
  • M. E. Taylor, “$L\sp p$-estimates on functions of the Laplace operator”, Duke Math. J. 58:3 (1989), 773–793.
  • S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes 42, Princeton University Press, 1993.
  • P. A. Tomas, “A restriction theorem for the Fourier transform”, Bull. Amer. Math. Soc. 81 (1975), 477–478.
  • H. Triebel, Theory of function spaces. II, Monographs in Mathematics 84, Birkhäuser, Basel, 1992.
  • A. Zygmund, “On Fourier coefficients and transforms of functions of two variables”, Studia Math. 50 (1974), 189–201.