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14
Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces, preprint
"... Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microl ..."
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Cited by 18 (7 self)
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Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microlocal analysis, and only takes a bit over a dozen pages after the statement of notation. It resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis, including semiclassical analysis. The rest of the paper is devoted to applications. Many natural applications arise in the setting of nonRiemannian bmetrics in the context of Melrose’s bstructures. These include asymptotically Minkowski metrics, asymptotically de Sittertype metrics on a blowup of the natural compactification and Kerrde Sittertype metrics. The simplest application, however, is to provide a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces). The results include, in particular, a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. For these results, only Section 2 and Section 4.44.9, starting with the paragraph of (4.8), are strictly needed. The appendix written by Dyatlov relates his analysis of resonances on exact Kerrde Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here. 1.
The wave equation on asymptotically de Sitterlike spaces
, 2007
"... Abstract. In this paper we obtain the asymptotic behavior of solutions of the KleinGordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Un ..."
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Cited by 10 (6 self)
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Abstract. In this paper we obtain the asymptotic behavior of solutions of the KleinGordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y−. 1.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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ASYMPTOTICS OF SOLUTIONS OF THE WAVE EQUATION ON DE SITTERSCHWARZSCHILD SPACE
"... Abstract. Solutions to the wave equation on de SitterSchwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal in nity, with corresponding uniform decay on the appropriately compacti ed space. 1. ..."
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Cited by 5 (3 self)
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Abstract. Solutions to the wave equation on de SitterSchwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal in nity, with corresponding uniform decay on the appropriately compacti ed space. 1.
A SUPPORT THEOREM FOR THE RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS
, 709
"... We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is ..."
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Cited by 4 (0 self)
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We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering
ASYMPTOTICS OF RADIATION FIELDS IN ASYMPTOTICALLY MINKOWSKI SPACE
"... Abstract. We consider a nontrapping ndimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen i ..."
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Cited by 2 (2 self)
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Abstract. We consider a nontrapping ndimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the “northern cap ” of the compactification. For small perturbations of Minkowski space that fit into our framework, we show a rate of decay that improves on the Klainerman–Sobolev estimates. 1.
RADIATION FIELDS ON SCHWARZSCHILD SPACETIME
"... Abstract. In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some reg ..."
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Cited by 1 (1 self)
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Abstract. In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field. 1.
Radiation fields for semilinear wave equations
 In preparation
, 2012
"... Abstract. We define the radiation fields of solutions to critical semilinear wave equations in R 3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to ..."
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Cited by 1 (1 self)
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Abstract. We define the radiation fields of solutions to critical semilinear wave equations in R 3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to this setting and can also be interpreted as a PaleyWiener theorem for the distorted nonlinear Fourier transform of radial functions. 1.
On Lars Hörmander’s remark on the characteristic Cauchy problem
, 2005
"... We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surfa ..."
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We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are L ∞ loc, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential, namely a C1 metric and a potential with continuous first order terms and locally L ∞ coefficients for the terms of order 0. Résumé Nous étendons des résultats dus à L. Hörmander [9] concernant la résolution du problème de Cauchy caractéristique pour des équations d’onde du second ordre avec un potentiel régulier du premier ordre. Le cadre géométrique de [9] était un espacetemps spatialement compact avec une métrique régulière. L’hypersurface sur laquelle les données initiales sont fixées était spatiale ou caractéristique en chaque point et simplement de régularité Lipschitz. Nous affaiblissons les hypothèses de régularité sur la métrique et le potentiel et nous obtenons des résultats analogues. Le problème de Cauchy pour une hypersurface spatiale est résolu dans le cas d’une métrique Lipschitz et pour un potentiel dont les coefficients sont localement L ∞ , avec le même espace de solutions que dans le cas régulier. Nous résolvons également le problème de Cauchy totalement caractéristique dans un cadre très légèrement plus régulier: une métrique C 1 et un potentiel dont les coefficients des termes du premier ordre sont continus et ceux des termes d’ordre 0 sont localement L ∞.
THE RADIATION FIELD IS A FOURIER INTEGRAL OPERATOR
, 2003
"... In this note, we exhibit explicitly the form of the “radiation field ” of F. G. Friedlander on two different types of manifolds: scattering manifolds, and asymptotically hyperbolic manifolds. The former class consists of manifolds with ends that look asymptotically like the large ends of cones, and ..."
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In this note, we exhibit explicitly the form of the “radiation field ” of F. G. Friedlander on two different types of manifolds: scattering manifolds, and asymptotically hyperbolic manifolds. The former class consists of manifolds with ends that look asymptotically like the large ends of cones, and