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We establish $$L^{q*}\rightarrow L^q$$ bounds for the resolvent of the Laplacian on compact Riemannian manifolds assuming only that the sectional curvatures of the manifold are uniformly bounded. When the resolvent parameter lies outside a parabolic neighborhood of $$[0,\infty )$$ , the operator norm of the resolvent is shown to depend only on upper bounds for the sectional curvature and diameter and lower bounds for the volume. The resolvent bounds are derived from square-function estimates for the wave equation, an approach that admits the use of paradifferential approximations in the parametrix construction.

We demonstrate a parametrix construction, together with associated pseudodifferential operator calculus, for an operator of sum-of-squares type with semiclassical parameter. The form of operator we consider includes the generator of kinetic Brownian motion on the cosphere bundle of a Riemannian manifold. Regularity estimates in semiclassical Sobolev spaces are proven by establishing mapping properties for the parametrix.

We establish space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded sectional curvature, with the same exponents as for $C^\infty$ metrics. The estimates are for bounded time intervals, so by finite propagation velocity the results apply also on non-compact manifolds under appropriate uniform conditions. We assume a priori that in local coordinates the metric tensor components satisfy ${\rm g}_{ij}\in W^{1,p}$ for some $p>d$, which ensures that the curvature tensor is well defined in the weak sense, but this can be relaxed to any assumption that suffices for the local harmonic coordinate calculations in the paper.

We establish $L^p$ bounds on $L^2$ normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all $2\le p\le\infty$, up to logarithmic losses for $6

We present a multi-scale solution scheme for hyperbolic evolution equations with curvelets. We assume, essentially, that the second-order derivatives of the symbol of the evolution operator are uniformly Lipschitz. The scheme is based on a solution construction introduced by Smith (1998) [1] and is composed of generating an approximate solution following a paradifferential decomposition of the mentioned symbol, here, with a second-order correction reminiscent of geometrical asymptotics involving a Hamilton–Jacobi system of equations and, subsequently, solving a particular Volterra equation. We analyze the regularity of the associated Volterra kernel, and then determine the optimal quadrature in the evolution parameter. Moreover, we provide an estimate for the spreading of (finite) sets of curvelets, enabling the multi-scale numerical computation with controlled error.

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out in \cite{HMSSZ} this case is more subtle than $n=3$ or 4 due to the fact that the arguments of the first two authors \cite{SmSo00}, Burq \cite{B} and Metcalfe \cite{M} showing that local Strichartz estimates for obstactles imply global ones require that the Sobolev index, $\gamma$, equal 1/2 when $n=2$. We overcome this difficulty by interpolating between energy estimates ($\gamma =0$) and ones for $\gamma=\frac12$ that are generalizations of Minkowski space estimates of Fang and the third author \cite{FaWa2}, \cite{FaWa}, the second author \cite{So08} and Sterbenz \cite{St05}., Comment: Final version, to appear in the Transactions of the AMS. 20 pages, 2 figures

We establish Strichartz estimates for the Schrodinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key $${L^{4}_{t}L^{\infty}_x}$$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrodinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrodinger equations on general compact manifolds with boundary.

We establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twice-differentiable Lame parameters. Precisely, we show that the P↔S components of the wave propagation operator are regularizing of order one on L 2 data, by establishing the diagonalization of the elastic system modulo a L 2-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.

We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagati...

We establish asymptotic bounds on the $L^p$ norms of spectrally localized functions in the case of two-dimensional Dirichlet forms with coefficients of Lipschitz regularity. These bounds are new for the range $6

We prove local Strichartz estimates with a loss of derivatives over compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.

We establish $L^q$ bounds on eigenfunctions, and more generally on spectrally localized functions (spectral clusters), associated to a self-adjoint elliptic operator on a compact manifold, under the assumption that the coefficients of the operator are of regularity $C^s$, where $0\le s\le 1$. We also produce examples which show that these bounds are best possible for the case $q=\infty$, and for $2\le q\le q_n$.

In this paper, we establish L p norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C 1,1 . Precisely, we show that the L p estimates proven by Sogge in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge that such estimates fail for C 1,α metrics if α < 1.

This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data. The new results obtained here are sharp in low dimension.

We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only uses the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as opposed to the more difficult $O(|t|^{-1})$ decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equations that gives $O(1/t)$ decay of solutions of the inhomomogeneous linear wave equation based in terms of $O(1/|x|)$ estimates for the forcing term., This revised version of our paper will appear in the Journal of the American Mathematical Society

On the other hand, in [3] there were constructed for each α < 1 examples of A with coefficients of regularity C for which the same estimates fail to hold. The first author then showed in [1] that, in space dimensions 2 and 3, the estimates do hold if the coefficients of A are C. The second author subsequently showed in [4] that the estimates hold for C metrics in all space dimensions, and that for operators with C coefficients, such estimates hold provided that γ is replaced by γ + σ/p, where σ = 1−α 3+α . Indeed, [5] showed that such estimates hold under the condition that 1 + α derivatives of the coefficients belong to LtL ∞ x , which is important for applications to quasilinear wave equations. The counterexamples of [3] do not coincide with the estimates established by [4], however. In this paper we remedy this gap, by producing examples of time independent C metrics which show that the results established in [4] are indeed best possible. (Strictly speaking, we produce a family of examples, one at each frequency, which show that if the Strichartz estimates hold with a constant C depending only on the C norm of the coefficients, and the assumption that the coefficients are pointwise close to the euclidean metric, then the loss of σ/p derivatives is sharp.) We remark that this construction also produces examples of C metrics which show that the closely related spectral projection estimates for C metrics established by the first author [2] are best possible. For the spectral projection estimates, however, the counterexamples of [3] were already sharp.

We show that, in odd dimensions, any real valued, bounded potential of compact support has at least one scattering resonance. In dimensions 3 and greater this was previously known only for sufficiently smooth potentials. The proof is based on an inverse result, which shows that the regularized trace of the associated heat kernel admits a full asymptotic expansion if and only if the potential is smooth.

We establish high energy $L^2$ estimates for the restriction of the free Green's function to hypersurfaces in $\mathbb{R}^d$. As an application, we estimate the size of a logarithmic resonance free region for scattering by potentials of the form $V\otimes \delta_{\Gamma}$, where $\Gamma \subset \mathbb{R}^d$ is a finite union of compact subsets of embedded hypersurfaces. In odd dimensions we prove a resonance expansion for solutions to the wave equation with such a potential., Comment: 24 pages

We use a wave packet transform and weighted norm estimates in phase space to establish propagation of singularities for solutions to time-dependent scalar hyperbolic equations that have coefficients of limited regularity. It is assumed that the second order derivatives of the principal coefficients belong to [math] , and that [math] is a solution to the homogeneous equation of global Sobolev regularity [math] or 1. It is then proven that the [math] wavefront set of [math] is a union of maximally extended null bicharacteristic curves.

We establish certain null form estimates of Klainerman–Machedon for parametrices of variable coefficient wave equations for the convex obstacle problem, and for wave equations with metrics of bounded curvature. These are then used to prove a local existence theorem for nonlinear Dirichlet-wave equations outside of convex obstacles.

(2000). Global strichartz estimates for nonthapping perturbations of the laplacian. Communications in Partial Differential Equations: Vol. 25, No. 11-12, pp. 2171-2183.

The authors prove global existence of small solutions to a semilinear wave equation outside of convex obstacles. This extends results of Christodoulou and Klainerman who handled the Minkowski space version. The proof is a compromise of the methods of Christodoulou and Klainerman. It relies on local estimates proved earlier by Smith and Sogge together with classical energy decay estimates for the wave equation of Morawetz, Lax and Phillips.

We introduce a new function space, denoted by H FIO 1 (ℝn), which is preserved by the algebra of Fourier integral operators of order 0 associated to canonical transformations. A subspace of L1 (ℝn), this space in many aspects resembles the real Hardy space of Fefferman-Stein. In particular, we obtain an atomic characterization of H FIO 1 (ℝn). In contrast to the standard Hardy space, these atoms are localized in frequency space as well as in real space.

We establish L p Sobolev mapping properties for averages over certain curves in R 3 , which improve upon the estimates obtained by L 2 L 1 interpolation. Let T be the operator given by convolution in R 3 against a smooth cutoff of arclength measure on the helix (t) = cost,sint,t � , Tf(x) = Z f(x1 − cost, x2 − sint, x3 −t)�(t)dt. For 1 < p < ∞, let H s,p (R 3 ) denote the nonhomogeneous Sobolev space consisting of functions in L p (R 3 ) whose fractional derivative of orders also lies in L p (R 3 ). We consider the following question

We prove optimal pointwise bounds on quasimodes of semiclassical Schrodinger operators with arbitrary smooth real potentials in dimension two. This end-point estimate was left open in the general study of semiclassical L bounds conducted by Koch-TataruZworski [2]. However, we show that the results of [2] imply the two dimensional end-point estimate by scaling and localization.

Weproduce,fordimensions n ≥ 3,examplesofwaveoperatorsforwhichtheStrichartzestimatesfail.TheexamplesincludebothLipschitzand C 1 ,α metrics,foreach0

We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space $\dot H_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial $\dot H_0^1$ initial data., 19 pages

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions., 16 pages. Couple of typos corrected, to appear in Annales de l'Institut Henri Poincare

The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with "abstract Strichartz" estimates for the free wave equation to prove corresponding estimates for the perturbed wave equation when $n\ge3$. As we shall see, we are only partially successful in the latter endeavor when the dimension is equal to two, and therefore, at present, our applications to nonlinear wave equations in this case are limited., 21 pages, simplified existence proof and corrected typos

We use microlocal and paradifferential techniques to obtain L8 norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal Lq bounds, in the range 2⩽q⩽∞, for L2 - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp Lq estimates in higher dimensions for a range of exponents q̅n⩽q⩽∞.

We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay., To appear in Journal d'Analyse. A paper on the quasilinear case will follow

A key challenge in the seismic imaging of reflectors using surface reflection data is the subsurface illumination produced by a given data set and for a given complexity of the background model (of wave speeds). The imaging is described here by the generalized Radon transform. To address the illumination challenge and enable (accurate) local parameter estimation, we develop a method for partial reconstruction. We make use of the curvelet transform, the structure of the associated matrix representation of the generalized Radon transform, which needs to be extended in the presence of caustics and phase linearization. We pair an image target with partial waveform reflection data, and develop a way to solve the matrix normal equations that connect their curvelet coefficients via diagonal approximation. Moreover, we develop an approximation, reminiscent of Gaussian beams, for the computation of the generalized Radon transform matrix elements only making use of multiplications and convolutions, given the underlying ray geometry; this leads to computational efficiency. Throughout, we exploit the (wave number) multi-scale features of the dyadic parabolic decomposition underlying the curvelet transform and establish approximations that are accurate for sufficiently fine scales. The analysis we develop here has its roots in and represents a unified framework for (double) beamforming and beam-stack imaging, parsimonious pre-stack Kirchhoff migration, pre-stack plane-wave (Kirchhoff) migration and delayed-shot pre-stack migration.

where P is a classical pseudodifferential operator in two dimensions, which is of principal type and satisfies (Til) of Nirenberg and TieYves [NT]. (We refer the reader to the bibliography for the precise meaning of these statements.) In [L], Lerner proves the estimate using a generalization of energy estimates of Nirenberg and Treves. Our proof results from the observation that, in two dimensions, the canonical form for such an operator equals the Calderon boundary operator for a boundary-value problem that can be inverted by simple integration.

We prove global existence of small-amplitude solutions of quasilinear Dirichlet-wave equations outside of star-shaped obstacles in (3+1)-dimensions. We use a variation of the conformal method of Christodoulou. Since the image of the space-time obstacle is not static in the Einstein diamond, our results do not follows directly from local existence theory as did Christodoulou's for the nonobstacle case. Instead, we develop weighted estimates that are adapted to the geometry. Using them and the energy-integral method we obtain solutions in the Einstein diamond minus the dime-dependent obstacle, which pull back to solutions in Minkowski space minus and obstacle.