Christopher D. Sogge

Concerning the LP norm of spectral clusters for second-order elliptic operators on compact manifolds

In this paper we study the (Lp, L2) mapping properties of a spectral projection operator for Riemannian manifolds. This operator is a generalization of the harmonic projection operator for spherical harmonics on S” (see C. D. Sogge, Duke Math. J. 53 (1986), 43-65). Among other things, we generalize the L2 restriction theorems of C. Fefferman, E. M. Stein, and P. Tomas (Bull. Amer. Math. Sot. 81 (1975), 477478) for the Fourier transform in R” to the setting of Riemannian manifolds. We obtain these results for the spectral projection operator as a corollary of a certain “Sobolev inequality” involving d +r2 for large r. This Sobolev inequality generalizes certain results for R” of C. Kenig, A. Ruiz, and the author (Duke Math. J. 55 (1987), 329-347). The main tools in the proof of the Sobolev inequalities for Riemannian manifolds are the Hadamard parametrix (cf. L. Hbrmander, Acta Mad 88 (1968), 341-370, and “The Analysis of Linear Partial Differential Equations,” Vol. III, Springer-Verlag, New York, 1985) and oscillatory integral theorems of L. Carleson and P. Sjolin (Studia Math. 44 (19X?), 287-299) and Stein (Arm. Math. Stud. 112 ( 1986), 307-357). 0 1988 Academic pnss, IW.

Global strichartz estimates for nonthapping perturbations of the laplacian

The authors prove global Strichartz estimates for compact perturbations of the wave operator in odd dimensions when a non-trapping assumption is satisfied.

Local smoothing of Fourier integral operators and Carleson-Sjölin estimates

The purpose of this paper is twofold. First, if Y and Z are smooth paracompact manifolds of dimensions n ~ 2 and n + 1 , respectively, we shall prove local regularity theorems for a certain class of Fourier integral operators !T E /f1.(Z , Y; ~) which naturally arise in the study of wave equations. Secondly, we want to apply these estimates to prove versions of the Carleson-Sjolin theorem on compact two-dimensional manifolds with periodic geodesic flow. The operators we shall study satisfy the curvature condition introduced in [32]. The main example occurs when Z = Y x JR and !T is the solution operator for the Cauchy problem for the wave equation

Almost global existence for some semilinear wave equations

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.

Almost global existence for quasilinear wave equations in three space dimensions

This article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichlet-wave equations in the exterior of a smooth, star shaped obstacle. We can also apply our methods to systems of quasilinear wave equations having different wave speeds. This extends our work [11] for the semilinear case. Previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non-obstacle case. In [9], John and Klainerman proved almost global existence on Minkowski space for quadratic, quasilinear equations using the Lorentz invariance of the wave operator in addition to the symmetries listed above. Subsequently, in [14], Klainerman and Sideris obtained the same result for a class of quadratic, divergence-form nonlinearities without relying on Lorentz invariance. This line of thought was refined and applied to prove global-in-time results for null-form equa tions related to the theory of elasticity in Sideris [22], [23], and for multiple-speed systems of null-form quasilinear equations in Sideris and Tu [24], and Yokoyama [29]. The main difference between our approach and the earlier ones is that we ex ploit the 0(|x|-1) decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0(t~l) decay. Here, of course, x = (x\,X2,x

Riemannian manifolds with maximal eigenfunction growth

) is the spatial component, and t the time component, of a space time vector (t, x) G M+ x E3. Establishing 0(|x|_1) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation. A weighted L2 space-time estimate for inhomogeneous wave equations (Proposition 3.1 below, from [11]) is important in making the spatial decay useful for the long-time existence argument. For semilinear systems, one can show almost global existence from small data using only this spatial decay [11]. For quasilinear systems, however, we also have to show that both first and second derivatives of u decay like 1/t. Fortunately, we can do this using a variant of some L1 ?> L?? estimates of John, H?rmander,

On the critical semilinear wave equation outside convex obstacles

On any compact Riemannian manifold

LONG-TIME EXISTENCE OF QUASILINEAR WAVE EQUATIONS EXTERIOR TO STAR-SHAPED OBSTACLES VIA ENERGY METHODS ∗

(M, g)

Wave front sets, local smoothing and Bourgain's circular maximal theorem

of dimension

Propagation of singularities and maximal functions in the plane

n