We prove spacetime weighted-L 2 estimates for the Schrödinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.

IntroductionWave maps are maps U from a Lorentzian manifold M"" (a manifold with a metric g of index 1) into a Riemannian manifold N" (one with a positive definite metric h ) that are the critical points (with respect to compactly supported variations) of the Lagrangian If M and N are rotationally symmetric, i.e., SO(n) and SO(K) act on M and N as isometries, then we can define equivariant maps by requiring that the orbit of any point in M under spatial rotations maps into the orbit of the image point in N . When this mapping of the orbits has degree 1, we call the map corotatzonaf.In a previous paper (see [ 171) we have shown that in the case M = [W2,I, the 2 + 1 -dimensional Minkowski space-time, and N , a rotationally symmetric surface diffeomorphic to I F! ', the Cauchy problem with smooth data of any size for the corotational wave map from M into N has globally smooth solutions, provided that the target N is geodesically convex. This condition was later laxed by Grillakis (see [9]), to include a certain class of nonconvex targets. Our proof of regularity was also simplified later by Shatah and Struwe; see [16]. In this paper we obtain several results for equivariant wave maps in the areas of existence and uniqueness, regularity, asymptotic behavior, development of singularities, and weak solutions:1. Existence and Uniqueness. We obtain a local-in-time existence and uniqueness theorem for equivariant wave maps that requires a minimal amount of

Wave maps are critical points U : M -N of the Lagrangian 2 [ U l = s , 11dLI1I2, where M is an Einsteinian manifold and N a Riemannian one. For the case M = R2.' and LI a spherically symmetric map, it is shown that the solution to the Cauchy problem for LI with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (I) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded and (2) there are two constants c and C such that the smallest eigenvalue X and the largest eigenvalue A of the second fundamental form km of any geodesic sphere X(p, s) of radius s centered at p E N satisfy sh 2 c and sA 5 C( 1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. 01993 John Wiley & Sons, Inc. T [ U ] = (dU, dU) = -/ gaPh,bd,U"dpUb dvM MIf the metric tensor g is positive definite, the resulting Euler-Lagrange equations for the critical point U of 2 will be elliptic, and U is called a harmonic map. If, however, g is Einsteinian, i.e., of index 1, then the equations will be hyperbolic, and the solutions are called wave maps. We study the case M = R2*', the 2 + 1dimensional Minkowski space-time, with the metric (gap) = diag(-1,1,1). Thus, in polar coordinates (t, r, 0) E R x W + x S', the line element of M isThe target N is a smooth, complete and connected Riemannian manifold of dimension n and positive definite metric h. In local coordinates, the equation for U is (1 .2) nu" + ~,(u)apuba~u' = 0,

We prove weighted L 2 estimates for the solutions of linear Schrödinger and wave equation with potentials that decay like |x| −2 for large x, by deducing them from estimates on the resolvent of the associated elliptic operator. We then deduce Strichartz estimates for these equations.

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