ESTIMATES ON SCATTERED WAVES

We derive estimates on solutions u(k,x) to a scattering problem with v ariable index of refraction in three space dimensions. To be precise, suppose n(x) is an element of C-infinity(R-3) is positive and n(x) = 1 for \x\ greater than or equal to R. We want to estimate solutions u ( k,x) to (0.1) (Delta + k(2)n(x)(2))u = 0, u = e(ikx.omega) + u(s), whe re u, satisfies the radiation condition. Here, k is an element of R de notes the frequency. There are two mechanisms that can make u(k, x) la rge. One is the presence of trapped rays. In this work we assume there are no trapped rays. The other mechanism is the focusing of waves, i. e., the formation of caustics. Our primary goal here is to estimate th e effect of this mechanism, without making any hypothesis on the geome trical nature of whatever caustics might arise. We show that (0.2) par allel to u(k,.)parallel to(L infinity(R3)) less than or equal to C[k], where [k] = (1 + k(2))(1/2).