Homogeneous estimates for oscillatory integrals

Abstract. Let u(x, t) be the solution to the free time-dependent Schrödinger equa-tion at the point (x, t) in space-time Rn+1 with initial data f. We characterize the size of u in terms Lp(Lq)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces Ḣs (Rn). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpolation of vector valued weighted Lebesgue spaces. 1