Weighted Estimates for the Helmholtz Equation and Some Applications

We characterize the radial functionsVin Rnfor which the a priori inequality ‖∇u1/2‖2+k ‖uV1/2‖2⩽C ‖V−1/2(Δ+k2) u‖2holds with constant independent ofk. The condition is forVto have the X-rays transform everywhere bounded. We apply these estimates to the well posedness of evolution Schrödinger equations with time dependent drift terms and to the restriction of the Fourier transform to Euclidean spheres.