WEIGHTED ESTIMATES FOR THE HELMHOLTZ-EQUATION AND SOME APPLICATIONS

We characterize the radial functions V in R-n for which the a priori i nequality \\del uV(1/2)\\(2) + k\\uV(1/2)\\(2) less than or equal to C \\V-1/2(Delta + k(2))u\\(2) holds with constant independent of k. The condition is for V to have the X-rays transform everywhere bounded. We apply these estimates to the well posedness of evolution Schrodinger equations with time dependent drift terms and to the restriction of th e Fourier transform to Euclidean spheres. (C) 1997 Academic Press.