Weighted and Mixed Norm Estimates for Oscillatory Fourier

In this working paper we suggest some lines of investigation regarding oscillatory Fourier transforms with homogeneous phase. A special case is the solution to the time-dependent free Schrr odinger equation. We consider time-global range-weighted L 2 ? R n+1-estimates. The main tools are Parseval's formula for Fourier transforms on R, orthogonality arguments arising from decomposing L 2 (R n) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments will be used to show that results are sharp with respect to regularity. We also consider a mixed norm estimate which arise in a natural way when applying Pitt's inequality instead of Parseval's formula. For a special case of one of the results below the proof has been carried out. See 6]. Otherwise results are preliminary in nature. All results are stated without proofs.