ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 69,   2   (2000)
pp.   151-171

HOMOGENEOUS ESTIMATES FOR OSCILLATORY INTEGRALS
B. G. WALTHER

Abstract.  Let $u(x,t)$ be the solution to the free time-dependent Schrodinger equation at the point $(x,t)$ in space-time $\R \sp n + 1$ with initial data $f$. We characterize the size of $u$ in terms $L \sp p (L \sp q)$-estimates with power weights. Our bounds are given by norms of $f$ in homogeneous Sobolev spaces $\sbsp n \dot s$. \endgraf Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpolation of vector valued weighted Lebesgue spaces.

AMS subject classification.  42B15, 42B99, 35J10, 35B40, 35B65, 35P25, 33C10, 46B70
Keywords.  Oscillatory Integrals, Weighted and Mixed Norm Inequalities, Global Smoothing and Decay, Time-dependent Schrodinger Equation, Bessel functions, Weighted interpolation spaces