Dispersive Estimate for the Wave Equation

Dispersive Estimate for the Wave Equation with the Inverse-Square Potential Fabrice Planchon John G. Stalker A. Shadi Tahvildar-Zadeh 12/27/01 Abstract We prove that spherically symmetric solutions of the Cauchy prob- lem for the linear wave equation with the inverse-square potential sat- isfy a modied dispersive inequality that bounds the L 1 norm of the solution in terms of certain Besov norms of the data, with a factor that decays in t for positive potentials. When the potential is negative we show that the decay is split between t and r, and the estimate blows up at r = 0. We also provide a counterexample showing that the use of Besov norms in dispersive inequalities for the wave equation are in general unavoidable. 1 Introduction Consider the following linear wave equation 8 < : n u+ a jxj 2 u = 0 u(0; x) = f(x) @ t u(0; x) = g(x) (1.1) where n = @ 2 t n is the D'Alembertian in R n+1 and a is a real number. The interest in this equation comes from the potential term being homogeneous of degree -2 and therefore scaling the same way as the D'Alembertian term.

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