On the Microlocal Smoothing Effect of Dispersive Partial Differential Equations, I: Second-Order Linear Equations

Abstract

A number of dispersive partial differential equations have local smoothing effect. The solutions of such equations have local regularity better than that of the Cauchy data, if the Cauchy data satisfy certain conditions. A dispersive equation, mean a non-Kowalevskian evolution equation on Rn whose evolution operators form a one-parameter group. This chapter describes a microlocal version of the regularizing effect of some systems of the second-order linear partial differential equations. Certain decay of the Cauchy data in some direction implies microlocal regularity of solutions at positive time. The principal symbol is assumed to be noncharacteristic or of principal type at the point. Furthermore, the decay of the Cauchy data is required only in some conic set, not in the whole space.