Microlocal partition of energy for fractional-type dispersive equations

This paper is devoted to the proof of the microlocal partition of energy for fractional-type dispersive equations including the Schrdinger equation, the linearized gravity or capillary water-wave equation and the half-Klein-Gordon equation.Roughly speaking, a quarter of the L 2 energy lies inside or outside the "light cone" |x| = |t P ( )| for large time.In addition, based on the study of the half-Klein-Gordon equation, the microlocal partition of energy will also be proved for the Klein-Gordon equation.