Global L2 estimates for a class of maximal operators associated to general dispersive equations

For a function phi satisfying some suitable growth conditions, consider the general dispersive equation defined by {i partial derivative tu + phi(root-Delta) u = 0, (x, t) is an element of R-n x R, u(x, 0) = f (x), f is an element of S(R-n). (*) In the present paper, we give some global L-2 estimate for the maximal operator S-phi*, which is defined by S-phi* f(x) = sup(0<t<1) vertical bar S(t,phi)f(x)vertical bar, x is an element of R-n, where S(t,phi)f is a formal solution of the equation (*). Especially, the estimates obtained in this paper can be applied to discuss the properties of solutions of the fractional Schrdinger equation, the fourth-order Schrdinger equation and the beam equation.