A maximal inequality associated to schrödinger type equation

In this note, we consider a maximal operator sup_t∈R |u(x, t)| = sup_t∈R |e^itΩ(D)f(x)|, where u is the solution to the initial value problem u_t = iΩ(D)u, u(0) = f for a C² function Ω with some growth rate at infinity. We prove that the operator sup_t∈R |u(x, t)| has a mapping property from a fractional Sobolev space H^1/4 with additional angular regularity in which the data lives to L²((1 + |x|)^−bdx) (b > 1). This mapping property implies the almost everywhere convergence of u(x, t) to f as t → 0, if the data f has an angular regularity as well as H^1/4 regularity.