On Schrodinger Oscillatory Integrals Associated with the Dunkl Transform

In the paper we study the Schrodinger oscillatory integrals T,atf(x) (0, a>1) associated with the one-dimensional Dunkl transform F. If a=2, the function u(x,t):=T,2tf(x) solves the free Schrodinger equation associated to the Dunkl operator, with f as the initial data. It is proved that, if f is in the Sobolev spaces Hs(R) associated with the Dunkl transform, with the exponents s not less than 1/4, then T,atf converges almost everywhere to f as t0. A counterexample is constructed to show that 1/4 can not be improved for a=2, and when 1/4s1/2, the Hausdorff dimension of the divergence set of T,atf for fHs(R) is proved to be 1-2s at most.