LAUSR

In the paper we study the Schrödinger oscillatory integrals $$T^t_{\lambda ,a}f(x)$$Tλ,atf(x) ($$\lambda \ge 0$$λ≥0, $$a>1$$a>1) associated with the one-dimensional Dunkl transform $${\mathscr {F}}_{\lambda }$$Fλ. If $$a=2$$a=2, the function $$u(x,t):=T^t_{\lambda ,2}f(x)$$u(x,t):=Tλ,2tf(x) solves the free Schrödinger equation associated to the Dunkl operator, with f as the initial data. It is proved that, if f is in the Sobolev spaces $$H^s_{\lambda }({\mathbb {R}})$$Hλs(R) associated with the Dunkl transform, with the exponents s not less than 1 / 4, then $$T^t_{\lambda ,a}f$$Tλ,atf converges almost everywhere to f as $$t\rightarrow 0$$t→0. A counterexample is constructed to show that 1 / 4 can not be improved for $$a=2$$a=2, and when $$1/4\le s\le 1/2$$1/4≤s≤1/2, the Hausdorff dimension of the divergence set of $$T^t_{\lambda ,a}f$$Tλ,atf for $$f\in H_{\lambda }^s({\mathbb {R}})$$f∈Hλs(R) is proved to be $$1-2s$$1-2s at most.