On Schrödinger Oscillatory Integrals Associated with the Dunkl Transform

In the paper we study the Schrodinger oscillatory integrals \(T^t_{\lambda ,a}f(x)\) (\(\lambda \ge 0\), \(a>1\)) associated with the one-dimensional Dunkl transform \({\mathscr {F}}_{\lambda }\). If \(a=2\), the function \(u(x,t):=T^t_{\lambda ,2}f(x)\) 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 \(H^s_{\lambda }({\mathbb {R}})\) associated with the Dunkl transform, with the exponents s not less than 1 / 4, then \(T^t_{\lambda ,a}f\) converges almost everywhere to f as \(t\rightarrow 0\). A counterexample is constructed to show that 1 / 4 can not be improved for \(a=2\), and when \(1/4\le s\le 1/2\), the Hausdorff dimension of the divergence set of \(T^t_{\lambda ,a}f\) for \(f\in H_{\lambda }^s({\mathbb {R}})\) is proved to be \(1-2s\) at most.