On Schrödinger Oscillatory Integrals Associated with the Dunkl Transform

Abstract

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.