On local dispersive and Strichartz estimates for the Grushin operator

Let $G=-\mathrm{\Delta }-\mathrm{\mid }x{\mathrm{\mid }}^2{\mathrm{\partial }}_t^2$ denote the Grushin operator on ${\mathbb{R}}^{n+1}$. The aim of this paper is two fold. In the first part, due to the non-dispersive phenomena of the Grushin-Schrödinger equation on ${\mathbb{R}}^{n+1}$, we establish a local dispersive estimate by defining the Grushin-Schrödinger kernel on a suitable domain. As a corollary we obtain a local Strichartz estimate for the Grushin-Schrödinger equation. In the next part, we prove a restriction theorem with respect to the scaled Hermite-Fourier transform on ${\mathbb{R}}^{n+2}$ for certain surfaces in ${\mathbb{N}}_0^n\times {\mathbb{R}}^{\ast }\times \mathbb{R}$ and derive anisotropic Strichartz estimates for the Grushin-Schrödinger equation and for the Grushin wave equation as well.