On local dispersive and Strichartz estimates for the Grushin operator

Let $G=-\Delta-|x|^2\partial_{t}^2$ 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-Schr\"odinger equation on $\mathbb{R}^{n+1}$, we establish a local dispersive estimate by defining the Grushin-Schr\"odinger kernel on a suitable domain. As a corollary we obtain a local Strichartz estimate for the Grushin-Schr\"odinger 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 $\mathbb{N}_0^n\times\mathbb{R^*}\times \mathbb{R}$ and derive anisotropic Strichartz estimates for the Grushin-Schr\"{o}dinger equation and for the Grushin wave equation as well.