Higher-dimensional Grushin wave-estimate conjecture

Let $G_n$ be the higher-dimensional Grushin operator on $\mathds{R}^d$, let $d:=n+1$ be the topological dimension, and define

$$S_{{\mathscr{C}}_1}:=\{(x,u)\in\mathds{R}^2;\ |x|\leq {\mathscr{C}}_1\}.$$

For $f$ in the Schwartz space $\mathscr{S}$, assume $\operatorname{supp}f\subseteq S_{{\mathscr{C}}_1}$.

Higher-dimensional Grushin wave-estimate conjecture. For every ${\mathscr{C}}_1>0$, $t>0$, $1\leq p\leq\infty$, and $\alpha>(d-1)|1/p-1/2|$, there exists a constant $C_{p,t,{\mathscr{C}}_1}^\alpha$ such that

$$\left\|\frac{\cos(t\sqrt{G_n})}{(1+G_n)^{\alpha/2}}f\right\|_{L_p(\mathds{R}^d)}\leq C_{p,t,{\mathscr{C}}_1}^\alpha\|f\|_{L_p(\mathds{R}^d)},$$

and

$$\left\|\frac{\sin(t\sqrt{G_n})}{\sqrt{G_n}(1+G_n)^{(\alpha-1)/2}}f\right\|_{L_p(\mathds{R}^d)}\leq C_{p,t,{\mathscr{C}}_1}^\alpha\|f\|_{L_p(\mathds{R}^d)}.$$

The paper proves the corresponding two-dimensional theorem with the threshold $\alpha>|1/p-1/2|$ and proposes this higher-dimensional extension.