Strichartz estimates for the Schrödinger equation on products of odd-dimensional spheres

We prove Strichartz estimates for the Schrödinger equation which are scale-invariant up to an $\epsilon$-loss on products of odd-dimensional spheres. Namely, for any product of odd-dimensional spheres $M={\mathbb{S}}^{d_1}\times \cdots \times {\mathbb{S}}^{d_r}$ (so that $M$ is of dimension $d=d_1+\cdots +d_r$ and rank $r$) equipped with rational metrics, the following Strichartz estimate \begin{equation*} |e^{itΔ}f|{L^p(I\times M)}\leq C\varepsilon|f|_{H^{\frac{d}{2}-\frac{d+2}{p}+\varepsilon}(M)} \end{equation*} holds for any $p\ge 2+\frac{8(s-1)}{sr}$, where $$s=\max \{\frac{2d_i}{d_i-1},i=1,\dots ,r\}\mathrm{.}$$