On smoothing estimates for Schr\"odinger equations on product spaces $\mathbb{T}^m\times \mathbb{R}^n$

Let $\Delta_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$\left\|e^{it\Delta_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{\alpha,p}(\mathbb{T}^m\times\mathbb{R}^n)}$$ holds if $p\geq 2(m+n+2)/(m+n)$ and $\alpha> (m+2n)(1/2-1/p)-2/p$. Furthermore, we apply the $\ell^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schr\"odinger operator $e^{it\Delta_{\mathbb{T}^m\times\mathbb{R}^n}}$ in modulation spaces $M_{p,q}^\alpha(\mathbb{T}^m\times\mathbb{R}^n)$: $$\|e^{it\Delta_{\mathbb{T}^m\times\mathbb{R}^n}}f\|_{L^p(\mathbb{T}^m\times\mathbb{R}^n\times [0,1])}\leq C \|f\|_{M_{p,q}^\alpha(\mathbb{T}^m\times\mathbb{R}^n)}$$ for some range of $\alpha$ and $p, q$. The smoothing estimates in $L^p$-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of $p$ and $q$.