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

Let ${\mathrm{\Delta }}_{{\mathbb{T}}^m\times {\mathbb{R}}^n}$ denote the Laplace-Beltrami operator on the product spaces ${\mathbb{T}}^m\times {\mathbb{R}}^n$. In this article we show that $${\parallel e^{it{\mathrm{\Delta }}_{{\mathbb{T}}^m\times {\mathbb{R}}^n}}f\parallel }_{L^p({\mathbb{T}}^m\times {\mathbb{R}}^n\times [0,1])}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{W^{\alpha ,p}({\mathbb{T}}^m\times {\mathbb{R}}^n)}$$ holds if $p\ge 2(m+n+2)\mathrm{/}(m+n)$ and $\alpha >(m+2n)(1\mathrm{/}2-1\mathrm{/}p)-2\mathrm{/}p$. Furthermore, we apply the ${\mathrm{\ell }}^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schrödinger operator $e^{it{\mathrm{\Delta }}_{{\mathbb{T}}^m\times {\mathbb{R}}^n}}$ in modulation spaces $M_{p,q}^{\alpha }({\mathbb{T}}^m\times {\mathbb{R}}^n)$: $$\mathrm{\parallel }{e}^{it{\mathrm{\Delta }}_{{\mathbb{T}}^m\times {\mathbb{R}}^n}}f{\mathrm{\parallel }}_{L^p({\mathbb{T}}^m\times {\mathbb{R}}^n\times [0,1])}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{M_{p,q}^{\alpha }({\mathbb{T}}^m\times {\mathbb{R}}^n)}$$ 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 $q$.