I want to study the stationary phase method in the following setting: Let $\varphi,f: \mathbb{T}^d \to \mathbb{R}$ be smooth functions on the $d$-torus, such that $\varphi'(x) \not= 0$ for all $x \in \text{supp}(f)$. It is true that the function $$ I(t) = \int_{\mathbb{T}^d} e^{it\varphi(x)} f(x) \, dx $$ Is in the Schwartz space $\mathcal{S}(\mathbb{R})$?
This can be easily reduced to the case where the functions are defined in $[0,1]^d$. I am under the impression that periodicity does not play any particularly relevant role in this and this is just the stationary phase method, but I have only found in the literature the cases where $\varphi$ does not have any critical points (which is bound to happen in the periodic setting) or when the integral is defined over all $\mathbb{R}^d$ and $f$ has compact support.
Edit: I also guess that there is a contribution from the boundary of $[0,1]^d$.