I'm trying to learn how to use the following result: Let $E\subset \mathbb R^d$ denote an open set and $\psi \in C_{c}^{\infty}(E).$ If $\phi \in C^{\infty}(E)$ and the rank of matrix $$(D_{x_j}D_{x_i}\phi(x))_{i,j=1}^{d}$$ is at least $\rho>0$ for all $x\in supp (\psi),$ then we have \begin{eqnarray} \left| \int_{\mathbb R^d} e^{i\lambda \phi (x)} \psi(x) dx \right| \lesssim |\lambda|^{-\frac{\rho}{2}} \|\psi\|_{C^{2d}}. \end{eqnarray}
Specifically: Let $\sigma$ be fixed nonnegative function in $\mathcal{S}(\mathbb R)$ (Schwartz space) support in the cube $[-\frac{4}{5}, \frac{4}{5}],$ and satisfies $\sigma(\xi)=1$ for any $\xi$ in the unit cube $[-\frac{1}{2}, \frac{1}{2}].$ Assume that $x\in \mathbb R, k\in \mathbb Z$ and $k\geq 100,$ and $t$ large real number. Put $B= \{x: 2t (|k|-2)\leq |x|\leq 2t (|k|+2) \}$
and $$A= \int_{\mathbb R} e^{i \Phi (x, \xi, k,t)} \sigma (\xi) d\xi$$ where $\Phi (x, \xi, k,t)= t|\xi+k|^2+ x\cdot k + x\cdot \xi.$
Question:Can we say that $| \chi_{B}(x) A| \leq C \frac{\chi_{B}(x) }{t^{1/2}}$ for some $C>0$?
Motivation: This fact has been used in the paper I have been reading here is link of paper, p.475. The author says using the above result one can get the above inequality.