Why the integral $\int_{\mathbb{R^n}}e^{-iAx} \hat{\phi} (-g(x))$ is well defined?

Question

Let $\phi$ be a smooth function on $\mathbb{R}^n$ with compact support. Let $ g : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth map such that there exists $c>0$ and $ k \in \mathbb{N} -\lbrace0\rbrace $ such that $ \forall x \in \mathbb{R}^n$, $||g(x)|| \geq c {||x||}^k$. Let $A$ be a square matrix. How can we show that the map $$ x \mapsto e^{-iAx} \hat{\phi} (-g(x))$$ decreases rapidly, which implies that the integral $$ \int_{\mathbb{R^n} }e^{-iAx} \hat{\phi} (-g(x)) dx$$ is well defined?