The function $\varphi(x)=(1+\vert x\vert^4)^{-1},$ where $x\in\mathbb{R}^d, d<4,$ is a good example of a function that is not the Schwartz space but it has a defined Fourier transform.
It satisfies that $$\varphi(x)\leq C(1+\vert x\vert)^{-(d+\delta)}, \delta=4-d. $$
I'm trying to proof or disproof that its Fourier transform has a similar property, ie I would like to know if there exists $\delta>0$ such that
$$\hat\varphi(\xi)=\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\varphi(x)d\xi\leq C(1+\vert \xi\vert)^{-(d+\delta)}. $$
I have the impression that the proof is not hard but I'm not an expert in Fourier analysis and I have no idea about where to start.