Fourier transform of a particular function

Question

In order to estimate the fundamental solution of some particular types of differential operators,I need estimates on some kind of oscillatory integrals.For simplicity, consider the Fourier transform of the following function on $\mathbb{R}^4$ $$ f(x)=(1+x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2+x_{1}^2x_{2}^2+x_{1}^2x_{3}^2+x_{1}^2x_{4}^2+x_{2}^2x_{3}^2+x_{2}^2x_{4}^2+x_{3}^2x_{4}^2)^{\alpha} $$ where $-1<\Re \alpha<0$.

Is there any way to estimate the decay of $\hat{f}(\xi)$ for large $\xi$ ?

I have tried to use a dyadic decomposition (write $\mathbb{R}^4$ as the union of disjoint rectangles)to treat the singularities,and then use integrating by parts.But it seems a little messy.I don't know if there were some papers already dealing with such kind of integrals, so I'm very apprieciated that if someone can show me.

Answer 1

Your function is in the space $\mathcal O_M(\mathbb R^4)$ (Notation from L. Schwartz). Fourier transform takes this space to $\mathcal O_C'(\mathbb R^4)$ (called rapidly decreasing distributions).

Shorter: Since $f$ is smooth and tempered, its Fourier transform $\hat f$ exists and is rapidly decreasing. But since $f$ is not rapidly decreasing, $\hat f$ is not smooth.