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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 $-1<\Re \alpha<0$.

My question is how

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.
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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 mathod

My question is how to evaluate this estimate $\hat{f}(\xi)$ ?

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.
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Fourier transform of a particular function

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 mathod to evaluate this ?

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.