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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 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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What sort of estimate do you want? What are the "singularities" you are talking about? Your function looks real-analytic to me. –  Alexandre Eremenko Nov 13 '12 at 13:28
    
I think "singularities" refers to directions of slow decay at infinity. –  Michael Renardy Nov 13 '12 at 14:32
    
@Alexandre Eremenko,I want to know how $\hat{f}(\xi)$ decay for large $\xi$,just as Renardy said,the problem is the slow decay of the functon $f$ at $\infty$ –  user23078 Nov 13 '12 at 15:07
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1 Answer

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.

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Thanks very much for this useful result.I think I should ask how does $\hat{f}$ behave near zero rather than at $\infty$. –  user23078 Nov 14 '12 at 11:17
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