Fourier transform of $ |x|^{\alpha - n}$

Question

I am attempting to understand the following proof (Theorem 5.9) from Lieb-Loss. They remark that by Fubini $$ \int_0^\infty\lambda^{-n/2}\lambda^{\alpha/2-1}\left (\int_{R^n}\exp[-\pi|x-y|²/\lambda]f(y)dy \right)d\lambda = c_{n-\alpha} \int_{R^n}|x-y|^{-n+\alpha}f(y)dy$$ with $c_\alpha = \pi^{-\alpha/2}\Gamma(\alpha/2). $ How does this equation follow from Fubini's theorem?

Answer 1

Since all the functions are non-negative we can indeed use Fubini's theorem to interchange the order of integration. To obtain the expression on the right-hand side we compute the integral over $\lambda$:

$$\int_0^{\infty} \lambda^{\alpha/2-n/2-1}\exp (-\pi |x-y|^2/\lambda)\, d\lambda.$$

We substitute $t = \pi|x-y|^2/\lambda$, then $d\lambda = -\pi|x-y|^2/t^2 \, dt$. After this substitution a term $|x-y|^{\alpha-d}$ will appear along with a power of $\pi$ and an integral which is exactly the gamma function. Can you proceed from here?