Fourier tranform of $\int_{R^n}\frac{u(x)-u(y)}{|y-x|^{n+2s}}dy$

Question

How to compute the Fourier transform of

$$F: x \mapsto\int_{R^n}\frac{u(x)-u(y)}{|y-x|^{n+2s}}dy$$

where $0<s<1$.

It is shown to be $\hat{F}(\xi)=|\xi|^{2s}\hat u(\xi)$.

Answer 1

I'm not sure your result is true. However, a result that is true and seems very close to your result is the following one :

$$\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} \ dx dy = C \int_{\mathbb{R}^n} |\xi|^{2s} |\hat{u}(\xi)|^2 \ d\xi$$

You can find a proof in https://arxiv.org/abs/1104.4345v3 at the prop 3.4.