Double integrals with kernels

Let $f, g$ be functions in the Schwartz space $\mathcal S(\mathbb R^n, \mathbb R)$. Let $a\in (0, 1)$ and consider the kernel $K(u, v)=|x-y|^{-N-a-1}$, being both $u, v, z\in\mathbb R^n$ vectors such that $x\neq y$ and $x\neq z$. Consider the integral$$(1)\qquad\int_{\mathbb R^n} \left(\int_{\mathbb R^n}(f(u)-f(v))(u-v) K(u-v) dv\right) \cdot \left(\int_{\mathbb R^n}(g(u)-g(z))(u-z) K(u-z) dz\right) du.$$

This integral comes up in a paper which discusses nonlocal equations applied to the theory of condensate materials. It is written that this ones reduces to$$(2)\qquad\int_{\mathbb R^n}\int_{\mathbb R^n} (f(u)-f(v)) (g(u)-g(v)) \overline K(u, v) du dv,$$where $\overline K(u, v)=|x-y|^{-N-2a}$.

I am not an expert, but I do have the feeling that the last integral is related to the fractional laplacian. However, I can not get from (1) to (2).

I have argued in this way: if $v=z$, then you get (2) but with the kernel $|x-y|^{-2N-2a}$. If $v\neq z$, then one can see that the internal integrals are symmetric with respect to $v$ and $z$. Thus, also in this case, one gets (2) with the kernel $|x-y|^{-2N-2a}$.

What am I doing wrong?