Suppose $f,g \in W^{2,\infty}(\mathbb{R}^n)$. How can I show that $$\mathcal{I} := \int_{\mathbb{R}^n} \frac{f(x+z)g(x+z)-f(x)g(x) - \mathbf{1}_{|z|<1}\nabla (f(x)g(x))\cdot z}{|z|^{n+2s}}dz, \quad s \in (0,1),$$ is well-defined and estimate it with the sum of $\Vert \cdot\Vert_{L^\infty}$ norm of $f$ and $g$ and their derivatives (like $\mathcal{I} \le C(\Vert D^2 f \Vert_{L^\infty} + \Vert \nabla f \Vert_{L^\infty} + \Vert f \Vert_{L^\infty} + \Vert D^2 g \Vert_{L^\infty} + \Vert \nabla g \Vert_{L^\infty} + \Vert g \Vert_{L^\infty})$?
If $s \in (0,1/2)$ and $g \equiv 1$, I see that we can use lagrange theorem and get $\mathcal{I} \le C(\Vert \nabla f \Vert_{L^\infty} + \Vert f \Vert_{L^\infty})$. What about the more general case?