Integral Inequality for $u\in\mathcal{S}(\mathbb{R}^n)$

Question

Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that: $$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$ where: $c=c(u,n,s)>0$ is a constant. I think that i have to use something like: $$ |u(x+y)+u(x-y)-2u(x)|\leq|D^2u(y)||y|^2,$$ but after i can't go on. Any help would be appreciated.

Answer 1

Hint: Write the Taylor formulas more carefully. You should get something with $|y|^2\,\sup_{|x|/2\leq |z| \leq 3|x|/2} |D^2 u(z)|$ on the right hand side. And this will decay fast since $u$ is a Schwartz function.