An inequality involving fractional Laplacian

Question

I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function): $$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$ for some $c_{n,s}>0$, where $$(-\Delta)^su(x):=-\frac{C(n,s)}{2}\int_{\mathbb{R}^n}\frac{u(x+y)+u(x-y)-2u(x)} {|y|^{n+2s}}\,dy,\quad\forall x\in\mathbb{R}^n, $$ is the fractional Laplacian. I have no idea. Any help would be appreciated.

Answer 1

Write $$\begin{aligned} -(-\Delta)^s u(x) & = \frac{C(n,s)}{2} \int_{B(x,1)} \frac{u(x + y) + u(x - y) - 2 u(x)}{|y|^{n + 2s}} \, dy \\ & \qquad + C(n,s) \int_{\mathbb{R}^n \setminus B(0,1)} \frac{u(x - y) - u(x)}{|y|^{n + 2s}} \, dy . \end{aligned}$$ Using Taylor's theorem and the fact that $u''$ is Schwartz class, we find that $$|u(x + y) + u(x - y) - 2 u(x)| \leqslant C_n |y|^2 \sup_{B(x, 1)} |u''| \leqslant C_{u,n} (1 + |x|)^{-n - 2s} |y|^2 $$ (here and below $C_p$ denotes some constant that only depends on the parameter $p$; the value of $C_p$ can be different each time it appears). Thus, $$\begin{aligned} \biggl| \int_{B(0,1)} \frac{u(x + y) + u(x - y) - 2 u(x)}{|y|^{n + 2s}} \, dy \biggr| & \leqslant \frac{C_{u,n}}{(1 + |x|)^{n + 2s}} \int_{B(0,1)} |y|^{2-n-2s} dy \\ & = \frac{C_{u,n,s}}{(1 + |x|)^{n+2s}} \end{aligned}$$ Furthermore, since $u$ is Schwartz class, we have $$|u(x + y) - u(x)| \leqslant |u(x - y)| + |u(x)| \leqslant C_u ((1 + |x - y|)^{-n-2s} + (1 + |x|)^{-n-2s}).$$ It is a nice exercise to show that $$ \int_{\mathbb{R}^n \setminus B(0,1)} \frac{1}{|y|^{n + 2s} (1 + |x - y|)^{n + 2 s}} \, dy \leqslant \frac{C_{n,s}}{(1 + |x|)^{n + 2s}} \, . $$ It follows that $$\begin{aligned} \biggl| \int_{\mathbb{R}^n \setminus B(0,1)} \frac{u(x - y) - u(x)}{|y|^{n + 2s}} \, dy \biggr| & \leqslant C_u \int_{\mathbb{R}^n \setminus B(0,1)} \frac{1}{|y|^{n + 2s} (1 + |x - y|)^{n + 2 s}} \, dy \\ & \qquad + \frac{C_u}{(1 + |x|)^{n + 2 s}} \int_{\mathbb{R}^n \setminus B(0,1)} \frac{1}{|y|^{n + 2s}} \, dy \\ & \leqslant \frac{C_{u,n,s}}{(1 + |x|)^{n + 2 s}} + \frac{C_{u,n,s}}{(1 + |x|)^{n + 2 s}} = \frac{C_{u,n,s}}{(1 + |x|)^{n + 2 s}} \, .\end{aligned} $$ The desired result follows: $$ |(-\Delta)^s u(x)| \leqslant \frac{C_{u,n,s}}{(1 + |x|)^{n + 2 s}} \, .$$