Let $$u(x) = \dfrac{1}{|x|^{N-2s}},N \neq 2s.$$ I want to show that $(-\Delta)^s u(x) = 0,\forall x \neq 0$, where $(-\Delta)^s $ is the $s-$fractional Laplacian, defined by
$$(-\Delta)^s u(x) := C(N,s) \lim_{\varepsilon \to 0} \int_{\mathbb{R}^N \backslash B(x,\varepsilon)} \dfrac{u(x)-u(y)}{|x-y|^{N+2s}}dy.$$ There are many proofs using Fourier transform. However, I am looking for a "more elementary" proof. Can someone help me?
Thank for your help.