Considering the following definition for the fractional Laplacian \begin{eqnarray} \label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{(u(x)-u(y))} {|x-y|^{d+2s}}dy \\ \nonumber & : = & a_{d,s}\lim_{\delta\rightarrow 0^{+}}\int_{\{y\in \mathbb{R}^d:\ \lvert x-y\rvert\} \ge \delta}\frac{(u(x)-u(y))} {|x-y|^{d+2s}}dy \end{eqnarray}
The Proposition 1.5 of LOCAL ELLIPTIC REGULARITY FOR THE DIRICHLET FRACTIONAL LAPLACIAN by Umberto Biccari et al., states that if $u, v$ be such that $(-\bigtriangleup)^{s}(u)$ and $(-\bigtriangleup)^{s}(v)$ exist and \begin{eqnarray*} \int_{\mathbb{R}^{d}} \frac{|(u(x)-u(y))(v(x)-v(y))|}{\vert x-y\vert^{d+2s}}dy < \infty. \end{eqnarray*} Then $(-\bigtriangleup)^{s}(uv)$ exist and is given by \begin{eqnarray}\label{formula} (-\bigtriangleup)^{s}(uv)=u(-\bigtriangleup)^{s}(v)+v(-\bigtriangleup)^{s}(u)-I_{s}(u,v) \end{eqnarray} where \begin{eqnarray*} I_{s}(u,v)(x)=a_{s,d}\int_{\mathbb{R}^{d}} \frac{(u(x)-u(y))(v(x)-v(y))}{\vert x-y\vert^{d+2s}}dy. (1) \end{eqnarray*}. The above identity can be proved using $ab-cd=a(b-d)+b(a-c)-(a-c)(b-d)$. Now, regarding to (1) , I have one question: Can the previous formula (1) be taken in a pointwise sense for $ u, v$ in a function space different from the Schwartz space?
