Does there exist a continuous bilinear form $\mathcal{B}$ on $\mathcal{S}(\mathbb{R})\times \mathcal{S}(\mathbb{R})$ such that
\begin{equation}
\mathcal{B}(\varphi\_1, \varphi\_2) =\int\_{\mathbb{R}\times\mathbb{R}} \frac{\varphi\_1(x) \varphi\_2(y)}{\lvert x - y \rvert} \mathrm{d}x\mathrm{d}y
\end{equation}
for every $\varphi\_1, \varphi\_2 \in \mathcal{S}(\mathbb{R})$ with disjoint supports? 


*Context:* 
A bilinear form $\mathcal{B}$ comes together with a kernel $K \in \mathcal{S}'(\mathbb{R}\times \mathbb{R})$ such that $\mathcal{B}(\varphi\_1 ,\varphi\_2) = \langle K , \varphi\_1 \otimes \varphi\_2 \rangle$ (by the kernel theorem). The restriction to functions with disjoint supports allows us to avoid possible diagonal problems for the kernel $K$. It is possible to define a valid kernel of the form $K(x,y) = \lvert x - y \rvert^{-\lambda}$ outside the diagonal for $0<\lambda<1$.
My guess is here that this is impossible for $\lambda = 1$, and therefore that it is impossible to find a bilinear form $\mathcal{B}$ satisfying the equation above when restricted to test functions with disjoint supports. Is this correct and if yes, how can one prove it?



No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?
Yes it is possible. Let $K\in S'(\mathbb{R}^2)$ be the distribution acting on test functions $f(x,y)$ by
$$
K(f)=\int\_{|x-y|\ge 1} \frac{f(x,y)}{|x-y|}\ dx\ dy
\ + \int\_{|x-y|< 1} \frac{f(x,y)-f(x,x)}{|x-y|}\ dx\ dy\ .
$$
What is impossible is to do this extension to the diagonal while preserving the degree -1 homogeneity. The only homogeneous distribution on $\mathbb{R}$ with this homogeneity is delta at the origin (up to scale).


A good reference for this type of questions is vol 1 of "Generalized Functions"
by Gelfand and Shilov. This becomes interesting in the context of probability theory see:
[Squaring random Schwartz distributions](https://mathoverflow.net/questions/201997/squaring-random-schwartz-distributions)


A related question is: can one interpret the energy field for the 2d Ising critical scaling limit as a random distribution?


Also a considerable amplification of this kind of methods for extending distributions to the diagonal is BPHZ renormalization. See, e.g., this lecture by Hairer: <http://www.mathtube.org/lecture/video/bphz-theorem-stochastic-pdes-0>


