# Characterisation of the square root of the Laplacian as a Dirichlet to Neumann mapping

I am looking for a (classical and/or oldest) reference giving the characterisation of the operator $(-\Delta)^{\frac 12}$ as the Dirichlet to Neumann map $w_y$ where $w$ is the harmonic extension on the upper-half plane.

It can be in domains or in the whole space. Does anyone know a citation? It seems very hard to find one.

I used this in my PhD thesis in 1995 almost as a folk result: it is elementary spectral theory. I recall reading, on the probabilistic side, papers related to the subject by P.A. Meyer from the '70s, and did not claim any originality, but rather pointed to the book on Littlewood-Paley theory by Stein. In short, after taking a Fourier transform the Laplace equation $$\partial_{tt}u+\Delta_xu=0$$ in $${\mathbb R}^{n+1}_+\ni(x,t)$$ space becomes $$\partial_{tt}\hat{u}(\xi,t)-|\xi|^2\hat{u}(\xi,t)=0$$, whose solutions are $$\hat{u}(\xi,t)=e^{-t|\xi|}\hat{f}(\xi)=0,$$ or $$u(x,t)=e^{-t\sqrt{-\Delta_x}}f(x).$$ The interpretation Dirichlet to Neumann immediately follows. Given the kind of argument, I would be very surprised if this was not commonplace already at the beginning of the 20th century, or even before.