  In the very influential paper \cite{CS07} Caffarelli and Silvestre studied
regularity of $(-\Delta)^s$, $0<s<1$, by identifying fractional powers with a
certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and
Gal\'e, Miana and Stinga \cite{GMS13} gave several more abstract versions of
this extension procedure. The purpose of this paper is to study precise
regularity properties of the Dirichlet and the Neumann problem in Hilbert
spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between
interpolation spaces and its part in the underlying Hilbert space is exactly
the fractional power.
