The authors relate the fractional Laplacian of a function \(f: \mathbb{R}^{n}\rightarrow \mathbb{R}\) to solutions \(u:\mathbb{R}^{n}\times [0,\infty )\rightarrow \mathbb{R}\) of the extension problem \[ \left\{ \begin{matrix} u(x,0)=f(x) \\ \Delta _{x}u+\frac{a}{y}u_{y}+u_{yy}=0. \end{matrix} \right. \] It is shown that \[ \lim_{y\rightarrow 0}y^{a}u_{y}(x,y)=u_{z}(x,0)=-(-\Delta )^{s}f(x) \] where \(s=\frac{1-a}{2}\) and \(z=\left( \frac{y}{1-a}\right) ^{1-a}.\) This work extends the well-known fact that the operator \((-\Delta )^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. Therefore, the present work generalizes this characterization to general fractional powers of the Laplacian. This is also done for other integro-differential operators and some properties of these integro-differential equations are derived.

Reviewer: Nasser-eddine Tatar (Dhahran)

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