The Square Root of the Laplacian

In this chapter, we study the simplest integro-differential elliptic operator: the square-root of the Laplacian, \(\sqrt {-\varDelta }\) . We start by establishing its basic properties, including the harmonic extension representation and the corresponding heat kernel and fundamental solution. We then prove the comparison principle, compute its Poisson kernel in a ball and find the corresponding mean value property, deduce the Harnack inequality, and establish interior regularity estimates. Finally, we construct some explicit solutions and develop the analogous results for the fractional Laplacian \((-\Delta )^s\) , with \(s \in (0, 1)\) .