In this topic, we are going to define the square root of the Laplacian in . Our approach makes use of the harmonic extension.
Let be a bounded continuous function in all of . There is a unique harmonic extension of in the half-space . That is,
Consider the operator . Since is still a harmonic function, if we apply the operator twice, we obtain
in . Thus, we see that the operator mapping the Dirichlet data to the Neumann data is actually a square root of the Laplacian, denoted by . It is worth noticing that it is only left to check that is indeed a positive operator, which follows by a simple integration by parts argument.
Example. In , function satisfies
However, it is no longer true for . In fact, it should be
in the harmonic extension of on . Therefore
Following is the main reference of this entry [here]. In addition, the book entitled “Foundations of Modern Potential Theory” (Springer-Verlag, 1972) due to N.S. Landkof is also a good choice.