Half-Laplacian has been discussed in this entry using harmonic extension. Today, we derive a more general form, called fractional laplacian denoted by , for a function where the parameter is a real number between 0 and 1 and .

It is worth recalling the following fact known as Newtonian potential in theory of PDEs. The Newtonian potential of a compactly supported integrable function , i.e. , is defined as the convolution

where the Newtonian kernel in dimension is defined by

.

Here is the volume of the unit in . Coefficients in , denoted by , are usually called normalization constants. The Newtonian potential of is a solution of the Poisson equation

.

In case , is actually the Riesz potential

with where the Riesz potential is defined by

where is some normalization constant given by

.

Observe that at least formally the Riezs potentials verify the following rule

which implies

.

Thus if we set to be (in fact, this is impossible) we get

which helps us to write down

.

Thus we wish to define the fractional Laplacian as follows

.

The above fractional Laplacian is also often called the Riesz fractional derivative [here]. In a paper entitled “An Extension Problem Related to the Fractional Laplacian” due to Luis Caffarelli et al. [here] published in *Comm. Partial Differential Equations* in 2007, the fractional Laplacian can also be defined using

.

Let us now study the terminology of weak solution to the following semilinear elliptic equation in the whole space

.

By a weak solution we mean a function such that

for any positive test function in the distribution sense. I will back to this stuff once I finish introducing the fractional Laplacian via pseudo-differential operators.

See also: Half-Laplacian in