The fractional Laplacian 
, with 
, is a nonlocal operator
defined, in particular, for Schwartz functions
on 
, where 
is a positive integer,
by
 |
where 
is the Euclidean norm and 
is the normalization constant.
Here, 
denotes the Cauchy principal value, obtained
by integrating over 
and letting 
. With the Fourier
transform convention using 
, its Fourier
multiplier is 
. Thus it is a fractional
power of the negative Laplacian (Kwaśnicki 2017).
When 
, the Riesz potential
of order 
gives its inverse on suitable functions. The fractional
Lane-Emden conjecture concerns a nonlinear system
of equations formed from this operator (Mei and
Wei 2026).
