Periodization of Two-Dimensional Fractional Riesz Operators
We consider the periodization of the Riesz fractional integrals
(Riesz potentials) of two variables and show that already in this
case we come across different effects, depending on whether we use
the repeated periodization, first in one variable, and afterwards
in another one, or the so called double periodization. We show that
the naturally introduced doubly-periodic Weyl-Riesz kernel of order
less than 2, in general coincides with the periodization of the
Riesz kernel, the repeated periodization being possible for all
orders , while the double one is applicable only for orders less
than 1. This is obtained as a realization of a certain general
scheme of periodization, both repeated and double versions. We
prove statements on coincidence of the corresponding periodic and
non-periodic convolutions and give an application to the case of
the Riesz kernel.
