ABSTRACT
The function f(y) is called a density of the potential Iaf. The operator Ia can be considered as a kind of generalization of the one-dimensional fractional integration operators (1.75)—(1.76).
Since Ia f is a convolution of its density / with the kernel
\x\a~n
it is natural to treat Iaf via Fourier transforms. By (1.91) we formally have
F(Iaf) = * « ( * ) / ( x ) , (2.4)
where
L E M M A 2.1 (Bochner). The Fourier transform of a radial function <p(\x\) is a radial function as well and
j e'* V( | J / | ) dy = | ^ y ^(p)p"/ 2 J „ / a _ i M x | ) dp, (2.6) i l n 0
for any function ip(p) such that
J p ^ i l + pf-^Mp^dpKco, (2.7)
provided that the integral on the left-hand side of (2.6) is interpreted as conventially convergent at infinity.
