ABSTRACT
We consider the integral operators on the sphere with a kernel depending on the inner product of arguments:
Kf= j k{x-a)f{a)d<ry x £ S n~x. (6.1)
Such operators play an important role in harmonic analysis and operator theory. Following Calderon and Zygmund [33] we call them spherical convolution operators. The operator (6.1) is well defined, for example, in the case
l
J \k{t)\{l - t2)(n~3V2 dt < oo (6.2)
- l
(see Theorem 6.8 below). We have already dealt with some special cases of the operators (6.1), see (4.32)
and (4.72)-(4.74). The well known Mikhlin's formula giving the symbol $(x) of the singular operator with the characteristic /(<r), is also of the same nature, adjoining the formulas (4.32) and (4.72)-(4.74) as the limiting case a = 0 (see the operator Ao in Subsection 2.5 of Chapter 4).
