Systems of singular integral operators on spheres
Daniel A. Levine · Transactions of the American Mathematical Society · 1969
This paper has two purposes : to develop a theory of special functions for SO («)/SO (« -2) ; and to apply it to the study of systems of singular integral operators on Sn ~1 having specified transformation properties under the action of rotations.The results on special functions for SO («)/SO (« -2) enable us to classify all "irreducible" systems and to decompose an arbitrary singular operator as a sum of operators equivalent (modulo the smoothing operators) to an element of one of the irreducible systems.To motivate the study of systems of operators, and to illustrate the point of view adopted, we ask the following question about operators on Lp(En).Which bounded operators on each L"(En), l 0, where (8nf)(x)=f(\'1x);(3) LaT=TLa for each rotation a, where (Laf)(x)=f(a~1x).Conditions (1) and (2) show that (Tf)~(Ç) = m(ii)f(li), where m(f) is a homogeneous function of degree 0, and/is the Fourier transform off.If condition (3) is satisfied then m must be constant, and A a scalar multiple of the identity.Therefore, to obtain a nontrivial answer we must relax at least one of the three conditions.To discuss translation-invariant singular operators we keep (1) and ( 2).(Keeping (1) and ( 3) leads to the theory of "variable-kernel" operators, to which we shall return shortly.)We relax (3) by asking instead if there is a (complex) vector space V of operators such that the action Aft» LaTL¿1 yields an (irreducible) representation a -> A0 of the rotation group SO («) on V.In other words, as an equality in V,for each TeV and a e SO («).If û-> Aa is the trivial representation given by A0 = 7for each a, we recover condition (3).If a -> P}a = a is the standard representation of SO («), the system spanned by the Riesz operators ft,..., An defined by (RifY'(Ç) = £Aè\~1f(è) satisfies (1), ( 2) and (3').Moreover, as a consideration of (A/ft(i) shows, any system satisfying (1), ( 2) and (3') for the standard representation must coincide with the span of the Riesz operators.As a generalization of the