Abstract
Introduction. An important technique in the use of singular integral operators is that known as regularization, which means roughly reduction to an ordinary integral. There seems to be no thorough investigation of this method for singular integral operators on manifolds, either for a single operator or for systems. This paper demonstrates that on compact manifolds of dimension two or more, for any singular integral operator H there is anl invertible operator H' such that H'H is the identity plus a completely continuous operator. Here elliptic means with symbol never zero. The well-developed theory of singular integrals along plane curves shows that this result is not true for the circle, so that the restriction to dimension two or more is essential.
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