A Priori Estimates for Singular Integral Operators

Abstract

In this paper we develop the theory of singular integral operators with finitely differentiable symbols and apply it to the derivation of a priori inequalities for these and pseudo-differential operators. In section 1 we treat singular integral operators and introduce the pseudo-differential operators as was done before they were given their name, namely as compositions of singular integral operators with powers of the operator ∧.

In our opinion this is the correct point of view since in the finitely differentiable case these do not form an algebra, but merely a module over the algebra of singular integral operators. The differentiability assumptions made here are designed to yield self-adjoint algebras which are specially suited for the treatment of the L² theory, and are not too far removed from the best possible. Relaxing these conditions substantially, as was done in [l], causes the loss of self-adjointness. In section 2 we discuss the action of our operators on rapidly oscillating functions with small support and obtain, as a byproduct, a new representation for the algebras under consideration which illuminates the negative, results on inequalities which are discussed in section. 4.