A criterion for boundedness of singular integral operators with complicated singularities

Singular integral operators acting on the space of (essentially) bounded functions defined on a smooth submanifold of $\mathbf R^n$ are considered. The singularities of the integrals given by the zeros of functions composing the integrand lie on smooth submanifolds. The basic result is as follows: if these submanifolds satisfy certain conditions of nondegeneracy type and the integral operator has no formal singularities (i.e., certain relations between the orders of the integrands and the dimensions of the manifolds of singularities are satisfied), then the singular integral operator is bounded on the space considered.
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