Singular integrals with flag kernels and analysis on quadratic CR manifolds

We study a class of operators on nilpotent Lie groups G given by convolutio n with flag kernels. These are special kinds of product-type distributions whose singularities are supported on an increasing subspace (0) subset of V -1 subset of ... subset ofV(k) subset of ... not subset of or equal to G. W e show that product kernels can he writtten as finite sums of nag kernels, that flag kernels can he characterized in terms of their Fourier transforms . and that flag kernels have good regularity. restriction, and composition properties. We then apply this theory to the study of the rectangle (b)-complex on cert ain quadratic CR submanifolds of C-n. We obtain L-p regularity for certain derivatives of the relative fundamental solution of rectangle (b) and for t he corresponding Szego projections onto the null space of rectangle (b) by showing that the distribution kernels of these operators are finite sums of flag kernels. (C) 2001 Academic Press.