Singular Integrals with Flag Kernels and Analysis on Quadratic CR Manifolds

We study a class of operators on nilpotent Lie groups G given by convolution with flag kernels. These are special kinds of product-type distributions whose singularities are supported on an increasing subspace (0)⊂V1⊂…⊂Vk⊂…⫋G. We show that product kernels can be written as finite sums of flag kernels, that flag kernels can be 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 □b-complex on certain quadratic CR submanifolds of Cn. We obtain Lp regularity for certain derivatives of the relative fundamental solution of □b and for the corresponding Szegö projections onto the null space of □b by showing that the distribution kernels of these operators are finite sums of flag kernels.