The fundamental solution to $$\Box _b$$ on quadric manifolds: part 2. $$L^p$$ regularity and invariant normal forms

Abstract

This paper is the second of a multi-part series in which we explore geometric and analytic properties of the Kohn–Laplacian and its inverse on general quadric submanifolds of $${\mathbb{C}}^n\times {\mathbb{C}}^m$$ . We have two goals in this paper. The first is to give useable sufficient conditions for a map T between quadrics to be a Lie group isomorphism that preserves $$\Box _b$$ , and the second is to establish a framework for which appropriate derivatives of the complex Green operator are continuous in $$L^p$$ and $$L^p$$ -Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension two quadrics in $${\mathbb{C}}^4$$ .