Intrinsic notion of principal symbol for the Heisenberg calculus

Abstract

In this paper we define an intrinsic notion of principal for the Hypoelliptic calculus on Heisenberg manifolds. More precisely, the principal symbol of a \psivdo appears as a homogeneous section over the linear dual of the tangent Lie algebra bundle of the manifold. This definition is an important step towards using global $K$-theoretic tools in the Heisenberg setting, such as those involved in the elliptic setting for proving the Atiyah-Singer index theorem or the regularity of the eta invariant. On the other hand, the intrinsic definition of the principal symbol enables us to give an intrinsic sense to the model operator of \psivdo at point and to give a definitive proof that the Heisenberg calculus is modelled at each point by the calculus of left-invariant \psidos on the tangent group at the point. This also allows us to define an intrinsic Rockland condition for \psivdos which is shown to be equivalent to the invertibility of the principal symbol, provided that the Levi form has constant rank. Furthermore, we review the main hypoellipticity results on Heisenberg manifolds in terms of the results of the paper. In particular, we complete the treatment of the Kohn Laplacian by Beals-Greiner and establish that for the horizontal sublaplacian the invertibility of the principal symbol is equivalent to some condition on the Levi form, called condition $X(k)$. Incidentally, this paper provides us with a relatively up-to-date overview of the main facts about the Heisenberg calculus.