CHAPTER 1

Introduction

This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, in-

cluding CR and contact manifolds. In this context the main differential operators at

stake include the Hormander's sum of squares, the Kohn Laplacian, the horizontal

sublaplacian, the CR conformal operators of Gover-Graham and the contact Lapla-

cian. These operators cannot be elliptic and the relevant pseudodifferential calculus

to study them is provided by the Heisenberg calculus of of Beals-Greiner [BG] and

Taylor [Tay].

The Heisenberg manifolds generalize CR and contact manifolds and their name

stems from the fact that the relevant notion of tangent space in this setting is

rather that of a bundle of graded two-step nilpotent Lie groups. Therefore, the

idea behind the Heisenberg calculus, which goes back to Stein, is to construct

a pseudodifferential calculus modelled on homogeneous left-invariant convolution

operators on nilpotent groups.

Our aim in this monograph is threefold. First, we give an intrinsic approach to

the Heisenberg calculus by defining an intrinsic notion of principal symbol in this

setting, in connection with the construction of the tangent groupoid in [Po6]. This

framework allows us to prove that the pointwise invertibility of a principal symbol,

which can be restated in terms of the so-called Rockland condition, actually implies

its global invertibility.

These results have been already used in [Po9] to produce new invariants for

CR and contact manifolds, extending previous results of Hirachi [Hi] and Boutet de

Monvel [Bo2]. Moreover, since our approach to the principal symbol connects nicely

with the construction of the tangent groupoid of a Heisenberg manifold in [Po6],

this presumably allows us to make use of global if-theoretic arguments in the

Heisenberg setting, as those involved in the proof of the (full) Atiyah-Singer index

theorem ([AS1], [AS2]). Therefore, this part of the memoir can also be seen as a

step towards a reformulation of the Index Theorem for hypoelliptic operators on

Heisenberg manifolds.

Second, we study complex powers of hypoelliptic operators on Heisenberg man-

ifolds in terms of the Heisenberg calculus. In particular, we show that complex pow-

ers of such operators give rise to holomorphic families in the Heisenberg calculus.

To this end, due to the lack of microlocality of the Heisenberg calculus, we cannot

make use of the standard approach of Seeley, so we rely on an alternative approach

based on the pseudodifferential representation of the heat kernel in [BGS]. This

has some interesting consequences related to hypoellipticity and allows us to con-

struct a scale of weighted Sobolev spaces providing us with sharp estimates for the

operators in the Heisenberg calculus.

These results are important ingredients in [Poll] to construct an analogue for

the Heisenberg calculus of the noncommutative residue trace of Wodzicki ([Wol],

1