1.2. INTRINSIC APPROAC H T O TH E HEISENBERG CALCULUS 3

(®fc/nAfl)©A^

1

0 A^

2

of horizontal forms. The contact Laplacian is a differential

operator of order 2 in degree k ^ n and of order 4 in degree n.

In the examples (a)-(c) the operators are instances of sublaplacians. More

precisely, a sublaplacian is a second order differential A : C°°(M,£) — » C°°(M,£)

which near any point a G M is of the form,

d d

(1.1.2) A = - J2 Xj ~ il*(x)X0 + ] T aj(x)Xj + 6(x),

where Xo, X i , . . . , Xj is a local frame of TM such that X\,..., X^ span H and the

coefficients /x(x) and ai(x),..., a^(x), 6(x) are local sections of End£.

1.2. Intrinsic approach to the Heisenberg calculus

Although the differential operators above may be hypoelliptic under some con-

ditions, they are definitely not elliptic. Therefore, we cannot rely on the standard

pseudodifferential calculus to study these operators.

The substitute to the standard pseudodifferential calculus is provided by the

Heisenberg calculus, independently introduced by Beals-Greiner [BG] and Tay-

lor [Tay] (see also [Bol], [CGGP], [Dyl], [Dy2], [EM], [FSl], [RS]). The idea

in the Heisenberg calculus, which goes back to Elias Stein, is the following. Since

the relevant notion of tangent structure for a Heisenberg manifold (M, H) is that

of a bundle GM of 2-step nilpotent graded Lie groups, it stands for reason to con-

struct a pseudodifferential calculus which at every point x G M is well modelled by

the calculus of convolution operators on the nilpotent tangent group GXM.

The result is a class of pseudodifferential operators, the ^HDOS, which are

locally ^DOs of type (|, | ) , but unlike the latter possess a full symbolic calculus

and makes sense on a general Heisenberg manifold. In particular, a \I/#DO admits

a parametrix in the Heisenberg calculus if, and only if, its principal symbol is

invertible, and then the \I/#DO is hypoelliptic with a gain of derivatives controlled

by its order (see Section 3.1 for a detailed review of the Heisenberg calculus).

1.2.1. Intrinsic notion of principal symbol. In [BG] and [Tay] the prin-

cipal symbol of a \£#E)0 is defined in local coordinates only, so the definition a

priori depends on the choice of these coordinates. In the special case of a contact

manifold, an intrinsic definition have been given in [EM] and [EMM] as a section

over a bundle of jets of vector fields representing the tangent group bundle of the

contact manifold. This approach is similar to that of Melrose [Me2] in the setting

of the 6-calculus for manifolds with boundary.

In this paper we give an intrinsic definition of the principal symbol, valid for

an arbitrary Heisenberg manifold, using the results of [Po6].

Let (M

d + 1

,if) be a Heisenberg manifold. As shown in [Po6] the tangent Lie

group bundle of (M, H) can be described as the bundle (TM/H) 0 H together with

the grading and group law such that, for sections XQ, YQ of TM/H and sections

X ; , Yf of H, we have

(1.2.1) t.(X0 + Xf) = t2X0 + tX\ t e R,

(1.2.2) (X0 + X').(Yo + Y') = X0 + ^o + \C(X',

Yf)

+ X' + Y',