Calculus on Heisenberg Manifolds. (AM-119)

Subelliptic SpinℂDirac operators, I

“…The operators we obtain on the boundary are neither classical, nor Heisenberg pseudodifferential operators, but rather operators belonging to the extended Heisenberg calculus introduced in [9]. Similar classes of operators were also introduced by Beals, Greiner and Stanton as well as Taylor; see [4], [3], [15]. In this paper we apply the analytic results obtained in [7] to obtain Hodge decompositions for each of the boundary conditions and (p, q)-types.…”

Noncommutative geometry of foliations

“…Recall that B denotes the algebra generated by the operators of the form δ k (a), a ∈ A, k ∈ N. Define the noncommutative integral determined by this spectral triple, setting The functional − is a trace on B, which is local in the sense of noncommutative geometry, since it vanishes for any element of B, which is a trace class operator in H. index theorem, Theorem 8.10. As it was already mentioned in Section 6.3, the study of these spectral triples makes an essential use of the hypoelliptic pseudodifferential calculus on Heisenberg manifolds developed by Beals and Greiner [10]. In particular, the noncommutative integral − determined by such a spectral triple coincides with the Wodzicki-Guillemin type residue trace τ defined on the Beals-Greiner algebra of pseudodifferential operators.…”

Analytic torsions on contact manifolds

“…However there is a (substantially more intricate) symbolic calculus that can be applied to it to obtain results on heat kernels qualitatively analogous to the elliptic case. This calculus is called the (Volterra)-Heisenberg calculus and was introduced by Beals-Greiner-Stanton [4,3] and Taylor [44]. A short account of its properties may be found in [21], and its use for the contact complex has been presented by Julg and Kasparov in [23, §5].…”