Noncommutative Residue for Heisenberg Manifolds. I.

In this paper we construct a noncommutative residue for the Heisenberg calculus, that is, for the hypoelliptic calculus on Heisenberg man-ifolds, including on CR and contact manifolds. This noncommutative residue as the residual induced on operators of integer orders by the analytic extension of the usual trace to operators of non-integer orders and it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the input operator. We also present applications of this constructions concerning traces and sum of commutators, zeta functions of hypoelliptic operators , logarithmic metric estimates for Green kernels of hypoelliptic operators and the Dixmier trace of the operators in the Heisenberg calculus.