A symbol calculus for foliations

Abstract

The classical Getzler rescaling theorem is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, as well as a Block-Fox calculus of asymptotic operators, is constructed for all transversely spin foliations. This calculus applies to operators of degree $m$ globally times degree $\ell$ in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations. The main result is that the composition of A$\Psi$DOs is again an A$\Psi$DO, and includes a formula for the leading symbol.