The classical Getzler rescaling theorem of  is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, , as well as a Block–Fox calculus of asymptotic pseudodifferential operators (ADOs), , is constructed for all transversely spin foliations. This calculus applies to operators of degree globally times degree 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 ADOs is again an ADO, and includes a formula for the leading symbol. Our formula is more complicated due to its wide generality but its form is essentially the same, and it simplifies notably for Riemannian foliations. In short, we construct an asymptotic pseudodifferential calculus for the “leaf space” of any foliation. Applications will be derived in [5,6] where we give a Getzler-like proof of a local topological formula for the Connes–Chern character of the Connes–Moscovici spectral triple of , as well as the (semi-finite) spectral triple given in , yielding an extension of the seminal Atiyah–Singer covering index theorem, , to coverings of “leaf spaces” of foliations.
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Moulay-Tahar Benameur, James L. Heitsch, A symbol calculus for foliations. J. Noncommut. Geom. 11 (2017), no. 3, pp. 1141–1194DOI 10.4171/JNCG/11-3-11