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This paper is concerned with pseudodifferential calculus on manifolds withfibred corners. Following work of Connes, Monthubert, Skandalis andAndroulidakis, we associate to every manifold with fibred corners alongitudinally smooth groupoid which algebraic and differential structure isexplicitely described. This groupoid has a natural geometric meaning as aholonomy groupoid of singular foliation, it is a singular leaf space in thesense of Androulidakis and Skandalis. We then show that the associatedcompactly supported pseudodifferential calculus coincides with Mazzeo andMelrose's phi-calculus and we introduce an extended algebra of smoothingoperators that is shown to be stable under holomorphic functional calculus.This result allows the interpretation of phi-calculus as the pseudodifferentialcalculus associated with the holonomy groupoid of the singular foliationdefined by the manifold with fibred corners. It is a key step to set the indextheory of those singular manifolds in the noncommutative geometry framework.

Pseudodifferential calculus on manifolds with fibred corners : the groupoid of phi-calculus