The Atiyah--Patodi--Singer index formula for measured foliations

Let X-0 be a compact Riemannian manifold with boundary endowed with an oriented, measured even dimensional foliation with purely transverse boundary. Let X be the manifold with cylinder attached and extended foliation. We prove that the L-2-measured index of a Dirac type operator is well defined and the following Atiyah-Patodi-Singer index formula is trueind(L2,Lambda) (D+) = <(A) over cap (X, del) Ch(E/S), C-Lambda > + 1/2[eta(Lambda)(D-F partial derivative) - h(Lambda)(+) + h(Lambda)(-)]Here Lambda is a holonomy invariant transverse measure, eta(Lambda)(D-F partial derivative) is the Ramachandran eta invariant (M. Ramachandran, 1993) [23] of the leafwise boundary operator and the Lambda-dimensions h(Lambda)(+/-) of the space of the limiting values of extended solutions is suitably defined using square integrable representations of the equivalence relation of the foliation with values on weighted Sobolev spaces on the leaves. (C) 2010 Published by Elsevier Masson SAS.