Invariant Sobolev calculus on the free loop space (Q1355860)

In this substantial and fundamental paper the differential calculus on the free loop space is studied. More precisely, a (scalar) Sobolev calculus invariant under the circle action is developed on the free loop space equipped with the Bismut/Høegh-Krohn measure which takes the role of the (otherwise absent) volume measure. Among other things an integration by parts formula is proved, the corresponding Ornstein-Uhlenbeck operator is analyzed, suitable connections are introduced, and various choices for the underlying tangent bundle over the loop space are discussed.