On circle diffeomorphisms with discontinuous derivatives and quasi-invariance subgroups of Malliavin–Shavgulidze measures

This note is devoted to construction of the domain of the regular representation of Diff+(S1).We consider a subgroup of diffeomorphism group such that the Malliavin–Shavgulidze measure is quasi-invariant with respect to the left action of the subgroup. The measure appears to be quasi-invariant with respect to the left action of diffeomorphisms with discontinuous second derivative. We derive an expression for quasiinvariance density and obtain an additional PSL(2,R) symmetry breaking term. © 2006 Elsevier Inc. All rights reserved.