  The periodic DNLS gauge is an anticipative map with singular generator which
revealed crucial in the study of the periodic derivative NLS. We prove
quasi-invariance of the Gaussian measure on $L^2(\T)$ with covariance
$[1+(-\D)^{s}]^{-1}$ under these transformations for any $s>\frac12$. This
extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani
(2011) and Genovese, Luc\`a and Valeri (2018), who proved the result for
integer values of the regularity parameter $s$.
