Sharp quasi-invariance threshold for the cubic Szegő equation

We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-{\mathrm{\partial }}_x^2{)}^{\frac{s}{2}}$, where $s>\frac{1}{2}$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\mathrm{\ne }\frac{3}{4}$, the transported measure and the initial Gaussian measure are mutually singular for almost every time. This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs.