Sharp quasi-invariance threshold for the cubic Szeg{o} equation

We consider the 1-dimensional cubic Szeg{o} equation with data distributed according to the Gaussian measure with inverse covariance operator (1-∂ₓ²)^ s2, where s>12. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, s≠ 34, 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.