We consider the 1d nonlinear Schrödinger equation (NLS) on the torus with initial data distributed according to the Gaussian measure with covariance operator \((1 - \Delta )^{-s}\) , where \(\Delta \) is the Laplace operator. We prove that the Gaussian measures are quasi-invariant along the flow of (NLS) for the full range \(s > \frac{3}{2}\) . This improves a previous result obtained by Planchon et al. (Math Ann 378:389–423, 2020) where the quasi-invariance is proven for \(s=2k\) for all integers \(k\ge 1\) . In our approach, to prove the quasi-invariance, we directly establish an explicit formula for the Radon–Nikodym derivative \(G_s(t,.)\) of the transported measures, which is obtained as the limit of truncated Radon–Nikodym derivatives \(G_{s,N}(t,.)\) for transported measures associated with a truncated system. We also prove that the Radon–Nikodym derivatives belong to \(L^p\) , \(p>1\) , with respect to \(H^1(\mathbb {T})\) -cutoff Gaussian measures, relying on the introduction of weighted Gaussian measures produced by a normal form reduction, following Sun and Tzvetkov (Quasi-invariance of Gaussian measures for the 3d energy critical nonlinear Schrödinger equation, 2023. arXiv:2308.12758). Additionally, we prove that the truncated densities \(G_{s,N}(t,.)\) converges to \(G_s(t,.)\) in \(L^p\) (with respect to the \(H^1(\mathbb {T})\) -cutoff Gaussian measures).
