We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by \(\begin{aligned} d\rho \propto \exp \big (\frac{1}{p}\int _{\mathbb T} |u|^p d x - \frac{1}{2}\int _{\mathbb T} |\nabla u|^2 d x - \frac{1}{2}\int _{\mathbb T} |u|^2 d x\big ) \mathbbm {1}_{\Vert u \Vert _{L^2(\mathbb T)}^2 \le K}dud\overline{u}. \end{aligned}\) When \(2 \le p \le 4\) , we show that these measures indeed satisfy a log-Sobolev inequality. When \(p> 4\) , we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure \(\rho \) .
