A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation

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 $$d\rho \propto \exp (\frac{1}{p}{\int }_{\mathbb{T}}\mathrm{\mid }u{\mathrm{\mid }}^{p}dx-\frac{1}{2}{\int }_{\mathbb{T}}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx-\frac{1}{2}{\int }_{\mathbb{T}}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx){\mathbf{1}}_{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathbb{T})}^2\le K}dud\bar{u}\mathrm{.}$$ When $2\le p\le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p>4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $\rho$.