Extend the mode-decomposition criterion to unbounded spaces with only bounded Hessian for W−

Determine whether the proof of Theorem 1.7 (the uniform log-Sobolev criterion based on strong convexity of the mode-projected free energy) can be adapted to unbounded domains by replacing the assumption that the non-convex interaction part W− admits bounded Lipschitz modes with the weaker condition that W− has uniformly bounded Hessian, i.e., sup_{x,y} ||He W−(x,y)||_op < ∞, without requiring a bounded mode decomposition.

Background

Theorem 1.7 establishes a uniform-in-N log-Sobolev inequality for the mean-field measure m^N_T when the non-convex interaction W− admits a mode decomposition with bounded, Lipschitz modes n_k and the projected free energy \hat{\mathcal F}_T is strongly convex. This assumption is natural on compact domains (torus or sphere) where Fourier or spherical harmonic mode decompositions are standard.

On unbounded spaces, the boundedness of modes is less compelling. The authors raise whether one could relax the mode-decomposition requirement and instead assume only that W− has bounded Hessian, thereby extending the applicability of their criterion beyond the compact setting.