Interpolation across the parameter space via endpoint cases v=0 and v=2

Ascertain whether an interpolation method (e.g., complex or real interpolation) can be developed that, starting from the endpoint cases v=0 (hard spheres) and v=2 (diffusive limit), proves the nonlocal differential log-Sobolev inequality (and thus Fisher-information monotonicity) for the full family of inverse-power-law interactions, covering all exponents s in (2,∞).

Background

For inverse power-law forces, the paper establishes Fisher-information decay at the two endpoints: the diffusive limit v=2 via curvature/Hessian arguments, and the hard-spheres limit v=0 via positivity/perturbation. Extending to all intermediate v would cover the whole physically relevant parameter region.

The author suggests interpolating the functional inequality between these endpoints but notes significant technical obstacles (nonconvexity of the functionals, analytic vs implicit kernel forms), leaving the viability of such an approach unresolved.