Optimal weighted Poincaré inequalities for the Barenblatt and uniform densities in Rⁿ

We prove weighted n -dimensional Poincaré inequalities for Barenblatt densities, which are stationary states of certain Fokker–Planck type equations with non-constant diffusion coefficients. The interest in Poincaré inequalities for Barenblatt densities stems from their deep connection with the trend-to-equilibrium problem for Fokker–Planck type equations that admit these densities as steady states.