A class of weighted Hardy inequalities and applications to evolution problems

We state the following weighted Hardy inequality: c_o, μ∫ _ℝ^Nφ ^2 /|x|^2 dμ≤∫ _ℝ^N |∇φ |^2 dμ + K ∫ _ℝ^Nφ ^2 dμ ∀ φ∈ H_μ ^1, in the context of the study of the Kolmogorov operators: Lu=Δ u+∇μ/μ·∇ u, perturbed by inverse square potentials and of the related evolution problems. The function μ in the drift term is a probability density on ℝ^N . We prove the optimality of the constant c_o, μ and state existence and nonexistence results following the Cabré–Martel’s approach (Cabré and Martel in C R Acad Sci Paris 329 (11): 973–978, 1999 ) extended to Kolmogorov operators.