Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential

Abstract

In this article, we give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardy's inequality with respect to an invariant measure μ, we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein–Uhlenbeck operator perturbed by an inverse-square potential in L ²(ℝ ^N , μ). In the case of the classical Ornstein–Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large.

Keywords:

• Hardy's inequality
• inverse-square potential
• Ornstein–Uhlenbeck operators
• invariant measure
• C ₀-semigroup
• positive weak solution

AMS Subject Classifications:

• 35K15
• 47D08
• 35B25
• 34G10