Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential
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. © 2012 Copyright Taylor and Francis Group, LLC.
Goldstein, G., Goldstein, J., & Rhandi, A. (2012). Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential. Applicable Analysis, 91 (11), 2057-2071. https://doi.org/10.1080/00036811.2011.587809