Weighted Hardy's inequalities and Kolmogorov-type operators. Issue 7 (19th May 2019)

• Record Type: Journal Article
• Authors: Canale, A.; Gregorio, F.; Rhandi, A.; Tacelli, C.
• Is Part Of: Applicable analysis. Volume 98:Issue 7(2019)
• Page Start: 1236
• Page End: 1254
• Publication Date: 2019-05-19
• Subjects: Hardy's inequality -- inverse-square potential -- Ornstein–Uhlenbeck operators -- invariant measure -- C0-semigroup; 35K15 -- 35K65 -- 35B25 -- 34G10 -- 47D03; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2017.1419200
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: We give general conditions to state the weighted Hardy inequalityc ∫ R N φ 2 | x | 2 d μ ≤ ∫ R N | ∇ φ | 2 d μ + C ∫ R N φ 2 d μ, φ ∈ C c ∞ ( R N ), c ≤ c 0, μ, with respect to a probability measured μ . Moreover, the optimality of the constantc 0, μ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potentialL u + V u = Δ u + ∇ μ μ · ∇ u + c | x | 2 u for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses ond μ allow the drift term to be of type∇ μ μ = - | x | m - 2 x withm > 0 .