Weighted Hardy’s inequalities and Kolmogorov-type operators

ABSTRACT

We give general conditions to state the weighted Hardy inequality $$\begin{array}{c}\hfill c{\int }_{{\mathbb{R}}^N}\frac{{\mathit{\phi }}^2}{{|x|}^2}\mathrm{d}\mathit{\mu }\le {\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }\mathit{\phi }|}^2\mathrm{d}\mathit{\mu }+C{\int }_{{\mathbb{R}}^N}{\mathit{\phi }}^2\mathrm{d}\mathit{\mu },\mathit{\phi }\in C_c^{\infty }({\mathbb{R}}^N),c\le c_{0,\mathit{\mu }},\end{array}$$

with respect to a probability measure $\mathrm{d}\mathit{\mu }$. Moreover, the optimality of the constant $c_{0,\mathit{\mu }}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential $$\begin{array}{c}\hfill Lu+Vu=\left(\mathrm{\Delta }u+\frac{\mathrm{\nabla }\mathit{\mu }}{\mathit{\mu }}·\mathrm{\nabla }u\right)+\frac{c}{{|x|}^2}u\end{array}$$

for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $\mathrm{d}\mathit{\mu }$ allow the drift term to be of type $\frac{\mathrm{\nabla }\mathit{\mu }}{\mathit{\mu }}=-{|x|}^{m-2}x$ with $m>0$.

KEYWORDS:

• Hardy’s inequality
• inverse-square potential
• Ornstein–Uhlenbeck operators
• invariant measure
• $C_0$-semigroup

AMS SUBJECT CLASSIFICATIONS:

• 35K15
• 35K65
• 35B25
• 34G10
• 47D03

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Notes

No potential conflict of interest was reported by the authors.