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Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 7
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Articles

Weighted Hardy’s inequalities and Kolmogorov-type operators

, , &
Pages 1236-1254 | Received 03 Aug 2017, Accepted 14 Dec 2017, Published online: 28 Dec 2017
 

ABSTRACT

We give general conditions to state the weighted Hardy inequality cRNφ2|x|2dμRN|φ|2dμ+CRNφ2dμ,φCc(RN),cc0,μ,

with respect to a probability measure dμ. Moreover, the optimality of the constant c0,μ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential Lu+Vu=Δu+μμ·u+c|x|2u

for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on dμ allow the drift term to be of type μμ=-|x|m-2x with m>0.

AMS SUBJECT CLASSIFICATIONS:

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.

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