Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

ABSTRACT

In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant on the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein–Uhlenbeck operator perturbed by a critical singular potential in a two-dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which are established in the work of Goldstein et al. [Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl Anal. 2012;91(11):2057–2071] and Hauer and Rhandi [A weighted Hardy inequality and nonexistence of positive solutions. Arch Math. 2013;100:273–287] in the non-critical cases, and are also considered as extensions of a result in Cabré and Martel [Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potential singulier. C R Acad Sci Paris Sér I Math. 1999;329:973–978] to the Kolmogorov operator case perturbed by a critical singular potential.

KEYWORDS:

• Weighted Hardy's inequality
• limiting case
• Kolmogorov operator
• singular potential

2010 Mathematics Subject Classifications:

• 35A23
• 26D10

Acknowledgements

The authors thank the anonymous referees for their careful reading the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Part of this work was supported by JSPS Grant-in-Aid for Fellows (DC2) [No. 16J07472 (M.S)] and JSPS Grant-in-Aid for Scientific Research (B) [No. 15H03631].