Weighted Hardy’s inequalities and Kolmogorov-type operators

Abstract

We give general conditions to state the weighted Hardy inequalitywith respect to a probability measure dµ. Moreover, the optimality of the constant c 0,µ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potentialfor 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 ∇µ µ = −|x| m−2 x with m > 0.

A class of weighted Hardy inequalities and applications to evolution problems

“…, is a necessary and sufficient condition for the existence of positive exponentially bounded in time solutions to the associated initial value problem. Later in [19,9] similar results have been extended to Kolmogorov operators. The proof uses some properties of the operator L and of its corresponding semigroup in…”

“…For weight functions µ satisfying assumption H 7 ) or H 8 ) there are some interesting properties regarding the semigroup {T (t)} t≥0 generated by the operator L. These properties listed in the Proposition below are well known under hypothesis H 7 ) (see [23]) and have been proved in [9] if µ satisfies H 8 ). (iii) T (t)L 2 µ ⊂ D(L) for all t > 0.…”

“…The following Theorem stated in [19] for functions µ satisfying condition H 7 ), was proved in [9] for functions µ under condition H 8 ). To get existence and nonexistence of solutions to (P ) we put together the weighted Hardy inequality (2.4), Theorem 3.1 and Theorem 4.2.…”

“…In [9] the authors state a weighted Hardy inequality using a different approach and improved Hardy inequalities. This requires suitable conditions on µ.…”

Weighted Hardy inequalities and Ornstein–Uhlenbeck type operators perturbed by multipolar inverse square potentials

“…A similar behaviour was obtained in [11] with the potential V = c |x| 2 and replacing the Laplacian by the Kolmogorov operator L . See also [6] where the hypotheses on µ allow the drift term to be of the type ∇µ µ = −|x| m−2 x, m > 0. In this paper we consider the generalized Ornstein-Uhlenbeck operator in R N…”