Optimal Hardy inequality for the fractional Laplacian on $L^p$

Abstract

For the fractional Laplacian we give Hardy inequality which is optimal in L p for 1 < p < ∞. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on L p .

On the supercritical fractional diffusion equation with Hardy-type drift

“…in the critical (α = 1) and the supercritical regimes (0 < α < 1). The terminology "critical" and "supercritical" refers to the fact that when α = 1 the drift term f • ∇ is of the same weight as the diffusion term (−∆) α 2 , while if α < 1 then, formally, f • ∇ dominates (−∆) α 2 , so the standard perturbation-theoretic techniques are not applicable.…”

“…[20]). Namely, multiplying the equation by u|u| r−2 and integrating, we have with the sharp constant c d,α,r (see [2]), we arrive at the condition |κ| d−α r < c d,α,r , which yields a constraint on κ < 0 from below. In fact, in the local case α = 2, some aspects of the regularity theory of the corresponding parabolic equation depend on this constraint, see [8].…”

“…(Clearly, the limit does not depend on the choice of {ε n } ↓ 0.) Extending T t 2 by continuity to L 2 , and then to all t > 0 by postulating the reproduction property, we obtain a C 0 semigroup on L 2 . Put e −tΛ 2 := T t 2 , t ≥ 0.…”