No 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 .

We obtain sharp fractional Hardy inequalities for the half-space and for convex domains. We extend the results of Bogdan and Dyda and of Loss and Sloan to the setting of Sobolev-Bregman forms.and the constant in (4) is the best possible, i.e. it cannot be replaced by a bigger one.

Let (pt) t≥0 be a convolution semigroup of probability measures on R d defined byand let Ω be an open subset of R d with finite Lebesgue measure. We consider the quantity HΩ(t) = Ω Ω−x pt(dy) dx, called the heat content. We study its asymptotic expansion under mild assumptions on ψ, in particular in the case of the α-stable semigroup.