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 $1<p<\infty$. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on $L^p$.

We added 3 figures, reformulated Theorem 3 and 4, and added acknowledgements and comments