Stability Estimates for Fractional Hardy-Schrödinger Operators

In this chapter, we derive optimal Hardy-Sobolev type improvements of fractional Hardy inequalities, formally written as Lsu≥wxxθu2∗−1, for the fractional Schrödinger operator Lsu=−Δsu−kn,sux2s associated with s-th powers of the Laplacian for s∈01, on bounded domains in Rn. Here, kn,s denotes the optimal constant in the fractional Hardy inequality, and 2∗=2n−θn−2s, for 0≤θ≤2s<n. The optimality refers to the singularity of the logarithmic correction w that has to be involved so that an improvement of this type is possible. It is interesting to note that Hardy inequalities related to two distinct fractional Laplacians on bounded domains admit the same optimal remainder terms of Hardy-Sobolev type. For deriving our results, we also discuss refined trace Hardy inequalities in the upper half space which are rather of independent interest.