Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds

In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the higher and fractional order Sobolev inequalities with the lower bounds. This extends to some extent the stability of the first order Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [18] to the higher and fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory. As another application of the stability of the HLS inequality, we also establish the stability of Beckner's [4] restrictive Sobolev inequalities of fractional order s with [Formula presented.] on the flat sub-manifold Rⁿ⁻¹ and the sphere Sⁿ⁻¹ with the explicit lower bound. When s=1, this implies the explicit lower bound for the stability of Escobar's first order Sobolev trace inequality [19] which has remained unknown in the literature.