  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 fractional Sobolev
inequalities with the lower bounds. This extends the stability of Sobolev
inequalities with the explicit lower bounds established by Dolbeault, Esteban,
Figalli, Frank and Loss in [16] to the 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.
