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

TL;DR

This paper proves the stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds, extending previous results to fractional orders using symmetrization and dual stability methods.

Contribution

It establishes the stability of fractional Sobolev inequalities with explicit lower bounds, linking it to HLS inequalities stability and extending prior work.

Findings

Stability of HLS inequalities with explicit bounds proven.

Stability results extended to fractional Sobolev inequalities.

Uses symmetrization and dual stability theory in proofs.

Abstract

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.