The interest for Sobolev’s inequalites on the Euclidean space started in the 20th century, after the works of Sobolev, Gagliardo, and Nirenberg. Optimal constants and functions were computed by Aubin and Talenti in the 80’s. Then, Brezis and Lieb conjectured that the deficit (say, the difference between the two sides) of the inequality could control some distance from the set of optimisers. A first result for Sobolev’s inequality, due to Bianchi and Egnell, appeared in the 90s. In the subsequent decades, more and more work has been devoted to stability issues, and in particular to Sobolev’s inequality. Sobolev’s inequality rewrites equivalently on the sphere, and the known stability results apply thoroughly. Much less was known about Gagliardo-Nirenberg inequalities on the sphere (i.e. the subcritical family interpolating between Sobolev’s and Poincaré inequalities), until a recent paper by Rupert Frank.

Inspired by the last contribution, a rather complete theory - with constructive stability estimates - has been established by Dolbeault, Simonov, and the speaker. The main results and techniques of such theory will be the subject of my talk.