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We consider Gagliardo-Nirenberg inequalities on the sphere which interpolatebetween the Poincar\'e inequality and the Sobolev inequality, and include thelogarithmic Sobolev inequality as a special case. We establish explicitstability results in the subcritical regime using spectral decompositiontechniques, and entropy and carr\'e du champ methods applied to nonlineardiffusion flows.

Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results