The sharp stability constant for the reverse Sobolev inequality on S^n equals 1 for s-n/2 in (1,2) and is not attained, while the range (0,1) has a positive constant with explicit upper bound.
A note on the Sobolev inequality
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Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere
The sharp stability constant for the reverse Sobolev inequality on S^n equals 1 for s-n/2 in (1,2) and is not attained, while the range (0,1) has a positive constant with explicit upper bound.