Published Paper

Inserted: 28 may 2024
Last Updated: 16 dec 2024

Journal: Int. Math. Res. Not.
Year: 2024
Doi: https://doi.org/10.1093/imrn/rnae269

Abstract:

We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls.