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Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence
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We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions.
Forward citations
Cited by 2 Pith papers
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The sharp $\sigma_2$-curvature inequality on the sphere in quantitative form
Near-minimizers of the normalized total σ2-curvature on the d-sphere are quantitatively close to the standard metric in two Sobolev norms, with optimal exponents 2 and 4.
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The paper reviews and re-exposes existing stability theorems for sharp Sobolev inequalities on manifolds with Ricci lower bounds.
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