Sharp gradient stability for the Sobolev inequality

Alessio Figalli; Yi Ru-Ya Zhang

doi:10.1215/00127094-2022-0051

1 September 2022 Sharp gradient stability for the Sobolev inequality

Alessio Figalli, Yi Ru-Ya Zhang

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Abstract

We prove a sharp quantitative version of the p-Sobolev inequality for any $1\textless p\textless n$ , with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p\le 2$ , while it depends on p for $p\textgreater 2$ .

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Alessio Figalli. Yi Ru-Ya Zhang. "Sharp gradient stability for the Sobolev inequality." Duke Math. J. 171 (12) 2407 - 2459, 1 September 2022. https://doi.org/10.1215/00127094-2022-0051

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Received: 9 March 2020; Revised: 13 June 2021; Published: 1 September 2022

First available in Project Euclid: 22 July 2022

MathSciNet: MR4484209

zbMATH: 1504.46040

Digital Object Identifier: 10.1215/00127094-2022-0051

Subjects:

Primary: 46E35

Secondary: 26D10

Keywords: Sobolev inequality , stability

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