Article
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Sharp Stability for LSI
Version 1
: Received: 26 April 2023 / Approved: 27 April 2023 / Online: 27 April 2023 (02:58:02 CEST)
How to cite: Indrei, E. Sharp Stability for LSI. Preprints 2023, 2023041008. https://doi.org/10.20944/preprints202304.1008.v1 Indrei, E. Sharp Stability for LSI. Preprints 2023, 2023041008. https://doi.org/10.20944/preprints202304.1008.v1
Abstract
A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier-Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability, the remainder is in terms of an entropy, not a metric. Recently, a stability result for H1 was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an Lp norm. Afterwards, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H1 convergence is identified in this paper. Moreover, an explicit H1 bound via a moment assumption is shown. Also, the Lp stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.
Keywords
Stability; Logarithmic Sobolev Inequality; Entropic Uncertainty; LSI; H^1
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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