Sharp Stability for LSI

Emanuel Indrei ()
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Emanuel Indrei: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA

Mathematics, 2023, vol. 11, issue 12, 1-13

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 inequality, the remainder is in terms of the entropy, not a metric. Recently, a stability result for H 1 was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an L p norm. Afterward, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H 1 convergence is identified in this paper. Moreover, an explicit H 1 bound via a moment assumption is shown. Additionally, the L p stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.

Keywords: stability; logarithmic Sobolev inequality; entropic uncertainty (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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