Sharp Stability for LSI

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 H1 was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an Lp 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 H1 convergence is identified in this paper. Moreover, an explicit H1 bound via a moment assumption is shown. Additionally, the Lp stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.