Sharp Stability for LSI

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 H1 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.