Poincar{é} inequalities and integrated curvature-dimension criterion for generalised Cauchy and convex measures
Baptiste Nicolas Huguet
TL;DR
The paper develops an integrated curvature-dimension framework to obtain sharp weighted Poincaré inequalities for generalised Cauchy and convex measures. It leverages a variance representation based on the carré du champ and its iterates to connect spectral gaps $\lambda_1(-L)$ with an integrated CD criterion and an explicit deficit term, enabling exact or near-exact characterisations of extremal functions across parameter regimes. For generalised Cauchy measures with density $d\mu_β \propto (1+|x|^2)^{-β}$ and $L f = \omega \Delta f - 2(β-1)\langle x, df\rangle$, the authors provide a complete piecewise description of the spectral gap in all dimensions, including the existence or nonexistence of eigenfunctions in various $β$-ranges, and they recover the optimal constants in the weighted Poincaré inequalities across these regimes. In 1D and, more generally, under strong convexity or bounded Hessian conditions, the results yield explicit lower bounds and extremal-function characterisations, unifying and extending prior approaches for heavy-tailed measures and offering a robust geometric-analytic method applicable to a broad class of convex measures.
Abstract
We obtain new sharp weighted Poincar{é} inequalities on Riemannian manifolds for a general class of measures. When specialised to generalised Cauchy measures, this gives a unified and simple proof of the weighted Poincar{é} inequality for the whole range of parameters, with the optimal spectral gap, the error term and the extremal functions.