Understanding two-scale criteria for Poincaré and log-Sobolev inequalities in the Euclidean case through Φ-entropies

We study settings in which mixture and joint distributions satisfy a Poincaré (or log-Sobolev) inequality induced by a marginal and a collection of conditional distributions that are assumed to satisfy Poincaré (or log-Sobolev, resp.) inequalities and supported over Euclidean spaces. In this note, we use the framework of $\mathrm{\Phi }$-Sobolev inequalities (Chafaï, 2004) to provide a unified approach to arriving at these inequalities in the Euclidean case. This results in a simpler proof technique for establishing these functional inequalities under certain two-scale criteria. We also discuss applications of these results to certain sampling algorithms.