Geometric logarithmic Hardy and Hardy–Poincaré inequalities on stratified groups

Abstract We develop a unified strategy to obtain the geometric logarithmic Hardy inequality on any open set of a stratified group , provided the validity of the Hardy inequality in this setting, where the so‐called “weight” is regarded to be any measurable nonnegative function on . Provided the legitimacy of the latter for some , we also show an inequality that is an extension of the ‘generalized Poincaré inequality’ introduced by Beckner with the addition of the weight , and this is referred to as the “geometric Hardy‐Poincaré inequality.” The aforesaid inequalities become explicit in the case where , the half‐space of , when , and in the case where , when is the “horizontal norm” on the first stratum of . For the second case, the semi‐Gaussian analog of the derived inequalities is proved, when the Gaussian measure is regarded with respect to the first stratum of . Applying our results to the case where (abelian case), we generalize the classical probabilistic Poincaré inequality by adding weights.