Geometric logarithmic Hardy and Hardy–Poincaré inequalities on stratified groups

We develop a unified strategy to obtain the geometric logarithmic Hardy inequality on any open set M⊂G$M\subset {\mathbb {G}}$ of a stratified group G${\mathbb {G}}$, provided the validity of the Hardy inequality in this setting, where the so‐called “weight” is regarded to be any measurable nonnegative function w$w$ on M$M$. Provided the legitimacy of the latter for some M,w$M,w$, we also show an inequality that is an extension of the ‘generalized Poincaré inequality’ introduced by Beckner with the addition of the weight w$w$, and this is referred to as the “geometric Hardy‐Poincaré inequality.” The aforesaid inequalities become explicit in the case where M=G+$M={\mathbb {G}}^{+}$, the half‐space of G${\mathbb {G}}$, when w(·)=dist(·,∂G+)$w(\cdot)={\text{dist}(\cdot,\partial \mathbb {G}^{+})}$, and in the case where M=G$M={\mathbb {G}}$, when w$w$ is the “horizontal norm” on the first stratum of G${\mathbb {G}}$. 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 G${\mathbb {G}}$. Applying our results to the case where G=Rn${\mathbb {G}}={\mathbb {R}}^n$ (abelian case), we generalize the classical probabilistic Poincaré inequality by adding weights.