On some strong Poincaré inequalities on Riemannian models and their improvements

Hardy-Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

“…Similarly, if we compare Corollary 2.2 with [28, Theorem 4.3], again, the corrections of the Rellich inequality provided there decays more rapidly than ours, either as r → 0 + and as r → +∞. As concerns the improved second order Poincaré inequalities given by Corollaries 2.3 and 2.4, the gain with respect to the inequalities already known in [6] is in the adding of a further remainder term.…”

“…the Rellich inequality which comes for l = 0 and the Hardy-Rellich inequality for l = 1. We recall that inequalities (1.5) are known from [27] and [30] with optimal constants, while improvements have been provided in [3]- [4] and, for radial operators, in [6]- [29]. Instead, inequalities (1.6) were firstly studied in [19] and in [31], where the optimality of the constants was proved together with the existence of some remainder terms.…”

On Higher order Poincaré Inequalities with radial derivatives and Hardy improvements on the hyperbolic space

“…Now using Lemma 3.4 and weighted Hardy inequality (3.6), we obtain the following weighted Rellich type inequality. Also note that, exploiting [9,Lemma 6.1], this version will become stronger than [37,Theorem 4.4] and will give a quick improvement of [33,Theorem 4.3], for the case p = 2. Theorem 3.4.…”

“…Also note that both constants N −1 2 2 and 1 4 are sharp in an obvious sense. Recently a sharper version of the above inequality considering only the radial part of the gradient has been obtained in [9,Theorem 2.1] which reads as follows (1.4)…”

“…As a matter of the fact, using Gauss's Lemma one has |∇ H N u| ≥ |∇ r,H N u| and this readily resolves the optimality issue of the constant N −1 2 2 . Also note that sharpness of the constant 1 4 has been proved by constructing minimizing sequence and through some delicate analysis in [9].…”