Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds

Abstract

We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold M , started in [17]. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold M . In particular, we obtain sharp constants for these inequalities on the hyperbolic space H n .

Weighted Hardy and Rellich inequality on Carnot groups

“…When p = 2, our inequality (4.1) is just the inequality (1.4) in [14], also it recovers the inequality (1.5) in [20] for Heisenberg groups.…”

“…Moreover, using the weighted L 2 Hardy inequality (1.3), we obtain a weighted L p Rellichtype inequality on general Carnot groups. Our results generalize some recent inequalities in [14,20]. The main idea of our proof is to consider a special class of weighted p-sub-Laplacian and the corresponding fundamental solution with singularity at the origin, then we obtain our inequality by using the fundamental solution and a careful choice of test functions.…”

Untitled

“…Hardy's inequalities were also studied for some subelliptic operators (see, e.g., [6,7,2,3,5,12,9]), in particular, for the sub-Laplacian on the Heisenberg group H. This group is the prime example in noncommutative harmonic analysis, and we refer to [13] for the background material.…”

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

“…Another important inequality is the following Hardy–Poincaré type inequality , which directly implies the weighted Hardy inequality (1.2), where and the weight function ρ satisfies in the sense of distribution. In this paper, under a new additional assumption on the weight functions and , we obtain two‐weight form of the Hardy–Poincaré inequality with a nonnegative remainder term.…”

“…In recent years, some improved versions of (1.1) and (1.5) have been obtained. For instance, Kombe and Özaydın obtained the following weighted L 2 Hardy type inequality involving the functions ρ and δ: where , , and the weight function ρ and δ satisfy in the sense of distribution. In Theorem , we present two‐weight version of the inequality (1.8) involving more nonnegative remainder terms.…”