Hardy inequalities on Riemannian manifolds and applications

Abstract

We prove a simple sufficient criteria to obtain some Hardy inequalities on Rie-\ud mannian manifolds related to quasilinear second-order differential operator ∆p u :=\ud div | u|p−2 u . Namely, if ρ is a nonnegative weight such that −∆p ρ ≥ 0, then\ud the Hardy inequality\ud c M\ud |u|p\ud | ρ|p dvg ≤\ud ρp\ud | u|p dvg ,\ud ∞\ud u ∈ C0 (M ).\ud M\ud holds. We show concrete examples specializing the function ρ.\ud Our approach allows to obtain a characterization of p-hyperbolic manifolds as\ud well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo-\ud Nirenberg inequalities, uncertain principle and first order Caffarelli-Kohn-Nirenberg\ud interpolation inequality

Hardy Inequalities with Best Constants on Finsler Metric Measure Manifolds

“…In the section, we study the Hardy inequalities for p-sub/superharmonic functions in the Finsler setting. Inspired by D'Ambrosio and Dipierro [13], we have the following result. (2) Additionally suppose F p (∇ρ) ρ p−α , ρ α ∈ L 1 loc (Ω) if α > p. Then we have the following weighted Hardy inequality…”

“…We also have a logarithmic Hardy inequality. [12,13,25]). Moreover, Theorem 1.3 can be generalized to the irreversible case.…”

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

“…On the other hand, there has been growing interest in establishing Hardy and Rellich type inequalities on Riemannian manifolds, e.g. , , , , –, –, , and the references therein. In an interesting paper, Carron developed a general principle to derive weighted L 2 Hardy type inequalities on complete Riemannian manifolds.…”

“…Recently, D'Ambrosio and Dipierro showed that if ρ is a nonnegative weight function such that then the following Hardy type inequality holds for all and .…”

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

“…Many authors consider generalized versions of the inequalities with remainder terms [1,4,28] as well as those expressed in Orlicz setting [15,17,44,46]. Recently, Hardy-type inequalities are investigated also on Riemannian manifolds [26].…”