On higher order Poincare inequalities with radial derivatives and Hardy improvements on the hyperbolic space

Abstract

In this paper we prove higher order PoincarÌ© inequalities involving radial derivatives namely, ‰öÇHN|‰öàkr,HNu|2dvHN‰ä´(N‰öÕ12)2(k‰öÕl)‰öÇHN|‰öàlr,HNu|2dvHN for all u‰ööHk(HN), where underlying space is N-dimensional hyperbolic space HN, 0‰ä?l<k are integers and the constant (N‰öÕ12)2(k‰öÕl) is sharp. Furthermore we improve the above inequalities by adding Hardy-type remainder terms and the sharpness of some constants is also discussed.