Improved Poincar\'e-Hardy inequalities on certain subspaces of the Sobolev space

TL;DR

This paper develops enhanced Poincaré-Hardy and Caffarelli-Kohn-Nirenberg inequalities on hyperbolic space, providing improved constants and new inequality forms using Bessel pairs, advancing the understanding of functional inequalities in non-Euclidean geometries.

Contribution

It introduces improved inequalities on hyperbolic space with better constants and new formulations, expanding the theoretical framework of Sobolev space inequalities.

Findings

Derived an improved Poincaré-Hardy inequality with a sharper constant.

Established a new Caffarelli-Kohn-Nirenberg inequality on hyperbolic space.

Demonstrated the effectiveness of Bessel pairs in inequality improvements.

Abstract

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space.