Rellich inequalities via Riccati pairs on model space forms

TL;DR

This paper introduces a straightforward approach using Riccati pairs and convexity arguments to establish Rellich inequalities on Riemannian manifolds with non-positive curvature, extending recent Hardy inequality results.

Contribution

It develops a new method leveraging Riccati pairs and convexity to prove Rellich inequalities on model space forms with constant non-positive curvature.

Findings

Established Rellich inequalities on non-positively curved manifolds.

Extended the Riccati pair approach from Hardy to Rellich inequalities.

Provided a unified framework for higher order inequalities on model spaces.

Abstract

We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kaj\'ant\'o, Krist\'aly, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.