Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature

TL;DR

This paper establishes sharp Hardy, Trudinger-Moser, and Caffarelli-Kohn-Nirenberg inequalities on negatively curved Riemannian manifolds, including hyperbolic spaces, and explores their equivalences and related uncertainty principles.

Contribution

It introduces new sharp inequalities on negatively curved manifolds and reveals their equivalence with Trudinger-Moser inequalities, advancing understanding of functional inequalities in geometric analysis.

Findings

Derived sharp inequalities with optimal constants on hyperbolic spaces

Established equivalences between different classes of inequalities

Obtained uncertainty principles related to the inequalities

Abstract

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.