Sufficient criteria for obtaining Hardy inequalities on Finsler manifolds

TL;DR

This paper establishes Hardy inequalities on Finsler manifolds using superharmonic weight functions, extending classical results and deriving related inequalities and principles in this geometric setting.

Contribution

It provides sufficient conditions for Hardy inequalities on Finsler manifolds based on superharmonicity, extending known Riemannian results to Finsler geometry.

Findings

Superharmonic weight functions yield Hardy inequalities on Finsler manifolds.

Derived weighted Caccioppoli, Gagliardo-Nirenberg, and uncertainty principles.

Extended Hardy inequalities to Finsler-Hadamard and bounded geometry manifolds.

Abstract

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if $\rho$ is a nonnegative function and $-\boldsymbol{\Delta} \rho \geq 0$ in weak sense, where $\boldsymbol{\Delta}$ is the Finsler-Laplace operator defined by $ \boldsymbol{\Delta} \rho = \mathrm{div}(\boldsymbol{\nabla} \rho)$, then we obtain the generalization of some Riemannian Hardy inequalities given in D'Ambrosio and Dipierro (Ann. Inst. H. Poincar\'e, 2013). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo-Nirenberg inequality and a Heisenberg-Pauli-Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler-Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy-inequality to a class of Finsler manifolds of bounded geometry.