Sufficient criteria for obtaining Hardy inequalities on Finsler 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.