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 $-\varvec{\Delta } \rho \ge 0$ in weak sense, where $\varvec{\Delta }$ is the Finsler–Laplace operator defined by $\varvec{\Delta } \rho = \mathrm {div}(\varvec{\nabla } \rho )$, then we obtain the generalization of some Riemannian Hardy inequalities given in D’Ambrosio and Dipierro (Ann Inst H Poincaré Anal Non Linéaire 31(3):449–475, 2014). 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.