A relativistic Hardy-type inequality with minimisers

Abstract

In this paper, we prove a sharp, weighted Hardy-type inequality for the Dirac operator. A key feature of our result is that the inequality is not only sharp but also attained, and we construct explicit minimizers that satisfy the equality case. This extends previous work on the spectral properties of Dirac operators, especially in the context of relativistic quantum mechanics and Coulomb-like potentials.