Sharp Constants, Threshold, and Failure of Uncertainty Principles on Non-trivial Flat Finsler Manifolds

Abstract

By uncertainty principles we mean the generalized Caffarelli--Kohn--Nirenberg (CKN) inequality and the Heisenberg--Pauli--Weyl (HPW) uncertainty principle, with the latter being an endpoint case and the Hardy inequality a further limiting case. We study these \(L^p\)-type inequalities \((1<p<n)\) on \(n\)-dimensional non-trivial flat Finsler manifolds, which arise from Hilbert&#39;s fourth problem. These manifolds are forward complete but not backward complete, projectively flat, with zero flag curvature, and have infinite reversibility, making them highly asymmetric. They are the simplest non-Riemannian, non-Minkowskian solutions to Hilbert&#39;s fourth problem, providing a natural framework in which classical inequalities exhibit new behavior. For the full class, we establish a sharp threshold at the parameter value \(s=2\): the generalized CKN inequality and the HPW uncertainty principle hold for \(s>2\) but fail completely for \(s\le2\). When the generating Minkowski norm is reversible (while the Finsler metric itself remains irreversible with infinite reversibility), we determine the sharp constants for \(s>2\), which were previously unknown. These constants are approached in the limit by compactly supported radial functions. As a corollary, our results cover Berwald&#39;s metric space and resolve the open questions on its sharp constants. These findings reveal that the threshold phenomenon is structural for the entire class and is a genuinely non-Riemannian feature with no analogue in the Riemannian flat case.