Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature

Abstract

The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive $S$-curvature, cf. Huang, Krist\'aly and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the $S$-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the $S$-curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the $S$-curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the $S$-curvature plays again a decisive role.