Hardy inequalities with best constants on Finsler metric measure manifolds

Abstract

The paper is devoted to weighted $L^p$-Hardy inequalities with best constants on Finsler metric measure manifolds. There are two major ingredients. The first, which is the main part of this paper, is the Hardy inequalities concerned with distance functions in the Finsler setting. In this case, we find that besides the flag curvature, the Ricci curvature together with two non-Riemannian quantities, i.e., reversibility and S-curvature, also play an important role. And we establish the optimal Hardy inequalities not only on noncompact manifolds, but also on closed manifolds. The second ingredient is the Hardy inequalities for Finsler $p$-sub/superharmonic functions, in which we also investigate the existence of extremals and the Brezis-V\'azquez improvement.