Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature

Michael Ruzhansky,Nurgissa Yessirkegenov

doi:10.1016/j.jmaa.2021.125795

Abstract

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.

Highlights

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces
On Riemannian manifolds, in [CX04] and [Mao15] the authors assuming that Caffarelli-Kohn-Nirenberg type inequalities hold, investigated the geometric property related to the volume of a geodesic ball on an n-dimensional (n ≥ 3) complete open manifold with non-negative Ricci curvature and on an n-dimensional (n ≥ 3) complete and noncompact smooth metric measure space with non-negative weighted Ricci curvature, respectively
On the hyperbolic space Hn with n ≥ 2, let us recall the following another recent result on Caffarelli-Kohn-Nirenberg type inequalities for radially symmetric functions [ST17]: Let 2 ≤ p ≤ ∞, there exists a positive constant cr = cr(n, p) such that for all f ∈ W01,r2ad(Hn) we have

Summary

Preliminaries

We briefly review some main concepts of Riemannian manifolds with negative curvature and refer to [GHL04], [Li93] and [SY94] for more detailed information. Let dVg be the volume form associated to the metric g, and ∇gf is the gradient with respect to the metric g. We will work on complete, connected Riemannian manifold with negative curvature. We will work on the Poincare ball model (coordinate map) of the hyperbolic space Hn (n ≥ 2), that is, when M has constant curvature equal to −1. This is the unit ball B in Rn centered at the origin and equipped with the Riemannian metric ds2 = 4 n i=1. The Riemannian measure, the gradient and the hyperbolic distance in the Poincare ball model are, respectively, 2n dVg

Hardy type inequalities on manifolds

Hardy type inequalities on hyperbolic spaces

Caffarelli-Kohn-Nirenberg inequalities on hyperbolic spaces