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

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.

Hardy inequalities on metric measure spaces

“…For the results in the case of X = R we can refer to [KP03,PS01], and also to [PSW07] for inequalities for q < p. For X = R n , the result has been proved in [Ver08], with related inequalities obtained in one dimension in [GKPW04,OK90]. For related works on hyperbolic spaces we can refer to [LY17,RY18a], and to [Ngu17,RY18a] for inequalities on Cartan-Hadamard manifolds, with the background analysis available in [GHL04,Hel01]. For the analysis of Hardy inequalities on homogeneous groups we can refer to [RS17,RSY18].…”

Rellich, Gagliardo-Nirenberg, Trudinger and Caffarelli-Kohn-Nirenberg inequalities for Dunkl operators and applications

“…For other extended Caffarelli-Kohn-Nirenberg inequalities, we refer to [20] and [21] for f P C 8 0 pGzt0uq on general homogeneous group, to [19] on stratified groups, to [26] for general hypoelliptic differential operators, to [24] on general Lie group, to [25] on Riemannian manifolds with negative curvature and references therein.…”

“…In the special case when γ " 0, hence ∆ k " ∆, the usual gradient in R N , the inequality (1.19) was investigated in [15] for p " 2, in [16] for p " N " 2, in [1] for p " N ě 2 and in [17] for p and N as (1.19). We also refer to [23] and [26] for hypoelliptic versions of Trudinger inequalities, and to [25] as well as to references therein on Riemannian manifolds with negative curvature. with nonlinearity satisfying the assumptions f p0q " 0 and |f paq ´f pbq| ď Cp|a| p´1 `|b| p´1 q|a ´b|, for some p ą 1.…”