Trudinger–Moser inequalities on harmonic [formula omitted] groups under Lorentz norms

Abstract

The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on harmonic AN groups S for functions whose gradient is in the Lorentz space L(n,q). Our results show that even if S is not with strictly negative sectional curvature, one can also replace the norm ∫S|∇Su|ndV+∫S|u|ndV by ∫S|∇Su|ndV in the Trudinger–Moser type inequalities.