Trudinger–Moser inequalities on hyperbolic spaces under Lorentz norms

Abstract

The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on hyperbolic spaces B n for functions whose hyperbolic gradient is in the Lorentz space L ( n , q ) , n ≥ 2 . Namely, if Ω ⊂ B n is bounded and 1 < q < ∞ , we will show that there exists a constant C = C ( n , q ) such that for all u ∈ C 0 ∞ ( Ω ) with ‖ ∇ H u ‖ L n , q ( Ω ) ≤ 1 , we have 1 | Ω | ∫ Ω e β q | u ( x ) | q ′ d V ≤ C , where β q = ( n ω n 1 / n ) q ′ and ∇ H is the hyperbolic gradient; if Ω = B n , we use only the norm ‖ ∇ H u ‖ L n , q ( B n ) rather than ‖ ∇ H u ‖ L n , q ( B n ) + ‖ u ‖ L n , q ( B n ) (unlike the case of Euclidean spaces) and will show that there exists a constant C = C ( n , q ) such that for all u ∈ C 0 ∞ ( B n ) with ‖ ∇ H u ‖ L n , q ( B n ) ≤ 1 , we have ∫ B n Φ q ( β q | u ( x ) | q ′ ) d V ≤ C , where Φ q ( t ) = e t − ∑ j = 0 j q − 2 t j j ! , j q = min ⁡ { j ∈ N : j ≥ 1 + n / q ′ } .