Sharp Hardy–Trudinger–Moser inequalities in any N-dimensional hyperbolic spaces. (October 2020) - Liang, Xining; Lu, Guozhen; Wang, Xumin; Yang, Qiaohua
Abstract: The first order Hardy–Trudinger–Moser type inequalities were only known in two dimensional hyperbolic space B 2 in the literature (Wang and Ye, 2012; Mancini et al., 2013; Lu and Yang, 2016). In this paper, we establish the Hardy–Trudinger–Moser inequalities in any N -dimensional hyperbolic spaces B N for N ≥ 2 . Namely, for any N ≥ 2 there exists a constant C = C ( N ) > 0 such that for all u ∈ W 0 1, N ( B N ) with ∫ B N | ∇ H u | N d V − N − 1 N N ∫ B N | u | N d V ≤ 1, there holds ∫ B N Φ N ( β N | u | N N − 1 ) d V ≤ C and ∫ B N e β N | u | N N − 1 d x ≤ C, where d V = 2 1 − | x | 2 N d x is the hyperbolic volume, ∇ H u is the hyperbolic gradient, β N = N N π N 2 Γ ( N 2 + 1 ) 1 N − 1 and Φ N ( t ) = e t − ∑ k = 0 N − 1 t k k ! . In the two dimensional case N = 2, we further establish that if and only if λ < 1 4 there exists a constant C ′ = C ′ ( λ ) > 0 such that for all u ∈ W 0 1, 2 ( B 2 ) with ∫ B 2 | ∇ H u | 2 d V − 1 4 ∫ B 2 | u | 2 d V − λ ∫ B 2 ( 1 − | x | 2 ) | u | 2 d V ≤ 1, there holds ∫ B 2 e 4 π u 2 − 1 − 4 π u 2 ( 1 − | x | 2 ) 2 d x ≤ C ′ and ∫ B 2 e 4 π u 2 d x ≤ C ′ . In fact, 1 4 is the sharp constant on the right hand side of the following inequality ∫ B 2 | ∇ H u | 2 d V − 1 4 ∫ B 2 | u | 2 d V ≥ 1 4 ∫ B 2 ( 1 − | x | 2 ) | u | 2 d V . These improve the known results in dimension two in the literature.