The first order Hardy–Trudinger–Moser type inequalities were only known in two dimensional hyperbolic space B2 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 BN for N≥2. Namely, for any N≥2 there exists a constant C=C(N)>0 such that for all u∈W01,N(BN) with ∫BN|∇Hu|NdV−N−1NN∫BN|u|NdV≤1, there holds ∫BNΦN(βN|u|NN−1)dV≤Cand∫BNeβN|u|NN−1dx≤C,where dV=21−|x|2Ndx is the hyperbolic volume, ∇Hu is the hyperbolic gradient, βN=NNπN2Γ(N2+1)1N−1 and ΦN(t)=et−∑k=0N−1tkk!.In the two dimensional case N=2, we further establish that if and only if λ<14 there exists a constant C′=C′(λ)>0 such that for all u∈W01,2(B2) with ∫B2|∇Hu|2dV−14∫B2|u|2dV−λ∫B2(1−|x|2)|u|2dV≤1, there holds ∫B2e4πu2−1−4πu2(1−|x|2)2dx≤C′and∫B2e4πu2dx≤C′.In fact, 14 is the sharp constant on the right hand side of the following inequality ∫B2|∇Hu|2dV−14∫B2|u|2dV≥14∫B2(1−|x|2)|u|2dV.These improve the known results in dimension two in the literature.
Read full abstract