Adams inequality on the hyperbolic space

In this article we establish the following Adams type inequality in the hyperbolic space HNHN:supu∈Cc∞(HN),∫HN(Pku)udvg≤1⁡∫HN(eβu2−1)dvg<∞ iff β≤β0(N,k)β≤β0(N,k) where 2k=N2k=N, PkPk is the critical GJMS operator in HNHN and β0(N,k)β0(N,k) is as defined in (1.3). As an application we prove the asymptotic behaviour of the best constants in Sobolev inequalities when 2k=N2k=N and also prove some existence results for the QkQk curvature type equation in HNHN.