Hardy–Adams Inequalities on ℍ2 × ℝn-2

Let ℍ2{\mathbb{H}^{2}} be the hyperbolic space of dimension 2. Denote by Mn=ℍ2×ℝn-2{M^{n}=\mathbb{H}^{2}\times\mathbb{R}^{n-2}} the product manifold of ℍ2{\mathbb{H}^{2}} and ℝn-2(n≥3){\mathbb{R}^{n-2}(n\geq 3)}. In this paper we establish some sharp Hardy–Adams inequalities on Mn{M^{n}}, though Mn{M^{n}} is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré–Sobolev inequality on Mn{M^{n}} coincides with the best Sobolev constant, which is of independent interest.