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.

中文翻译：

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y2×ℝn-2的Hardy-Adams不等式

令ℍ2{\ mathbb {H} ^ {2}}为维度2的双曲空间。用Mn =ℍ2×ℝn-2{M ^ {n} = \ mathbb {H} ^ {2} \ times \ mathbb表示{R} ^ {n-2}}ℍ2{\ mathbb {H} ^ {2}}和ℝn-2（n≥3）{\ mathbb {R} ^ {n-2}（n \ geq 3）}。在本文中，尽管Mn {M ^ {n}}并非严格地具有负的截面曲率，但我们在Mn {M ^ {n}}上建立了一些尖锐的Hardy-Adams不等式。我们还表明，Mn {M ^ {n}}上Poincaré-Sobolev不等式中的尖锐常数与具有独立利益的最佳Sobolev常数一致。