Hardy-Adams inequalities on \(\mathbb{H}^2 \times \mathbb{R}^{n-2} \). (English) Zbl 1480.46043
Summary: Let \(\mathbb{H}^2\) be the hyperbolic space of dimension 2. Denote by \(M^n=\mathbb{H}^2\times\mathbb{R}^{n-2}\) the product manifold of \(\mathbb{H}^2\) and \(\mathbb{R}^{n-2}\) \( (n\geq 3)\). In this paper we establish some sharp Hardy-Adams inequalities on \(M^n \), though \(M^n\) is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré-Sobolev inequality on \(M^n\) coincides with the best Sobolev constant, which is of independent interest.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35J20 | Variational methods for second-order elliptic equations |
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