On higher order Poincaré inequalities with radial derivatives and Hardy improvements on the hyperbolic space. (English) Zbl 1477.31038
Summary: In this paper we prove higher order Poincaré inequalities involving radial derivatives namely,
\[ \int_{\mathbb{H}^N} |\nabla_{r,\mathbb{H}^N}^k u|^2 \, \text{d}v_{\mathbb{H}^N} \ge \bigg (\frac{N-1}{2}\bigg )^{2(k-l)} \int_{\mathbb{H}^N} |\nabla_{r,\mathbb{H}^N}^l u|^2 \, \text{d}v_{\mathbb{H}^N} \text{ for all } u\in H^k(\mathbb{H}^N), \]
where underlying space is \(N\)-dimensional hyperbolic space \(\mathbb{H}^N\), \(0\le l<k\) are integers and the constant \(\big (\frac{N-1}{2}\big )^{2(k-l)}\) is sharp. Furthermore we improve the above inequalities by adding Hardy-type remainder terms and the sharpness of some constants is also discussed.
MSC:
| 31C12 | Potential theory on Riemannian manifolds and other spaces |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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