论文标题
在更高阶段的庞加莱不平等现象，具有径向衍生物和双曲线空间的壮大改善
On Higher order Poincaré Inequalities with radial derivatives and Hardy improvements on the hyperbolic space
论文作者
论文摘要
在本文中，我们证明了涉及径向衍生物的高级庞加莱不平等现象，即 \ begin {equation*} \ int _ {\ Mathbb {h}^{n}} | \ nabla_ {r，\ Mathbb { \ bigG（\ frac {n-1} {2} \ bigG）^{2（k-l）} \ int _ {\ Mathbb {\ MathBB { d} v _ {\ mathbb {h}^{n}}} \ \ \ \ \ \ \ text {for All} u \ in H^k（\ Mathbb {h}^n}）， \ end {equation*}，其中基础空间为$ n $ - 二维空间$ \ mathbb {h}^{n} $，$ 0 \ leq l <k $是整数和常数$ \ big（\ frac {n-1} {n-1} {2} {2} {2} {2} \ big）^（k-lig）^{2（k-l-ls）} $。此外，我们还通过添加硬性型剩余术语和某些常数的清晰度来改善上述不平等现象。
In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, \begin{equation*} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{k} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \geq \bigg(\frac{N-1}{2}\bigg)^{2(k-l)} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{l} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \ \ \text{ for all } u\in H^k(\mathbb{H}^{N}), \end{equation*} where underlying space is $N$-dimensional hyperbolic space $\mathbb{H}^{N}$, $0\leq l<k$ 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.
