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Improved higher order Poincar'e inequalities on the hyperbolic space via Hardy-type remainder terms | | Elvise Berchio
; Debdip Ganguly
; | | Date: |
2 Nov 2015 | | Abstract: | The paper deals about Hardy-type inequalities associated with the following
higher order Poincar’e inequality:
$$
left( frac{N-1}{2}
ight)^{2(k -l)} := inf_{ u in C_{c}^{infty}
setminus {0}} frac{int_{mathbb{H}^{N}} |
abla_{mathbb{H}^{N}}^{k} u|^2
dv_{mathbb{H}^{N}}}{int_{mathbb{H}^{N}} |
abla_{mathbb{H}^{N}}^{l} u|^2
dv_{mathbb{H}^{N}} },,
$$ where $0 leq l < k$ are integers and $mathbb{H}^{N}$ denotes the
hyperbolic space. More precisely, we improve the Poincar’e inequality
associated with the above ratio by showing the existence of $k$ Hardy-type
remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further
remainder terms are provided and the sharpness of some constants is also
discussed. As an application, we derive improved Rellich type inequalities on
upper half space of the Euclidean space with non-standard remainder terms. | | Source: | arXiv, 1511.0474 | | Services: | Forum | Review | PDF | Favorites | |
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