低维中的尖锐离散Hardy常数

中文摘要 AI 辅助

对于$d\ge3$，设$\lambda_d$为$\mathbb Z^d$上在原点处消失的函数具有欧几里得逆平方权的最近邻Hardy不等式中的尖锐常数。近期工作确立了连续Hardy系数\\[ C_d=\frac{(d-2)^2}{4} \\]在三维和四维中是尖锐的，而对于$d\ge9$则不是尖锐的。我们证明对于剩余情形$5\le d\le8$，有$\lambda_d=C_d$。证明使用了互易边场以及相关顶点权的逐点估计。外部估计由多项式不等式得出，而原点附近的权通过显式估计和局部修正来处理。

英文摘要

For $d\ge3$, let $λ_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on $\mathbb Z^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient \[ C_d=\frac{(d-2)^2}{4} \] is sharp in dimensions three and four, while it is not sharp for $d\ge9$. We prove $λ_d=C_d$ for the remaining cases $5\le d\le8$. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.

发表机构

• University of California, Irvine（加州大学尔湾分校）

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