发表机构
JIS University（JIS大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在任意实区间上刻画加权双线性 Hardy 不等式，通过有向紧限制和端点安全输入分布，给出全局常数与局部特征的等价刻画，并恢复经典 $A_1,\ldots,A_7$ 条件。
AI 中文摘要
我们研究了任意实区间 $I$ 上同向乘积 $H_I f H_I g$ 的加权双线性 Hardy 不等式，其中 $0<q<\infty$，$1\le p_1,p_2\le\infty$，且可测权重允许取 $0$ 和 $+\infty$ 值。该刻画通过有向紧限制 $J\Subset I$ 来表述。在每个紧区间上，端点安全输入分布在 $p_i=1$ 边界贡献处保留为 Stieltjes 原子，而低三角约化将闭 Hardy 部分与严格 Copson 部分分离，从而公共原子被精确计数一次。上部和混合情形通过精确冻结和紧线性 Hardy 估计获得。下部情形通过幂提升和测度值 Hardy-Copson 定理得到。在对权重进行同时正则化后，紧最优常数与一个可操作局部特征在仅依赖指数的常数意义下等价，且全局常数满足 $C_I\asymp\sup_{J\Subset I} A_J^{\mathrm{op}}$。在正则内部范围内，所得条件恢复了经典的 $A_1,\ldots,A_7$ 刻画。
英文摘要
We study the weighted bilinear Hardy inequality for the same-direction product $H_I f H_I g$ on an arbitrary real interval $I$, with $0<q<\infty$, $1\le p_1,p_2\le\infty$, and measurable weights allowed to take the values $0$ and $+\infty$. The characterisation is formulated through directed compact restrictions $J\Subset I$. On each compact interval, endpoint-safe input profiles retain the $p_i=1$ boundary contribution as a Stieltjes atom, while the lower-triangle reduction separates a closed Hardy part from a strict Copson part so that common atoms are counted exactly once. Upper and mixed regimes follow from exact freezing and compact linear Hardy estimates. The lower regimes are obtained by power lifting and a measure-valued Hardy-Copson theorem. After simultaneous regularisation of the weights, the compact optimal constant is equivalent, with exponent-only constants, to an operational local characteristic, and the global constant satisfies $C_I\asymp\sup_{J\Subset I} A_J^{\mathrm{op}}$. In the regular interior range, the resulting conditions recover the classical $A_1,\ldots,A_7$ characterisations.
Comments78 pages, no figures, 5 appendices
