Endpoint-Safe Direct Characterisations of Weighted Bilinear Hardy Inequalities on Arbitrary Intervals
Abstract
We study the weighted bilinear Hardy inequality for the same-direction product HI f HI g on an arbitrary real interval I, with 0<q<∞, 1≤ p1,p2≤∞, and measurable weights allowed to take the values $0$ and +∞. The characterisation is formulated through directed compact restrictions JSubset I. On each compact interval, endpoint-safe input profiles retain the pi=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 CIasympsupJSubset I AJop. In the regular interior range, the resulting conditions recover the classical A1,…,A7 characterisations.
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BibTeX
@misc{kanjilal2026endpointsafe,
title = {{Endpoint-Safe Direct Characterisations of Weighted Bilinear Hardy Inequalities on Arbitrary Intervals}},
author = {Saikat Kanjilal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34110},
eprint = {2609.34110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34110},
}
RIS
TY - UNPB
TI - Endpoint-Safe Direct Characterisations of Weighted Bilinear Hardy Inequalities on Arbitrary Intervals
AU - Saikat Kanjilal
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34110
UR - https://arxiv.org/abs/2609.34110
AB - 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<∞, 1≤ p_1,p_2≤∞, and measurable weights allowed to take the values $0$ and +∞. The characterisation is formulated through directed compact restrictions JSubset 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_Iasympsup_JSubset I A_J^op. In the regular interior range, the resulting conditions recover the classical A_1,…,A_7 characterisations.
ER -