Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators
Abstract
We study compactness of the product of two weighted Hardy primitives into Lq(u), 0<q<∞, on the input boundary 1≤ p1,p2≤∞, where at least one input exponent is $1$ or ∞. For L1× Lp, 1<p<∞, boundedness implies compactness when q<p/(p+1). For L1× L1, the corresponding range is q<1/2. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when q<1. For a finite left endpoint, ordinary L∞ inputs admit an exact scalar transfer completed by linear nodal approximation.
Related
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BibTeX
@misc{kanjilal2026inputendpoint,
title = {{Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators}},
author = {Saikat Kanjilal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02649},
eprint = {2610.02649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02649},
}
RIS
TY - UNPB
TI - Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators
AU - Saikat Kanjilal
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02649
UR - https://arxiv.org/abs/2610.02649
AB - We study compactness of the product of two weighted Hardy primitives into L^q(u), 0<q<∞, on the input boundary 1≤ p_1,p_2≤∞, where at least one input exponent is $1$ or ∞. For L^1× L^p, 1<p<∞, boundedness implies compactness when q<p/(p+1). For L^1× L^1, the corresponding range is q<1/2. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when q<1. For a finite left endpoint, ordinary L^∞ inputs admit an exact scalar transfer completed by linear nodal approximation.
ER -