Compactness and Essential Distances of Weighted Bilinear Hardy-Steklov Operators
Abstract
We study weighted bilinear Hardy-Steklov operators on the half-line for 1<p1,p2<∞ and 0<q<∞. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into Lq, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions I,II1,II2,III1, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region III2, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary P=Q and the summable regime P>Q. The quasi-Banach range 0<q<1 is treated throughout with the powered metric dq(f,g)=|f−g|qq.
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BibTeX
@misc{kanjilal2026compactness,
title = {{Compactness and Essential Distances of Weighted Bilinear Hardy-Steklov Operators}},
author = {Saikat Kanjilal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02937},
eprint = {2610.02937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02937},
}
RIS
TY - UNPB
TI - Compactness and Essential Distances of Weighted Bilinear Hardy-Steklov Operators
AU - Saikat Kanjilal
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02937
UR - https://arxiv.org/abs/2610.02937
AB - We study weighted bilinear Hardy-Steklov operators on the half-line for 1<p_1,p_2<∞ and 0<q<∞. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into L^q, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions I,II_1,II_2,III_1, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region III_2, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary P=Q and the summable regime P>Q. The quasi-Banach range 0<q<1 is treated throughout with the powered metric d_q(f,g)=|f−g|_q^q.
ER -