Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals
Abstract
Let I=(a,b) be an open interval with finite or infinite endpoints. For 1<p1,p2<∞ and 0<q<∞, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from Lp1(v1;I)× Lp2(v2;I) to Lq(u;I), including the quasi-Banach target range 0<q<1. With ϑ=min{1,q}, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the ϑ-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for q<1 explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex L1/2(0,1) example for which am+1(2)(ℋ2)=O(m−1/2).
Related
- Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators
- Compactness and Essential Distances of Weighted Bilinear Hardy-Steklov Operators
- Endpoint-Safe Direct Characterisations of Weighted Bilinear Hardy Inequalities on Arbitrary Intervals
- Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces
- Weighted decoupling inequalities via fractal bilinear Kakeya estimates
BibTeX
@misc{kanjilal2026compactness,
title = {{Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals}},
author = {Saikat Kanjilal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37998},
eprint = {2609.37998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37998},
}
RIS
TY - UNPB
TI - Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals
AU - Saikat Kanjilal
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37998
UR - https://arxiv.org/abs/2609.37998
AB - Let I=(a,b) be an open interval with finite or infinite endpoints. For 1<p_1,p_2<∞ and 0<q<∞, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from L^p_1(v_1;I)× L^p_2(v_2;I) to L^q(u;I), including the quasi-Banach target range 0<q<1. With ϑ=min{1,q}, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the ϑ-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for q<1 explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex L^1/2(0,1) example for which a_m+1^(2)(ℋ_2)=O(m^−1/2).
ER -