Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
Abstract
We establish Bourgain--Brezis selection maps for Hodge systems in new ranges of Besov and Triebel--Lizorkin spaces. Our approach combines the trace-free Beurling--Ahlfors transform with a nonlinear duality method. To control concentrations, we use tools developed by Stolyarov in the study of Maz'ya's Φ-inequalities. These methods yield selection maps in Besov spaces Ḃn/pp,q(ℝn) for every 1≤ q≤ p≤ 2, and in Triebel--Lizorkin spaces Ḟn/pp,q(ℝn) for every 1<p≤ q≤ 2. The theorems hold for every dimension n≥ 2 and every form degree 1≤ l≤ n−1. Although our results overlap with some earlier work, they considerably extend the known range of solutions to the Bourgain--Brezis problem and underscore the importance of the trace-free Beurling--Ahlfors transform in its resolution.
Related
- Preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with operators
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, II: Pointwise Multiplication, Localization, Differences
- Triebel - Lizorkin spaces of dominating mixed smoothness are algebras with respect to pointwise multiplication
- On Besov and Triebel-Lizorkin spaces associated with the Grushin operator
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, III: the Spaces F∞,qα
BibTeX
@misc{arsenio2026bourgainbrezis,
title = {{Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces}},
author = {Diogo Arsénio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33876},
eprint = {2609.33876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33876},
}
RIS
TY - UNPB
TI - Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
AU - Diogo Arsénio
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33876
UR - https://arxiv.org/abs/2609.33876
AB - We establish Bourgain--Brezis selection maps for Hodge systems in new ranges of Besov and Triebel--Lizorkin spaces. Our approach combines the trace-free Beurling--Ahlfors transform with a nonlinear duality method. To control concentrations, we use tools developed by Stolyarov in the study of Maz'ya's Φ-inequalities. These methods yield selection maps in Besov spaces Ḃ^n/p_p,q(ℝ^n) for every 1≤ q≤ p≤ 2, and in Triebel--Lizorkin spaces Ḟ^n/p_p,q(ℝ^n) for every 1<p≤ q≤ 2. The theorems hold for every dimension n≥ 2 and every form degree 1≤ l≤ n−1. Although our results overlap with some earlier work, they considerably extend the known range of solutions to the Bourgain--Brezis problem and underscore the importance of the trace-free Beurling--Ahlfors transform in its resolution.
ER -