Carleson duality and endpoint structure of Euclidean flag Triebel--Lizorkin spaces
Abstract
The main purpose of this paper is to develop the duality theory of the homogeneous Triebel--Lizorkin spaces associated with the Euclidean flag, using the discrete Littlewood--Paley--Stein analysis. After establishing bounded analysis and synthesis maps, we describe the continuous duals by Carleson conditions. In the Hardy spatial range, the representing distributions satisfy an open-set Carleson condition when the sequence exponent is greater than one and a coefficient supremum condition when the sequence exponent is at most one; in the Banach spatial range the dual is a conjugate-exponent or a supremum space. At the supremum sequence endpoint, the closure of the flag test functions is characterized by a pointwise scale tail measured in Lp, and a coefficient pairing decomposes every functional on the full space into a regular part and a part that annihilates the closure. By means of a fixed-envelope modulation construction we show that this closure is proper for every finite spatial exponent and that the annihilator is nonzero when the spatial exponent is at least one. The same analysis and synthesis framework also yields complex interpolation, real interpolation in the spatial exponent, the lifting property and the standard mixed-norm embeddings.
Related
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups with Polynomial Growth, I: Inclusions, Discretization, Duality
- Endpoint Estimates for Stein's Purely Quadratic Carleson Operator
- Endpoint Schatten class properties of commutators on quantum Euclidean spaces
- Coefficient duality and interpolation of Euclidean flag Besov spaces
- Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality
BibTeX
@misc{li2026carleson,
title = {{Carleson duality and endpoint structure of Euclidean flag Triebel--Lizorkin spaces}},
author = {Chang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35055},
eprint = {2610.35055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35055},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Carleson duality and endpoint structure of Euclidean flag Triebel--Lizorkin spaces
AU - Chang Li
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35055
UR - https://arxiv.org/abs/2610.35055
AB - The main purpose of this paper is to develop the duality theory of the homogeneous Triebel--Lizorkin spaces associated with the Euclidean flag, using the discrete Littlewood--Paley--Stein analysis. After establishing bounded analysis and synthesis maps, we describe the continuous duals by Carleson conditions. In the Hardy spatial range, the representing distributions satisfy an open-set Carleson condition when the sequence exponent is greater than one and a coefficient supremum condition when the sequence exponent is at most one; in the Banach spatial range the dual is a conjugate-exponent or a supremum space. At the supremum sequence endpoint, the closure of the flag test functions is characterized by a pointwise scale tail measured in L^p, and a coefficient pairing decomposes every functional on the full space into a regular part and a part that annihilates the closure. By means of a fixed-envelope modulation construction we show that this closure is proper for every finite spatial exponent and that the annihilator is nonzero when the spatial exponent is at least one. The same analysis and synthesis framework also yields complex interpolation, real interpolation in the spatial exponent, the lifting property and the standard mixed-norm embeddings.
ER -