Coefficient duality and interpolation of Euclidean flag Besov spaces
Abstract
The main purpose of this paper is to develop the duality theory of the homogeneous Besov spaces associated with the Euclidean flag, using the discrete Littlewood--Paley--Stein analysis. We first establish an exact discrete sampling formula together with bounded analysis and synthesis maps, by which the space is identified with a sequence space of coefficients. The continuous dual is then computed through a coefficient norm. In this computation the geometry of the sampling rectangles plays an essential role: when the spatial exponent is below one, the volume of a dyadic rectangle forces an explicit shift of both smoothness indices. We devote particular attention to the supremum sequence exponent. At that endpoint the closure of the flag test functions is characterized by vanishing scale tails, and the limiting tail is comparable to the distance to this closure; every continuous functional on the full space decomposes into a canonical regular part and a part that annihilates the closure, and the latter may be nonzero. For finite exponents the dual is again a flag Besov space. As applications, finite Calderón sums give a quantitative approximation theorem, and the common analysis and synthesis framework yields real and complex interpolation as well as the lifting property. The arguments are based on the exact sampling formula, scalar maximal estimates and the transpose of one bounded coefficient projection.
Related
- Besov and Triebel-Lizorkin Spaces on Filtered Lie Groups with Polynomial Growth, II: Interpolation, Localization, Pointwise Multiplication
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups with Polynomial Growth, I: Inclusions, Discretization, Duality
- Limits of Besov Norms on Quantum Euclidean Spaces
- Matrix-weighted Z-spaces and tent spaces:characterizations, duality and interpolation
- Carleson duality and endpoint structure of Euclidean flag Triebel--Lizorkin spaces
BibTeX
@misc{li2026coefficient,
title = {{Coefficient duality and interpolation of Euclidean flag Besov spaces}},
author = {Chang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32204},
eprint = {2610.32204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32204},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coefficient duality and interpolation of Euclidean flag Besov spaces
AU - Chang Li
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32204
UR - https://arxiv.org/abs/2610.32204
AB - The main purpose of this paper is to develop the duality theory of the homogeneous Besov spaces associated with the Euclidean flag, using the discrete Littlewood--Paley--Stein analysis. We first establish an exact discrete sampling formula together with bounded analysis and synthesis maps, by which the space is identified with a sequence space of coefficients. The continuous dual is then computed through a coefficient norm. In this computation the geometry of the sampling rectangles plays an essential role: when the spatial exponent is below one, the volume of a dyadic rectangle forces an explicit shift of both smoothness indices. We devote particular attention to the supremum sequence exponent. At that endpoint the closure of the flag test functions is characterized by vanishing scale tails, and the limiting tail is comparable to the distance to this closure; every continuous functional on the full space decomposes into a canonical regular part and a part that annihilates the closure, and the latter may be nonzero. For finite exponents the dual is again a flag Besov space. As applications, finite Calderón sums give a quantitative approximation theorem, and the common analysis and synthesis framework yields real and complex interpolation as well as the lifting property. The arguments are based on the exact sampling formula, scalar maximal estimates and the transpose of one bounded coefficient projection.
ER -