Schatten--Lorentz properties of commutators with matrix weights
Abstract
We study the membership of commutators [b,T] of Calderón--Zygmund operators in the Schatten--Lorentz classes in the matrix-weighted setting. Working on a space of homogeneous type (X,ρ,μ) with a matrix A2 weight W, we characterise, for p∈(1,∞) and q∈[1,∞), the Schatten--Lorentz norm |[b,T]|Sp,q(L2(W)) in terms of an oscillation (Besov-type) norm |b|Oscp,q of the symbol: for all p∈(1,∞), finiteness of |b|Oscp,q is necessary; and for p>p(η), where p(η) depends on the regularity of the kernel of T and a dimensional parameter of the space, the two-sided estimate |[b,T]|Sp,q(L2(W))≈|b|Oscp,q holds. Extending a recent unified framework for unweighted and scalar-weighted versions of such results by one of us to the matrix-weighted setting, our approach combines the dyadic representation of singular integral operators with the nearly weakly orthogonal (NWO) sequences of Rochberg and Semmes.
Related
BibTeX
@misc{hytonen2026schattenlorentz,
title = {{Schatten--Lorentz properties of commutators with matrix weights}},
author = {Tuomas Hytönen and Xingyan Quan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16476},
eprint = {2610.16476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16476},
}
RIS
TY - UNPB
TI - Schatten--Lorentz properties of commutators with matrix weights
AU - Tuomas Hytönen
AU - Xingyan Quan
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16476
UR - https://arxiv.org/abs/2610.16476
AB - We study the membership of commutators [b,T] of Calderón--Zygmund operators in the Schatten--Lorentz classes in the matrix-weighted setting. Working on a space of homogeneous type (X,ρ,μ) with a matrix A_2 weight W, we characterise, for p∈(1,∞) and q∈[1,∞), the Schatten--Lorentz norm |[b,T]|_S^p,q(L^2(W)) in terms of an oscillation (Besov-type) norm |b|_Osc^p,q of the symbol: for all p∈(1,∞), finiteness of |b|_Osc^p,q is necessary; and for p>p(η), where p(η) depends on the regularity of the kernel of T and a dimensional parameter of the space, the two-sided estimate |[b,T]|_S^p,q(L^2(W))≈|b|_Osc^p,q holds. Extending a recent unified framework for unweighted and scalar-weighted versions of such results by one of us to the matrix-weighted setting, our approach combines the dyadic representation of singular integral operators with the nearly weakly orthogonal (NWO) sequences of Rochberg and Semmes.
ER -