Commutators with two matrix weights

Abstract

Let U,V be matrix 𝒜_p weights, 1<p<∞, and let B be a matrix-valued function. For Calderón--Zygmund operators T with scalar standard kernels, we characterize boundedness and compactness of [M_B,T⊗ I_m]:L^p(U)→ L^p(V) by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix 𝒜₂ weights U,V and a fixed Riesz transform R_j on ℝ^d, d≥2, we characterize the Schatten class membership of C_{B,R_j}:L²(U)→ L²(V). We characterize 𝒮^r membership for r>d and 𝒮^d,∞ membership by ℓ^r and weak ℓ^d conditions on the two Hilbert--Schmidt cube oscillations. For 0<r≤ d, 𝒮^r membership is equivalent to B being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for C¹ symbols, with a converse for constant or uniformly elliptic weights.

Schatten--Lorentz properties of commutators with matrix weights
Tuomas Hytönen, Xingyan Quan
arXiv12 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
The reverse Hölder inequality for 𝒜_p(·) weights with applications to matrix weights
David V. Cruz-Uribe, Michael Penrod
Publicacions Matemàtiques17 Jun 2026Peer-reviewedLicence not stated10 citedGeneral mathematicsOpenAlex
Equality cases for matrix spaces with bounded-rank commutators
Zhi-Lin Zhang
arXiv1 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
From matrix inversion to constraints: provably tighter confidence regions for importance weights in label shift
Mushan Li, Kihyun Han, Yanyuan Ma
arXiv13 Sep 2026PreprintarXiv licenceFull textStatisticsarXiv
Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents
Tuomas Hytönen, Dachun Yang, Wen Yuan +1
arXiv17 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{li2026commutators,
  title = {{Commutators with two matrix weights}},
  author = {Ji Li and Chong-Wei Liang and Chun-Yen Shen and Brett D. Wick},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17054},
  eprint = {2609.17054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17054},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Commutators with two matrix weights
AU  - Ji Li
AU  - Chong-Wei Liang
AU  - Chun-Yen Shen
AU  - Brett D. Wick
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17054
UR  - https://arxiv.org/abs/2609.17054
AB  - Let U,V be matrix 𝒜_p weights, 1<p<∞, and let B be a matrix-valued function. For Calderón--Zygmund operators T with scalar standard kernels, we characterize boundedness and compactness of [M_B,T⊗ I_m]:L^p(U)→ L^p(V) by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix 𝒜_2 weights U,V and a fixed Riesz transform R_j on ℝ^d, d≥2, we characterize the Schatten class membership of C_B,R_j:L^2(U)→ L^2(V). We characterize 𝒮^r membership for r>d and 𝒮^d,∞ membership by ℓ^r and weak ℓ^d conditions on the two Hilbert--Schmidt cube oscillations. For 0<r≤ d, 𝒮^r membership is equivalent to B being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for C^1 symbols, with a converse for constant or uniformly elliptic weights.
ER  -

Abstract

Related

Cite