Sharp Matrix-Weighted Lower Bounds for Non-Degenerate Convolution Calderón--Zygmund Operators

Abstract

Let T be a convolution Calderón--Zygmund operator satisfying Hytönen's non-degeneracy condition. We prove that, for every p∈(1,∞) and m≥3, there exists a positive constant C such that, for any t∈[1,∞), sup_{[W]_{𝒜_p}≤ t} |T|_{L^p(W)→ L^p(W)} ≥ C t^{1+1/(p(p−1))}, where the supremum is taken over all matrix weights W: ℝⁿ→ℂ^{m× m} with their weight constants [W]_{𝒜_p}≤ t. If, in addition, the odd part of the kernel is non-degenerate (such as the Hilbert transform and the Riesz transforms), the same conclusion holds for every m≥2.

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BibTeX RIS

BibTeX

@misc{bu2026sharp,
  title = {{Sharp Matrix-Weighted Lower Bounds for Non-Degenerate Convolution Calderón--Zygmund Operators}},
  author = {Fan Bu and Dachun Yang and Wen Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.33560},
  eprint = {2609.33560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33560},
}

RIS

TY  - UNPB
TI  - Sharp Matrix-Weighted Lower Bounds for Non-Degenerate Convolution Calderón--Zygmund Operators
AU  - Fan Bu
AU  - Dachun Yang
AU  - Wen Yuan
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.33560
UR  - https://arxiv.org/abs/2609.33560
AB  - Let T be a convolution Calderón--Zygmund operator satisfying Hytönen's non-degeneracy condition. We prove that, for every p∈(1,∞) and m≥3, there exists a positive constant C such that, for any t∈[1,∞), sup_[W]_𝒜_p≤ t |T|_L^p(W)→ L^p(W) ≥ C t^1+1/(p(p−1)), where the supremum is taken over all matrix weights W: ℝ^n→ℂ^m× m with their weight constants [W]_𝒜_p≤ t. If, in addition, the odd part of the kernel is non-degenerate (such as the Hilbert transform and the Riesz transforms), the same conclusion holds for every m≥2.
ER  -

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