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Sparse domination and the bi-parameter Hilbert transform
Abstract
We study domination of the bi-parameter Hilbert transform by (1,1)-type sparse forms over axis-parallel rectangles, and show that such domination fails even for smooth test functions compactly supported in (0,1)2. Our proof combines an explicit construction of two finite lattices based on the arithmetic of ℤ[√2] with a randomization argument.
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BibTeX
@misc{hu2026sparse,
title = {{Sparse domination and the bi-parameter Hilbert transform}},
author = {Bingyang Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04812},
eprint = {2610.04812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04812},
}
RIS
TY - UNPB
TI - Sparse domination and the bi-parameter Hilbert transform
AU - Bingyang Hu
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04812
UR - https://arxiv.org/abs/2610.04812
AB - We study domination of the bi-parameter Hilbert transform by (1,1)-type sparse forms over axis-parallel rectangles, and show that such domination fails even for smooth test functions compactly supported in (0,1)^2. Our proof combines an explicit construction of two finite lattices based on the arithmetic of ℤ[√2] with a randomization argument.
ER -