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arXiv · 2610.04812

Sparse domination and the bi-parameter Hilbert transform

Bingyang Hu
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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 $\mathbb Z[\sqrt2]$ with a randomization argument.

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Bingyang Hu. 2026-10-03. Sparse domination and the bi-parameter Hilbert transform. https://arxiv.org/abs/2610.04812

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