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arXiv · 2610.20339

Weak (1,1) Shur multipliers and the one-dimensional Hörmander--Mikhlin--Schur theorem

Yongqiang Tian
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Abstract

Let $S_m$ be the Schur multiplier with symbol $m$. We characterize the boundedness of $S_m:\mathcal S^1\to\mathcal S^{1,\infty}$ by approximation of weighted columns in a common finite-dimensional subspace. If $W(m)$ is the weak $(1,1)$ bound and $\fD(m)$ is the column approximation constant, then $\tfrac12W(m)\le\fD(m)\le2W(m)$. These estimates hold uniformly under amplification. Our second main result is the one-dimensional Hörmander--Mikhlin--Schur weak $(1,1)$ estimate. This settles the one-dimensional case of the weak $(1,1)$ problem stated in Parcet's ICM 2026 survey. The proof uses two adjacent dyadic grids and block decompositions indexed by finite binary trees. We apply Cuculescu's projections to the associated block-diagonal filtration. The resulting tree estimate is independent of the depth and grows as $2^{r/2}$ with the descendant complexity $r$.

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Yongqiang Tian. 2026-10-14. Weak (1,1) Shur multipliers and the one-dimensional Hörmander--Mikhlin--Schur theorem. https://arxiv.org/abs/2610.20339

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