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

Let $S_m$ be the Schur multiplier with symbol $m$. We characterize the boundedness of $S_m:{\mathcal{S}}^1\to {\mathcal{S}}^{1,\mathrm{\infty }}$ by approximation of weighted columns in a common finite-dimensional subspace. If $W(m)$ is the weak $(1,1)$ bound and $\text{fD}(m)$ is the column approximation constant, then $\frac{1}{2}W(m)\le \text{fD}(m)\le 2W(m)$. 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 $2^{r\mathrm{/}2}$ with the descendant complexity $r$.