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Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem
Abstract
Let Sm be the Schur multiplier with symbol m. We characterize the boundedness of Sm:𝒮1→𝒮1,∞ 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 1/2W(m)≤fD(m)≤2W(m). These estimates hold uniformly under amplification. We also prove the Hörmander--Mikhlin--Schur weak (1,1) estimate in every Euclidean dimension under the usual integer-order condition. The proof combines Cuculescu's projections, adjacent dyadic grids and tensor polynomial interpolation on a binary block decomposition.
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BibTeX
@misc{tian2026weak,
title = {{Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem}},
author = {Yongqiang Tian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20339},
eprint = {2610.20339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20339},
}
RIS
TY - UNPB
TI - Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem
AU - Yongqiang Tian
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.20339
UR - https://arxiv.org/abs/2610.20339
AB - Let S_m be the Schur multiplier with symbol m. We characterize the boundedness of S_m:𝒮^1→𝒮^1,∞ 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 1/2W(m)≤fD(m)≤2W(m). These estimates hold uniformly under amplification. We also prove the Hörmander--Mikhlin--Schur weak (1,1) estimate in every Euclidean dimension under the usual integer-order condition. The proof combines Cuculescu's projections, adjacent dyadic grids and tensor polynomial interpolation on a binary block decomposition.
ER -