A sharp spectral multiplier theorem on the Engel group
Abstract
We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local Hs condition with s>2. The multiplier operators F(ℒ) are of weak type (1,1) and bounded on Lp for every 1<p<∞. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in (0,∞), we also determine the leading term and first correction for the localized kernels. Whether the weak type (1,1) estimate holds under the local H2 condition remains open.
Related
- Sharp bounds in Ionescu--Wainger multiplier theorem for canonical fractions
- Spectral complexes and currents on the Engel group
- A Hörmander spectral multiplier theorem for fractional Hardy operators
- A Sharp Spectral Mantel Theorem for Quantum Transport
- The spectral Erdős book theorem: sharp bounds and stability
BibTeX
@misc{han2026sharp,
title = {{A sharp spectral multiplier theorem on the Engel group}},
author = {Zhicheng Han and Zhipeng Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08976},
eprint = {2610.08976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08976},
}
RIS
TY - UNPB
TI - A sharp spectral multiplier theorem on the Engel group
AU - Zhicheng Han
AU - Zhipeng Yang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08976
UR - https://arxiv.org/abs/2610.08976
AB - We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local H^s condition with s>2. The multiplier operators F(ℒ) are of weak type (1,1) and bounded on L^p for every 1<p<∞. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in (0,∞), we also determine the leading term and first correction for the localized kernels. Whether the weak type (1,1) estimate holds under the local H^2 condition remains open.
ER -