Sharp convex body sparse bounds and quantitative matrix weighted estimates for Hilbert transforms along curves
Abstract
We prove convex body sparse bounds for coordinatewise Hilbert transforms along monomial curves and local Cd curves with uniformly nonvanishing torsion on the closed integration interval. Adapting the scalar estimates of Cladek--Ou \cite{CladekOu2018} and the lifting method of Laukkarinen--Lorist \cite{LaukkarinenLorist2026} to the anisotropic geometry, we obtain one sparse family of genuine curve-adapted boxes and quantitative Lp(W) bounds for matrix weights. The convex body sparse exponent range is sharp up to its boundary: estimates hold in the interior of Ωd, while any fixed finite number of sparse families fails outside Ωd̅ when 1/r+1/s>1. We also establish an explicit endpoint-contact obstruction. For every interior planar exponent pair, a polynomial curve with positive torsion on the open integration interval violates the full-interval sparse estimate.
Related
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BibTeX
@misc{cen2026sharp,
title = {{Sharp convex body sparse bounds and quantitative matrix weighted estimates for Hilbert transforms along curves}},
author = {Xi Cen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40820},
eprint = {2610.40820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40820},
}
RIS
TY - UNPB
TI - Sharp convex body sparse bounds and quantitative matrix weighted estimates for Hilbert transforms along curves
AU - Xi Cen
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40820
UR - https://arxiv.org/abs/2610.40820
AB - We prove convex body sparse bounds for coordinatewise Hilbert transforms along monomial curves and local C^d curves with uniformly nonvanishing torsion on the closed integration interval. Adapting the scalar estimates of Cladek--Ou \cite{CladekOu2018} and the lifting method of Laukkarinen--Lorist \cite{LaukkarinenLorist2026} to the anisotropic geometry, we obtain one sparse family of genuine curve-adapted boxes and quantitative L^p(W) bounds for matrix weights. The convex body sparse exponent range is sharp up to its boundary: estimates hold in the interior of Ω_d, while any fixed finite number of sparse families fails outside Ω_d̅ when 1/r+1/s>1. We also establish an explicit endpoint-contact obstruction. For every interior planar exponent pair, a polynomial curve with positive torsion on the open integration interval violates the full-interval sparse estimate.
ER -