PreprintarXiv licencearXiv
Hardy space estimates for the parabolic maximal operator below p=1
Abstract
We study the parabolic maximal operator Mpar along the moment curve (t,t2). In 1988, Christ proved that Mpar maps the parabolic Hardy space Hpar1(ℝ2) into L1,∞(ℝ2). We show that this result is sharp at p=1: for every 0<p<1, the natural extension from Hparp(ℝ2) to Lp,∞(ℝ2) fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space Hparp,*(ℝ2) and a weak tendril space 𝒯p,∞(ℝ2), and prove that Mpar: Hparp,*(ℝ2) ⟶ 𝒯p,∞(ℝ2), 0<p<1, is bounded. At p=1, these spaces recover those in Christ's theorem: Hpar1,*(ℝ2)=Hpar1(ℝ2) and 𝒯1,∞(ℝ2)=L1,∞(ℝ2). Thus, our result provides a natural extension of Christ's work to the range 0<p<1.
Related
- Maximal velocity estimates for Hardy operators
- The norm of the centered Hardy--Littlewood maximal operator
- Muckenhoupt weights for the Hardy-Littlewood maximal operator on hyperbolic spaces
- Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations
- Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators
BibTeX
@misc{han2026hardy,
title = {{Hardy space estimates for the parabolic maximal operator below $p=1$}},
author = {Yongsheng Han and Bingyang Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.15427},
eprint = {2607.15427},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.15427},
}
RIS
TY - UNPB
TI - Hardy space estimates for the parabolic maximal operator below p=1
AU - Yongsheng Han
AU - Bingyang Hu
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2607.15427
UR - https://arxiv.org/abs/2607.15427
AB - We study the parabolic maximal operator M_par along the moment curve (t,t^2). In 1988, Christ proved that M_par maps the parabolic Hardy space H_par^1(ℝ^2) into L^1,∞(ℝ^2). We show that this result is sharp at p=1: for every 0<p<1, the natural extension from H_par^p(ℝ^2) to L^p,∞(ℝ^2) fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space H_par^p,*(ℝ^2) and a weak tendril space 𝒯^p,∞(ℝ^2), and prove that M_par: H_par^p,*(ℝ^2) ⟶ 𝒯^p,∞(ℝ^2), 0<p<1, is bounded. At p=1, these spaces recover those in Christ's theorem: H_par^1,*(ℝ^2)=H_par^1(ℝ^2) and 𝒯^1,∞(ℝ^2)=L^1,∞(ℝ^2). Thus, our result provides a natural extension of Christ's work to the range 0<p<1.
ER -