Poisson area and non-tangential maximal functions on manifolds with ends and applications
Abstract
Let M=ℝm♯(ℝn×𝕊m−n), where m>n≥3. The full-cone Poisson area and non-tangential maximal functions fail to be of weak type (p,p) for 1<p<2. Let K be the compact part joining the two ends and put ρ(x)=1+d(x,K). For fixed small c>0, we restrict the horizontal distance from the vertex to cρ(x) and retain every positive time. The resulting maximal function is of weak type (1,1) and bounded on Lp for 1<p≤∞. The corresponding area function uses the full spatial and time gradient. It satisfies a Fefferman--Stein distribution inequality. Its Lp norm is equivalent to that of f for 1<p≤2. This area function defines a global Hardy space. We prove its Δ-atomic description, its injective embedding into L1, its characterization by the restricted maximal function, and its real interpolation identity with L2. Direct estimates for spectral-band kernels then give a Mihlin multiplier theorem from this Hardy space to L1. Interpolation and duality give Lp boundedness for every 1<p<∞.
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BibTeX
@misc{chen2026poisson,
title = {{Poisson area and non-tangential maximal functions on manifolds with ends and applications}},
author = {Peng Chen and Ji Li and Lixin Yan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04177},
eprint = {2611.04177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04177},
}
RIS
TY - UNPB
TI - Poisson area and non-tangential maximal functions on manifolds with ends and applications
AU - Peng Chen
AU - Ji Li
AU - Lixin Yan
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04177
UR - https://arxiv.org/abs/2611.04177
AB - Let M=ℝ^m♯(ℝ^n×𝕊^m−n), where m>n≥3. The full-cone Poisson area and non-tangential maximal functions fail to be of weak type (p,p) for 1<p<2. Let K be the compact part joining the two ends and put ρ(x)=1+d(x,K). For fixed small c>0, we restrict the horizontal distance from the vertex to cρ(x) and retain every positive time. The resulting maximal function is of weak type (1,1) and bounded on L^p for 1<p≤∞. The corresponding area function uses the full spatial and time gradient. It satisfies a Fefferman--Stein distribution inequality. Its L^p norm is equivalent to that of f for 1<p≤2. This area function defines a global Hardy space. We prove its Δ-atomic description, its injective embedding into L^1, its characterization by the restricted maximal function, and its real interpolation identity with L^2. Direct estimates for spectral-band kernels then give a Mihlin multiplier theorem from this Hardy space to L^1. Interpolation and duality give L^p boundedness for every 1<p<∞.
ER -