Sharp boundedness criteria for nonnegative kernels on regular curves in the Heisenberg group
Abstract
In this paper we study singular integrals associated with the nonnegative kernels Kα(x,y,z)=|z|α/2/(|(x,y,z)|α+1), α>0, in the first Heisenberg group ℍ, where |·| denotes the Korányi norm. This family was introduced by Chousionis and Li. We first show that there exists a compact $1$-Ahlfors regular horizontal C1 curve E⊂ℍ such that the singular integrals associated with Kα are unbounded on L2(E) for every 0<α<4. This extends the counterexamples of Chousionis, Li and Zhang from 0<α<2 to the full range 0<α<4. Combined with the boundedness theorem of Fässler and Orponen for α≥4, our result determines the exact range of exponents for boundedness on every $1$-Ahlfors regular curve. We also consider the more general kernels KΦ(p)=1/|p| Φ (|z|1/2/|p|), p=(x,y,z)≠0, where Φ:[0,1]→[0,∞) is continuous and nondecreasing, with Φ(0)=0. We show that the associated singular integral is L2-bounded on every $1$-Ahlfors regular curve if and only if Φ(u)=O(u4) as u→0. When this condition fails, we construct a compact $1$-Ahlfors regular horizontal C1 curve on which the operator is unbounded. The curves in both constructions can be chosen as horizontal lifts of planar C1 graphs with arbitrarily small slope.
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BibTeX
@misc{zhang2026sharp,
title = {{Sharp boundedness criteria for nonnegative kernels on regular curves in the Heisenberg group}},
author = {Lingxiao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27730},
eprint = {2610.27730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27730},
}
RIS
TY - UNPB
TI - Sharp boundedness criteria for nonnegative kernels on regular curves in the Heisenberg group
AU - Lingxiao Zhang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27730
UR - https://arxiv.org/abs/2610.27730
AB - In this paper we study singular integrals associated with the nonnegative kernels K_α(x,y,z)=|z|^α/2/(|(x,y,z)|^α+1), α>0, in the first Heisenberg group ℍ, where |·| denotes the Korányi norm. This family was introduced by Chousionis and Li. We first show that there exists a compact $1$-Ahlfors regular horizontal C^1 curve E⊂ℍ such that the singular integrals associated with K_α are unbounded on L^2(E) for every 0<α<4. This extends the counterexamples of Chousionis, Li and Zhang from 0<α<2 to the full range 0<α<4. Combined with the boundedness theorem of Fässler and Orponen for α≥4, our result determines the exact range of exponents for boundedness on every $1$-Ahlfors regular curve. We also consider the more general kernels K_Φ(p)=1/|p| Φ (|z|^1/2/|p|), p=(x,y,z)≠0, where Φ:[0,1]→[0,∞) is continuous and nondecreasing, with Φ(0)=0. We show that the associated singular integral is L^2-bounded on every $1$-Ahlfors regular curve if and only if Φ(u)=O(u^4) as u→0. When this condition fails, we construct a compact $1$-Ahlfors regular horizontal C^1 curve on which the operator is unbounded. The curves in both constructions can be chosen as horizontal lifts of planar C^1 graphs with arbitrarily small slope.
ER -