On singular integrals with non-negative kernels in the Heisenberg group
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913137456840704 |
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| author | Chousionis, Vasileios Li, Sean Zhang, Lingxiao |
| author_facet | Chousionis, Vasileios Li, Sean Zhang, Lingxiao |
| contents | In this paper we revisit nonnegative kernels in the first Heisenberg group $\He$, and in particular we further study the family $$K_α(x,y,z)= \frac{|z|^{α/2}}{\|(x,y,z)\|_{H}^{α+1}}, \quad α>0,$$ which was introduced in \cite{CL}.
We first show that if $E \subset \He$ is a $1$-Ahlfors regular set and the SIO associated with the kernel $K_4$ is $L^2(E)$-bounded, then $E$ is contained in a $1$-Ahlfors regular curve. Combined with the converse implication which was obtained by Fässler and Orponen in \cite{FO1dim}, our result provides a characterization of uniform $1$-rectifiability in the Heisenberg group via the $L^2$-boundedness of a singular integral.
We also give a negative answer to a question of Fässler and Orponen from \cite{FO1dim} by showing that for any $α\in (0,2)$ there exists a $1$-Ahlfors regular curve $E_a$ such that the operators associated with the kernels $K_α$ are not bounded in $L^2(E_α)$.
We finally show that there exists a $1$-Ahlfors regular and purely $1$-unrectifiable set $E$ such that the singular integral associated with $|x| \|(x,y,z)\|^{-2}$ is $L^2(E)$ -bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17680 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On singular integrals with non-negative kernels in the Heisenberg group Chousionis, Vasileios Li, Sean Zhang, Lingxiao Classical Analysis and ODEs Metric Geometry In this paper we revisit nonnegative kernels in the first Heisenberg group $\He$, and in particular we further study the family $$K_α(x,y,z)= \frac{|z|^{α/2}}{\|(x,y,z)\|_{H}^{α+1}}, \quad α>0,$$ which was introduced in \cite{CL}. We first show that if $E \subset \He$ is a $1$-Ahlfors regular set and the SIO associated with the kernel $K_4$ is $L^2(E)$-bounded, then $E$ is contained in a $1$-Ahlfors regular curve. Combined with the converse implication which was obtained by Fässler and Orponen in \cite{FO1dim}, our result provides a characterization of uniform $1$-rectifiability in the Heisenberg group via the $L^2$-boundedness of a singular integral. We also give a negative answer to a question of Fässler and Orponen from \cite{FO1dim} by showing that for any $α\in (0,2)$ there exists a $1$-Ahlfors regular curve $E_a$ such that the operators associated with the kernels $K_α$ are not bounded in $L^2(E_α)$. We finally show that there exists a $1$-Ahlfors regular and purely $1$-unrectifiable set $E$ such that the singular integral associated with $|x| \|(x,y,z)\|^{-2}$ is $L^2(E)$ -bounded. |
| title | On singular integrals with non-negative kernels in the Heisenberg group |
| topic | Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2605.17680 |