Restricted weak type endpoint estimate for the spherical maximal operators on the Heisenberg group
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910850555576320 |
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| author | Jeon, Hyunwoo Kim, Joonil |
| author_facet | Jeon, Hyunwoo Kim, Joonil |
| contents | Let $\mathbb{H}^n$ denote the Heisenberg group, identified with $\mathbb{R}^d \times \mathbb{R}$, where $d = 2n$ and $n \in \mathbb{N}$. We consider the spherical maximal operator $\mathcal{M}$ associated with the sphere $S^{d-1}$ embedded in the horizontal subspace $\mathbb{R}^d \times \{0\}$ of $\mathbb{H}^n$. It is known that $\mathcal{M}$ is bounded on $L^p(\mathbb{H}^n)$ if and only if $p \in (\tfrac{d}{d-1}, \infty]$. In this paper, we establish a restricted weak type $(p,p)$ estimate at the endpoint $p = \tfrac{d}{d-1}$ for $\mathcal{M}$, provided $d \ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20932 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restricted weak type endpoint estimate for the spherical maximal operators on the Heisenberg group Jeon, Hyunwoo Kim, Joonil Classical Analysis and ODEs 42B15, 42B30 Let $\mathbb{H}^n$ denote the Heisenberg group, identified with $\mathbb{R}^d \times \mathbb{R}$, where $d = 2n$ and $n \in \mathbb{N}$. We consider the spherical maximal operator $\mathcal{M}$ associated with the sphere $S^{d-1}$ embedded in the horizontal subspace $\mathbb{R}^d \times \{0\}$ of $\mathbb{H}^n$. It is known that $\mathcal{M}$ is bounded on $L^p(\mathbb{H}^n)$ if and only if $p \in (\tfrac{d}{d-1}, \infty]$. In this paper, we establish a restricted weak type $(p,p)$ estimate at the endpoint $p = \tfrac{d}{d-1}$ for $\mathcal{M}$, provided $d \ge 3$. |
| title | Restricted weak type endpoint estimate for the spherical maximal operators on the Heisenberg group |
| topic | Classical Analysis and ODEs 42B15, 42B30 |
| url | https://arxiv.org/abs/2502.20932 |