A new proof of maximal theorem on Heisenberg groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913139879051264 |
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| author | Sun, Chuhan Wang, Zipeng |
| author_facet | Sun, Chuhan Wang, Zipeng |
| contents | Given $0\leqα<1$, we define \[\begin{array}{lr} \mathbf{M}_αf(u,v,t) = \sup_{ \mathbf{R} \ni (0,0,0)} {\rm vol} \{\mathbf{R}\}^{α-1} \iiint_\mathbf{R}\left|f [(u,v,t)\odot(ξ,η,τ)^{-1}]\right|dξdηdτ\end{array}\] where $\mathbf{R}\subset\mathbb{R}^{2n+1}$ is a rectangle parallel to the coordinates. Moreover, $\odot$ denotes the multiplication law on a real Heisenberg group.
The $\mathbf{L}^p$-boundedness of $\mathbf{M}_0$ has been previously proved by M. Christ. We show $\mathbf{M}_α\colon\mathbf{L}^p(\mathbb{R}^{2n+1}) \to \mathbf{L}^q(\mathbb{R}^{2n+1})$ for $α={1\over p}-{1\over q},~ 1<p\leq q<\infty$ by applying a geometric covering lemma due to Córdoba and Fefferman. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14961 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new proof of maximal theorem on Heisenberg groups Sun, Chuhan Wang, Zipeng Classical Analysis and ODEs Given $0\leqα<1$, we define \[\begin{array}{lr} \mathbf{M}_αf(u,v,t) = \sup_{ \mathbf{R} \ni (0,0,0)} {\rm vol} \{\mathbf{R}\}^{α-1} \iiint_\mathbf{R}\left|f [(u,v,t)\odot(ξ,η,τ)^{-1}]\right|dξdηdτ\end{array}\] where $\mathbf{R}\subset\mathbb{R}^{2n+1}$ is a rectangle parallel to the coordinates. Moreover, $\odot$ denotes the multiplication law on a real Heisenberg group. The $\mathbf{L}^p$-boundedness of $\mathbf{M}_0$ has been previously proved by M. Christ. We show $\mathbf{M}_α\colon\mathbf{L}^p(\mathbb{R}^{2n+1}) \to \mathbf{L}^q(\mathbb{R}^{2n+1})$ for $α={1\over p}-{1\over q},~ 1<p\leq q<\infty$ by applying a geometric covering lemma due to Córdoba and Fefferman. |
| title | A new proof of maximal theorem on Heisenberg groups |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2605.14961 |