Fractional type operators on the Heisenberg group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909167650865152 |
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| author | Rocha, Pablo |
| author_facet | Rocha, Pablo |
| contents | Let $ρ(\cdot)$ be the Koranyi norm on the Heisenberg group $\mathbb{H}^{n} \equiv (\mathbb{R}^{2n} \times \mathbb{R}, \, \cdot \, )$ defined by \[ ρ(x,t) = \left( |x|^{4} + 16 t^{2} \right)^{1/4}, \,\,\,\, (x,t) \in \mathbb{H}^{n}. \] For $0 \leq α< Q:=2n+2$, $m \in \mathbb{N} \cap \left(1 - \fracα{Q}, \infty \right)$, and $m$ positive constants $α_1, ..., α_m$ such that $α_1 + \cdot \cdot \cdot + α_m = Q - α$, we consider the following generalization of the Riesz potential on $\mathbb{H}^{n}$ \[ T_{α, \, m}f(x,t) = \int_{\mathbb{H}^{n}} f(y,s) \prod_{j=1}^{m} ρ\left((A_j y, r_j^{-2} s)^{-1} \cdot ( x, t)\right)^{-α_j} \, dy \, ds, \] where, in the case $0 < α< Q$, the $A_j$'s are matrices belonging to $Sp (2n, \mathbb{R}) \cap SO(2n)$ and $r_j = 1$ for every $j=1, ..., m$; for $α= 0$, we consider $A_j = r_j^{-1} \, I_{2n \times 2n}$ for every $j=1, ..., m$, where the $r_j$'s are positive constants such that $r_{i}^{2} - r_{j}^{2} \neq 0$ if $i \neq j$. In this note we study the behavior of these operators on variable Hardy spaces in $\mathbb{H}^{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional type operators on the Heisenberg group Rocha, Pablo Classical Analysis and ODEs Let $ρ(\cdot)$ be the Koranyi norm on the Heisenberg group $\mathbb{H}^{n} \equiv (\mathbb{R}^{2n} \times \mathbb{R}, \, \cdot \, )$ defined by \[ ρ(x,t) = \left( |x|^{4} + 16 t^{2} \right)^{1/4}, \,\,\,\, (x,t) \in \mathbb{H}^{n}. \] For $0 \leq α< Q:=2n+2$, $m \in \mathbb{N} \cap \left(1 - \fracα{Q}, \infty \right)$, and $m$ positive constants $α_1, ..., α_m$ such that $α_1 + \cdot \cdot \cdot + α_m = Q - α$, we consider the following generalization of the Riesz potential on $\mathbb{H}^{n}$ \[ T_{α, \, m}f(x,t) = \int_{\mathbb{H}^{n}} f(y,s) \prod_{j=1}^{m} ρ\left((A_j y, r_j^{-2} s)^{-1} \cdot ( x, t)\right)^{-α_j} \, dy \, ds, \] where, in the case $0 < α< Q$, the $A_j$'s are matrices belonging to $Sp (2n, \mathbb{R}) \cap SO(2n)$ and $r_j = 1$ for every $j=1, ..., m$; for $α= 0$, we consider $A_j = r_j^{-1} \, I_{2n \times 2n}$ for every $j=1, ..., m$, where the $r_j$'s are positive constants such that $r_{i}^{2} - r_{j}^{2} \neq 0$ if $i \neq j$. In this note we study the behavior of these operators on variable Hardy spaces in $\mathbb{H}^{n}$. |
| title | Fractional type operators on the Heisenberg group |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2404.05195 |