A sharp multiplier theorem for solvable extensions of Heisenberg and related groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911909504090112 |
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| author | Martini, Alessio Plewa, Paweł |
| author_facet | Martini, Alessio Plewa, Paweł |
| contents | Let $G$ be the semidirect product $N \rtimes \mathbb{R}$, where $N$ is a stratified Lie group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum $Δ$ is a left-invariant sub-Laplacian on $G$. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin--Hörmander type was proved for $Δ$, showing that an operator of the form $F(Δ)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ provided $F$ satisfies a scale-invariant smoothness condition of order $s > (Q+1)/2$, where $Q$ is the homogeneous dimension of $N$. Here we show that, if $N$ is a group of Heisenberg type, or more generally a direct product of Métivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold $s>(d+1)/2$, where $d$ is the topological dimension of $N$. The proof is based on lifting to $G$ weighted Plancherel estimates on $N$ and exploits a relation between the functional calculi for $Δ$ and analogous operators on semidirect extensions of Bessel--Kingman hypergroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03467 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A sharp multiplier theorem for solvable extensions of Heisenberg and related groups Martini, Alessio Plewa, Paweł Classical Analysis and ODEs Functional Analysis 22E30, 42B15, 42B20, 43A22 Let $G$ be the semidirect product $N \rtimes \mathbb{R}$, where $N$ is a stratified Lie group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum $Δ$ is a left-invariant sub-Laplacian on $G$. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin--Hörmander type was proved for $Δ$, showing that an operator of the form $F(Δ)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ provided $F$ satisfies a scale-invariant smoothness condition of order $s > (Q+1)/2$, where $Q$ is the homogeneous dimension of $N$. Here we show that, if $N$ is a group of Heisenberg type, or more generally a direct product of Métivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold $s>(d+1)/2$, where $d$ is the topological dimension of $N$. The proof is based on lifting to $G$ weighted Plancherel estimates on $N$ and exploits a relation between the functional calculi for $Δ$ and analogous operators on semidirect extensions of Bessel--Kingman hypergroups. |
| title | A sharp multiplier theorem for solvable extensions of Heisenberg and related groups |
| topic | Classical Analysis and ODEs Functional Analysis 22E30, 42B15, 42B20, 43A22 |
| url | https://arxiv.org/abs/2305.03467 |