Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces
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| Format: | Preprint |
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2024
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| _version_ | 1866918124641583104 |
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| author | Dewan, Utsav Ray, Swagato K. |
| author_facet | Dewan, Utsav Ray, Swagato K. |
| contents | For $f \in \mathscr{S}^2(\mathcal S)_{o}$, the collection of radial $L^2$-Schwartz class functions on Damek--Ricci spaces $\mathcal S$, we consider the Schrödinger maximal function, \begin{equation*} S^* f(x):= \displaystyle\sup_{0<t<4/Q^2} \left|S_tf(x)\right|\:,\:\:\:\:\:\:x\in\mathcal S\:, \end{equation*} corresponding to the Laplace--Beltrami operator $Δ$ with initial data $f$. We first obtain the complete description of the pairs $(q, α) \in [1, \infty] \times [0,\infty)$ for which the estimate \begin{equation*} {\|S^*f\|}_{L^q\left(B_R\right)} \le C_R\: {\|f\|}_{H^α(\mathcal S)}\:, \end{equation*} holds on geodesic balls $B_R$, for all $f \in \mathscr{S}^2(\mathcal S)_{o}$. Our results are sharp and agree with the Euclidean case. We also prove that for all $f \in \mathscr{S}^2(\mathcal S)_{o}$, the following global estimate \begin{equation*} {\|S^*f\|}_{L^{2,\infty}(\mathcal S)} \le C\: {\|f\|}_{H^α(\mathcal S)},\:\:\:\:α>1/2, \end{equation*} holds true. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces Dewan, Utsav Ray, Swagato K. Functional Analysis Primary 35J10, 43A85, Secondary 22E30, 43A90 For $f \in \mathscr{S}^2(\mathcal S)_{o}$, the collection of radial $L^2$-Schwartz class functions on Damek--Ricci spaces $\mathcal S$, we consider the Schrödinger maximal function, \begin{equation*} S^* f(x):= \displaystyle\sup_{0<t<4/Q^2} \left|S_tf(x)\right|\:,\:\:\:\:\:\:x\in\mathcal S\:, \end{equation*} corresponding to the Laplace--Beltrami operator $Δ$ with initial data $f$. We first obtain the complete description of the pairs $(q, α) \in [1, \infty] \times [0,\infty)$ for which the estimate \begin{equation*} {\|S^*f\|}_{L^q\left(B_R\right)} \le C_R\: {\|f\|}_{H^α(\mathcal S)}\:, \end{equation*} holds on geodesic balls $B_R$, for all $f \in \mathscr{S}^2(\mathcal S)_{o}$. Our results are sharp and agree with the Euclidean case. We also prove that for all $f \in \mathscr{S}^2(\mathcal S)_{o}$, the following global estimate \begin{equation*} {\|S^*f\|}_{L^{2,\infty}(\mathcal S)} \le C\: {\|f\|}_{H^α(\mathcal S)},\:\:\:\:α>1/2, \end{equation*} holds true. |
| title | Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces |
| topic | Functional Analysis Primary 35J10, 43A85, Secondary 22E30, 43A90 |
| url | https://arxiv.org/abs/2411.04084 |