Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
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| Format: | Preprint |
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2025
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| author | Dewan, Utsav |
| author_facet | Dewan, Utsav |
| contents | We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases}
i\frac{\partial u}{\partial t} +Ψ(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:,
\end{cases}
\end{equation*} where $\mathcal{L}= Δ$, the Laplace-Beltrami operator or $\tildeΔ$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $ψ$ satisfies for some $a \in (0,1)$, the large frequency asymptotic: \begin{equation*} ψ(λ)=λ^a + \mathcal{O}(1)\:,\:\: λ\gg 1\:. \end{equation*} For almost everywhere pointwise convergence of the solution $u$ to its radial initial data $f$, we obtain the almost sharp regularity threshold $β>a/4$. This result is new even for $\mathbb{R}^n$ and in the special case of the fractional Schrödinger equations, generalizes classical Euclidean results of Walther. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces Dewan, Utsav Analysis of PDEs Primary 35J10, 43A85, Secondary 22E30, 43A90 We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +Ψ(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:, \end{cases} \end{equation*} where $\mathcal{L}= Δ$, the Laplace-Beltrami operator or $\tildeΔ$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $ψ$ satisfies for some $a \in (0,1)$, the large frequency asymptotic: \begin{equation*} ψ(λ)=λ^a + \mathcal{O}(1)\:,\:\: λ\gg 1\:. \end{equation*} For almost everywhere pointwise convergence of the solution $u$ to its radial initial data $f$, we obtain the almost sharp regularity threshold $β>a/4$. This result is new even for $\mathbb{R}^n$ and in the special case of the fractional Schrödinger equations, generalizes classical Euclidean results of Walther. |
| title | Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces |
| topic | Analysis of PDEs Primary 35J10, 43A85, Secondary 22E30, 43A90 |
| url | https://arxiv.org/abs/2506.00881 |