Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces
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| Formato: | Preprint |
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2024
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| _version_ | 1866917727054069760 |
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| author | Dewan, Utsav |
| author_facet | Dewan, Utsav |
| contents | One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schrödinger equation given by \begin{equation*}\begin{cases} i\frac{\partial u}{\partial t} =Δu\:,\: (x,t) \in \mathbb{R}^n \times \mathbb{R} \\ u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases}\end{equation*} in terms of the index $α$ such that $f$ belongs to the inhomogeneous Sobolev space $H^α(\mathbb{R}^n)$ , so that the solution of the Schrödinger operator $u$ converges pointwise to $f$, $\lim_{t \to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we consider the Carleson's problem for the Schrödinger equation with radial initial data on Damek-Ricci spaces and obtain the sharp bound up to the endpoint $α\ge 1/4$, which agrees with the classical Euclidean case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13736 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces Dewan, Utsav Classical Analysis and ODEs Analysis of PDEs 35J10, 43A85 One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schrödinger equation given by \begin{equation*}\begin{cases} i\frac{\partial u}{\partial t} =Δu\:,\: (x,t) \in \mathbb{R}^n \times \mathbb{R} \\ u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases}\end{equation*} in terms of the index $α$ such that $f$ belongs to the inhomogeneous Sobolev space $H^α(\mathbb{R}^n)$ , so that the solution of the Schrödinger operator $u$ converges pointwise to $f$, $\lim_{t \to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we consider the Carleson's problem for the Schrödinger equation with radial initial data on Damek-Ricci spaces and obtain the sharp bound up to the endpoint $α\ge 1/4$, which agrees with the classical Euclidean case. |
| title | Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces |
| topic | Classical Analysis and ODEs Analysis of PDEs 35J10, 43A85 |
| url | https://arxiv.org/abs/2407.13736 |