A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914730184015872 |
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| author | Minguillón, Javier |
| author_facet | Minguillón, Javier |
| contents | In this short note, we give an easy proof of the following result: for $ n\geq 2, $ $\underset{t\to0}{\lim} \,e^{itΔ}f\left(x+γ(t)\right) = f(x) $ almost everywhere whenever $ γ$ is an $ α- $Hölder curve with $ \frac12\leq α\leq 1 $ and $ f\in H^s(\mathbb{R}^n) $, with $ s > \frac{n}{2(n+1)} $. This is the optimal range of regularity up to the endpoint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18032 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum Minguillón, Javier Classical Analysis and ODEs 35Q41, 42B25, 42B37 In this short note, we give an easy proof of the following result: for $ n\geq 2, $ $\underset{t\to0}{\lim} \,e^{itΔ}f\left(x+γ(t)\right) = f(x) $ almost everywhere whenever $ γ$ is an $ α- $Hölder curve with $ \frac12\leq α\leq 1 $ and $ f\in H^s(\mathbb{R}^n) $, with $ s > \frac{n}{2(n+1)} $. This is the optimal range of regularity up to the endpoint. |
| title | A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum |
| topic | Classical Analysis and ODEs 35Q41, 42B25, 42B37 |
| url | https://arxiv.org/abs/2403.18032 |