Propagation of singularities for equations with $C^{r}$ coefficients for $r>1$
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| Format: | Preprint |
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2025
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| _version_ | 1866914158234042368 |
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| author | Rozendaal, Jan |
| author_facet | Rozendaal, Jan |
| contents | We observe that, for $r>1$, $s$ in an $r$-dependent interval, $p$ a homogeneous pseudodifferential symbol of order $m$ having $C^{r}$ regularity in space, and $u\in H^{s+m-r}(\mathbb{R}^{n})$ such that $p(x,D)u\in H^{s}(\mathbb{R}^{n})$, each point in the $H^{s+m-1}$ wavefront set of $u$ lies on a maximally extended null bicharacteristic of $p$ which is contained in the $H^{s+m-1}$ wavefront set of $u$. In fact, for $r=2$ slightly less than $C^{1,1}$ regularity suffices, and here the results apply to manifolds with bounded Ricci curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Propagation of singularities for equations with $C^{r}$ coefficients for $r>1$ Rozendaal, Jan Analysis of PDEs Primary 58J47. Secondary 35A18, 35S50, 58J45 We observe that, for $r>1$, $s$ in an $r$-dependent interval, $p$ a homogeneous pseudodifferential symbol of order $m$ having $C^{r}$ regularity in space, and $u\in H^{s+m-r}(\mathbb{R}^{n})$ such that $p(x,D)u\in H^{s}(\mathbb{R}^{n})$, each point in the $H^{s+m-1}$ wavefront set of $u$ lies on a maximally extended null bicharacteristic of $p$ which is contained in the $H^{s+m-1}$ wavefront set of $u$. In fact, for $r=2$ slightly less than $C^{1,1}$ regularity suffices, and here the results apply to manifolds with bounded Ricci curvature. |
| title | Propagation of singularities for equations with $C^{r}$ coefficients for $r>1$ |
| topic | Analysis of PDEs Primary 58J47. Secondary 35A18, 35S50, 58J45 |
| url | https://arxiv.org/abs/2510.16182 |