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| Natura: | Preprint |
| Pubblicazione: |
2019
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| Accesso online: | https://arxiv.org/abs/1912.10511 |
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| _version_ | 1866914730855104512 |
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| author | Winterrose, David Scott |
| author_facet | Winterrose, David Scott |
| contents | Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $μ_0$. We give an integral representation of $μ_0$ in terms of the resolutions supplied by Hironaka's theorem. This $μ_0$ is weakly approximated in $H^t_{\mathrm{loc}}(\mathbb{R}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_10511 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A structure theorem for fundamental solutions of analytic multipliers in $\mathbb{R}^n$ Winterrose, David Scott Analysis of PDEs 35A08 (Primary) 35E05, 35C05, 35A17 (Secondary) Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $μ_0$. We give an integral representation of $μ_0$ in terms of the resolutions supplied by Hironaka's theorem. This $μ_0$ is weakly approximated in $H^t_{\mathrm{loc}}(\mathbb{R}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols. |
| title | A structure theorem for fundamental solutions of analytic multipliers in $\mathbb{R}^n$ |
| topic | Analysis of PDEs 35A08 (Primary) 35E05, 35C05, 35A17 (Secondary) |
| url | https://arxiv.org/abs/1912.10511 |