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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2408.11130 |
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| _version_ | 1866916865633157120 |
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| author | Trapasso, S. Ivan |
| author_facet | Trapasso, S. Ivan |
| contents | We perform a phase space analysis of evolution equations associated with the Weyl quantization $q^{\mathrm{w}}$ of a complex quadratic form $q$ on $\mathbb{R}^{2d}$ with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup $e^{tq^{\mathrm{w}}}$ if $\mathrm{Re} (q) \le 0$ and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of $e^{tq^{\mathrm{w}}}$ with $\mathrm{Re} (q) \le 0$, thereby extending the $L^2$ analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces $M^p(\mathbb{R}^d)$, $1 \le p \le \infty$, with optimal explicit bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wave packet analysis of semigroups generated by quadratic differential operators Trapasso, S. Ivan Analysis of PDEs Functional Analysis 35S10, 42B37, 47D06, 35K05, 42B35, 35H99 We perform a phase space analysis of evolution equations associated with the Weyl quantization $q^{\mathrm{w}}$ of a complex quadratic form $q$ on $\mathbb{R}^{2d}$ with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup $e^{tq^{\mathrm{w}}}$ if $\mathrm{Re} (q) \le 0$ and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of $e^{tq^{\mathrm{w}}}$ with $\mathrm{Re} (q) \le 0$, thereby extending the $L^2$ analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces $M^p(\mathbb{R}^d)$, $1 \le p \le \infty$, with optimal explicit bounds. |
| title | Wave packet analysis of semigroups generated by quadratic differential operators |
| topic | Analysis of PDEs Functional Analysis 35S10, 42B37, 47D06, 35K05, 42B35, 35H99 |
| url | https://arxiv.org/abs/2408.11130 |