Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth
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| Format: | Preprint |
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2024
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| _version_ | 1866911977424551936 |
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| author | Pilipović, Stevan Prangoski, Bojan Vučković, Đorđe |
| author_facet | Pilipović, Stevan Prangoski, Bojan Vučković, Đorđe |
| contents | In the Gelfand-Shilov setting, the localisation operator $A^{φ_1,φ_2}_a$ is equal to the Weyl operator whose symbol is the convolution of $a$ with the Wigner transform of the windows $φ_2$ and $φ_1$. We employ this fact, to extend the definition of localisation operators to symbols $a$ having very fast super-exponential growth by allowing them to be mappings from ${\mathcal D}^{\{M_p\}}(\mathbb R^d)$ into ${\mathcal D}'^{\{M_p\}}(\mathbb R^d)$, where $M_p$, $p\in\mathbb N$, is a non-quasi-analytic Gevrey type sequence. By choosing the windows $φ_1$ and $φ_2$ appropriately, our main results show that one can consider symbols with growth in position space of the form $\exp(\exp(l|\cdot|^q))$, $l,q>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth Pilipović, Stevan Prangoski, Bojan Vučković, Đorđe Functional Analysis 47G30, 46F05, 35S05 In the Gelfand-Shilov setting, the localisation operator $A^{φ_1,φ_2}_a$ is equal to the Weyl operator whose symbol is the convolution of $a$ with the Wigner transform of the windows $φ_2$ and $φ_1$. We employ this fact, to extend the definition of localisation operators to symbols $a$ having very fast super-exponential growth by allowing them to be mappings from ${\mathcal D}^{\{M_p\}}(\mathbb R^d)$ into ${\mathcal D}'^{\{M_p\}}(\mathbb R^d)$, where $M_p$, $p\in\mathbb N$, is a non-quasi-analytic Gevrey type sequence. By choosing the windows $φ_1$ and $φ_2$ appropriately, our main results show that one can consider symbols with growth in position space of the form $\exp(\exp(l|\cdot|^q))$, $l,q>0$. |
| title | Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth |
| topic | Functional Analysis 47G30, 46F05, 35S05 |
| url | https://arxiv.org/abs/2408.02437 |