Extension of localisation operators to ultradistributional symbols with super-exponential growth. (English) Zbl 1514.46031
Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116, No. 4, Paper No. 172, 16 p. (2022).
Summary: In the Gelfand-Shilov setting, the localisation operator \(A^{\varphi_1, \varphi_2}_a\) is equal to the Weyl operator whose symbol is the convolution of \(a\) with the Wigner transform of the windows \(\varphi_2\) and \(\varphi_1\). 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 \(\mathcal{D}^{\prime\{M_p\}}(\mathbb{R}^d)\), where \(M_p\), \(p\in \mathbb N\), is a non-quasi-analytic Gevrey type sequence. By choosing the windows \(\varphi_1\) and \(\varphi_2\) appropriately, our main results show that one can consider symbols with growth in position space of the form \(\exp(\exp(l|\cdot|^q))\), \(l,q>0\).
MSC:
| 46F12 | Integral transforms in distribution spaces |
| 47G30 | Pseudodifferential operators |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
Keywords:
localisation operators; anti-Wick quantisation; extensions with ultradistribution symbols; ultradistributionsReferences:
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