Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth

Abstract

In the Gelfand-Shilov setting, the localisation operator $A_a^{{\phi }_1,{\phi }_2}$ is equal to the Weyl operator whose symbol is the convolution of $a$ with the Wigner transform of the windows ${\phi }_2$ and ${\phi }_1$. 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 ${\phi }_1$ and ${\phi }_2$ appropriately, our main results show that one can consider symbols with growth in position space of the form $\exp (\exp (l|\cdot {|}^q))$, $l,q>0$.